diff --git a/apps/2_3DGrid/2_3DGrid.cpp b/apps/2_3DGrid/2_3DGrid.cpp
index 8b69b90..7197e9c 100644
--- a/apps/2_3DGrid/2_3DGrid.cpp
+++ b/apps/2_3DGrid/2_3DGrid.cpp
@@ -95,11 +95,13 @@ int main(int argc, char* argv[]) {
     }
 
     monad::Hex8Grid grid({nx, ny, nz}, {lx, ly, lz});
-    grid.setDensitiesRandom(seed);
 
+    monad::DensityField3d densityField({nx, ny, nz});
+    densityField.setRandom(static_cast<unsigned int>(seed));
+ 
     const auto file = std::filesystem::path(__FILE__).parent_path() / "output.msh";
 
-    monad::saveGrid(grid, file.string(), true);
+    monad::saveGridAndDensityField(grid, densityField, file.string());
 
     std::cout << "Saved to " + file.string() << std::endl;
 
diff --git a/apps/3_DensityFunction/3_DensityFunction.cpp b/apps/3_DensityFunction/3_DensityFunction.cpp
index 7c5c897..6f2a0dd 100644
--- a/apps/3_DensityFunction/3_DensityFunction.cpp
+++ b/apps/3_DensityFunction/3_DensityFunction.cpp
@@ -41,11 +41,11 @@ int main() {
         return (xComponent + yComponent + zComponent) / 3;
     };
 
-    grid.setDensitiesFunction(f);
+    const auto densityField = monad::makeDensityFieldFromFunction(grid, f);
 
     const auto file = std::filesystem::path(__FILE__).parent_path() / "output.msh";
 
-    monad::saveGrid(grid, file.string(), true);
+    monad::saveGridAndDensityField(grid, densityField, file.string());
 
     std::cout << "Saved to " + file.string() << std::endl;
 
diff --git a/apps/4_LinearElasticity/4_LinearElasticity.cpp b/apps/4_LinearElasticity/4_LinearElasticity.cpp
index 11017aa..42a8f50 100644
--- a/apps/4_LinearElasticity/4_LinearElasticity.cpp
+++ b/apps/4_LinearElasticity/4_LinearElasticity.cpp
@@ -70,32 +70,32 @@ int main(int argc, char* argv[]) {
     const auto folder = std::filesystem::path(__FILE__).parent_path();
     const auto csvFile = folder / "data.csv";
 
-    grid.setDensitiesFile(csvFile.string());
+    const auto densityField = monad::makeDensityFieldFromCsv(csvFile);
 
     monad::SolverOptions options;
     options.fields = monad::FieldSave::All;
 
-    const monad::LinearElasticSolver solver(grid, material);
+    const monad::LinearElasticSolver<monad::Quad8Grid> solver;
 
-    const auto results = solver.solve(options);
+    const auto results = solver.solve(grid, densityField, material, options);
 
     std::cout << "---Homogenized stiffness tensor---\n" << results.CBar << std::endl;
 
     auto file = folder / "density.msh";
 
-    monad::saveGrid(grid, file.string(), true);
+    monad::saveGridAndDensityField(grid, densityField, file.string());
 
     file = folder / "uMacro.msh";
 
-    monad::saveGridAndField(grid, results.uMacro[0], file.string(), "Macro displacement");
+    monad::saveGridAndNodalField(grid, results.uMacro[0], file.string(), "Macro displacement");
 
     file = folder / "uMicro.msh";
 
-    monad::saveGridAndField(grid, results.uMicro[0], file.string(), "Micro displacement");
+    monad::saveGridAndNodalField(grid, results.uMicro[0], file.string(), "Micro displacement");
 
     file = folder / "u.msh";
 
-    monad::saveGridAndField(grid, results.u[0], file.string(), "Displacement");
+    monad::saveGridAndNodalField(grid, results.u[0], file.string(), "Displacement");
 
     std::cout << "Saved to " + file.string() << std::endl;
 
diff --git a/apps/5_LinearDielectric/5_LinearDielectric.cpp b/apps/5_LinearDielectric/5_LinearDielectric.cpp
index 5629182..d7cd52e 100644
--- a/apps/5_LinearDielectric/5_LinearDielectric.cpp
+++ b/apps/5_LinearDielectric/5_LinearDielectric.cpp
@@ -60,37 +60,37 @@ int main(int argc, char* argv[]) {
     const auto folder = std::filesystem::path(__FILE__).parent_path();
     const auto csvFile = folder / "data.csv";
 
-    grid.setDensitiesFile(csvFile.string());
+    const auto densityField = monad::makeDensityFieldFromCsv(csvFile);
 
     monad::SolverOptions options;
     options.fields = monad::FieldSave::All;
 
-    const monad::LinearDielectricSolver solver(grid, material);
+    const monad::LinearDielectricSolver<monad::Quad8Grid> solver;
 
     // Results follow the generic transport naming:
-    // results.KBar → ε̄
+    // results.KBar → ϵ̄
     // results.phiMacro → φ̄
     // results.phiMicro → φ̃
     // results.phi → φ
-    const auto results = solver.solve(options);
+    const auto results = solver.solve(grid, densityField, material, options);
 
     std::cout << "---Homogenized permittivity tensor---\n" << results.KBar << std::endl;
 
     auto file = folder / "density.msh";
 
-    monad::saveGrid(grid, file.string(), true);
+    monad::saveGridAndDensityField(grid, densityField, file.string());
 
     file = folder / "phiMacro.msh";
 
-    monad::saveGridAndField(grid, results.phiMacro[0], file.string(), "Macro electric potential");
+    monad::saveGridAndNodalField(grid, results.phiMacro[0], file.string(), "Macro electric potential");
 
     file = folder / "phiMicro.msh";
 
-    monad::saveGridAndField(grid, results.phiMicro[0], file.string(), "Micro electric potential");
+    monad::saveGridAndNodalField(grid, results.phiMicro[0], file.string(), "Micro electric potential");
 
     file = folder / "phi.msh";
 
-    monad::saveGridAndField(grid, results.phi[0], file.string(), "Electric potential");
+    monad::saveGridAndNodalField(grid, results.phi[0], file.string(), "Electric potential");
 
     std::cout << "Saved to " + file.string() << std::endl;
 
diff --git a/apps/6_LinearPiezoelectricity/6_LinearPiezoelectricity.cpp b/apps/6_LinearPiezoelectricity/6_LinearPiezoelectricity.cpp
index c8db59f..ce28bd3 100644
--- a/apps/6_LinearPiezoelectricity/6_LinearPiezoelectricity.cpp
+++ b/apps/6_LinearPiezoelectricity/6_LinearPiezoelectricity.cpp
@@ -14,10 +14,12 @@
  *
  * The base material's piezoelectric coupling tensor is set to
  *
+ * ```text
  * ⎡d  0  0⎤
  * ⎣0 -d -d⎦
+ * ```
  *
- * where d=E/10.
+ * where d=⅒E.
  *
  * The Gmsh files are written to:
  *
@@ -96,23 +98,26 @@ int main(int argc, char* argv[]) {
     const monad::LinearElasticMaterial2d elasticMaterial(E, nu, monad::LinearElasticMaterial2d::PlaneCondition::PlaneStress);
     const monad::LinearDielectricMaterial2d dielectricMaterial(epsilon);
 
+    const auto c = elasticMaterial.materialTensor();
+    const auto epsilonTensor = dielectricMaterial.materialTensor();
     const Eigen::Matrix<double, 2, 3> d {
         {E / 10.0, 0.0, 0.0},
         {0.0, E / 10.0, E / 10.0}
     };
 
-    const monad::LinearPiezoelectricMaterial2d material(elasticMaterial, dielectricMaterial, d);
+    const monad::LinearPiezoelectricMaterial2d material(c, epsilonTensor, d);
 
     const auto folder = std::filesystem::path(__FILE__).parent_path();
     const auto csvFile = folder / "data.csv";
 
-    grid.setDensitiesFile(csvFile.string());
+    const auto densityField = monad::makeDensityFieldFromCsv(csvFile);
 
     monad::SolverOptions options;
     options.fields = monad::FieldSave::All;
 
-    const monad::LinearPiezoelectricSolver solver(grid, material);
-    const auto results = solver.solve(options);
+    const monad::LinearPiezoelectricSolver<monad::Quad8Grid> solver;
+
+    const auto results = solver.solve(grid, densityField, material, options);
 
     std::cout << "---Homogenized stiffness tensor---\n" << results.cBar << std::endl;
     std::cout << "\n---Homogenized permittivity tensor---\n" << results.epsilonBar << std::endl;
@@ -120,31 +125,31 @@ int main(int argc, char* argv[]) {
 
     auto file = folder / "density.msh";
 
-    monad::saveGrid(grid, file.string(), true);
+    monad::saveGridAndDensityField(grid, densityField, file.string());
 
     file = folder / "uMacro.msh";
 
-    monad::saveGridAndField(grid, results.uMacro[0], file.string(), "Macro displacement");
+    monad::saveGridAndNodalField(grid, results.uMacro[0], file.string(), "Macro displacement");
 
     file = folder / "uMicro.msh";
 
-    monad::saveGridAndField(grid, results.uMicro[0], file.string(), "Micro displacement");
+    monad::saveGridAndNodalField(grid, results.uMicro[0], file.string(), "Micro displacement");
 
     file = folder / "u.msh";
 
-    monad::saveGridAndField(grid, results.u[0], file.string(), "Displacement");
+    monad::saveGridAndNodalField(grid, results.u[0], file.string(), "Displacement");
 
     file = folder / "phiMacro.msh";
 
-    monad::saveGridAndField(grid, results.phiMacro[0], file.string(), "Macro electric potential");
+    monad::saveGridAndNodalField(grid, results.phiMacro[0], file.string(), "Macro electric potential");
 
     file = folder / "phiMicro.msh";
 
-    monad::saveGridAndField(grid, results.phiMicro[0], file.string(), "Micro electric potential");
+    monad::saveGridAndNodalField(grid, results.phiMicro[0], file.string(), "Micro electric potential");
 
     file = folder / "phi.msh";
 
-    monad::saveGridAndField(grid, results.phi[0], file.string(), "Electric potential");
+    monad::saveGridAndNodalField(grid, results.phi[0], file.string(), "Electric potential");
 
     std::cout << "Saved to " + file.string() << std::endl;
 
diff --git a/assets/create_teaser.py b/assets/create_teaser.py
index 0ca79a2..9d42745 100644
--- a/assets/create_teaser.py
+++ b/assets/create_teaser.py
@@ -12,8 +12,8 @@
     'custom',
     [
         (0.0, 'white'),
-        (0.5, '#e6d0d1'),
-        (1.0, '#9e5457')
+        (0.5, '#c9e3f6'),
+        (1.0, 'C0')
     ]
 )
 
diff --git a/assets/images/teaser.png b/assets/images/teaser.png
index a27b455..24cf0ba 100644
Binary files a/assets/images/teaser.png and b/assets/images/teaser.png differ
diff --git a/assets/images/teaser.svg b/assets/images/teaser.svg
index f573801..26ad8a4 100644
--- a/assets/images/teaser.svg
+++ b/assets/images/teaser.svg
@@ -6,11 +6,11 @@
   <rdf:RDF xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:cc="http://creativecommons.org/ns#" xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
    <cc:Work>
     <dc:type rdf:resource="http://purl.org/dc/dcmitype/StillImage"/>
-    <dc:date>2026-01-26T10:36:06.317221</dc:date>
+    <dc:date>2026-06-12T09:00:35.521131</dc:date>
     <dc:format>image/svg+xml</dc:format>
     <dc:creator>
      <cc:Agent>
-      <dc:title>Matplotlib v3.9.1, https://matplotlib.org/</dc:title>
+      <dc:title>Matplotlib v3.9.2, https://matplotlib.org/</dc:title>
      </cc:Agent>
     </dc:creator>
    </cc:Work>
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+  <clipPath id="p67bfec989b">
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diff --git a/build.sh b/build.sh
index 811be59..f2248fe 100755
--- a/build.sh
+++ b/build.sh
@@ -6,5 +6,5 @@ fi
 rm -rf build
 mkdir build
 cd build
-cmake .. -DMONAD_USE_OPENMP=ON -DMONAD_BUILD_APPS=ON -DMONAD_BUILD_TESTS=ON
+cmake .. -DMONAD_USE_OPENMP=ON -DMONAD_BUILD_APPS=ON -DMONAD_BUILD_TESTS=OFF
 make -j8
diff --git a/include/monad/detail/constants.hpp b/include/monad/detail/constants.hpp
index 5a394b6..f8f356b 100644
--- a/include/monad/detail/constants.hpp
+++ b/include/monad/detail/constants.hpp
@@ -2,7 +2,7 @@
 
 namespace monad {
 
-    /// @brief Numerical tolerance used to treat small floating-point values as zero.
+    /// @brief Numerical tolerance for treating small floating-point values as zero.
     constexpr double NUMERICAL_ZERO = 1e-9;
 
 } // namespace monad
diff --git a/include/monad/detail/eigen_utils.hpp b/include/monad/detail/eigen_utils.hpp
index 6cdd3b9..9ce44be 100644
--- a/include/monad/detail/eigen_utils.hpp
+++ b/include/monad/detail/eigen_utils.hpp
@@ -10,32 +10,12 @@ namespace monad {
 
     namespace detail {
 
-        /**
-         * @brief Converts an array to an Eigen vector.
-         *
-         * @tparam N Size of array.
-         *
-         * @param[in] array Array.
-         *
-         * @returns Eigen vector of `array`.
-         */
-        template <std::size_t N>
-        inline Eigen::Vector<int, N> arrayToEigen(const std::array<std::size_t, N> &array) noexcept {
-            Eigen::Vector<int, N> vector;
-
-            for (std::size_t i = 0; i < N; ++i) {
-                vector[static_cast<int>(i)] = static_cast<int>(array[i]);
-            }
-
-            return vector;
-        }
-
         /**
          * @brief Symmetrizes a square matrix.
          *
-         * This function replaces A with ½(A+Aᵀ), removing numerical
-         * asymmetry introduced by floating-point error in computations
-         * that theoretically produce symmetric matrices.
+         * Replaces A with ½(A+Aᵀ) to remove numerical asymmetry from
+         * floating-point error in computations that should produce a
+         * symmetric matrix.
          *
          * @tparam Derived Eigen matrix type.
          *
@@ -43,7 +23,7 @@ namespace monad {
          *
          * @throws std::invalid_argument if `A` is not square.
          *
-         * @note Use this only to clean up numerical noise.
+         * @note Use this only to remove numerical noise.
          */
         template <typename Derived>
         inline void symmetrize(Eigen::MatrixBase<Derived> &A) {
@@ -75,8 +55,8 @@ namespace monad {
         /**
          * @brief Checks if a matrix is positive definite.
          *
-         * This function checks if a matrix is positive definite
-         * by attempting to perform a Cholesky factorization.
+         * Uses a Cholesky factorization to test positive
+         * definiteness.
          *
          * @tparam Derived Eigen matrix type.
          *
@@ -99,8 +79,8 @@ namespace monad {
         /**
          * @brief Checks if a matrix is positive semi-definite.
          *
-         * This function checks if a matrix is positive semi-definite
-         * by verifying that all its eigenvalues are non-negative.
+         * Checks positive semi-definiteness by verifying that all
+         * eigenvalues are non-negative up to numerical tolerance.
          *
          * @tparam Derived Eigen matrix type.
          *
diff --git a/include/monad/detail/mean.hpp b/include/monad/detail/mean.hpp
index aff6a25..8265f3f 100644
--- a/include/monad/detail/mean.hpp
+++ b/include/monad/detail/mean.hpp
@@ -7,30 +7,46 @@ namespace monad {
     namespace detail {
 
         /**
-         * @brief Arithmetic mean for a list of scalar values.
+         * @brief Arithmetic mean of a list of scalar values.
          *
-         * Given a collection of values {x₁,x₂,...,xₙ}, the arithmetic mean is defined as:
+         * Given values
          *
+         * ```text
+         * {x₁, x₂, ..., xₙ}
+         * ```
+         * 
+         * the arithmetic mean is defined as:
+         *
+         * ```text
          * (1/n)∑ᵢxᵢ
+         * ```
          *
-         * @param[in] x List of scalar values.
+         * @param[in] x Values.
          *
-         * @returns Arithmetic mean for a list of scalar values.
+         * @returns Arithmetic mean of `x`.
          */
         double arithmeticMean(const std::vector<double> &x) noexcept;
 
         /**
-         * @brief Harmonic mean for a list of scalar values.
+         * @brief Harmonic mean of a list of scalar values.
+         *
+         * Given values
+         *
+         * ```text
+         * {x₁,x₂,...,xₙ}
+         * ```
          *
-         * Given a collection of values {x₁,x₂,...,xₙ}, the harmonic mean is defined as:
+         * the harmonic mean is defined as:
          *
+         * ```text
          * n/∑ᵢ(1/xᵢ)
+         * ```
          *
-         * @param[in] x List of scalar values.
+         * @param[in] x Values.
          *
-         * @returns Harmonic mean for a list of scalar values.
+         * @returns Harmonic mean of `x`.
          *
-         * @throws std::invalid_argument if any entry in `x` is zero.
+         * @throws std::invalid_argument if any entry of `x` is zero.
          */
         double harmonicMean(const std::vector<double> &x);
 
diff --git a/include/monad/fem/element/element_base.hpp b/include/monad/fem/element/element_base.hpp
index e9f299b..dfc2150 100644
--- a/include/monad/fem/element/element_base.hpp
+++ b/include/monad/fem/element/element_base.hpp
@@ -14,12 +14,12 @@ namespace monad {
         /**
          * @brief Curiously recurring template pattern (CRTP) base class for FEM elements.
          *
-         * @tparam Derived Concrete element class (for CRTP).
+         * @tparam Derived Concrete element type for CRTP.
          * @tparam D Spatial dimension (2 or 3).
-         * @tparam K Number of nodes in the element.
-         * @tparam N Number of integration points in the element.
+         * @tparam K Number of element nodes.
+         * @tparam N Number of integration points.
          *
-         * @note `Derived` classes must provide:
+         * @note `Derived` must provide:
          *
          * - `static NodesMatrix localNodes() noexcept`
          *
@@ -66,11 +66,17 @@ namespace monad {
             /**
              * @brief Shape functions evaluated at a local point.
              *
-             * For a D-dimensional element:
+             * - 2D element:
              *
-             * - D=2: [ξ η]ᵀ → [N₁(ξ,η) ... Nₖ(ξ,η)]ᵀ
+             * ```
+             * [ξ η]ᵀ → [N₁(ξ,η) ... Nₖ(ξ,η)]ᵀ
+             * ```
+             *
+             * - 3D element:
              *
-             * - D=3: [ξ η ζ]ᵀ → [N₁(ξ,η,ζ) ... Nₖ(ξ,η,ζ)]ᵀ
+             * ```
+             * [ξ η ζ]ᵀ → [N₁(ξ,η,ζ) ... Nₖ(ξ,η,ζ)]ᵀ
+             * ```
              *
              * @param[in] point Local point.
              *
@@ -83,21 +89,19 @@ namespace monad {
             /**
              * @brief Shape function gradients evaluated at a local point.
              *
-             * For a D-dimensional element:
-             *
-             * - D=2: [ξ η]ᵀ →
+             * - 2D element
              *
              * ```text
-             * ⎡∂N₁/∂ξ ... ∂Nₖ/∂ξ⎤
-             * ⎣∂N₁/∂η ... ∂Nₖ/∂η⎦
+             * [ξ η]ᵀ → ⎡∂N₁/∂ξ ... ∂Nₖ/∂ξ⎤
+             *          ⎣∂N₁/∂η ... ∂Nₖ/∂η⎦
              * ```
              *
-             * - D=3: [ξ η ζ]ᵀ →
+             * - 3D element:
              *
              * ```text
-             * ⎡∂N₁/∂ξ ... ∂Nₖ/∂ξ⎤
-             * ⎪∂N₁/∂η ... ∂Nₖ/∂η⎪
-             * ⎣∂N₁/∂ζ ... ∂Nₖ/∂ζ⎦
+             *            ⎡∂N₁/∂ξ ... ∂Nₖ/∂ξ⎤
+             * [ξ η ζ]ᵀ → ⎢∂N₁/∂η ... ∂Nₖ/∂η⎥
+             *            ⎣∂N₁/∂ζ ... ∂Nₖ/∂ζ⎦
              * ```
              *
              * @param[in] point Local point.
@@ -111,21 +115,19 @@ namespace monad {
             /**
              * @brief Jacobian matrix evaluated at a local point.
              *
-             * For a D-dimensional element:
-             *
-             * - D=2: [ξ η]ᵀ →
+             * - 2D element:
              *
              * ```text
-             * ⎡∂x/∂ξ ∂x/∂η⎤
-             * ⎣∂y/∂ξ ∂y/∂η⎦
+             * [ξ η]ᵀ → ⎡∂x/∂ξ ∂x/∂η⎤
+             *          ⎣∂y/∂ξ ∂y/∂η⎦
              * ```
              *
-             * - D=3: [ξ η ζ]ᵀ →
+             * - 3D element:
              *
              * ```text
-             * ⎡∂x/∂ξ ∂x/∂η ∂x/∂ζ⎤
-             * ⎪∂y/∂ξ ∂y/∂η ∂y/∂ζ⎪
-             * ⎣∂z/∂ξ ∂z/∂η ∂z/∂ζ⎦
+             *            ⎡∂x/∂ξ ∂x/∂η ∂x/∂ζ⎤
+             * [ξ η ζ]ᵀ → ⎢∂y/∂ξ ∂y/∂η ∂y/∂ζ⎥
+             *            ⎣∂z/∂ξ ∂z/∂η ∂z/∂ζ⎦
              * ```
              *
              * @param[in] point Local point.
@@ -137,7 +139,7 @@ namespace monad {
                 return gradShapeFunctions(point) * nodes;
             }
 
-            /// @brief Quadrature rule for integration.
+            /// @brief Quadrature rule.
             static QuadratureRule quadratureRule() noexcept {
                 return Derived::quadratureRule();
             }
@@ -145,9 +147,11 @@ namespace monad {
             /**
              * @brief Area (2D) or volume (3D).
              *
-             * Computes area/volume via:
+             * Computes the measure as
              *
+             * ```text
              * ∫_Ω|det(J)|dΩ
+             * ```
              *
              * @param[in] nodes Element nodes.
              *
diff --git a/include/monad/fem/element/hex20.hpp b/include/monad/fem/element/hex20.hpp
index 623a115..40607e2 100644
--- a/include/monad/fem/element/hex20.hpp
+++ b/include/monad/fem/element/hex20.hpp
@@ -42,7 +42,9 @@ namespace monad {
             /**
              * @brief Shape functions evaluated at a local point.
              *
+             * ```text
              * [ξ η ζ]ᵀ → [N₁(ξ,η,ζ) ... N₂₀(ξ,η,ζ)]ᵀ
+             * ```
              *
              * @param[in] point Local point.
              *
@@ -53,12 +55,10 @@ namespace monad {
             /**
              * @brief Shape function gradients evaluated at a local point.
              *
-             * [ξ η ζ]ᵀ →
-             *
              * ```text
-             * ⎡∂N₁/∂ξ ... ∂N₂₀/∂ξ⎤
-             * ⎪∂N₁/∂η ... ∂N₂₀/∂η⎪
-             * ⎣∂N₁/∂ζ ... ∂N₂₀/∂ζ⎦
+             *            ⎡∂N₁/∂ξ ... ∂N₂₀/∂ξ⎤
+             * [ξ η ζ]ᵀ → ⎢∂N₁/∂η ... ∂N₂₀/∂η⎥
+             *            ⎣∂N₁/∂ζ ... ∂N₂₀/∂ζ⎦
              * ```
              *
              * @param[in] point Local point.
@@ -67,7 +67,7 @@ namespace monad {
              */
             static ShapeFuncGradMatrix gradShapeFunctions(const Point &point) noexcept;
 
-            /// @brief Quadrature rule for integration.
+            /// @brief Quadrature rule.
             static QuadratureRule quadratureRule() noexcept;
 
             /// @brief Gmsh element type ID.
diff --git a/include/monad/fem/element/hex8.hpp b/include/monad/fem/element/hex8.hpp
index b562105..dba21e7 100644
--- a/include/monad/fem/element/hex8.hpp
+++ b/include/monad/fem/element/hex8.hpp
@@ -42,7 +42,9 @@ namespace monad {
             /**
              * @brief Shape functions evaluated at a local point.
              *
+             * ```text
              * [ξ η ζ]ᵀ → [N₁(ξ,η,ζ) ... N₈(ξ,η,ζ)]ᵀ
+             * ```
              *
              * @param[in] point Local point.
              *
@@ -53,12 +55,10 @@ namespace monad {
             /**
              * @brief Shape function gradients evaluated at a local point.
              *
-             * [ξ η ζ]ᵀ →
-             *
              * ```text
-             * ⎡∂N₁/∂ξ ... ∂N₈/∂ξ⎤
-             * ⎪∂N₁/∂η ... ∂N₈/∂η⎪
-             * ⎣∂N₁/∂ζ ... ∂N₈/∂ζ⎦
+             *            ⎡∂N₁/∂ξ ... ∂N₈/∂ξ⎤
+             * [ξ η ζ]ᵀ → ⎢∂N₁/∂η ... ∂N₈/∂η⎥
+             *            ⎣∂N₁/∂ζ ... ∂N₈/∂ζ⎦
              * ```
              *
              * @param[in] point Local point.
@@ -67,7 +67,7 @@ namespace monad {
              */
             static ShapeFuncGradMatrix gradShapeFunctions(const Point &point) noexcept;
 
-            /// @brief Quadrature rule for integration.
+            /// @brief Quadrature rule.
             static QuadratureRule quadratureRule() noexcept;
 
             /// @brief Gmsh element type ID.
diff --git a/include/monad/fem/element/quad4.hpp b/include/monad/fem/element/quad4.hpp
index f226f87..3609fab 100644
--- a/include/monad/fem/element/quad4.hpp
+++ b/include/monad/fem/element/quad4.hpp
@@ -7,7 +7,7 @@ namespace monad {
     namespace fem {
 
         /**
-         * @brief 4-node quadrilateral (Quad4) finite element.
+         * @brief 4-node quadrilateral (Quad4) element.
          *
          * The Quad4 element uses bilinear shape-functions on ξ,η∈[-1,1].
          *
@@ -34,7 +34,9 @@ namespace monad {
             /**
              * @brief Shape functions evaluated at a local point.
              *
+             * ```text
              * [ξ η]ᵀ → [N₁(ξ,η) ... N₄(ξ,η)]ᵀ
+             * ```
              *
              * @param[in] point Local point.
              *
@@ -45,11 +47,9 @@ namespace monad {
             /**
              * @brief Shape function gradients evaluated at a local point.
              *
-             * [ξ η]ᵀ →
-             *
              * ```text
-             * ⎡∂N₁/∂ξ ... ∂N₄/∂ξ⎤
-             * ⎣∂N₁/∂η ... ∂N₄/∂η⎦
+             * [ξ η]ᵀ → ⎡∂N₁/∂ξ ... ∂N₄/∂ξ⎤
+             *          ⎣∂N₁/∂η ... ∂N₄/∂η⎦
              * ```
              *
              * @param[in] point Local point.
@@ -58,7 +58,7 @@ namespace monad {
              */
             static ShapeFuncGradMatrix gradShapeFunctions(const Point &point) noexcept;
 
-            /// @brief Quadrature rule for integration.
+            /// @brief Quadrature rule.
             static QuadratureRule quadratureRule() noexcept;
 
             /// @brief Gmsh element type ID.
diff --git a/include/monad/fem/element/quad8.hpp b/include/monad/fem/element/quad8.hpp
index 099b719..56e7658 100644
--- a/include/monad/fem/element/quad8.hpp
+++ b/include/monad/fem/element/quad8.hpp
@@ -34,7 +34,9 @@ namespace monad {
             /**
              * @brief Shape functions evaluated at a local point.
              *
+             * ```text
              * [ξ η]ᵀ → [N₁(ξ,η) ... N₈(ξ,η)]ᵀ
+             * ```
              *
              * @param[in] point Local point.
              *
@@ -45,11 +47,9 @@ namespace monad {
             /**
              * @brief Shape function gradients evaluated at a local point.
              *
-             * [ξ η]ᵀ →
-             *
              * ```text
-             * ⎡∂N₁/∂ξ ... ∂N₈/∂ξ⎤
-             * ⎣∂N₁/∂η ... ∂N₈/∂η⎦
+             * [ξ η]ᵀ → ⎡∂N₁/∂ξ ... ∂N₈/∂ξ⎤
+             *          ⎣∂N₁/∂η ... ∂N₈/∂η⎦
              * ```
              *
              * @param[in] point Local point.
@@ -58,7 +58,7 @@ namespace monad {
              */
             static ShapeFuncGradMatrix gradShapeFunctions(const Point &point) noexcept;
 
-            /// @brief Quadrature rule for integration.
+            /// @brief Quadrature rule.
             static QuadratureRule quadratureRule() noexcept;
 
             /// @brief Gmsh element type ID.
diff --git a/include/monad/fem/kernel/mechanical/linear_elastic_kernel.hpp b/include/monad/fem/kernel/mechanical/linear_elastic_kernel.hpp
index 54d4e6b..6eed3dd 100644
--- a/include/monad/fem/kernel/mechanical/linear_elastic_kernel.hpp
+++ b/include/monad/fem/kernel/mechanical/linear_elastic_kernel.hpp
@@ -3,7 +3,6 @@
 #include <stdexcept>
 #include <Eigen/Core>
 #include "monad/material/mechanical/linear_elastic_material.hpp"
-#include "monad/fem/kernel/mechanical/linear_elastic_kernel_traits.hpp"
 #include "monad/integration/integrate_matrix.hpp"
 #include "monad/detail/eigen_utils.hpp"
 
@@ -14,17 +13,22 @@ namespace monad {
         namespace mechanical {
 
             /**
-             * @brief Provides core FEM computations for a linear elastic element.
+             * @brief Core FEM computations for a linear elastic element.
              *
-             * This kernel implements the weak form of the linear elastic PDE:
+             * This kernel implements the weak form of the linear elasticity:
              *
+             * ```text
              * ∇·σ=∇·(Cε)=0
+             * ```
              *
-             * where the displacements are decomposed into macroscopic and microscopic components:
+             * where the displacements u∈ℝᵈ are decomposed into macroscopic and
+             * microscopic components:
              *
+             * ```text
              * u=ū+ũ
+             * ```
              *
-             * @tparam ElementT Element class (e.g. Quad4).
+             * @tparam ElementT Element type (e.g. Quad4).
              */
             template <class Element>
             struct LinearElasticKernel {
@@ -33,8 +37,7 @@ namespace monad {
                 /// @brief Number of dofs in the element.
                 static constexpr int NumDofs = Element::Dim * Element::NumNodes;
 
-                using Material = LinearElasticMaterial<Element::Dim>;
-                using Traits = LinearElasticKernelTraits<Element::Dim>;
+                using Material = material::LinearElasticMaterial<Element::Dim>;
 
                 using Point = typename Element::Point;
                 using NodesMatrix = typename Element::NodesMatrix;
@@ -49,23 +52,20 @@ namespace monad {
                 /**
                  * @brief Element B matrix evaluated at a local point.
                  *
-                 * The B matrix relates an element's nodal displacements u to strains ε in Voigt
-                 * notation at a local `point`:
+                 * The B matrix maps nodal displacements u∈ℝᵈ to
+                 * strains ε∈ℝᵛ at a local `point`:
                  *
+                 * ```text
                  * ε=Bu
-                 *
-                 * For a D-dimensional element:
-                 *
-                 * - D=2: point=[ξ η]ᵀ
-                 *
-                 * - D=3: point=[ξ η ζ]ᵀ
+                 * ```
                  *
                  * @param[in] point Local point.
                  * @param[in] nodes Element nodes.
                  *
                  * @returns Element B matrix evaluated at `point`.
                  *
-                 * @throws std::invalid_argument if `nodes` define a degenerate or inverted element geometry.
+                 * @throws std::invalid_argument if `nodes` define a degenerate element.
+                 * @throws std::invalid_argument if `nodes` define an inverted element.
                  */
                 static BMatrix bMatrix(const Point &point, const NodesMatrix &nodes) {
                     const auto J = Element::jacobian(point, nodes);
@@ -82,25 +82,61 @@ namespace monad {
                     const auto dN = Element::gradShapeFunctions(point);
                     const auto dNGlobal = J.inverse() * dN;
 
-                    BMatrix B;
-                    Traits::fillB(B, dNGlobal);
+                    BMatrix B = BMatrix::Zero();
+
+                    if constexpr (Element::Dim == 2) {
+                        // Normal strain ε₁₁
+                        B(0, Eigen::seq(0, Eigen::indexing::last, 2)) = dNGlobal.row(0);
+
+                        // Normal strain ε₂₂
+                        B(1, Eigen::seq(1, Eigen::indexing::last, 2)) = dNGlobal.row(1);
+
+                        // Shear strain ε₁₂
+                        B(2, Eigen::seq(0, Eigen::indexing::last, 2)) = dNGlobal.row(1);
+                        B(2, Eigen::seq(1, Eigen::indexing::last, 2)) = dNGlobal.row(0);
+                    }
+                    else {
+                        // Normal strain ε₁₁
+                        B(0, Eigen::seq(0, Eigen::indexing::last, 3)) = dNGlobal.row(0);
+
+                        // Normal strain ε₂₂
+                        B(1, Eigen::seq(1, Eigen::indexing::last, 3)) = dNGlobal.row(1);
+
+                        // Normal strain ε₃₃
+                        B(2, Eigen::seq(2, Eigen::indexing::last, 3)) = dNGlobal.row(2);
+
+                        // Shear strain ε₁₂
+                        B(3, Eigen::seq(0, Eigen::indexing::last, 3)) = dNGlobal.row(1);
+                        B(3, Eigen::seq(1, Eigen::indexing::last, 3)) = dNGlobal.row(0);
+
+                        // Shear strain ε₁₃
+                        B(4, Eigen::seq(0, Eigen::indexing::last, 3)) = dNGlobal.row(2);
+                        B(4, Eigen::seq(2, Eigen::indexing::last, 3)) = dNGlobal.row(0);
+
+                        // Shear strain ε₂₃
+                        B(5, Eigen::seq(1, Eigen::indexing::last, 3)) = dNGlobal.row(2);
+                        B(5, Eigen::seq(2, Eigen::indexing::last, 3)) = dNGlobal.row(1);
+                    }
 
                     return B;
                 }
 
                 /**
-                 * @brief Element mechanical stiffness matrix (left-hand side of the discretized weak form).
+                 * @brief Element mechanical stiffness matrix evaluated at a local point.
                  *
-                 * Weak form lhs for an element e:
+                 * For an element e:
                  *
+                 * ```text
                  * Kₑ=∫_ΩₑBᵀCBdΩₑ
+                 * ```
                  *
-                 * @param[in] point Local point.
+                 * @param[in] material Linear elastic material.
                  * @param[in] nodes Element nodes.
                  *
                  * @returns Element mechanical stiffness matrix evaluated at `point`.
                  *
-                 * @throws std::invalid_argument if `nodes` define a degenerate or inverted element geometry.
+                 * @throws std::invalid_argument if `nodes` define a degenerate element.
+                 * @throws std::invalid_argument if `nodes` define an inverted element.
                  */
                 static StiffnessMatrix lhs(const Material &material, const NodesMatrix &nodes) {
                     const auto rule = Element::quadratureRule();
@@ -116,25 +152,28 @@ namespace monad {
 
                     StiffnessMatrix K = integration::integrateMatrix(integrand, rule);
 
-                    // Remove numerical artifacts
+                    // Remove numerical asymmetry
                     detail::symmetrize(K);
 
                     return K;
                 }
 
                 /**
-                 * @brief Element force matrix (right-hand side of the discretized weak form).
+                 * @brief Element force matrix evaluated at a local point.
                  *
-                 * Weak form rhs for an element e:
+                 * For an element e:
                  *
+                 * ```text
                  * Fₑ=-∫_ΩₑBᵀCdΩₑε̄
+                 * ```
                  *
-                 * @param[in] point Local point.
+                 * @param[in] material Linear transport material.
                  * @param[in] nodes Element nodes.
                  *
                  * @returns Element force matrix evaluated at `point`.
                  *
-                 * @throws std::invalid_argument if `nodes` define a degenerate or inverted element geometry.
+                 * @throws std::invalid_argument if `nodes` define a degenerate element.
+                 * @throws std::invalid_argument if `nodes` define an inverted element.
                  */
                 static FieldMatrix rhs(const Material &material, const NodesMatrix &nodes) {
                     const auto rule = Element::quadratureRule();
@@ -148,6 +187,7 @@ namespace monad {
                         return B.transpose() * C * J.determinant();
                     };
 
+                    // No need to multiply by ε̄=I for unit macroscopic strains
                     return -integration::integrateMatrix(integrand, rule);
                 }
             };
diff --git a/include/monad/fem/kernel/mechanical/linear_elastic_kernel_traits.hpp b/include/monad/fem/kernel/mechanical/linear_elastic_kernel_traits.hpp
deleted file mode 100644
index b751993..0000000
--- a/include/monad/fem/kernel/mechanical/linear_elastic_kernel_traits.hpp
+++ /dev/null
@@ -1,113 +0,0 @@
-#pragma once
-
-#include <Eigen/Core>
-
-namespace monad {
-
-    namespace fem {
-
-        namespace mechanical {
-
-            /**
-             * @brief Linear elastic kernel traits.
-             *
-             * This struct isolates dimension-specific logic so that the finite element kernel
-             * can be written generically in terms of spatial dimension.
-             *
-             * @tparam D Spatial dimension (2 or 3).
-             */
-            template <int D>
-            struct LinearElasticKernelTraits;
-
-            /**
-             * @brief 2D linear elastic kernel traits.
-             *
-             * This struct isolates 2D-specific logic so that the finite element kernel
-             * can be written generically in terms of spatial dimension.
-             */
-            template <>
-            struct LinearElasticKernelTraits<2> {
-                /**
-                 * @brief 2D element B matrix evaluated at a local point.
-                 *
-                 * The B matrix relates an element's nodal displacements u to strains ε in Voigt
-                 * notation at a local point:
-                 *
-                 * ε=Bu
-                 *
-                 * @tparam BMatrix B matrix type.
-                 * @tparam ShapeFuncGradMatrix Shape function gradient matrix type.
-                 *
-                 * @param[in,out] B Element B matrix evaluated at a point.
-                 * @param[in] dN Element shape function gradients evaluated at a point.
-                 */
-                template <class BMatrix, class ShapeFuncGradMatrix>
-                static void fillB(BMatrix &B, const ShapeFuncGradMatrix &dN) {
-                    B.setZero();
-
-                    // Normal strain ε₁₁
-                    B(0, Eigen::seq(0, Eigen::indexing::last, 2)) = dN.row(0);
-
-                    // Normal strain ε₂₂
-                    B(1, Eigen::seq(1, Eigen::indexing::last, 2)) = dN.row(1);
-
-                    // Shear strain ε₁₂
-                    B(2, Eigen::seq(0, Eigen::indexing::last, 2)) = dN.row(1);
-                    B(2, Eigen::seq(1, Eigen::indexing::last, 2)) = dN.row(0);
-                }
-            };
-
-            /**
-             * @brief 3D linear elastic kernel traits.
-             *
-             * This struct isolates 3D-specific logic so that the finite element kernel
-             * can be written generically in terms of spatial dimension.
-             */
-            template <>
-            struct LinearElasticKernelTraits<3> {
-                /**
-                 * @brief 3D element B matrix evaluated at a local point.
-                 *
-                 * The B matrix relates an element's nodal displacements u to strains ε in Voigt
-                 * notation at a local point:
-                 *
-                 * ε=Bu
-                 *
-                 * @tparam BMatrix B matrix type.
-                 * @tparam ShapeFuncGradMatrix Shape function gradient matrix type.
-                 *
-                 * @param[in,out] B Element B matrix evaluated at a point.
-                 * @param[in] dN Element shape function gradients evaluated at a point.
-                 */
-                template <class BMatrix, class ShapeFuncGradMatrix>
-                static void fillB(BMatrix &B, const ShapeFuncGradMatrix &dN) {
-                    B.setZero();
-
-                    // Normal strain ε₁₁
-                    B(0, Eigen::seq(0, Eigen::indexing::last, 3)) = dN.row(0);
-
-                    // Normal strain ε₂₂
-                    B(1, Eigen::seq(1, Eigen::indexing::last, 3)) = dN.row(1);
-
-                    // Normal strain ε₃₃
-                    B(2, Eigen::seq(2, Eigen::indexing::last, 3)) = dN.row(2);
-
-                    // Shear strain ε₁₂
-                    B(3, Eigen::seq(0, Eigen::indexing::last, 3)) = dN.row(1);
-                    B(3, Eigen::seq(1, Eigen::indexing::last, 3)) = dN.row(0);
-
-                    // Shear strain ε₁₃
-                    B(4, Eigen::seq(0, Eigen::indexing::last, 3)) = dN.row(2);
-                    B(4, Eigen::seq(2, Eigen::indexing::last, 3)) = dN.row(0);
-
-                    // Shear strain ε₂₃
-                    B(5, Eigen::seq(1, Eigen::indexing::last, 3)) = dN.row(2);
-                    B(5, Eigen::seq(2, Eigen::indexing::last, 3)) = dN.row(1);
-                }
-            };
-
-        } // namespace mechanical
-
-    } // namespace fem
-
-} // namespace monad
diff --git a/include/monad/fem/kernel/multiphysics/linear_piezoelectric_kernel.hpp b/include/monad/fem/kernel/multiphysics/linear_piezoelectric_kernel.hpp
index 1004a42..8527748 100644
--- a/include/monad/fem/kernel/multiphysics/linear_piezoelectric_kernel.hpp
+++ b/include/monad/fem/kernel/multiphysics/linear_piezoelectric_kernel.hpp
@@ -8,6 +8,7 @@
 #include "monad/fem/kernel/mechanical/linear_elastic_kernel.hpp"
 #include "monad/fem/kernel/scalar/linear_scalar_diffusive_kernel.hpp"
 #include "monad/integration/integrate_matrix.hpp"
+#include "monad/detail/eigen_utils.hpp"
 
 namespace monad {
 
@@ -16,19 +17,24 @@ namespace monad {
         namespace multiphysics {
 
             /**
-             * @brief Provides core FEM computations for a linear piezoelectric element.
+             * @brief Core FEM computations for a linear piezoelectric element.
              *
-             * This kernel implements the weak form of the linear piezoelectric PDE:
+             * This kernel implements the weak form of the linear piezoelectricity:
              *
-             * ∇·S=∇·(cT-dᵀE)=0
-             * ∇·(-D)=∇·(-dT-εE)=0
+             * ```text
+             * ∇·T=∇·(cS-dᵀE)=0
+             * ∇·(D)=∇·(dS+ϵE)=0
+             * ```
              *
-             * where the displacements and electric potentials are decomposed into macroscopic and microscopic components:
+             * where the displacements u∈ℝᵈ and electric potentials φ∈ℝ are decomposed into
+             * macroscopic and microscopic components:
              *
+             * ```text
              * u=ū+ũ
              * φ=φ̄+φ̃
+             * ```
              *
-             * @tparam Element Element class (e.g. Quad4).
+             * @tparam Element Element type (e.g. Quad4).
              */
             template <class Element>
             struct LinearPiezoelectricKernel {
@@ -40,7 +46,9 @@ namespace monad {
                 /// @brief Number of dofs in the element.
                 static constexpr int NumDofs = ElectricalKernel::NumDofs + MechanicalKernel::NumDofs;
 
-                using Material = LinearPiezoelectricMaterial<LinearElasticMaterial<Element::Dim>, LinearTransportMaterial<Element::Dim>>;
+                using MechanicalMaterial = material::LinearElasticMaterial<Element::Dim>;
+                using ElectricalMaterial = material::LinearTransportMaterial<Element::Dim>;
+                using Material = material::LinearPiezoelectricMaterial<Element::Dim>;
 
                 using Point = typename Element::Point;
                 using NodesMatrix = typename Element::NodesMatrix;
@@ -51,37 +59,50 @@ namespace monad {
                 /// @brief Element piezoelectric stiffness matrix type.
                 using StiffnessMatrix = Eigen::Matrix<double, NumDofs, NumDofs>;
 
-                /// @brief Element mechanical field (induced by macroscopic electrical loading) matrix type.
+                /// @brief Element mechanical field matrix type induced by macroscopic electrical loading.
                 using UPhiCouplingFieldMatrix = Eigen::Matrix<double, MechanicalKernel::NumDofs, Element::Dim>;
 
-                /// @brief Element electrical field (induced by macroscopic mechanical loading) matrix type.
+                /// @brief Element electrical field matrix type induced by macroscopic mechanical loading.
                 using PhiUCouplingFieldMatrix = Eigen::Matrix<double, ElectricalKernel::NumDofs, Material::VoigtSize>;
 
                 /// @brief Element electromechanical field matrix type.
                 using FieldMatrix = Eigen::Matrix<double, NumDofs, Material::VoigtSize + Element::Dim>;
 
                 /**
-                 * @brief Element piezoelectric stiffness matrix (left-hand side of the discretized weak form).
+                 * @brief Element piezoelectric stiffness matrix evaluated at a local point.
                  *
-                 * Weak form lhs for an element e:
+                 * For an element e:
                  *
                  * ```text
                  * Kₑ = ⎡ (Kᵤᵤ)ₑ  -(Kᵤᵩ)ₑ⎤
                  *      ⎣-(Kᵩᵤ)ₑ  -(Kᵩᵩ)ₑ⎦
                  * ```
                  *
-                 * - (Kᵤᵤ)ₑ=∫_ΩₑBᵤᵀsBᵤdΩₑ (element mechanical stiffness matrix)
+                 * - Element mechanical stiffness matrix:
                  *
-                 * - (Kᵩᵩ)ₑ=∫_ΩᵩBᵤᵀεBᵩdΩₑ (element electrical stiffness matrix)
+                 * ```text
+                 * (Kᵤᵤ)ₑ=∫_ΩₑBᵤᵀcBᵤdΩₑ
+                 * ```
+                 *
+                 * - Element electrical stiffness matrix:
+                 *
+                 * ```text
+                 * (Kᵩᵩ)ₑ=∫_ΩᵩBᵤᵀεBᵩdΩₑ
+                 * ```
+                 *
+                 * - Element piezoelectric coupling stiffness matrix:
                  *
-                 * - (Kᵩᵤ)ₑ=(Kᵤᵩ)ₑᵀ=∫_ΩᵩBᵩᵀdBᵤdΩₑ=(∫_ΩᵩBᵤᵀdᵀBᵩdΩₑ)ᵀ (element coupling stiffness matrix)
+                 * ```text
+                 * (Kᵩᵤ)ₑ=(Kᵤᵩ)ₑᵀ=∫_ΩᵩBᵩᵀdBᵤdΩₑ=(∫_ΩᵩBᵤᵀdᵀBᵩdΩₑ)ᵀ
+                 * ```
                  *
-                 * @param[in] point Local point.
+                 * @param[in] material Linear piezoelectric material.
                  * @param[in] nodes Element nodes.
                  *
                  * @returns Element piezoelectric stiffness matrix evaluated at `point`.
                  *
-                 * @throws std::invalid_argument if `nodes` define a degenerate or inverted element geometry.
+                 * @throws std::invalid_argument if `nodes` define a degenerate element.
+                 * @throws std::invalid_argument if `nodes` define an inverted element.
                  */
                 static StiffnessMatrix lhs(const Material &material, const NodesMatrix &nodes) {
                     const auto rule = Element::quadratureRule();
@@ -96,37 +117,44 @@ namespace monad {
                         return Bphi.transpose() * d * Bu * J.determinant();
                     };
 
-                    const auto Kuu = MechanicalKernel::lhs(material.elasticMaterial(), nodes);
-                    const auto Kphiphi = ElectricalKernel::lhs(material.dielectricMaterial(), nodes);
+                    const MechanicalMaterial mechanicalMaterial(material.stiffnessTensor());
+                    const ElectricalMaterial electricalMaterial(material.permittivityTensor());
+
+                    const auto Kuu = MechanicalKernel::lhs(mechanicalMaterial, nodes);
+                    const auto Kphiphi = ElectricalKernel::lhs(electricalMaterial, nodes);
                     const CouplingStiffnessMatrix Kphiu = integration::integrateMatrix(integrand, rule);
 
                     StiffnessMatrix K;
                     K << Kuu, -Kphiu.transpose(),
                          -Kphiu, -Kphiphi;
 
+                    // Remove numerical asymmetry
+                    detail::symmetrize(K);
+
                     return K;
                 }
 
                 /**
-                 * @brief Element source matrix (right-hand side of the discretized weak form).
+                 * @brief Element source matrix evaluated at a local point.
                  *
-                 * Weak form rhs for an element e:
+                 * For an element e:
                  *
                  * ```text
                  * Fₑ = ⎡ (Fᵤ)ₑ⎤ = ⎡ (Fᵤᵤ)ₑ  (Fᵤᵩ)ₑ⎤
                  *      ⎣-(Fᵩ)ₑ⎦   ⎣-(Fᵩᵤ)ₑ -(Fᵩᵩ)ₑ⎦
                  * ```
                  *
-                 * - (Fᵤ)ₑ=[-∫_ΩₑBᵤᵀsdΩₑT̄  ∫_ΩₑBᵤᵀdᵀdΩₑĒ]
+                 * - (Fᵤ)ₑ=[-∫_ΩₑBᵤᵀcdΩₑT̄  ∫_ΩₑBᵤᵀdᵀdΩₑĒ]
                  *
                  * - (Fᵩ)ₑ=[-∫_ΩₑBᵩᵀdᵀdΩₑT̄ -∫_ΩₑBᵩᵀεdΩₑĒ]
                  *
-                 * @param[in] point Local point.
+                 * @param[in] material Linear piezoelectric material.
                  * @param[in] nodes Element nodes.
                  *
                  * @returns Element source matrix evaluated at `point`.
                  *
-                 * @throws std::invalid_argument if `nodes` define a degenerate or inverted element geometry.
+                 * @throws std::invalid_argument if `nodes` define a degenerate element.
+                 * @throws std::invalid_argument if `nodes` define an inverted element.
                  */
                 static FieldMatrix rhs(const Material &material, const NodesMatrix &nodes) {
                     const auto rule = Element::quadratureRule();
@@ -147,9 +175,16 @@ namespace monad {
                         return B.transpose() * d.transpose() * J.determinant();
                     };
 
-                    const auto Fuu = MechanicalKernel::rhs(material.elasticMaterial(), nodes);
-                    const auto Fphiphi = ElectricalKernel::rhs(material.dielectricMaterial(), nodes);
+                    const MechanicalMaterial mechanicalMaterial(material.stiffnessTensor());
+                    const ElectricalMaterial electricalMaterial(material.permittivityTensor());
+
+                    const auto Fuu = MechanicalKernel::rhs(mechanicalMaterial, nodes);
+                    const auto Fphiphi = ElectricalKernel::rhs(electricalMaterial, nodes);
+
+                    // No need to multiply by T̄=I for unit macroscopic strains
                     const PhiUCouplingFieldMatrix Fphiu = -integration::integrateMatrix(integrandPhiU, rule);
+
+                    // No need to multiply by Ē=I for unit macroscopic electric fields
                     const UPhiCouplingFieldMatrix Fuphi = integration::integrateMatrix(integrandUPhi, rule);
 
                     FieldMatrix F;
diff --git a/include/monad/fem/kernel/scalar/linear_scalar_diffusive_kernel.hpp b/include/monad/fem/kernel/scalar/linear_scalar_diffusive_kernel.hpp
index 05487e6..2a2bdea 100644
--- a/include/monad/fem/kernel/scalar/linear_scalar_diffusive_kernel.hpp
+++ b/include/monad/fem/kernel/scalar/linear_scalar_diffusive_kernel.hpp
@@ -2,9 +2,9 @@
 
 #include <stdexcept>
 #include <Eigen/Core>
+#include "monad/material/transport/linear_transport_material.hpp"
 #include "monad/integration/integrate_matrix.hpp"
 #include "monad/detail/eigen_utils.hpp"
-#include "monad/material/transport/linear_transport_material.hpp"
 
 namespace monad {
 
@@ -12,26 +12,31 @@ namespace monad {
 
         namespace scalar {
 
-            /// @brief Specifies the sign convention relating the physical field G to the scalar potential gradient ∇φ.
+            /// @brief Sign convention for the derived field ∇φ.
             enum class GradientConvention {
-                /// @brief Physical field is defined as field G=−∇φ.
+                /// @brief Derived field is defined as −∇φ.
                 Negative,
-                /// @brief Physical field is defined as field G=∇φ.
+                /// @brief Derived field is defined as ∇φ.
                 Positive
             };
 
             /**
-             * @brief Provides core FEM computations for a linear scalar diffusive element.
+             * @brief Core FEM computations for a linear scalar diffusive element.
              *
-             * This kernel implements the weak form of the scalar diffusion PDE:
+             * This kernel implements the weak form of scalar diffusion:
              *
+             * ```text
              * ∇·J=∇·(K∇φ)=0
+             * ```
              *
-             * where the scalar potentials are decomposed into macroscopic and microscopic components:
+             * where the scalar potentials φ∈ℝ are decomposed into macroscopic and
+             * microscopic components:
              *
+             * ```text
              * φ=φ̄+φ̃
+             * ```
              *
-             * @tparam ElementT Element class (e.g. Quad4).
+             * @tparam ElementT Element type (e.g. Quad4).
              * @tparam C Gradient sign convention (GradientConvention::Negative or Positive).
              */
             template <class ElementT, GradientConvention C>
@@ -46,7 +51,7 @@ namespace monad {
                 /// @brief Gradient sign.
                 static constexpr double GradSign = (C == GradientConvention::Negative ? -1.0 : 1.0);
 
-                using Material = LinearTransportMaterial<Element::Dim>;
+                using Material = material::LinearTransportMaterial<Element::Dim>;
 
                 using Point = typename Element::Point;
                 using NodesMatrix = typename Element::NodesMatrix;
@@ -61,23 +66,20 @@ namespace monad {
                 /**
                  * @brief Element B matrix evaluated at a local point.
                  *
-                 * The B matrix relates an element's nodal scalar potentials φ to scalar gradients at a
-                 * local `point`:
+                 * The B matrix masp nodal scalar potentials φ∈ℝ to
+                 * scalar potential gradients ∇φ∈ℝᵈ at a local `point`:
                  *
+                 * ```text
                  * ∇φ=Bφ
-                 *
-                 * For a D-dimensional element:
-                 *
-                 * - D=2: point=[ξ η]ᵀ
-                 *
-                 * - D=3: point=[ξ η ζ]ᵀ
+                 * ```
                  *
                  * @param[in] point Local point.
                  * @param[in] nodes Element nodes.
                  *
                  * @returns Element B matrix evaluated at `point`.
                  *
-                 * @throws std::invalid_argument if `nodes` define a degenerate or inverted element geometry.
+                 * @throws std::invalid_argument if `nodes` define a degenerate element.
+                 * @throws std::invalid_argument if `nodes` define an inverted element.
                  */
                 static BMatrix bMatrix(const Point &point, const NodesMatrix &nodes) {
                     const auto J = Element::jacobian(point, nodes);
@@ -97,18 +99,21 @@ namespace monad {
                 }
 
                 /**
-                 * @brief Element diffusive stiffness matrix (left-hand side of the discretized weak form) evaluated at a local point.
+                 * @brief Element diffusive stiffness matrix evaluated at a local point.
                  *
-                 * Weak form lhs for an element e:
+                 * For an element e:
                  *
-                 * Kₑ=∫_ΩₑBᵀABdΩₑ
+                 * ```text
+                 * Kₑ=∫_ΩₑBᵀKBdΩₑ
+                 * ```
                  *
                  * @param[in] material Linear transport material.
                  * @param[in] nodes Element nodes.
                  *
                  * @returns Element diffusive stiffness matrix evaluated at `point`.
                  *
-                 * @throws std::invalid_argument if `nodes` define a degenerate or inverted element geometry.
+                 * @throws std::invalid_argument if `nodes` define a degenerate element.
+                 * @throws std::invalid_argument if `nodes` define an inverted element.
                  */
                 static StiffnessMatrix lhs(const Material &material, const NodesMatrix &nodes) {
                     const auto rule = Element::quadratureRule();
@@ -124,25 +129,28 @@ namespace monad {
 
                     StiffnessMatrix K = integration::integrateMatrix(integrand, rule);
 
-                    // Remove numerical artifacts
+                    // Remove numerical asymmetry
                     detail::symmetrize(K);
 
                     return K;
                 }
 
                 /**
-                 * @brief Element source matrix (right-hand side of the discretized weak form) evaluated at a local point.
+                 * @brief Element source matrix evaluated at a local point.
                  *
-                 * Weak form rhs for an element e:
+                 * For an element e:
                  *
+                 * ```text
                  * Fₑ=-∫_ΩₑBᵀAdΩₑ∇φ̄
+                 * ```
                  *
                  * @param[in] material Linear transport material.
                  * @param[in] nodes Element nodes.
                  *
                  * @returns Element source matrix evaluated at `point`.
                  *
-                 * @throws std::invalid_argument if `nodes` define a degenerate or inverted element geometry.
+                 * @throws std::invalid_argument if `nodes` define a degenerate element.
+                 * @throws std::invalid_argument if `nodes` define an inverted element.
                  */
                 static FieldMatrix rhs(const Material &material, const NodesMatrix &nodes) {
                     const auto rule = Element::quadratureRule();
@@ -156,6 +164,7 @@ namespace monad {
                         return B.transpose() * A * J.determinant();
                     };
 
+                    // No need to multiply by ∇φ̄=I for unit macroscopic scalar potential gradients
                     return -integration::integrateMatrix(integrand, rule);
                 }
             };
diff --git a/include/monad/fem/operator/dof_map.hpp b/include/monad/fem/operator/dof_map.hpp
new file mode 100644
index 0000000..eebaec6
--- /dev/null
+++ b/include/monad/fem/operator/dof_map.hpp
@@ -0,0 +1,140 @@
+#pragma once
+
+#include <array>
+#include <cstddef>
+#include <stdexcept>
+#include <string>
+#include <vector>
+
+namespace monad {
+
+    namespace fem {
+
+        /**
+         * @brief Maps element periodic dofs into the reduced periodic dof space.
+         *
+         * Fixed dofs are stored as `-1`.
+         *
+         * @tparam Grid Grid type (e.g. Quad4Grid).
+         * @tparam Traits Dof traits type (e.g. LinearElasticDofTraits).
+         */
+        template <class Grid, class Traits>
+        class DofMap {
+        public:
+            using Element = typename Grid::Element;
+            using ElementDofs = std::array<int, Element::NumNodes * Traits::NumNodeDofs>;
+            using ElementDofList = std::vector<ElementDofs>;
+
+            /**
+             * @brief Constructs a dof map.
+             *
+             * @param[in] grid Grid.
+             */
+            explicit DofMap(const Grid &grid) {
+                numPeriodicNodes_ = grid.numPeriodicNodes();
+                numReducedDofs_ = Traits::NumNodeDofs * numPeriodicNodes_ - Traits::NumFixedDofs;
+
+                elementDofs_.reserve(grid.numElements());
+
+                for (std::size_t i = 0; i < grid.numElements(); ++i) {
+                    const auto element = grid.periodicElement(i);
+                    const auto periodicDofs = Traits::elementDofs(element, numPeriodicNodes_);
+
+                    ElementDofs reducedDofs;
+
+                    for (std::size_t j = 0; j < periodicDofs.size(); ++j) {
+                        const std::size_t periodicDof = periodicDofs[j];
+
+                        if (Traits::isFixedPeriodicDof(periodicDof, numPeriodicNodes_)) {
+                            reducedDofs[j] = -1;
+                        }
+                        else {
+                            const std::size_t reducedDof = Traits::periodicToReducedDof(periodicDof, numPeriodicNodes_); 
+                            reducedDofs[j] = static_cast<int>(reducedDof);
+                        }
+                    }
+
+                    elementDofs_.push_back(reducedDofs);
+                }
+            }
+
+            /// @brief Number of elements in the map.
+            std::size_t size() const noexcept {
+                return elementDofs_.size();
+            }
+
+            /// @brief Number of reduced periodic dofs.
+            std::size_t numReducedDofs() const noexcept {
+                return numReducedDofs_;
+            }
+
+            /**
+             * @brief Reduced periodic dofs of an element.
+             *
+             * @param[in] index Element index.
+             *
+             * @returns Reduced periodic dofs of an element.
+             *
+             * @throws std::out_of_range if `index` is outside the range [0, `size()`).
+             */
+            const ElementDofs &reducedDofs(std::size_t index) const {
+                if (index >= elementDofs_.size()) {
+                    throw std::out_of_range("Index (" + std::to_string(index) + ") is out of range [0," + std::to_string(elementDofs_.size()) + ").");
+                }
+
+                return elementDofs_[index];
+            }
+
+            /**
+             * @brief Returns `true` if a periodic dof is fixed.
+             *
+             * @param[in] dof Periodic dof.
+             *
+             * @returns `true` if `dof` is fixed, `false` otherwise.
+             */
+            bool isFixedPeriodicDof(std::size_t dof) const noexcept {
+                return Traits::isFixedPeriodicDof(dof, numPeriodicNodes_);
+            }
+
+            /**
+             * @brief Maps a periodic dof to a reduced periodic dof.
+             *
+             * The reduced periodic dof space is formed by removing fixed
+             * periodic dofs from the full periodic dof numbering.
+             *
+             * @param[in] dof Periodic dof.
+             *
+             * @returns Reduced periodic dof.
+             */
+            std::size_t periodicToReducedDof(std::size_t dof) const noexcept {
+                return Traits::periodicToReducedDof(dof, numPeriodicNodes_);
+            }
+
+            /**
+             * @brief Maps a reduced periodic dof to a periodic dof.
+             *
+             * The reduced periodic dof space is formed by removing fixed
+             * periodic dofs from the full periodic dof numbering.
+             *
+             * @param[in] dof Reduced periodic dof.
+             *
+             * @returns Periodic dof.
+             */
+            std::size_t reducedToPeriodicDof(std::size_t dof) const noexcept {
+                return Traits::reducedToPeriodicDof(dof, numPeriodicNodes_);
+            }
+
+        private:
+            /// @brief Number of periodic nodes.
+            std::size_t numPeriodicNodes_;
+
+            /// @brief Number of reduced periodic dofs.
+            std::size_t numReducedDofs_;
+
+            /// @brief Per-element mapping into the reduced periodic dof space.
+            ElementDofList elementDofs_;
+        };
+
+    } // namespace fem
+
+} // namespace monad
diff --git a/include/monad/fem/operator/matrix_free_operator.hpp b/include/monad/fem/operator/matrix_free_operator.hpp
index 6ea505e..19a6ccd 100644
--- a/include/monad/fem/operator/matrix_free_operator.hpp
+++ b/include/monad/fem/operator/matrix_free_operator.hpp
@@ -1,46 +1,40 @@
 #pragma once
 
 #include <cstddef>
-#include <array>
-#include <vector>
-#include <cmath>
 #ifdef _OPENMP
 #include <omp.h>
 #endif
 #include <Eigen/Core>
 #include <Eigen/SparseCore>
-#include "monad/detail/constants.hpp"
+#include "monad/field/density_field.hpp"
+#include "monad/fem/operator/dof_map.hpp"
 #include "monad/detail/eigen_utils.hpp"
-#include "monad/grid/grid_base.hpp"
 
 namespace monad {
 
     namespace fem {
 
         /**
-         * @brief Matrix-free operator for the global stiffness matrix.
+         * @brief Matrix-free operator for the reduced global stiffness matrix.
          *
-         * The operator represents the action of the global stiffness matrix
-         * on a vector without explicitly assembling the matrix and is defined
-         * on the reduced set of unconstrained periodic dofs.
+         * This class applies the global stiffness operator without explicitly
+         * assembling the global matrix.
          *
-         * @tparam Grid Grid class (e.g. Quad4Grid).
-         * @tparam Element Element class (e.g. Quad4).
-         * @tparam Traits Matrix free operator traits class (e.g. LinearElasticMatrixFreeOperatorTraits<2>).
+         * @tparam Grid Grid type (e.g. Quad4Grid).
+         * @tparam Traits Dof traits type (e.g. LinearElasticDofTraits<2>).
          *
          * @note Inspiration: https://libeigen.gitlab.io/eigen/docs-nightly/group__MatrixfreeSolverExample.html
          */
-        template <class Grid, class Element, class Traits>
-        class MatrixFreeOperator : public Eigen::EigenBase<MatrixFreeOperator<Grid, Element, Traits>> {
+        template <class Grid, class Traits>
+        class MatrixFreeOperator : public Eigen::EigenBase<MatrixFreeOperator<Grid, Traits>> {
         public:
-            static_assert(Element::Dim == Grid::Dim, "Element spatial dimension must equal grid spatial dimension.");
+            using Element = typename Grid::Element;
 
             /// @brief Number of dofs per element.
             static constexpr int NumElementDofs = Element::NumNodes * Traits::NumNodeDofs;
 
-            using ElementsList = typename GridBase<Grid, Element>::ElementsList;
-            using DensityList = typename GridBase<Grid, Element>::DensityList;
-            using ElementDofList = std::vector<std::array<int, NumElementDofs>>;
+            /// @brief Element dof vector type.
+            using DofVector = Eigen::Vector<double, NumElementDofs>;
 
             /// @brief Element stiffness matrix type.
             using StiffnessMatrix = Eigen::Matrix<double, NumElementDofs, NumElementDofs>;
@@ -55,73 +49,29 @@ namespace monad {
             };
 
             /**
-             * @brief Constructs the matrix-free operator.
+             * @brief Constructs a matrix-free operator.
              *
-             * @param[in] grid Periodic unit cell grid.
+             * @param[in] grid Grid.
+             * @param[in] densityField Per-element density field defined on `grid`.
              * @param[in] elementKReference Reference element stiffness matrix for unit density.
-             */
-            MatrixFreeOperator(const GridBase<Grid, Element> &grid, const StiffnessMatrix &elementKReference)
-                : elementKReference_(elementKReference), elements_(grid.periodicElements()), densities_(grid.densities()) {
-                const std::size_t numPeriodicNodes = grid.numPeriodicNodes();
-                const std::size_t numReducedDofs = Traits::NumNodeDofs * numPeriodicNodes - Traits::NumFixedDofs;
-
-                numRows_ = static_cast<int>(numReducedDofs);
-                numCols_ = numRows_;
-
-                // Precompute global-reduced dof mapping
-                // Fixed dofs are marked with -1 and skipped during gather/scatter
-                const std::size_t numElements = elements_.size();
-                elementDofs_.resize(numElements);
-
-                for (std::size_t i = 0; i < numElements; ++i) {
-                    auto &element = elements_[i];
-                    const auto dofs = Traits::dofs(element, numPeriodicNodes);
-
-                    for (std::size_t j = 0; j < dofs.size(); ++j) {
-                        const std::size_t dof = dofs[j];
-                        if (Traits::isFixedDof(dof, numPeriodicNodes)) {
-                            elementDofs_[i][j] = -1;
-                        }
-                        else {
-                            const std::size_t reducedDof = Traits::reducedDof(dof, numPeriodicNodes);
-                            elementDofs_[i][j] = static_cast<int>(reducedDof);
-                        }
-                    }
-                }
-            }
-
-            /// @brief Reference element stiffness matrix for unit density.
-            const StiffnessMatrix &elementKReference() const noexcept {
-                return elementKReference_;
-            }
-
-            /// @brief Periodic node indices for all elements.
-            const ElementsList &elements() const noexcept {
-                return elements_;
-            }
-
-            /// @brief Material densities for all elements.
-            const DensityList &densities() const noexcept {
-                return densities_;
-            }
-
-            /**
-             * @brief Reduced dofs for all elements.
              *
-             * @note Fixed dofs are set to -1.
+             * @throws std::invalid_argument if `grid` and `densityField` do not have the same resolution.
              */
-            const ElementDofList &elementDofs() const noexcept {
-                return elementDofs_;
+            MatrixFreeOperator(const Grid &grid, const field::DensityField<Grid::Dim> &densityField, const StiffnessMatrix &elementKReference)
+                : densityField_(densityField), dofMap_(grid), elementKReference_(elementKReference) {
+                if (grid.resolution() != densityField_.resolution()) {
+                    throw std::invalid_argument( "Grid resolution must match density field resolution.");
+                }
             }
 
             /// @brief Number of rows.
             int rows() const noexcept {
-                return numRows_;
+                return static_cast<int>(dofMap_.numReducedDofs());
             }
 
             /// @brief Number of columns.
             int cols() const noexcept {
-                return numCols_;
+                return static_cast<int>(dofMap_.numReducedDofs());
             }
 
             /// @brief `true` if the matrix-free operator is symmetric, `false` otherwise.
@@ -134,43 +84,95 @@ namespace monad {
                 return detail::isPSD(elementKReference_);
             }
 
+            /// @brief Per-element density field.
+            const field::DensityField<Grid::Dim> &densityField() const noexcept {
+                return densityField_;
+            };
+
+            /// @brief Reduced periodic dof map.
+            const DofMap<Grid, Traits> &dofMap() const noexcept {
+                return dofMap_;
+            }
+
+            /// @brief Reference element stiffness matrix for unit density.
+            const StiffnessMatrix& elementKReference() const noexcept {
+                return elementKReference_;
+            }
+
             /**
-             * @brief Overloads the multiplication operator to define the matrix-vector product y=Kx.
+             * @brief Matrix-vector multiplication Kx=y.
              *
-             * @tparam Rhs Type of the right-hand side vector.
+             * @tparam Lhs Type of the left-hand side vector x.
+             * @tparam Rhs Type of the right-hand side vector y.
              *
-             * @param[in] x Right-hand side vector.
-             *
-             * @returns Product Kx.
+             * @param[in] lhs Left-hand side vector x.
+             * @param[in,out] rhs Right-hand side vector y.
              */
-            template <typename Rhs>
-            Eigen::Product<MatrixFreeOperator, Rhs, Eigen::AliasFreeProduct>
-            operator*(const Eigen::MatrixBase<Rhs> &x) const {
-                return Eigen::Product<MatrixFreeOperator, Rhs, Eigen::AliasFreeProduct>(*this, x.derived());
-            }
+            template <typename Lhs, typename Rhs>
+            void apply(const Eigen::MatrixBase<Lhs> &lhs, Rhs &rhs) const {
+#ifdef _OPENMP
+#ifdef DEFAULT_OMP_NUM_THREADS
+                omp_set_num_threads(DEFAULT_OMP_NUM_THREADS);
+#endif
+                #pragma omp parallel for schedule(static)
+#endif
+                for (std::size_t i = 0; i < dofMap_.size(); ++i) {
+                    const auto &reducedDofs = dofMap_.reducedDofs(i);
+                    const double density = densityField_.getDensity(i);
 
-        private:
-            /// @brief Reference element stiffness matrix for unit density.
-            const StiffnessMatrix &elementKReference_;
+                    // Gather the element dof vector from the reduced periodic vector.
+                    // Fixed dofs are represented by -1 in the map and are treated as zero.
+                    DofVector x;
 
-            /// @brief Number of rows.
-            int numRows_;
+                    for (std::size_t j = 0; j < reducedDofs.size(); ++j) {
+                        const int reducedDof = reducedDofs[j];
+                        const int localReducedDof = static_cast<int>(j);
 
-            /// @brief Number of columns.
-            int numCols_;
+                        x(localReducedDof) = (reducedDof >= 0) ? lhs(reducedDof) : 0.0;
+                    }
 
-            /// @brief Periodic node indices for all elements.
-            ElementsList elements_;
+                    // Apply the density-scaled reference element stiffness
+                    const DofVector y = density * (elementKReference_ * x);
 
-            /// @brief Material densities for all elements.
-            DensityList densities_;
+                    // Scatter the local contribution back into the reduced global vector
+                    for (std::size_t j = 0; j < reducedDofs.size(); ++j) {
+                        const int reducedDof = reducedDofs[j];
+                        const int localReducedDofs = static_cast<int>(j);
+
+                        if (reducedDof >= 0) {
+#ifdef _OPENMP
+                            #pragma omp atomic
+#endif
+                            rhs(reducedDof) += y(localReducedDofs);
+                        }
+                    }
+                }
+            }
 
             /**
-             * @brief Reduced dofs for all elements.
+             * @brief Overloaded operator for matrix-vector multiplication y=Kx.
              *
-             * @note Fixed dofs are set to -1.
+             * @tparam Lhs Type of the right-hand side vector x.
+             *
+             * @param[in] x Left-hand side vector x.
+             *
+             * @returns Matrix-vector product Kx.
              */
-            ElementDofList elementDofs_;
+            template <typename Lhs>
+            Eigen::Product<MatrixFreeOperator, Lhs, Eigen::AliasFreeProduct>
+            operator*(const Eigen::MatrixBase<Lhs> &x) const {
+                return Eigen::Product<MatrixFreeOperator, Lhs, Eigen::AliasFreeProduct>(*this, x.derived());
+            }
+
+        private:
+            /// @brief Per-element density field.
+            const field::DensityField<Grid::Dim> &densityField_;
+
+            /// @brief Reduced periodic dof map.
+            DofMap<Grid, Traits> dofMap_;
+
+            /// @brief Reference element stiffness matrix for unit density.
+            const StiffnessMatrix &elementKReference_;
         };
 
     } // namespace fem
@@ -182,109 +184,58 @@ namespace Eigen {
     namespace internal {
 
         /**
-         * @brief This step informs Eigen's expression system that MatrixFreeOperator should be treated similarly to Eigen::SparseMatrix.
+         * @brief Eigen traits specialization for `MatrixFreeOperator`.
+         *
+         * This tells Eigen to treat `MatrixFreeOperator` like a sparse matrix in
+         * expression templates.
          *
-         * @tparam Grid Grid class (e.g. Quad4Grid).
-         * @tparam Element Element class (e.g. Quad4).
-         * @tparam Traits Matrix free operator traits class (e.g. LinearElasticMatrixFreeOperatorTraits).
+         * @tparam Grid Grid type (e.g. Quad4Grid).
+         * @tparam Traits Dof traits type (e.g. LinearElasticDofTraits<2>).
          *
          * @note Inspiration: https://libeigen.gitlab.io/eigen/docs-nightly/group__MatrixfreeSolverExample.html
          */
-        template <class Grid, class Element, class Traits>
-        struct traits<monad::fem::MatrixFreeOperator<Grid, Element, Traits>> : public Eigen::internal::traits<Eigen::SparseMatrix<double>> {};
+        template <class Grid, class Traits>
+        struct traits<monad::fem::MatrixFreeOperator<Grid, Traits>> : public Eigen::internal::traits<Eigen::SparseMatrix<double>> {};
 
         /**
-         * @brief Specializes Eigen's product implementation for the matrix-vector multiplication Kx.
+         * @brief  Eigen product specialization for matrix-vector multiplication.
          *
-         * @tparam Grid Grid class (e.g. Quad4Grid).
-         * @tparam Element Element class (e.g. Quad4).
-         * @tparam Traits Matrix free operator traits class (e.g. LinearElasticMatrixFreeOperatorTraits).
-         * @tparam Rhs Type of the right-hand side vector.
+         * @tparam Grid Grid type (e.g. Quad4Grid).
+         * @tparam Traits Dof traits type (e.g. LinearElasticDofTraits<2>).
+         * @tparam Lhs Type of the left-hand side vector.
          *
          * @note Inspiration: https://libeigen.gitlab.io/eigen/docs-nightly/group__MatrixfreeSolverExample.html
          */
-        template <class Grid, class Element, class Traits, typename Rhs>
+        template <class Grid, class Traits, typename Lhs>
         struct generic_product_impl<
-            monad::fem::MatrixFreeOperator<Grid, Element, Traits>,
-            Rhs,
+            monad::fem::MatrixFreeOperator<Grid, Traits>,
+            Lhs,
             SparseShape,
             DenseShape,
             GemvProduct // General matrix-vector product
-            > : generic_product_impl_base<monad::fem::MatrixFreeOperator<Grid, Element, Traits>, Rhs,
-                                          generic_product_impl<monad::fem::MatrixFreeOperator<Grid, Element, Traits>, Rhs>> {
-            using Operator = typename monad::fem::MatrixFreeOperator<Grid, Element, Traits>;
-            using Scalar = typename Product<Operator, Rhs>::Scalar;
+            > : generic_product_impl_base<monad::fem::MatrixFreeOperator<Grid, Traits>, Lhs,
+                                          generic_product_impl<monad::fem::MatrixFreeOperator<Grid, Traits>, Lhs>> {
+            using Operator = typename monad::fem::MatrixFreeOperator<Grid, Traits>;
+            using Scalar = typename Product<Operator, Lhs>::Scalar;
 
             /**
-             * @brief Implements the core matrix-vector multiplication y=Kx.
+             * @brief Matrix-vector multiplication Kx=y.
              *
-             * @tparam Dest The type of the destination vector y.
+             * @tparam Rhs Type of the right-hand side vector y.
              *
-             * @param[in,out] dst Destination vector y.
-             * @param[in] lhs Matrix-free operator K.
-             * @param[in] rhs Right-hand side vector x.
+             * @param[in,out] rhs Right-hand side vector y.
+             * @param[in] K Matrix-free operator.
+             * @param[in] lhs Left-hand side vector x.
              * @param[in] alpha Scaling factor (must be 1).
              *
              * @note Inspiration: https://libeigen.gitlab.io/eigen/docs-nightly/group__MatrixfreeSolverExample.html
              */
-            template <typename Dest>
-            static void scaleAndAddTo(Dest &dst, const Operator &lhs, const Rhs &rhs, const Scalar &alpha) {
+            template <typename Rhs>
+            static void scaleAndAddTo(Rhs &rhs, const Operator &K, const Lhs &lhs, const Scalar &alpha) {
                 eigen_assert(alpha == Scalar(1) && "scaling is not implemented");
                 EIGEN_ONLY_USED_FOR_DEBUG(alpha);
 
-                using DofVector = Eigen::Vector<double, Operator::NumElementDofs>;
-
-                const auto &elementKReference = lhs.elementKReference();
-                const auto &elements = lhs.elements();
-                const auto &densities = lhs.densities();
-                const auto &elementDofs = lhs.elementDofs();
-
-#ifdef _OPENMP
-#ifdef DEFAULT_OMP_NUM_THREADS
-                omp_set_num_threads(DEFAULT_OMP_NUM_THREADS);
-#endif
-                #pragma omp parallel for schedule(static)
-#endif
-                for (std::size_t i = 0; i < elements.size(); ++i) {
-                    const double density = densities[i];
-                    const auto &dofs = elementDofs[i];
-
-                    // --- Gather ---
-                    // Assemble the local vector x from the reduced global vector rhs.
-                    // Fixed dofs are treated as zero.
-                    DofVector x;
-
-                    for (std::size_t j = 0; j < dofs.size(); ++j) {
-                        const int globalDof = dofs[j];
-                        const int localDof = static_cast<int>(j);
-
-                        if (globalDof >= 0) {
-                            x(localDof) = rhs(globalDof);
-                        }
-                        else {
-                            x(localDof) = 0.0;
-                        }
-                    }
-
-                    // --- Apply ---
-                    // Apply the element stiffness Kx (scaled by material density).
-                    const DofVector y = density * (elementKReference * x);
-
-                    // --- Scatter ---
-                    // Accumulate local element Kx contributions into the reduced global vector dst.
-                    for (std::size_t j = 0; j < dofs.size(); ++j) {
-                        const int globalDof = dofs[j];
-                        const int localDof = static_cast<int>(j);
-
-                        if (globalDof >= 0) {
-
-#ifdef _OPENMP
-                            #pragma omp atomic
-#endif
-                            dst(globalDof) += y(localDof);
-                        }
-                    }
-                }
+                K.apply(lhs, rhs);
             }
         };
 
diff --git a/include/monad/fem/operator/mechanical/linear_elastic_dof_traits.hpp b/include/monad/fem/operator/mechanical/linear_elastic_dof_traits.hpp
new file mode 100644
index 0000000..4a1484f
--- /dev/null
+++ b/include/monad/fem/operator/mechanical/linear_elastic_dof_traits.hpp
@@ -0,0 +1,132 @@
+#pragma once
+
+#include <array>
+#include <cstddef>
+
+namespace monad {
+
+    namespace fem {
+
+        namespace mechanical {
+
+            /**
+             * @brief Dof numbering rules for periodic linear elasticity problems.
+             *
+             * @tparam D Spatial dimension (2 or 3).
+             */
+            template <int D>
+            struct LinearElasticDofTraits {
+                /// @brief Number of dofs per node.
+                static constexpr int NumNodeDofs = D;
+
+                /// @brief Number of fixed periodic dofs to remove rigid-body transformations.
+                static constexpr int NumFixedDofs = D;
+
+                /**
+                 * @brief Expands element node indices into dofs.
+                 *
+                 * Given an element with node indices [n₁, ..., nₖ], the
+                 * associated dofs are
+                 *
+                 * - 2D: [2n₁, 2n₁+1, ..., 2nₖ, 2nₖ+1]
+                 *
+                 * - 3D: [3n₁, 3n₁+1, 3n₁+2, ..., 3nₖ, 3nₖ+1, 3nₖ+2]
+                 *
+                 * @tparam K Number of element nodes.
+                 *
+                 * @param[in] element Element node indices.
+                 * @param[in] numNodes Total number of nodes.
+                 *
+                 * @returns Dofs associated with `element`.
+                 *
+                 * @note `numNodes` is unused here. It is kept only to match
+                 * the interface of other dof traits that require node-based offsets.
+                 */
+                template <std::size_t K>
+                static auto elementDofs(const std::array<std::size_t, K> &element, std::size_t numNodes) noexcept {
+                    (void)numNodes;
+
+                    std::array<std::size_t, NumNodeDofs * K> dofs;
+
+                    std::size_t index = 0;
+
+                    for (std::size_t i = 0; i < K; ++i) {
+                        const std::size_t node = element[i];
+
+                        for (std::size_t j = 0; j < NumNodeDofs; ++j) {
+                            dofs[index++] = NumNodeDofs * node + j;
+                        }
+                    }
+
+                    return dofs;
+                }
+
+                /**
+                 * @brief Returns `true` if a periodic dof is fixed.
+                 *
+                 * The first `D` periodic dofs are fixed.
+                 *
+                 * - 2D: [0, 1]
+                 *
+                 * - 3D: [0, 1, 2]
+                 *
+                 * @param[in] dof Periodic dof.
+                 * @param[in] numNodes Total number of periodic nodes.
+                 *
+                 * @returns `true` if `dof` is fixed, `false` otherwise.
+                 *
+                 * @note `numNodes` is unused here. It is kept only to match
+                 * the interface of other dof traits that require node-based offsets.
+                 */
+                static bool isFixedPeriodicDof(std::size_t dof, std::size_t numNodes) noexcept {
+                    (void)numNodes;
+
+                    return dof < NumFixedDofs;
+                }
+
+                /**
+                 * @brief Maps a periodic dof to a reduced periodic dof.
+                 *
+                 * The reduced periodic dof space is formed by removing fixed
+                 * periodic dofs from the full periodic dof numbering.
+                 *
+                 * @param[in] dof Periodic dof.
+                 * @param[in] numNodes Total number of periodic nodes.
+                 *
+                 * @returns Reduced periodic dof.
+                 *
+                 * @note `numNodes` is unused here. It is kept only to match
+                 * the interface of other dof traits that require node-based offsets.
+                 */
+                static std::size_t periodicToReducedDof(std::size_t dof, std::size_t numNodes) noexcept {
+                    (void)numNodes;
+
+                    return dof - NumFixedDofs;
+                }
+
+                /**
+                 * @brief Maps a reduced periodic dof to a periodic dof.
+                 *
+                 * The reduced periodic dof space is formed by removing fixed
+                 * periodic dofs from the full periodic dof numbering.
+                 *
+                 * @param[in] dof Reduced periodic dof.
+                 * @param[in] numNodes Total number of periodic nodes.
+                 *
+                 * @returns Periodic dof.
+                 *
+                 * @note `numNodes` is unused here. It is kept only to match
+                 * the interface of other dof traits that require node-based offsets.
+                 */
+                static std::size_t reducedToPeriodicDof(std::size_t dof, std::size_t numNodes) noexcept {
+                    (void)numNodes;
+
+                    return dof + NumFixedDofs;
+                }
+            };
+
+        } // namespace mechanical
+
+    } // namespace fem
+
+} // namespace monad
diff --git a/include/monad/fem/operator/mechanical/linear_elastic_matrix_free_operator_traits.hpp b/include/monad/fem/operator/mechanical/linear_elastic_matrix_free_operator_traits.hpp
deleted file mode 100644
index 0f62608..0000000
--- a/include/monad/fem/operator/mechanical/linear_elastic_matrix_free_operator_traits.hpp
+++ /dev/null
@@ -1,141 +0,0 @@
-#pragma once
-
-#include <array>
-#include <cstddef>
-
-namespace monad {
-
-    namespace fem {
-
-        namespace mechanical {
-
-            /**
-             * @brief Linear elastic matrix free operator traits.
-             *
-             * This struct isolates physics- and dimension-specific logic
-             * so that the matrix free operator can be written generically.
-             *
-             * @tparam D Spatial dimension (2 or 3).
-             */
-            template <int D>
-            struct LinearElasticMatrixFreeOperatorTraits {
-                /// @brief Number of dofs per node.
-                static constexpr int NumNodeDofs = D;
-
-                /// @brief Number of fixed dofs to remove rigid body transformations.
-                static constexpr int NumFixedDofs = D;
-
-                /**
-                 * @brief List of dofs associated with an element's nodes.
-                 *
-                 * Given an element with node indices [n₁, ..., nₖ], its dofs are:
-                 *
-                 * - D=2: [2n₁, 2n₁+1, ..., 2nₖ, 2nₖ+1]
-                 *
-                 * - D=3: [3n₁, 3n₁+1, 3n₁+2, ..., 3nₖ, 3nₖ+1, 3nₖ+2]
-                 *
-                 * @tparam K Number of nodes in the element.
-                 *
-                 * @param[in] element Node indices for a specific element.
-                 * @param[in] numNodes Number of nodes.
-                 *
-                 * @returns List of dofs associated with `element`'s nodes.
-                 *
-                 * @note The `numNodes` argument is included for multi-physics
-                 * operators where dof offsets may depend on the total number
-                 * of nodes.
-                 */
-                template <std::size_t K>
-                static auto dofs(const std::array<std::size_t, K> &element, std::size_t numNodes) noexcept {
-                    (void)numNodes;
-
-                    std::array<std::size_t, NumNodeDofs * K> data;
-
-                    std::size_t index = 0;
-
-                    for (std::size_t i = 0; i < K; ++i) {
-                        const std::size_t node = element[i];
-
-                        for (std::size_t j = 0; j < NumNodeDofs; ++j) {
-                            data[index++] = NumNodeDofs * node + j;
-                        }
-                    }
-
-                    return data;
-                }
-
-                /**
-                 * @brief Checks if a global dof is fixed.
-                 *
-                 * The fixed dofs are:
-                 *
-                 * - D=2: [0, 1]
-                 *
-                 * - D=3: [0, 1, 2]
-                 *
-                 * @param[in] dof Global dof.
-                 * @param[in] numNodes Number of nodes.
-                 *
-                 * @returns `true` if `dof` is fixed, `false` otherwise.
-                 *
-                 * @note The `numNodes` argument is included for multi-physics
-                 * operators where dof offsets may depend on the total number
-                 * of nodes.
-                 */
-                static bool isFixedDof(std::size_t dof, std::size_t numNodes) noexcept
-                {
-                    (void)numNodes;
-
-                    return dof < NumFixedDofs;
-                }
-
-                /**
-                 * @brief Maps a global dof to its reduced version.
-                 *
-                 * Matrix-free operators act only on unconstrained periodic dofs.
-                 * This function maps dofs from a space where fixed dofs are
-                 * included to a space where they are removed.
-                 *
-                 * @param[in] dof Global dof.
-                 * @param[in] numNodes Number of nodes.
-                 *
-                 * @returns Reduced dof.
-                 *
-                 * @note The `numNodes` argument is included for multi-physics
-                 * operators where dof offsets may depend on the total number
-                 * of nodes.
-                 */
-                static std::size_t reducedDof(std::size_t dof, std::size_t numNodes) noexcept {
-                    (void)numNodes;
-
-                    return dof - NumFixedDofs;
-                }
-
-                /**
-                 * @brief Maps a reduced dof to its global version.
-                 *
-                 * Matrix-free operators act only on unconstrained periodic dofs.
-                 * This function maps dofs from a space where fixed dofs
-                 * are removed to a space where they are included.
-                 *
-                 * @param[in] dof Reduced dof.
-                 * @param[in] numNodes Number of nodes.
-                 *
-                 * @returns Global dof.
-                 *
-                 * @note The `numNodes` argument is included for multi-physics
-                 * operators where dof offsets may depend on the total number
-                 * of nodes.
-                 */
-                static std::size_t expandedDof(std::size_t dof, std::size_t numNodes) noexcept {
-                    (void)numNodes;
-
-                    return dof + NumFixedDofs;
-                }
-            };
-
-        } // namespace mechanical
-
-    } // namespace fem
-
-} // namespace monad
diff --git a/include/monad/fem/operator/multiphysics/linear_piezoelectric_dof_traits.hpp b/include/monad/fem/operator/multiphysics/linear_piezoelectric_dof_traits.hpp
new file mode 100644
index 0000000..63d444c
--- /dev/null
+++ b/include/monad/fem/operator/multiphysics/linear_piezoelectric_dof_traits.hpp
@@ -0,0 +1,159 @@
+#pragma once
+
+#include <array>
+#include <cstddef>
+#include "monad/fem/operator/mechanical/linear_elastic_dof_traits.hpp"
+#include "monad/fem/operator/scalar/linear_scalar_diffusive_dof_traits.hpp"
+
+namespace monad {
+
+    namespace fem {
+
+        namespace multiphysics {
+
+            /**
+             * @brief Dof numbering rules for periodic linear piezoelectricity problems.
+             *
+             * @tparam D Spatial dimension (2 or 3).
+             */
+            template <int D>
+            struct LinearPiezoelectricDofTraits {
+                using MechanicalTraits = mechanical::LinearElasticDofTraits<D>;
+                using ElectricalTraits = scalar::LinearScalarDiffusiveDofTraits;
+
+                /// @brief Number of dofs per node.
+                static constexpr int NumNodeDofs = MechanicalTraits::NumNodeDofs + ElectricalTraits::NumNodeDofs;
+
+                /// @brief Number of fixed periodic dofs to remove rigid-body transformations.
+                static constexpr int NumFixedDofs = MechanicalTraits::NumFixedDofs + ElectricalTraits::NumFixedDofs;
+
+                /**
+                 * @brief Expands element node indices into dofs.
+                 *
+                 * Given an element with node indices [n₁, ..., nₖ], the
+                 * associated dofs are
+                 *
+                 * - 2D: [2n₁, 2n₁+1, ..., 2nₖ, 2nₖ+1, n₁+2k, ..., nₖ+2k]
+                 *
+                 * - 3D: [3n₁, 3n₁+1, 3n₁+2, ..., 3nₖ, 3nₖ+1, 3nₖ+2, n₁+3k, ..., nₖ+3k]
+                 *
+                 * @tparam K Number of element nodes.
+                 *
+                 * @param[in] element Element node indices.
+                 * @param[in] numNodes Total number of nodes.
+                 *
+                 * @returns Dofs associated with `element`.
+                 */
+                template <std::size_t K>
+                static auto elementDofs(const std::array<std::size_t, K> &element, std::size_t numNodes) noexcept {
+                    const std::size_t numMechanicalDofs = numNodes * MechanicalTraits::NumNodeDofs;
+
+                    // Mechanical dofs
+                    const auto mechanicalDofs = MechanicalTraits::elementDofs(element, numNodes);
+
+                    // Electrical dofs
+                    auto electricalDofs = ElectricalTraits::elementDofs(element, numNodes);
+
+                    // Offset electrical dofs by number of mechanical dofs
+                    for (auto &dof : electricalDofs) {
+                        dof += numMechanicalDofs;
+                    }
+
+                    // Concatenate
+                    std::array<std::size_t, NumNodeDofs * K> dofs;
+                    std::size_t index = 0;
+
+                    for (auto dof : mechanicalDofs) {
+                        dofs[index++] = dof;
+                    }
+                    for (auto dof : electricalDofs) {
+                        dofs[index++] = dof;
+                    }
+
+                    return dofs;
+                }
+
+                /**
+                 * @brief Returns `true` if a periodic dof is fixed.
+                 *
+                 * The first `D` mechanical dofs and the first electrical
+                 * dof are fixed:
+                 *
+                 * - 2D: [0, 1, 2m]
+                 *
+                 * - 3D: [0, 1, 2, 3m]
+                 *
+                 * @param[in] dof Periodic dof.
+                 * @param[in] numNodes Total number of periodic nodes.
+                 *
+                 * @returns `true` if `dof` is fixed, `false` otherwise.
+                 */
+                static bool isFixedPeriodicDof(std::size_t dof, std::size_t numNodes) noexcept {
+                    const std::size_t numMechanicalDofs = numNodes * MechanicalTraits::NumNodeDofs;
+
+                    // Fixed mechanical periodic dofs
+                    if (dof < numMechanicalDofs) {
+                        return MechanicalTraits::isFixedPeriodicDof(dof, numNodes);
+                    }
+
+                    // Fixed electrical periodic dof
+                    // Electrical periodic dofs are offset by number of mechanical periodic dofs
+                    return ElectricalTraits::isFixedPeriodicDof(dof - numMechanicalDofs, numNodes);
+                }
+
+                /**
+                 * @brief Maps a periodic dof to a reduced periodic dof.
+                 *
+                 * The reduced periodic dof space is formed by removing fixed
+                 * periodic dofs from the full periodic dof numbering.
+                 *
+                 * @param[in] dof Periodic dof.
+                 * @param[in] numNodes Total number of periodic nodes.
+                 *
+                 * @returns Reduced periodic dof.
+                 */
+                static std::size_t periodicToReducedDof(std::size_t dof, std::size_t numNodes) noexcept {
+                    const std::size_t numMechanicalDofs = numNodes * MechanicalTraits::NumNodeDofs;
+                    const std::size_t numReducedMechanicalDofs = numMechanicalDofs - MechanicalTraits::NumFixedDofs;
+
+                    // Mechanical periodic dofs
+                    if (dof < numMechanicalDofs) {
+                        return MechanicalTraits::periodicToReducedDof(dof, numNodes);
+                    }
+
+                    // Electrical periodic dofs
+                    // Electrical periodic dofs are offset by number of mechanical periodic dofs
+                    return ElectricalTraits::periodicToReducedDof(dof - numMechanicalDofs, numNodes) + numReducedMechanicalDofs;
+                }
+
+                /**
+                 * @brief Maps a reduced periodic dof to a periodic dof.
+                 *
+                 * The reduced periodic dof space is formed by removing fixed
+                 * periodic dofs from the full periodic dof numbering.
+                 *
+                 * @param[in] dof Reduced periodic dof.
+                 * @param[in] numNodes Total number of periodic nodes.
+                 *
+                 * @returns Periodic dof.
+                 */
+                static std::size_t reducedToPeriodicDof(std::size_t dof, std::size_t numNodes) noexcept {
+                    const std::size_t numMechanicalDofs = numNodes * MechanicalTraits::NumNodeDofs;
+                    const std::size_t numReducedMechanicalDofs = numMechanicalDofs - MechanicalTraits::NumFixedDofs;
+
+                    // Mechanical reduced periodic dofs
+                    if (dof < numReducedMechanicalDofs) {
+                        return MechanicalTraits::reducedToPeriodicDof(dof, numNodes);
+                    }
+
+                    // Electrical reduced periodic dofs
+                    // Electrical reduced periodic dofs are offset by number of mechanical reduced periodic dofs
+                    return ElectricalTraits::reducedToPeriodicDof(dof - numReducedMechanicalDofs, numNodes) + numMechanicalDofs;
+                }
+            };
+
+        } // namespace multiphysics
+
+    } // namespace fem
+
+} // namespace monad
diff --git a/include/monad/fem/operator/multiphysics/linear_piezoelectric_matrix_free_operator_traits.hpp b/include/monad/fem/operator/multiphysics/linear_piezoelectric_matrix_free_operator_traits.hpp
deleted file mode 100644
index 3fb9e34..0000000
--- a/include/monad/fem/operator/multiphysics/linear_piezoelectric_matrix_free_operator_traits.hpp
+++ /dev/null
@@ -1,178 +0,0 @@
-#pragma once
-
-#include <array>
-#include <cstddef>
-#include "monad/fem/operator/mechanical/linear_elastic_matrix_free_operator_traits.hpp"
-#include "monad/fem/operator/scalar/linear_scalar_diffusive_matrix_free_operator_traits.hpp"
-
-namespace monad {
-
-    namespace fem {
-
-        namespace multiphysics {
-
-            /**
-             * @brief Linear piezoelectric matrix free operator traits.
-             *
-             * This struct isolates physics- and dimension-specific logic
-             * so that the matrix free operator can be written generically.
-             *
-             * @tparam D Spatial dimension (2 or 3).
-             */
-            template <int D>
-            struct LinearPiezoelectricMatrixFreeOperatorTraits {
-                using MechanicalTraits = mechanical::LinearElasticMatrixFreeOperatorTraits<D>;
-                using ElectricalTraits = scalar::LinearScalarDiffusiveMatrixFreeOperatorTraits;
-
-                /// @brief Number of dofs per node.
-                static constexpr int NumNodeDofs = MechanicalTraits::NumNodeDofs + ElectricalTraits::NumNodeDofs;
-
-                /// @brief Number of fixed dofs to remove rigid body transformations.
-                static constexpr int NumFixedDofs = MechanicalTraits::NumFixedDofs + ElectricalTraits::NumFixedDofs;
-
-                /**
-                 * @brief List of dofs associated with an element's nodes.
-                 *
-                 * Given an element with node indices [n₁, ..., nₖ], its dofs are:
-                 *
-                 * - D=2: [2n₁, 2n₁+1, ..., 2nₖ, 2nₖ+1, n₁+2k, ..., nₖ+2k]
-                 *
-                 * - D=3: [3n₁, 3n₁+1, 3n₁+2, ..., 3nₖ, 3nₖ+1, 3nₖ+2, n₁+3k, ..., nₖ+3k]
-                 *
-                 * @tparam K Number of nodes in the element.
-                 *
-                 * @param[in] element Node indices for a specific element.
-                 * @param[in] numNodes Number of nodes.
-                 *
-                 * @returns List of dofs associated with `element`'s nodes.
-                 *
-                 * @note The `numNodes` argument is included for multi-physics
-                 * operators where dof offsets may depend on the total number
-                 * of nodes.
-                 */
-                template <std::size_t K>
-                static auto dofs(const std::array<std::size_t, K> &element, std::size_t numNodes) noexcept {
-                    const std::size_t numMechanicalDofs = numNodes * MechanicalTraits::NumNodeDofs;
-
-                    // Mechanical dofs
-                    const auto mechanicalDofs = MechanicalTraits::dofs(element, numNodes);
-
-                    // Electrical dofs
-                    auto electricalDofs = ElectricalTraits::dofs(element, numNodes);
-
-                    // Offset electrical dofs by number of mechanical dofs
-                    for (auto &dof : electricalDofs) {
-                        dof += numMechanicalDofs;
-                    }
-
-                    // Concatenate
-                    std::array<std::size_t, NumNodeDofs * K> data;
-                    std::size_t index = 0;
-
-                    for (auto dof : mechanicalDofs) {
-                        data[index++] = dof;
-                    }
-                    for (auto dof : electricalDofs) {
-                        data[index++] = dof;
-                    }
-
-                    return data;
-                }
-
-                /**
-                 * @brief Checks if a global dof is fixed.
-                 *
-                 * The fixed dofs are (assuming m total nodes):
-                 *
-                 * - D=2: [0, 1, 2m]
-                 *
-                 * - D=3: [0, 1, 2, 3m]
-                 *
-                 * @param[in] dof Global dof.
-                 * @param[in] numNodes Number of nodes.
-                 *
-                 * @returns `true` if `dof` is fixed, `false` otherwise.
-                 *
-                 * @note The `numNodes` argument is included for multi-physics
-                 * operators where dof offsets may depend on the total number
-                 * of nodes.
-                 */
-                static bool isFixedDof(std::size_t dof, std::size_t numNodes) noexcept {
-                    const std::size_t numMechanicalDofs = numNodes * MechanicalTraits::NumNodeDofs;
-
-                    // Fixed mechanical dofs
-                    if (dof < numMechanicalDofs) {
-                        return MechanicalTraits::isFixedDof(dof, numNodes);
-                    }
-
-                    // Fixed electrical dof
-                    // Electrical dofs are offset by number of mechanical dofs
-                    return ElectricalTraits::isFixedDof(dof - numMechanicalDofs, numNodes);
-                }
-
-                /**
-                 * @brief Maps a global dof to its reduced version.
-                 *
-                 * Matrix-free operators act only on unconstrained periodic dofs.
-                 * This function maps dofs from a space where fixed dofs are
-                 * included to a space where they are removed.
-                 *
-                 * @param[in] dof Global dof.
-                 * @param[in] numNodes Number of nodes.
-                 *
-                 * @returns Reduced dof.
-                 *
-                 * @note The `numNodes` argument is included for multi-physics
-                 * operators where dof offsets may depend on the total number
-                 * of nodes.
-                 */
-                static std::size_t reducedDof(std::size_t dof, std::size_t numNodes) noexcept {
-                    const std::size_t numMechanicalDofs = numNodes * MechanicalTraits::NumNodeDofs;
-                    const std::size_t numReducedMechanicalDofs = numMechanicalDofs - MechanicalTraits::NumFixedDofs;
-
-                    // Shifting for mechanical dofs.
-                    if (dof < numMechanicalDofs) {
-                        return MechanicalTraits::reducedDof(dof, numNodes);
-                    }
-
-                    // Shifting for electrical dof
-                    // Electrical dofs are offset by number of mechanical dofs
-                    return ElectricalTraits::reducedDof(dof - numMechanicalDofs, numNodes) + numReducedMechanicalDofs;
-                }
-
-                /**
-                 * @brief Maps a reduced dof to its global version.
-                 *
-                 * Matrix-free operators act only on unconstrained periodic dofs.
-                 * This function maps dofs from a space where fixed dofs
-                 * are removed to a space where they are included.
-                 *
-                 * @param[in] dof Reduced dof.
-                 * @param[in] numNodes Number of nodes.
-                 *
-                 * @returns Global dof.
-                 *
-                 * @note The `numNodes` argument is included for multi-physics
-                 * operators where dof offsets may depend on the total number
-                 * of nodes.
-                 */
-                static std::size_t expandedDof(std::size_t dof, std::size_t numNodes) noexcept {
-                    const std::size_t numMechanicalDofs = numNodes * MechanicalTraits::NumNodeDofs;
-                    const std::size_t numReducedMechanicalDofs = numMechanicalDofs - MechanicalTraits::NumFixedDofs;
-
-                    // Shifting for mechanical dofs
-                    if (dof < numReducedMechanicalDofs) {
-                        return MechanicalTraits::expandedDof(dof, numNodes);
-                    }
-
-                    // Shifting for electrical dof
-                    // Electrical dofs are offset by number of mechanical dofs
-                    return ElectricalTraits::expandedDof(dof - numReducedMechanicalDofs, numNodes) + numMechanicalDofs;
-                }
-            };
-
-        } // namespace multiphysics
-
-    } // namespace fem
-
-} // namespace monad
diff --git a/include/monad/fem/operator/scalar/linear_scalar_diffusive_dof_traits.hpp b/include/monad/fem/operator/scalar/linear_scalar_diffusive_dof_traits.hpp
new file mode 100644
index 0000000..985e532
--- /dev/null
+++ b/include/monad/fem/operator/scalar/linear_scalar_diffusive_dof_traits.hpp
@@ -0,0 +1,107 @@
+#pragma once
+
+#include <array>
+#include <cstddef>
+
+namespace monad {
+
+    namespace fem {
+
+        namespace scalar {
+
+            /// @brief Dof numbering rules for periodic linear scalar diffusive problems.
+            struct LinearScalarDiffusiveDofTraits {
+                /// @brief Number of dofs per node.
+                static constexpr int NumNodeDofs = 1;
+
+                /// @brief Number of fixed periodic dofs to remove rigid-body transformations.
+                static constexpr int NumFixedDofs = 1;
+
+                /**
+                 * @brief Expands element node indices into dofs.
+                 *
+                 * Given an element with node indices [n₁, ..., nₖ], the
+                 * associated dofs are [n₁, ..., nₖ].
+                 *
+                 * @tparam K Number of element nodes.
+                 *
+                 * @param[in] element Element node indices.
+                 * @param[in] numNodes Total number of nodes.
+                 *
+                 * @returns Dofs associated with `element`.
+                 *
+                 * @note `numNodes` is unused here. It is kept only to match
+                 * the interface of other dof traits that require node-based offsets.
+                 */
+                template <std::size_t K>
+                static auto elementDofs(const std::array<std::size_t, K> &element, std::size_t numNodes) noexcept {
+                    (void)numNodes;
+
+                    return element;
+                }
+
+                /**
+                 * @brief Returns `true` if a periodic dof is fixed.
+                 *
+                 * The first periodic dof is fixed.
+                 *
+                 * @param[in] dof Periodic dof.
+                 * @param[in] numNodes Total number of periodic nodes.
+                 *
+                 * @returns `true` if `dof` is fixed, `false` otherwise.
+                 *
+                 * @note `numNodes` is unused here. It is kept only to match
+                 * the interface of other dof traits that require node-based offsets.
+                 */
+                static bool isFixedPeriodicDof(std::size_t dof, std::size_t numNodes) noexcept {
+                    (void)numNodes;
+
+                    return dof == 0;
+                }
+
+                /**
+                 * @brief Maps a periodic dof to a reduced periodic dof.
+                 *
+                 * The reduced periodic dof space is formed by removing fixed
+                 * periodic dofs from the full periodic dof numbering.
+                 *
+                 * @param[in] dof Periodic dof.
+                 * @param[in] numNodes Total number of periodic nodes.
+                 *
+                 * @returns Reduced periodic dof.
+                 *
+                 * @note `numNodes` is unused here. It is kept only to match
+                 * the interface of other dof traits that require node-based offsets.
+                 */
+                static std::size_t periodicToReducedDof(std::size_t dof, std::size_t numNodes) noexcept {
+                    (void)numNodes;
+
+                    return dof - NumFixedDofs;
+                }
+
+                /**
+                 * @brief Maps a reduced periodic dof to a periodic dof.
+                 *
+                 * The reduced periodic dof space is formed by removing fixed
+                 * periodic dofs from the full periodic dof numbering.
+                 *
+                 * @param[in] dof Reduced periodic dof.
+                 * @param[in] numNodes Total number of periodic nodes.
+                 *
+                 * @returns Periodic dof.
+                 *
+                 * @note `numNodes` is unused here. It is kept only to match
+                 * the interface of other dof traits that require node-based offsets.
+                 */
+                static std::size_t reducedToPeriodicDof(std::size_t dof, std::size_t numNodes) noexcept {
+                    (void)numNodes;
+
+                    return dof + NumFixedDofs;
+                }
+            };
+
+        } // namespace scalar
+
+    } // namespace fem
+
+} // namespace monad
diff --git a/include/monad/fem/operator/scalar/linear_scalar_diffusive_matrix_free_operator_traits.hpp b/include/monad/fem/operator/scalar/linear_scalar_diffusive_matrix_free_operator_traits.hpp
deleted file mode 100644
index 10ed857..0000000
--- a/include/monad/fem/operator/scalar/linear_scalar_diffusive_matrix_free_operator_traits.hpp
+++ /dev/null
@@ -1,123 +0,0 @@
-#pragma once
-
-#include <array>
-#include <cstddef>
-
-namespace monad {
-
-    namespace fem {
-
-        namespace scalar {
-
-            /**
-             * @brief Linear scalar diffusive matrix free operator traits.
-             *
-             * This struct isolates physics- and dimension-specific logic
-             * so that the matrix free operator can be written generically.
-             */
-            struct LinearScalarDiffusiveMatrixFreeOperatorTraits {
-                /// @brief Number of dofs per node.
-                static constexpr int NumNodeDofs = 1;
-
-                /// @brief Number of fixed dofs to remove rigid body transformations.
-                static constexpr int NumFixedDofs = 1;
-
-                /**
-                 * @brief List of dofs associated with an element's nodes.
-                 *
-                 * Given an element with node indices [n₁, ..., nₖ], its dofs are:
-                 *
-                 * [n₁, ..., nₖ]
-                 *
-                 * @tparam K Number of nodes in the element.
-                 *
-                 * @param[in] element Node indices for a specific element.
-                 * @param[in] numNodes Number of nodes.
-                 *
-                 * @returns List of dofs associated with `element`'s nodes.
-                 *
-                 * @note The `numNodes` argument is included for multi-physics
-                 * operators where dof offsets may depend on the total number
-                 * of nodes.
-                 */
-                template <std::size_t K>
-                static auto dofs(const std::array<std::size_t, K> &element, std::size_t numNodes) noexcept {
-                    (void)numNodes;
-
-                    auto dofs = element;
-
-                    return dofs;
-                }
-
-                /**
-                 * @brief Checks if a global dof is fixed.
-                 *
-                 * The fixed dofs are:
-                 *
-                 * [0]
-                 *
-                 * @param[in] dof Global dof.
-                 * @param[in] numNodes Number of nodes.
-                 *
-                 * @returns `true` if `dof` is fixed, `false` otherwise.
-                 *
-                 * @note The `numNodes` argument is included for multi-physics
-                 * operators where dof offsets may depend on the total number
-                 * of nodes.
-                 */
-                static bool isFixedDof(std::size_t dof, std::size_t numNodes) noexcept {
-                    (void)numNodes;
-
-                    return dof == 0;
-                }
-
-                /**
-                 * @brief Maps a global dof to its reduced version.
-                 *
-                 * Matrix-free operators act only on unconstrained periodic dofs.
-                 * This function maps dofs from a space where fixed dofs are
-                 * included to a space where they are removed.
-                 *
-                 * @param[in] dof Global dof.
-                 * @param[in] numNodes Number of nodes.
-                 *
-                 * @returns Reduced dof.
-                 *
-                 * @note The `numNodes` argument is included for multi-physics
-                 * operators where dof offsets may depend on the total number
-                 * of nodes.
-                 */
-                static std::size_t reducedDof(std::size_t dof, std::size_t numNodes) noexcept {
-                    (void)numNodes;
-
-                    return dof - NumFixedDofs;
-                }
-
-                /**
-                 * @brief Maps a reduced dof to its global version.
-                 *
-                 * Matrix-free operators act only on unconstrained periodic dofs.
-                 * This function maps dofs from a space where fixed dofs
-                 * are removed to a space where they are included.
-                 *
-                 * @param[in] dof Reduced dof.
-                 * @param[in] numNodes Number of nodes.
-                 *
-                 * @returns Global dof.
-                 *
-                 * @note The `numNodes` argument is included for multi-physics
-                 * operators where dof offsets may depend on the total number
-                 * of nodes.
-                 */
-                static std::size_t expandedDof(std::size_t dof, std::size_t numNodes) noexcept {
-                    (void)numNodes;
-
-                    return dof + NumFixedDofs;
-                }
-            };
-
-        } // namespace scalar
-
-    } // namespace fem
-
-} // namespace monad
diff --git a/include/monad/field/density_field.hpp b/include/monad/field/density_field.hpp
new file mode 100644
index 0000000..c3b15e6
--- /dev/null
+++ b/include/monad/field/density_field.hpp
@@ -0,0 +1,271 @@
+#pragma once
+
+#include <algorithm>
+#include <array>
+#include <cstddef>
+#include <random>
+#include <stdexcept>
+#include <string>
+#include <vector>
+#include "monad/detail/constants.hpp"
+
+namespace monad {
+
+    namespace field {
+
+        /**
+         * @brief Per-element density field.
+         *
+         * @tparam D Spatial dimension (2 or 3).
+         */
+        template <int D>
+        class DensityField {
+        public:
+            static_assert(D == 2 || D == 3, "Density field spatial dimension must be 2 or 3.");
+
+            static constexpr int Dim = D;
+
+            using Resolution = std::array<std::size_t, D>;
+            using DensityList = std::vector<double>;
+
+            /**
+             * @brief Constructs a density field.
+             *
+             * @param[in] resolution Number of elements in each dimension.
+             *
+             * @throws std::invalid_argument if any entry in `resolution` is zero.
+             *
+             * @note Densities are initalized to zero.
+             */
+            explicit DensityField(const Resolution& resolution)
+                : resolution_(resolution) {
+                for (std::size_t i = 0; i < D; ++i) {
+                    if (resolution_[i] == 0) {
+                        throw std::invalid_argument("Resolution in dimension " + std::to_string(i + 1) + " must be positive.");
+                    }
+                }
+
+                densities_.resize(numElements());
+                setZeros();
+            }
+
+            /// @brief Number of elements in each dimension.
+            const Resolution& resolution() const noexcept {
+                return resolution_;
+            }
+
+            /// @brief Number of elements.
+            std::size_t numElements() const noexcept {
+                std::size_t total = 1;
+
+                for (std::size_t n : resolution_) {
+                    total *= n;
+                }
+
+                return total;
+            }
+
+            /**
+             * @brief Per-element material densities.
+             *
+             * @note Densities are stored in row-major order.
+             */
+            const DensityList& densities() const noexcept {
+                return densities_;
+            }
+
+            /**
+             * @brief Material density of an element.
+             *
+             * @param[in] index Element index.
+             *
+             * @returns Material density of an element.
+             *
+             * @throws std::out_of_range if `index` is outside the range [0,`numElements()`).
+             */
+            double getDensity(std::size_t index) const {
+                if (index >= numElements()) {
+                    throw std::out_of_range("Index (" + std::to_string(index) + ") is out of range [0," + std::to_string(numElements()) + ").");
+                }
+
+                return densities_[index];
+            }
+
+            /**
+             * @brief Set the material density of an element.
+             *
+             * @param[in] index Element index.
+             * @param[in] density Density.
+             *
+             * @throws std::invalid_argument if `density` is outside the range [0,1].
+             * @throws std::out_of_range if `index` is outside the range [0,`numElements()`).
+             *
+             * @note Densities are clamped to a minimum value of `ZERO_TOLERANCE` to improve
+             * numerical stability.
+             */
+            void setDensity(std::size_t index, double density) {
+                if (index >= numElements()) {
+                    throw std::out_of_range("Index (" + std::to_string(index) + ") is out of range [0," + std::to_string(numElements()) + ").");
+                }
+
+                if (density < 0.0 || density > 1.0) {
+                    throw std::invalid_argument("Density (" + std::to_string(density) + ") is out of range [0,1].");
+                }
+
+                densities_[index] = std::max(NUMERICAL_ZERO, density);
+            }
+
+            /**
+             * @brief Set the material densities.
+             *
+             * @param[in] densities Densities.
+             *
+             * @throws std::invalid_argument if the size of `densities` does not equal the number of grid elements.
+             * @throws std::invalid_argument if any value in `densities` is outside the range [0,1].
+             *
+             * @note Densities are stored in row-major order.
+             * @note Densities are clamped to a minimum value of `ZERO_TOLERANCE` to improve numerical stability.
+             */
+            void setDensities(const DensityList& densities) {
+                if (densities.size() != numElements()) {
+                    throw std::invalid_argument("Densities size (" + std::to_string(densities.size()) + ") must equal number of elements (" + std::to_string(numElements()) + ").");
+                }
+
+                for (std::size_t i = 0; i < densities.size(); ++i) {
+                    setDensity(i, densities[i]);
+                }
+            }
+
+            /**
+             * @brief Sets all material densities to a constant value.
+             *
+             * @param[in] density Density.
+             *
+             * @throws std::invalid_argument if `density` is outside the range [0,1].
+             *
+             * @note Densities are clamped to a minimum value of `ZERO_TOLERANCE` to
+             * improve numerical stability.
+             */
+            void setConstant(double density) {
+                if (density < 0.0 || density > 1.0) {
+                    throw std::invalid_argument("Density (" + std::to_string(density) + ") is out of range [0,1].");
+                }
+
+                for (std::size_t i = 0; i < numElements(); ++i) {
+                    setDensity(i, density);
+                }
+            }
+
+            /// @brief Sets all material densities to zero.
+            void setZeros() noexcept {
+                setConstant(0.0);
+            }
+
+            /// @brief Sets all material densities to one.
+            void setOnes() noexcept {
+                setConstant(1.0);
+            }
+
+            /**
+             * @brief Set the material densities to random values.
+             *
+             * @param[in] seed RNG seed (optional).
+             *
+             * @note Densities are clamped to a minimum value of
+             * `ZERO_TOLERANCE` to improve numerical stability.
+             */
+            void setRandom(unsigned int seed = std::random_device{}()) noexcept {
+                std::mt19937 rng(seed);
+
+                for (std::size_t i = 0; i < numElements(); ++i) {
+                    // std::uniform_real_distribution is not portably defined across compilers.
+                    // Use the standardized mt19937 engine to guarantee consistency across Mac,
+                    // Windows, and Linux.
+                    const double density = static_cast<double>(rng()) / static_cast<double>(std::mt19937::max());
+
+                    setDensity(i, density);
+                }
+            }
+
+            /**
+             * @brief Periodically translates the element densities.
+             *
+             * @param[in] shift Shift in each directions.
+             */
+            void translate(const Resolution& shift) noexcept {
+                DensityList shifted(numElements());
+
+                if constexpr (D == 2) {
+                    const std::size_t nx = resolution_[0];
+                    const std::size_t ny = resolution_[1];
+
+                    for (std::size_t i = 0; i < nx; ++i) {
+                        for (std::size_t j = 0; j < ny; ++j) {
+                            const std::size_t oldIndex = j * nx + i;
+                            const std::size_t iNew = (i + shift[0]) % nx;
+                            const std::size_t jNew = (j + shift[1]) % ny;
+                            const std::size_t newIndex = jNew * nx + iNew;
+                            
+                            shifted[newIndex] = densities_[oldIndex];
+                        }
+                    }
+                }
+                else {
+                    const std::size_t nx = resolution_[0];
+                    const std::size_t ny = resolution_[1];
+                    const std::size_t nz = resolution_[2];
+                    const std::size_t elementsPerPlane = nx * ny;
+
+                    for (std::size_t i = 0; i < nx; ++i) {
+                        for (std::size_t j = 0; j < ny; ++j) {
+                            for (std::size_t k = 0; k < nz; ++k) {
+                                const std::size_t oldIndex = k * elementsPerPlane + j * nx + i;
+                                const std::size_t iNew = (i + shift[0]) % nx;
+                                const std::size_t jNew = (j + shift[1]) % ny;
+                                const std::size_t kNew = (k + shift[2]) % nz;
+                                const std::size_t newIndex = kNew * elementsPerPlane + jNew * nx + iNew;
+                                
+                                shifted[newIndex] = densities_[oldIndex];
+                            }
+                        }
+                    }
+                }
+
+                setDensities(shifted);
+            }
+
+            /// @brief Equality comparison.
+            bool operator==(const DensityField& other) const noexcept {
+                auto vectorEqual = [](const DensityList& v1, const DensityList& v2) noexcept {
+                    if (v1.size() != v2.size()) {
+                        return false;
+                    }
+
+                    for (std::size_t i = 0; i < v1.size(); ++i) {
+                        if (std::fabs(v1[i] - v2[i]) > NUMERICAL_ZERO) {
+                            return false;
+                        }
+                    }
+
+                    return true;
+                };
+
+                return resolution_ == other.resolution_ && vectorEqual(densities_, other.densities_);
+            }
+
+            /// @brief Inequality comparison.
+            bool operator!=(const DensityField& other) const noexcept {
+                return !(*this == other);
+            }
+            
+        private:
+            /// @brief Number of elements in each dimension.
+            Resolution resolution_;
+
+            /// @brief Per-element material densities.
+            DensityList densities_;
+        };
+
+    } // namespace field
+
+} // namespace monad
diff --git a/include/monad/field/field_aliases.hpp b/include/monad/field/field_aliases.hpp
new file mode 100644
index 0000000..d223bf4
--- /dev/null
+++ b/include/monad/field/field_aliases.hpp
@@ -0,0 +1,13 @@
+#pragma once
+
+#include "monad/field/density_field.hpp"
+
+namespace monad {
+    
+    /// @brief Per-element density field for 2d grids.
+    using DensityField2d = field::DensityField<2>;
+
+    /// @brief Per-element density field for 2d grids.
+    using DensityField3d = field::DensityField<3>;
+
+} // namespace monad
diff --git a/include/monad/field/make_density_field_from_csv.hpp b/include/monad/field/make_density_field_from_csv.hpp
new file mode 100644
index 0000000..6c3a9c9
--- /dev/null
+++ b/include/monad/field/make_density_field_from_csv.hpp
@@ -0,0 +1,26 @@
+#pragma once
+
+#include <string>
+#include "monad/field/field_aliases.hpp" 
+    
+namespace monad {
+
+    /**
+     * @brief Constructs a 2D density field from a CSV file.
+     *
+     * @param[in] file Path to the CSV file.
+     *
+     * @returns 2D density field defined by the `file` entries.
+     *
+     * @throws std::runtime_error if `file` cannot be opened.
+     * @throws std::runtime_error if `file` is empty.
+     * @throws std::runtime_error if `file` contains non-numeric values.
+     * @throws std::runtime_error if `file` contains any value outside the range [0,1].
+     * @throws std::runtime_error if `file` contains rows with inconsistent lengths.
+     *
+     * @note CSV entries are interpreted as a 2D grid with the origin at the
+     * bottom-left corner.
+     */
+    DensityField2d makeDensityFieldFromCsv(const std::string& file);
+
+} // namespace monad
diff --git a/include/monad/field/make_density_field_from_function.hpp b/include/monad/field/make_density_field_from_function.hpp
new file mode 100644
index 0000000..e3fbe4d
--- /dev/null
+++ b/include/monad/field/make_density_field_from_function.hpp
@@ -0,0 +1,63 @@
+#pragma once
+
+#include <cstddef>
+#include <functional>
+#include <cmath>
+#include <stdexcept>
+#include "monad/field/density_field.hpp" 
+#include "monad/integration/integrate_scalar.hpp"
+
+namespace monad {
+
+    /**
+     * @brief Constructs a density field from a continuous function.
+     *
+     * Element densities are computed by averaging a continuous function
+     *
+     * ```text
+     * f:ℝᵈ→[0,1]
+     * ```
+     *
+     * over each grid element using numerical quadrature.
+     *
+     * @tparam Grid Grid type (e.g. Quad4Grid).
+     *
+     * @param[in] grid Grid.
+     * @param[in] f Function.
+     *
+     * @returns Density field defined on the grid elements.
+     *
+     * @throws std::invalid_argument if `f` returns a value outside the range [0,1].
+     *
+     * @note Densities are clamped to a minimum value of `ZERO_TOLERANCE` to improve
+     * numerical stability.
+     */
+    template <class Grid>
+    inline field::DensityField<Grid::Dim> makeDensityFieldFromFunction(const Grid &grid, const std::function<double(const typename Grid::Point&)>& f) {
+        field::DensityField<Grid::Dim> densityField(grid.resolution());
+
+        for (std::size_t i = 0; i < grid.numElements(); ++i) {
+            const auto nodes = grid.elementNodes(i);
+
+            auto integrand = [&](const typename Grid::Point& point) -> double {
+                const auto N = Grid::Element::shapeFunctions(point);
+                const typename Grid::Point globalPoint = N.transpose() * nodes;
+
+                const double density = f(globalPoint);
+                if (density < 0.0 || density > 1.0) {
+                    throw std::invalid_argument("Function value (" + std::to_string(density) + ") is outside range [0,1].");
+                }
+
+                const auto J = Grid::Element::jacobian(point, nodes);
+
+                return density * std::abs(J.determinant());
+            };
+
+            const double density = integration::integrateScalar(integrand, Grid::Element::quadratureRule()) / Grid::Element::measure(nodes);
+            densityField.setDensity(i, density);
+        }
+
+        return densityField;
+    }
+
+} // namespace monad
diff --git a/include/monad/grid/grid_2d_base.hpp b/include/monad/grid/grid_2d_base.hpp
deleted file mode 100644
index 06070a3..0000000
--- a/include/monad/grid/grid_2d_base.hpp
+++ /dev/null
@@ -1,154 +0,0 @@
-#pragma once
-
-#include <fstream>
-#include <stdexcept>
-#include <cstddef>
-#include <vector>
-#include <string>
-#include <sstream>
-#include "monad/grid/grid_base.hpp"
-
-namespace monad {
-
-    /**
-     * @brief Curiously recurring template pattern (CRTP) base class for 2D grid meshes.
-     *
-     * @tparam Derived Concrete grid mesh class (CRTP).
-     * @tparam Element Concrete element class (e.g. Quad4).
-     *
-     * @note `Derived` classes must provide:
-     *
-     * - `std::size_t numNodes() const noexcept`
-     *
-     * - `std::size_t numPeriodicNodes() const noexcept`
-     *
-     * - `Point node(std::size_t index) const`
-     *
-     * - `ElementList element(std::size_t index) const`
-     *
-     * - `ElementList periodicElement(std::size_t index) const`
-     */
-    template <class Derived, class Element>
-    class Grid2dBase : public GridBase<Derived, Element> {
-    public:
-        static_assert(Element::Dim == 2, "Element spatial dimension must be 2.");
-
-        using GridBase<Derived, Element>::GridBase;
-        using Resolution = typename GridBase<Derived, Element>::Resolution;
-        using DensityList = typename GridBase<Derived, Element>::DensityList;
-
-        /**
-         * @brief Translates the element densities periodically.
-         *
-         * @param[in] shift Shift in each directions.
-         */
-        void translate(const Resolution &shift) noexcept {
-            DensityList shiftedDensities(this->numElements());
-
-            const std::size_t nx = this->resolution_[0];
-            const std::size_t ny = this->resolution_[1];
-
-            for (std::size_t i = 0; i < nx; ++i) {
-                for (std::size_t j = 0; j < ny; ++j) {
-                    const std::size_t oldIndex = j * nx + i;
-
-                    const std::size_t iNew = (i + shift[0]) % nx;
-                    const std::size_t jNew = (j + shift[1]) % ny;
-
-                    std::size_t newIndex = jNew * nx + iNew;
-
-                    shiftedDensities[newIndex] = this->densities_[oldIndex];
-                }
-            }
-
-            this->setDensities(shiftedDensities);
-        }
-
-        /// @brief Grid area.
-        double area() const noexcept {
-            return this->measure();
-        }
-
-        /**
-         * @brief Set the material densities using a CSV file.
-         *
-         * @param[in] file CSV file.
-         *
-         * @throws std::runtime_error if `file` cannot be opened.
-         * @throws std::runtime_error if `file` contains no data.
-         * @throws std::runtime_error if `file` contains non-numeric data.
-         * @throws std::runtime_error if `file` contains numbers outside the range [0, 1].
-         * @throws std::runtime_error if `file` contains rows/columns not matching the grid resolution.
-         *
-         * @note The CSV data is interpreted as a 2D array with the origin
-         * located at the bottom-left corner.
-         */
-        void setDensitiesFile(const std::string &file) {
-            std::ifstream ifs(file, std::ios::in | std::ios::binary);
-            if (!ifs.is_open()) {
-                throw std::runtime_error("Could not open " + file + " for reading.");
-            }
-
-            const std::size_t nx = this->resolution_[0];
-            const std::size_t ny = this->resolution_[1];
-
-            std::vector<std::vector<double>> rows;
-
-            std::string line;
-
-            // Load csv data
-            while (std::getline(ifs, line)) {
-                if (line.empty()) {
-                    continue;
-                }
-
-                std::istringstream iss(line);
-                std::string cell;
-
-                std::vector<double> row;
-                row.reserve(nx);
-
-                while (std::getline(iss, cell, ',')) {
-                    if (cell.empty()) {
-                        continue;
-                    }
-
-                    double value;
-                    try {
-                        value = std::stod(cell);
-                    }
-                    catch (const std::invalid_argument &) {
-                        throw std::runtime_error("File " + file + " contains non-numeric data.");
-                    }
-                    if (value < 0.0 || value > 1.0) {
-                        throw std::runtime_error("File " + file + " contains data outside the range [0,1].");
-                    }
-                    row.push_back(value);
-                }
-
-                if (row.empty()) {
-                    continue;
-                }
-
-                if (row.size() != nx) {
-                    throw std::runtime_error("File " + file + " number of columns (" + std::to_string(row.size()) + ") does not equal grid x-resolution (" + std::to_string(nx) + ").");
-                }
-
-                rows.push_back(row);
-            }
-
-            if (rows.size() != ny) {
-                throw std::runtime_error("File " + file + " number of rows (" + std::to_string(rows.size()) + ") does not equal grid y-resolution (" + std::to_string(ny) + ").");
-            }
-
-            // Flatten csv data
-            for (std::size_t i = 0; i < ny; ++i) {
-                for (std::size_t j = 0; j < nx; ++j) {
-                    std::size_t rowStart = nx * (ny - 1 - i);
-                    this->setDensity(rowStart + j, rows[i][j]);
-                }
-            }
-        }
-    };
-
-} // namespace monad
diff --git a/include/monad/grid/grid_3d_base.hpp b/include/monad/grid/grid_3d_base.hpp
deleted file mode 100644
index e0cdc36..0000000
--- a/include/monad/grid/grid_3d_base.hpp
+++ /dev/null
@@ -1,73 +0,0 @@
-#pragma once
-
-#include "monad/grid/grid_base.hpp"
-
-namespace monad {
-
-    /**
-     * @brief Curiously recurring template pattern (CRTP) base class for 3D grid meshes.
-     *
-     * @tparam Derived Concrete grid mesh class (CRTP).
-     * @tparam Element Concrete element class (e.g. Hex8).
-     *
-     * @note `Derived` classes must provide:
-     *
-     * - `std::size_t numNodes() const noexcept`
-     *
-     * - `std::size_t numPeriodicNodes() const noexcept`
-     *
-     * - `Point node(std::size_t index) const`
-     *
-     * - `ElementList element(std::size_t index) const`
-     *
-     * - `ElementList periodicElement(std::size_t index) const`
-     */
-    template <class Derived, class Element>
-    class Grid3dBase : public GridBase<Derived, Element> {
-    public:
-        static_assert(Element::Dim == 3, "Element spatial dimension must be 3.");
-
-        using GridBase<Derived, Element>::GridBase;
-        using Resolution = typename GridBase<Derived, Element>::Resolution;
-        using DensityList = typename GridBase<Derived, Element>::DensityList;
-
-        /**
-         * @brief Translates the element densities periodically.
-         *
-         * @param[in] shift Shift in each directions.
-         */
-        void translate(const Resolution &shift) noexcept {
-            DensityList shiftedDensities(this->numElements());
-
-            const std::size_t nx = this->resolution_[0];
-            const std::size_t ny = this->resolution_[1];
-            const std::size_t nz = this->resolution_[2];
-
-            const std::size_t elementsPerPlane = nx * ny;
-
-            for (std::size_t i = 0; i < nx; ++i) {
-                for (std::size_t j = 0; j < ny; ++j) {
-                    for (std::size_t k = 0; k < nz; ++k) {
-                        const std::size_t oldIndex = k * elementsPerPlane + j * nx + i;
-
-                        const std::size_t iNew = (i + shift[0]) % nx;
-                        const std::size_t jNew = (j + shift[1]) % ny;
-                        const std::size_t kNew = (k + shift[2]) % nz;
-
-                        std::size_t newIndex = kNew * elementsPerPlane + jNew * nx + iNew;
-
-                        shiftedDensities[newIndex] = this->densities_[oldIndex];
-                    }
-                }
-            }
-
-            this->setDensities(shiftedDensities);
-        }
-
-        /// @brief Grid volume.
-        double volume() const noexcept {
-            return this->measure();
-        }
-    };
-
-} // namespace monad
diff --git a/include/monad/grid/grid_aliases.hpp b/include/monad/grid/grid_aliases.hpp
new file mode 100644
index 0000000..bbdd253
--- /dev/null
+++ b/include/monad/grid/grid_aliases.hpp
@@ -0,0 +1,139 @@
+#pragma once
+
+#include "monad/grid/structured_grid.hpp"
+#include "monad/grid/quad4_topology.hpp"
+#include "monad/grid/quad8_topology.hpp"
+#include "monad/grid/hex8_topology.hpp"
+#include "monad/grid/hex20_topology.hpp"
+
+namespace monad {
+
+    /**
+     * @brief 2D structured grid of 4-node quadrilateral (Quad4) elements.
+     *
+     * Example 3x2 grid:
+     *
+     * ```text
+     *      o───────o───────o───────o
+     *      │       │       │       │
+     *      │       │       │       │
+     *      │       │       │       │
+     *      o───────o───────o───────o
+     *      │       │       │       │
+     *      │       │       │       │
+     *      │       │       │       │
+     *      o───────o───────o───────o
+     * ```
+     *
+     * Global axes:
+     *
+     * - x runs left→right.
+     *
+     * - y runs bottom→top.
+     */
+    using Quad4Grid = grid::StructuredGrid<grid::Quad4Topology>;
+
+    /**
+     * @brief 2D structured grid of 8-node quadrilateral (Quad8) elements.
+     *
+     * Example 3x2 grid:
+     *
+     * ```text
+     *      o───o───o───o───o───o───o
+     *      │       │       │       │
+     *      o       o       o       o
+     *      │       │       │       │
+     *      o───o───o───o───o───o───o
+     *      │       │       │       │
+     *      o       o       o       o
+     *      │       │       │       │
+     *      o───o───o───o───o───o───o
+     * ```
+     *
+     * Global axes:
+     *
+     * - x runs left→right.
+     *
+     * - y runs bottom→top.
+     */
+    using Quad8Grid = grid::StructuredGrid<grid::Quad8Topology>;
+
+        /**
+     * @brief 3D structured grid of 8-node hexahedral (Hex8) elements.
+     *
+     * Example 3x2x2 grid:
+     *
+     * ```text
+     *               o─────────o─────────o─────────o
+     *              /         /         /         /│
+     *             /         /         /         / │
+     *            /         /         /         /  │
+     *           o─────────o─────────o─────────o   │
+     *          /         /         /         /│   │
+     *         /         /         /         / │   o
+     *        /         /         /         /  │  /│
+     *       o─────────o─────────o─────────o   │ / │
+     *       │         │         │         │   │/  │
+     *       │         │         │         │   o   │
+     *       │         │         │         │  /│   │
+     *       │         │         │         │ / │   o
+     *       │         │         │         │/  │  /
+     *       o─────────o─────────o─────────o   │ /
+     *       │         │         │         │   │/
+     *       │         │         │         │   o
+     *       │         │         │         │  /
+     *       │         │         │         │ /
+     *       │         │         │         │/
+     *       o─────────o─────────o─────────o
+     * ```
+     *
+     * Global axes:
+     *
+     * - x runs left→right.
+     *
+     * - y runs front→back.
+     *
+     * - z runs bottom→top.
+     */
+    using Hex8Grid = grid::StructuredGrid<grid::Hex8Topology>;
+
+    /**
+     * @brief 3D structured grid of 20-node hexahedral (Hex20) elements.
+     *
+     * Example 3x2x2 grid:
+     *
+     * ```text
+     *               o────o────o────o────o────o────o
+     *              /         /         /         /│
+     *             o         o         o         o │
+     *            /         /         /         /  o
+     *           o────o────o────o────o────o────o   │
+     *          /         /         /         /│   │
+     *         o         o         o         o │   o
+     *        /         /         /         /  o  /│
+     *       o────o────o────o────o────o────o   │ o │
+     *       │         │         │         │   │/  o
+     *       │         │         │         │   o   │
+     *       o         o         o         o  /│   │
+     *       │         │         │         │ o │   o
+     *       │         │         │         │/  o  /
+     *       o────o────o────o────o────o────o   │ o
+     *       │         │         │         │   │/
+     *       │         │         │         │   o
+     *       o         o         o         o  /
+     *       │         │         │         │ o
+     *       │         │         │         │/
+     *       o────o────o────o────o────o────o
+     * ```
+     *
+     * Global axes:
+     *
+     * - x runs left→right.
+     *
+     * - y runs front→back.
+     *
+     * - z runs bottom→top.
+     */
+    using Hex20Grid = grid::StructuredGrid<grid::Hex20Topology>;
+
+} // namespace monad
diff --git a/include/monad/grid/grid_base.hpp b/include/monad/grid/grid_base.hpp
deleted file mode 100644
index 0cb26e8..0000000
--- a/include/monad/grid/grid_base.hpp
+++ /dev/null
@@ -1,424 +0,0 @@
-#pragma once
-
-#include <array>
-#include <cstddef>
-#include <vector>
-#include <stdexcept>
-#include <string>
-#include <algorithm>
-#include <random>
-#include <functional>
-#include "monad/detail/constants.hpp"
-#include "monad/integration/integrate_scalar.hpp"
-
-namespace monad {
-
-    /**
-     * @brief Curiously recurring template pattern (CRTP) base class for grid meshes.
-     *
-     * @tparam Derived Concrete grid mesh class (CRTP).
-     * @tparam ElementT Concrete element class (e.g. Quad4).
-     *
-     * @note `Derived` classes must provide:
-     *
-     * - `std::size_t numNodes() const noexcept`
-     *
-     * - `std::size_t numPeriodicNodes() const noexcept`
-     *
-     * - `Point node(std::size_t index) const`
-     *
-     * - `ElementList element(std::size_t index) const`
-     *
-     * - `ElementList periodicElement(std::size_t index) const`
-     *
-     * - `void translate(const Resolution &shift) noexcept`
-     */
-    template <class Derived, class ElementT>
-    class GridBase {
-    public:
-        static_assert(ElementT::Dim == 2 || ElementT::Dim == 3, "Element spatial dimension must be 2 or 3.");
-
-        using Element = ElementT;
-
-        /// @brief Spatial dimension (2 or 3).
-        static constexpr int Dim = Element::Dim;
-
-        using Resolution = std::array<std::size_t, Dim>;
-        using Size = std::array<double, Dim>;
-
-        using Point = typename Element::Point;
-        using NodesMatrix = typename Element::NodesMatrix;
-
-        using NodesList = std::vector<Point>;
-        using ElementList = std::array<std::size_t, Element::NumNodes>;
-        using ElementsList = std::vector<ElementList>;
-        using DensityList = std::vector<double>;
-
-        /**
-         * @brief Constructs a grid mesh.
-         *
-         * @param[in] resolution How many cells in each dimension.
-         * @param[in] size Physical lengths in each dimension.
-         *
-         * @throws std::invalid_argument if any entry in `resolution` is zero.
-         * @throws std::invalid_argument if any entry in `size` is non-positive.
-         *
-         * @note Densities are initalized to 0.
-         */
-        GridBase(const Resolution &resolution, const Size &size)
-            : resolution_(resolution), size_(size) {
-            for (std::size_t i = 0; i < Dim; ++i) {
-                if (resolution_[i] == 0) {
-                    throw std::invalid_argument("Resolution in dimension " + std::to_string(i + 1) + " must be positive.");
-                }
-
-                if (size_[i] <= 0) {
-                    throw std::invalid_argument("Size in dimension " + std::to_string(i + 1) + " (" + std::to_string(size_[i]) + ") must be positive.");
-                }
-            }
-
-            densities_.resize(numElements());
-            setDensitiesZeros();
-        }
-
-        /// @brief How many cells in each dimension.
-        const Resolution &resolution() const noexcept {
-            return resolution_;
-        }
-
-        /// @brief Physical lengths in each dimension.
-        const Size &size() const noexcept {
-            return size_;
-        }
-
-        /**
-         * @brief Material density at each element.
-         *
-         * @note Densities are stored in row-major order.
-         */
-        const DensityList &densities() const noexcept {
-            return densities_;
-        }
-
-        /// @brief Number of elements.
-        std::size_t numElements() const noexcept {
-            std::size_t product = 1;
-
-            for (std::size_t entry : resolution_) {
-                product *= entry;
-            }
-
-            return product;
-        }
-
-        /// @brief Number of nodes.
-        std::size_t numNodes() const noexcept {
-            return derived().numNodes();
-        }
-
-        /// @brief Number of periodic nodes.
-        std::size_t numPeriodicNodes() const noexcept {
-            return derived().numPeriodicNodes();
-        }
-
-        /**
-         * @brief Coordinates for a specific node.
-         *
-         * @param[in] index Node index.
-         *
-         * @returns Coordinates.
-         *
-         * @throws std::out_of_range if `index` is outside the range [0,`numNodes()`).
-         */
-        Point node(std::size_t index) const {
-            return derived().node(index);
-        }
-
-        /// @brief Coordinates for all nodes.
-        NodesList nodes() const noexcept {
-            NodesList data;
-            data.reserve(numNodes());
-
-            for (std::size_t i = 0; i < numNodes(); ++i) {
-                data.push_back(node(i));
-            }
-
-            return data;
-        }
-
-        /**
-         * @brief Node indices for a specific element.
-         *
-         * @param[in] index Element index.
-         *
-         * @returns Node indices.
-         *
-         * @throws std::out_of_range if `index` is outside the range [0,`numElements()`).
-         */
-        ElementList element(std::size_t index) const {
-            return derived().element(index);
-        }
-
-        /// @brief Node indices for all elements.
-        ElementsList elements() const noexcept {
-            ElementsList data;
-            data.reserve(numElements());
-
-            for (std::size_t i = 0; i < numElements(); ++i) {
-                data.push_back(element(i));
-            }
-
-            return data;
-        }
-
-        /**
-         * @brief Periodic node indices for a specific element.
-         *
-         * @param[in] index Element index.
-         *
-         * @returns Periodic node indices.
-         *
-         * @throws std::out_of_range if `index` is outside the range [0,`numElements()`).
-         */
-        ElementList periodicElement(std::size_t index) const {
-            return derived().periodicElement(index);
-        }
-
-        /// @brief Periodic node indices for all elements.
-        ElementsList periodicElements() const noexcept {
-            ElementsList data;
-            data.reserve(numElements());
-
-            for (std::size_t i = 0; i < numElements(); ++i) {
-                data.push_back(periodicElement(i));
-            }
-
-            return data;
-        }
-
-        /**
-         * @brief Material density for a specific element.
-         *
-         * @param[in] index Element index.
-         *
-         * @returns Material density.
-         *
-         * @throws std::out_of_range if `index` is outside the range [0,`numElements()`).
-         */
-        double getDensity(std::size_t index) const {
-            if (index >= numElements()) {
-                throw std::out_of_range("Index (" + std::to_string(index) + ") is out of range [0," + std::to_string(numElements()) + ").");
-            }
-
-            return densities_[index];
-        }
-
-        /**
-         * @brief Set the material density for a specific element.
-         *
-         * @param[in] index Element index.
-         * @param[in] density Element density.
-         *
-         * @throws std::invalid_argument if `density` is outside the range [0,1].
-         * @throws std::out_of_range if `index` is outside the range [0,`numElements()`).
-         *
-         * @note Densities are clamped to a minimum value `NUMERICAL_ZERO` to ensure numerical stability.
-         */
-        void setDensity(std::size_t index, double density) {
-            if (index >= numElements()) {
-                throw std::out_of_range("Index (" + std::to_string(index) + ") is out of range [0," + std::to_string(numElements()) + ").");
-            }
-
-            if (density < 0 || density > 1) {
-                throw std::invalid_argument("Density (" + std::to_string(density) + ") is out of range [0,1].");
-            }
-
-            densities_[index] = std::max(NUMERICAL_ZERO, density);
-        }
-
-        /**
-         * @brief Set the material densities.
-         *
-         * @param[in] densities Material densities.
-         *
-         * @throws std::invalid_argument if the size of `densities` does not equal the number of grid elements.
-         * @throws std::invalid_argument if any value in `densities` is outside the range [0,1].
-         *
-         * @note Densities are stored in row-major order.
-         * @note Densities are clamped to a minimum value `NUMERICAL_ZERO` to ensure numerical stability.
-         */
-        void setDensities(DensityList densities) {
-            if (densities.size() != numElements()) {
-                throw std::invalid_argument("Densities size (" + std::to_string(densities.size()) + ") must equal number of grid elements (" + std::to_string(numElements()) + ").");
-            }
-
-            for (std::size_t i = 0; i < densities.size(); ++i) {
-                setDensity(i, densities[i]);
-            }
-        }
-
-        /**
-         * @brief Set the material densities to a specific value.
-         *
-         * @param[in] density Element density.
-         *
-         * @throws std::invalid_argument if `density` is outside the range [0,1].
-         *
-         * @note Densities are clamped to a minimum value `NUMERICAL_ZERO` to ensure numerical stability.
-         */
-        void setDensitiesConstant(double density) {
-            if (density < 0 || density > 1) {
-                throw std::invalid_argument("Density (" + std::to_string(density) + ") is out of range [0,1].");
-            }
-
-            for (std::size_t i = 0; i < numElements(); ++i) {
-                setDensity(i, density);
-            }
-        }
-
-        /// @brief Set the material densities to zeros.
-        void setDensitiesZeros() noexcept {
-            setDensitiesConstant(0.0);
-        }
-
-        /// @brief Set the material densities to ones.
-        void setDensitiesOnes() noexcept {
-            setDensitiesConstant(1.0);
-        }
-
-        /**
-         * @brief Set the material densities to random values.
-         *
-         * @param[in] seed RNG seed (default: -1 for random).
-         *
-         * @note Densities are clamped to a minimum value `NUMERICAL_ZERO` to ensure numerical stability.
-         */
-        void setDensitiesRandom(int seed = -1) noexcept {
-            std::mt19937 rng;
-            if (seed >= 0) {
-                rng.seed(static_cast<unsigned int>(seed));
-            }
-
-            // std::uniform_real_distribution generates values in [0, 1), so setting
-            // Use the next representable double after 1 to make the range effectively inclusive
-            const double upperBound = std::nextafter(1.0, 2.0);
-
-            std::uniform_real_distribution<double> dist(NUMERICAL_ZERO, upperBound);
-
-            for (std::size_t i = 0; i < numElements(); ++i) {
-                setDensity(i, dist(rng));
-            }
-        }
-
-        /**
-         * @brief Set the material densities using a continuous function.
-         *
-         * Element values are computed as the element-wise average of the continuous function.
-         *
-         * @param[in] f Continuous function from ℝᵈ→[0,1] evaluated at element nodes.
-         *
-         * @throws std::invalid_argument if `f` is outside the range [0,1].
-         *
-         * @note Densities are clamped to a minimum value `NUMERICAL_ZERO` to ensure numerical stability.
-         */
-        void setDensitiesFunction(const std::function<double(const Point &)> &f) {
-            for (std::size_t i = 0; i < numElements(); ++i) {
-                const auto nodes = elementNodes(i);
-
-                auto integrand = [&](const Point &point) -> double {
-                    const auto N = Element::shapeFunctions(point);
-                    const Point globalPoint = N.transpose() * nodes;
-
-                    const double density = f(globalPoint);
-
-                    if (density < 0 || density > 1) {
-                        throw std::invalid_argument("Function value (" + std::to_string(density) + ") is outside range [0,1].");
-                    }
-
-                    const auto J = Element::jacobian(point, nodes);
-
-                    return density * std::abs(J.determinant());
-                };
-
-                double density = integration::integrateScalar(integrand, Element::quadratureRule()) / Element::measure(nodes);
-
-                setDensity(i, density);
-            }
-        }
-
-        /**
-         * @brief Translates the element densities periodically.
-         *
-         * @param[in] shift Shift in each directions.
-         */
-        void translate(const Resolution &shift) noexcept {
-            derived().translate(shift);
-        }
-
-        /**
-         * @brief Nodal coordinates for a specific element.
-         *
-         * @param[in] index Element index.
-         *
-         * @returns Nodal coordinates.
-         *
-         * @throws std::out_of_range if `index` is outside the range [0,`numElements()`).
-         */
-        NodesMatrix elementNodes(std::size_t index) const {
-            if (index >= numElements()) {
-                throw std::out_of_range("Index (" + std::to_string(index) + ") is out of range [0," + std::to_string(numElements()) + ").");
-            }
-
-            NodesMatrix data;
-
-            int j = 0;
-            for (std::size_t i : element(index)) {
-                data.row(j++) = node(i);
-            }
-
-            return data;
-        }
-
-        /// @brief Grid area (2D) or volume (3D).
-        double measure() const noexcept {
-            const auto referenceElementNodes = elementNodes(0);
-            const double referenceElementMeasure = Element::measure(referenceElementNodes);
-
-            return referenceElementMeasure * static_cast<double>(numElements());
-        }
-
-        /// @brief Equality comparison.
-        bool operator==(const GridBase<Derived, Element> &other) const noexcept {
-            return resolution_ == other.resolution_
-                && size_ == other.size_
-                && densities_ == other.densities_;
-        }
-
-        /// @brief Inequality comparison.
-        bool operator!=(const GridBase<Derived, Element> &other) const noexcept {
-            return !(*this == other);
-        }
-
-    protected:
-        /// @brief How many cells in each dimension.
-        Resolution resolution_;
-
-        /// @brief Physical lengths in each dimension.
-        Size size_;
-
-        /**
-         * @brief Material density at each element.
-         *
-         * @note Densities are stored in row-major order.
-         */
-        DensityList densities_;
-
-    private:
-        /// @brief Utility method to safely cast the base class pointer to the derived class reference (CRTP).
-        const Derived &derived() const noexcept {
-            // The static_cast is safe because Derived must inherit from GridBase<Derived, Element>
-            return *static_cast<const Derived *>(this);
-        }
-    };
-
-} // namespace monad
diff --git a/include/monad/grid/hex20_grid.hpp b/include/monad/grid/hex20_grid.hpp
deleted file mode 100644
index 33b11de..0000000
--- a/include/monad/grid/hex20_grid.hpp
+++ /dev/null
@@ -1,167 +0,0 @@
-#pragma once
-
-#include "monad/grid/grid_3d_base.hpp"
-#include "monad/fem/element/hex20.hpp"
-
-namespace monad {
-
-    /**
-     * @brief 2D grid mesh of 4-node quadrilaterl (Hex20) elements.
-     *
-     * Example 3x2x2 grid:
-     *
-     * ```text
-     *               o────o────o────o────o────o────o
-     *              /         /         /         /│
-     *             o         o         o         o │
-     *            /         /         /         /  o
-     *           o────o────o────o────o────o────o   │
-     *          /         /         /         /│   │
-     *         o         o         o         o │   o
-     *        /         /         /         /  o  /│
-     *       o────o────o────o────o────o────o   │ o │
-     *       │         │         │         │   │/  o
-     *       │         │         │         │   o   │
-     *       o         o         o         o  /│   │
-     *       │         │         │         │ o │   o
-     *       │         │         │         │/  o  /
-     *       o────o────o────o────o────o────o   │ o
-     *       │         │         │         │   │/
-     *       │         │         │         │   o
-     *       o         o         o         o  /
-     *       │         │         │         │ o
-     *       │         │         │         │/
-     *       o────o────o────o────o────o────o
-     * ```
-     *
-     * Global axes:
-     *
-     * - x runs left→right.
-     *
-     * - y runs front→back.
-     *
-     * - z runs bottom→top.
-     */
-    class Hex20Grid : public Grid3dBase<Hex20Grid, fem::Hex20> {
-    public:
-        using Grid3dBase<Hex20Grid, fem::Hex20>::Grid3dBase;
-
-        /// @brief Number of nodes.
-        std::size_t numNodes() const noexcept;
-
-        /// @brief Number of periodic nodes.
-        std::size_t numPeriodicNodes() const noexcept;
-
-        /**
-         * @brief Coordinates for a specific node.
-         *
-         * Example 3x2x2 grid:
-         *
-         * ```text
-         *               32───60───33───61───34───62───35
-         *              /         /         /         /│
-         *             83        84        85        86│
-         *            /         /         /         / 110
-         *           28───57───29───58───30───59───31  │
-         *          /         /         /         /│   │
-         *         79        80        81        82│   23
-         *        /         /         /         / 106 /│
-         *       24───54───25───55───26───56───27  │ 78│
-         *       │         │         │         │   │/  98
-         *       │         │         │         │   19  │
-         *       99       100       101       102 /│   │
-         *       │         │         │         │ 74│   11
-         *       │         │         │         │/  94 /
-         *       12───45───13───46───14───47───15  │ 70
-         *       │         │         │         │   │/
-         *       │         │         │         │   7
-         *       87        88        89        90 /
-         *       │         │         │         │ 66
-         *       │         │         │         │/
-         *       0────36───1────37───2────38───3
-         * ```
-         *
-         * @param[in] index Node index.
-         *
-         * @returns Coordinates.
-         *
-         * @throws std::out_of_range if `index` is outside the range [0,`numNodes()`).
-         */
-        Point node(std::size_t index) const;
-
-        /**
-         * @brief Node indices for a specific element.
-         *
-         * Example 3x2x2 grid:
-         *
-         * ```text
-         *               32───60───33───61───34───62───35
-         *              /         /         /         /│
-         *             83   𝟿    84   𝟷𝟶   85   𝟷𝟷   86│
-         *            /         /         /         / 110
-         *           28───57───29───58───30───59───31  │
-         *          /         /         /         /│   │
-         *         79        80        81        82│   23
-         *        /         /         /         / 106 /│
-         *       24───54───25───55───26───56───27  │ 78│
-         *       │         │         │         │   │/  98
-         *       │         │         │         │   19  │
-         *       99   𝟼   100   𝟽   101   𝟾   102 /│   │
-         *       │         │         │         │ 74│   11
-         *       │         │         │         │/  94 /
-         *       12───45───13───46───14───47───15  │ 70
-         *       │         │         │         │   │/
-         *       │         │         │         │   7
-         *       87   𝟶    88   𝟷    89   𝟸    90 /
-         *       │         │         │         │ 66
-         *       │         │         │         │/
-         *       0────36───1────37───2────38───3
-         * ```
-         *
-         * @param[in] index Element index.
-         *
-         * @returns Node indices.
-         *
-         * @throws std::out_of_range if `index` is outside the range [0,`numElements()`).
-         */
-        ElementList element(std::size_t index) const;
-
-        /**
-         * @brief Periodic node indices for a specific element.
-         *
-         * Example 3x2x2 grid:
-         *
-         * ```text
-         *               0────12───1────13───2────14───0
-         *              /         /         /         /│
-         *             27   𝟿    28   𝟷𝟶   29   𝟷𝟷   27│
-         *            /         /         /         /  42
-         *           3────15───4────16───5────17───3   │
-         *          /         /         /         /│   │
-         *         24        25        26        24│   6
-         *        /         /         /         /  45 /│
-         *       0────12───1────13───2────14───0   │ 33│
-         *       │         │         │         │   │/  36
-         *       │         │         │         │   9   │
-         *       42   𝟼    43   𝟽    44   𝟾    42 /│   │
-         *       │         │         │         │ 30│   0
-         *       │         │         │         │/  39 /
-         *       6────18───7────19───8────20───6   │ 27
-         *       │         │         │         │   │/
-         *       │         │         │         │   3
-         *       36   𝟶    37   𝟷    38   𝟸    36 /
-         *       │         │         │         │ 24
-         *       │         │         │         │/
-         *       0────12───1────13───2────14───0
-         * ```
-         *
-         * @param[in] index Element index.
-         *
-         * @returns Periodic node indices.
-         *
-         * @throws std::out_of_range if `index` is outside the range [0,`numElements()`).
-         */
-        ElementList periodicElement(std::size_t index) const;
-    };
-
-} // namespace monad
diff --git a/include/monad/grid/hex20_topology.hpp b/include/monad/grid/hex20_topology.hpp
new file mode 100644
index 0000000..fae8126
--- /dev/null
+++ b/include/monad/grid/hex20_topology.hpp
@@ -0,0 +1,109 @@
+#pragma once
+
+#include <array>
+#include <cstddef>
+#include "monad/fem/element/hex20.hpp"
+
+namespace monad {
+
+    namespace grid {
+
+        /**
+         * @brief Topology rules for structured 3D Hex20 grids.
+         */
+        struct Hex20Topology {
+            using Element = fem::Hex20;
+            using Resolution = std::array<std::size_t, Element::Dim>;
+            using Size = std::array<double, Element::Dim>;
+            using Point = typename Element::Point;
+            using ElementList = std::array<std::size_t, Element::NumNodes>;
+
+            /**
+             * @brief Number of corner nodes.
+             *
+             * @param[in] resolution Number of elements in each dimension.
+             *
+             * @returns Number of corner nodes.
+             */
+            static std::size_t numCornerNodes(const Resolution& resolution) noexcept;
+
+            /**
+             * @brief Number of x-edge midpoint nodes.
+             *
+             * @param[in] resolution Number of elements in each dimension.
+             *
+             * @returns Number of x-edge midpoint nodes.
+             */
+            static std::size_t numXMidNodes(const Resolution& resolution) noexcept;
+
+            /**
+             * @brief Number of y-edge midpoint nodes.
+             *
+             * @param[in] resolution Number of elements in each dimension.
+             *
+             * @returns Number of y-edge midpoint nodes.
+             */
+            static std::size_t numYMidNodes(const Resolution& resolution) noexcept;
+
+            /**
+             * @brief Number of z-edge midpoint nodes.
+             *
+             * @param[in] resolution Number of elements in each dimension.
+             *
+             * @returns Number of z-edge midpoint nodes.
+             */
+            static std::size_t numZMidNodes(const Resolution& resolution) noexcept;
+
+            /**
+             * @brief Number of nodes.
+             *
+             * @param[in] resolution Number of elements in each dimension.
+             *
+             * @returns Number of nodes.
+             */
+            static std::size_t numNodes(const Resolution& resolution) noexcept;
+
+            /**
+             * @brief Number of periodic nodes.
+             *
+             * @param[in] resolution Number of elements in each dimension.
+             *
+             * @returns Number of periodic nodes.
+             */
+            static std::size_t numPeriodicNodes(const Resolution& resolution) noexcept;
+
+            /**
+             * @brief Coordinates of a node.
+             *
+             * @param[in] index Node index.
+             * @param[in] resolution Number of elements in each dimension.
+             * @param[in] size Physical lengths in each dimension.
+             *
+             * @returns Coordinates of a node.
+             */
+            static Point node(std::size_t index, const Resolution& resolution, const Size& size) noexcept;
+
+            /**
+             * @brief Node indices of an element.
+             *
+             * @param[in] index Element index.
+             * @param[in] resolution Number of elements in each dimension.
+             *
+             * @returns Node indices of an element.
+             */
+            static ElementList element(std::size_t index, const Resolution& resolution) noexcept;
+
+            /**
+             * @brief Periodic node indices of an element.
+             *
+             * @param[in] index Element index.
+             * @param[in] resolution Number of elements in each dimension.
+             *
+             * @returns Periodic node indices of an element.
+             */
+            static ElementList periodicElement(std::size_t index, const Resolution& resolution) noexcept;
+        };
+
+    } // namespace grid
+
+} // namespace monad
diff --git a/include/monad/grid/hex8_grid.hpp b/include/monad/grid/hex8_grid.hpp
deleted file mode 100644
index 2360a19..0000000
--- a/include/monad/grid/hex8_grid.hpp
+++ /dev/null
@@ -1,167 +0,0 @@
-#pragma once
-
-#include "monad/grid/grid_3d_base.hpp"
-#include "monad/fem/element/hex8.hpp"
-
-namespace monad {
-
-    /**
-     * @brief 2D grid mesh of 4-node quadrilaterl (Hex8) elements.
-     *
-     * Example 3x2x2 grid:
-     *
-     * ```text
-     *               o─────────o─────────o─────────o
-     *              /         /         /         /│
-     *             /         /         /         / │
-     *            /         /         /         /  │
-     *           o─────────o─────────o─────────o   │
-     *          /         /         /         /│   │
-     *         /         /         /         / │   o
-     *        /         /         /         /  │  /│
-     *       o─────────o─────────o─────────o   │ / │
-     *       │         │         │         │   │/  │
-     *       │         │         │         │   o   │
-     *       │         │         │         │  /│   │
-     *       │         │         │         │ / │   o
-     *       │         │         │         │/  │  /
-     *       o─────────o─────────o─────────o   │ /
-     *       │         │         │         │   │/
-     *       │         │         │         │   o
-     *       │         │         │         │  /
-     *       │         │         │         │ /
-     *       │         │         │         │/
-     *       o─────────o─────────o─────────o
-     * ```
-     *
-     * Global axes:
-     *
-     * - x runs left→right.
-     *
-     * - y runs front→back.
-     *
-     * - z runs bottom→top.
-     */
-    class Hex8Grid : public Grid3dBase<Hex8Grid, fem::Hex8> {
-    public:
-        using Grid3dBase<Hex8Grid, fem::Hex8>::Grid3dBase;
-
-        /// @brief Number of nodes.
-        std::size_t numNodes() const noexcept;
-
-        /// @brief Number of periodic nodes.
-        std::size_t numPeriodicNodes() const noexcept;
-
-        /**
-         * @brief Coordinates for a specific node.
-         *
-         * Example 3x2x2 grid:
-         *
-         * ```text
-         *               32────────33────────34────────35
-         *              /         /         /         /│
-         *             /         /         /         / │
-         *            /         /         /         /  │
-         *           28────────29────────30────────31  │
-         *          /         /         /         /│   │
-         *         /         /         /         / │   23
-         *        /         /         /         /  │  /│
-         *       24────────25────────26────────27  │ / │
-         *       │         │         │         │   │/  │
-         *       │         │         │         │   19  │
-         *       │         │         │         │  /│   │
-         *       │         │         │         │ / │   11
-         *       │         │         │         │/  │  /
-         *       12────────13────────14────────15  │ /
-         *       │         │         │         │   │/
-         *       │         │         │         │   7
-         *       │         │         │         │  /
-         *       │         │         │         │ /
-         *       │         │         │         │/
-         *       0─────────1─────────2─────────3
-         * ```
-         *
-         * @param[in] index Node index.
-         *
-         * @returns Coordinates.
-         *
-         * @throws std::out_of_range if `index` is outside the range [0,`numNodes()`).
-         */
-        Point node(std::size_t index) const;
-
-        /**
-         * @brief Node indices for a specific element.
-         *
-         * Example 3x2x2 grid:
-         *
-         * ```text
-         *               32────────33────────34────────35
-         *              /         /         /         /│
-         *             /    𝟿    /    𝟷𝟶   /    𝟷𝟷   / │
-         *            /         /         /         /  │
-         *           28────────29────────30────────31  │
-         *          /         /         /         /│   │
-         *         /         /         /         / │   23
-         *        /         /         /         /  │  /│
-         *       24────────25────────26────────27  │ / │
-         *       │         │         │         │   │/  │
-         *       │         │         │         │   19  │
-         *       │    𝟼    │    𝟽    │    𝟾    │  /│   │
-         *       │         │         │         │ / │   11
-         *       │         │         │         │/  │  /
-         *       12────────13────────14────────15  │ /
-         *       │         │         │         │   │/
-         *       │         │         │         │   7
-         *       │    𝟶    │    𝟷    │    𝟸    │  /
-         *       │         │         │         │ /
-         *       │         │         │         │/
-         *       0─────────1─────────2─────────3
-         * ```
-         *
-         * @param[in] index Element index.
-         *
-         * @returns Node indices.
-         *
-         * @throws std::out_of_range if `index` is outside the range [0,`numElements()`).
-         */
-        ElementList element(std::size_t index) const;
-
-        /**
-         * @brief Periodic node indices for a specific element.
-         *
-         * Example 3x2x2 grid:
-         *
-         * ```text
-         *               0─────────1─────────2─────────0
-         *              /         /         /         /│
-         *             /    𝟿    /    𝟷𝟶   /    𝟷𝟷   / │
-         *            /         /         /         /  │
-         *           3─────────4─────────5─────────3   │
-         *          /         /         /         /│   │
-         *         /         /         /         / │   6
-         *        /         /         /         /  │  /│
-         *       0─────────1─────────2─────────0   │ / │
-         *       │         │         │         │   │/  │
-         *       │         │         │         │   9   │
-         *       │    𝟼    │    𝟽    │    𝟾    │  /│   │
-         *       │         │         │         │ / │   0
-         *       │         │         │         │/  │  /
-         *       6─────────7─────────8─────────6   │ /
-         *       │         │         │         │   │/
-         *       │         │         │         │   3
-         *       │    𝟶    │    𝟷    │    𝟸    │  /
-         *       │         │         │         │ /
-         *       │         │         │         │/
-         *       0─────────1─────────2─────────0
-         * ```
-         *
-         * @param[in] index Element index.
-         *
-         * @returns Periodic node indices.
-         *
-         * @throws std::out_of_range if `index` is outside the range [0,`numElements()`).
-         */
-        ElementList periodicElement(std::size_t index) const;
-    };
-
-} // namespace monad
diff --git a/include/monad/grid/hex8_topology.hpp b/include/monad/grid/hex8_topology.hpp
new file mode 100644
index 0000000..532e714
--- /dev/null
+++ b/include/monad/grid/hex8_topology.hpp
@@ -0,0 +1,73 @@
+#pragma once
+
+#include <array>
+#include <cstddef>
+#include "monad/fem/element/hex8.hpp"
+
+namespace monad {
+
+    namespace grid {
+
+        /**
+         * @brief Topology rules for structured 3D Hex8 grids.
+         */
+        struct Hex8Topology {
+            using Element = fem::Hex8;
+            using Resolution = std::array<std::size_t, Element::Dim>;
+            using Size = std::array<double, Element::Dim>;
+            using Point = typename Element::Point;
+            using ElementList = std::array<std::size_t, Element::NumNodes>;
+
+            /**
+             * @brief Number of nodes.
+             *
+             * @param[in] resolution Number of elements in each dimension.
+             *
+             * @returns Number of nodes.
+             */
+            static std::size_t numNodes(const Resolution& resolution) noexcept;
+
+            /**
+             * @brief Number of periodic nodes.
+             *
+             * @param[in] resolution Number of elements in each dimension.
+             *
+             * @returns Number of periodic nodes.
+             */
+            static std::size_t numPeriodicNodes(const Resolution& resolution) noexcept;
+
+            /**
+             * @brief Coordinates of a node.
+             *
+             * @param[in] index Node index.
+             * @param[in] resolution Number of elements in each dimension.
+             * @param[in] size Physical lengths in each dimension.
+             *
+             * @returns Coordinates of a node.
+             */
+            static Point node(std::size_t index, const Resolution& resolution, const Size& size) noexcept;
+
+            /**
+             * @brief Node indices of an element.
+             *
+             * @param[in] index Element index.
+             * @param[in] resolution Number of elements in each dimension.
+             *
+             * @returns Node indices of an element.
+             */
+            static ElementList element(std::size_t index, const Resolution& resolution) noexcept;
+
+            /**
+             * @brief Periodic node indices of an element.
+             *
+             * @param[in] index Element index.
+             * @param[in] resolution Number of elements in each dimension.
+             *
+             * @returns Periodic node indices of an element.
+             */
+            static ElementList periodicElement(std::size_t index, const Resolution& resolution) noexcept;
+        };
+
+    } // namespace grid
+
+} // namespace monad
diff --git a/include/monad/grid/quad4_grid.hpp b/include/monad/grid/quad4_grid.hpp
deleted file mode 100644
index 0380ccf..0000000
--- a/include/monad/grid/quad4_grid.hpp
+++ /dev/null
@@ -1,117 +0,0 @@
-#pragma once
-
-#include "monad/grid/grid_2d_base.hpp"
-#include "monad/fem/element/quad4.hpp"
-
-namespace monad {
-
-    /**
-     * @brief 2D grid mesh of 4-node quadrilaterl (Quad4) elements.
-     *
-     * Example 3x2 grid:
-     *
-     * ```text
-     *      o───────o───────o───────o
-     *      │       │       │       │
-     *      │       │       │       │
-     *      │       │       │       │
-     *      o───────o───────o───────o
-     *      │       │       │       │
-     *      │       │       │       │
-     *      │       │       │       │
-     *      o───────o───────o───────o
-     * ```
-     *
-     * Global axes:
-     *
-     * - x runs left→right.
-     *
-     * - y runs bottom→top.
-     */
-    class Quad4Grid : public Grid2dBase<Quad4Grid, fem::Quad4> {
-    public:
-        using Grid2dBase<Quad4Grid, fem::Quad4>::Grid2dBase;
-
-        /// @brief Number of nodes.
-        std::size_t numNodes() const noexcept;
-
-        /// @brief Number of periodic nodes.
-        std::size_t numPeriodicNodes() const noexcept;
-
-        /**
-         * @brief Coordinates for a specific node.
-         *
-         * Example 3x2 grid:
-         *
-         * ```text
-         *      8───────9───────10──────11
-         *      │       │       │       │
-         *      │       │       │       │
-         *      │       │       │       │
-         *      4───────5───────6───────7
-         *      │       │       │       │
-         *      │       │       │       │
-         *      │       │       │       │
-         *      0───────1───────2───────3
-         * ```
-         *
-         * @param[in] index Node index.
-         *
-         * @returns Coordinates.
-         *
-         * @throws std::out_of_range if `index` is outside the range [0,`numNodes()`).
-         */
-        Point node(std::size_t index) const;
-
-        /**
-         * @brief Node indices for a specific element.
-         *
-         * Example 3x2 grid:
-         *
-         * ```text
-         *      8───────9───────10──────11
-         *      │       │       │       │
-         *      │   𝟹   │   𝟺   │   𝟻   │
-         *      │       │       │       │
-         *      4───────5───────6───────7
-         *      │       │       │       │
-         *      │   𝟶   │   𝟷   │   𝟸   │
-         *      │       │       │       │
-         *      0───────1───────2───────3
-         * ```
-         *
-         * @param[in] index Element index.
-         *
-         * @returns Node indices.
-         *
-         * @throws std::out_of_range if `index` is outside the range [0,`numElements()`).
-         */
-        ElementList element(std::size_t index) const;
-
-        /**
-         * @brief Periodic node indices for a specific element.
-         *
-         * Example 3x2 grid:
-         *
-         * ```text
-         *      0───────1───────2───────0
-         *      │       │       │       │
-         *      │   𝟹   │   𝟺   │   𝟻   │
-         *      │       │       │       │
-         *      3───────4───────5───────3
-         *      │       │       │       │
-         *      │   𝟶   │   𝟷   │   𝟸   │
-         *      │       │       │       │
-         *      0───────1───────2───────0
-         * ```
-         *
-         * @param[in] index Element index.
-         *
-         * @returns Periodic node indices.
-         *
-         * @throws std::out_of_range if `index` is outside the range [0,`numElements()`).
-         */
-        ElementList periodicElement(std::size_t index) const;
-    };
-
-} // namespace monad
diff --git a/include/monad/grid/quad4_topology.hpp b/include/monad/grid/quad4_topology.hpp
new file mode 100644
index 0000000..caaa00d
--- /dev/null
+++ b/include/monad/grid/quad4_topology.hpp
@@ -0,0 +1,73 @@
+#pragma once
+
+#include <array>
+#include <cstddef>
+#include "monad/fem/element/quad4.hpp"
+
+namespace monad {
+
+    namespace grid {
+
+        /**
+         * @brief Topology rules for structured 2D Quad4 grids.
+         */
+        struct Quad4Topology {
+            using Element = fem::Quad4;
+            using Resolution = std::array<std::size_t, Element::Dim>;
+            using Size = std::array<double, Element::Dim>;
+            using Point = typename Element::Point;
+            using ElementList = std::array<std::size_t, Element::NumNodes>;
+
+            /**
+             * @brief Number of nodes.
+             *
+             * @param[in] resolution Number of elements in each dimension.
+             *
+             * @returns Number of nodes.
+             */
+            static std::size_t numNodes(const Resolution& resolution) noexcept;
+
+            /**
+             * @brief Number of periodic nodes.
+             *
+             * @param[in] resolution Number of elements in each dimension.
+             *
+             * @returns Number of periodic nodes.
+             */
+            static std::size_t numPeriodicNodes(const Resolution& resolution) noexcept;
+
+            /**
+             * @brief Coordinates of a node.
+             *
+             * @param[in] index Node index.
+             * @param[in] resolution Number of elements in each dimension.
+             * @param[in] size Physical lengths in each dimension.
+             *
+             * @returns Coordinates of a node.
+             */
+            static Point node(std::size_t index, const Resolution& resolution, const Size& size) noexcept;
+
+            /**
+             * @brief Node indices of an element.
+             *
+             * @param[in] index Element index.
+             * @param[in] resolution Number of elements in each dimension.
+             *
+             * @returns Node indices of an element.
+             */
+            static ElementList element(std::size_t index, const Resolution& resolution) noexcept;
+
+            /**
+             * @brief Periodic node indices of an element.
+             *
+             * @param[in] index Element index.
+             * @param[in] resolution Number of elements in each dimension.
+             *
+             * @returns Periodic node indices of an element.
+             */
+            static ElementList periodicElement(std::size_t index, const Resolution& resolution) noexcept;
+        };
+
+    } // namespace grid
+
+} // namespace monad
diff --git a/include/monad/grid/quad8_grid.hpp b/include/monad/grid/quad8_grid.hpp
deleted file mode 100644
index a00b30f..0000000
--- a/include/monad/grid/quad8_grid.hpp
+++ /dev/null
@@ -1,117 +0,0 @@
-#pragma once
-
-#include "monad/grid/grid_2d_base.hpp"
-#include "monad/fem/element/quad8.hpp"
-
-namespace monad {
-
-    /**
-     * @brief 2D grid mesh of 8-node quadrilaterl (Quad8) elements.
-     *
-     * Example 3x2 grid:
-     *
-     * ```text
-     *      o───o───o───o───o───o───o
-     *      │       │       │       │
-     *      o       o       o       o
-     *      │       │       │       │
-     *      o───o───o───o───o───o───o
-     *      │       │       │       │
-     *      o       o       o       o
-     *      │       │       │       │
-     *      o───o───o───o───o───o───o
-     * ```
-     *
-     * Global axes:
-     *
-     * - x runs left→right.
-     *
-     * - y runs bottom→top.
-     */
-    class Quad8Grid : public Grid2dBase<Quad8Grid, fem::Quad8> {
-    public:
-        using Grid2dBase<Quad8Grid, fem::Quad8>::Grid2dBase;
-
-        /// @brief Number of nodes.
-        std::size_t numNodes() const noexcept;
-
-        /// @brief Number of periodic nodes.
-        std::size_t numPeriodicNodes() const noexcept;
-
-        /**
-         * @brief Coordinates for a specific node.
-         *
-         * Example 3x2 grid:
-         *
-         * ```text
-         *      8───18──9───19──10──20──11
-         *      │       │       │       │
-         *      25      26      27      28
-         *      │       │       │       │
-         *      4───15──5───16──6───17──7
-         *      │       │       │       │
-         *      21      22      23      24
-         *      │       │       │       │
-         *      0───12──1───13──2───14──3
-         * ```
-         *
-         * @param[in] index Node index.
-         *
-         * @returns Coordinates.
-         *
-         * @throws std::out_of_range if `index` is outside the range [0,`numNodes()`).
-         */
-        Point node(std::size_t index) const;
-
-        /**
-         * @brief Node indices for a specific element.
-         *
-         * Example 3x2 grid:
-         *
-         * ```text
-         *      8───18──9───19──10──20──11
-         *      │       │       │       │
-         *      25  𝟹   26  𝟺   27  𝟻   28
-         *      │       │       │       │
-         *      4───15──5───16──6───17──7
-         *      │       │       │       │
-         *      21  𝟶   22  𝟷   23  𝟸   24
-         *      │       │       │       │
-         *      0───12──1───13──2───14──3
-         * ```
-         *
-         * @param[in] index Element index.
-         *
-         * @returns Node indices.
-         *
-         * @throws std::out_of_range if `index` is outside the range [0,`numElements()`).
-         */
-        ElementList element(std::size_t index) const;
-
-        /**
-         * @brief Periodic node indices for a specific element.
-         *
-         * Example 3x2 grid:
-         *
-         * ```text
-         *      0───6───1───7───2───8───0
-         *      │       │       │       │
-         *      15  𝟹   16  𝟺   17  𝟻   15
-         *      │       │       │       │
-         *      3───9───4───10──5───11──3
-         *      │       │       │       │
-         *      12  𝟶   13  𝟷   14  𝟸   13
-         *      │       │       │       │
-         *      0───6───1───7───2───8───0
-         * ```
-         *
-         * @param[in] index Element index.
-         *
-         * @returns Periodic node indices.
-         *
-         * @throws std::out_of_range if `index` is outside the range [0,`numElements()`).
-         */
-        ElementList periodicElement(std::size_t index) const;
-    };
-
-} // namespace monad
diff --git a/include/monad/grid/quad8_topology.hpp b/include/monad/grid/quad8_topology.hpp
new file mode 100644
index 0000000..4f4de58
--- /dev/null
+++ b/include/monad/grid/quad8_topology.hpp
@@ -0,0 +1,100 @@
+#pragma once
+
+#include <array>
+#include <cstddef>
+#include "monad/fem/element/quad8.hpp"
+
+namespace monad {
+
+    namespace grid {
+
+        /**
+         * @brief Topology rules for 2D structured Quad8 grids.
+         */
+        struct Quad8Topology {
+            using Element = fem::Quad8;
+            using Resolution = std::array<std::size_t, Element::Dim>;
+            using Size = std::array<double, Element::Dim>;
+            using Point = typename Element::Point;
+            using ElementList = std::array<std::size_t, Element::NumNodes>;
+
+            /**
+             * @brief Number of corner nodes.
+             *
+             * @param[in] resolution Number of elements in each dimension.
+             *
+             * @returns Number of corner nodes.
+             */
+            static std::size_t numCornerNodes(const Resolution& resolution) noexcept;
+
+            /**
+             * @brief Number of x-edge midpoint nodes.
+             *
+             * @param[in] resolution Number of elements in each dimension.
+             *
+             * @returns Number of x-edge midpoint nodes.
+             */
+            static std::size_t numXMidNodes(const Resolution& resolution) noexcept;
+
+            /**
+             * @brief Number of y-edge midpoint nodes.
+             *
+             * @param[in] resolution Number of elements in each dimension.
+             *
+             * @returns Number of y-edge midpoint nodes.
+             */
+            static std::size_t numYMidNodes(const Resolution& resolution) noexcept;
+
+            /**
+             * @brief Number of nodes.
+             *
+             * @param[in] resolution Number of elements in each dimension.
+             *
+             * @returns Number of nodes.
+             */
+            static std::size_t numNodes(const Resolution& resolution) noexcept;
+
+            /**
+             * @brief Number of periodic nodes.
+             *
+             * @param[in] resolution Number of elements in each dimension.
+             *
+             * @returns Number of periodic nodes.
+             */
+            static std::size_t numPeriodicNodes(const Resolution& resolution) noexcept;
+
+            /**
+             * @brief Coordinates of a node.
+             *
+             * @param[in] index Node index.
+             * @param[in] resolution Number of elements in each dimension.
+             * @param[in] size Physical lengths in each dimension.
+             *
+             * @returns Coordinates of a node.
+             */
+            static Point node(std::size_t index, const Resolution& resolution, const Size& size) noexcept;
+
+            /**
+             * @brief Node indices of an element.
+             *
+             * @param[in] index Element index.
+             * @param[in] resolution Number of elements in each dimension.
+             *
+             * @returns Node indices of an element.
+             */
+            static ElementList element(std::size_t index, const Resolution& resolution) noexcept;
+
+            /**
+             * @brief Periodic node indices of an element.
+             *
+             * @param[in] index Element index.
+             * @param[in] resolution Number of elements in each dimension.
+             *
+             * @returns Periodic node indices of an element.
+             */
+            static ElementList periodicElement(std::size_t index, const Resolution& resolution) noexcept;
+        };
+
+    } // namespace grid
+
+} // namespace monad
diff --git a/include/monad/grid/structured_grid.hpp b/include/monad/grid/structured_grid.hpp
new file mode 100644
index 0000000..c876d55
--- /dev/null
+++ b/include/monad/grid/structured_grid.hpp
@@ -0,0 +1,239 @@
+#pragma once
+
+#include <array>
+#include <cstddef>
+#include <vector>
+#include <stdexcept>
+#include <string>
+#include <type_traits>
+
+namespace monad {
+
+    namespace grid {
+
+        /**
+         * @brief Structured grid mesh.
+         *
+         * @tparam TopologyT Concrete topology type (e.g. Quad4Topology).
+         */
+        template <class TopologyT>
+        class StructuredGrid {
+        public:
+            using Topology = TopologyT;
+            using Element = typename Topology::Element;
+
+            static constexpr int Dim = Element::Dim;
+
+            using Resolution = std::array<std::size_t, Dim>;
+            using Size = std::array<double, Dim>;
+            using Point = typename Element::Point;
+            using NodesMatrix = typename Element::NodesMatrix;
+            using ElementList = std::array<std::size_t, Element::NumNodes>;
+            using ElementsList = std::vector<ElementList>;
+            using NodesList = std::vector<Point>;
+
+            /**
+             * @brief Constructs a structured grid.
+             *
+             * @param[in] resolution Number of elements in each dimension.
+             * @param[in] size Physical lengths in each dimension.
+             *
+             * @throws std::invalid_argument if any entry in `resolution` is zero.
+             * @throws std::invalid_argument if any entry in `size` is non-positive.
+             */
+            StructuredGrid(const Resolution &resolution, const Size &size)
+                : resolution_(resolution), size_(size) {
+                for (std::size_t i = 0; i < Dim; ++i) {
+                    if (resolution_[i] == 0) {
+                        throw std::invalid_argument("Resolution in dimension " + std::to_string(i + 1) + " must be positive.");
+                    }
+
+                    if (size_[i] <= 0) {
+                        throw std::invalid_argument("Size in dimension " + std::to_string(i + 1) + " (" + std::to_string(size_[i]) + ") must be positive.");
+                    }
+                }
+            }
+
+            /// @brief Number of elements in each dimension.
+            const Resolution &resolution() const noexcept {
+                return resolution_;
+            }
+
+            /// @brief Physical lengths in each dimension.
+            const Size &size() const noexcept {
+                return size_;
+            }
+
+            /// @brief Number of elements.
+            std::size_t numElements() const noexcept {
+                std::size_t total = 1;
+
+                for (std::size_t n : resolution_) {
+                    total *= n;
+                }
+
+                return total;
+            }
+
+            /// @brief Number of nodes.
+            std::size_t numNodes() const noexcept {
+                return Topology::numNodes(resolution_);
+            }
+
+            /// @brief Number of periodic nodes.
+            std::size_t numPeriodicNodes() const noexcept {
+                return Topology::numPeriodicNodes(resolution_);
+            }
+
+            /**
+             * @brief Coordinates of a node
+             *
+             * @param[in] index Node index.
+             *
+             * @returns Coordinates of a node.
+             *
+             * @throws std::out_of_range if `index` is outside the range [0,`numNodes()`).
+             */
+            Point node(std::size_t index) const {
+                if (index >= numNodes()) {
+                    throw std::out_of_range("Index (" + std::to_string(index) + ") is out of range [0," + std::to_string(numNodes()) + ").");
+                }
+
+                return Topology::node(index, resolution_, size_);
+            }
+
+            /// @brief Coordinates of all nodes.
+            NodesList nodes() const noexcept {
+                NodesList out;
+                out.reserve(numNodes());
+
+                for (std::size_t i = 0; i < numNodes(); ++i) {
+                    out.push_back(node(i));
+                }
+
+                return out;
+            }
+
+            /**
+             * @brief Node indices of an element.
+             *
+             * @param[in] index Element index.
+             *
+             * @returns Node indices of an element.
+             *
+             * @throws std::out_of_range if `index` is outside the range [0,`numElements()`).
+             */
+            ElementList element(std::size_t index) const {
+                if (index >= numElements()) {
+                    throw std::out_of_range("Index (" + std::to_string(index) + ") is out of range [0," + std::to_string(numElements()) + ").");
+                }
+
+                return Topology::element(index, resolution_);
+            }
+
+            /// @brief Node indices of all elements.
+            ElementsList elements() const noexcept {
+                ElementsList out;
+                out.reserve(numElements());
+
+                for (std::size_t i = 0; i < numElements(); ++i) {
+                    out.push_back(element(i));
+                }
+
+                return out;
+            }
+
+            /**
+             * @brief Periodic node indices of an element.
+             *
+             * @param[in] index Element index.
+             *
+             * @returns Periodic node indices of an element.
+             *
+             * @throws std::out_of_range if `index` is outside the range [0,`numElements()`).
+             */
+            ElementList periodicElement(std::size_t index) const {
+                if (index >= numElements()) {
+                    throw std::out_of_range("Index (" + std::to_string(index) + ") is out of range [0," + std::to_string(numElements()) + ").");
+                }
+
+                return Topology::periodicElement(index, resolution_);
+            }
+
+            /// @brief Periodic node indices of all elements.
+            ElementsList periodicElements() const noexcept {
+                ElementsList out;
+                out.reserve(numElements());
+
+                for (std::size_t i = 0; i < numElements(); ++i) {
+                    out.push_back(periodicElement(i));
+                }
+
+                return out;
+            }
+
+            /**
+             * @brief Nodal coordinates of an element.
+             *
+             * @param[in] index Element index.
+             *
+             * @returns Nodal coordinates of an element.
+             *
+             * @throws std::out_of_range if `index` is outside the range [0,`numElements()`).
+             */
+            NodesMatrix elementNodes(std::size_t index) const {
+                if (index >= numElements()) {
+                    throw std::out_of_range("Index (" + std::to_string(index) + ") is out of range [0," + std::to_string(numElements()) + ").");
+                }
+
+                NodesMatrix out;
+
+                int row = 0;
+                for (std::size_t nodeIndex : element(index)) {
+                    out.row(row++) = node(nodeIndex);
+                }
+
+                return out;
+            }
+
+            /// @brief Grid area (2D) or volume (3D).
+            double measure() const noexcept {
+                const auto referenceElementNodes = elementNodes(0);
+                const double referenceElementMeasure = Element::measure(referenceElementNodes);
+
+                return referenceElementMeasure * static_cast<double>(numElements());
+            }
+
+            /// @brief Grid area.
+            template <int EnableDim = Dim>
+            std::enable_if_t<EnableDim == 2, double> area() const noexcept {
+                return measure();
+            }
+
+            /// @brief Grid volume.
+            template <int EnableDim = Dim>
+            std::enable_if_t<EnableDim == 3, double> volume() const noexcept {
+                return measure();
+            }
+
+            /// @brief Equality comparison.
+            bool operator==(const StructuredGrid<Topology> &other) const noexcept {
+                return resolution_ == other.resolution_ && size_ == other.size_;
+            }
+
+            /// @brief Inequality comparison.
+            bool operator!=(const StructuredGrid<Topology> &other) const noexcept {
+                return !(*this == other);
+            }
+
+        private:
+            /// @brief Number of elements in each dimension.
+            Resolution resolution_;
+
+            /// @brief Physical lengths in each dimension.
+            Size size_;
+        };
+
+    } // namespace grid
+
+} // namespace monad
diff --git a/include/monad/integration/integrate_matrix.hpp b/include/monad/integration/integrate_matrix.hpp
index 9c1d283..25bf43d 100644
--- a/include/monad/integration/integrate_matrix.hpp
+++ b/include/monad/integration/integrate_matrix.hpp
@@ -9,15 +9,19 @@ namespace monad {
     namespace integration {
 
         /**
-         * @brief Performs numerical integration (quadrature) of a vector integrand function.
+         * @brief Performs numerical integration (quadrature) of a matrix-valued integrand.
          *
-         * The integrand is a scalar function from ℝᵈ→ℝᵐˣᵐ.
+         * The integrand is a function of the form:
+         *
+         * ```text
+         * f:ℝᵈ→ℝᵐˣⁿ
+         * ```
          *
          * @tparam F Type of the integrand.
          * @tparam D Spatial dimension of the integration domain.
          * @tparam N Number of integration points.
          *
-         * @param[in] integrand Function to be integrated.
+         * @param[in] integrand Integrand function.
          * @param[in] rule Quadrature rule.
          *
          * @returns Approximate value of the integral.
@@ -27,7 +31,7 @@ namespace monad {
             using DomainType = typename QuadratureRule<D, N>::Point;
             using RangeType = std::invoke_result_t<F, DomainType>;
 
-            static_assert(std::is_invocable_r_v<RangeType, F, DomainType>, "Integrand must be callable as PointRange(PointDomain).");
+            static_assert(std::is_invocable_r_v<RangeType, F, DomainType>, "Integrand must be callable as f(x).");
 
             RangeType result = RangeType::Zero();
 
diff --git a/include/monad/integration/integrate_scalar.hpp b/include/monad/integration/integrate_scalar.hpp
index 1db0475..88d57c1 100644
--- a/include/monad/integration/integrate_scalar.hpp
+++ b/include/monad/integration/integrate_scalar.hpp
@@ -9,15 +9,19 @@ namespace monad {
     namespace integration {
 
         /**
-         * @brief Performs numerical integration (quadrature) of a scalar integrand function.
+         * @brief Performs numerical integration (quadrature) of a scalar-valued integrand.
          *
-         * The integrand is a scalar function from ℝᵈ→ℝ.
+         * The integrand is a function of the form:
+         *
+         * ```text
+         * f:ℝᵈ→ℝ
+         * ```
          *
          * @tparam F Type of the integrand.
          * @tparam D Spatial dimension of the integration domain.
          * @tparam N Number of integration points.
          *
-         * @param[in] integrand Function to be integrated.
+         * @param[in] integrand Integrand function.
          * @param[in] rule Quadrature rule.
          *
          * @returns Approximate value of the integral.
@@ -26,7 +30,7 @@ namespace monad {
         double integrateScalar(const F &integrand, const QuadratureRule<D, N> &rule) {
             using DomainType = typename QuadratureRule<D, N>::Point;
 
-            static_assert(std::is_invocable_r_v<double, F, DomainType>, "Integrand must be callable as double(Point).");
+            static_assert(std::is_invocable_r_v<double, F, DomainType>, "Integrand must be callable as f(x).");
 
             double result = 0.0;
 
diff --git a/include/monad/integration/quadrature_rule.hpp b/include/monad/integration/quadrature_rule.hpp
index 4ca7033..cbab247 100644
--- a/include/monad/integration/quadrature_rule.hpp
+++ b/include/monad/integration/quadrature_rule.hpp
@@ -3,6 +3,7 @@
 #include <array>
 #include <cstddef>
 #include <Eigen/Core>
+#include "monad/detail/constants.hpp"
 
 namespace monad {
 
@@ -11,13 +12,10 @@ namespace monad {
         /**
          * @brief Quadrature rule for numerical integration.
          *
-         * A quadrature rule contains:
+         * A quadrature rule consists of a set of integration
+         * points and the corresponding integration weights.
          *
-         * 1. A list of integration points.
-         *
-         * 2. A corresponding list of weights.
-         *
-         * @tparam D Spatial dimension of the integration domain.
+         * @tparam D Spatial dimension.
          * @tparam N Number of integration points.
          */
         template <int D, int N>
@@ -25,7 +23,7 @@ namespace monad {
             static_assert(D > 0, "Spatial dimension of the integration domain must be positive.");
             static_assert(N > 0, "Number of integration points must be positive.");
 
-            /// @brief Spatial dimension of the integration domain.
+            /// @brief Spatial dimension.
             static constexpr int Dim = D;
 
             /// @brief Number of integration points.
@@ -38,13 +36,13 @@ namespace monad {
             /// @brief Integration points.
             PointsList points;
 
-            /// @brief Integration point weights.
+            /// @brief Integration weights.
             WeightsList weights;
 
             /// @brief Equality comparison.
-            bool operator==(QuadratureRule const &other) const {
+            bool operator==(const QuadratureRule &other) const noexcept {
                 for (std::size_t i = 0; i < NumPoints; ++i) {
-                    if (!points[i].isApprox(other.points[i])) {
+                    if (!points[i].isApprox(other.points[i], NUMERICAL_ZERO)) {
                         return false;
                     }
                 }
@@ -52,7 +50,7 @@ namespace monad {
             }
 
             /// @brief Inequality comparison.
-            bool operator!=(QuadratureRule const &rhs) const {
+            bool operator!=(const QuadratureRule &rhs) const noexcept {
                 return !(*this == rhs);
             }
         };
diff --git a/include/monad/io/gmsh/write_gmsh_densities.hpp b/include/monad/io/gmsh/write_gmsh_densities.hpp
index c28c00a..3d147b4 100644
--- a/include/monad/io/gmsh/write_gmsh_densities.hpp
+++ b/include/monad/io/gmsh/write_gmsh_densities.hpp
@@ -3,6 +3,7 @@
 #include <cstddef>
 #include <string>
 #include <ostream>
+#include "monad/field/density_field.hpp"
 #include "monad/detail/constants.hpp"
 
 namespace monad {
@@ -12,17 +13,19 @@ namespace monad {
         namespace gmsh {
 
             /**
-             * @brief Writes the Gmsh file `$ElementData` section containing per-element material densities.
+             * @brief Writes the Gmsh `$ElementData` section for per-element material densities.
              *
-             * @tparam Grid Grid class (e.g. Quad4Grid).
+             * @tparam D Spatial dimension (2 or 3).
              *
              * @param[in,out] os Output stream.
-             * @param[in] grid Periodic unit cell grid.
+             * @param[in] densityField Per-element density field.
              *
              * @note Gmsh documentation: https://gmsh.info/doc/texinfo/gmsh.html#MSH-file-format
              */
-            template <typename Grid>
-            void writeGmshDensities(std::ostream &os, const Grid &grid) noexcept {
+            template <int D>
+            void writeGmshDensities(std::ostream &os, const field::DensityField<D> &densityField) noexcept {
+                static_assert(D == 2 || D == 3, "Density field spatial dimension must be 2 or 3.");
+
                 const int numStringTags = 1;
                 const std::string stringTag = "Density";
                 const int numRealTags = 0;
@@ -37,15 +40,15 @@ namespace monad {
                    << numIntegerTags << "\n"
                    << "0\n"
                    << "1\n"
-                   << static_cast<int>(grid.numElements()) << "\n";
+                   << static_cast<int>(densityField.numElements()) << "\n";
 
                 // Section body
-                for (std::size_t i = 0; i < grid.numElements(); ++i) {
+                for (std::size_t i = 0; i < densityField.numElements(); ++i) {
                     const std::size_t elementTag = i + 1;
 
                     os << elementTag;
 
-                    double value = grid.getDensity(i);
+                    double value = densityField.getDensity(i);
 
                     // Report 0 rather than NUMERICAL_ZERO for cleaner output
                     if (value <= NUMERICAL_ZERO) {
diff --git a/include/monad/io/gmsh/write_gmsh_elements.hpp b/include/monad/io/gmsh/write_gmsh_elements.hpp
index fa947a6..3aee46e 100644
--- a/include/monad/io/gmsh/write_gmsh_elements.hpp
+++ b/include/monad/io/gmsh/write_gmsh_elements.hpp
@@ -10,12 +10,12 @@ namespace monad {
         namespace gmsh {
 
             /**
-             * @brief Writes the Gmsh file `$Elements` section.
+             * @brief Writes the Gmsh `$Elements` section.
              *
-             * @tparam Grid Grid class (e.g. Quad4Grid).
+             * @tparam Grid Grid type (e.g. Quad4Grid).
              *
              * @param[in,out] os Output stream.
-             * @param[in] grid Periodic unit cell grid.
+             * @param[in] grid Grid.
              *
              * @note Gmsh documentation: https://gmsh.info/doc/texinfo/gmsh.html#MSH-file-format
              */
diff --git a/include/monad/io/gmsh/write_gmsh_header.hpp b/include/monad/io/gmsh/write_gmsh_header.hpp
index 227ee05..30402f4 100644
--- a/include/monad/io/gmsh/write_gmsh_header.hpp
+++ b/include/monad/io/gmsh/write_gmsh_header.hpp
@@ -9,7 +9,7 @@ namespace monad {
         namespace gmsh {
 
             /**
-             * @brief Writes the Gmsh file `$MeshFormat` section.
+             * @brief Writes the Gmsh `$MeshFormat` section.
              *
              * @param[in,out] os Output stream.
              *
diff --git a/include/monad/io/gmsh/write_gmsh_nodes.hpp b/include/monad/io/gmsh/write_gmsh_nodes.hpp
index 8450972..8d44e7b 100644
--- a/include/monad/io/gmsh/write_gmsh_nodes.hpp
+++ b/include/monad/io/gmsh/write_gmsh_nodes.hpp
@@ -10,12 +10,12 @@ namespace monad {
         namespace gmsh {
 
             /**
-             * @brief Writes the Gmsh file `$Nodes` section.
+             * @brief Writes the Gmsh `$Nodes` section.
              *
-             * @tparam Grid Grid class (e.g. Quad4Grid).
+             * @tparam Grid Grid type (e.g. Quad4Grid).
              *
              * @param[in,out] os Output stream.
-             * @param[in] grid Periodic unit cell grid.
+             * @param[in] grid Grid.
              *
              * @note Gmsh documentation: https://gmsh.info/doc/texinfo/gmsh.html#MSH-file-format
              */
diff --git a/include/monad/io/save_grid.hpp b/include/monad/io/save_grid.hpp
index e60bcf6..bd45d3c 100644
--- a/include/monad/io/save_grid.hpp
+++ b/include/monad/io/save_grid.hpp
@@ -13,17 +13,16 @@ namespace monad {
     /**
      * @brief Writes a grid to a Gmsh file.
      *
-     * @tparam Grid Grid class (e.g. Quad4Grid).
+     * @tparam Grid Grid type (e.g. Quad4Grid).
      *
-     * @param[in] grid Periodic unit cell grid.
-     * @param[in] file File.
-     * @param[in] saveDensities Set to `true` to save the material densities (default=`false`).
+     * @param[in] grid Grid.
+     * @param[in] file Path to the Gmsh file.
      *
-     * @throws std::invalid_argument if the `file` extension is not `.msh`.
+     * @throws std::invalid_argument if `file` does not have the `.msh` extension.
      * @throws std::runtime_error if the `file` cannot be opened for writing.
      */
     template <typename Grid>
-    void saveGrid(const Grid &grid, const std::string &file, bool saveDensities = false) {
+    void saveGrid(const Grid &grid, const std::string &file) {
         static_assert(Grid::Dim == 2 || Grid::Dim == 3, "Grid spatial dimension must be 2 or 3.");
 
         if (std::filesystem::path(file).extension() != ".msh") {
@@ -40,12 +39,6 @@ namespace monad {
         io::gmsh::writeGmshNodes(ofs, grid);
         ofs << "\n\n";
         io::gmsh::writeGmshElements(ofs, grid);
-
-        if (saveDensities) {
-            ofs << "\n\n";
-            io::gmsh::writeGmshDensities(ofs, grid);
-        }
-
         ofs << "\n";
     }
 
diff --git a/include/monad/io/save_grid_and_density_field.hpp b/include/monad/io/save_grid_and_density_field.hpp
new file mode 100644
index 0000000..f4c2f34
--- /dev/null
+++ b/include/monad/io/save_grid_and_density_field.hpp
@@ -0,0 +1,55 @@
+#pragma once
+
+#include <string>
+#include <filesystem>
+#include <stdexcept>
+#include <fstream>
+#include "monad/field/density_field.hpp"
+#include "monad/io/gmsh/write_gmsh_header.hpp"
+#include "monad/io/gmsh/write_gmsh_nodes.hpp"
+#include "monad/io/gmsh/write_gmsh_elements.hpp"
+#include "monad/io/gmsh/write_gmsh_densities.hpp"
+
+namespace monad {
+
+    /**
+     * @brief Writes a grid and its element density field to a Gmsh file.
+     *
+     * @tparam Grid Grid type (e.g. Quad4Grid).
+     *
+     * @param[in] grid Grid.
+     * @param[in] densityField Per-element density field defined on `grid`.
+     * @param[in] file Path to the Gmsh file.
+     *
+     * @throws std::invalid_argument if `grid` and `densityField` do not have the same resolution.
+     * @throws std::invalid_argument if `file` does not have the `.msh` extension.
+     * @throws std::runtime_error if the `file` cannot be opened for writing.
+     */
+    template <typename Grid>
+    void saveGridAndDensityField(const Grid &grid, const field::DensityField<Grid::Dim> &densityField, const std::string &file) {
+        static_assert(Grid::Dim == 2 || Grid::Dim == 3, "Grid spatial dimension must be 2 or 3.");
+
+        if (grid.resolution() != densityField.resolution()) {
+            throw std::invalid_argument( "Grid resolution must match density field resolution.");
+        }
+
+        if (std::filesystem::path(file).extension() != ".msh") {
+            throw std::invalid_argument("File extension must be \".msh\".");
+        }
+
+        std::ofstream ofs(file, std::ios::trunc);
+        if (!ofs.is_open()) {
+            throw std::runtime_error("Could not open " + file + " for writing.");
+        }
+
+        io::gmsh::writeGmshHeader(ofs);
+        ofs << "\n\n";
+        io::gmsh::writeGmshNodes(ofs, grid);
+        ofs << "\n\n";
+        io::gmsh::writeGmshElements(ofs, grid);
+        ofs << "\n\n";
+        io::gmsh::writeGmshDensities(ofs, densityField);
+        ofs << "\n";
+    }
+
+} // namespace monad
diff --git a/include/monad/io/save_grid_and_field.hpp b/include/monad/io/save_grid_and_nodal_field.hpp
similarity index 73%
rename from include/monad/io/save_grid_and_field.hpp
rename to include/monad/io/save_grid_and_nodal_field.hpp
index 0a3f9e4..e78efa1 100644
--- a/include/monad/io/save_grid_and_field.hpp
+++ b/include/monad/io/save_grid_and_nodal_field.hpp
@@ -11,33 +11,34 @@
 #include "monad/io/gmsh/write_gmsh_nodal_field.hpp"
 
 namespace monad {
+
     /**
-     * @brief Writes a grid and nodal field to a Gmsh file.
+     * @brief Writes a grid and its nodal field to a Gmsh file.
      *
-     * @tparam Grid Grid class (e.g. Quad4Grid).
+     * @tparam Grid Grid type (e.g. Quad4Grid).
      * @tparam Derived Eigen matrix type.
      *
-     * @param[in] grid Periodic unit cell grid.
-     * @param[in] file File.
+     * @param[in] grid Grid.
      * @param[in] field An Nx1 (scalar), Nx2 or Nx3 (vector) nodal field.
-     * @param[in] name Optional name tag for the nodal field.
+     * @param[in] file Path to the Gmsh file.
+     * @param[in] name Optional name for the nodal field.
      *
-     * @throws std::invalid_argument if the `file` extension is not `.msh`.
-     * @throws std::invalid_argument if the `field` size (N) does not equal the number of `grid` nodes.
+     * @throws std::invalid_argument if `field` size does not equal the number of `grid` nodes.
+     * @throws std::invalid_argument if `file` does not have the `.msh` extension.
      * @throws std::runtime_error if the `file` cannot be opened for writing.
      */
     template <typename Grid, typename Derived>
-    void saveGridAndField(const Grid &grid, const Eigen::MatrixBase<Derived>& field, const std::string &file, const std::string &name = "") {
+    void saveGridAndNodalField(const Grid &grid, const Eigen::MatrixBase<Derived>& field, const std::string &file, const std::string &name = "") {
         static_assert(Grid::Dim == 2 || Grid::Dim == 3, "Grid spatial dimension must be 2 or 3.");
 
-        if (std::filesystem::path(file).extension() != ".msh") {
-            throw std::invalid_argument("File extension must be \".msh\".");
-        }
-
         if (field.rows() != static_cast<int>(grid.numNodes())) {
             throw std::invalid_argument("Field size (" + std::to_string(field.rows()) + ") must equal number of grid nodes (" + std::to_string(grid.numNodes()) + ").");
         }
 
+        if (std::filesystem::path(file).extension() != ".msh") {
+            throw std::invalid_argument("File extension must be \".msh\".");
+        }
+
         std::ofstream ofs(file, std::ios::trunc);
         if (!ofs.is_open()) {
             throw std::runtime_error("Could not open " + file + " for writing.");
diff --git a/include/monad/material/bounds.hpp b/include/monad/material/bounds.hpp
new file mode 100644
index 0000000..0c63973
--- /dev/null
+++ b/include/monad/material/bounds.hpp
@@ -0,0 +1,42 @@
+#pragma once
+
+#include "monad/field/density_field.hpp"
+#include "monad/detail/mean.hpp"
+
+namespace monad {
+    
+    /**
+     * @brief Voigt upper bound for the homogenized material tensor.
+     *
+     * @tparam Material Material type (e.g. LinearElasticMaterial2d).
+     *
+     * @param[in] material Base material.
+     * @param[in] densityField Per-element density field.
+     *
+     * @returns Voigt upper bound for the homogenized material tensor.
+     */
+    template <class Material>
+    auto voigtBound(const Material& material, const field::DensityField<Material::Dim>& densityField) {
+        const double density = detail::arithmeticMean(densityField.densities());
+
+        return density * material.materialTensor();
+    }
+
+    /**
+     * @brief Reuss lower bound for the homogenized material tensor.
+     *
+     * @tparam Material Material type (e.g. LinearElasticMaterial2d).
+     *
+     * @param[in] material Base material.
+     * @param[in] densityField Per-element density field.
+     *
+     * @returns Reuss lower bound for the homogenized material tensor.
+     */
+    template <class Material>
+    auto reussBound(const Material& material, const field::DensityField<Material::Dim>& densityField) {
+        const double density = detail::harmonicMean(densityField.densities());
+
+        return density * material.materialTensor();
+    }
+
+} // namespace monad
diff --git a/include/monad/material/material_aliases.hpp b/include/monad/material/material_aliases.hpp
new file mode 100644
index 0000000..3d228b7
--- /dev/null
+++ b/include/monad/material/material_aliases.hpp
@@ -0,0 +1,240 @@
+#pragma once
+
+#include "monad/material/transport/linear_transport_material.hpp"
+#include "monad/material/multiphysics/linear_piezoelectric_material.hpp"
+
+namespace monad {
+
+    /**
+     * @brief 2D linear dielectric material model.
+     *
+     * By Gauss's law, the electric displacement D∈ℝ² is a linear function of
+     * the electric field E∈ℝ²:
+     *
+     * ```text
+     * D=ϵE=-ϵ∇φ
+     * ```
+     *
+     * - φ∈ℝ is the electric potential.
+     *
+     * - ϵ∈Sym₂(ℝ) is the permittivity tensor.
+     */
+    using LinearDielectricMaterial2d = material::LinearTransportMaterial<2>;
+
+    /**
+     * @brief 3D linear dielectric material model.
+     *
+     * By Gauss's law, the electric displacement D∈ℝ³ is a linear function of
+     * the electric field E∈ℝ³:
+     *
+     * ```text
+     * D=ϵE=-ϵ∇φ
+     * ```
+     *
+     * - φ∈ℝ is the electric potential.
+     *
+     * - ϵ∈Sym₃(ℝ) is the permittivity tensor.
+     */
+    using LinearDielectricMaterial3d = material::LinearTransportMaterial<3>;
+
+    /**
+     * @brief 2D linear electrical conductive material model.
+     *
+     * By Ohm's law, the current density J∈ℝ² is a linear function of
+     * the electric field E∈ℝ²:
+     *
+     * ```text
+     * J=σE=-σ∇φ
+     * ```
+     *
+     * - φ∈ℝ is the electric potential.
+     *
+     * - σ∈Sym₂(ℝ) is the conductivity tensor.
+     */
+    using LinearElectricalConductiveMaterial2d = material::LinearTransportMaterial<2>;
+
+    /**
+     * @brief 3D linear electrical conductive material model.
+     *
+     * By Ohm's law, the current density J∈ℝ³ is a linear function of
+     * the electric field E∈ℝ³:
+     *
+     * ```text
+     * J=σE=-σ∇φ
+     * ```
+     *
+     * - φ∈ℝ is the electric potential.
+     *
+     * - σ∈Sym₃(ℝ) is the conductivity tensor.
+     */
+    using LinearElectricalConductiveMaterial3d = material::LinearTransportMaterial<3>;
+
+    /**
+     * @brief 2D linear magnetic material model
+     *
+     * In a linear magnetic constitutive law, the magnetic flux density B∈ℝ² is a linear function of
+     * the magnetic field H∈ℝ²:
+     *
+     * ```text
+     * B=μH=-μ∇φ
+     * ```
+     *
+     * - φ∈ℝ is the magnetic potential.
+     *
+     * - μ∈Sym₂(ℝ) is the permeability tensor.
+     */
+    using LinearMagneticMaterial2d = material::LinearTransportMaterial<2>;
+
+    /**
+     * @brief 3D linear magnetic material model
+     *
+     * In a linear magnetic constitutive law, the magnetic flux density B∈ℝ³ is a linear function of
+     * the magnetic field H∈ℝ³:
+     *
+     * ```text
+     * B=μH=-μ∇φ
+     * ```
+     *
+     * - φ∈ℝ is the magnetic potential.
+     *
+     * - μ∈Sym₃(ℝ) is the permeability tensor.
+     */
+    using LinearMagneticMaterial3d = material::LinearTransportMaterial<3>;
+
+    /**
+     * @brief 2D linear mass diffusive material model.
+     *
+     * By Fick's law, the diffusion flux J∈ℝ² is a linear function of
+     * the mass concentration gradient ∇c∈ℝ²:
+     *
+     * ```text
+     * J=-D∇c
+     * ```
+     *
+     * - c∈ℝ is the mass concentration.
+     *
+     * - D∈Sym₂(ℝ) is the diffusivity tensor.
+     */
+    using LinearMassDiffusiveMaterial2d = material::LinearTransportMaterial<2>;
+
+    /**
+     * @brief 3D linear mass diffusive material model.
+     *
+     * By Fick's law, the diffusion flux J∈ℝ³ is a linear function of
+     * the mass concentration gradient ∇c∈ℝ³:
+     *
+     * ```text
+     * J=-D∇c
+     * ```
+     *
+     * - c∈ℝ is the mass concentration.
+     *
+     * - D∈Sym₃(ℝ) is the diffusivity tensor.
+     */
+    using LinearMassDiffusiveMaterial3d = material::LinearTransportMaterial<3>;
+
+    /**
+     * @brief 2D linear porous material model.
+     *
+     * By Darcy's law, the volumetric flux q∈ℝ² is a linear function of
+     * the pressure gradient ∇p∈ℝ²:
+     *
+     * ```text
+     * q=-K∇p
+     * ```
+     *
+     * - p∈ℝ is the pressure.
+     *
+     * - K∈Sym₂(ℝ) is the permeability tensor.
+     */
+    using LinearPorousMaterial2d = material::LinearTransportMaterial<2>;
+
+    /**
+     * @brief 3D linear porous material model.
+     *
+     * By Darcy's law, the volumetric flux q∈ℝ³ is a linear function of
+     * the pressure gradient ∇p∈ℝ³:
+     *
+     * ```text
+     * q=-K∇p
+     * ```
+     *
+     * - p∈ℝ is the pressure.
+     *
+     * - K∈Sym₃(ℝ) is the permeability tensor.
+     */
+    using LinearPorousMaterial3d = material::LinearTransportMaterial<3>;
+
+    /**
+     * @brief 2D linear thermal conductive material model.
+     *
+     * By Fourier's law, the heat flux q∈ℝ² is a linear function of
+     * the temperature gradient ∇T∈ℝ²:
+     *
+     * ```text
+     * q=-κ∇T
+     * ```
+     *
+     * - T∈ℝ is the temperature.
+     *
+     * - κ∈Sym₂(ℝ) is the conductivity tensor.
+     */
+    using LinearThermalConductiveMaterial2d = material::LinearTransportMaterial<2>;
+
+    /**
+     * @brief 3D linear thermal conductive material model.
+     *
+     * By Fourier's law, the heat flux q∈ℝ³ is a linear function of
+     * the temperature gradient ∇T∈ℝ³:
+     *
+     * ```text
+     * q=-κ∇T
+     * ```
+     *
+     * - T∈ℝ is the temperature.
+     *
+     * - κ∈Sym₃(ℝ) is the conductivity tensor.
+     */
+    using LinearThermalConductiveMaterial3d = material::LinearTransportMaterial<3>;
+
+    /**
+     * @brief 2D linear piezoelectric material model.
+     *
+     * In the stress-charge form, mechanical fields (stress T∈ℝ³ and strain S∈ℝ³)
+     * and electrical fields (electric displacement D∈ℝ² and electric field E∈ℝ²)
+     * are coupled by:
+     *
+     * ```text
+     * T=cS-dᵀE
+     * D=dT+ϵE
+     * ```
+     *
+     * - c∈Sym₃(ℝ) is the stiffness tensor in Voigt notation.
+     *
+     * - ϵ∈Sym₂(ℝ) is the permittivity tensor.
+     *
+     * - d∈ℝ²ˣ³ is the piezoelectric coupling tensor.
+     */
+    using LinearPiezoelectricMaterial2d = material::LinearPiezoelectricMaterial<2>;
+
+    /**
+     * @brief 3D linear piezoelectric material model.
+     *
+     * In the stress-charge form, mechanical fields (stress T∈ℝ⁶ and strain S∈ℝ⁶)
+     * and electrical fields (electric displacement D∈ℝ³ and electric field E∈ℝ³)
+     * are coupled by:
+     *
+     * ```text
+     * T=cS-dᵀE
+     * D=dT+ϵE
+     * ```
+     *
+     * - c∈Sym₆(ℝ) is the stiffness tensor in Voigt notation.
+     *
+     * - ϵ∈Sym₃(ℝ) is the permittivity tensor.
+     *
+     * - d∈ℝ³ˣ⁶ is the piezoelectric coupling tensor.
+     */
+    using LinearPiezoelectricMaterial3d = material::LinearPiezoelectricMaterial<3>;
+
+} // namespace monad
diff --git a/include/monad/material/mechanical/linear_elastic_material.hpp b/include/monad/material/mechanical/linear_elastic_material.hpp
index b33de7c..d768a61 100644
--- a/include/monad/material/mechanical/linear_elastic_material.hpp
+++ b/include/monad/material/mechanical/linear_elastic_material.hpp
@@ -2,106 +2,78 @@
 
 #include <stdexcept>
 #include <Eigen/Core>
-#include "monad/detail/mean.hpp"
 #include "monad/detail/eigen_utils.hpp"
-#include "monad/grid/grid_base.hpp"
+#include "monad/detail/constants.hpp"
 
 namespace monad {
 
-    /**
-     * @brief Represents a linear elastic material model.
-     *
-     * By Hooke's law, stress σ∈ℝᵛ is a linear function of strain ε∈ℝᵛ:
-     *
-     * σ=Cε
-     *
-     * - C∈Symᵥ(ℝ) is the stiffness tensor (in Voigt notation).
-     *
-     * @tparam D Spatial dimension (2 or 3).
-     */
-    template <int D>
-    class LinearElasticMaterial {
-    public:
-        static_assert(D == 2 || D == 3, "Spatial dimension D must be 2 or 3.");
-
-        /// @brief Spatial dimension (2 or 3).
-        static constexpr int Dim = D;
-
-        /// @brief Number of components in Voigt notation.
-        static constexpr int VoigtSize = (Dim == 2) ? 3 : 6;
-
-        /// @brief Stiffness tensor (in Voigt notation) type.
-        using MaterialTensor = Eigen::Matrix<double, VoigtSize, VoigtSize>;
-
-        /// @brief Default constructor.
-        LinearElasticMaterial() = default;
+    namespace material {
 
         /**
-         * @brief Constructs a linear elastic material.
-         *
-         * @param[in] C Stiffness tensor (in Voigt notation).
+         * @brief Linear elastic material model.
          *
-         * @throws std::invalid_argument if `C` is not positive definite.
-         */
-        explicit LinearElasticMaterial(const MaterialTensor &C)
-            : C_(C) {
-            if (!detail::isPD(C_)) {
-                throw std::invalid_argument("Stiffness tensor is not positive definite.");
-            }
-        }
-
-        /// @brief Stiffness tensor C (in Voigt notation).
-        const MaterialTensor &materialTensor() const noexcept {
-            return C_;
-        }
-
-        /**
-         * @brief Voigt upper bound for the homogenized stiffness tensor.
+         * By Hooke's law, the stress σ∈ℝᵛ is a linear function of
+         * the strain ε∈ℝᵛ:
          *
-         * @tparam Grid Grid class (e.g. Quad4Grid).
-         * @tparam Element Element class (e.g. Quad4).
+         * ```text
+         * σ=Cε
+         * ```
          *
-         * @param[in] grid Periodic unit cell grid.
+         * - C∈Symᵥ(ℝ) is the stiffness tensor in Voigt notation.
          *
-         * @returns Voigt upper bound for the homogenized stiffness tensor.
+         * @tparam D Spatial dimension (2 or 3).
          */
-        template <class Grid, class Element>
-        MaterialTensor voigt(const GridBase<Grid, Element> &grid) const noexcept {
-            const double density = detail::arithmeticMean(grid.densities());
-
-            return density * C_;
-        }
+        template <int D>
+        class LinearElasticMaterial {
+        public:
+            static_assert(D == 2 || D == 3, "Spatial dimension D must be 2 or 3.");
+
+            /// @brief Spatial dimension (2 or 3).
+            static constexpr int Dim = D;
+
+            /// @brief Number of components in Voigt notation.
+            static constexpr int VoigtSize = (Dim == 2) ? 3 : 6;
+
+            /// @brief Stiffness tensor type.
+            using MaterialTensor = Eigen::Matrix<double, VoigtSize, VoigtSize>;
+
+            /// @brief Default constructor.
+            LinearElasticMaterial() = default;
+
+            /**
+             * @brief Constructs a linear elastic material.
+             *
+             * @param[in] C Stiffness tensor.
+             *
+             * @throws std::invalid_argument if `C` is not positive definite.
+             */
+            explicit LinearElasticMaterial(const MaterialTensor &C)
+                : C_(C) {
+                if (!detail::isPD(C_)) {
+                    throw std::invalid_argument("Stiffness tensor is not positive definite.");
+                }
+            }
 
-        /**
-         * @brief Reuss lower bound for the homogenized stiffness tensor.
-         *
-         * @tparam Grid Grid class (e.g. Quad4Grid).
-         * @tparam Element Element class (e.g. Quad4).
-         *
-         * @param[in] grid Periodic unit cell grid.
-         *
-         * @returns Reuss lower bound for the homogenized stiffness tensor.
-         */
-        template <class Grid, class Element>
-        MaterialTensor reuss(const GridBase<Grid, Element> &grid) const noexcept {
-            const double density = detail::harmonicMean(grid.densities());
+            /// @brief Stiffness tensor.
+            const MaterialTensor &materialTensor() const noexcept {
+                return C_;
+            }
 
-            return density * C_;
-        }
+            /// @brief Equality comparison.
+            bool operator==(const LinearElasticMaterial &other) const noexcept {
+                return C_.isApprox(other.C_, NUMERICAL_ZERO);
+            }
 
-        /// @brief Equality comparison.
-        bool operator==(const LinearElasticMaterial &other) const noexcept {
-            return C_.isApprox(other.C_);
-        }
+            /// @brief Inequality comparison.
+            bool operator!=(const LinearElasticMaterial &other) const noexcept {
+                return !(*this == other);
+            }
 
-        /// @brief Inequality comparison.
-        bool operator!=(const LinearElasticMaterial &other) const noexcept {
-            return !(*this == other);
-        }
+        protected:
+            /// @brief Stiffness tensor.
+            MaterialTensor C_;
+        };
 
-    protected:
-        /// @brief Stiffness tensor (in Voigt notation).
-        MaterialTensor C_;
-    };
+    } // namespace material
 
 } // namespace monad
diff --git a/include/monad/material/mechanical/linear_elastic_material_2d.hpp b/include/monad/material/mechanical/linear_elastic_material_2d.hpp
index c91821a..cb40077 100644
--- a/include/monad/material/mechanical/linear_elastic_material_2d.hpp
+++ b/include/monad/material/mechanical/linear_elastic_material_2d.hpp
@@ -5,24 +5,27 @@
 namespace monad {
 
     /**
-     * @brief Represents a 2D linear elastic material model.
+     * @brief 2D linear elastic material model.
      *
-     * By Hooke's law, stress σ∈ℝ² is a linear function of strain ε∈ℝ²:
+     * By Hooke's law, the stress σ∈ℝ³ is a linear function of
+     * the strain ε∈ℝ³:
      *
+     * ```text
      * σ=Cε
+     * ```
      *
-     * - C∈Sym₂(ℝ) is the stiffness tensor (in Voigt notation).
+     * - C∈Sym₃(ℝ) is the stiffness tensor in Voigt notation.
      */
-    class LinearElasticMaterial2d : public LinearElasticMaterial<2> {
+    class LinearElasticMaterial2d : public material::LinearElasticMaterial<2> {
     public:
-        using LinearElasticMaterial<2>::LinearElasticMaterial;
+        using material::LinearElasticMaterial<2>::LinearElasticMaterial;
 
-        /// @brief Defines the plane conditions for 2D linear elasticity.
+        /// @brief Plane condition for 2D linear elasticity.
         enum class PlaneCondition {
-            /// @brief Assumes zero out-of-plane stress.
+            /// @brief Zero out-of-plane stress.
             PlaneStress = 0,
 
-            /// @brief Assumes zero out-of-plane strain.
+            /// @brief Zero out-of-plane strain.
             PlaneStrain
         };
 
@@ -31,7 +34,7 @@ namespace monad {
          *
          * @param[in] E Young's modulus.
          * @param[in] nu Poisson's ratio.
-         * @param[in] condition 2D plane condition.
+         * @param[in] condition Plane condition.
          *
          * @throws std::invalid_argument if `E` is non-positive.
          * @throws std::invalid_argument if `nu` is not in range (-1,0.5).
diff --git a/include/monad/material/mechanical/linear_elastic_material_3d.hpp b/include/monad/material/mechanical/linear_elastic_material_3d.hpp
index c1ee246..fd3cc3e 100644
--- a/include/monad/material/mechanical/linear_elastic_material_3d.hpp
+++ b/include/monad/material/mechanical/linear_elastic_material_3d.hpp
@@ -5,17 +5,20 @@
 namespace monad {
 
     /**
-     * @brief Represents a 3D linear elastic material model.
+     * @brief 3D linear elastic material model.
      *
-     * By Hooke's law, stress σ∈ℝ³ is a linear function of strain ε∈ℝ³:
+     * By Hooke's law, the stress σ∈ℝ⁶ is a linear function of
+     * the strain ε∈ℝ⁶:
      *
+     * ```text
      * σ=Cε
+     * ```
      *
-     * - C∈Sym₃(ℝ) is the stiffness tensor (in Voigt notation).
+     * - C∈Sym₆(ℝ) is the stiffness tensor in Voigt notation.
      */
-    class LinearElasticMaterial3d : public LinearElasticMaterial<3> {
+    class LinearElasticMaterial3d : public material::LinearElasticMaterial<3> {
     public:
-        using LinearElasticMaterial<3>::LinearElasticMaterial;
+        using material::LinearElasticMaterial<3>::LinearElasticMaterial;
 
         /**
          * @brief Constructs a 3D isotropic linear elastic material.
diff --git a/include/monad/material/multiphysics/linear_piezoelectric_material.hpp b/include/monad/material/multiphysics/linear_piezoelectric_material.hpp
index 7894808..7cdcb3e 100644
--- a/include/monad/material/multiphysics/linear_piezoelectric_material.hpp
+++ b/include/monad/material/multiphysics/linear_piezoelectric_material.hpp
@@ -2,197 +2,177 @@
 
 #include <stdexcept>
 #include <Eigen/Core>
-#include "monad/material/mechanical/linear_elastic_material.hpp"
-#include "monad/material/transport/linear_transport_material.hpp"
 #include "monad/detail/eigen_utils.hpp"
-#include "monad/grid/grid_base.hpp"
 
 namespace monad {
 
-    /**
-     * @brief Represents a linear piezoelectric material model.
-     *
-     * By the stress-charge form, mechanical (stress S∈ℝᵛ and strain T∈ℝᵛ) and
-     * electrical (electric displacement D∈ℝᵈ and electric field E∈ℝᵈ) fields
-     * are coupled:
-     *
-     * S=cT-dᵀE
-     *
-     * -D=-dT-εE
-     *
-     * - c∈Symᵥ(ℝ) is the stiffness tensor (in Voigt notation).
-     *
-     * - ε∈Symᴅ(ℝ) is the permittivity tensor (analogous to transport tensor K).
-     *
-     * - d∈ℝᵈˣᵛ is the piezoelectric coupling tensor.
-     *
-     * @tparam MechanicalMaterial Linear elastic material class (e.g. LinearElasticMaterial2d).
-     * @tparam ElectricalMaterial Linear dielectric material class (e.g. LinearDielectricMaterial2d).
-     */
-    template <class MechanicalMaterial, class ElectricalMaterial>
-    class LinearPiezoelectricMaterial {
-    public:
-        static_assert(MechanicalMaterial::Dim == ElectricalMaterial::Dim, "Spatial dimension of materials must be equal.");
-        static_assert(MechanicalMaterial::Dim == 2 || MechanicalMaterial::Dim == 3, "Spatial dimension D must be 2 or 3.");
-
-        /// @brief Spatial dimension (2 or 3).
-        static constexpr int Dim = MechanicalMaterial::Dim;
-
-        /// @brief Number of components in Voigt notation.
-        static constexpr int VoigtSize = MechanicalMaterial::VoigtSize;
-
-        using CouplingTensor = Eigen::Matrix<double, Dim, VoigtSize>;
-
-        /** @brief Coupled constitutive operator type.
+    namespace material {
+
+        /**
+         * @brief Linear piezoelectric material model.
+         *
+         * In the stress-charge form, mechanical fields (stress T∈ℝᵛ and strain S∈ℝᵛ)
+         * and electrical fields (electric displacement D∈ℝᵈ and electric field E∈ℝᵈ)
+         * are coupled by:
          *
          * ```text
-         * ⎡ c  -dᵀ⎤
-         * ⎣-d  -ε ⎦
+         * T=cS-dᵀE
+         * D=dT+ϵE
          * ```
-         */
-        using MaterialTensor = Eigen::Matrix<double, VoigtSize + Dim, VoigtSize + Dim>;
-
-        /**
-         * @brief Constructs a linear piezoelectric material.
          *
-         * @param[in] elasticMaterial The linear elastic material.
-         * @param[in] dielectricMaterial The linear dielectric material.
-         * @param[in] d The piezoelectric coupling tensor.
+         * - c∈Symᵥ(ℝ) is the stiffness tensor in Voigt notation.
          *
-         * @throws std::invalid_argument if Schur complement
+         * - ϵ∈Symᴅ(ℝ) is the permittivity tensor.
          *
-         * c-dᵀε⁻¹d
+         * - d∈ℝᵈˣᵛ is the piezoelectric coupling tensor.
          *
-         * is not positive definite.
+         * @tparam D Spatial dimension (2 or 3).
          */
-        LinearPiezoelectricMaterial(const MechanicalMaterial &elasticMaterial, const ElectricalMaterial &dielectricMaterial, const CouplingTensor &d)
-            : elasticMaterial_(elasticMaterial), dielectricMaterial_(dielectricMaterial), d_(d) {
-            const auto &c = elasticMaterial_.materialTensor();
-            const auto &epsilon = dielectricMaterial_.materialTensor();
-
-            // Schur complement
-            // Must be PD for thermodynamic stability
-            auto S = c - d_.transpose() * epsilon.inverse() * d_;
-            if (!detail::isPD(S)) {
-                throw std::invalid_argument("Schur complement is not positive definite.");
+        template <int D>
+        class LinearPiezoelectricMaterial {
+        public:
+            static_assert(D == 2 || D == 3, "Spatial dimension D must be 2 or 3.");
+
+            /// @brief Spatial dimension (2 or 3).
+            static constexpr int Dim = D;
+
+            /// @brief Number of components in Voigt notation.
+            static constexpr int VoigtSize = (Dim == 2) ? 3 : 6;
+
+            /// @brief Stiffness tensor type.
+            using StiffnessTensor = Eigen::Matrix<double, VoigtSize, VoigtSize>;
+
+            /// @brief Permittivity tensor type.
+            using PermittivityTensor = Eigen::Matrix<double, Dim, Dim>;
+
+            /// @brief Piezoelectric coupling tensor type.
+            using CouplingTensor = Eigen::Matrix<double, Dim, VoigtSize>;
+
+            /** @brief Coupled constitutive operator type.
+             *
+             * ```text
+             * ⎡ c  -dᵀ⎤
+             * ⎣-d  -ϵ ⎦
+             * ```
+             */
+            using MaterialTensor = Eigen::Matrix<double, VoigtSize + Dim, VoigtSize + Dim>;
+
+            /**
+             * @brief Constructs a linear piezoelectric material.
+             *
+             * @param[in] c Stiffness tensor.
+             * @param[in] epsilon Permittivity tensor.
+             * @param[in] d Piezoelectric coupling tensor.
+             *
+             * @throws std::invalid_argument if `c` is not positive definite.
+             * @throws std::invalid_argument if `epsilon` is not positive definite.
+             * @throws std::invalid_argument if the Schur complement
+             *
+             * ```
+             * c-dᵀϵ⁻¹d
+             * ```
+             *
+             * is not positive definite.
+             */
+            LinearPiezoelectricMaterial(const StiffnessTensor &c, const PermittivityTensor &epsilon, const CouplingTensor &d)
+                : c_(c), epsilon_(epsilon), d_(d) {
+
+                if (!detail::isPD(c_)) {
+                    throw std::invalid_argument("Stiffness tensor is not positive definite.");
+                }
+
+                if (!detail::isPD(epsilon_)) {
+                    throw std::invalid_argument("Permittivity tensor is not positive definite.");
+                }
+
+                op_ << c_, -d_.transpose(),
+                       -d_, -epsilon_;
+
+                // Schur complement must be positive definite for thermodynamic stability.
+                const auto schur = c_ - d_.transpose() * epsilon_.inverse() * d_;
+                if (!detail::isPD(schur)) {
+                    throw std::invalid_argument("Schur complement is not positive definite.");
+                }
             }
 
-            op_ << c, -d_.transpose(),
-                   -d_, -epsilon;
-        }
+            // /**
+            //  * @brief Converts from a compatible piezoelectric material type.
+            //  *
+            //  * This constructor allows piezoelectric materials built from compatible
+            //  * mechanical and electrical material types to be converted into this
+            //  * instantiation.
+            //  *
+            //  * This is needed when solver code expects a canonical
+            //  * `LinearPiezoelectricMaterial<...>` type, but the input uses derived
+            //  * or aliased component material types.
+            //  *
+            //  * @tparam OtherMechanicalMaterial Compatible mechanical material type.
+            //  * @tparam OtherElectricalMaterial Compatible electrical material type.
+            //  *
+            //  * @param[in] other Piezoelectric material to convert from.
+            //  */
+            // template <typename OtherMechanicalMaterial, typename OtherElectricalMaterial>
+            // LinearPiezoelectricMaterial(const LinearPiezoelectricMaterial<OtherMechanicalMaterial, OtherElectricalMaterial> &other)
+            //     : LinearPiezoelectricMaterial(other.elasticMaterial(), other.dielectricMaterial(), other.couplingTensor()) {}
+
+            /// @brief Stiffness tensor.
+            const StiffnessTensor &stiffnessTensor() const noexcept {
+                return c_;
+            }
 
-        /**
-         * @brief Converting constructor for compatible piezoelectric material types.
-         *
-         * Converts any derived instantiation of `LinearPiezoelectricMaterial` into the
-         * canonical form expected by the solver.
-         *
-         * Example:
-         *
-         * LinearPiezoelectricMaterial2d<LinearElasticMaterial2d, LinearDielectricMaterial2d>
-         *
-         * ↳ LinearPiezoelectricMaterial<LinearElasticMaterial<2>, LinearTransportMaterial<2>>
-         *
-         * @tparam OtherMechanicalMaterial Derived mechanical material type.
-         * @tparam OtherElectricalMaterial Derived electrical material type.
-         *
-         * @param[in] other Linear piezoelectric material.
-         *
-         * @note Unlike single-physics materials, the piezoelectric class is a nested
-         * template over two material types, which prevents implicit type deduction
-         * for the solver. This constructor explicitly bridges that gap by forwarding
-         * the material to the canonical base type.
-         */
-        template <typename OtherMechanicalMaterial, typename OtherElectricalMaterial>
-        LinearPiezoelectricMaterial(const LinearPiezoelectricMaterial<OtherMechanicalMaterial, OtherElectricalMaterial> &other)
-            : LinearPiezoelectricMaterial(other.elasticMaterial(), other.dielectricMaterial(), other.couplingTensor()) {}
-
-        /// @brief Linear elastic material.
-        const MechanicalMaterial &elasticMaterial() const noexcept {
-            return elasticMaterial_;
-        }
-
-        /// @brief Linear dielectric material.
-        const ElectricalMaterial &dielectricMaterial() const noexcept {
-            return dielectricMaterial_;
-        }
-
-        /// @brief Piezoelectric coupling tensor d.
-        const CouplingTensor &couplingTensor() const noexcept {
-            return d_;
-        }
-
-        /// @brief Coupled constitutive operator.
-        const MaterialTensor &materialTensor() const noexcept {
-            return op_;
-        }
-
-        /// @brief Equality comparison.
-        bool operator==(const LinearPiezoelectricMaterial &other) const noexcept {
-            return op_ == other.op_;
-        }
-
-        /// @brief Inequality comparison.
-        bool operator!=(const LinearPiezoelectricMaterial &other) const noexcept {
-            return !(*this == other);
-        }
-
-    private:
-        /// @brief Linear elastic material.
-        const MechanicalMaterial elasticMaterial_;
-
-        /// @brief Linear dielectric material.
-        const ElectricalMaterial dielectricMaterial_;
-
-        /// @brief Piezoelectric coupling tensor.
-        const CouplingTensor d_;
-
-        /** @brief Coupled constitutive operator.
-         *
-         * ```text
-         * ⎡ c  -dᵀ⎤
-         * ⎣-d  -ε ⎦
-         * ```
-         */
-        MaterialTensor op_;
-    };
-
-    /**
-     * @brief Represents a 2D linear piezoelectric material model.
-     *
-     * By the stress-charge form, mechanical (stress S∈ℝ³ and strain T∈ℝ³) and
-     * electrical (electric displacement D∈ℝ² and electric field E∈ℝ²) fields
-     * are coupled:
-     *
-     * S=cT-dᵀE
-     *
-     * -D=-dT-εE
-     *
-     * - c∈Sym₃(ℝ) is the stiffness tensor (in Voigt notation).
-     *
-     * - ε∈Sym₂(ℝ) is the dielectric tensor.
-     *
-     * - d∈ℝ²ˣ³ is the piezoelectric coupling tensor.
-     */
-    using LinearPiezoelectricMaterial2d = LinearPiezoelectricMaterial<LinearElasticMaterial<2>, LinearTransportMaterial<2>>;
-
-    /**
-     * @brief Represents a 3D linear piezoelectric material model.
-     *
-     * By the stress-charge form, mechanical (stress S∈ℝ⁶ and strain T∈ℝ⁶) and
-     * electrical (electric displacement D∈ℝ³ and electric field E∈ℝ³) fields
-     * are coupled:
-     *
-     * S=cT-dᵀE
-     *
-     * -D=-dT-εE
-     *
-     * - c∈Sym₆(ℝ) is the stiffness tensor (in Voigt notation).
-     *
-     * - ε∈Sym₃(ℝ) is the dielectric tensor.
-     *
-     * - d∈ℝ³ˣ⁶ is the piezoelectric coupling tensor.
-     */
-    using LinearPiezoelectricMaterial3d = LinearPiezoelectricMaterial<LinearElasticMaterial<3>, LinearTransportMaterial<3>>;
+            /// @brief Permittivity tensor.
+            const PermittivityTensor &permittivityTensor() const noexcept {
+                return epsilon_;
+            }
+
+            /// @brief Piezoelectric coupling tensor.
+            const CouplingTensor &couplingTensor() const noexcept {
+                return d_;
+            }
+
+            /** @brief Coupled constitutive operator.
+             *
+             * ```text
+             * ⎡ c  -dᵀ⎤
+             * ⎣-d  -ϵ ⎦
+             * ```
+             */
+            const MaterialTensor &materialTensor() const noexcept {
+                return op_;
+            }
+
+            /// @brief Equality comparison.
+            bool operator==(const LinearPiezoelectricMaterial &other) const noexcept {
+                return c_.isApprox(other.c_, NUMERICAL_ZERO)
+                    && epsilon_.isApprox(other.epsilon_, NUMERICAL_ZERO)
+                    && d_.isApprox(other.d_, NUMERICAL_ZERO);
+            }
+
+            /// @brief Inequality comparison.
+            bool operator!=(const LinearPiezoelectricMaterial &other) const noexcept {
+                return !(*this == other);
+            }
+
+        private:
+            /// @brief Stiffness tensor.
+            const StiffnessTensor c_;
+
+            /// @brief Permittivity tensor.
+            const PermittivityTensor epsilon_;
+
+            /// @brief Piezoelectric coupling tensor.
+            const CouplingTensor d_;
+
+            /** @brief Coupled constitutive operator.
+             *
+             * ```text
+             * ⎡ c  -dᵀ⎤
+             * ⎣-d  -ϵ ⎦
+             * ```
+             */
+            MaterialTensor op_;
+        };
+
+    } // namespace material
 
 } // namespace monad
diff --git a/include/monad/material/transport/linear_transport_material.hpp b/include/monad/material/transport/linear_transport_material.hpp
index 71e2541..2507769 100644
--- a/include/monad/material/transport/linear_transport_material.hpp
+++ b/include/monad/material/transport/linear_transport_material.hpp
@@ -3,115 +3,87 @@
 #include <stdexcept>
 #include <string>
 #include <Eigen/Core>
-#include "monad/detail/mean.hpp"
 #include "monad/detail/eigen_utils.hpp"
-#include "monad/grid/grid_base.hpp"
+#include "monad/detail/constants.hpp"
 
 namespace monad {
 
-    /**
-     * @brief Represents a linear transport material model.
-     *
-     * By a linear transport constitutive law, a flux J∈ℝᵈ is a linear function of a scalar potential gradient ∇φ∈ℝᵈ:
-     *
-     * J=-K∇φ
-     *
-     * - K∈Symᴅ(ℝ) is the transport tensor.
-     *
-     * @tparam D Spatial dimension (2 or 3).
-     */
-    template <int D>
-    class LinearTransportMaterial {
-    public:
-        static_assert(D == 2 || D == 3, "Spatial dimension D must be 2 or 3.");
-
-        /// @brief Spatial dimension (2 or 3).
-        static constexpr int Dim = D;
-
-        /// @brief Transport tensor type.
-        using MaterialTensor = Eigen::Matrix<double, Dim, Dim>;
+    namespace material {
 
         /**
-         * @brief Constructs a linear transport material.
+         * @brief Linear transport material model.
          *
-         * @param[in] K Transport tensor.
+         * In a linear transport constitutive law, the flux J∈ℝᵈ is
+         * a linear function of a scalar potential gradient ∇φ∈ℝᵈ:
          *
-         * @throws std::invalid_argument if `K` is not positive definite.
-         */
-        explicit LinearTransportMaterial(const MaterialTensor &K)
-            : K_(K) {
-            if (!detail::isPD(K_)) {
-                throw std::invalid_argument("Transport tensor is not positive definite.");
-            }
-        }
-
-        /**
-         * @brief Constructs an isotropic linear transport material.
+         * ```text
+         * J=-K∇φ
+         * ```
          *
-         * @param[in] K Transport constant.
+         * - K∈Symᴅ(ℝ) is the transport tensor.
          *
-         * @throws std::invalid_argument if `K` is non-positive.
+         * @tparam D Spatial dimension (2 or 3).
          */
-        explicit LinearTransportMaterial(double K) {
-            if (K <= 0.0) {
-                throw std::invalid_argument("K (" + std::to_string(K) + ") must be positive.");
+        template <int D>
+        class LinearTransportMaterial {
+        public:
+            static_assert(D == 2 || D == 3, "Spatial dimension D must be 2 or 3.");
+
+            /// @brief Spatial dimension (2 or 3).
+            static constexpr int Dim = D;
+
+            /// @brief Transport tensor type.
+            using MaterialTensor = Eigen::Matrix<double, Dim, Dim>;
+
+            /**
+             * @brief Constructs a linear transport material.
+             *
+             * @param[in] K Transport tensor.
+             *
+             * @throws std::invalid_argument if `K` is not positive definite.
+             */
+            explicit LinearTransportMaterial(const MaterialTensor &K)
+                : K_(K) {
+                if (!detail::isPD(K_)) {
+                    throw std::invalid_argument("Transport tensor is not positive definite.");
+                }
             }
 
-            K_ = MaterialTensor::Identity() * K;
-        }
-
-        /// @brief Transport tensor K.
-        const MaterialTensor &materialTensor() const noexcept {
-            return K_;
-        }
-
-        /**
-         * @brief Voigt upper bound for the homogenized transport tensor.
-         *
-         * @tparam Grid Grid class (e.g. Quad4Grid).
-         * @tparam Element Element class (e.g. Quad4).
-         *
-         * @param[in] grid Periodic unit cell grid.
-         *
-         * @returns Voigt upper bound for the homogenized transport tensor.
-         */
-        template <class Grid, class Element>
-        MaterialTensor voigt(const GridBase<Grid, Element> &grid) const noexcept {
-            const double density = detail::arithmeticMean(grid.densities());
-
-            return density * K_;
-        }
+            /**
+             * @brief Constructs an isotropic linear transport material.
+             *
+             * @param[in] K Transport constant.
+             *
+             * @throws std::invalid_argument if `K` is non-positive.
+             */
+            explicit LinearTransportMaterial(double K) {
+                if (K <= 0.0) {
+                    throw std::invalid_argument("K (" + std::to_string(K) + ") must be positive.");
+                }
+
+                K_ = MaterialTensor::Identity() * K;
+            }
 
-        /**
-         * @brief Reuss lower bound for the homogenized transport tensor.
-         *
-         * @tparam Grid Grid class (e.g. Quad4Grid).
-         * @tparam Element Element class (e.g. Quad4).
-         *
-         * @param[in] grid Periodic unit cell grid.
-         *
-         * @returns Reuss lower bound for the homogenized transport tensor.
-         */
-        template <class Grid, class Element>
-        MaterialTensor reuss(const GridBase<Grid, Element> &grid) const noexcept {
-            const double density = detail::harmonicMean(grid.densities());
+            /// @brief Transport tensor.
+            const MaterialTensor &materialTensor() const noexcept {
+                return K_;
+            }
 
-            return density * K_;
-        }
+            /// @brief Equality comparison.
+            bool operator==(const LinearTransportMaterial &other) const noexcept {
+                return K_.isApprox(other.K_, NUMERICAL_ZERO);
+            }
 
-        /// @brief Equality comparison.
-        bool operator==(const LinearTransportMaterial &other) const noexcept {
-            return K_.isApprox(other.K_);
-        }
+            /// @brief Inequality comparison.
+            bool operator!=(const LinearTransportMaterial &other) const noexcept {
+                return !(*this == other);
+            }
 
-        /// @brief Inequality comparison.
-        bool operator!=(const LinearTransportMaterial &other) const noexcept {
-            return !(*this == other);
-        }
+        private:
+            /// @brief Transport tensor.
+            MaterialTensor K_;
+        };
 
-    private:
-        /// @brief Transport tensor K.
-        MaterialTensor K_;
-    };
+    } // namespace material
 
 } // namespace monad
diff --git a/include/monad/material/transport/linear_transport_material_aliases.hpp b/include/monad/material/transport/linear_transport_material_aliases.hpp
deleted file mode 100644
index 1cd9bd7..0000000
--- a/include/monad/material/transport/linear_transport_material_aliases.hpp
+++ /dev/null
@@ -1,259 +0,0 @@
-#pragma once
-
-#include "monad/material/transport/linear_transport_material.hpp"
-
-namespace monad {
-
-    /**
-     * @brief Represents a 2D linear dielectric material model.
-     *
-     * This type is a physics-specific alias of a general linear transport material,
-     * whose constitutive relation takes the form:
-     *
-     * J=-K∇φ
-     *
-     * For dielectric materials, Gauss's law provides the constitutive relation:
-     *
-     * D=εE
-     *
-     * - D∈ℝ² is the electric displacement (analogous to flux J)..
-     *
-     * - E=-∇φ∈ℝ² is the electric field (analogous to scalar potential gradient ∇φ)..
-     *
-     * - φ∈ℝ is the electric potential (analogous to scalar potential φ)..
-     *
-     * - ε∈Sym₂(ℝ) is the permittivity tensor (analogous to transport tensor K)..
-     */
-    using LinearDielectricMaterial2d = LinearTransportMaterial<2>;
-
-    /**
-     * @brief Represents a 3D linear dielectric material model.
-     *
-     * This type is a physics-specific alias of a general linear transport material,
-     * whose constitutive relation takes the form:
-     *
-     * J=-K∇φ
-     *
-     * For dielectric materials, Gauss's law provides the constitutive relation:
-     *
-     * D=εE
-     *
-     * - D∈ℝ³ is the electric displacement (analogous to flux J).
-     *
-     * - E=-∇φ∈ℝ³ is the electric field (analogous to scalar potential gradient ∇φ).
-     *
-     * - φ∈ℝ is the electric potential (analogous to scalar potential φ).
-     *
-     * - ε∈Sym₃(ℝ) is the permittivity tensor (analogous to transport tensor K).
-     */
-    using LinearDielectricMaterial3d = LinearTransportMaterial<3>;
-
-    /**
-     * @brief Represents a 2D linear electrical conductive material model.
-     *
-     * This type is a physics-specific alias of a general linear transport material,
-     * whose constitutive relation takes the form:
-     *
-     * J=-K∇φ
-     *
-     * For electrical conductive materials, Ohm's law provides the constitutive relation:
-     *
-     * J=σE
-     *
-     * - J∈ℝ² is the current density (analogous to flux J).
-     *
-     * - E=-∇φ∈ℝ² is the electric field (analogous to scalar potential gradient ∇φ).
-     *
-     * - φ∈ℝ is the electric potential (analogous to scalar potential φ).
-     *
-     * - σ∈Sym₂(ℝ) is the conductivity tensor (analogous to transport tensor K).
-     */
-    using LinearElectricalConductiveMaterial2d = LinearTransportMaterial<2>;
-
-    /**
-     * @brief Represents a 3D linear electrical conductive material model.
-     *
-     * This type is a physics-specific alias of a general linear transport material,
-     * whose constitutive relation takes the form:
-     *
-     * J=-K∇φ
-     *
-     * For electrical conductive materials, Ohm's law provides the constitutive relation:
-     *
-     * J=σE
-     *
-     * - J∈ℝ³ is the current density (analogous to flux J).
-     *
-     * - E=-∇φ∈ℝ³ is the electric field (analogous to scalar potential gradient ∇φ).
-     *
-     * - φ∈ℝ is the electric potential (analogous to scalar potential φ).
-     *
-     * - σ∈Sym₃(ℝ) is the conductivity tensor (analogous to transport tensor K).
-     */
-    using LinearElectricalConductiveMaterial3d = LinearTransportMaterial<3>;
-
-    /**
-     * @brief Represents a 2D linear magnetic material model.
-     *
-     * This type is a physics-specific alias of a general linear transport material,
-     * whose constitutive relation takes the form:
-     *
-     * J=-K∇φ
-     *
-     * For magnetic materials, the constitutive relation is:
-     *
-     * B=μH
-     *
-     * - B∈ℝ² is the magnetic flux density (analogous to flux J).
-     *
-     * - H=-∇φ∈ℝ² is the magnetic field (analogous to scalar potential gradient ∇φ).
-     *
-     * - φ∈ℝ is the magnetic potential (analogous to scalar potential φ).
-     *
-     * - μ∈Sym₂(ℝ) is the permeability tensor (analogous to transport tensor K).
-     */
-    using LinearMagneticMaterial2d = LinearTransportMaterial<2>;
-
-    /**
-     * @brief Represents a 3D linear magnetic material model.
-     *
-     * This type is a physics-specific alias of a general linear transport material,
-     * whose constitutive relation takes the form:
-     *
-     * J=-K∇φ
-     *
-     * For magnetic materials, the constitutive relation is:
-     *
-     * B=μH
-     *
-     * - B∈ℝ³ is the magnetic flux density (analogous to flux J).
-     *
-     * - H=-∇φ∈ℝ³ is the magnetic field (analogous to scalar potential gradient ∇φ).
-     *
-     * - φ∈ℝ is the magnetic potential (analogous to scalar potential φ).
-     *
-     * - μ∈Sym₃(ℝ) is the permeability tensor (analogous to transport tensor K).
-     */
-    using LinearMagneticMaterial3d = LinearTransportMaterial<3>;
-
-    /**
-     * @brief Represents a 2D linear mass diffusive material model.
-     *
-     * This type is a physics-specific alias of a general linear transport material,
-     * whose constitutive relation takes the form:
-     *
-     * J=-K∇φ
-     *
-     * For mass diffusive materials, Fick's law provides the constitutive relation:
-     *
-     * J=-D∇c
-     *
-     * - J∈ℝ² is the diffusion flux (analogous to flux J).
-     *
-     * - ∇c∈ℝ² is the mass concentration gradient (analogous to scalar potential gradient ∇φ).
-     *
-     * - D∈Sym₂(ℝ) is the diffusivity tensor (analogous to transport tensor K).
-     */
-    using LinearMassDiffusiveMaterial2d = LinearTransportMaterial<2>;
-
-    /**
-     * @brief Represents a 3D linear mass diffusive material model.
-     *
-     * This type is a physics-specific alias of a general linear transport material,
-     * whose constitutive relation takes the form:
-     *
-     * J=-K∇φ
-     *
-     * For mass diffusive materials, Fick's law provides the constitutive relation:
-     *
-     * J=-D∇c
-     *
-     * - J∈ℝ³ is the diffusion flux (analogous to flux J).
-     *
-     * - ∇c∈ℝ³ is the mass concentration gradient (analogous to scalar potential gradient ∇φ).
-     *
-     * - D∈Sym₃(ℝ) is the diffusivity tensor (analogous to transport tensor K).
-     */
-    using LinearMassDiffusiveMaterial3d = LinearTransportMaterial<3>;
-
-    /**
-     * @brief Represents a 2D linear porous material model.
-     *
-     * This type is a physics-specific alias of a general linear transport material,
-     * whose constitutive relation takes the form:
-     *
-     * J=-K∇φ
-     *
-     * For porous materials, Darcy's law provides the constitutive relation:
-     *
-     * q=-K∇p
-     *
-     * - q∈ℝ² is the volumetric flux (analogous to flux J).
-     *
-     * - ∇p∈ℝ² is the pressure gradient (analogous to scalar potential gradient ∇φ).
-     *
-     * - K∈Sym₂(ℝ) is the permeability tensor (analogous to transport tensor K).
-     */
-    using LinearPorousMaterial2d = LinearTransportMaterial<2>;
-
-    /**
-     * @brief Represents a 3D linear porous material model.
-     *
-     * This type is a physics-specific alias of a general linear transport material,
-     * whose constitutive relation takes the form:
-     *
-     * J=-K∇φ
-     *
-     * For porous materials, Darcy's law provides the constitutive relation:
-     *
-     * q=-K∇p
-     *
-     * - D∈ℝ³ is the volumetric flux (analogous to flux J).
-     *
-     * - ∇p∈ℝ³ is the pressure gradient (analogous to scalar potential gradient ∇φ).
-     *
-     * - K∈Sym₃(ℝ) is the permeability tensor (analogous to transport tensor K).
-     */
-    using LinearPorousMaterial3d = LinearTransportMaterial<3>;
-
-    /**
-     * @brief Represents a 2D linear thermal conductive material model.
-     *
-     * This type is a physics-specific alias of a general linear transport material,
-     * whose constitutive relation takes the form:
-     *
-     * J=-K∇φ
-     *
-     * For thermal conductive materials, Fourier's law provides the constitutive relation:
-     *
-     * q=-κ∇T
-     *
-     * - q∈ℝ² is the heat flux (analogous to flux J).
-     *
-     * - ∇T∈ℝ² is the temperature gradient (analogous to scalar potential gradient ∇φ).
-     *
-     * - κ∈Sym₂(ℝ) is the conductivity tensor (analogous to transport tensor K).
-     */
-    using LinearThermalConductiveMaterial2d = LinearTransportMaterial<2>;
-
-    /**
-     * @brief Represents a 3D linear thermal conductive material model.
-     *
-     * This type is a physics-specific alias of a general linear transport material,
-     * whose constitutive relation takes the form:
-     *
-     * J=-K∇φ
-     *
-     * For thermal conductive materials, Fourier's law provides the constitutive relation:
-     *
-     * q=-κ∇T
-     *
-     * - q∈ℝ³ is the heat flux (analogous to flux J).
-     *
-     * - ∇T∈ℝ³ is the temperature gradient (analogous to scalar potential gradient ∇φ).
-     *
-     * - κ∈Sym₃(ℝ) is the conductivity tensor (analogous to transport tensor K).
-     */
-    using LinearThermalConductiveMaterial3d = LinearTransportMaterial<3>;
-
-} // namespace monad
diff --git a/include/monad/monad.hpp b/include/monad/monad.hpp
index 44a9082..6cdb979 100644
--- a/include/monad/monad.hpp
+++ b/include/monad/monad.hpp
@@ -1,23 +1,24 @@
 #pragma once
 
+// Fields
+#include "monad/field/field_aliases.hpp"
+#include "monad/field/make_density_field_from_csv.hpp"
+#include "monad/field/make_density_field_from_function.hpp"
+
 // Grids
-#include "monad/grid/hex8_grid.hpp"
-#include "monad/grid/hex20_grid.hpp"
-#include "monad/grid/quad4_grid.hpp"
-#include "monad/grid/quad8_grid.hpp"
+#include "monad/grid/grid_aliases.hpp"
 
 // IO
 #include "monad/io/save_grid.hpp"
-#include "monad/io/save_grid_and_field.hpp"
+#include "monad/io/save_grid_and_density_field.hpp"
+#include "monad/io/save_grid_and_nodal_field.hpp"
 
 // Material
 #include "monad/material/mechanical/linear_elastic_material_2d.hpp"
 #include "monad/material/mechanical/linear_elastic_material_3d.hpp"
-#include "monad/material/transport/linear_transport_material_aliases.hpp"
-#include "monad/material/multiphysics/linear_piezoelectric_material.hpp"
+#include "monad/material/bounds.hpp"
+#include "monad/material/material_aliases.hpp"
 
 // Solvers
 #include "monad/solver/solver_options.hpp"
-#include "monad/solver/mechanical/linear_elastic_solver.hpp"
-#include "monad/solver/scalar/linear_scalar_diffusive_solver_aliases.hpp"
-#include "monad/solver/multiphysics/linear_piezoelectric_solver.hpp"
+#include "monad/solver/solver_aliases.hpp"
diff --git a/include/monad/solver/identity_preconditioner.hpp b/include/monad/solver/identity_preconditioner.hpp
new file mode 100644
index 0000000..a5d5bf1
--- /dev/null
+++ b/include/monad/solver/identity_preconditioner.hpp
@@ -0,0 +1,23 @@
+#pragma once
+
+#include <Eigen/Sparse>
+
+namespace monad {
+
+    namespace solver {
+
+        /**
+         * @brief Identity preconditioner for a matrix-free stiffness operator.
+         *
+         * This preconditioner approximates the inverse of the reduced global
+         * stiffness operator A by the identity matrix:
+         *
+         * ```text
+         * A⁻¹≈I
+         * ```
+         */
+        using IdentityPreconditioner = Eigen::IdentityPreconditioner;
+
+    } // namespace solver
+
+} // namespace monad
diff --git a/include/monad/fem/operator/jacobi_preconditioner.hpp b/include/monad/solver/jacobi_preconditioner.hpp
similarity index 52%
rename from include/monad/fem/operator/jacobi_preconditioner.hpp
rename to include/monad/solver/jacobi_preconditioner.hpp
index e03e59f..f50c12b 100644
--- a/include/monad/fem/operator/jacobi_preconditioner.hpp
+++ b/include/monad/solver/jacobi_preconditioner.hpp
@@ -4,17 +4,19 @@
 
 namespace monad {
 
-    namespace fem {
+    namespace solver {
 
         /**
-         * @brief Jacobi (diagonal) preconditioner for the matrix-free operator.
+         * @brief Jacobi preconditioner for a matrix-free stiffness operator.
          *
-         * This preconditioner approximates the inverse of the global stiffness matrix A
-         * by inverting only its diagonal:
+         * This preconditioner approximates the inverse of the reduced global
+         * stiffness operator A by inverting only its diagonal:
          *
+         * ```text
          * A⁻¹≈diag(A)⁻¹
+         * ```
          *
-         * @tparam Operator Matrix-free operator class (e.g. MatrixFreeOperator<Quad4Grid, Quad4, LinearElasticMatrixFreeOperatorTraits<2>>).
+         * @tparam Operator Matrix-free operator type (e.g. MatrixFreeOperator<Quad4Grid, LinearElasticDofTraits<2>>).
          */
         template <class Operator>
         class JacobiPreconditioner {
@@ -35,7 +37,7 @@ namespace monad {
             }
 
             /**
-             * @brief Computes the diagonal of the global stiffness matrix.
+             * @brief Computes the diagonal preconditioner from a matrix-free operator.
              *
              * @param[in] op Matrix-free operator.
              *
@@ -46,24 +48,33 @@ namespace monad {
                 diagonal_.setZero(n);
 
                 const auto &elementKReference = op.elementKReference();
-                const auto &elementDofs = op.elementDofs();
-                const auto &densities = op.densities();
+                const auto &dofMap = op.dofMap();
+                const auto &densityField = op.densityField();
 
-                // Loop over all elements and accumulate their diagonal contributions
-                for (std::size_t i = 0; i < elementDofs.size(); ++i) {
-                    const double density = densities[i];
-                    const auto &dofs = elementDofs[i];
+                // Assemble the reduced global diagonal by accumulating the
+                // density-scaled diagonal entries of each element matrix
+                for (std::size_t i = 0; i < dofMap.size(); ++i) {
+                    const double density = densityField.getDensity(i);
+                    const auto &reducedDofs = dofMap.reducedDofs(i);
 
-                    for (std::size_t j = 0; j < dofs.size(); ++j) {
-                        const int globalDof = dofs[j];
-                        const int localDof = static_cast<int>(j);
+                    for (std::size_t j = 0; j < reducedDofs.size(); ++j) {
+                        const int reducedDof = reducedDofs[j];
+                        const int localReducedDof = static_cast<int>(j);
 
-                        if (globalDof >= 0) {
-                            diagonal_(globalDof) += density * elementKReference(localDof, localDof);
+                        if (reducedDof >= 0) {
+                            diagonal_(reducedDof) += density * elementKReference(localReducedDof, localReducedDof);
                         }
                     }
                 }
 
+                // Jacobi requires invertible diagonal entries
+                for (int i = 0; i < diagonal_.size(); ++i) {
+                    if (diagonal_(i) == Scalar(0)) {
+                        info_ = Eigen::NumericalIssue;
+                        break;
+                    }
+                }
+
                 return *this;
             }
 
@@ -72,9 +83,11 @@ namespace monad {
              *
              * Computes
              *
+             * ```text
              * x=A⁻¹b≈diag(A)⁻¹b
+             * ```
              *
-             * via element-wise division.
+             * by element-wise division.
              *
              * @tparam Rhs Type of the right-hand side vector.
              *
@@ -83,20 +96,23 @@ namespace monad {
              * @returns Preconditioned vector.
              */
             template <typename Rhs>
-            inline Vector solve(const Eigen::MatrixBase<Rhs> &b) const noexcept {
+            inline Vector solve(const Eigen::MatrixBase<Rhs> &b) const {
                 return b.cwiseQuotient(diagonal_);
             }
 
             /// @brief Status of the preconditioner.
             Eigen::ComputationInfo info() const noexcept {
-                return Eigen::Success;
+                return info_;
             }
 
         private:
             /// @brief Diagonal of the global stiffness matrix.
             Vector diagonal_;
+
+            /// @brief Status of the preconditioner.
+            Eigen::ComputationInfo info_ = Eigen::Success;
         };
 
-    } // namespace fem
+    } // namespace solver
 
 } // namespace monad
diff --git a/include/monad/solver/mechanical/linear_elastic_physics.hpp b/include/monad/solver/mechanical/linear_elastic_physics.hpp
deleted file mode 100644
index 67402d4..0000000
--- a/include/monad/solver/mechanical/linear_elastic_physics.hpp
+++ /dev/null
@@ -1,154 +0,0 @@
-#pragma once
-
-#include <array>
-#include <cstddef>
-#include "monad/fem/kernel/mechanical/linear_elastic_kernel.hpp"
-#include "monad/fem/operator/mechanical/linear_elastic_matrix_free_operator_traits.hpp"
-#include "monad/solver/solver_options.hpp"
-#include "monad/solver/mechanical/linear_elastic_physics_traits.hpp"
-
-namespace monad {
-
-    namespace solver {
-
-        namespace mechanical {
-
-            /**
-             * @brief Physics policy for linear elastic homogenization.
-             *
-             * @tparam Element Element class (e.g. Quad4).
-             */
-            template <class Element>
-            struct LinearElasticPhysics {
-                using Kernel = fem::mechanical::LinearElasticKernel<Element>;
-                using Material = typename Kernel::Material;
-
-                /// @brief Spatial dimension (2 or 3).
-                static constexpr int Dim = Element::Dim;
-
-                /**
-                 * @brief Number of macroscopic fields:
-                 *
-                 * - D=2: ε̄₁₁, ε̄₂₂, ε̄₁₂.
-                 *
-                 * - D=3: ε̄₁₁, ε̄₂₂, ε̄₃₃, ε̄₁₂, ε̄₁₃, ε̄₂₃.
-                 */
-                static constexpr int NumMacroFields = Material::VoigtSize;
-
-                using OperatorTraits = fem::mechanical::LinearElasticMatrixFreeOperatorTraits<Dim>;
-
-                using Traits = LinearElasticPhysicsTraits<Dim>;
-
-                /// @brief Global force matrix type.
-                using FieldMatrix = Eigen::Matrix<double, Eigen::Dynamic, NumMacroFields>;
-
-                /// @brief Homogenized stiffness tensor type.
-                using MaterialTensor = typename Material::MaterialTensor;
-
-                /**
-                 * @brief Results from linear elastic homogenization.
-                 *
-                 * @note Displacement fields are stored in arrays
-                 * ordered by the prescribed macroscopic strain
-                 * loading directions:
-                 *
-                 * - D=2: ε̄₁₁, ε̄₂₂, ε̄₁₂.
-                 *
-                 * - D=3: ε̄₁₁, ε̄₂₂, ε̄₃₃, ε̄₁₂, ε̄₁₃, ε̄₂₃.
-                 *
-                 * Each array entry corresponds to one independent macroscopic loading case.
-                 */
-                struct Results {
-                    using NodalField = Eigen::Matrix<double, Eigen::Dynamic, Dim>;
-
-                    /// @brief Homogenized stiffness tensor C̄.
-                    MaterialTensor CBar;
-
-                    /// @brief Total nodal displacement fields u=ū+ũ.
-                    std::array<NodalField, NumMacroFields> u;
-
-                    /// @brief Macroscopic nodal displacement fields ū.
-                    std::array<NodalField, NumMacroFields> uMacro;
-
-                    /// @brief Microscopic nodal displacement fields ũ.
-                    std::array<NodalField, NumMacroFields> uMicro;
-                };
-
-                /**
-                 * @brief Macroscopic nodal displacements.
-                 *
-                 * The macroscopic displacements Ū are the displacements
-                 * resulting from the macroscopic strains ε̄:
-                 *
-                 * Ū=ε̄x
-                 *
-                 * @tparam Grid Grid class (e.g. Quad4Grid).
-                 *
-                 * @param[in] grid Periodic unit cell grid.
-                 *
-                 * @returns Macroscopic nodal displacements.
-                 */
-                template <class Grid>
-                static FieldMatrix macroscopicField(const GridBase<Grid, Element> &grid) {
-                    return Traits::macroDisplacements(grid);
-                }
-
-                /**
-                 * @brief Constructs linear elastic homogenization results.
-                 *
-                 * @param[in] CBar Homogenized stiffness tensor.
-                 * @param[in] U Total nodal displacement fields.
-                 * @param[in] UMacro Macroscopic nodal displacement fields.
-                 * @param[in] UMicro Microscopic nodal displacement fields.
-                 * @param[in] fields Controls which nodal fields are stored in the solver results.
-                 *
-                 * @returns Linear elastic homogenization results.
-                 */
-                static Results makeResults(const MaterialTensor &CBar, const FieldMatrix &U, const FieldMatrix &UMacro, const FieldMatrix &UMicro, FieldSave fields) noexcept {
-                    using NodalField = typename Results::NodalField;
-
-                    Results results;
-
-                    results.CBar = CBar;
-
-                    const auto numNodes = U.rows() / Dim;
-
-                    if (wants(fields, FieldSave::Total)) {
-                        std::array<NodalField, NumMacroFields> temp;
-
-                        for (std::size_t i = 0; i < NumMacroFields; ++i) {
-                            temp[i] = U.col(static_cast<int>(i)).template reshaped<Eigen::RowMajor>(numNodes, Dim);
-                        }
-
-                        results.u = temp;
-                    }
-
-                    if (wants(fields, FieldSave::Macro)) {
-                        std::array<NodalField, NumMacroFields> temp;
-
-                        for (std::size_t i = 0; i < NumMacroFields; ++i) {
-                            temp[i] = UMacro.col(static_cast<int>(i)).template reshaped<Eigen::RowMajor>(numNodes, Dim);
-                        }
-
-                        results.uMacro = temp;
-                    }
-
-                    if (wants(fields, FieldSave::Micro)) {
-                        std::array<NodalField, NumMacroFields> temp;
-
-                        for (std::size_t i = 0; i < NumMacroFields; ++i) {
-                            temp[i] = UMicro.col(static_cast<int>(i)).template reshaped<Eigen::RowMajor>(numNodes, Dim);
-                        }
-
-                        results.uMicro = temp;
-                    }
-
-                    return results;
-                }
-            };
-
-        } // namespace mechanical
-
-    } // namespace solver
-
-} // namespace monad
diff --git a/include/monad/solver/mechanical/linear_elastic_physics_traits.hpp b/include/monad/solver/mechanical/linear_elastic_physics_traits.hpp
deleted file mode 100644
index c7f4755..0000000
--- a/include/monad/solver/mechanical/linear_elastic_physics_traits.hpp
+++ /dev/null
@@ -1,144 +0,0 @@
-#pragma once
-
-#include <cstddef>
-#include <Eigen/Core>
-#include "monad/grid/grid_base.hpp"
-#include "monad/fem/operator/mechanical/linear_elastic_matrix_free_operator_traits.hpp"
-
-namespace monad {
-
-    namespace solver {
-
-        namespace mechanical {
-
-            /**
-             * @brief Linear elastic physics traits.
-             *
-             * This struct isolates dimension-specific logic so that linear elastic physics
-             * can be written generically in terms of spatial dimension.
-             *
-             * @tparam D Spatial dimension (2 or 3).
-             */
-            template <int D>
-            struct LinearElasticPhysicsTraits;
-
-            /**
-             * @brief 2D linear elastic physics traits.
-             *
-             * This struct isolates 2D-specific logic so that linear elastic physics
-             * can be written generically in terms of spatial dimension.
-             */
-            template <>
-            struct LinearElasticPhysicsTraits<2> {
-                using OperatorTraits = fem::mechanical::LinearElasticMatrixFreeOperatorTraits<2>;
-
-                /**
-                 * @brief Macroscopic nodal displacements.
-                 *
-                 * The macroscopic displacements Ū are the displacements
-                 * resulting from the macroscopic strains ε̄:
-                 *
-                 * Ū=ε̄x
-                 *
-                 * @tparam Grid Grid class (e.g. Quad4Grid).
-                 * @tparam Element Element class (e.g. Quad4).
-                 *
-                 * @param[in] grid Periodic unit cell grid.
-                 *
-                 * @returns Macroscopic nodal displacements.
-                 */
-                template <class Grid, class Element>
-                static Eigen::MatrixX3d macroDisplacements(const GridBase<Grid, Element> &grid) {
-                    static_assert(Element::Dim == Grid::Dim, "Element spatial dimension must equal grid spatial dimension.");
-
-                    const std::size_t numNodes = grid.numNodes();
-                    const std::size_t numDofs = OperatorTraits::NumNodeDofs * numNodes;
-
-                    Eigen::MatrixX3d U = Eigen::MatrixX3d::Zero(static_cast<int>(numDofs), 3);
-
-                    for (std::size_t i = 0; i < grid.numNodes(); ++i) {
-                        auto node = grid.node(i);
-
-                        const double x = node(0);
-                        const double y = node(1);
-
-                        const int u = static_cast<int>(2 * i);
-                        const int v = static_cast<int>(2 * i + 1);
-
-                        U(u, 0) = x;
-                        U(v, 1) = y;
-                        U(u, 2) = 0.5 * y;
-                        U(v, 2) = 0.5 * x;
-                    }
-
-                    return U;
-                }
-            };
-
-            /**
-             * @brief 3D linear elastic physics traits.
-             *
-             * This struct isolates 3D-specific logic so that linear elastic physics
-             * can be written generically in terms of spatial dimension.
-             */
-            template <>
-            struct LinearElasticPhysicsTraits<3> {
-                using OperatorTraits = fem::mechanical::LinearElasticMatrixFreeOperatorTraits<3>;
-
-                using MatrixX6d = Eigen::Matrix<double, Eigen::Dynamic, 6>;
-
-                /**
-                 * @brief Macroscopic nodal displacements.
-                 *
-                 * The macroscopic displacements Ū are the displacements
-                 * resulting from the macroscopic strains ε̄:
-                 *
-                 * Ū=ε̄x
-                 *
-                 * @tparam Grid Grid class (e.g. Hex8Grid).
-                 * @tparam Element Element class (e.g. Hex8).
-                 *
-                 * @param[in] grid Periodic unit cell grid.
-                 *
-                 * @returns Macroscopic nodal displacements.
-                 */
-                template <class Grid, class Element>
-                static MatrixX6d macroDisplacements(const GridBase<Grid, Element> &grid) {
-                    static_assert(Element::Dim == Grid::Dim, "Element spatial dimension must equal grid spatial dimension.");
-
-                    const std::size_t numNodes = grid.numNodes();
-                    const std::size_t numDofs = OperatorTraits::NumNodeDofs * numNodes;
-
-                    MatrixX6d U = MatrixX6d::Zero(static_cast<int>(numDofs), 6);
-
-                    for (std::size_t i = 0; i < grid.numNodes(); ++i) {
-                        const auto node = grid.node(i);
-
-                        const double x = node(0);
-                        const double y = node(1);
-                        const double z = node(2);
-
-                        const int u = static_cast<int>(3 * i);
-                        const int v = static_cast<int>(3 * i + 1);
-                        const int w = static_cast<int>(3 * i + 2);
-
-                        U(u, 0) = x;
-                        U(v, 1) = y;
-                        U(w, 2) = z;
-                        U(u, 3) = 0.5 * y;
-                        U(v, 3) = 0.5 * x;
-                        U(u, 4) = 0.5 * z;
-                        U(w, 4) = 0.5 * x;
-                        U(w, 5) = 0.5 * y;
-                        U(v, 5) = 0.5 * z;
-                    }
-
-                    return U;
-                }
-            };
-
-        } // namespace mechanical
-
-    } // namespace solver
-
-} // namespace monad
diff --git a/include/monad/solver/mechanical/linear_elastic_policy.hpp b/include/monad/solver/mechanical/linear_elastic_policy.hpp
new file mode 100644
index 0000000..ee66736
--- /dev/null
+++ b/include/monad/solver/mechanical/linear_elastic_policy.hpp
@@ -0,0 +1,199 @@
+#pragma once
+
+#include <array>
+#include <cstddef>
+#include <Eigen/Core>
+#include "monad/fem/kernel/mechanical/linear_elastic_kernel.hpp"
+#include "monad/fem/operator/mechanical/linear_elastic_dof_traits.hpp"
+#include "monad/material/mechanical/linear_elastic_material.hpp"
+#include "monad/solver/solver_options.hpp"
+
+namespace monad {
+
+    namespace solver {
+
+        namespace mechanical {
+
+            /**
+             * @brief Physics policy for periodic linear elastic homogenization.
+             *
+             * @tparam Element Element type (e.g. Quad4).
+             */
+            template <class Element>
+            struct LinearElasticPolicy {
+                using Kernel = fem::mechanical::LinearElasticKernel<Element>;
+                using DofTraits = fem::mechanical::LinearElasticDofTraits<Element::Dim>;
+                using Material = material::LinearElasticMaterial<Element::Dim>;
+                using MaterialTensor = typename Material::MaterialTensor;
+
+                /// @brief Spatial dimension (2 or 3).
+                static constexpr int Dim = Element::Dim;
+
+                /**
+                 * @brief Number of macroscopic strain load cases.
+                 *
+                 * The load cases are ordered in Voigt notation:
+                 *
+                 * - 2D: ε̄₁₁, ε̄₂₂, ε̄₁₂.
+                 *
+                 * - 3D: ε̄₁₁, ε̄₂₂, ε̄₃₃, ε̄₁₂, ε̄₁₃, ε̄₂₃.
+                 */
+                static constexpr int NumLoadCases = Material::VoigtSize;
+
+                /// @brief Dof-major matrix type storing one global field per load case.
+                using DofFieldMatrix = Eigen::Matrix<double, Eigen::Dynamic, NumLoadCases>;
+                
+                /// @brief Nodal field type.
+                using NodalFieldMatrix = Eigen::Matrix<double, Eigen::Dynamic, Dim>;
+
+                /// @brief List of nodal fields, one per load case.
+                using NodalFieldList = std::array<NodalFieldMatrix, NumLoadCases>;
+
+                /**
+                 * @brief Linear elastic homogenization results.
+                 *
+                 * The homogenized stiffness tensor is always returned.
+                 * Nodal displacements are stored only when requested
+                 * through `FieldSave`.
+                 */
+                struct Results {
+                    /// @brief Homogenized stiffness tensor.
+                    MaterialTensor CBar;
+
+                    /// @brief Total nodal displacements.
+                    NodalFieldList u;
+
+                    /// @brief Macroscopic nodal displacements.
+                    NodalFieldList uMacro;
+
+                    /// @brief Microscopic nodal displacements.
+                    NodalFieldList uMicro;
+                };
+
+                /**
+                 * @brief Macroscopic nodal displacements.
+                 *
+                 * For each prescribed macroscopic strain ε̄,
+                 * the macroscopic displacements are:
+                 *
+                 * ```text
+                 * ū=ε̄x
+                 * ```
+                 *
+                 * @tparam Grid Grid type (e.g. Quad4Grid).
+                 *
+                 * @param[in] grid Grid.
+                 *
+                 * @returns Macroscopic nodal displacements.
+                 */
+                template <class Grid>
+                static DofFieldMatrix makeMacroscopicFields(const Grid &grid) noexcept {
+                    const std::size_t numNodes = grid.numNodes();
+                    const std::size_t numDofs = DofTraits::NumNodeDofs * numNodes;
+
+                    DofFieldMatrix UBar = DofFieldMatrix::Zero(static_cast<int>(numDofs), NumLoadCases);
+
+                    for (std::size_t i = 0; i < numNodes; ++i) {
+                        const auto node = grid.node(i);
+
+                        if constexpr (Dim == 2) {
+                            const double x = node(0);
+                            const double y = node(1);
+
+                            const int u = static_cast<int>(2 * i);
+                            const int v = static_cast<int>(2 * i + 1);
+
+                            // ū⁽¹¹⁾=ε̄⁽¹¹⁾x
+                            UBar(u, 0) = x;
+                            // ū⁽²²⁾=ε̄⁽²²⁾x
+                            UBar(v, 1) = y;
+                            // ū⁽¹²⁾=ε̄⁽¹²⁾x
+                            UBar(u, 2) = 0.5 * y;
+                            UBar(v, 2) = 0.5 * x;
+                        }
+                        else {
+                            const double x = node(0);
+                            const double y = node(1);
+                            const double z = node(2);
+
+                            const int u = static_cast<int>(3 * i);
+                            const int v = static_cast<int>(3 * i + 1);
+                            const int w = static_cast<int>(3 * i + 2);
+
+                            // ū⁽¹¹⁾=ε̄⁽¹¹⁾x
+                            UBar(u, 0) = x;
+                            // ū⁽²²⁾=ε̄⁽²²⁾x
+                            UBar(v, 1) = y;
+                            // ū⁽³³⁾=ε̄⁽³³⁾x
+                            UBar(w, 2) = z;
+                            // ū⁽¹²⁾=ε̄⁽¹²⁾x
+                            UBar(u, 3) = 0.5 * y;
+                            UBar(v, 3) = 0.5 * x;
+                            // ū⁽¹³⁾=ε̄⁽¹³⁾x
+                            UBar(u, 4) = 0.5 * z;
+                            UBar(w, 4) = 0.5 * x;
+                            // ū⁽²³⁾=ε̄⁽²³⁾x
+                            UBar(v, 5) = 0.5 * z;
+                            UBar(w, 5) = 0.5 * y;
+                        }
+                    }
+
+                    return UBar;
+                }
+
+                /**
+                 * @brief Packages solver outputs into linear elastic results.
+                 *
+                 * @param[in] CBar Homogenized stiffness tensor.
+                 * @param[in] U Total nodal displacements.
+                 * @param[in] UMacro Macroscopic nodal displacements.
+                 * @param[in] UMicro Microscopic nodal displacements.
+                 * @param[in] fields Controls which nodal fields are stored.
+                 *
+                 * @returns Homogenization results.
+                 */
+                static Results makeResults(const MaterialTensor &CBar, const DofFieldMatrix &U, const DofFieldMatrix &UMacro, const DofFieldMatrix &UMicro, FieldSave fields) noexcept {
+                    Results results;
+                    results.CBar = CBar;
+
+                    if (wants(fields, FieldSave::Total)) {
+                        results.u = makeStoredFields_(U);
+                    }
+
+                    if (wants(fields, FieldSave::Macro)) {
+                        results.uMacro = makeStoredFields_(UMacro);
+                    }
+
+                    if (wants(fields, FieldSave::Micro)) {
+                        results.uMicro = makeStoredFields_(UMicro);
+                    }
+
+                    return results;
+                }
+
+            private:
+                /**
+                 * @brief Converts a nodal displacement block matrix into a list of nodal displacements.
+                 *
+                 * @param[in] U Nodal displacement block matrix.
+                 *
+                 * @return List of nodal displacements.
+                 */
+                static NodalFieldList makeStoredFields_(const DofFieldMatrix &U) {
+                    const int numNodes = static_cast<int>(U.rows()) / Dim;
+                    NodalFieldList out;
+
+                    for (std::size_t i = 0; i < NumLoadCases; ++i) {
+                        const NodalFieldMatrix u = U.col(static_cast<int>(i)).template reshaped<Eigen::RowMajor>(numNodes, Dim);
+                        out[i] = u;
+                    }
+
+                    return out;
+                }
+            };
+
+        } // namespace mechanical
+
+    } // namespace solver
+
+} // namespace monad
diff --git a/include/monad/solver/mechanical/linear_elastic_solver.hpp b/include/monad/solver/mechanical/linear_elastic_solver.hpp
deleted file mode 100644
index 4a0fef2..0000000
--- a/include/monad/solver/mechanical/linear_elastic_solver.hpp
+++ /dev/null
@@ -1,27 +0,0 @@
-#pragma once
-
-#include "monad/grid/grid_base.hpp"
-#include "monad/solver/mechanical/linear_elastic_physics.hpp"
-#include "monad/solver/periodic_cell_solver.hpp"
-#include "monad/material/mechanical/linear_elastic_material.hpp"
-
-namespace monad {
-
-    /**
-     * @brief Periodic unit cell solver for linear elastic problems.
-     *
-     * @tparam Grid Grid class (e.g. Quad4Grid).
-     * @tparam Element Element class (e.g. Quad4).
-     */
-    template <class Grid, class Element>
-    class LinearElasticSolver : public solver::PeriodicCellSolver<Grid, Element, solver::mechanical::LinearElasticPhysics<Element>> {
-    public:
-        using Base = solver::PeriodicCellSolver<Grid, Element, solver::mechanical::LinearElasticPhysics<Element>>;
-        using Base::Base;
-    };
-
-    // Template arugment deduction
-    template <class Grid, class Element>
-    LinearElasticSolver(const GridBase<Grid, Element> &, const LinearElasticMaterial<Element::Dim> &) -> LinearElasticSolver<Grid, Element>;
-
-} // namespace monad
diff --git a/include/monad/solver/multiphysics/linear_piezoelectric_physics.hpp b/include/monad/solver/multiphysics/linear_piezoelectric_physics.hpp
deleted file mode 100644
index dc2466d..0000000
--- a/include/monad/solver/multiphysics/linear_piezoelectric_physics.hpp
+++ /dev/null
@@ -1,217 +0,0 @@
-#pragma once
-
-#include <array>
-#include <cstddef>
-#include "monad/fem/kernel/multiphysics/linear_piezoelectric_kernel.hpp"
-#include "monad/fem/operator/multiphysics/linear_piezoelectric_matrix_free_operator_traits.hpp"
-#include "monad/solver/mechanical/linear_elastic_physics.hpp"
-#include "monad/solver/scalar/linear_scalar_diffusive_physics.hpp"
-#include "monad/solver/solver_options.hpp"
-
-namespace monad {
-
-    namespace solver {
-
-        namespace multiphysics {
-
-            /**
-             * @brief Physics policy for linear piezoelectric homogenization.
-             *
-             * @tparam Element Element class (e.g. Quad4).
-             */
-            template <class Element>
-            struct LinearPiezoelectricPhysics {
-                using Kernel = fem::multiphysics::LinearPiezoelectricKernel<Element>;
-                using Material = typename Kernel::Material;
-
-                /// @brief Spatial dimension (2 or 3).
-                static constexpr int Dim = Element::Dim;
-
-                /**
-                 * @brief Number of macroscopic fields:
-                 *
-                 * - D=2: ε̄₁₁, ε̄₂₂, ε̄₁₂, Ē₁₁, Ē₂₂.
-                 *
-                 * - D=3: ε̄₁₁, ε̄₂₂, ε̄₃₃, ε̄₁₂, ε̄₁₃, ε̄₂₃, Ē₁₁, Ē₂₂, Ē₃₃.
-                 */
-                static constexpr int NumMacroFields = Material::VoigtSize + Dim;
-
-                using OperatorTraits = fem::multiphysics::LinearPiezoelectricMatrixFreeOperatorTraits<Dim>;
-
-                using MechanicalPhysics = solver::mechanical::LinearElasticPhysics<Element>;
-                using ElectricalPhysics = solver::scalar::LinearScalarDiffusivePhysics<Element, fem::scalar::GradientConvention::Negative>;
-
-                /// @brief Global electromechanical field matrix type.
-                using FieldMatrix = Eigen::Matrix<double, Eigen::Dynamic, NumMacroFields>;
-
-                /// @brief Homogenized coupled constitutive operator type.
-                using MaterialTensor = typename Material::MaterialTensor;
-
-                /**
-                 * @brief Results from linear piezoelectric homogenization.
-                 *
-                 * @note Displacement fields are stored in arrays
-                 * ordered by the prescribed macroscopic strain
-                 * loading directions:
-                 *
-                 * - D=2: ε̄₁₁, ε̄₂₂, ε̄₁₂, Ē₁₁, Ē₂₂.
-                 *
-                 * - D=3: ε̄₁₁, ε̄₂₂, ε̄₃₃, ε̄₁₂, ε̄₁₃, ε̄₂₃, Ē₁₁, Ē₂₂, Ē₃₃.
-                 *
-                 * Each array entry corresponds to one independent macroscopic loading case.
-                 */
-                struct Results {
-                    using MechanicalNodalField = Eigen::Matrix<double, Eigen::Dynamic, Dim>;
-                    using ElectricalNodalField = Eigen::VectorXd;
-
-                    using LinearElasticMaterialTensor = typename MechanicalPhysics::Material::MaterialTensor;
-                    using LinearDielectricMaterialTensor = typename ElectricalPhysics::Material::MaterialTensor;
-                    using CouplingTensor = typename Material::CouplingTensor;
-
-                    /// @brief Homogenized stiffness tensor c̄.
-                    LinearElasticMaterialTensor cBar;
-
-                    /// @brief Homogenized permittivity tensor ε̄.
-                    LinearDielectricMaterialTensor epsilonBar;
-
-                    /// @brief Homogenized piezoelectric coupling tensor d̄.
-                    CouplingTensor dBar;
-
-                    /// @brief Total nodal displacement fields u=ū+ũ.
-                    std::array<MechanicalNodalField, NumMacroFields> u;
-
-                    /// @brief Macroscopic nodal displacement fields ū.
-                    std::array<MechanicalNodalField, NumMacroFields> uMacro;
-
-                    /// @brief Microscopic nodal displacement fields ũ.
-                    std::array<MechanicalNodalField, NumMacroFields> uMicro;
-
-                    /// @brief Total nodal electric potential fields φ=φ̄+φ̃.
-                    std::array<ElectricalNodalField, NumMacroFields> phi;
-
-                    /// @brief Macroscopic nodal electric potential fields φ̄.
-                    std::array<ElectricalNodalField, NumMacroFields> phiMacro;
-
-                    /// @brief Microscopic nodal electric potential fields φ̃.
-                    std::array<ElectricalNodalField, NumMacroFields> phiMicro;
-                };
-
-                /**
-                 * @brief Macroscopic electromechanical fields.
-                 *
-                 * The macroscopic electromechanical fields X̄ are:
-                 *
-                 * ```text
-                 * ⎡Ū 0⎤
-                 * ⎣0 φ̄⎦
-                 * ```
-                 *
-                 * - Ū are the displacements resulting from the macroscopic strains ε̄.
-                 * - φ̄ are the electric potentials resulting from macroscopic electric fields Ē.
-                 *
-                 * @tparam Grid Grid class (e.g. Quad4Grid).
-                 *
-                 * @param[in] grid Periodic unit cell grid.
-                 *
-                 * @returns Macroscopic electromechanical fields.
-                 */
-                template <class Grid>
-                static FieldMatrix macroscopicField(const GridBase<Grid, Element> &grid) {
-                    const std::size_t numNodes = grid.numNodes();
-                    const std::size_t numDofs = OperatorTraits::NumNodeDofs * numNodes;
-
-                    FieldMatrix X = FieldMatrix::Zero(static_cast<int>(numDofs), NumMacroFields);
-
-                    // Mechanical block
-                    const auto U = MechanicalPhysics::macroscopicField(grid);
-                    X.topLeftCorner(U.rows(), MechanicalPhysics::NumMacroFields) = U;
-
-                    // Electrical block
-                    const auto Phi = ElectricalPhysics::macroscopicField(grid);
-                    X.bottomRightCorner(Phi.rows(), ElectricalPhysics::NumMacroFields) = Phi;
-
-                    return X;
-                }
-
-                /**
-                 * @brief Constructs linear piezoelectric homogenization results.
-                 *
-                 * @param[in] opBar Homogenized coupled constitutive operator.
-                 * @param[in] X Total nodal fields.
-                 * @param[in] XMacro Macroscopic nodal fields.
-                 * @param[in] XMicro Microscopic nodal fields.
-                 * @param[in] fields Controls which nodal fields are stored in the solver results.
-                 *
-                 * @returns Linear piezoelectric homogenization results.
-                 */
-                static Results makeResults(const MaterialTensor &opBar, const FieldMatrix &X, const FieldMatrix &XMacro, const FieldMatrix &XMicro, FieldSave fields) noexcept {
-                    using MechanicalNodalField = typename Results::MechanicalNodalField;
-                    using ElectricalNodalField = typename Results::ElectricalNodalField;
-
-                    Results results;
-
-                    results.cBar = opBar.topLeftCorner(Material::VoigtSize, Material::VoigtSize);
-                    results.epsilonBar = -opBar.bottomRightCorner(Dim, Dim);
-                    results.dBar = -opBar.bottomLeftCorner(Dim, Material::VoigtSize);
-
-                    const auto numNodes = X.rows() / OperatorTraits::NumNodeDofs;
-                    const auto numMechanicalDofs = numNodes * OperatorTraits::MechanicalTraits::NumNodeDofs;
-                    const auto numElectricalDofs = numNodes * OperatorTraits::ElectricalTraits::NumNodeDofs;
-
-                    if (wants(fields, FieldSave::Total)) {
-                        const auto U = X.topRows(numMechanicalDofs);
-                        const auto Phi = X.bottomRows(numElectricalDofs);
-
-                        std::array<MechanicalNodalField, NumMacroFields> tempU;
-                        std::array<ElectricalNodalField, NumMacroFields> tempPhi;
-
-                        for (std::size_t i = 0; i < NumMacroFields; ++i) {
-                            tempU[i] = U.col(static_cast<int>(i)).template reshaped<Eigen::RowMajor>(numNodes, Dim);
-                            tempPhi[i] = Phi.col(static_cast<int>(i));
-                        }
-
-                        results.u = tempU;
-                        results.phi = tempPhi;
-                    }
-
-                    if (wants(fields, FieldSave::Macro)) {
-                        const auto UMacro = XMacro.topRows(numMechanicalDofs);
-                        const auto PhiMacro = XMacro.bottomRows(numElectricalDofs);
-
-                        std::array<MechanicalNodalField, NumMacroFields> tempU;
-                        std::array<ElectricalNodalField, NumMacroFields> tempPhi;
-
-                        for (std::size_t i = 0; i < NumMacroFields; ++i) {
-                            tempU[i] = UMacro.col(static_cast<int>(i)).template reshaped<Eigen::RowMajor>(numNodes, Dim);
-                            tempPhi[i] = PhiMacro.col(static_cast<int>(i));
-                        }
-
-                        results.uMacro = tempU;
-                        results.phiMacro = tempPhi;
-                    }
-
-                    if (wants(fields, FieldSave::Micro)) {
-                        const auto UMicro = XMicro.topRows(numMechanicalDofs);
-                        const auto PhiMicro = XMicro.bottomRows(numElectricalDofs);
-
-                        std::array<MechanicalNodalField, NumMacroFields> tempU;
-                        std::array<ElectricalNodalField, NumMacroFields> tempPhi;
-
-                        for (std::size_t i = 0; i < NumMacroFields; ++i) {
-                            tempU[i] = UMicro.col(static_cast<int>(i)).template reshaped<Eigen::RowMajor>(numNodes, Dim);
-                            tempPhi[i] = PhiMicro.col(static_cast<int>(i));
-                        }
-
-                        results.uMicro = tempU;
-                        results.phiMicro = tempPhi;
-                    }
-
-                    return results;
-                }
-            };
-
-        } // namespace multiphysics
-
-    } // namespace solver
-
-} // namespace monad
diff --git a/include/monad/solver/multiphysics/linear_piezoelectric_policy.hpp b/include/monad/solver/multiphysics/linear_piezoelectric_policy.hpp
new file mode 100644
index 0000000..b473256
--- /dev/null
+++ b/include/monad/solver/multiphysics/linear_piezoelectric_policy.hpp
@@ -0,0 +1,223 @@
+#pragma once
+
+#include <array>
+#include <cstddef>
+#include <utility>
+#include <Eigen/Core>
+#include "monad/fem/kernel/multiphysics/linear_piezoelectric_kernel.hpp"
+#include "monad/fem/operator/multiphysics/linear_piezoelectric_dof_traits.hpp"
+#include "monad/material/multiphysics/linear_piezoelectric_material.hpp"
+#include "monad/solver/mechanical/linear_elastic_policy.hpp"
+#include "monad/solver/scalar/linear_scalar_diffusive_policy.hpp"
+#include "monad/solver/solver_options.hpp"
+
+namespace monad {
+
+    namespace solver {
+
+        namespace multiphysics {
+
+            /**
+             * @brief Physics policy for periodic linear piezoelectric homogenization.
+             *
+             * @tparam Element Element type (e.g. Quad4).
+             */
+            template <class Element>
+            struct LinearPiezoelectricPolicy {
+                using Kernel = fem::multiphysics::LinearPiezoelectricKernel<Element>;
+                using DofTraits = fem::multiphysics::LinearPiezoelectricDofTraits<Element::Dim>;
+                using Material = material::LinearPiezoelectricMaterial<Element::Dim>;
+                using MaterialTensor = typename Material::MaterialTensor;
+                using StiffnessTensor = typename Material::StiffnessTensor;
+                using PermittivityTensor = typename Material::PermittivityTensor;
+                using CouplingTensor = typename Material::CouplingTensor;
+                using MechanicalPolicy = solver::mechanical::LinearElasticPolicy<Element>;
+                using ElectricalPolicy = solver::scalar::LinearScalarDiffusivePolicy<Element, fem::scalar::GradientConvention::Negative>;
+
+                /// @brief Spatial dimension (2 or 3).
+                static constexpr int Dim = Element::Dim;
+
+                /**
+                 * @brief Number of macroscopic electromechanical load cases.
+                 *
+                 * The load cases are ordered as mechanical strain cases in
+                 * Voigt order followed by electrical field cases in Cartesian
+                 * component order:
+                 *
+                 * - 2D: S̄₁₁, S̄₂₂, S̄₁₂, Ē₁₁, Ē₂₂.
+                 *
+                 * - 3D: S̄₁₁, S̄₂₂, S̄₃₃, S̄₁₂, S̄₁₃, S̄₂₃, Ē₁₁, Ē₂₂, Ē₃₃.
+                 */
+                static constexpr int NumLoadCases = Material::VoigtSize + Dim;
+
+                /// @brief Dof-major matrix type storing one global field per load case.
+                using DofFieldMatrix = Eigen::Matrix<double, Eigen::Dynamic, NumLoadCases>;
+
+                /// @brief Nodal mechanical field type.
+                using NodalFieldMatrix = Eigen::Matrix<double, Eigen::Dynamic, Dim>;
+
+                /// @brief Nodal electrical field type.
+                using NodalFieldVector = Eigen::Vector<double, Eigen::Dynamic>;
+
+                /// @brief List of nodal mechanical fields, one per load case.
+                using MechanicalNodalFieldList = std::array<NodalFieldMatrix, NumLoadCases>;
+
+                /// @brief List of nodal electrical fields, one per load case.
+                using ElectricalNodalFieldList = std::array<NodalFieldVector, NumLoadCases>;
+
+                /**
+                 * @brief Linear piezoelectric homogenization results.
+                 *
+                 * The homogenized material tensors are always returned.
+                 * Nodal fields are stored only when requested
+                 * through `FieldSave`.
+                 */
+                struct Results {
+                    /// @brief Homogenized coupled constitutive operator.
+                    MaterialTensor opBar;
+
+                    /// @brief Homogenized stiffness tensor.
+                    StiffnessTensor cBar;
+
+                    /// @brief Homogenized permittivity tensor.
+                    PermittivityTensor epsilonBar;
+
+                    /// @brief Homogenized piezoelectric coupling tensor.
+                    CouplingTensor dBar;
+
+                    /// @brief Total nodal displacements.
+                    MechanicalNodalFieldList u;
+
+                    /// @brief Macroscopic nodal displacements.
+                    MechanicalNodalFieldList uMacro;
+
+                    /// @brief Microscopic nodal displacements.
+                    MechanicalNodalFieldList uMicro;
+
+                    /// @brief Total nodal electric potentials.
+                    ElectricalNodalFieldList phi;
+
+                    /// @brief Macroscopic nodal electric potentials.
+                    ElectricalNodalFieldList phiMacro;
+
+                    /// @brief Microscopic nodal electric potentials.
+                    ElectricalNodalFieldList phiMicro;
+                };
+
+                /**
+                 * @brief Macroscopic electromechanical fields.
+                 *
+                 * The macroscopic field is assembled blockwise from:
+                 *
+                 * - Macroscopic displacements for unit macroscopic strains
+                 *
+                 * - Macroscopic electric potentials for unit macroscopic electric fields
+                 *
+                 * ```text
+                 * ⎡Ū 0⎤
+                 * ⎣0 Φ̄⎦
+                 * ```
+                 *
+                 * @tparam Grid Grid type (e.g. Quad4Grid).
+                 *
+                 * @param[in] grid Grid.
+                 *
+                 * @returns Macroscopic electromechanical fields.
+                 */
+                template <class Grid>
+                static DofFieldMatrix makeMacroscopicFields(const Grid &grid) noexcept {
+                    const std::size_t numNodes = grid.numNodes();
+                    const std::size_t numDofs = DofTraits::NumNodeDofs * numNodes;
+
+                    DofFieldMatrix XBar = DofFieldMatrix::Zero(static_cast<int>(numDofs), NumLoadCases);
+
+                    // Mechanical block
+                    const auto UBar = MechanicalPolicy::makeMacroscopicFields(grid);
+                    XBar.topLeftCorner(UBar.rows(), MechanicalPolicy::NumLoadCases) = UBar;
+
+                    // Electrical block
+                    const auto PhiBar = ElectricalPolicy::makeMacroscopicFields(grid);
+                    XBar.bottomRightCorner(PhiBar.rows(), ElectricalPolicy::NumLoadCases) = PhiBar;
+
+                    return XBar;
+                }
+
+                /**
+                 * @brief Packages solver outputs into linear elastic results.
+                 *
+                 * @param[in] opBar Homogenized coupled constitutive operator.
+                 * @param[in] X Total nodal fields.
+                 * @param[in] XMacro Macroscopic nodal fields.
+                 * @param[in] XMicro Microscopic nodal fields.
+                 * @param[in] fields Controls which nodal fields are stored.
+                 *
+                 * @returns Homogenization results.
+                 */
+                static Results makeResults(const MaterialTensor &opBar, const DofFieldMatrix &X, const DofFieldMatrix &XMacro, const DofFieldMatrix &XMicro, FieldSave fields) noexcept {
+                    Results results;
+                    results.opBar = opBar;
+                    results.cBar = opBar.topLeftCorner(Material::VoigtSize, Material::VoigtSize);
+                    results.epsilonBar = -opBar.bottomRightCorner(Dim, Dim);
+                    results.dBar = -opBar.bottomLeftCorner(Dim, Material::VoigtSize);
+
+                    if (wants(fields, FieldSave::Total)) {
+                        auto [u, phi] = makeStoredFields_(X);
+
+                        results.u = u;
+                        results.phi = phi;
+                    }
+
+                    if (wants(fields, FieldSave::Macro)) {
+                        auto [uMacro, phiMacro] = makeStoredFields_(XMacro);
+
+                        results.uMacro = uMacro;
+                        results.phiMacro = phiMacro;
+                    }
+
+                    if (wants(fields, FieldSave::Micro)) {
+                        auto [uMicro, phiMicro] = makeStoredFields_(XMicro);
+
+                        results.uMicro = uMicro;
+                        results.phiMicro = phiMicro;
+                    }
+
+                    return results;
+                }
+
+            private:
+                /**
+                 * @brief Splits a nodal coupled field block matrix into lists of nodal displacements and 
+                 * and electric potentials.
+                 *
+                 * @param[in] X Nodal coupled field block matrix.
+                 *
+                 * @return Pair of lists for nodal displacements and nodal electric potentials.
+                 */
+                static std::pair<MechanicalNodalFieldList, ElectricalNodalFieldList> makeStoredFields_(const DofFieldMatrix &X) {
+                    const int numNodes = static_cast<int>(X.rows()) / (Dim + 1);
+                    const int numMechanicalDofs = numNodes * Dim;
+                    const int numElectricalDofs = numNodes;
+
+                    const auto U = X.topRows(numMechanicalDofs);
+                    const auto Phi = X.bottomRows(numElectricalDofs);
+
+                    MechanicalNodalFieldList outMechanical;
+                    ElectricalNodalFieldList outElectrical;
+
+                    for (std::size_t i = 0; i < NumLoadCases; ++i) {
+                        const NodalFieldMatrix u = U.col(static_cast<int>(i)).template reshaped<Eigen::RowMajor>(numNodes, Dim);
+                        const NodalFieldVector phi = Phi.col(static_cast<int>(i));
+
+                        outMechanical[i] = u;
+                        outElectrical[i] = phi;
+                    }
+
+                    return {outMechanical, outElectrical};
+                }
+            };
+
+        } // namespace multiphysics
+
+    } // namespace solver
+
+} // namespace monad
diff --git a/include/monad/solver/multiphysics/linear_piezoelectric_solver.hpp b/include/monad/solver/multiphysics/linear_piezoelectric_solver.hpp
deleted file mode 100644
index 9127d87..0000000
--- a/include/monad/solver/multiphysics/linear_piezoelectric_solver.hpp
+++ /dev/null
@@ -1,23 +0,0 @@
-#pragma once
-
-#include "monad/grid/grid_base.hpp"
-#include "monad/solver/multiphysics/linear_piezoelectric_physics.hpp"
-#include "monad/solver/periodic_cell_solver.hpp"
-#include "monad/material/mechanical/linear_elastic_material.hpp"
-#include "monad/material/multiphysics/linear_piezoelectric_material.hpp"
-#include "monad/material/transport/linear_transport_material.hpp"
-
-namespace monad {
-
-    template <class Grid, class Element>
-    class LinearPiezoelectricSolver : public solver::PeriodicCellSolver<Grid, Element, solver::multiphysics::LinearPiezoelectricPhysics<Element>> {
-    public:
-        using Base = solver::PeriodicCellSolver<Grid, Element, solver::multiphysics::LinearPiezoelectricPhysics<Element>>;
-        using Base::Base;
-    };
-
-    // Template arugment deduction
-    template <class Grid, class Element, typename MechanicalMaterial, typename ElectricalMaterial>
-    LinearPiezoelectricSolver(const GridBase<Grid, Element> &, const LinearPiezoelectricMaterial<MechanicalMaterial, ElectricalMaterial> &) -> LinearPiezoelectricSolver<Grid, Element>;
-
-} // namespace monad
diff --git a/include/monad/solver/periodic_cell_solver.hpp b/include/monad/solver/periodic_cell_solver.hpp
index abb732d..1057c81 100644
--- a/include/monad/solver/periodic_cell_solver.hpp
+++ b/include/monad/solver/periodic_cell_solver.hpp
@@ -3,36 +3,38 @@
 #include <cstddef>
 #include <stdexcept>
 #include <string>
+#include <iostream>
+#include <chrono>
 #include <Eigen/Core>
 #include <Eigen/IterativeLinearSolvers>
-#include "monad/detail/constants.hpp"
+#include "monad/field/density_field.hpp"
+#include "monad/fem/operator/matrix_free_operator.hpp"
 #include "monad/detail/eigen_utils.hpp"
-#include "monad/grid/grid_base.hpp"
 #include "monad/solver/solver_options.hpp"
-#include "monad/fem/operator/matrix_free_operator.hpp"
-#include "monad/fem/operator/jacobi_preconditioner.hpp"
+#include "monad/solver/jacobi_preconditioner.hpp"
 
 namespace monad {
 
     namespace solver {
 
         /**
-         * @brief Periodic unit cell solver for linear physics problems.
-         *
-         * This class provides a dimension- and physics-agnostic implementation of a
-         * periodic unit cell solver using a matrix-free finite element operator.
+         * @brief Stateless periodic-cell solver for structured linear homogenization problems.
          *
-         * @tparam Grid Grid class (e.g. Quad4Grid).
-         * @tparam Element Element class (e.g. Quad4).
-         * @tparam Physics Physics class (e.g. LinearElasticPhysics2d).
+         * @tparam Grid Grid type (e.g. Quad4Grid).
+         * @tparam Policy Physics policy type (e.g. LinearElasticPolicy<Quad4>).
          */
-        template <class Grid, class Element, class Physics>
+        template <class Grid, class Policy>
         class PeriodicCellSolver {
         public:
-            using Material = typename Physics::Material;
-            using Kernel = typename Physics::Kernel;
-            using OperatorTraits = typename Physics::OperatorTraits;
-            using Operator = fem::MatrixFreeOperator<Grid, Element, OperatorTraits>;
+            using Material = typename Policy::Material;
+            using Kernel = typename Policy::Kernel;
+            using DofTraits = typename Policy::DofTraits;
+            using DensityField = field::DensityField<Grid::Dim>;
+            using Operator = fem::MatrixFreeOperator<Grid, DofTraits>;
+            using DofMap = fem::DofMap<Grid, DofTraits>;
+            using Preconditioner = JacobiPreconditioner<Operator>;
+
+            using MaterialTensor = typename Material::MaterialTensor;
 
             /// @brief Element stiffness matrix type.
             using ElementStiffnessMatrix = typename Kernel::StiffnessMatrix;
@@ -41,126 +43,113 @@ namespace monad {
             using ElementFieldMatrix = typename Kernel::FieldMatrix;
 
             /// @brief Global field matrix type.
-            using FieldMatrix = typename Physics::FieldMatrix;
-
-            /// @brief Homogenized material tensor type.
-            using MaterialTensor = typename Material::MaterialTensor;
+            using FieldMatrix = typename Policy::DofFieldMatrix;
 
             /// @brief Homogenized results type.
-            using Results = typename Physics::Results;
-
-            /**
-             * @brief Constructs a periodic unit cell solver.
-             *
-             * @param[in] grid Periodic unit cell grid.
-             * @param[in] material Base material model.
-             */
-            PeriodicCellSolver(const GridBase<Grid, Element> &grid, const Material &material)
-                : grid_(grid), material_(material) {
-                const auto nodes = grid_.elementNodes(0);
-
-                elementKReference_ = Kernel::lhs(material_, nodes);
-                elementFReference_ = Kernel::rhs(material_, nodes);
-            }
-
-            /// @brief Periodic unit cell grid_.
-            const GridBase<Grid, Element> &grid() const noexcept {
-                return grid_;
-            }
-
-            /// @brief Base material model.
-            const Material &material() const noexcept {
-                return material_;
-            }
+            using Results = typename Policy::Results;
 
             /**
-             * @brief Solves the periodic unit cell problem and computes the homogenized material tensor.
-             *
-             * The solver computes the total field X decomposed as:
-             *
-             * X=X̄+X̃
-             *
-             * - X̄ is the macroscopic field prescribed via uniform macroscopic loading.
-             *
-             * - X̃ is the microscopic field enforcing equilibrium under periodic boundary conditions.
-             *
-             * The homogenized material tensor M̄ is then computed via the physics-specific
-             * homogenization functional:
-             *
-             * M̄=1/V∑ₑXₑᵀKₑXₑ
+             * @brief Solves the periodic cell problem.
              *
+             * @param[in] grid Grid.
+             * @param[in] densityField Per-element density field defined on `grid`.
+             * @param[in] material Base material.
              * @param[in] options Solver options.
              *
              * @returns Homogenized material tensor.
              *
-             * @throws std::runtime_error if the linear solver fails to converge.
+             * @throws std::invalid_argument if `grid` and `densityField` do not
+             * have the same resolution.
+             * @throws std::runtime_error if the iterative linear solver fails.
              */
-            Results solve(const SolverOptions &options = SolverOptions::defaults()) const {
-                const FieldMatrix XMacro = Physics::macroscopicField(grid_);
-                const FieldMatrix XMicro = microscopicField_(options);
-                const FieldMatrix X = XMacro + XMicro;
+            Results solve(const Grid &grid, const DensityField &densityField, const Material &material, const SolverOptions &options = {}) const {
+                if (grid.resolution() != densityField.resolution()) {
+                    throw std::invalid_argument("Grid resolution must match density field resolution.");
+                }
 
-                const MaterialTensor MBar = homogenize_(X);
+                const auto referenceNodes = grid.elementNodes(0);
+                const ElementStiffnessMatrix elementKReference = Kernel::lhs(material, referenceNodes);
+                const ElementFieldMatrix elementFReference = Kernel::rhs(material, referenceNodes);
+
+                const Operator K(grid, densityField, elementKReference);
+                const DofMap dofMap = K.dofMap();
+
+                if (options.verbose) {
+                    std::cout << "[PeriodicCellSolver]\n"
+                              << "\t[Problem]\n"
+                              << "\t\tElements: " << grid.numElements() << std::endl
+                              << "\t\tNodes: " << grid.numNodes() << std::endl
+                              << "\t\tPeriodic nodes: " << grid.numPeriodicNodes() << std::endl
+                              << "\t\tGlobal dofs: " << grid.numNodes() * DofTraits::NumNodeDofs << std::endl
+                              << "\t\tReduced periodic dofs: " << dofMap.numReducedDofs() << std::endl
+                              << "\t\tLoad cases: " << Policy::NumLoadCases << std::endl
+                              << "\t[Solver]\n"
+                              << "\t\tIterations: " << options.maxIterations << std::endl
+                              << "\t\tTolerance: " << options.tolerance << std::endl;
+                }
 
-                return Physics::makeResults(MBar, X, XMacro, XMicro, options.fields);
-            }
+                int iterations = 0;
+                double error = 0.0;
 
-            /// @brief Equality comparison.
-            bool operator==(const PeriodicCellSolver &other) const noexcept {
-                return material_ == other.material_ && grid_ == other.grid_;
-            }
+                const FieldMatrix reducedF = buildReducedRhs_(grid, densityField, dofMap, elementFReference);
 
-            /// @brief Inequality comparison.
-            bool operator!=(const PeriodicCellSolver &other) const noexcept {
-                return !(*this == other);
-            }
+                const auto start = std::chrono::steady_clock::now();
+                const FieldMatrix reducedWMicro = solveReducedSystem_(K, reducedF, options, iterations, error);
+                const auto stop = std::chrono::steady_clock::now();
+                auto duration = std::chrono::duration_cast<std::chrono::milliseconds>(stop - start);
 
-        private:
-            /// @brief Periodic unit cell grid_.
-            GridBase<Grid, Element> grid_;
+                if (options.verbose) {
+                    std::cout << "\t[Results]\n"
+                              << "\t\tIterations: " << iterations << std::endl
+                              << "\t\tError: " << error << std::endl
+                              << "\t\tSolve time (ms): " << duration.count() << std::endl;
+                }
 
-            /// @brief Base material model.
-            Material material_;
+                const FieldMatrix WMicro = expandReducedLhs_(grid, dofMap, reducedWMicro);
+                const FieldMatrix WMacro = Policy::makeMacroscopicFields(grid);
+                const FieldMatrix W = WMacro + WMicro;
 
-            /// @brief Reference element stiffness matrix for unit density.
-            ElementStiffnessMatrix elementKReference_;
+                const MaterialTensor ABar = homogenize_(grid, densityField, elementKReference, W);
 
-            /// @brief Reference element source matrix for unit density.
-            ElementFieldMatrix elementFReference_;
+                return Policy::makeResults(ABar, W, WMacro, WMicro, options.fields);
+            }
 
+        private:
             /**
-             * @brief Builds the reduced right-hand side source matrix.
+             * @brief Builds the reduced source matrix for the microscopic solve.
              *
-             * Constructs the source matrix F in the reduced linear system
+             * Constructs the right-hand side matrix F in
              *
-             * KX̃=F
+             * ```text
+             * KW̃=F
+             * ```
              *
-             * where K acts only on a reduced set of unconstrained periodic dofs.
+             * on the reduced periodic dof space.
              *
-             * @param[in] K Matrix-free stiffness operator.
+             * @param[in] grid Grid.
+             * @param[in] densityField Per-element density field defined on `grid`.
+             * @param[in] dofMap Reduced periodic dof map.
+             * @param[in] elementFReference Reference element source matrix for unit density.
              *
-             * @returns Reduced right-hand side source matrix.
+             * @returns Reduced periodic source matrix.
              */
-            FieldMatrix buildReducedRhs_(const Operator &K) const noexcept {
-                const std::size_t numPeriodicNodes = grid_.numPeriodicNodes();
-                const std::size_t numPeriodicDofs = OperatorTraits::NumNodeDofs * numPeriodicNodes;
-                const std::size_t numReducedDofs = numPeriodicDofs - OperatorTraits::NumFixedDofs;
-
-                FieldMatrix reducedF = FieldMatrix::Zero(static_cast<int>(numReducedDofs), static_cast<int>(Physics::NumMacroFields));
+            FieldMatrix buildReducedRhs_(const Grid &grid, const DensityField &densityField, const DofMap &dofMap, const ElementFieldMatrix &elementFReference) const noexcept {
+                const int numReducedDofs = static_cast<int>(dofMap.numReducedDofs());
+                const int numLoadCases = static_cast<int>(Policy::NumLoadCases);
 
-                const auto &elementDofs = K.elementDofs();
+                FieldMatrix reducedF = FieldMatrix::Zero(numReducedDofs, numLoadCases);
 
-                for (std::size_t i = 0; i < grid_.numElements(); ++i) {
-                    const double density = grid_.getDensity(i);
-                    const ElementFieldMatrix elementF = density * elementFReference_;
+                for (std::size_t i = 0; i < grid.numElements(); ++i) {
+                    const auto &reducedDofs = dofMap.reducedDofs(i);
+                    const double density = densityField.getDensity(i);
+                    const ElementFieldMatrix elementF = density * elementFReference;
 
-                    const auto &dofs = elementDofs[i];
-
-                    for (std::size_t j = 0; j < dofs.size(); ++j) {
-                        const int dof = dofs[j];
+                    for (std::size_t j = 0; j < reducedDofs.size(); ++j) {
+                        const int reducedDof = reducedDofs[j];
+                        const int localReducedDof = static_cast<int>(j);
 
-                        if (dof >= 0) {
-                            reducedF(dof, Eigen::indexing::all) += elementF.row(static_cast<int>(j));
+                        if (reducedDof >= 0) {
+                            reducedF.row(reducedDof) += elementF.row(localReducedDof);
                         }
                     }
                 }
@@ -169,136 +158,154 @@ namespace monad {
             }
 
             /**
-             * @brief Builds the expanded left-hand side microscopic field.
+             * @brief Solves the reduced microscopic system.
              *
-             * Expands the microscopic field matrix X̃ in the reduced linear system
+             * Solves
              *
-             * KX̃=F
+             * ```text
+             * KW̃=F
+             * ```
              *
-             * from a reduced set of unconstrained periodic dofs to global dofs.
+             * on the reduced periodic dof space.
              *
-             * @param[in] reducedX Reduced microscopic field.
+             * @param[in] K Matrix-free operator for the reduced periodic stiffness matrix.
+             * @param[in] reducedF Reduced periodic source matrix.
+             * @param[in] options Solver options.
+             * @param[out] iterations Number of iterations used by the solver.
+             * @param[out] error Final relative residual reported by the solver.
+             *
+             * @returns Reduced microscopic field matrix.
              *
-             * @returns Expanded left-hand side microscopic field.
+             * @throws std::runtime_error if the iterative solver fails.
              */
-            FieldMatrix buildExpandedLhs_(const FieldMatrix &reducedX) const noexcept {
-                // Expand reduced to periodic
-                const std::size_t numPeriodicNodes = grid_.numPeriodicNodes();
-                const std::size_t numPeriodicDofs = OperatorTraits::NumNodeDofs * numPeriodicNodes;
-                const std::size_t numReducedDofs = numPeriodicDofs - OperatorTraits::NumFixedDofs;
-
-                FieldMatrix periodicX = FieldMatrix::Zero(static_cast<int>(numPeriodicDofs), static_cast<int>(Physics::NumMacroFields));
-
-                for (std::size_t i = 0; i < numReducedDofs; ++i) {
-                    const std::size_t iExpanded = OperatorTraits::expandedDof(i, numPeriodicNodes);
-                    periodicX.row(static_cast<int>(iExpanded)) = reducedX.row(static_cast<int>(i));
-                }
-
-                // Expand periodic to global
-                const std::size_t numNodes = grid_.numNodes();
-                const std::size_t numDofs = OperatorTraits::NumNodeDofs * numNodes;
+            FieldMatrix solveReducedSystem_(const Operator &K, const FieldMatrix &reducedF, const SolverOptions &options, int &iterations, double &error) const {
+                Eigen::ConjugateGradient<Operator, Eigen::Lower | Eigen::Upper, Preconditioner> cg;
 
-                FieldMatrix X = FieldMatrix::Zero(static_cast<int>(numDofs), static_cast<int>(Physics::NumMacroFields));
+                cg.setMaxIterations(options.maxIterations);
+                cg.setTolerance(options.tolerance);
+                cg.compute(K);
 
-                for (std::size_t i = 0; i < grid_.numElements(); ++i) {
-                    const auto element = grid_.element(i);
-                    const auto periodicElement = grid_.periodicElement(i);
+                FieldMatrix reducedXMicro = cg.solve(reducedF);
 
-                    const auto dofs = OperatorTraits::dofs(element, numNodes);
-                    const auto periodicDofs = OperatorTraits::dofs(periodicElement, numPeriodicNodes);
+                iterations = static_cast<int>(cg.iterations());
+                error = cg.error();
 
-                    for (std::size_t j = 0; j < dofs.size(); ++j) {
-                        const std::size_t dof = dofs[j];
-                        const std::size_t periodicDof = periodicDofs[j];
+                if (cg.info() != Eigen::Success) {
+                    std::string message = "Iterative solver failed";
 
-                        X.row(static_cast<int>(dof)) = periodicX.row(static_cast<int>(periodicDof));
+                    if (cg.info() == Eigen::NoConvergence) {
+                        message += ": No convergence before the iteration limit.";
                     }
+                    else if (cg.info() == Eigen::NumericalIssue) {
+                        message += ": Numerical issue encountered during the solve.";
+                    }
+
+                    throw std::runtime_error(message);
                 }
 
-                return X;
+                return reducedXMicro;
             }
 
             /**
-             * @brief Microscopic correction field.
-             *
-             * Solves the reduced linear system:
+             * @brief Expands the reduced periodic microscopic field matrix to the global dof space.
              *
-             * KX̃=F
+             * Expands the left-hand side matrix W̃ in
              *
-             * using a matrix-free solver then expands
-             * the solution to global dofs.
+             * ```text
+             * KW̃=F
+             * ```
              *
-             * @param[in] options Solver options.
-             *
-             * @returns Microscopic correction field.
+             * to the global dof space.
              *
-             * @throws std::runtime_error if the solver fails to converge.
+             * @param[in] grid Grid.
+             * @param[in] dofMap Reduced periodic dof map.
+             * @param[in] reducedX Reduced microscopic field matrix.
              *
-             * @note If the global stiffness matric is symmetric, conjugate gradient
-             * is used, otherwise biconjugate gradient stabilized (BiCSSTAB) is used.
+             * @returns Global microscopic field matrix.
              */
-            FieldMatrix microscopicField_(const SolverOptions &options) const {
-                const Operator K(grid_, elementKReference_);
-                const FieldMatrix reducedF = buildReducedRhs_(K);
+            FieldMatrix expandReducedLhs_(const Grid &grid, const DofMap &dofMap, const FieldMatrix &reducedX) const noexcept {
+                // Expand reduced periodic dofs to periodic dofs
+                const std::size_t numPeriodicNodes = grid.numPeriodicNodes();
+                const std::size_t numPeriodicDofs = DofTraits::NumNodeDofs * numPeriodicNodes;
 
-                using preconditioner = fem::JacobiPreconditioner<Operator>;
+                FieldMatrix periodicX = FieldMatrix::Zero(static_cast<int>(numPeriodicDofs), static_cast<int>(Policy::NumLoadCases));
 
-                Eigen::ConjugateGradient<Operator, Eigen::Lower | Eigen::Upper, preconditioner> cg;
-                cg.setMaxIterations(options.maxIterations);
-                cg.setTolerance(options.tolerance);
-                cg.compute(K);
+                for (std::size_t i = 0; i < dofMap.numReducedDofs(); ++i) {
+                    const std::size_t reducedDof = i;
+                    const std::size_t periodicDof = dofMap.reducedToPeriodicDof(i);
 
-                FieldMatrix reducedX = cg.solve(reducedF);
+                    periodicX.row(static_cast<int>(periodicDof)) = reducedX.row(static_cast<int>(reducedDof));
+                }
 
-                if (cg.info() != Eigen::Success) {
-                    std::string error_message = "Solver failed to converge.";
-                    if (cg.info() == Eigen::NoConvergence) {
-                        error_message += " No Convergence - Max iterations reached or tolerance not met.";
-                    }
-                    else if (cg.info() == Eigen::NumericalIssue) {
-                        error_message += " Numerical Issue - Solver encountered stability problems.";
-                    }
+                // Expand periodic dofs to global dofs
+                const std::size_t numNodes = grid.numNodes();
+                const std::size_t numDofs = DofTraits::NumNodeDofs * numNodes;
+
+                FieldMatrix X = FieldMatrix::Zero(static_cast<int>(numDofs), static_cast<int>(Policy::NumLoadCases));
+
+                for (std::size_t i = 0; i < grid.numElements(); ++i) {
+                    const auto element = grid.element(i);
+                    const auto periodicElement = grid.periodicElement(i);
 
-                    throw std::runtime_error(error_message);
+                    const auto globalDofs = DofTraits::elementDofs(element, numNodes);
+                    const auto periodicDofs = DofTraits::elementDofs(periodicElement, numPeriodicNodes);
+
+                    for (std::size_t j = 0; j < globalDofs.size(); ++j) {
+                        const std::size_t globalDof = globalDofs[j];
+                        const std::size_t periodicDof = periodicDofs[j];
+
+                        X.row(static_cast<int>(globalDof)) = periodicX.row(static_cast<int>(periodicDof));
+                    }
                 }
 
-                return buildExpandedLhs_(reducedX);
+                return X;
             }
 
             /**
              * @brief Homogenized material tensor.
              *
-             * The homogenized material tensor M̄ is obtained via the Hill-Mandel lemma,
+             * Using the total field matrix W, the homogenized material tensor Ā
+             * is computed  via the Hill-Mandel lemma:
              *
-             * M̄=1/V∑ₑXₑᵀKₑXₑ
+             * ```text
+             * Ā=1/V∑ₑWₑᵀKₑWₑ
+             * ```
              *
-             * where the total field X is the sum of the macroscopic and microscopic components, X=X̄+X̃.
-             *
-             * @param[in] X Nodal total field.
+             * @param[in] grid Grid.
+             * @param[in] densityField Per-element density field defined on `grid`.
+             * @param[in] elementKReference Reference element stiffness matrix for unit density.
+             * @param[in] W Global total field matrix.
              *
              * @returns Homogenized material tensor.
              */
-            MaterialTensor homogenize_(const FieldMatrix &X) const {
-                MaterialTensor MBar = MaterialTensor::Zero();
+            MaterialTensor homogenize_(const Grid &grid, const DensityField &densityField, const ElementStiffnessMatrix &elementKReference, const FieldMatrix &W) const {
+                MaterialTensor ABar = MaterialTensor::Zero();
+
+                const std::size_t numNodes = grid.numNodes();
 
-                const std::size_t numNodes = grid_.numNodes();
+                for (std::size_t i = 0; i < grid.numElements(); ++i) {
+                    const double density = densityField.getDensity(i);
+                    const auto elementK = density * elementKReference;
+                    const auto element = grid.element(i);
+                    const auto dofs = DofTraits::elementDofs(element, numNodes);
 
-                for (std::size_t i = 0; i < grid_.numElements(); ++i) {
-                    const double density = grid_.getDensity(i);
-                    const auto elementK = density * elementKReference_;
-                    const auto element = grid_.element(i);
-                    const auto dofs = OperatorTraits::dofs(element, numNodes);
-                    const auto elementX = X(detail::arrayToEigen(dofs), Eigen::indexing::all);
+                    ElementFieldMatrix elementW;
+
+                    for (std::size_t j = 0; j < dofs.size(); ++j) {
+                        const std::size_t dof = dofs[j];
+                        const std::size_t localDof = j;
+                        elementW.row(static_cast<int>(localDof)) = W.row(static_cast<int>(dof));
+                    }
 
-                    MBar.noalias() += elementX.transpose() * elementK * elementX;
+                    ABar.noalias() += elementW.transpose() * elementK * elementW;
                 }
 
-                MBar /= grid_.measure();
+                ABar /= grid.measure();
 
-                // Remove numerical artifacts
-                detail::symmetrize(MBar);
+                // Remove numerical asymmetry
+                detail::symmetrize(ABar);
 
-                return MBar;
+                return ABar;
             }
         };
 
diff --git a/include/monad/solver/scalar/linear_scalar_diffusive_physics.hpp b/include/monad/solver/scalar/linear_scalar_diffusive_physics.hpp
deleted file mode 100644
index 46bf66e..0000000
--- a/include/monad/solver/scalar/linear_scalar_diffusive_physics.hpp
+++ /dev/null
@@ -1,156 +0,0 @@
-#pragma once
-
-#include <array>
-#include <cstddef>
-#include "monad/fem/kernel/scalar/linear_scalar_diffusive_kernel.hpp"
-#include "monad/fem/operator/scalar/linear_scalar_diffusive_matrix_free_operator_traits.hpp"
-#include "monad/solver/solver_options.hpp"
-
-namespace monad {
-
-    namespace solver {
-
-        namespace scalar {
-
-            /**
-             * @brief Physics policy for linear scalar diffusive homogenization.
-             *
-             * @tparam Element Element class (e.g. Quad4).
-             * @tparam C Gradient sign convention (GradientConvention::Negative or Positive).
-             */
-            template <class Element, fem::scalar::GradientConvention C>
-            struct LinearScalarDiffusivePhysics {
-                using Kernel = fem::scalar::LinearScalarDiffusiveKernel<Element, C>;
-                using Material = typename Kernel::Material;
-
-                /**
-                 * @brief Number of macroscopic fields:
-                 *
-                 * - D=2: ∇φ̄₁₁, ∇φ̄₂₂.
-                 *
-                 * - D=3: ∇φ̄₁₁, ∇φ̄₂₂, ∇φ̄₃₃.
-                 */
-                static constexpr int NumMacroFields = Element::Dim;
-
-                using OperatorTraits = fem::scalar::LinearScalarDiffusiveMatrixFreeOperatorTraits;
-
-                /// @brief Global field matrix type.
-                using FieldMatrix = Eigen::Matrix<double, Eigen::Dynamic, NumMacroFields>;
-
-                /// @brief Homogenized material tensor type.
-                using MaterialTensor = typename Material::MaterialTensor;
-
-                /**
-                 * @brief Linear scalar diffusive homogenization results.
-                 *
-                 * @note Scalar potential fields are stored in arrays
-                 * ordered by the prescribed macroscopic scalar field
-                 * gradient loading directions:
-                 *
-                 * - D=2: ∇φ̄₁₁, ∇φ̄₂₂.
-                 *
-                 * - D=3: ∇φ̄₁₁, ∇φ̄₂₂, ∇φ̄₃₃.
-                 *
-                 * Each array entry corresponds to one independent macroscopic loading case.
-                 */
-                struct Results {
-                    using NodalField = Eigen::VectorXd;
-
-                    /// @brief Homogenized transport tensor K̄.
-                    MaterialTensor KBar;
-
-                    /// @brief Total nodal scalar potential fields φ=φ̄+φ̃.
-                    std::array<NodalField, NumMacroFields> phi;
-
-                    /// @brief Macroscopic nodal scalar potential fields φ̄.
-                    std::array<NodalField, NumMacroFields> phiMacro;
-
-                    /// @brief Microscopic nodal scalar potential fields φ̃.
-                    std::array<NodalField, NumMacroFields> phiMicro;
-                };
-
-                /**
-                 * @brief Macroscopic nodal scalar potentials.
-                 *
-                 * The macroscopic scalar potentials φ̄ are the scalar potentials
-                 * resulting from macroscopic scalar potential gradients ∇φ̄:
-                 *
-                 * φ̄=(∇φ̄)·x
-                 *
-                 * @tparam Grid Grid class (e.g. Quad4Grid).
-                 *
-                 * @param[in] grid Periodic unit cell grid.
-                 *
-                 * @returns Macroscopic nodal scalar potentials.
-                 */
-                template <class Grid>
-                static FieldMatrix macroscopicField(const GridBase<Grid, Element> &grid) {
-                    const std::size_t numNodes = grid.numNodes();
-                    const std::size_t numDofs = OperatorTraits::NumNodeDofs * numNodes;
-
-                    FieldMatrix Phi = FieldMatrix::Zero(static_cast<int>(numDofs), NumMacroFields);
-
-                    for (std::size_t i = 0; i < grid.numNodes(); ++i) {
-                        Phi.row(static_cast<int>(i)) = Kernel::GradSign * grid.node(i);
-                    }
-
-                    return Phi;
-                }
-
-                /**
-                 * @brief Constructs linear scalar diffusive homogenization results.
-                 *
-                 * @param[in] KBar Homogenized transport tensor.
-                 * @param[in] Phi Total nodal scalar potential fields.
-                 * @param[in] PhiMacro Macroscopic nodal scalar potential fields.
-                 * @param[in] PhiMicro Microscopic nodal scalar potential fields.
-                 * @param[in] fields Controls which nodal fields are stored in the solver results.
-                 *
-                 * @returns Linear scalar diffusive homogenization results.
-                 */
-                static Results makeResults(const MaterialTensor &KBar, const FieldMatrix &Phi, const FieldMatrix &PhiMacro, const FieldMatrix &PhiMicro, FieldSave fields) noexcept {
-                    using NodalField = typename Results::NodalField;
-
-                    Results results;
-
-                    results.KBar = KBar;
-
-                    if (wants(fields, FieldSave::Total)) {
-                        std::array<NodalField, NumMacroFields> temp;
-
-                        for (std::size_t i = 0; i < NumMacroFields; ++i) {
-                            temp[i] = Phi.col(static_cast<int>(i));
-                        }
-
-                        results.phi = temp;
-                    }
-
-                    if (wants(fields, FieldSave::Macro)) {
-                        std::array<NodalField, NumMacroFields> temp;
-
-                        for (std::size_t i = 0; i < NumMacroFields; ++i) {
-                            temp[i] = PhiMacro.col(static_cast<int>(i));
-                        }
-
-                        results.phiMacro = temp;
-                    }
-
-                    if (wants(fields, FieldSave::Micro)) {
-                        std::array<NodalField, NumMacroFields> temp;
-
-                        for (std::size_t i = 0; i < NumMacroFields; ++i) {
-                            temp[i] = PhiMicro.col(static_cast<int>(i));
-                        }
-
-                        results.phiMicro = temp;
-                    }
-
-                    return results;
-                }
-            };
-
-        } // namespace scalar
-
-    } // namespace solver
-
-} // namespace monad
diff --git a/include/monad/solver/scalar/linear_scalar_diffusive_policy.hpp b/include/monad/solver/scalar/linear_scalar_diffusive_policy.hpp
new file mode 100644
index 0000000..5cdfcb6
--- /dev/null
+++ b/include/monad/solver/scalar/linear_scalar_diffusive_policy.hpp
@@ -0,0 +1,159 @@
+#pragma once
+
+#include <array>
+#include <cstddef>
+#include "monad/fem/kernel/scalar/linear_scalar_diffusive_kernel.hpp"
+#include "monad/fem/operator/scalar/linear_scalar_diffusive_dof_traits.hpp"
+#include "monad/material/transport/linear_transport_material.hpp"
+#include "monad/solver/solver_options.hpp"
+
+namespace monad {
+
+    namespace solver {
+
+        namespace scalar {
+
+            /**
+             * @brief Physics policy for periodic linear scalar diffusive homogenization.
+             *
+             * @tparam Element Element type (e.g. Quad4).
+             * @tparam C Gradient sign convention (GradientConvention::Negative or Positive).
+             */
+            template <class Element, fem::scalar::GradientConvention C>
+            struct LinearScalarDiffusivePolicy {
+                using Kernel = fem::scalar::LinearScalarDiffusiveKernel<Element, C>;
+                using DofTraits = fem::scalar::LinearScalarDiffusiveDofTraits;
+                using Material = material::LinearTransportMaterial<Element::Dim>;
+                using MaterialTensor = typename Material::MaterialTensor;
+
+                /// @brief Spatial dimension (2 or 3).
+                static constexpr int Dim = Element::Dim;
+
+                /**
+                 * @brief Number of macroscopic scalar potential gradient load cases.
+                 *
+                 * The load cases are ordered in Cartesian component order:
+                 *
+                 * - 2D: ∇φ̄₁₁, ∇φ̄₂₂.
+                 *
+                 * - 3D: ∇φ̄₁₁, ∇φ̄₂₂, ∇φ̄₃₃.
+                 */
+                static constexpr int NumLoadCases = Dim;
+
+                /// @brief Dof-major matrix type storing one global field per load case.
+                using DofFieldMatrix = Eigen::Matrix<double, Eigen::Dynamic, NumLoadCases>;
+
+                /// @brief Nodal field type.
+                using NodalFieldVector = Eigen::Vector<double, Eigen::Dynamic>;
+
+                /// @brief List of nodal fields, one per load case.
+                using NodalFieldList = std::array<NodalFieldVector, NumLoadCases>;
+
+                /**
+                 * @brief Linear scalar diffusive homogenization results.
+                 *
+                 * The homogenized transport tensor is always returned.
+                 * Nodal scalar potentials are stored only when requested
+                 * through `FieldSave`.
+                 *
+                 * Each array entry corresponds to one independent macroscopic loading case.
+                 */
+                struct Results {
+                    /// @brief Homogenized transport tensor.
+                    MaterialTensor KBar;
+
+                    /// @brief Total nodal scalar potential fields.
+                    NodalFieldList phi;
+
+                    /// @brief Macroscopic nodal scalar potential fields.
+                    NodalFieldList phiMacro;
+
+                    /// @brief Microscopic nodal scalar potential fields.
+                    NodalFieldList phiMicro;
+                };
+
+                /**
+                 * @brief Macroscopic nodal scalar potentials.
+                 *
+                 * For each prescribed macroscopic scalar potential gradient ∇φ̄,
+                 * the macroscopic scalar potentials are:
+                 *
+                 * ```text
+                 * φ̄=(∇φ̄)·x
+                 * ```
+                 *
+                 * @tparam Grid Grid type (e.g. Quad4Grid).
+                 *
+                 * @param[in] grid Grid.
+                 *
+                 * @returns Macroscopic nodal scalar potentials.
+                 */
+                template <class Grid>
+                static DofFieldMatrix makeMacroscopicFields(const Grid &grid) {
+                    const std::size_t numNodes = grid.numNodes();
+                    const std::size_t numDofs = DofTraits::NumNodeDofs * numNodes;
+
+                    DofFieldMatrix PhiBar = DofFieldMatrix::Zero(static_cast<int>(numDofs), NumLoadCases);
+
+                    for (std::size_t i = 0; i < numNodes; ++i) {
+                        PhiBar.row(static_cast<int>(i)) = Kernel::GradSign * grid.node(i);
+                    }
+
+                    return PhiBar;
+                }
+
+                /**
+                 * @brief Packages solver outputs into linear elastic results.
+                 *
+                 * @param[in] KBar Homogenized transport tensor.
+                 * @param[in] Phi Total nodal scalar potentials.
+                 * @param[in] PhiMacro Macroscopic nodal scalar potentials.
+                 * @param[in] PhiMicro Microscopic nodal scalar potentials.
+                 * @param[in] fields Controls which nodal fields are stored.
+                 *
+                 * @returns Homogenization results.
+                 */
+                static Results makeResults(const MaterialTensor &KBar, const DofFieldMatrix &Phi, const DofFieldMatrix &PhiMacro, const DofFieldMatrix &PhiMicro, FieldSave fields) noexcept {
+                    Results results;
+                    results.KBar = KBar;
+
+                    if (wants(fields, FieldSave::Total)) {
+                        results.phi = makeStoredFields_(Phi);
+                    }
+
+                    if (wants(fields, FieldSave::Macro)) {
+                        results.phiMacro = makeStoredFields_(PhiMacro);
+                    }
+
+                    if (wants(fields, FieldSave::Micro)) {
+                        results.phiMicro = makeStoredFields_(PhiMicro);
+                    }
+
+                    return results;
+                }
+            
+            private:
+                /**
+                 * @brief Converts a nodal scalar potential block matrix into a list of nodal scalar potentials.
+                 *
+                 * @param[in] Phi Nodal scalar potential block matrix.
+                 *
+                 * @return List of nodal scalar potentials.
+                 */
+                static NodalFieldList makeStoredFields_(const DofFieldMatrix &Phi) {
+                    NodalFieldList out;
+
+                    for (std::size_t i = 0; i < NumLoadCases; ++i) {
+                        const NodalFieldVector phi = Phi.col(static_cast<int>(i));
+                        out[i] = phi;
+                    }
+
+                    return out;
+                }
+            };
+
+        } // namespace scalar
+
+    } // namespace solver
+
+} // namespace monad
diff --git a/include/monad/solver/scalar/linear_scalar_diffusive_solver.hpp b/include/monad/solver/scalar/linear_scalar_diffusive_solver.hpp
deleted file mode 100644
index bd4fe87..0000000
--- a/include/monad/solver/scalar/linear_scalar_diffusive_solver.hpp
+++ /dev/null
@@ -1,22 +0,0 @@
-#pragma once
-
-#include "monad/solver/scalar/linear_scalar_diffusive_physics.hpp"
-#include "monad/solver/periodic_cell_solver.hpp"
-
-namespace monad {
-
-    /**
-     * @brief Periodic unit cell solver for linear scalar diffusive problems.
-     *
-     * @tparam Grid Grid class (e.g. Quad4Grid).
-     * @tparam Element Element class (e.g. Quad4).
-     * @tparam C Gradient sign convention (GradientConvention::Negative or Positive).
-     */
-    template <class Grid, class Element, fem::scalar::GradientConvention C>
-    class LinearScalarDiffusiveSolver : public solver::PeriodicCellSolver<Grid, Element, solver::scalar::LinearScalarDiffusivePhysics<Element, C>> {
-    public:
-        using Base = solver::PeriodicCellSolver<Grid, Element, solver::scalar::LinearScalarDiffusivePhysics<Element, C>>;
-        using Base::Base;
-    };
-
-} // namespace monad
diff --git a/include/monad/solver/scalar/linear_scalar_diffusive_solver_aliases.hpp b/include/monad/solver/scalar/linear_scalar_diffusive_solver_aliases.hpp
deleted file mode 100644
index ae39379..0000000
--- a/include/monad/solver/scalar/linear_scalar_diffusive_solver_aliases.hpp
+++ /dev/null
@@ -1,111 +0,0 @@
-#pragma once
-
-#include "monad/grid/grid_base.hpp"
-#include "monad/solver/scalar/linear_scalar_diffusive_solver.hpp"
-#include "monad/material/transport/linear_transport_material.hpp"
-
-namespace monad {
-
-    /**
-     * @brief Periodic unit cell solver for linear dielectric problems.
-     *
-     * @tparam Grid Grid class (e.g. Quad4Grid).
-     * @tparam Element Element class (e.g. Quad4).
-     */
-    template <class Grid, class Element>
-    class LinearDielectricSolver : public LinearScalarDiffusiveSolver<Grid, Element, fem::scalar::GradientConvention::Negative> {
-    public:
-        using Base = LinearScalarDiffusiveSolver<Grid, Element, fem::scalar::GradientConvention::Negative>;
-        using Base::Base;
-    };
-
-    // Template arugment deduction
-    template <class Grid, class Element>
-    LinearDielectricSolver(const GridBase<Grid, Element> &, const LinearTransportMaterial<Element::Dim> &) -> LinearDielectricSolver<Grid, Element>;
-
-    /**
-     * @brief Periodic unit cell solver for linear electrical conductive problems.
-     *
-     * @tparam Grid Grid class (e.g. Quad4Grid).
-     * @tparam Element Element class (e.g. Quad4).
-     */
-    template <class Grid, class Element>
-    class LinearElectricalConductiveSolver : public LinearScalarDiffusiveSolver<Grid, Element, fem::scalar::GradientConvention::Negative> {
-    public:
-        using Base = LinearScalarDiffusiveSolver<Grid, Element, fem::scalar::GradientConvention::Negative>;
-        using Base::Base;
-    };
-
-    // Template arugment deduction
-    template <class Grid, class Element>
-    LinearElectricalConductiveSolver(const GridBase<Grid, Element> &, const LinearTransportMaterial<Element::Dim> &) -> LinearElectricalConductiveSolver<Grid, Element>;
-
-    /**
-     * @brief Periodic unit cell solver for linear magnetic problems.
-     *
-     * @tparam Grid Grid class (e.g. Quad4Grid).
-     * @tparam Element Element class (e.g. Quad4).
-     */
-    template <class Grid, class Element>
-    class LinearMagneticSolver : public LinearScalarDiffusiveSolver<Grid, Element, fem::scalar::GradientConvention::Negative> {
-    public:
-        using Base = LinearScalarDiffusiveSolver<Grid, Element, fem::scalar::GradientConvention::Negative>;
-        using Base::Base;
-    };
-
-    // Template arugment deduction
-    template <class Grid, class Element>
-    LinearMagneticSolver(const GridBase<Grid, Element> &, const LinearTransportMaterial<Element::Dim> &) -> LinearMagneticSolver<Grid, Element>;
-
-    /**
-     * @brief Periodic unit cell solver for linear mass diffusive problems.
-     *
-     * @tparam Grid Grid class (e.g. Quad4Grid).
-     * @tparam Element Element class (e.g. Quad4).
-     */
-    template <class Grid, class Element>
-    class LinearMassDiffusiveSolver : public LinearScalarDiffusiveSolver<Grid, Element, fem::scalar::GradientConvention::Positive> {
-    public:
-        using Base = LinearScalarDiffusiveSolver<Grid, Element, fem::scalar::GradientConvention::Positive>;
-        using Base::Base;
-    };
-
-    // Template arugment deduction
-    template <class Grid, class Element>
-    LinearMassDiffusiveSolver(const GridBase<Grid, Element> &, const LinearTransportMaterial<Element::Dim> &) -> LinearMassDiffusiveSolver<Grid, Element>;
-
-    /**
-     * @brief Periodic unit cell solver for linear porous problems.
-     *
-     * @tparam Grid Grid class (e.g. Quad4Grid).
-     * @tparam Element Element class (e.g. Quad4).
-     */
-    template <class Grid, class Element>
-    class LinearPorousSolver : public LinearScalarDiffusiveSolver<Grid, Element, fem::scalar::GradientConvention::Positive> {
-    public:
-        using Base = LinearScalarDiffusiveSolver<Grid, Element, fem::scalar::GradientConvention::Positive>;
-        using Base::Base;
-    };
-
-    // Template arugment deduction
-    template <class Grid, class Element>
-    LinearPorousSolver(const GridBase<Grid, Element> &, const LinearTransportMaterial<Element::Dim> &) -> LinearPorousSolver<Grid, Element>;
-
-    /**
-     * @brief Periodic unit cell solver for linear thermal conductive problems.
-     *
-     * @tparam Grid Grid class (e.g. Quad4Grid).
-     * @tparam Element Element class (e.g. Quad4).
-     */
-    template <class Grid, class Element>
-    class LinearThermalConductiveSolver : public LinearScalarDiffusiveSolver<Grid, Element, fem::scalar::GradientConvention::Positive> {
-    public:
-        using Base = LinearScalarDiffusiveSolver<Grid, Element, fem::scalar::GradientConvention::Positive>;
-        using Base::Base;
-    };
-
-    // Template arugment deduction
-    template <class Grid, class Element>
-    LinearThermalConductiveSolver(const GridBase<Grid, Element> &, const LinearTransportMaterial<Element::Dim> &) -> LinearThermalConductiveSolver<Grid, Element>;
-
-} // namespace monad
diff --git a/include/monad/solver/solver_aliases.hpp b/include/monad/solver/solver_aliases.hpp
new file mode 100644
index 0000000..7ad54be
--- /dev/null
+++ b/include/monad/solver/solver_aliases.hpp
@@ -0,0 +1,74 @@
+#pragma once
+
+#include "monad/solver/mechanical/linear_elastic_policy.hpp"
+#include "monad/solver/scalar/linear_scalar_diffusive_policy.hpp"
+#include "monad/solver/multiphysics/linear_piezoelectric_policy.hpp"
+#include "monad/solver/periodic_cell_solver.hpp"
+
+namespace monad {
+    
+    /**
+     * @brief Stateless periodic-cell solver for structured linear elastic homogenization problems.
+     *
+     * @tparam Grid Grid type (e.g. Quad4Grid).
+     */
+    template <class Grid>
+    using LinearElasticSolver = solver::PeriodicCellSolver<Grid, solver::mechanical::LinearElasticPolicy<typename Grid::Element>>;
+
+    /**
+     * @brief Stateless periodic-cell solver for structured linear dielectric homogenization problems.
+     *
+     * @tparam Grid Grid type (e.g. Quad4Grid).
+     */
+    template <class Grid>
+    using LinearDielectricSolver = solver::PeriodicCellSolver<Grid, solver::scalar::LinearScalarDiffusivePolicy<typename Grid::Element, fem::scalar::GradientConvention::Negative>>;
+
+    /**
+     * @brief Stateless periodic-cell solver for structured linear electrical conductive homogenization problems.
+     *
+     * @tparam Grid Grid type (e.g. Quad4Grid).
+     */
+    template <class Grid>
+    using LinearElectricalConductiveSolver = solver::PeriodicCellSolver<Grid, solver::scalar::LinearScalarDiffusivePolicy<typename Grid::Element, fem::scalar::GradientConvention::Negative>>;
+
+    /**
+     * @brief Stateless periodic-cell solver for structured linear magnetic homogenization problems.
+     *
+     * @tparam Grid Grid type (e.g. Quad4Grid).
+     */
+    template <class Grid>
+    using LinearMagneticSolver = solver::PeriodicCellSolver<Grid, solver::scalar::LinearScalarDiffusivePolicy<typename Grid::Element, fem::scalar::GradientConvention::Negative>>;
+
+    /**
+     * @brief Stateless periodic-cell solver for structured linear mass diffusive homogenization problems.
+     *
+     * @tparam Grid Grid type (e.g. Quad4Grid).
+     */
+    template <class Grid>
+    using LinearMassDiffusiveSolver = solver::PeriodicCellSolver<Grid, solver::scalar::LinearScalarDiffusivePolicy<typename Grid::Element, fem::scalar::GradientConvention::Positive>>;
+
+    /**
+     * @brief Stateless periodic-cell solver for structured linear porous homogenization problems.
+     *
+     * @tparam Grid Grid type (e.g. Quad4Grid).
+     */
+    template <class Grid>
+    using LinearPorousSolver = solver::PeriodicCellSolver<Grid, solver::scalar::LinearScalarDiffusivePolicy<typename Grid::Element, fem::scalar::GradientConvention::Negative>>;
+
+    /**
+     * @brief Stateless periodic-cell solver for structured linear thermal conductive homogenization problems.
+     *
+     * @tparam Grid Grid type (e.g. Quad4Grid).
+     */
+    template <class Grid>
+    using LinearThermalConductiveSolver = solver::PeriodicCellSolver<Grid, solver::scalar::LinearScalarDiffusivePolicy<typename Grid::Element, fem::scalar::GradientConvention::Positive>>;
+
+    /**
+     * @brief Stateless periodic-cell solver for structured linear piezoelectric homogenization problems.
+     *
+     * @tparam Grid Grid type (e.g. Quad4Grid).
+     */
+    template <class Grid>
+    using LinearPiezoelectricSolver = solver::PeriodicCellSolver<Grid, solver::multiphysics::LinearPiezoelectricPolicy<typename Grid::Element>>;
+
+} // namespace monad
diff --git a/include/monad/solver/solver_options.hpp b/include/monad/solver/solver_options.hpp
index a4d4a4b..6755f23 100644
--- a/include/monad/solver/solver_options.hpp
+++ b/include/monad/solver/solver_options.hpp
@@ -4,51 +4,51 @@ namespace monad {
 
     /// @brief Bitmask controlling which nodal fields are stored in solver results.
     enum class FieldSave : unsigned int {
-        /// @brief Do not store any noal fields.
+        /// @brief Do not store any nodal fields.
         None = 0,
 
-        /// @brief Store total nodal fields X=X̄+X̃.
+        /// @brief Store total nodal fields.
         Total = 1 << 0,
 
-        /// @brief Store macroscopic nodal fields X̄.
+        /// @brief Store macroscopic nodal fields.
         Macro = 1 << 1,
 
-        /// @brief Store microscopic nodal fields X̃.
+        /// @brief Store microscopic nodal fields.
         Micro = 1 << 2,
 
         /// @brief Store all nodal fields.
         All = Total | Macro | Micro
     };
 
-    /// @brief Bitwise OR operator.
+    /// @brief Bitwise OR for field-save flags.
     FieldSave operator|(FieldSave a, FieldSave b) noexcept;
 
-    /// @brief Bitwise AND operator.
+    /// @brief Bitwise AND for field-save flags.
     FieldSave operator&(FieldSave a, FieldSave b) noexcept;
 
     /**
-     * @brief Checks whether a specific field-save flag is enabled.
+     * @brief Returns `true` if a field-save flag is enabled, `false` otherwise.
      *
      * @param[in] flags Combined field-save flags.
-     * @param[in] bit Specific field-save option to test.
+     * @param[in] bit Flag to test.
      *
      * @returns `true` if `bit` is enabled in `flags`, `false` otherwise.
      */
     bool wants(FieldSave flags, FieldSave bit) noexcept;
 
-    /// @brief Solver options.
+    /// @brief Options controlling the iterative solves and result storage.
     struct SolverOptions {
         /// @brief Maximum number of solver iterations.
         int maxIterations = 1000;
 
-        /// @brief Convergence tolerance for the solver's residual norm.
+        /// @brief Relative residual tolerance.
         double tolerance = 1e-6;
 
-        /// @brief Controls which nodal fields are stored in the solver results.
+        /// @brief Nodal fields to store in the results.
         FieldSave fields = FieldSave::None;
 
-        /// @brief Default solver options.
-        static SolverOptions defaults();
+        /// @brief Set to `true` to print solver diagnostics.
+        bool verbose = false;
 
         /// @brief Equality comparison.
         bool operator==(const SolverOptions &other) const;
diff --git a/src/CMakeLists.txt b/src/CMakeLists.txt
index 9aecef1..07cf168 100644
--- a/src/CMakeLists.txt
+++ b/src/CMakeLists.txt
@@ -4,10 +4,11 @@ set(MONAD_SOURCES
     fem/element/hex20.cpp
     fem/element/quad4.cpp
     fem/element/quad8.cpp
-    grid/hex8_grid.cpp
-    grid/hex20_grid.cpp
-    grid/quad4_grid.cpp
-    grid/quad8_grid.cpp
+    field/make_density_field_from_csv.cpp
+    grid/hex8_topology.cpp
+    grid/hex20_topology.cpp
+    grid/quad4_topology.cpp
+    grid/quad8_topology.cpp
     io/gmsh/write_gmsh_header.cpp
     material/mechanical/linear_elastic_2d.cpp
     material/mechanical/linear_elastic_3d.cpp
diff --git a/src/field/make_density_field_from_csv.cpp b/src/field/make_density_field_from_csv.cpp
new file mode 100644
index 0000000..61ed676
--- /dev/null
+++ b/src/field/make_density_field_from_csv.cpp
@@ -0,0 +1,82 @@
+#include <cstddef>
+#include <fstream>
+#include <sstream>
+#include <stdexcept>
+#include <vector>
+#include "monad/field/make_density_field_from_csv.hpp"
+
+namespace monad {
+
+    DensityField2d makeDensityFieldFromCsv(const std::string& file) {
+        std::ifstream ifs(file, std::ios::in | std::ios::binary);
+        if (!ifs.is_open()) {
+            throw std::runtime_error("Could not open " + file + " for reading.");
+        }
+
+        std::size_t nx = 0;
+        std::vector<std::vector<double>> rows;
+        std::string line;
+
+        while (std::getline(ifs, line)) {
+            if (line.empty()) {
+                continue;
+            }
+
+            std::istringstream iss(line);
+            std::string cell;
+            std::vector<double> row;
+
+            while (std::getline(iss, cell, ',')) {
+                if (cell.empty()) {
+                    continue;
+                }
+
+                double value = 0.0;
+                try {
+                    value = std::stod(cell);
+                }
+                catch (const std::invalid_argument&) {
+                    throw std::runtime_error("File " + file + " contains non-numeric values.");
+                }
+
+                if (value < 0.0 || value > 1.0) {
+                    throw std::runtime_error("File " + file + " contains values outside the range [0,1].");
+                }
+
+                row.push_back(value);
+            }
+
+            if (row.empty()) {
+                continue;
+            }
+
+            if (rows.empty()) {
+                nx = row.size();
+            }
+
+            if (row.size() != nx) {
+                throw std::runtime_error("File " + file + " number of columns (" + std::to_string(row.size()) + ") does not equal density field x-resolution (" + std::to_string(nx) + ").");
+            }
+
+            rows.push_back(row);
+        }
+
+        const std::size_t ny = rows.size();
+
+        if (ny == 0) {
+            throw std::runtime_error("File " + file + " is empty.");
+        }
+
+        DensityField2d densityField({nx, ny});
+
+        for (std::size_t i = 0; i < ny; ++i) {
+            for (std::size_t j = 0; j < nx; ++j) {
+                const std::size_t rowStart = nx * (ny - 1 - i);
+                densityField.setDensity(rowStart + j, rows[i][j]);
+            }
+        }
+
+        return densityField;
+    }
+
+} // namespace monad
diff --git a/src/grid/hex20_grid.cpp b/src/grid/hex20_grid.cpp
deleted file mode 100644
index 3c20986..0000000
--- a/src/grid/hex20_grid.cpp
+++ /dev/null
@@ -1,262 +0,0 @@
-#include <stdexcept>
-#include <string>
-#include "monad/grid/hex20_grid.hpp"
-
-namespace monad {
-
-    std::size_t Hex20Grid::numNodes() const noexcept {
-        const std::size_t nx = resolution_[0];
-        const std::size_t ny = resolution_[1];
-        const std::size_t nz = resolution_[2];
-
-        const std::size_t numCornerNodes = (nx + 1) * (ny + 1) * (nz + 1);
-        const std::size_t numXMidNodes = nx * (ny + 1) * (nz + 1);
-        const std::size_t numYMidNodes = (nx + 1) * ny * (nz + 1);
-        const std::size_t numZMidNodes = (nx + 1) * (ny + 1) * nz;
-
-        return numCornerNodes + numXMidNodes + numYMidNodes + numZMidNodes;
-    }
-
-    std::size_t Hex20Grid::numPeriodicNodes() const noexcept {
-        return 4 * numElements();
-    }
-
-    Hex20Grid::Point Hex20Grid::node(std::size_t index) const {
-        if (index >= numNodes()) {
-            throw std::out_of_range("Index (" + std::to_string(index) + ") is out of range [0," + std::to_string(numNodes()) + ").");
-        }
-
-        const std::size_t nx = resolution_[0];
-        const std::size_t ny = resolution_[1];
-        const std::size_t nz = resolution_[2];
-
-        const std::size_t numCornerNodes = (nx + 1) * (ny + 1) * (nz + 1);
-        const std::size_t numXMidNodes = nx * (ny + 1) * (nz + 1);
-        const std::size_t numYMidNodes = (nx + 1) * ny * (nz + 1);
-
-        const double lx = size_[0];
-        const double ly = size_[1];
-        const double lz = size_[2];
-
-        const double dx = lx / static_cast<double>(nx);
-        const double dy = ly / static_cast<double>(ny);
-        const double dz = lz / static_cast<double>(nz);
-
-        double x;
-        double y;
-        double z;
-
-        if (index < numCornerNodes) {
-            const std::size_t nodesPerPlane = (nx + 1) * (ny + 1);
-            const std::size_t indexInPlane = index % nodesPerPlane;
-
-            const std::size_t i = indexInPlane % (nx + 1);
-            const std::size_t j = indexInPlane / (nx + 1);
-            const std::size_t k = index / nodesPerPlane;
-
-            x = static_cast<double>(i) * dx;
-            y = static_cast<double>(j) * dy;
-            z = static_cast<double>(k) * dz;
-        }
-        else if (index < numCornerNodes + numXMidNodes) {
-            // x-midpoints: grid size (i: nx, j: ny+1, k: nz+1)
-            index -= numCornerNodes;
-
-            std::size_t nodesPerPlane = nx * (ny + 1);
-            std::size_t indexInPlane = index % nodesPerPlane;
-
-            std::size_t i = indexInPlane % nx;
-            std::size_t j = indexInPlane / nx;
-            std::size_t k = index / nodesPerPlane;
-
-            x = (static_cast<double>(i) + 0.5) * dx;
-            y = static_cast<double>(j) * dy;
-            z = static_cast<double>(k) * dz;
-        }
-        else if (index < numCornerNodes + numXMidNodes + numYMidNodes) {
-            // y-midpoints: grid size (i: nx+1, j: ny, k: nz+1)
-            index -= numCornerNodes + numXMidNodes;
-
-            const std::size_t nodesPerPlane = (nx + 1) * ny;
-            const std::size_t indexInPlane = index % nodesPerPlane;
-
-            const std::size_t i = indexInPlane % (nx + 1);
-            const std::size_t j = indexInPlane / (nx + 1);
-            const std::size_t k = index / nodesPerPlane;
-
-            x = static_cast<double>(i) * dx;
-            y = (static_cast<double>(j) + 0.5) * dy;
-            z = static_cast<double>(k) * dz;
-        }
-        else {
-            // z-midpoints: grid size (i: nx+1, j: ny+1, k: nz)
-            index -= numCornerNodes + numXMidNodes + numYMidNodes;
-
-            const std::size_t nodesPerPlane = (nx + 1) * (ny + 1);
-            const std::size_t indexInPlane = index % nodesPerPlane;
-
-            const std::size_t i = indexInPlane % (nx + 1);
-            const std::size_t j = indexInPlane / (nx + 1);
-            const std::size_t k = index / nodesPerPlane;
-
-            x = static_cast<double>(i) * dx;
-            y = static_cast<double>(j) * dy;
-            z = (static_cast<double>(k) + 0.5) * dz;
-        }
-
-        return Point(x, y, z);
-    }
-
-    Hex20Grid::ElementList Hex20Grid::element(std::size_t index) const {
-        if (index >= numElements()) {
-            throw std::out_of_range("Index (" + std::to_string(index) + ") is out of range [0," + std::to_string(numElements()) + ").");
-        }
-
-        const std::size_t nx = resolution_[0];
-        const std::size_t ny = resolution_[1];
-        const std::size_t nz = resolution_[2];
-
-        const std::size_t elementsPerPlane = nx * ny;
-        const std::size_t numCornerNodes = (nx + 1) * (ny + 1) * (nz + 1);
-        const std::size_t numXMidNodes = nx * (ny + 1) * (nz + 1);
-        const std::size_t numYMidNodes = (nx + 1) * ny * (nz + 1);
-
-        const std::size_t i = index % nx;
-        const std::size_t j = (index / nx) % ny;
-        const std::size_t k = index / elementsPerPlane;
-
-        auto cornerNodeIndex = [&](std::size_t i, std::size_t j, std::size_t k) {
-            return k * ((nx + 1) * (ny + 1)) + j * (nx + 1) + i;
-        };
-
-        auto xMidNodeIndex = [&](std::size_t i, std::size_t j, std::size_t k) {
-            return numCornerNodes + k * nx * (ny + 1) + j * nx + i;
-        };
-
-        auto yMidNodeIndex = [&](std::size_t i, std::size_t j, std::size_t k) {
-            return numCornerNodes + numXMidNodes + k * (nx + 1) * ny + j * (nx + 1) + i;
-        };
-
-        auto zMidNodeIndex = [&](std::size_t i, std::size_t j, std::size_t k) {
-            return numCornerNodes + numXMidNodes + numYMidNodes + k * (nx + 1) * (ny + 1) + j * (nx + 1) + i;
-        };
-
-        // Bottom face corners
-        const std::size_t n1 = cornerNodeIndex(i, j, k);
-        const std::size_t n2 = cornerNodeIndex(i + 1, j, k);
-        const std::size_t n3 = cornerNodeIndex(i + 1, j + 1, k);
-        const std::size_t n4 = cornerNodeIndex(i, j + 1, k);
-
-        // Top face corners
-        const std::size_t n5 = cornerNodeIndex(i, j, k + 1);
-        const std::size_t n6 = cornerNodeIndex(i + 1, j, k + 1);
-        const std::size_t n7 = cornerNodeIndex(i + 1, j + 1, k + 1);
-        const std::size_t n8 = cornerNodeIndex(i, j + 1, k + 1);
-
-        // Bottom face midpoints
-        const std::size_t n9 = xMidNodeIndex(i, j, k); // between n1-n2
-        const std::size_t n10 = yMidNodeIndex(i + 1, j, k); // between n2-n3
-        const std::size_t n11 = xMidNodeIndex(i, j + 1, k); // between n3-n4
-        const std::size_t n12 = yMidNodeIndex(i, j, k); // between n1-n4
-
-        // Top face edge midpoints
-        const std::size_t n13 = xMidNodeIndex(i, j, k + 1); // between n5-n6
-        const std::size_t n14 = yMidNodeIndex(i + 1, j, k + 1); // between n6-n7
-        const std::size_t n15 = xMidNodeIndex(i, j + 1, k + 1); // between n7-n8
-        const std::size_t n16 = yMidNodeIndex(i, j, k + 1); // between n5-n8
-
-        // Vertical midpoints
-        const std::size_t n17 = zMidNodeIndex(i, j, k); // between n1-n5
-        const std::size_t n18 = zMidNodeIndex(i + 1, j, k); // between n2-n6
-        const std::size_t n19 = zMidNodeIndex(i + 1, j + 1, k); // between n3-n7
-        const std::size_t n20 = zMidNodeIndex(i, j + 1, k); // between n4-n8
-
-        return {n1, n2, n3, n4, n5, n6, n7, n8, n9, n10, n11, n12, n13, n14, n15, n16, n17, n18, n19, n20};
-    }
-
-    Hex20Grid::ElementList Hex20Grid::periodicElement(std::size_t index) const {
-        if (index >= numElements()) {
-            throw std::out_of_range("Index (" + std::to_string(index) + ") is out of range [0," + std::to_string(numElements()) + ").");
-        }
-
-        const std::size_t nx = resolution_[0];
-        const std::size_t ny = resolution_[1];
-        const std::size_t nz = resolution_[2];
-
-        const std::size_t elementsPerPlane = nx * ny;
-        const std::size_t numElements = nx * ny * nz;
-
-        const std::size_t i = index % nx;
-        const std::size_t j = (index / nx) % ny;
-        const std::size_t k = index / elementsPerPlane;
-
-        auto cornerNodeIndex = [&](std::size_t i, std::size_t j, std::size_t k) {
-            // Periodic wrapping
-            const std::size_t iP = (i + nx) % nx;
-            const std::size_t jP = (j + ny) % ny;
-            const std::size_t kP = (k + nz) % nz;
-
-            return kP * nx * ny + jP * nx + iP;
-        };
-
-        auto xMidNodeIndex = [&](std::size_t i, std::size_t j, std::size_t k) {
-            // Periodic wrapping
-            const std::size_t iP = (i + nx) % nx;
-            const std::size_t jP = (j + ny) % ny;
-            const std::size_t kP = (k + nz) % nz;
-
-            return numElements + kP * nx * ny + jP * nx + iP;
-        };
-
-        auto yMidNodeIndex = [&](std::size_t i, std::size_t j, std::size_t k) {
-            // Periodic wrapping
-            const std::size_t iP = (i + nx) % nx;
-            const std::size_t jP = (j + ny) % ny;
-            const std::size_t kP = (k + nz) % nz;
-
-            return numElements + numElements + kP * nx * ny + jP * nx + iP;
-        };
-
-        auto zMidNodeIndex = [&](std::size_t i, std::size_t j, std::size_t k) {
-            // Periodic wrapping
-            const std::size_t iP = (i + nx) % nx;
-            const std::size_t jP = (j + ny) % ny;
-            const std::size_t kP = (k + nz) % nz;
-
-            return numElements + numElements + numElements + kP * nx * ny + jP * nx + iP;
-        };
-
-        // Bottom face corners
-        const std::size_t n1 = cornerNodeIndex(i, j, k);
-        const std::size_t n2 = cornerNodeIndex(i + 1, j, k);
-        const std::size_t n3 = cornerNodeIndex(i + 1, j + 1, k);
-        const std::size_t n4 = cornerNodeIndex(i, j + 1, k);
-
-        // Top face corners
-        const std::size_t n5 = cornerNodeIndex(i, j, k + 1);
-        const std::size_t n6 = cornerNodeIndex(i + 1, j, k + 1);
-        const std::size_t n7 = cornerNodeIndex(i + 1, j + 1, k + 1);
-        const std::size_t n8 = cornerNodeIndex(i, j + 1, k + 1);
-
-        // Bottom face midpoints
-        const std::size_t n9 = xMidNodeIndex(i, j, k); // between n1-n2
-        const std::size_t n10 = yMidNodeIndex(i + 1, j, k); // between n2-n3
-        const std::size_t n11 = xMidNodeIndex(i, j + 1, k); // between n3-n4
-        const std::size_t n12 = yMidNodeIndex(i, j, k); // between n1-n4
-
-        // Top face edge midpoints
-        const std::size_t n13 = xMidNodeIndex(i, j, k + 1); // between n5-n6
-        const std::size_t n14 = yMidNodeIndex(i + 1, j, k + 1); // between n6-n7
-        const std::size_t n15 = xMidNodeIndex(i, j + 1, k + 1); // between n7-n8
-        const std::size_t n16 = yMidNodeIndex(i, j, k + 1); // between n5-n8
-
-        // Vertical midpoints
-        const std::size_t n17 = zMidNodeIndex(i, j, k); // between n1-n5
-        const std::size_t n18 = zMidNodeIndex(i + 1, j, k); // between n2-n6
-        const std::size_t n19 = zMidNodeIndex(i + 1, j + 1, k); // between n3-n7
-        const std::size_t n20 = zMidNodeIndex(i, j + 1, k); // between n4-n8
-
-        return {n1, n2, n3, n4, n5, n6, n7, n8, n9, n10, n11, n12, n13, n14, n15, n16, n17, n18, n19, n20};
-    }
-
-} // namespace monad
diff --git a/src/grid/hex20_topology.cpp b/src/grid/hex20_topology.cpp
new file mode 100644
index 0000000..b8c2766
--- /dev/null
+++ b/src/grid/hex20_topology.cpp
@@ -0,0 +1,206 @@
+#include "monad/grid/hex20_topology.hpp"
+
+namespace monad {
+
+    namespace grid {
+
+        std::size_t Hex20Topology::numCornerNodes(const Resolution& resolution) noexcept {
+            return (resolution[0] + 1) * (resolution[1] + 1) * (resolution[2] + 1);
+        }
+
+        std::size_t Hex20Topology::numXMidNodes(const Resolution& resolution) noexcept {
+            return resolution[0] * (resolution[1] + 1) * (resolution[2] + 1);
+        }
+
+        std::size_t Hex20Topology::numYMidNodes(const Resolution& resolution) noexcept {
+            return (resolution[0] + 1) * resolution[1] * (resolution[2] + 1);
+        }
+
+        std::size_t Hex20Topology::numZMidNodes(const Resolution& resolution) noexcept {
+            return (resolution[0] + 1) * (resolution[1] + 1) * resolution[2];
+        }
+
+        std::size_t Hex20Topology::numNodes(const Resolution& resolution) noexcept {
+            return numCornerNodes(resolution) + numXMidNodes(resolution) + numYMidNodes(resolution) + numZMidNodes(resolution);
+        }
+
+        std::size_t Hex20Topology::numPeriodicNodes(const Resolution& resolution) noexcept {
+            return 4 * resolution[0] * resolution[1] * resolution[2];
+        }
+
+        Hex20Topology::Point Hex20Topology::node(std::size_t index, const Resolution& resolution, const Size& size) noexcept {
+            const std::size_t nx = resolution[0];
+            const std::size_t ny = resolution[1];
+            const std::size_t nz = resolution[2];
+            const std::size_t corners = numCornerNodes(resolution);
+            const std::size_t xMids = numXMidNodes(resolution);
+            const std::size_t yMids = numYMidNodes(resolution);
+
+            const double dx = size[0] / static_cast<double>(nx);
+            const double dy = size[1] / static_cast<double>(ny);
+            const double dz = size[2] / static_cast<double>(nz);
+
+            double x;
+            double y;
+            double z;
+
+            if (index < corners) {
+                const std::size_t nodesPerPlane = (nx + 1) * (ny + 1);
+                const std::size_t indexInPlane = index % nodesPerPlane;
+                const std::size_t i = indexInPlane % (nx + 1);
+                const std::size_t j = indexInPlane / (nx + 1);
+                const std::size_t k = index / nodesPerPlane;
+
+                x = static_cast<double>(i) * dx;
+                y = static_cast<double>(j) * dy;
+                z = static_cast<double>(k) * dz;
+            }
+            else if (index < corners + xMids) {
+                index -= corners;
+
+                const std::size_t nodesPerPlane = nx * (ny + 1);
+                const std::size_t indexInPlane = index % nodesPerPlane;
+                const std::size_t i = indexInPlane % nx;
+                const std::size_t j = indexInPlane / nx;
+                const std::size_t k = index / nodesPerPlane;
+
+                x = (static_cast<double>(i) + 0.5) * dx;
+                y = static_cast<double>(j) * dy;
+                z = static_cast<double>(k) * dz;
+            }
+            else if (index < corners + xMids + yMids) {
+                index -= corners + xMids;
+
+                const std::size_t nodesPerPlane = (nx + 1) * ny;
+                const std::size_t indexInPlane = index % nodesPerPlane;
+                const std::size_t i = indexInPlane % (nx + 1);
+                const std::size_t j = indexInPlane / (nx + 1);
+                const std::size_t k = index / nodesPerPlane;
+
+                x = static_cast<double>(i) * dx;
+                y = (static_cast<double>(j) + 0.5) * dy;
+                z = static_cast<double>(k) * dz;
+            }
+            else {
+                index -= corners + xMids + yMids;
+
+                const std::size_t nodesPerPlane = (nx + 1) * (ny + 1);
+                const std::size_t indexInPlane = index % nodesPerPlane;
+                const std::size_t i = indexInPlane % (nx + 1);
+                const std::size_t j = indexInPlane / (nx + 1);
+                const std::size_t k = index / nodesPerPlane;
+
+                x = static_cast<double>(i) * dx;
+                y = static_cast<double>(j) * dy;
+                z = (static_cast<double>(k) + 0.5) * dz;
+            }
+            
+            return Point(x, y, z);
+        }
+
+        Hex20Topology::ElementList Hex20Topology::element(std::size_t index, const Resolution& resolution) noexcept {
+            const std::size_t nx = resolution[0];
+            const std::size_t ny = resolution[1];
+            const std::size_t elementsPerPlane = nx * ny;
+            const std::size_t corners = numCornerNodes(resolution);
+            const std::size_t xMids = numXMidNodes(resolution);
+            const std::size_t yMids = numYMidNodes(resolution);
+
+            const std::size_t i = index % nx;
+            const std::size_t j = (index / nx) % ny;
+            const std::size_t k = index / elementsPerPlane;
+
+            const auto cornerNodeIndex = [nx, ny](std::size_t ii, std::size_t jj, std::size_t kk) {
+                return kk * ((nx + 1) * (ny + 1)) + jj * (nx + 1) + ii;
+            };
+
+            const auto xMidNodeIndex = [nx, ny, corners](std::size_t ii, std::size_t jj, std::size_t kk) {
+                return corners + kk * nx * (ny + 1) + jj * nx + ii;
+            };
+
+            const auto yMidNodeIndex = [nx, ny, corners, xMids](std::size_t ii, std::size_t jj, std::size_t kk) {
+                return corners + xMids + kk * (nx + 1) * ny + jj * (nx + 1) + ii;
+            };
+
+            const auto zMidNodeIndex = [nx, ny, corners, xMids, yMids](std::size_t ii, std::size_t jj, std::size_t kk) {
+                return corners + xMids + yMids + kk * (nx + 1) * (ny + 1) + jj * (nx + 1) + ii;
+            };
+
+            return {
+                cornerNodeIndex(i, j, k),
+                cornerNodeIndex(i + 1, j, k),
+                cornerNodeIndex(i + 1, j + 1, k),
+                cornerNodeIndex(i, j + 1, k),
+                cornerNodeIndex(i, j, k + 1),
+                cornerNodeIndex(i + 1, j, k + 1),
+                cornerNodeIndex(i + 1, j + 1, k + 1),
+                cornerNodeIndex(i, j + 1, k + 1),
+                xMidNodeIndex(i, j, k),
+                yMidNodeIndex(i + 1, j, k),
+                xMidNodeIndex(i, j + 1, k),
+                yMidNodeIndex(i, j, k),
+                xMidNodeIndex(i, j, k + 1),
+                yMidNodeIndex(i + 1, j, k + 1),
+                xMidNodeIndex(i, j + 1, k + 1),
+                yMidNodeIndex(i, j, k + 1),
+                zMidNodeIndex(i, j, k),
+                zMidNodeIndex(i + 1, j, k),
+                zMidNodeIndex(i + 1, j + 1, k),
+                zMidNodeIndex(i, j + 1, k)
+            };
+        }
+
+        Hex20Topology::ElementList Hex20Topology::periodicElement(std::size_t index, const Resolution& resolution) noexcept {
+            const std::size_t nx = resolution[0];
+            const std::size_t ny = resolution[1];
+            const std::size_t nz = resolution[2];
+            const std::size_t numElements = nx * ny * nz;
+            const std::size_t elementsPerPlane = nx * ny;
+
+            const std::size_t i = index % nx;
+            const std::size_t j = (index / nx) % ny;
+            const std::size_t k = index / elementsPerPlane;
+
+            const auto cornerNodeIndex = [nx, ny, nz](std::size_t ii, std::size_t jj, std::size_t kk) {
+                return (kk % nz) * (nx * ny) + (jj % ny) * nx + (ii % nx);
+            };
+
+            const auto xMidNodeIndex = [nx, ny, nz, numElements](std::size_t ii, std::size_t jj, std::size_t kk) {
+                return numElements + (kk % nz) * (nx * ny) + (jj % ny) * nx + (ii % nx);
+            };
+
+            const auto yMidNodeIndex = [nx, ny, nz, numElements](std::size_t ii, std::size_t jj, std::size_t kk) {
+                return 2 * numElements + (kk % nz) * (nx * ny) + (jj % ny) * nx + (ii % nx);
+            };
+
+            const auto zMidNodeIndex = [nx, ny, nz, numElements](std::size_t ii, std::size_t jj, std::size_t kk) {
+                return 3 * numElements + (kk % nz) * (nx * ny) + (jj % ny) * nx + (ii % nx);
+            };
+
+            return {
+                cornerNodeIndex(i, j, k),
+                cornerNodeIndex(i + 1, j, k),
+                cornerNodeIndex(i + 1, j + 1, k),
+                cornerNodeIndex(i, j + 1, k),
+                cornerNodeIndex(i, j, k + 1),
+                cornerNodeIndex(i + 1, j, k + 1),
+                cornerNodeIndex(i + 1, j + 1, k + 1),
+                cornerNodeIndex(i, j + 1, k + 1),
+                xMidNodeIndex(i, j, k),
+                yMidNodeIndex(i + 1, j, k),
+                xMidNodeIndex(i, j + 1, k),
+                yMidNodeIndex(i, j, k),
+                xMidNodeIndex(i, j, k + 1),
+                yMidNodeIndex(i + 1, j, k + 1),
+                xMidNodeIndex(i, j + 1, k + 1),
+                yMidNodeIndex(i, j, k + 1),
+                zMidNodeIndex(i, j, k),
+                zMidNodeIndex(i + 1, j, k),
+                zMidNodeIndex(i + 1, j + 1, k),
+                zMidNodeIndex(i, j + 1, k)
+            };
+        }
+
+    } // namespace grid
+
+} // namespace monad
diff --git a/src/grid/hex8_grid.cpp b/src/grid/hex8_grid.cpp
deleted file mode 100644
index 1f9e467..0000000
--- a/src/grid/hex8_grid.cpp
+++ /dev/null
@@ -1,121 +0,0 @@
-#include <stdexcept>
-#include <string>
-#include "monad/grid/hex8_grid.hpp"
-
-namespace monad {
-
-    std::size_t Hex8Grid::numNodes() const noexcept {
-        const std::size_t nx = resolution_[0];
-        const std::size_t ny = resolution_[1];
-        const std::size_t nz = resolution_[2];
-
-        return (nx + 1) * (ny + 1) * (nz + 1);
-    }
-
-    std::size_t Hex8Grid::numPeriodicNodes() const noexcept {
-        return numElements();
-    }
-
-    Hex8Grid::Point Hex8Grid::node(std::size_t index) const {
-        if (index >= numNodes()) {
-            throw std::out_of_range("Index (" + std::to_string(index) + ") is out of range [0," + std::to_string(numNodes()) + ").");
-        }
-
-        const std::size_t nx = resolution_[0];
-        const std::size_t ny = resolution_[1];
-        const std::size_t nz = resolution_[2];
-
-        const std::size_t nodesPerPlane = (nx + 1) * (ny + 1);
-        const std::size_t indexInPlane = index % nodesPerPlane;
-
-        const double lx = size_[0];
-        const double ly = size_[1];
-        const double lz = size_[2];
-
-        const double dx = lx / static_cast<double>(nx);
-        const double dy = ly / static_cast<double>(ny);
-        const double dz = lz / static_cast<double>(nz);
-
-        const std::size_t i = indexInPlane % (nx + 1);
-        const std::size_t j = indexInPlane / (nx + 1);
-        const std::size_t k = index / nodesPerPlane;
-
-        const double x = static_cast<double>(i) * dx;
-        const double y = static_cast<double>(j) * dy;
-        const double z = static_cast<double>(k) * dz;
-
-        return Point(x, y, z);
-    }
-
-    Hex8Grid::ElementList Hex8Grid::element(std::size_t index) const {
-        if (index >= numElements()) {
-            throw std::out_of_range("Index (" + std::to_string(index) + ") is out of range [0," + std::to_string(numElements()) + ").");
-        }
-
-        const std::size_t nx = resolution_[0];
-        const std::size_t ny = resolution_[1];
-
-        const std::size_t elementsPerPlane = nx * ny;
-
-        const std::size_t i = index % nx;
-        const std::size_t j = (index / nx) % ny;
-        const std::size_t k = index / elementsPerPlane;
-
-        auto nodeIndex = [&](std::size_t i, std::size_t j, std::size_t k) {
-            return k * ((nx + 1) * (ny + 1)) + j * (nx + 1) + i;
-        };
-
-        // Bottom face
-        const std::size_t n1 = nodeIndex(i, j, k);
-        const std::size_t n2 = nodeIndex(i + 1, j, k);
-        const std::size_t n3 = nodeIndex(i + 1, j + 1, k);
-        const std::size_t n4 = nodeIndex(i, j + 1, k);
-
-        // Top face
-        const std::size_t n5 = nodeIndex(i, j, k + 1);
-        const std::size_t n6 = nodeIndex(i + 1, j, k + 1);
-        const std::size_t n7 = nodeIndex(i + 1, j + 1, k + 1);
-        const std::size_t n8 = nodeIndex(i, j + 1, k + 1);
-
-        return {n1, n2, n3, n4, n5, n6, n7, n8};
-    }
-
-    Hex8Grid::ElementList Hex8Grid::periodicElement(std::size_t index) const {
-        if (index >= numElements()) {
-            throw std::out_of_range("Index (" + std::to_string(index) + ") is out of range [0," + std::to_string(numElements()) + ").");
-        }
-
-        const std::size_t nx = resolution_[0];
-        const std::size_t ny = resolution_[1];
-        const std::size_t nz = resolution_[2];
-
-        const std::size_t elementsPerPlane = nx * ny;
-
-        const std::size_t i = index % nx;
-        const std::size_t j = (index / nx) % ny;
-        const std::size_t k = index / elementsPerPlane;
-
-        auto nodeIndex = [&](std::size_t i, std::size_t j, std::size_t k) {
-            const std::size_t iP = i % nx;
-            const std::size_t jP = j % ny;
-            const std::size_t kP = k % nz;
-
-            return kP * (nx * ny) + jP * nx + iP;
-        };
-
-        // Bottom face
-        const std::size_t n1 = nodeIndex(i, j, k);
-        const std::size_t n2 = nodeIndex(i + 1, j, k);
-        const std::size_t n3 = nodeIndex(i + 1, j + 1, k);
-        const std::size_t n4 = nodeIndex(i, j + 1, k);
-
-        // Top face
-        const std::size_t n5 = nodeIndex(i, j, k + 1);
-        const std::size_t n6 = nodeIndex(i + 1, j, k + 1);
-        const std::size_t n7 = nodeIndex(i + 1, j + 1, k + 1);
-        const std::size_t n8 = nodeIndex(i, j + 1, k + 1);
-
-        return {n1, n2, n3, n4, n5, n6, n7, n8};
-    }
-
-} // namespace monad
diff --git a/src/grid/hex8_topology.cpp b/src/grid/hex8_topology.cpp
new file mode 100644
index 0000000..a2a6431
--- /dev/null
+++ b/src/grid/hex8_topology.cpp
@@ -0,0 +1,90 @@
+#include "monad/grid/hex8_topology.hpp"
+
+namespace monad {
+
+    namespace grid {
+
+        std::size_t Hex8Topology::numNodes(const Resolution& resolution) noexcept {
+            return (resolution[0] + 1) * (resolution[1] + 1) * (resolution[2] + 1);
+        }
+
+        std::size_t Hex8Topology::numPeriodicNodes(const Resolution& resolution) noexcept {
+            return resolution[0] * resolution[1] * resolution[2];
+        }
+
+        Hex8Topology::Point Hex8Topology::node(std::size_t index, const Resolution& resolution, const Size& size) noexcept {
+            const std::size_t nx = resolution[0];
+            const std::size_t ny = resolution[1];
+            const std::size_t nz = resolution[2];
+            const std::size_t nodesPerPlane = (nx + 1) * (ny + 1);
+            const std::size_t indexInPlane = index % nodesPerPlane;
+
+            const std::size_t i = indexInPlane % (nx + 1);
+            const std::size_t j = indexInPlane / (nx + 1);
+            const std::size_t k = index / nodesPerPlane;
+
+            const double dx = size[0] / static_cast<double>(nx);
+            const double dy = size[1] / static_cast<double>(ny);
+            const double dz = size[2] / static_cast<double>(nz);
+
+            const double x = static_cast<double>(i) * dx;
+            const double y = static_cast<double>(j) * dy;
+            const double z = static_cast<double>(k) * dz;
+
+            return Point(x, y, z);
+        }
+
+        Hex8Topology::ElementList Hex8Topology::element(std::size_t index, const Resolution& resolution) noexcept {
+            const std::size_t nx = resolution[0];
+            const std::size_t ny = resolution[1];
+            const std::size_t elementsPerPlane = nx * ny;
+
+            const std::size_t i = index % nx;
+            const std::size_t j = (index / nx) % ny;
+            const std::size_t k = index / elementsPerPlane;
+
+            const auto nodeIndex = [nx, ny](std::size_t ii, std::size_t jj, std::size_t kk) {
+                return kk * ((nx + 1) * (ny + 1)) + jj * (nx + 1) + ii;
+            };
+
+            return {
+                nodeIndex(i, j, k),
+                nodeIndex(i + 1, j, k),
+                nodeIndex(i + 1, j + 1, k),
+                nodeIndex(i, j + 1, k),
+                nodeIndex(i, j, k + 1),
+                nodeIndex(i + 1, j, k + 1),
+                nodeIndex(i + 1, j + 1, k + 1),
+                nodeIndex(i, j + 1, k + 1)
+            };
+        }
+
+        Hex8Topology::ElementList Hex8Topology::periodicElement(std::size_t index, const Resolution& resolution) noexcept {
+            const std::size_t nx = resolution[0];
+            const std::size_t ny = resolution[1];
+            const std::size_t nz = resolution[2];
+            const std::size_t elementsPerPlane = nx * ny;
+
+            const std::size_t i = index % nx;
+            const std::size_t j = (index / nx) % ny;
+            const std::size_t k = index / elementsPerPlane;
+
+            const auto nodeIndex = [nx, ny, nz](std::size_t ii, std::size_t jj, std::size_t kk) {
+                return (kk % nz) * (nx * ny) + (jj % ny) * nx + (ii % nx);
+            };
+
+            return {
+                nodeIndex(i, j, k),
+                nodeIndex(i + 1, j, k),
+                nodeIndex(i + 1, j + 1, k),
+                nodeIndex(i, j + 1, k),
+                nodeIndex(i, j, k + 1),
+                nodeIndex(i + 1, j, k + 1),
+                nodeIndex(i + 1, j + 1, k + 1),
+                nodeIndex(i, j + 1, k + 1)
+            };
+        }
+
+    } // namespace grid
+
+} // namespace monad
diff --git a/src/grid/quad4_grid.cpp b/src/grid/quad4_grid.cpp
deleted file mode 100644
index b19612b..0000000
--- a/src/grid/quad4_grid.cpp
+++ /dev/null
@@ -1,89 +0,0 @@
-#include <stdexcept>
-#include <string>
-#include "monad/grid/quad4_grid.hpp"
-
-namespace monad {
-
-    std::size_t Quad4Grid::numNodes() const noexcept {
-        const std::size_t nx = resolution_[0];
-        const std::size_t ny = resolution_[1];
-
-        return (nx + 1) * (ny + 1);
-    }
-
-    std::size_t Quad4Grid::numPeriodicNodes() const noexcept {
-        return numElements();
-    }
-
-    Quad4Grid::Point Quad4Grid::node(std::size_t index) const {
-        if (index >= numNodes()) {
-            throw std::out_of_range("Index (" + std::to_string(index) + ") is out of range [0," + std::to_string(numNodes()) + ").");
-        }
-
-        const std::size_t nx = resolution_[0];
-        const std::size_t ny = resolution_[1];
-
-        const double lx = size_[0];
-        const double ly = size_[1];
-
-        const double dx = lx / static_cast<double>(nx);
-        const double dy = ly / static_cast<double>(ny);
-
-        const std::size_t i = index % (nx + 1);
-        const std::size_t j = index / (nx + 1);
-
-        const double x = static_cast<double>(i) * dx;
-        const double y = static_cast<double>(j) * dy;
-
-        return Point(x, y);
-    }
-
-    Quad4Grid::ElementList Quad4Grid::element(std::size_t index) const {
-        if (index >= numElements()) {
-            throw std::out_of_range("Index (" + std::to_string(index) + ") is out of range [0," + std::to_string(numElements()) + ").");
-        }
-
-        const std::size_t nx = resolution_[0];
-
-        const std::size_t i = index % nx;
-        const std::size_t j = index / nx;
-
-        auto nodeIndex = [&](std::size_t i, std::size_t j) {
-            return j * (nx + 1) + i;
-        };
-
-        const std::size_t n1 = nodeIndex(i, j);
-        const std::size_t n2 = nodeIndex(i + 1, j);
-        const std::size_t n3 = nodeIndex(i + 1, j + 1);
-        const std::size_t n4 = nodeIndex(i, j + 1);
-
-        return {n1, n2, n3, n4};
-    }
-
-    Quad4Grid::ElementList Quad4Grid::periodicElement(std::size_t index) const {
-        if (index >= numElements()) {
-            throw std::out_of_range("Index (" + std::to_string(index) + ") is out of range [0," + std::to_string(numElements()) + ").");
-        }
-
-        const std::size_t nx = resolution_[0];
-        const std::size_t ny = resolution_[1];
-
-        const std::size_t i = index % nx;
-        const std::size_t j = index / nx;
-
-        auto nodeIndex = [&](std::size_t i, std::size_t j) {
-            const std::size_t iP = i % nx;
-            const std::size_t jP = j % ny;
-
-            return jP * nx + iP;
-        };
-
-        const std::size_t n1 = nodeIndex(i, j);
-        const std::size_t n2 = nodeIndex(i + 1, j);
-        const std::size_t n3 = nodeIndex(i + 1, j + 1);
-        const std::size_t n4 = nodeIndex(i, j + 1);
-
-        return {n1, n2, n3, n4};
-    }
-
-} // namespace monad
diff --git a/src/grid/quad4_topology.cpp b/src/grid/quad4_topology.cpp
new file mode 100644
index 0000000..9848843
--- /dev/null
+++ b/src/grid/quad4_topology.cpp
@@ -0,0 +1,67 @@
+#include "monad/grid/quad4_topology.hpp"
+
+namespace monad {
+
+    namespace grid {
+
+        std::size_t Quad4Topology::numNodes(const Resolution& resolution) noexcept {
+            return (resolution[0] + 1) * (resolution[1] + 1);
+        }
+
+        std::size_t Quad4Topology::numPeriodicNodes(const Resolution& resolution) noexcept {
+            return resolution[0] * resolution[1];
+        }
+
+        Quad4Topology::Point Quad4Topology::node(std::size_t index, const Resolution& resolution, const Size& size) noexcept {
+            const std::size_t nx = resolution[0];
+            const std::size_t ny = resolution[1];
+            const std::size_t i = index % (nx + 1);
+            const std::size_t j = index / (nx + 1);
+
+            const double dx = size[0] / static_cast<double>(nx);
+            const double dy = size[1] / static_cast<double>(ny);
+
+            const double x = static_cast<double>(i) * dx;
+            const double y = static_cast<double>(j) * dy;
+
+            return Point(x, y);
+        }
+
+        Quad4Topology::ElementList Quad4Topology::element(std::size_t index, const Resolution& resolution) noexcept {
+            const std::size_t nx = resolution[0];
+            const std::size_t i = index % nx;
+            const std::size_t j = index / nx;
+
+            const auto nodeIndex = [nx](std::size_t ii, std::size_t jj) {
+                return jj * (nx + 1) + ii;
+            };
+
+            return {
+                nodeIndex(i, j),
+                nodeIndex(i + 1, j),
+                nodeIndex(i + 1, j + 1),
+                nodeIndex(i, j + 1)
+            };
+        }
+
+        Quad4Topology::ElementList Quad4Topology::periodicElement(std::size_t index, const Resolution& resolution) noexcept {
+            const std::size_t nx = resolution[0];
+            const std::size_t ny = resolution[1];
+            const std::size_t i = index % nx;
+            const std::size_t j = index / nx;
+
+            const auto nodeIndex = [nx, ny](std::size_t ii, std::size_t jj) {
+                return (jj % ny) * nx + (ii % nx);
+            };
+
+            return {
+                nodeIndex(i, j),
+                nodeIndex(i + 1, j),
+                nodeIndex(i + 1, j + 1),
+                nodeIndex(i, j + 1)
+            };
+        }
+
+    } // namespace grid
+
+} // namespace monad
diff --git a/src/grid/quad8_grid.cpp b/src/grid/quad8_grid.cpp
deleted file mode 100644
index 31508c5..0000000
--- a/src/grid/quad8_grid.cpp
+++ /dev/null
@@ -1,165 +0,0 @@
-#include <stdexcept>
-#include <string>
-#include "monad/grid/quad8_grid.hpp"
-
-namespace monad {
-
-    std::size_t Quad8Grid::numNodes() const noexcept {
-        const std::size_t nx = resolution_[0];
-        const std::size_t ny = resolution_[1];
-
-        const std::size_t numCornerNodes = (nx + 1) * (ny + 1);
-        const std::size_t numXMidNodes = nx * (ny + 1);
-        const std::size_t numYMidNodes = (nx + 1) * ny;
-
-        return numCornerNodes + numXMidNodes + numYMidNodes;
-    }
-
-    std::size_t Quad8Grid::numPeriodicNodes() const noexcept {
-        return 3 * numElements();
-    }
-
-    Quad8Grid::Point Quad8Grid::node(std::size_t index) const {
-        if (index >= numNodes()) {
-            throw std::out_of_range("Index (" + std::to_string(index) + ") is out of range [0," + std::to_string(numNodes()) + ").");
-        }
-
-        const std::size_t nx = resolution_[0];
-        const std::size_t ny = resolution_[1];
-
-        const std::size_t numCornerNodes = (nx + 1) * (ny + 1);
-        const std::size_t numXMidNodes = nx * (ny + 1);
-
-        const double lx = size_[0];
-        const double ly = size_[1];
-
-        const double dx = lx / static_cast<double>(nx);
-        const double dy = ly / static_cast<double>(ny);
-
-        double x;
-        double y;
-
-        if (index < numCornerNodes) {
-            const std::size_t i = index % (nx + 1);
-            const std::size_t j = index / (nx + 1);
-
-            x = static_cast<double>(i) * dx;
-            y = static_cast<double>(j) * dy;
-        }
-        else if (index < numCornerNodes + numXMidNodes) {
-            // x-midpoints are laid out in nx columns (i) and (ny + 1) rows (j).
-            index -= numCornerNodes;
-
-            const std::size_t i = index % nx;
-            const std::size_t j = index / nx;
-
-            x = (static_cast<double>(i) + 0.5) * dx;
-            y = static_cast<double>(j) * dy;
-        }
-        else {
-            // y-midpoints are laid out in (nx + 1) columns (i) and ny rows (j).
-            index -= numCornerNodes + numXMidNodes;
-
-            const std::size_t i = index % (nx + 1);
-            const std::size_t j = index / (nx + 1);
-
-            x = static_cast<double>(i) * dx;
-            y = (static_cast<double>(j) + 0.5) * dy;
-        }
-
-        return Point(x, y);
-    }
-
-    Quad8Grid::ElementList Quad8Grid::element(std::size_t index) const {
-        if (index >= numElements()) {
-            throw std::out_of_range("Index (" + std::to_string(index) + ") is out of range [0," + std::to_string(numElements()) + ").");
-        }
-
-        const std::size_t nx = resolution_[0];
-        const std::size_t ny = resolution_[1];
-
-        const std::size_t numCornerNodes = (nx + 1) * (ny + 1);
-        const std::size_t numXMidNodes = nx * (ny + 1);
-
-        const std::size_t i = index % nx;
-        const std::size_t j = index / nx;
-
-        auto cornerNodeIndex = [&](std::size_t i, std::size_t j) {
-            return j * (nx + 1) + i;
-        };
-
-        auto xMidNodeIndex = [&](std::size_t i, std::size_t j) {
-            return numCornerNodes + j * nx + i;
-        };
-
-        auto yMidNodeIndex = [&](std::size_t i, std::size_t j) {
-            return numCornerNodes + numXMidNodes + j * (nx + 1) + i;
-        };
-
-        // Corner nodes
-        const std::size_t n1 = cornerNodeIndex(i, j);
-        const std::size_t n2 = cornerNodeIndex(i + 1, j);
-        const std::size_t n3 = cornerNodeIndex(i + 1, j + 1);
-        const std::size_t n4 = cornerNodeIndex(i, j + 1);
-
-        // Midpoint nodes
-        const std::size_t n5 = xMidNodeIndex(i, j); // between n1-n2
-        const std::size_t n6 = yMidNodeIndex(i + 1, j); // between n2-n3
-        const std::size_t n7 = xMidNodeIndex(i, j + 1); // between n3-n4
-        const std::size_t n8 = yMidNodeIndex(i, j); // between n1-n4
-
-        return {n1, n2, n3, n4, n5, n6, n7, n8};
-    }
-
-    Quad8Grid::ElementList Quad8Grid::periodicElement(std::size_t index) const {
-        if (index >= numElements()) {
-            throw std::out_of_range("Index (" + std::to_string(index) + ") is out of range [0," + std::to_string(numElements()) + ").");
-        }
-
-        const std::size_t nx = resolution_[0];
-        const std::size_t ny = resolution_[1];
-
-        const std::size_t numElements = nx * ny;
-
-        const std::size_t i = index % nx;
-        const std::size_t j = index / nx;
-
-        auto cornerNodeIndex = [&](std::size_t i, std::size_t j) {
-            const std::size_t iP = i % nx;
-            const std::size_t jP = j % ny;
-
-            return jP * nx + iP;
-        };
-
-        auto xMidNodeIndex = [&](std::size_t i, std::size_t j) {
-            // Periodic wrapping
-            const std::size_t iP = i % nx;
-            const std::size_t jP = j % ny;
-
-            return numElements + jP * nx + iP;
-        };
-
-        auto yMidNodeIndex = [&](std::size_t i, std::size_t j) {
-            // Periodic wrapping
-            const std::size_t iP = i % nx;
-            const std::size_t jP = j % ny;
-
-            return numElements + numElements + jP * nx + iP;
-        };
-
-        // Corner nodes
-        const std::size_t n1 = cornerNodeIndex(i, j);
-        const std::size_t n2 = cornerNodeIndex(i + 1, j);
-        const std::size_t n3 = cornerNodeIndex(i + 1, j + 1);
-        const std::size_t n4 = cornerNodeIndex(i, j + 1);
-
-        // Midpoint nodes
-        const std::size_t n5 = xMidNodeIndex(i, j); // between n1-n2
-        const std::size_t n6 = yMidNodeIndex(i + 1, j); // between n2-n3
-        const std::size_t n7 = xMidNodeIndex(i, j + 1); // between n3-n4
-        const std::size_t n8 = yMidNodeIndex(i, j); // between n1-n4
-
-        return {n1, n2, n3, n4, n5, n6, n7, n8};
-    }
-
-} // namespace monad
diff --git a/src/grid/quad8_topology.cpp b/src/grid/quad8_topology.cpp
new file mode 100644
index 0000000..b916b6d
--- /dev/null
+++ b/src/grid/quad8_topology.cpp
@@ -0,0 +1,134 @@
+#include "monad/grid/quad8_topology.hpp"
+
+namespace monad {
+
+    namespace grid {
+
+        std::size_t Quad8Topology::numCornerNodes(const Resolution& resolution) noexcept {
+            return (resolution[0] + 1) * (resolution[1] + 1);
+        }
+
+        std::size_t Quad8Topology::numXMidNodes(const Resolution& resolution) noexcept {
+            return resolution[0] * (resolution[1] + 1);
+        }
+
+        std::size_t Quad8Topology::numYMidNodes(const Resolution& resolution) noexcept {
+            return (resolution[0] + 1) * resolution[1];
+        }
+
+        std::size_t Quad8Topology::numNodes(const Resolution& resolution) noexcept {
+            return numCornerNodes(resolution) + numXMidNodes(resolution) + numYMidNodes(resolution);
+        }
+
+        std::size_t Quad8Topology::numPeriodicNodes(const Resolution& resolution) noexcept {
+            return 3 * resolution[0] * resolution[1];
+        }
+
+        Quad8Topology::Point Quad8Topology::node(std::size_t index, const Resolution& resolution, const Size& size) noexcept {
+            const std::size_t nx = resolution[0];
+            const std::size_t ny = resolution[1];
+            const std::size_t corners = numCornerNodes(resolution);
+            const std::size_t xMids = numXMidNodes(resolution);
+
+            const double dx = size[0] / static_cast<double>(nx);
+            const double dy = size[1] / static_cast<double>(ny);
+
+            double x;
+            double y;
+
+            if (index < corners) {
+                const std::size_t i = index % (nx + 1);
+                const std::size_t j = index / (nx + 1);
+
+                x = static_cast<double>(i) * dx;
+                y = static_cast<double>(j) * dy;
+            }
+            else if (index < corners + xMids) {
+                index -= corners;
+
+                const std::size_t i = index % nx;
+                const std::size_t j = index / nx;
+
+                x = (static_cast<double>(i) + 0.5) * dx;
+                y = static_cast<double>(j) * dy;
+            }
+            else {
+                index -= corners + xMids;
+
+                const std::size_t i = index % (nx + 1);
+                const std::size_t j = index / (nx + 1);
+
+                x = static_cast<double>(i) * dx;
+                y = (static_cast<double>(j) + 0.5) * dy;
+            }
+
+            return Point(x, y);
+        }
+
+        Quad8Topology::ElementList Quad8Topology::element(std::size_t index, const Resolution& resolution) noexcept {
+            const std::size_t nx = resolution[0];
+            const std::size_t corners = numCornerNodes(resolution);
+            const std::size_t xMids = numXMidNodes(resolution);
+
+            const std::size_t i = index % nx;
+            const std::size_t j = index / nx;
+
+            const auto cornerNodeIndex = [nx](std::size_t ii, std::size_t jj) {
+                return jj * (nx + 1) + ii;
+            };
+
+            const auto xMidNodeIndex = [nx, corners](std::size_t ii, std::size_t jj) {
+                return corners + jj * nx + ii;
+            };
+
+            const auto yMidNodeIndex = [nx, corners, xMids](std::size_t ii, std::size_t jj) {
+                return corners + xMids + jj * (nx + 1) + ii;
+            };
+
+            return {
+                cornerNodeIndex(i, j),
+                cornerNodeIndex(i + 1, j),
+                cornerNodeIndex(i + 1, j + 1),
+                cornerNodeIndex(i, j + 1),
+                xMidNodeIndex(i, j),
+                yMidNodeIndex(i + 1, j),
+                xMidNodeIndex(i, j + 1),
+                yMidNodeIndex(i, j)
+            };
+        }
+
+        Quad8Topology::ElementList Quad8Topology::periodicElement(std::size_t index, const Resolution& resolution) noexcept {
+            const std::size_t nx = resolution[0];
+            const std::size_t ny = resolution[1];
+            const std::size_t numElements = nx * ny;
+
+            const std::size_t i = index % nx;
+            const std::size_t j = index / nx;
+
+            const auto cornerNodeIndex = [nx, ny](std::size_t ii, std::size_t jj) {
+                return (jj % ny) * nx + (ii % nx);
+            };
+
+            const auto xMidNodeIndex = [nx, ny, numElements](std::size_t ii, std::size_t jj) {
+                return numElements + (jj % ny) * nx + (ii % nx);
+            };
+
+            const auto yMidNodeIndex = [nx, ny, numElements](std::size_t ii, std::size_t jj) {
+                return 2 * numElements + (jj % ny) * nx + (ii % nx);
+            };
+
+            return {
+                cornerNodeIndex(i, j),
+                cornerNodeIndex(i + 1, j),
+                cornerNodeIndex(i + 1, j + 1),
+                cornerNodeIndex(i, j + 1),
+                xMidNodeIndex(i, j),
+                yMidNodeIndex(i + 1, j),
+                xMidNodeIndex(i, j + 1),
+                yMidNodeIndex(i, j)
+            };
+        }
+
+    } // namespace grid
+
+} // namespace monad
diff --git a/src/solver/solver_options.cpp b/src/solver/solver_options.cpp
index 8e2474b..5b3329d 100644
--- a/src/solver/solver_options.cpp
+++ b/src/solver/solver_options.cpp
@@ -14,14 +14,11 @@ namespace monad {
         return (flags & bit) != FieldSave::None;
     }
 
-    SolverOptions SolverOptions::defaults() {
-        return SolverOptions{};
-    }
-
     bool SolverOptions::operator==(const SolverOptions &other) const {
         return maxIterations == other.maxIterations
             && tolerance == other.tolerance
-            && fields == other.fields;
+            && fields == other.fields
+            && verbose == other.verbose;
     }
 
     bool SolverOptions::operator!=(const SolverOptions &other) const {
diff --git a/tests/CMakeLists.txt b/tests/CMakeLists.txt
index 44c3677..de89344 100644
--- a/tests/CMakeLists.txt
+++ b/tests/CMakeLists.txt
@@ -1,6 +1,44 @@
 cmake_minimum_required(VERSION 3.10)
 
-file(GLOB_RECURSE TEST_SOURCES ${CMAKE_SOURCE_DIR}/tests/*.cpp)
+set(TEST_SOURCES
+    detail/eigen_utils_test.cpp
+    detail/mean_test.cpp
+    fem/element/element_test.cpp
+    fem/kernel/mechanical/linear_elastic_kernel_test.cpp
+    fem/kernel/multiphysics/linear_piezoelectric_kernel_test.cpp
+    fem/kernel/scalar/linear_scalar_diffusion_kernel_test.cpp
+    fem/operator/mechanical/linear_elastic_dof_traits_test.cpp
+    fem/operator/multiphysics/linear_piezoelectric_dof_traits_test.cpp
+    fem/operator/scalar/linear_scalar_diffusion_dof_traits_test.cpp
+    field/density_field_test.cpp
+    field/make_density_field_from_csv_test.cpp
+    field/make_density_field_from_function_test.cpp
+    grid/hex8_topology_test.cpp
+    grid/hex20_topology_test.cpp
+    grid/quad4_topology_test.cpp
+    grid/quad8_topology_test.cpp
+    grid/structured_grid_test.cpp
+    integration/integrate_matrix_test.cpp
+    integration/integrate_scalar_test.cpp
+    integration/quadrature_rule_test.cpp
+    io/gmsh/write_gmsh_densities_test.cpp
+    io/gmsh/write_gmsh_elements_test.cpp
+    io/gmsh/write_gmsh_header_test.cpp
+    io/gmsh/write_gmsh_nodal_field_test.cpp
+    io/gmsh/write_gmsh_nodes_test.cpp
+    io/save_grid_and_density_field_test.cpp
+    io/save_grid_and_nodal_field_test.cpp
+    io/save_grid_test.cpp
+    material/mechanical/linear_elastic_material_2d_test.cpp
+    material/mechanical/linear_elastic_material_3d_test.cpp
+    material/multiphysics/linear_piezoelectric_material_test.cpp
+    material/transport/linear_transport_material_test.cpp
+    material/bounds_test.cpp
+    solver/mechanical/linear_elastic_solver_test.cpp
+    solver/multiphysics/linear_piezoelectric_solver_test.cpp
+    solver/scalar/linear_scalar_diffusion_solver_test.cpp
+    solver/solver_options_test.cpp
+)
 
 add_executable(monad_tests ${TEST_SOURCES})
 
diff --git a/tests/detail/eigen_utils_test.cpp b/tests/detail/eigen_utils_test.cpp
index 291b57e..1ba67b1 100644
--- a/tests/detail/eigen_utils_test.cpp
+++ b/tests/detail/eigen_utils_test.cpp
@@ -6,16 +6,11 @@
 
 using namespace monad::detail;
 
-TEST_CASE("monad::detail: Test arrayToEigen", "[monad]") {
-    const std::array<std::size_t, 3> array{1, 2, 3};
-
-    const Eigen::Vector3i expected(1, 2, 3);
-
-    REQUIRE(arrayToEigen(array) == expected);
-}
-
 TEST_CASE("monad::detail: Test symmetrize/isSymmetric", "[monad]") {
-    Eigen::Matrix3d A = Eigen::Matrix3d::Random();
+    Eigen::Matrix3d A;
+    A << 1, 2, 3,
+         2, 3, 4.001,
+         3, 4, 5;
 
     REQUIRE(!isSymmetric(A));
 
diff --git a/tests/fem/element/element_test.cpp b/tests/fem/element/element_test.cpp
index 2cf6236..0dfc322 100644
--- a/tests/fem/element/element_test.cpp
+++ b/tests/fem/element/element_test.cpp
@@ -1,8 +1,8 @@
 #include <tuple>
+#include <type_traits>
 #include <catch2/catch_test_macros.hpp>
 #include <catch2/catch_template_test_macros.hpp>
 #include <catch2/matchers/catch_matchers_floating_point.hpp>
-#include <catch2/matchers/catch_matchers_vector.hpp>
 #include "monad/fem/element/quad4.hpp"
 #include "monad/fem/element/quad8.hpp"
 #include "monad/fem/element/hex8.hpp"
@@ -118,7 +118,7 @@ TEMPLATE_LIST_TEST_CASE("monad::fem::ElementBase Test quadratureRule", "[monad]"
             expected = analyticIntegral(p, p, p);
         }
 
-        auto integrand = [p](const Point &point) -> double {
+        auto integrand = [&](const Point &point) -> double {
             if constexpr (TestType::Dim == 2) {
                 return numericIntegrand(point, p, p);
             }
diff --git a/tests/fem/kernel/mechanical/linear_elastic_kernel_2d_test.cpp b/tests/fem/kernel/mechanical/linear_elastic_kernel_2d_test.cpp
deleted file mode 100644
index d728673..0000000
--- a/tests/fem/kernel/mechanical/linear_elastic_kernel_2d_test.cpp
+++ /dev/null
@@ -1,132 +0,0 @@
-#include <tuple>
-#include <stdexcept>
-#include <Eigen/Core>
-#include <catch2/catch_template_test_macros.hpp>
-#include "monad/fem/element/quad4.hpp"
-#include "monad/fem/element/quad8.hpp"
-#include "monad/fem/kernel/mechanical/linear_elastic_kernel.hpp"
-#include "monad/material/mechanical/linear_elastic_material_2d.hpp"
-#include "monad/detail/eigen_utils.hpp"
-#include "monad/detail/constants.hpp"
-
-using namespace monad;
-using namespace monad::fem;
-using namespace monad::fem::mechanical;
-using namespace monad::detail;
-
-using Types = std::tuple<Quad4, Quad8>;
-
-TEMPLATE_LIST_TEST_CASE("monad::fem::mechanical::LinearElasticKernel (2d): Test bMatrix", "[monad]", Types) {
-    using FieldMatrix = typename LinearElasticKernel<TestType>::FieldMatrix;
-
-    auto nodes = TestType::localNodes();
-    const auto points = TestType::quadratureRule().points;
-
-    SECTION("BU=ε for unit strains") {
-        const auto expected = Eigen::Matrix3d::Identity();
-
-        for (auto point : points) {
-            const auto B = LinearElasticKernel<TestType>::bMatrix(point, nodes);
-
-            FieldMatrix U;
-            U.setZero();
-
-            // Displacements of unit axial strain ε₁₁
-            U(Eigen::seq(0, Eigen::indexing::last, 2), 0) = nodes.col(0);
-
-            // Displacements of unit axial strain ε₂₂
-            U(Eigen::seq(1, Eigen::indexing::last, 2), 1) = nodes.col(1);
-
-            // Displacements of unit shear strain ε₁₂
-            U(Eigen::seq(0, Eigen::indexing::last, 2), 2) = 0.5 * nodes.col(1);
-            U(Eigen::seq(1, Eigen::indexing::last, 2), 2) = 0.5 * nodes.col(0);
-
-            REQUIRE((B * U).isApprox(expected, NUMERICAL_ZERO));
-        }
-    }
-
-    SECTION("BU=0 for rigid body transformations") {
-        for (auto point : points) {
-            const auto B = LinearElasticKernel<TestType>::bMatrix(point, nodes);
-
-            FieldMatrix U;
-            U.setZero();
-
-            // Unit x translation
-            U(Eigen::seq(0, Eigen::indexing::last, 2), 0).setOnes();
-
-            // Unit y translation
-            U(Eigen::seq(1, Eigen::indexing::last, 2), 1).setOnes();
-
-            // 90° ccw rotation
-            U(Eigen::seq(0, Eigen::indexing::last, 2), 2) = -nodes.col(1);
-            U(Eigen::seq(1, Eigen::indexing::last, 2), 2) = nodes.col(0);
-
-            REQUIRE((B * U).isZero(NUMERICAL_ZERO));
-        }
-    }
-
-    SECTION("Invalid element - degenerate element") {
-        nodes = 0.0 * nodes;
-
-        const auto point = nodes.row(0);
-
-        REQUIRE_THROWS_AS(LinearElasticKernel<TestType>::bMatrix(point, nodes), std::invalid_argument);
-    }
-
-    SECTION("Invalid element - reverse node ordering") {
-        nodes = nodes.rowwise().reverse();
-
-        const auto point = nodes.row(0);
-
-        REQUIRE_THROWS_AS(LinearElasticKernel<TestType>::bMatrix(point, nodes), std::invalid_argument);
-    }
-}
-
-TEMPLATE_LIST_TEST_CASE("monad::fem::mechanical::LinearElasticKernel (2d): Test lhs/rhs", "[monad]", Types) {
-    using FieldMatrix = typename LinearElasticKernel<TestType>::FieldMatrix;
-
-    const LinearElasticMaterial2d material(1.0, 0.3, LinearElasticMaterial2d::PlaneCondition::PlaneStress);
-    auto nodes = TestType::localNodes();
-
-    SECTION("No errors") {
-        const auto K = LinearElasticKernel<TestType>::lhs(material, nodes);
-
-        REQUIRE(isSymmetric(K));
-        REQUIRE(isPSD(K));
-
-        SECTION("UᵀKU=UᵀF=0 for rigid body transformations") {
-            const FieldMatrix F = LinearElasticKernel<TestType>::rhs(material, nodes);
-
-            FieldMatrix U;
-            U.setZero();
-
-            // Unit x translation
-            U(Eigen::seq(0, Eigen::indexing::last, 2), 0).setOnes();
-
-            // Unit y translation
-            U(Eigen::seq(1, Eigen::indexing::last, 2), 1).setOnes();
-
-            // 90° ccw rotation
-            U(Eigen::seq(0, Eigen::indexing::last, 2), 2) = -nodes.col(1);
-            U(Eigen::seq(1, Eigen::indexing::last, 2), 2) = nodes.col(0);
-
-            REQUIRE((U.transpose() * K * U).isZero(NUMERICAL_ZERO));
-            REQUIRE((U.transpose() * F).isZero(NUMERICAL_ZERO));
-        }
-    }
-
-    SECTION("Invalid element - degenerate element") {
-        nodes = 0.0 * nodes;
-
-        REQUIRE_THROWS_AS(LinearElasticKernel<TestType>::lhs(material, nodes), std::invalid_argument);
-        REQUIRE_THROWS_AS(LinearElasticKernel<TestType>::rhs(material, nodes), std::invalid_argument);
-    }
-
-    SECTION("Invalid element - reverse node ordering") {
-        nodes = nodes.rowwise().reverse();
-
-        REQUIRE_THROWS_AS(LinearElasticKernel<TestType>::lhs(material, nodes), std::invalid_argument);
-        REQUIRE_THROWS_AS(LinearElasticKernel<TestType>::rhs(material, nodes), std::invalid_argument);
-    }
-}
diff --git a/tests/fem/kernel/mechanical/linear_elastic_kernel_3d_test.cpp b/tests/fem/kernel/mechanical/linear_elastic_kernel_test.cpp
similarity index 56%
rename from tests/fem/kernel/mechanical/linear_elastic_kernel_3d_test.cpp
rename to tests/fem/kernel/mechanical/linear_elastic_kernel_test.cpp
index 43a00f3..389dd06 100644
--- a/tests/fem/kernel/mechanical/linear_elastic_kernel_3d_test.cpp
+++ b/tests/fem/kernel/mechanical/linear_elastic_kernel_test.cpp
@@ -4,7 +4,10 @@
 #include <catch2/catch_template_test_macros.hpp>
 #include "monad/fem/element/hex8.hpp"
 #include "monad/fem/element/hex20.hpp"
+#include "monad/fem/element/quad4.hpp"
+#include "monad/fem/element/quad8.hpp"
 #include "monad/fem/kernel/mechanical/linear_elastic_kernel.hpp"
+#include "monad/material/mechanical/linear_elastic_material_2d.hpp"
 #include "monad/material/mechanical/linear_elastic_material_3d.hpp"
 #include "monad/detail/eigen_utils.hpp"
 #include "monad/detail/constants.hpp"
@@ -14,9 +17,75 @@ using namespace monad::fem;
 using namespace monad::fem::mechanical;
 using namespace monad::detail;
 
-using Types = std::tuple<Hex8, Hex20>;
+using Types2d = std::tuple<Quad4, Quad8>;
+using Types3d = std::tuple<Hex8, Hex20>;
 
-TEMPLATE_LIST_TEST_CASE("monad::fem::mechanical::LinearElasticKernel (3d): Test bMatrix", "[monad]", Types) {
+TEMPLATE_LIST_TEST_CASE("monad::fem::mechanical::LinearElasticKernel (2d): Test bMatrix", "[monad]", Types2d) {
+    using FieldMatrix = typename LinearElasticKernel<TestType>::FieldMatrix;
+
+    auto nodes = TestType::localNodes();
+    const auto points = TestType::quadratureRule().points;
+
+    SECTION("BU=ε for unit strains") {
+        const auto expected = Eigen::Matrix3d::Identity();
+
+        for (auto point : points) {
+            const auto B = LinearElasticKernel<TestType>::bMatrix(point, nodes);
+
+            FieldMatrix U = FieldMatrix::Zero();
+
+            // Displacements of unit axial strain ε₁₁
+            U(Eigen::seq(0, Eigen::indexing::last, 2), 0) = nodes.col(0);
+
+            // Displacements of unit axial strain ε₂₂
+            U(Eigen::seq(1, Eigen::indexing::last, 2), 1) = nodes.col(1);
+
+            // Displacements of unit shear strain ε₁₂
+            U(Eigen::seq(0, Eigen::indexing::last, 2), 2) = 0.5 * nodes.col(1);
+            U(Eigen::seq(1, Eigen::indexing::last, 2), 2) = 0.5 * nodes.col(0);
+
+            REQUIRE((B * U).isApprox(expected, NUMERICAL_ZERO));
+        }
+    }
+
+    SECTION("BU=0 for rigid body transformations") {
+        for (auto point : points) {
+            const auto B = LinearElasticKernel<TestType>::bMatrix(point, nodes);
+
+            FieldMatrix U = FieldMatrix::Zero();
+
+            // Unit x translation
+            U(Eigen::seq(0, Eigen::indexing::last, 2), 0).setOnes();
+
+            // Unit y translation
+            U(Eigen::seq(1, Eigen::indexing::last, 2), 1).setOnes();
+
+            // 90° ccw rotation
+            U(Eigen::seq(0, Eigen::indexing::last, 2), 2) = -nodes.col(1);
+            U(Eigen::seq(1, Eigen::indexing::last, 2), 2) = nodes.col(0);
+
+            REQUIRE((B * U).isZero(NUMERICAL_ZERO));
+        }
+    }
+
+    SECTION("Invalid element - degenerate element") {
+        nodes = 0.0 * nodes;
+
+        const auto point = nodes.row(0);
+
+        REQUIRE_THROWS_AS(LinearElasticKernel<TestType>::bMatrix(point, nodes), std::invalid_argument);
+    }
+
+    SECTION("Invalid element - reverse node ordering") {
+        nodes = nodes.rowwise().reverse();
+
+        const auto point = nodes.row(0);
+
+        REQUIRE_THROWS_AS(LinearElasticKernel<TestType>::bMatrix(point, nodes), std::invalid_argument);
+    }
+}
+
+TEMPLATE_LIST_TEST_CASE("monad::fem::mechanical::LinearElasticKernel (3d): Test bMatrix", "[monad]", Types3d) {
     using FieldMatrix = typename LinearElasticKernel<TestType>::FieldMatrix;
 
     auto nodes = TestType::localNodes();
@@ -28,8 +97,7 @@ TEMPLATE_LIST_TEST_CASE("monad::fem::mechanical::LinearElasticKernel (3d): Test
         for (auto point : points) {
             const auto B = LinearElasticKernel<TestType>::bMatrix(point, nodes);
 
-            FieldMatrix U;
-            U.setZero();
+            FieldMatrix U = FieldMatrix::Zero();
 
             // Displacements of unit axial strain ε₁₁
             U(Eigen::seq(0, Eigen::indexing::last, 3), 0) = nodes.col(0);
@@ -60,8 +128,7 @@ TEMPLATE_LIST_TEST_CASE("monad::fem::mechanical::LinearElasticKernel (3d): Test
         for (auto point : points) {
             auto B = LinearElasticKernel<TestType>::bMatrix(point, nodes);
 
-            FieldMatrix U;
-            U.setZero();
+            FieldMatrix U = FieldMatrix::Zero();
 
             // Unit x translation
             U(Eigen::seq(0, Eigen::indexing::last, 3), 0).setOnes();
@@ -105,7 +172,54 @@ TEMPLATE_LIST_TEST_CASE("monad::fem::mechanical::LinearElasticKernel (3d): Test
     }
 }
 
-TEMPLATE_LIST_TEST_CASE("monad::fem::mechanical::LinearElasticKernel (3d): Test lhs", "[monad]", Types) {
+TEMPLATE_LIST_TEST_CASE("monad::fem::mechanical::LinearElasticKernel (2d): Test lhs/rhs", "[monad]", Types2d) {
+    using FieldMatrix = typename LinearElasticKernel<TestType>::FieldMatrix;
+
+    const LinearElasticMaterial2d material(1.0, 0.3, LinearElasticMaterial2d::PlaneCondition::PlaneStress);
+    auto nodes = TestType::localNodes();
+
+    SECTION("No errors") {
+        const auto K = LinearElasticKernel<TestType>::lhs(material, nodes);
+
+        REQUIRE(isSymmetric(K));
+        REQUIRE(isPSD(K));
+
+        SECTION("UᵀKU=UᵀF=0 for rigid body transformations") {
+            const FieldMatrix F = LinearElasticKernel<TestType>::rhs(material, nodes);
+
+            FieldMatrix U = FieldMatrix::Zero();
+
+            // Unit x translation
+            U(Eigen::seq(0, Eigen::indexing::last, 2), 0).setOnes();
+
+            // Unit y translation
+            U(Eigen::seq(1, Eigen::indexing::last, 2), 1).setOnes();
+
+            // 90° ccw rotation
+            U(Eigen::seq(0, Eigen::indexing::last, 2), 2) = -nodes.col(1);
+            U(Eigen::seq(1, Eigen::indexing::last, 2), 2) = nodes.col(0);
+
+            REQUIRE((U.transpose() * K * U).isZero(NUMERICAL_ZERO));
+            REQUIRE((U.transpose() * F).isZero(NUMERICAL_ZERO));
+        }
+    }
+
+    SECTION("Invalid element - degenerate element") {
+        nodes = 0.0 * nodes;
+
+        REQUIRE_THROWS_AS(LinearElasticKernel<TestType>::lhs(material, nodes), std::invalid_argument);
+        REQUIRE_THROWS_AS(LinearElasticKernel<TestType>::rhs(material, nodes), std::invalid_argument);
+    }
+
+    SECTION("Invalid element - reverse node ordering") {
+        nodes = nodes.rowwise().reverse();
+
+        REQUIRE_THROWS_AS(LinearElasticKernel<TestType>::lhs(material, nodes), std::invalid_argument);
+        REQUIRE_THROWS_AS(LinearElasticKernel<TestType>::rhs(material, nodes), std::invalid_argument);
+    }
+}
+
+TEMPLATE_LIST_TEST_CASE("monad::fem::mechanical::LinearElasticKernel (3d): Test lhs", "[monad]", Types3d) {
     using FieldMatrix = typename LinearElasticKernel<TestType>::FieldMatrix;
 
     const LinearElasticMaterial3d material(1.0, 0.3);
@@ -120,8 +234,7 @@ TEMPLATE_LIST_TEST_CASE("monad::fem::mechanical::LinearElasticKernel (3d): Test
         SECTION("UᵀKU=UᵀF=0 for rigid body transformations") {
             const FieldMatrix F = LinearElasticKernel<TestType>::rhs(material, nodes);
 
-            FieldMatrix U;
-            U.setZero();
+            FieldMatrix U = FieldMatrix::Zero();
 
             // Unit x translation
             U(Eigen::seq(0, Eigen::indexing::last, 3), 0).setOnes();
diff --git a/tests/fem/kernel/multiphysics/linear_piezoelectric_kernel_2d_test.cpp b/tests/fem/kernel/multiphysics/linear_piezoelectric_kernel_2d_test.cpp
deleted file mode 100644
index d285a96..0000000
--- a/tests/fem/kernel/multiphysics/linear_piezoelectric_kernel_2d_test.cpp
+++ /dev/null
@@ -1,98 +0,0 @@
-#include <tuple>
-#include <stdexcept>
-#include <Eigen/Core>
-#include <catch2/catch_template_test_macros.hpp>
-#include "monad/fem/element/quad4.hpp"
-#include "monad/fem/element/quad8.hpp"
-#include "monad/fem/kernel/multiphysics/linear_piezoelectric_kernel.hpp"
-#include "monad/material/mechanical/linear_elastic_material_2d.hpp"
-#include "monad/material/transport/linear_transport_material_aliases.hpp"
-#include "monad/material/multiphysics/linear_piezoelectric_material.hpp"
-#include "monad/detail/eigen_utils.hpp"
-#include "monad/detail/constants.hpp"
-
-using namespace monad;
-using namespace monad::fem;
-using namespace monad::fem::mechanical;
-using namespace monad::fem::scalar;
-using namespace monad::fem::multiphysics;
-using namespace monad::detail;
-
-using Types = std::tuple<Quad4, Quad8>;
-
-TEMPLATE_LIST_TEST_CASE("monad::fem::electrical::LinearPiezoelectricKernel (2d): Test lhs/rhs", "[monad]", Types) {
-    using CouplingTensor = typename LinearPiezoelectricMaterial2d::CouplingTensor;
-
-    const LinearElasticMaterial2d elasticMaterial(1.0, 0.3, monad::LinearElasticMaterial2d::PlaneCondition::PlaneStress);
-    const LinearDielectricMaterial2d dielectricMaterial(2.1);
-
-    const CouplingTensor d {
-        {0.01, 0.0, 0.0},
-        {0.0, 0.01, 0.01}
-    };
-
-    const LinearPiezoelectricMaterial2d material(elasticMaterial, dielectricMaterial, d);
-
-    auto nodes = TestType::localNodes();
-
-    SECTION("No errors") {
-        const auto K = LinearPiezoelectricKernel<TestType>::lhs(material, nodes);
-
-        REQUIRE(isSymmetric(K));
-        REQUIRE(!isPSD(K));
-
-        SECTION("xᵀKx=xᵀF=0 for rigid body transformations") {
-            using FieldVector = Eigen::Vector<double, LinearPiezoelectricKernel<TestType>::NumDofs>;
-            using MechanicalKernel = typename LinearPiezoelectricKernel<TestType>::MechanicalKernel;
-
-            const auto F = LinearPiezoelectricKernel<TestType>::rhs(material, nodes);
-
-            FieldVector x;
-
-            const int numMechanicalDofs = MechanicalKernel::NumDofs;
-
-            // Unit x translation
-            x.setZero();
-            x(Eigen::seq(0, numMechanicalDofs - 1, 2)).setOnes();
-
-            REQUIRE((x.transpose() * K * x).isZero(NUMERICAL_ZERO));
-            REQUIRE((x.transpose() * F).isZero(NUMERICAL_ZERO));
-
-            // Unit y translation
-            x.setZero();
-            x(Eigen::seq(1, numMechanicalDofs - 1, 2)).setOnes();
-
-            REQUIRE((x.transpose() * K * x).isZero(NUMERICAL_ZERO));
-            REQUIRE((x.transpose() * F).isZero(NUMERICAL_ZERO));
-
-            // 90° ccw rotation
-            x.setZero();
-            x(Eigen::seq(0, numMechanicalDofs - 1, 2)) = -nodes.col(1);
-            x(Eigen::seq(1, numMechanicalDofs - 1, 2)) = nodes.col(0);
-
-            REQUIRE((x.transpose() * K * x).isZero(NUMERICAL_ZERO));
-            REQUIRE((x.transpose() * F).isZero(NUMERICAL_ZERO));
-
-            // Unit electrical translation
-            x.setZero();
-            x(Eigen::seq(numMechanicalDofs, Eigen::indexing::last)).setOnes();
-
-            REQUIRE((x.transpose() * K * x).isZero(NUMERICAL_ZERO));
-            REQUIRE((x.transpose() * F).isZero(NUMERICAL_ZERO));
-        }
-    }
-
-    SECTION("Invalid element - degenerate element") {
-        nodes = 0.0 * nodes;
-
-        REQUIRE_THROWS_AS(LinearPiezoelectricKernel<TestType>::lhs(material, nodes), std::invalid_argument);
-        REQUIRE_THROWS_AS(LinearPiezoelectricKernel<TestType>::rhs(material, nodes), std::invalid_argument);
-    }
-
-    SECTION("Invalid element - reverse node ordering") {
-        nodes = nodes.rowwise().reverse();
-
-        REQUIRE_THROWS_AS(LinearPiezoelectricKernel<TestType>::lhs(material, nodes), std::invalid_argument);
-        REQUIRE_THROWS_AS(LinearPiezoelectricKernel<TestType>::rhs(material, nodes), std::invalid_argument);
-    }
-}
diff --git a/tests/fem/kernel/multiphysics/linear_piezoelectric_kernel_3d_test.cpp b/tests/fem/kernel/multiphysics/linear_piezoelectric_kernel_3d_test.cpp
deleted file mode 100644
index 2504960..0000000
--- a/tests/fem/kernel/multiphysics/linear_piezoelectric_kernel_3d_test.cpp
+++ /dev/null
@@ -1,122 +0,0 @@
-#include <tuple>
-#include <stdexcept>
-#include <Eigen/Core>
-#include <catch2/catch_template_test_macros.hpp>
-#include "monad/fem/element/hex8.hpp"
-#include "monad/fem/element/hex20.hpp"
-#include "monad/fem/kernel/multiphysics/linear_piezoelectric_kernel.hpp"
-#include "monad/material/mechanical/linear_elastic_material_3d.hpp"
-#include "monad/material/transport/linear_transport_material_aliases.hpp"
-#include "monad/material/multiphysics/linear_piezoelectric_material.hpp"
-#include "monad/detail/eigen_utils.hpp"
-#include "monad/detail/constants.hpp"
-
-using namespace monad;
-using namespace monad::fem;
-using namespace monad::fem::mechanical;
-using namespace monad::fem::scalar;
-using namespace monad::fem::multiphysics;
-using namespace monad::detail;
-
-using Types = std::tuple<Hex8, Hex20>;
-
-TEMPLATE_LIST_TEST_CASE("monad::fem::electrical::LinearPiezoelectricKernel (3d): Test lhs/rhs", "[monad]", Types) {
-    using CouplingTensor = typename LinearPiezoelectricMaterial3d::CouplingTensor;
-
-    const LinearElasticMaterial3d elasticMaterial(1.0, 0.3);
-    const LinearDielectricMaterial3d dielectricMaterial(2.1);
-
-    const CouplingTensor d {
-        {0.0, 0.0, 0.0, 0.01, 0.0, 0.0},
-        {0.0, 0.0, 0.01, 0.0, 0.0, 0.0},
-        {0.01, 0.01, 0.01, 0.0, 0.0, 0.0}
-    };
-
-    const LinearPiezoelectricMaterial3d material(elasticMaterial, dielectricMaterial, d);
-
-    auto nodes = TestType::localNodes();
-
-    SECTION("No errors") {
-        const auto K = LinearPiezoelectricKernel<TestType>::lhs(material, nodes);
-
-        REQUIRE(isSymmetric(K));
-        REQUIRE(!isPSD(K));
-
-        SECTION("xᵀKx=xᵀF=0 for rigid body transformations") {
-            using FieldVector = Eigen::Vector<double, LinearPiezoelectricKernel<TestType>::NumDofs>;
-            using MechanicalKernel = typename LinearPiezoelectricKernel<TestType>::MechanicalKernel;
-
-            const auto F = LinearPiezoelectricKernel<TestType>::rhs(material, nodes);
-
-            FieldVector x;
-
-            const int numMechanicalDofs = MechanicalKernel::NumDofs;
-
-            // Unit x translation
-            x.setZero();
-            x(Eigen::seq(0, numMechanicalDofs - 1, 3)).setOnes();
-
-            REQUIRE((x.transpose() * K * x).isZero(NUMERICAL_ZERO));
-            REQUIRE((x.transpose() * F).isZero(NUMERICAL_ZERO));
-
-            // Unit y translation
-            x.setZero();
-            x(Eigen::seq(1, numMechanicalDofs - 1, 3)).setOnes();
-
-            REQUIRE((x.transpose() * K * x).isZero(NUMERICAL_ZERO));
-            REQUIRE((x.transpose() * F).isZero(NUMERICAL_ZERO));
-
-            // Unit z translation
-            x.setZero();
-            x(Eigen::seq(2, numMechanicalDofs - 1, 3)).setOnes();
-
-            REQUIRE((x.transpose() * K * x).isZero(NUMERICAL_ZERO));
-            REQUIRE((x.transpose() * F).isZero(NUMERICAL_ZERO));
-
-            // 90° ccw rotation in xy-plane
-            x.setZero();
-            x(Eigen::seq(0, numMechanicalDofs - 1, 3)) = -nodes.col(1);
-            x(Eigen::seq(1, numMechanicalDofs - 1, 3)) = nodes.col(0);
-
-            REQUIRE((x.transpose() * K * x).isZero(NUMERICAL_ZERO));
-            REQUIRE((x.transpose() * F).isZero(NUMERICAL_ZERO));
-
-            // 90° ccw rotation in xz-plane
-            x.setZero();
-            x(Eigen::seq(0, numMechanicalDofs - 1, 3)) = nodes.col(2);
-            x(Eigen::seq(2, numMechanicalDofs - 1, 3)) = -nodes.col(0);
-
-            REQUIRE((x.transpose() * K * x).isZero(NUMERICAL_ZERO));
-            REQUIRE((x.transpose() * F).isZero(NUMERICAL_ZERO));
-
-            // 90° ccw rotation in yz-plane
-            x.setZero();
-            x(Eigen::seq(1, numMechanicalDofs - 1, 3)) = -nodes.col(2);
-            x(Eigen::seq(2, numMechanicalDofs - 1, 3)) = nodes.col(1);
-
-            REQUIRE((x.transpose() * K * x).isZero(NUMERICAL_ZERO));
-            REQUIRE((x.transpose() * F).isZero(NUMERICAL_ZERO));
-
-            // Unit electrical translation
-            x.setZero();
-            x(Eigen::seq(numMechanicalDofs, Eigen::indexing::last)).setOnes();
-
-            REQUIRE((x.transpose() * K * x).isZero(NUMERICAL_ZERO));
-            REQUIRE((x.transpose() * F).isZero(NUMERICAL_ZERO));
-        }
-    }
-
-    SECTION("Invalid element - degenerate element") {
-        nodes = 0.0 * nodes;
-
-        REQUIRE_THROWS_AS(LinearPiezoelectricKernel<TestType>::lhs(material, nodes), std::invalid_argument);
-        REQUIRE_THROWS_AS(LinearPiezoelectricKernel<TestType>::rhs(material, nodes), std::invalid_argument);
-    }
-
-    SECTION("Invalid element - reverse node ordering") {
-        nodes = nodes.rowwise().reverse();
-
-        REQUIRE_THROWS_AS(LinearPiezoelectricKernel<TestType>::lhs(material, nodes), std::invalid_argument);
-        REQUIRE_THROWS_AS(LinearPiezoelectricKernel<TestType>::rhs(material, nodes), std::invalid_argument);
-    }
-}
diff --git a/tests/fem/kernel/multiphysics/linear_piezoelectric_kernel_test.cpp b/tests/fem/kernel/multiphysics/linear_piezoelectric_kernel_test.cpp
new file mode 100644
index 0000000..a4ec24a
--- /dev/null
+++ b/tests/fem/kernel/multiphysics/linear_piezoelectric_kernel_test.cpp
@@ -0,0 +1,219 @@
+#include <tuple>
+#include <stdexcept>
+#include <Eigen/Core>
+#include <catch2/catch_template_test_macros.hpp>
+#include "monad/fem/element/hex8.hpp"
+#include "monad/fem/element/hex20.hpp"
+#include "monad/fem/element/quad4.hpp"
+#include "monad/fem/element/quad8.hpp"
+#include "monad/fem/kernel/multiphysics/linear_piezoelectric_kernel.hpp"
+#include "monad/detail/eigen_utils.hpp"
+#include "monad/detail/constants.hpp"
+
+using namespace monad;
+using namespace monad::fem;
+using namespace monad::fem::mechanical;
+using namespace monad::fem::scalar;
+using namespace monad::fem::multiphysics;
+using namespace monad::detail;
+using namespace monad::material;
+
+using Types2d = std::tuple<Quad4, Quad8>;
+using Types3d = std::tuple<Hex8, Hex20>;
+
+TEMPLATE_LIST_TEST_CASE("monad::fem::electrical::LinearPiezoelectricKernel (2d): Test lhs/rhs", "[monad]", Types2d) {
+    using Kernel = LinearPiezoelectricKernel<TestType>;
+    using Material = typename Kernel::Material;
+    using StiffnessTensor = typename Material::StiffnessTensor;
+    using PermittivityTensor = typename Material::PermittivityTensor;
+    using CouplingTensor = typename Material::CouplingTensor;
+
+    StiffnessTensor c = StiffnessTensor::Random();
+    // Make PSD
+    c = c.transpose() * c;
+    // Make PD
+    c += StiffnessTensor::Identity();
+
+    PermittivityTensor epsilon = PermittivityTensor::Random();
+    // Make PSD
+    epsilon = epsilon.transpose() * epsilon;
+    // Make PD
+    epsilon += PermittivityTensor::Identity();
+
+    const CouplingTensor d = CouplingTensor::Constant(0.01);
+
+    const Material material(c, epsilon, d);
+
+    auto nodes = TestType::localNodes();
+
+    SECTION("No errors") {
+        const auto K = Kernel::lhs(material, nodes);
+
+        REQUIRE(isSymmetric(K));
+        REQUIRE(!isPSD(K));
+
+        SECTION("xᵀKx=xᵀF=0 for rigid body transformations") {
+            using FieldVector = Eigen::Vector<double, Kernel::NumDofs>;
+            using MechanicalKernel = typename Kernel::MechanicalKernel;
+
+            const auto F = Kernel::rhs(material, nodes);
+
+            FieldVector x;
+
+            const int numMechanicalDofs = MechanicalKernel::NumDofs;
+
+            // Unit x translation
+            x.setZero();
+            x(Eigen::seq(0, numMechanicalDofs - 1, 2)).setOnes();
+
+            REQUIRE((x.transpose() * K * x).isZero(NUMERICAL_ZERO));
+            REQUIRE((x.transpose() * F).isZero(NUMERICAL_ZERO));
+
+            // Unit y translation
+            x.setZero();
+            x(Eigen::seq(1, numMechanicalDofs - 1, 2)).setOnes();
+
+            REQUIRE((x.transpose() * K * x).isZero(NUMERICAL_ZERO));
+            REQUIRE((x.transpose() * F).isZero(NUMERICAL_ZERO));
+
+            // 90° ccw rotation
+            x.setZero();
+            x(Eigen::seq(0, numMechanicalDofs - 1, 2)) = -nodes.col(1);
+            x(Eigen::seq(1, numMechanicalDofs - 1, 2)) = nodes.col(0);
+
+            REQUIRE((x.transpose() * K * x).isZero(NUMERICAL_ZERO));
+            REQUIRE((x.transpose() * F).isZero(NUMERICAL_ZERO));
+
+            // Unit electrical translation
+            x.setZero();
+            x(Eigen::seq(numMechanicalDofs, Eigen::indexing::last)).setOnes();
+
+            REQUIRE((x.transpose() * K * x).isZero(NUMERICAL_ZERO));
+            REQUIRE((x.transpose() * F).isZero(NUMERICAL_ZERO));
+        }
+    }
+
+    SECTION("Invalid element - degenerate element") {
+        nodes = 0.0 * nodes;
+
+        REQUIRE_THROWS_AS(Kernel::lhs(material, nodes), std::invalid_argument);
+        REQUIRE_THROWS_AS(Kernel::rhs(material, nodes), std::invalid_argument);
+    }
+
+    SECTION("Invalid element - reverse node ordering") {
+        nodes = nodes.rowwise().reverse();
+
+        REQUIRE_THROWS_AS(Kernel::lhs(material, nodes), std::invalid_argument);
+        REQUIRE_THROWS_AS(Kernel::rhs(material, nodes), std::invalid_argument);
+    }
+}
+
+TEMPLATE_LIST_TEST_CASE("monad::fem::electrical::LinearPiezoelectricKernel (3d): Test lhs/rhs", "[monad]", Types3d) {
+    using Kernel = LinearPiezoelectricKernel<TestType>;
+    using Material = typename Kernel::Material;
+    using StiffnessTensor = typename Material::StiffnessTensor;
+    using PermittivityTensor = typename Material::PermittivityTensor;
+    using CouplingTensor = typename Material::CouplingTensor;
+
+    StiffnessTensor c = StiffnessTensor::Random();
+    // Make PSD
+    c = c.transpose() * c;
+    // Make PD
+    c += StiffnessTensor::Identity();
+
+    PermittivityTensor epsilon = PermittivityTensor::Random();
+    // Make PSD
+    epsilon = epsilon.transpose() * epsilon;
+    // Make PD
+    epsilon += PermittivityTensor::Identity();
+
+    const CouplingTensor d = CouplingTensor::Constant(0.01);
+
+    const Material material(c, epsilon, d);
+
+    auto nodes = TestType::localNodes();
+
+    SECTION("No errors") {
+        const auto K = Kernel::lhs(material, nodes);
+
+        REQUIRE(isSymmetric(K));
+        REQUIRE(!isPSD(K));
+
+        SECTION("xᵀKx=xᵀF=0 for rigid body transformations") {
+            using FieldVector = Eigen::Vector<double, Kernel::NumDofs>;
+            using MechanicalKernel = typename Kernel::MechanicalKernel;
+
+            const auto F = Kernel::rhs(material, nodes);
+
+            FieldVector x;
+
+            const int numMechanicalDofs = MechanicalKernel::NumDofs;
+
+            // Unit x translation
+            x.setZero();
+            x(Eigen::seq(0, numMechanicalDofs - 1, 3)).setOnes();
+
+            REQUIRE((x.transpose() * K * x).isZero(NUMERICAL_ZERO));
+            REQUIRE((x.transpose() * F).isZero(NUMERICAL_ZERO));
+
+            // Unit y translation
+            x.setZero();
+            x(Eigen::seq(1, numMechanicalDofs - 1, 3)).setOnes();
+
+            REQUIRE((x.transpose() * K * x).isZero(NUMERICAL_ZERO));
+            REQUIRE((x.transpose() * F).isZero(NUMERICAL_ZERO));
+
+            // Unit z translation
+            x.setZero();
+            x(Eigen::seq(2, numMechanicalDofs - 1, 3)).setOnes();
+
+            REQUIRE((x.transpose() * K * x).isZero(NUMERICAL_ZERO));
+            REQUIRE((x.transpose() * F).isZero(NUMERICAL_ZERO));
+
+            // 90° ccw rotation in xy-plane
+            x.setZero();
+            x(Eigen::seq(0, numMechanicalDofs - 1, 3)) = -nodes.col(1);
+            x(Eigen::seq(1, numMechanicalDofs - 1, 3)) = nodes.col(0);
+
+            REQUIRE((x.transpose() * K * x).isZero(NUMERICAL_ZERO));
+            REQUIRE((x.transpose() * F).isZero(NUMERICAL_ZERO));
+
+            // 90° ccw rotation in xz-plane
+            x.setZero();
+            x(Eigen::seq(0, numMechanicalDofs - 1, 3)) = nodes.col(2);
+            x(Eigen::seq(2, numMechanicalDofs - 1, 3)) = -nodes.col(0);
+
+            REQUIRE((x.transpose() * K * x).isZero(NUMERICAL_ZERO));
+            REQUIRE((x.transpose() * F).isZero(NUMERICAL_ZERO));
+
+            // 90° ccw rotation in yz-plane
+            x.setZero();
+            x(Eigen::seq(1, numMechanicalDofs - 1, 3)) = -nodes.col(2);
+            x(Eigen::seq(2, numMechanicalDofs - 1, 3)) = nodes.col(1);
+
+            REQUIRE((x.transpose() * K * x).isZero(NUMERICAL_ZERO));
+            REQUIRE((x.transpose() * F).isZero(NUMERICAL_ZERO));
+
+            // Unit electrical translation
+            x.setZero();
+            x(Eigen::seq(numMechanicalDofs, Eigen::indexing::last)).setOnes();
+
+            REQUIRE((x.transpose() * K * x).isZero(NUMERICAL_ZERO));
+            REQUIRE((x.transpose() * F).isZero(NUMERICAL_ZERO));
+        }
+    }
+
+    SECTION("Invalid element - degenerate element") {
+        nodes = 0.0 * nodes;
+
+        REQUIRE_THROWS_AS(Kernel::lhs(material, nodes), std::invalid_argument);
+        REQUIRE_THROWS_AS(Kernel::rhs(material, nodes), std::invalid_argument);
+    }
+
+    SECTION("Invalid element - reverse node ordering") {
+        nodes = nodes.rowwise().reverse();
+
+        REQUIRE_THROWS_AS(Kernel::lhs(material, nodes), std::invalid_argument);
+        REQUIRE_THROWS_AS(Kernel::rhs(material, nodes), std::invalid_argument);
+    }
+}
diff --git a/tests/fem/kernel/scalar/linear_scalar_diffusion_kernel_test.cpp b/tests/fem/kernel/scalar/linear_scalar_diffusion_kernel_test.cpp
index a48b722..6c1b796 100644
--- a/tests/fem/kernel/scalar/linear_scalar_diffusion_kernel_test.cpp
+++ b/tests/fem/kernel/scalar/linear_scalar_diffusion_kernel_test.cpp
@@ -15,6 +15,7 @@ using namespace monad;
 using namespace monad::fem;
 using namespace monad::fem::scalar;
 using namespace monad::detail;
+using namespace monad::material;
 
 using Types = std::tuple<
     LinearScalarDiffusiveKernel<Quad4, GradientConvention::Negative>,
diff --git a/tests/fem/operator/mechanical/linear_elastic_matrix_free_operator_test.cpp b/tests/fem/operator/mechanical/linear_elastic_dof_traits_test.cpp
similarity index 64%
rename from tests/fem/operator/mechanical/linear_elastic_matrix_free_operator_test.cpp
rename to tests/fem/operator/mechanical/linear_elastic_dof_traits_test.cpp
index cedaa94..7e0b0be 100644
--- a/tests/fem/operator/mechanical/linear_elastic_matrix_free_operator_test.cpp
+++ b/tests/fem/operator/mechanical/linear_elastic_dof_traits_test.cpp
@@ -1,25 +1,25 @@
 #include <tuple>
 #include <catch2/catch_template_test_macros.hpp>
-#include "monad/grid/quad4_grid.hpp"
-#include "monad/grid/quad8_grid.hpp"
-#include "monad/grid/hex8_grid.hpp"
-#include "monad/grid/hex20_grid.hpp"
+#include "monad/grid/grid_aliases.hpp"
+#include "monad/field/density_field.hpp"
 #include "monad/fem/kernel/mechanical/linear_elastic_kernel.hpp"
+#include "monad/fem/operator/mechanical/linear_elastic_dof_traits.hpp"
 #include "monad/fem/operator/matrix_free_operator.hpp"
-#include "monad/fem/operator/mechanical/linear_elastic_matrix_free_operator_traits.hpp"
 
 using namespace monad;
+using namespace monad::field;
 using namespace monad::fem;
 using namespace monad::fem::mechanical;
 
 using Types = std::tuple<Quad4Grid, Quad8Grid, Hex8Grid, Hex20Grid>;
 
-TEMPLATE_LIST_TEST_CASE("monad::fem::mechanical::LinearElasticMatrixFreeOperator: Test isSymmetric/isPSD", "[monad]", Types) {
+TEMPLATE_LIST_TEST_CASE("monad::fem::mechanical: Test LinearElasticDofTraits", "[monad]", Types) {
     using Resolution = typename TestType::Resolution;
     using Size = typename TestType::Size;
     using Element = typename TestType::Element;
     using Kernel = LinearElasticKernel<Element>;
-    using Operator = MatrixFreeOperator<TestType, Element, LinearElasticMatrixFreeOperatorTraits<Element::Dim>>;
+    using Traits = LinearElasticDofTraits<Element::Dim>;
+    using Operator = MatrixFreeOperator<TestType, Traits>;
     using Material = typename Kernel::Material;
     using StiffnessTensor = typename Material::MaterialTensor;
 
@@ -29,8 +29,10 @@ TEMPLATE_LIST_TEST_CASE("monad::fem::mechanical::LinearElasticMatrixFreeOperator
     Size size;
     size.fill(0.5);
 
-    TestType grid(resolution, size);
-    grid.setDensitiesRandom(1234);
+    const TestType grid(resolution, size);
+
+    DensityField<Element::Dim> densityField(resolution);
+    densityField.setRandom(1234);
 
     StiffnessTensor C = StiffnessTensor::Random();
     // Make PSD
@@ -43,7 +45,7 @@ TEMPLATE_LIST_TEST_CASE("monad::fem::mechanical::LinearElasticMatrixFreeOperator
     const auto nodes = grid.elementNodes(0);
     const auto elementKReference = Kernel::lhs(material, nodes);
 
-    const Operator K(grid, elementKReference);
+    const Operator K(grid, densityField, elementKReference);
 
     REQUIRE(K.isSymmetric());
     REQUIRE(K.isPSD());
diff --git a/tests/fem/operator/multiphysics/linear_piezoelectric_dof_traits_test.cpp b/tests/fem/operator/multiphysics/linear_piezoelectric_dof_traits_test.cpp
new file mode 100644
index 0000000..b898d7d
--- /dev/null
+++ b/tests/fem/operator/multiphysics/linear_piezoelectric_dof_traits_test.cpp
@@ -0,0 +1,62 @@
+#include <tuple>
+#include <catch2/catch_template_test_macros.hpp>
+#include "monad/grid/grid_aliases.hpp"
+#include "monad/field/density_field.hpp"
+#include "monad/fem/kernel/multiphysics/linear_piezoelectric_kernel.hpp"
+#include "monad/fem/operator/multiphysics/linear_piezoelectric_dof_traits.hpp"
+#include "monad/fem/operator/matrix_free_operator.hpp"
+
+using namespace monad;
+using namespace monad::field;
+using namespace monad::fem;
+using namespace monad::fem::multiphysics;
+
+using Types = std::tuple<Quad4Grid, Quad8Grid, Hex8Grid, Hex20Grid>;
+
+TEMPLATE_LIST_TEST_CASE("monad::fem::multiphysics: Test LinearPiezoelectricDofTraits", "[monad]", Types) {
+    using Resolution = typename TestType::Resolution;
+    using Size = typename TestType::Size;
+    using Element = typename TestType::Element;
+    using Kernel = LinearPiezoelectricKernel<Element>;
+    using Traits = LinearPiezoelectricDofTraits<Element::Dim>;
+    using Operator = MatrixFreeOperator<TestType, Traits>;
+    using Material = typename Kernel::Material;
+    using StiffnessTensor = typename Material::StiffnessTensor;
+    using PermittivityTensor = typename Material::PermittivityTensor;
+    using CouplingTensor = typename Material::CouplingTensor;
+
+    Resolution resolution;
+    resolution.fill(2);
+
+    Size size;
+    size.fill(0.5);
+
+    const TestType grid(resolution, size);
+
+    DensityField<Element::Dim> densityField(resolution);
+    densityField.setRandom(1234);
+
+    StiffnessTensor c = StiffnessTensor::Random();
+    // Make PSD
+    c = c.transpose() * c;
+    // Make PD
+    c += StiffnessTensor::Identity();
+
+    PermittivityTensor epsilon = PermittivityTensor::Random();
+    // Make PSD
+    epsilon = epsilon.transpose() * epsilon;
+    // Make PD
+    epsilon += PermittivityTensor::Identity();
+
+    const CouplingTensor d = CouplingTensor::Constant(0.01);
+
+    const Material material(c, epsilon, d);
+
+    const auto nodes = grid.elementNodes(0);
+    const auto elementKReference = Kernel::lhs(material, nodes);
+
+    const Operator K(grid, densityField, elementKReference);
+
+    REQUIRE(K.isSymmetric());
+    REQUIRE(!K.isPSD());
+}
diff --git a/tests/fem/operator/multiphysics/linear_piezoelectric_matrix_free_operator_test.cpp b/tests/fem/operator/multiphysics/linear_piezoelectric_matrix_free_operator_test.cpp
deleted file mode 100644
index 7d900af..0000000
--- a/tests/fem/operator/multiphysics/linear_piezoelectric_matrix_free_operator_test.cpp
+++ /dev/null
@@ -1,61 +0,0 @@
-#include <tuple>
-#include <catch2/catch_template_test_macros.hpp>
-#include "monad/grid/quad4_grid.hpp"
-#include "monad/grid/quad8_grid.hpp"
-#include "monad/grid/hex8_grid.hpp"
-#include "monad/grid/hex20_grid.hpp"
-#include "monad/fem/kernel/multiphysics/linear_piezoelectric_kernel.hpp"
-#include "monad/fem/operator/matrix_free_operator.hpp"
-#include "monad/fem/operator/multiphysics/linear_piezoelectric_matrix_free_operator_traits.hpp"
-#include "monad/material/mechanical/linear_elastic_material_2d.hpp"
-#include "monad/material/mechanical/linear_elastic_material.hpp"
-#include "monad/material/transport/linear_transport_material.hpp"
-
-using namespace monad;
-using namespace monad::fem;
-using namespace monad::fem::multiphysics;
-
-using Types = std::tuple<Quad4Grid, Quad8Grid, Hex8Grid, Hex20Grid>;
-
-TEMPLATE_LIST_TEST_CASE("monad::fem::multiphysics::LinearPiezoelectricMatrixFreeOperator: Test isSymmetric/isPSD", "[monad]", Types) {
-    using Resolution = typename TestType::Resolution;
-    using Size = typename TestType::Size;
-    using Element = typename TestType::Element;
-    using Kernel = LinearPiezoelectricKernel<Element>;
-    using Operator = MatrixFreeOperator<TestType, Element, LinearPiezoelectricMatrixFreeOperatorTraits<Element::Dim>>;
-    using MechanicalMaterial = LinearElasticMaterial<Element::Dim>;
-    using ElectricalMaterial = LinearTransportMaterial<Element::Dim>;
-    using Material = typename Kernel::Material;
-    using StiffnessTensor = typename MechanicalMaterial::MaterialTensor;
-    using CouplingTensor = typename Material::CouplingTensor;
-
-    Resolution resolution;
-    resolution.fill(2);
-
-    Size size;
-    size.fill(0.5);
-
-    TestType grid(resolution, size);
-    grid.setDensitiesRandom(1234);
-
-    StiffnessTensor c = StiffnessTensor::Random();
-    // Make PSD
-    c = c.transpose() * c;
-    // Make PD
-    c += StiffnessTensor::Identity();
-
-    const MechanicalMaterial elasticMaterial(c);
-    const ElectricalMaterial dielectricMaterial(2.1);
-
-    const CouplingTensor d = 0.1 * CouplingTensor::Random();
-
-    const Material material(elasticMaterial, dielectricMaterial, d);
-
-    const auto nodes = grid.elementNodes(0);
-    const auto elementKReference = Kernel::lhs(material, nodes);
-
-    const Operator K(grid, elementKReference);
-
-    REQUIRE(K.isSymmetric());
-    REQUIRE(!K.isPSD());
-}
diff --git a/tests/fem/operator/scalar/linear_scalar_diffusion_matrix_free_operator_traits_test.cpp b/tests/fem/operator/scalar/linear_scalar_diffusion_dof_traits_test.cpp
similarity index 71%
rename from tests/fem/operator/scalar/linear_scalar_diffusion_matrix_free_operator_traits_test.cpp
rename to tests/fem/operator/scalar/linear_scalar_diffusion_dof_traits_test.cpp
index 8a14a71..ce69051 100644
--- a/tests/fem/operator/scalar/linear_scalar_diffusion_matrix_free_operator_traits_test.cpp
+++ b/tests/fem/operator/scalar/linear_scalar_diffusion_dof_traits_test.cpp
@@ -1,14 +1,13 @@
 #include <tuple>
 #include <catch2/catch_template_test_macros.hpp>
-#include "monad/grid/quad4_grid.hpp"
-#include "monad/grid/quad8_grid.hpp"
-#include "monad/grid/hex8_grid.hpp"
-#include "monad/grid/hex20_grid.hpp"
+#include "monad/grid/grid_aliases.hpp"
+#include "monad/field/density_field.hpp"
 #include "monad/fem/kernel/scalar/linear_scalar_diffusive_kernel.hpp"
 #include "monad/fem/operator/matrix_free_operator.hpp"
-#include "monad/fem/operator/scalar/linear_scalar_diffusive_matrix_free_operator_traits.hpp"
+#include "monad/fem/operator/scalar/linear_scalar_diffusive_dof_traits.hpp"
 
 using namespace monad;
+using namespace monad::field;
 using namespace monad::fem;
 using namespace monad::fem::scalar;
 
@@ -29,13 +28,14 @@ using Types = std::tuple<
     TypePair<Hex20Grid, GradientConvention::Positive>
 >;
 
-TEMPLATE_LIST_TEST_CASE("monad::fem::scalar::LinearScalarDiffusiveMatrixFreeOperator: Test isSymmetric/isPSD", "[monad]", Types) {
+TEMPLATE_LIST_TEST_CASE("monad::fem::scalar: Test LinearScalarDiffusiveDofTraits", "[monad]", Types) {
     using Grid = typename TestType::Grid;
     using Resolution = typename Grid::Resolution;
     using Size = typename Grid::Size;
     using Element = typename Grid::Element;
     using Kernel = LinearScalarDiffusiveKernel<Element, TestType::Convention>;
-    using Operator = MatrixFreeOperator<Grid, Element, LinearScalarDiffusiveMatrixFreeOperatorTraits>;
+    using Traits = LinearScalarDiffusiveDofTraits;
+    using Operator = MatrixFreeOperator<Grid, Traits>;
     using Material = typename Kernel::Material;
 
     Resolution resolution;
@@ -44,15 +44,17 @@ TEMPLATE_LIST_TEST_CASE("monad::fem::scalar::LinearScalarDiffusiveMatrixFreeOper
     Size size;
     size.fill(0.5);
 
-    Grid grid(resolution, size);
-    grid.setDensitiesRandom(1234);
+    const Grid grid(resolution, size);
+
+    DensityField<Element::Dim> densityField(resolution);
+    densityField.setRandom(1234);
 
     const Material material(2.1);
 
     const auto nodes = grid.elementNodes(0);
     const auto elementKReference = Kernel::lhs(material, nodes);
 
-    const Operator K(grid, elementKReference);
+    const Operator K(grid, densityField, elementKReference);
 
     REQUIRE(K.isSymmetric());
     REQUIRE(K.isPSD());
diff --git a/tests/grid/csv/bad1.csv b/tests/field/csv/bad1.csv
similarity index 100%
rename from tests/grid/csv/bad1.csv
rename to tests/field/csv/bad1.csv
diff --git a/tests/grid/csv/bad2.csv b/tests/field/csv/bad2.csv
similarity index 100%
rename from tests/grid/csv/bad2.csv
rename to tests/field/csv/bad2.csv
diff --git a/tests/grid/csv/bad3.csv b/tests/field/csv/bad3.csv
similarity index 100%
rename from tests/grid/csv/bad3.csv
rename to tests/field/csv/bad3.csv
diff --git a/tests/grid/csv/bad4.csv b/tests/field/csv/bad4.csv
similarity index 100%
rename from tests/grid/csv/bad4.csv
rename to tests/field/csv/bad4.csv
diff --git a/tests/grid/csv/bad5.csv b/tests/field/csv/bad5.csv
similarity index 100%
rename from tests/grid/csv/bad5.csv
rename to tests/field/csv/bad5.csv
diff --git a/tests/grid/csv/bad6.csv b/tests/field/csv/bad6.csv
similarity index 100%
rename from tests/grid/csv/bad6.csv
rename to tests/field/csv/bad6.csv
diff --git a/tests/grid/csv/good.csv b/tests/field/csv/good.csv
similarity index 100%
rename from tests/grid/csv/good.csv
rename to tests/field/csv/good.csv
diff --git a/tests/field/density_field_test.cpp b/tests/field/density_field_test.cpp
new file mode 100644
index 0000000..fda5957
--- /dev/null
+++ b/tests/field/density_field_test.cpp
@@ -0,0 +1,282 @@
+#include <tuple>
+#include <stdexcept>
+#include <cmath>
+#include <cstddef>
+#include <catch2/catch_template_test_macros.hpp>
+#include <catch2/matchers/catch_matchers_vector.hpp>
+#include <catch2/matchers/catch_matchers_floating_point.hpp>
+#include "monad/field/field_aliases.hpp"
+#include "monad/detail/constants.hpp"
+
+using namespace monad;
+
+using Types = std::tuple<DensityField2d, DensityField3d>;
+
+TEMPLATE_LIST_TEST_CASE("monad::DensityField: Test initalization", "[monad]", Types) {
+    using Resolution = typename TestType::Resolution;
+
+    Resolution resolution;
+    resolution.fill(2);
+
+    SECTION("Invalid resolution") {
+        for (std::size_t i = 0; i < TestType::Dim; ++i) {
+            Resolution resolutionInvalid = resolution;
+            resolutionInvalid[i] = 0;
+
+            REQUIRE_THROWS_AS(TestType(resolutionInvalid), std::invalid_argument);
+        }
+    }
+}
+
+TEMPLATE_LIST_TEST_CASE("monad::DensityField: Test resolution", "[monad]", Types) {
+    using Resolution = typename TestType::Resolution;
+
+    Resolution resolution;
+    resolution.fill(2);
+
+    const TestType densityField(resolution);
+
+    const Resolution actual = densityField.resolution();
+
+    REQUIRE(actual == resolution);
+}
+
+TEMPLATE_LIST_TEST_CASE("monad::DensityField: Test numElements", "[monad]", Types) {
+    using Resolution = typename TestType::Resolution;
+
+    Resolution resolution;
+    resolution.fill(2);
+
+    const TestType densityField(resolution);
+
+    const std::size_t actual = static_cast<std::size_t>(std::pow(2.0, TestType::Dim));
+    const std::size_t expected = densityField.numElements();
+
+    REQUIRE(actual == expected);
+}
+
+TEMPLATE_LIST_TEST_CASE("monad::DensityField: Test densities", "[monad]", Types) {
+    using Resolution = typename TestType::Resolution;
+    using DensityList = typename TestType::DensityList;
+
+    Resolution resolution;
+    resolution.fill(2);
+
+    const TestType densityField(resolution);
+
+    const DensityList actual(densityField.numElements(), NUMERICAL_ZERO);
+    const DensityList expected = densityField.densities();
+
+    REQUIRE_THAT(actual, Catch::Matchers::Approx(expected).margin(NUMERICAL_ZERO));
+}
+
+TEMPLATE_LIST_TEST_CASE("monad::DensityField: Test getDensity/setDensity", "[monad]", Types) {
+    using Resolution = typename TestType::Resolution;
+
+    Resolution resolution;
+    resolution.fill(2);
+
+    TestType densityField(resolution);
+
+    SECTION("No errors") {
+        REQUIRE_THAT(densityField.getDensity(1), Catch::Matchers::WithinAbs(NUMERICAL_ZERO, NUMERICAL_ZERO));
+
+        densityField.setDensity(1, 0.4);
+
+        REQUIRE_THAT(densityField.getDensity(1), Catch::Matchers::WithinAbs(0.4, NUMERICAL_ZERO));
+    }
+
+    SECTION("Invalid index") {
+        const std::size_t badIndex = densityField.numElements();
+
+        REQUIRE_THROWS_AS(densityField.getDensity(badIndex), std::out_of_range);
+        REQUIRE_THROWS_AS(densityField.setDensity(badIndex, 0.4), std::out_of_range);
+    }
+
+    SECTION("Invalid density") {
+        REQUIRE_THROWS_AS(densityField.setDensity(0, -0.1), std::invalid_argument);
+        REQUIRE_THROWS_AS(densityField.setDensity(0, 1.1), std::invalid_argument);
+    }
+}
+
+TEMPLATE_LIST_TEST_CASE("monad::DensityField: Test setDensities", "[monad]", Types) {
+    using Resolution = typename TestType::Resolution;
+    using DensityList = typename TestType::DensityList;
+
+    Resolution resolution;
+    resolution.fill(2);
+
+    TestType densityField(resolution);
+
+    DensityList densities(densityField.numElements(), 0.4);
+
+    SECTION("No errors") {
+        densityField.setDensities(densities);
+
+        REQUIRE_THAT(densityField.densities(), Catch::Matchers::Approx(densities).margin(NUMERICAL_ZERO));
+    }
+
+    SECTION("Invalid density") {
+        densities[0] = -0.1;
+        
+        REQUIRE_THROWS_AS(densityField.setDensities(densities), std::invalid_argument);
+        
+        densities[0] = 1.1;
+        
+        REQUIRE_THROWS_AS(densityField.setDensities(densities), std::invalid_argument);
+    }
+
+    SECTION("Invalid size") {
+        densities.push_back(0.4);
+
+        REQUIRE_THROWS_AS(densityField.setDensities(densities), std::invalid_argument);
+    }
+}
+
+TEMPLATE_LIST_TEST_CASE("monad::DensityField: Test setConstant", "[monad]", Types) {
+    using Resolution = typename TestType::Resolution;
+    using DensityList = typename TestType::DensityList;
+
+    Resolution resolution;
+    resolution.fill(2);
+
+    TestType densityField(resolution);
+
+    SECTION("No errors") {
+        DensityList expected(densityField.numElements(), 0.4);
+
+        densityField.setConstant(0.4);
+
+        REQUIRE_THAT(densityField.densities(), Catch::Matchers::Approx(expected).margin(NUMERICAL_ZERO));
+    }
+
+    SECTION("Invalid density") {        
+        REQUIRE_THROWS_AS(densityField.setConstant(-0.1), std::invalid_argument);  
+        REQUIRE_THROWS_AS(densityField.setConstant(1.1), std::invalid_argument);
+    }
+}
+
+TEMPLATE_LIST_TEST_CASE("monad::DensityField: Test setZeros", "[monad]", Types) {
+    using Resolution = typename TestType::Resolution;
+
+    Resolution resolution;
+    resolution.fill(2);
+
+    TestType densityField1(resolution);
+    TestType densityField2(resolution);
+
+    densityField1.setConstant(0.0);
+    densityField2.setZeros();
+
+    REQUIRE(densityField1 == densityField2);
+}
+
+TEMPLATE_LIST_TEST_CASE("monad::DensityField: Test setOnes", "[monad]", Types) {
+    using Resolution = typename TestType::Resolution;
+
+    Resolution resolution;
+    resolution.fill(2);
+
+    TestType densityField1(resolution);
+    TestType densityField2(resolution);
+
+    densityField1.setConstant(1.0);
+    densityField2.setOnes();
+
+    REQUIRE(densityField1 == densityField2);
+}
+
+TEMPLATE_LIST_TEST_CASE("monad::DensityField: Test setRandom", "[monad]", Types) {
+    using Resolution = typename TestType::Resolution;
+    using DensityList = typename TestType::DensityList;
+    
+    Resolution resolution;
+    resolution.fill(2);
+
+    TestType densityField(resolution);
+
+    densityField.setRandom(1234);
+
+    if constexpr (TestType::Dim == 2) {
+        DensityList expected{
+            0.19151945020806033,
+            0.49766366637723142,
+            0.62210876648829061,
+            0.8178384429351051,
+        };
+
+        REQUIRE_THAT(densityField.densities(), Catch::Matchers::Approx(expected).margin(NUMERICAL_ZERO));
+    }
+
+    else {
+        DensityList expected{
+            0.19151945020806033,
+            0.49766366637723142,
+            0.62210876648829061,
+            0.8178384429351051,
+            0.43772773734240972,
+            0.61211189362502472,
+            0.78535858373748102,
+            0.77135991905149071
+        };
+
+        REQUIRE_THAT(densityField.densities(), Catch::Matchers::Approx(expected).margin(NUMERICAL_ZERO));
+    }
+}
+
+TEMPLATE_LIST_TEST_CASE("monad::DensityField: Test translate", "[monad]", Types) {
+    using Resolution = typename TestType::Resolution;
+    
+    Resolution resolution;
+    resolution.fill(2);
+
+    TestType densityField(resolution);
+
+    densityField.setDensity(1, 0.4);
+
+    if constexpr (TestType::Dim == 2) {
+        // Pre-shift: Point=(1,0) → Index=1
+        // Post-shift: Point=(0,0) → Index=0
+        densityField.translate({1, 2});
+
+        REQUIRE_THAT(densityField.getDensity(0), Catch::Matchers::WithinAbs(0.4, NUMERICAL_ZERO));
+    }
+
+    else {
+        // Pre-shift: Point=(1,0,0) → Index=1
+        // Post-shift: Point=(0,0,1) → Index=4
+        densityField.translate({1, 2, 3});
+
+        REQUIRE_THAT(densityField.getDensity(4), Catch::Matchers::WithinAbs(0.4, NUMERICAL_ZERO));
+    }
+}
+
+TEMPLATE_LIST_TEST_CASE("monad::DensityField: Test operator==", "[monad]", Types) {
+    using Resolution = typename TestType::Resolution;
+
+    Resolution resolution;
+    resolution.fill(2);
+
+    TestType densityField1(resolution);
+    densityField1.setConstant(0.4);
+
+    TestType densityField2(resolution);
+    densityField2.setConstant(0.4);
+
+    REQUIRE(densityField1 == densityField2);
+}
+
+TEMPLATE_LIST_TEST_CASE("monad::DensityField: Test operator!=", "[monad]", Types) {
+    using Resolution = typename TestType::Resolution;
+
+    Resolution resolution;
+    resolution.fill(2);
+
+    TestType densityField1(resolution);
+    densityField1.setConstant(0.4);
+
+    TestType densityField2(resolution);
+    densityField2.setConstant(0.5);
+
+    REQUIRE(densityField1 != densityField2);
+}
diff --git a/tests/field/make_density_field_from_csv_test.cpp b/tests/field/make_density_field_from_csv_test.cpp
new file mode 100644
index 0000000..d7407f8
--- /dev/null
+++ b/tests/field/make_density_field_from_csv_test.cpp
@@ -0,0 +1,66 @@
+#include <tuple>
+#include <stdexcept>
+#include <filesystem>
+#include <catch2/catch_test_macros.hpp>
+#include <catch2/matchers/catch_matchers_vector.hpp>
+#include "monad/field/field_aliases.hpp"
+#include "monad/field/make_density_field_from_csv.hpp"
+
+using namespace monad;
+
+TEST_CASE("monad: Test makeDensityFieldFromCsv", "[monad]") {
+    using DensityList = typename DensityField2d::DensityList;
+
+    DensityField2d densityField({2, 3});
+
+    const auto folder = std::filesystem::path(__FILE__).parent_path() / "csv";
+
+    SECTION("No errors") {
+        const auto file = folder / "good.csv";
+
+        auto densityField = makeDensityFieldFromCsv(file.string());
+
+        const DensityList expected {
+            0.1,
+            0.2,
+            0.3,
+            0.4,
+            0.5,
+            0.6
+        };
+
+        REQUIRE_THAT(densityField.densities(), Catch::Matchers::Approx(expected).margin(NUMERICAL_ZERO));
+    }
+
+    SECTION("Invalid csv data") {
+        // data < 0
+        auto file = folder / "bad1.csv";
+
+        REQUIRE_THROWS_AS(makeDensityFieldFromCsv(file.string()), std::runtime_error);
+
+        // data > 1
+        file = folder / "bad2.csv";
+
+        REQUIRE_THROWS_AS(makeDensityFieldFromCsv(file.string()), std::runtime_error);
+
+        // not enough data
+        file = folder / "bad3.csv";
+
+        REQUIRE_THROWS_AS(makeDensityFieldFromCsv(file.string()), std::runtime_error);
+
+        // too much data
+        file = folder / "bad4.csv";
+
+        REQUIRE_THROWS_AS(makeDensityFieldFromCsv(file.string()), std::runtime_error);
+
+        // no data
+        file = folder / "bad5.csv";
+
+        REQUIRE_THROWS_AS(makeDensityFieldFromCsv(file.string()), std::runtime_error);
+
+        // non-numeric data
+        file = folder / "bad6.csv";
+
+        REQUIRE_THROWS_AS(makeDensityFieldFromCsv(file.string()), std::runtime_error);
+    }
+}
diff --git a/tests/field/make_density_field_from_function_test.cpp b/tests/field/make_density_field_from_function_test.cpp
new file mode 100644
index 0000000..2487e9e
--- /dev/null
+++ b/tests/field/make_density_field_from_function_test.cpp
@@ -0,0 +1,49 @@
+#include <tuple>
+#include <stdexcept>
+#include <catch2/catch_template_test_macros.hpp>
+#include <catch2/matchers/catch_matchers_floating_point.hpp>
+#include "monad/grid/grid_aliases.hpp"
+#include "monad/field/make_density_field_from_function.hpp"
+
+using namespace monad;
+
+using Types = std::tuple<Quad4Grid, Quad8Grid, Hex8Grid, Hex20Grid>;
+
+TEMPLATE_LIST_TEST_CASE("monad: Test makeDensityFieldFromFunction", "[monad]", Types) {
+    using Resolution = typename TestType::Resolution;
+    using Size = typename TestType::Size;
+    using Point = typename TestType::Point;
+
+    Resolution resolution;
+    resolution.fill(2);
+
+    Size size;
+    size.fill(0.4);
+
+    const TestType grid(resolution, size);
+
+    SECTION("No errors") {
+        // f(x)=0.1Σᵢxᵢ
+        auto f = [](const Point &x) -> double {
+            return 0.1 * x.sum();
+        };
+
+        const auto densityField = makeDensityFieldFromFunction(grid, f);
+
+        // The average of a linear function over a convex region equals its value at the centroid
+        const auto nodes = grid.elementNodes(1);
+        const Point centroid = nodes.colwise().mean();
+        const double expected = f(centroid);
+
+        REQUIRE_THAT(densityField.getDensity(1), Catch::Matchers::WithinAbs(expected, NUMERICAL_ZERO));
+    }
+
+    SECTION("Bad function") {
+        // f(x)=exp(Σᵢxᵢ)
+        auto f = [](const Point &x) -> double {
+            return std::exp(x.sum());
+        };
+
+        REQUIRE_THROWS_AS(makeDensityFieldFromFunction(grid, f), std::invalid_argument);
+    }
+}
diff --git a/tests/grid/grid_2d_test.cpp b/tests/grid/grid_2d_test.cpp
deleted file mode 100644
index 841db9a..0000000
--- a/tests/grid/grid_2d_test.cpp
+++ /dev/null
@@ -1,620 +0,0 @@
-#include <tuple>
-#include <stdexcept>
-#include <cmath>
-#include <cstddef>
-#include <filesystem>
-#include <catch2/catch_test_macros.hpp>
-#include <catch2/catch_template_test_macros.hpp>
-#include <catch2/matchers/catch_matchers_vector.hpp>
-#include <catch2/matchers/catch_matchers_floating_point.hpp>
-#include "monad/grid/quad4_grid.hpp"
-#include "monad/grid/quad8_grid.hpp"
-#include "monad/detail/constants.hpp"
-#include "test_common.hpp"
-
-using namespace monad;
-using namespace monad::testing;
-
-using Types = std::tuple<Quad4Grid, Quad8Grid>;
-
-TEMPLATE_LIST_TEST_CASE("monad::Grid2d: Test initalization", "[monad]", Types) {
-    SECTION("Invalid resolution") {
-        REQUIRE_THROWS_AS(TestType({0, 3}, {0.5, 1.5}), std::invalid_argument);
-        REQUIRE_THROWS_AS(TestType({2, 0}, {0.5, 1.5}), std::invalid_argument);
-    }
-
-    SECTION("Invalid size") {
-        REQUIRE_THROWS_AS(TestType({2, 3}, {0.0, 1.5}), std::invalid_argument);
-        REQUIRE_THROWS_AS(TestType({2, 3}, {-0.5, 1.5}), std::invalid_argument);
-        REQUIRE_THROWS_AS(TestType({2, 3}, {0.5, 0.0}), std::invalid_argument);
-        REQUIRE_THROWS_AS(TestType({2, 3}, {0.5, -1.5}), std::invalid_argument);
-    }
-}
-
-TEMPLATE_LIST_TEST_CASE("monad::Grid2d: Test resolution", "[monad]", Types) {
-    using Resolution = typename TestType::Resolution;
-
-    const TestType grid({2, 3}, {0.5, 1.5});
-
-    const Resolution expected{2, 3};
-
-    REQUIRE(grid.resolution() == expected);
-}
-
-TEMPLATE_LIST_TEST_CASE("monad::Grid2d: Test size", "[monad]", Types) {
-    using Size = typename TestType::Size;
-
-    const TestType grid({2, 3}, {0.5, 1.5});
-
-    const Size expected{0.5, 1.5};
-
-    REQUIRE(grid.size() == expected);
-}
-
-TEMPLATE_LIST_TEST_CASE("monad::Grid2d: Test densities", "[monad]", Types) {
-    using DensityList = typename TestType::DensityList;
-
-    const TestType grid({2, 3}, {0.5, 1.5});
-
-    const DensityList expected(6, 0.0);
-
-    REQUIRE_THAT(grid.densities(), Catch::Matchers::Approx(expected).margin(NUMERICAL_ZERO));
-}
-
-TEMPLATE_LIST_TEST_CASE("monad::Grid2d: Test numElements", "[monad]", Types) {
-    const TestType grid({2, 3}, {0.5, 1.5});
-
-    REQUIRE(grid.numElements() == 6);
-}
-
-TEST_CASE("monad::Grid2d: Test numNodes", "[monad]") {
-    SECTION("Quad4Grid") {
-        const Quad4Grid grid({2, 3}, {0.5, 1.5});
-
-        REQUIRE(grid.numNodes() == 12);
-    }
-
-    SECTION("Quad8Grid") {
-        const Quad8Grid grid({2, 3}, {0.5, 1.5});
-
-        REQUIRE(grid.numNodes() == 29);
-    }
-}
-
-TEST_CASE("monad::Grid2d: Test numPeriodicNodes", "[monad]") {
-    SECTION("Quad4Grid") {
-        const Quad4Grid grid({2, 3}, {0.5, 1.5});
-
-        REQUIRE(grid.numPeriodicNodes() == 6);
-    }
-
-    SECTION("Quad8Grid") {
-        const Quad8Grid grid({2, 3}, {0.5, 1.5});
-
-        REQUIRE(grid.numPeriodicNodes() == 18);
-    }
-}
-
-TEST_CASE("monad::Grid2d: Test node", "[monad]") {
-    SECTION("Quad4Grid") {
-        const Quad4Grid grid({2, 3}, {0.5, 1.5});
-
-        SECTION("No errors") {
-            const Quad4Grid::Point expected(0.25, 0.0);
-
-            REQUIRE(grid.node(1).isApprox(expected, NUMERICAL_ZERO));
-        }
-
-        SECTION("Invalid index") {
-            REQUIRE_THROWS_AS(grid.node(grid.numNodes()), std::out_of_range);
-        }
-    }
-
-    SECTION("Quad8Grid") {
-        const Quad8Grid grid({2, 3}, {0.5, 1.5});
-
-        SECTION("No errors") {
-            const Quad8Grid::Point expected(0.25, 0.0);
-
-            REQUIRE(grid.node(1).isApprox(expected, NUMERICAL_ZERO));
-        }
-
-        SECTION("Invalid index") {
-            REQUIRE_THROWS_AS(grid.node(grid.numNodes()), std::out_of_range);
-        }
-    }
-}
-
-TEST_CASE("monad::Grid2d: Test nodes", "[monad]") {
-    SECTION("Quad4Grid") {
-        using NodesList = typename Quad4Grid::NodesList;
-
-        const Quad4Grid grid({2, 3}, {0.5, 1.5});
-
-        const auto actual = grid.nodes();
-
-        const NodesList expected {
-            {0.0, 0.0},
-            {0.25, 0.0},
-            {0.5, 0.0},
-            {0.0, 0.5},
-            {0.25, 0.5},
-            {0.5, 0.5},
-            {0.0, 1.0},
-            {0.25, 1.0},
-            {0.5, 1.0},
-            {0.0, 1.5},
-            {0.25, 1.5},
-            {0.5, 1.5}
-        };
-
-        REQUIRE(actual.size() == expected.size());
-
-        for (std::size_t i = 0; i < actual.size(); ++i) {
-            REQUIRE(actual[i].isApprox(expected[i], NUMERICAL_ZERO));
-        }
-    }
-
-    SECTION("Quad8Grid") {
-        using NodesList = typename Quad8Grid::NodesList;
-
-        const Quad8Grid grid({2, 3}, {0.5, 1.5});
-
-        const auto actual = grid.nodes();
-
-        const NodesList expected {
-            {0.0, 0.0},
-            {0.25, 0.0},
-            {0.5, 0.0},
-            {0.0, 0.5},
-            {0.25, 0.5},
-            {0.5, 0.5},
-            {0.0, 1.0},
-            {0.25, 1.0},
-            {0.5, 1.0},
-            {0.0, 1.5},
-            {0.25, 1.5},
-            {0.5, 1.5},
-            {0.125, 0.0},
-            {0.375, 0.0},
-            {0.125, 0.5},
-            {0.375, 0.5},
-            {0.125, 1.0},
-            {0.375, 1.0},
-            {0.125, 1.5},
-            {0.375, 1.5},
-            {0.0, 0.25},
-            {0.25, 0.25},
-            {0.5, 0.25},
-            {0.0, 0.75},
-            {0.25, 0.75},
-            {0.5, 0.75},
-            {0.0, 1.25},
-            {0.25, 1.25},
-            {0.5, 1.25}
-        };
-
-        REQUIRE(actual.size() == expected.size());
-
-        for (std::size_t i = 0; i < actual.size(); ++i) {
-            REQUIRE(actual[i].isApprox(expected[i], NUMERICAL_ZERO));
-        }
-    }
-}
-
-TEST_CASE("monad::Grid2d: Test element", "[monad]") {
-    SECTION("Quad4Grid") {
-        const Quad4Grid grid({2, 3}, {0.5, 1.5});
-
-        SECTION("No errors") {
-            const Quad4Grid::ElementList expected{1, 2, 5, 4};
-
-            REQUIRE(grid.element(1) == expected);
-        }
-
-        SECTION("Invalid index") {
-            REQUIRE_THROWS_AS(grid.element(grid.numElements()), std::out_of_range);
-        }
-    }
-
-    SECTION("Quad8Grid") {
-        const Quad8Grid grid({2, 3}, {0.5, 1.5});
-
-        SECTION("No errors") {
-            const Quad8Grid::ElementList expected{1, 2, 5, 4, 13, 22, 15, 21};
-
-            REQUIRE(grid.element(1) == expected);
-        }
-
-        SECTION("Invalid index") {
-            REQUIRE_THROWS_AS(grid.element(grid.numElements()), std::out_of_range);
-        }
-    }
-}
-
-TEST_CASE("monad::Grid2d: Test elements", "[monad]") {
-    SECTION("Quad4Grid") {
-        const Quad4Grid grid({2, 3}, {0.5, 1.5});
-
-        const Quad4Grid::ElementsList expected {
-            {0, 1, 4, 3},
-            {1, 2, 5, 4},
-            {3, 4, 7, 6},
-            {4, 5, 8, 7},
-            {6, 7, 10, 9},
-            {7, 8, 11, 10}
-        };
-
-        REQUIRE(grid.elements() == expected);
-    }
-
-    SECTION("Quad8Grid") {
-        const Quad8Grid grid({2, 3}, {0.5, 1.5});
-
-        const Quad8Grid::ElementsList expected {
-            {0, 1, 4, 3, 12, 21, 14, 20},
-            {1, 2, 5, 4, 13, 22, 15, 21},
-            {3, 4, 7, 6, 14, 24, 16, 23},
-            {4, 5, 8, 7, 15, 25, 17, 24},
-            {6, 7, 10, 9, 16, 27, 18, 26},
-            {7, 8, 11, 10, 17, 28, 19, 27}
-        };
-
-        REQUIRE(grid.elements() == expected);
-    }
-}
-
-TEST_CASE("monad::Grid2d: Test periodicElement", "[monad]") {
-    SECTION("Quad4Grid") {
-        const Quad4Grid grid({2, 3}, {0.5, 1.5});
-
-        SECTION("No errors") {
-            const Quad4Grid::ElementList expected{1, 0, 2, 3};
-
-            REQUIRE(grid.periodicElement(1) == expected);
-        }
-
-        SECTION("Invalid index") {
-            REQUIRE_THROWS_AS(grid.periodicElement(grid.numElements()), std::out_of_range);
-        }
-    }
-
-    SECTION("Quad8Grid") {
-        const Quad8Grid grid({2, 3}, {0.5, 1.5});
-
-        SECTION("No errors") {
-            const Quad8Grid::ElementList expected{1, 0, 2, 3, 7, 12, 9, 13};
-
-            REQUIRE(grid.periodicElement(1) == expected);
-        }
-
-        SECTION("Invalid index") {
-            REQUIRE_THROWS_AS(grid.periodicElement(grid.numElements()), std::out_of_range);
-        }
-    }
-}
-
-TEST_CASE("monad::Grid2d: Test periodicElements", "[monad]") {
-    SECTION("Quad4Grid") {
-        const Quad4Grid grid({2, 3}, {0.5, 1.5});
-
-        const Quad4Grid::ElementsList expected {
-            {0, 1, 3, 2},
-            {1, 0, 2, 3},
-            {2, 3, 5, 4},
-            {3, 2, 4, 5},
-            {4, 5, 1, 0},
-            {5, 4, 0, 1}
-        };
-
-        REQUIRE(grid.periodicElements() == expected);
-    }
-
-    SECTION("Quad8Grid") {
-        const Quad8Grid grid({2, 3}, {0.5, 1.5});
-
-        const Quad8Grid::ElementsList expected {
-            {0, 1, 3, 2, 6, 13, 8, 12},
-            {1, 0, 2, 3, 7, 12, 9, 13},
-            {2, 3, 5, 4, 8, 15, 10, 14},
-            {3, 2, 4, 5, 9, 14, 11, 15},
-            {4, 5, 1, 0, 10, 17, 6, 16},
-            {5, 4, 0, 1, 11, 16, 7, 17}
-        };
-
-        REQUIRE(grid.periodicElements() == expected);
-    }
-}
-
-TEMPLATE_LIST_TEST_CASE("monad::Grid2d: Test getDensity/setDensity", "[monad]", Types) {
-    TestType grid({2, 3}, {0.5, 1.5});
-
-    SECTION("No errors") {
-        REQUIRE_THAT(grid.getDensity(1), Catch::Matchers::WithinAbs(0.0, NUMERICAL_ZERO));
-
-        grid.setDensity(1, 0.1);
-
-        REQUIRE_THAT(grid.getDensity(1), Catch::Matchers::WithinAbs(0.1, NUMERICAL_ZERO));
-    }
-
-    SECTION("Invalid index") {
-        REQUIRE_THROWS_AS(grid.getDensity(grid.numElements()), std::out_of_range);
-
-        REQUIRE_THROWS_AS(grid.setDensity(grid.numElements(), 0.1), std::out_of_range);
-    }
-
-    SECTION("Invalid density") {
-        REQUIRE_THROWS_AS(grid.setDensity(1, -0.1), std::invalid_argument);
-        REQUIRE_THROWS_AS(grid.setDensity(1, 1.1), std::invalid_argument);
-    }
-}
-
-TEMPLATE_LIST_TEST_CASE("monad::Grid2d: Test setDensities", "[monad]", Types) {
-    using DensityList = typename TestType::DensityList;
-
-    TestType grid({2, 3}, {0.5, 1.5});
-
-    SECTION("No errors") {
-        const DensityList densities(grid.numElements(), 0.5);
-
-        grid.setDensities(densities);
-
-        REQUIRE_THAT(grid.densities(), Catch::Matchers::Approx(densities).margin(NUMERICAL_ZERO));
-    }
-
-    SECTION("Invalid size") {
-        const DensityList densities(grid.numElements() + 1, 0.5);
-
-        REQUIRE_THROWS_AS(grid.setDensities(densities), std::invalid_argument);
-    }
-
-    SECTION("Invalid density") {
-        DensityList densities(grid.numElements(), 0.5);
-        densities[1] *= -1;
-
-        REQUIRE_THROWS_AS(grid.setDensities(densities), std::invalid_argument);
-    }
-}
-
-TEMPLATE_LIST_TEST_CASE("monad::Grid2d: Test setDensitiesConstant", "[monad]", Types) {
-    using DensityList = typename TestType::DensityList;
-
-    TestType grid({2, 3}, {0.5, 1.5});
-
-    SECTION("No errors") {
-        grid.setDensitiesConstant(0.5);
-
-        const DensityList expected(grid.numElements(), 0.5);
-
-        REQUIRE_THAT(grid.densities(), Catch::Matchers::Approx(expected).margin(NUMERICAL_ZERO));
-    }
-
-    SECTION("Invalid density") {
-        REQUIRE_THROWS_AS(grid.setDensitiesConstant(-0.1), std::invalid_argument);
-        REQUIRE_THROWS_AS(grid.setDensitiesConstant(1.1), std::invalid_argument);
-    }
-}
-
-TEMPLATE_LIST_TEST_CASE("monad::Grid2d: Test setDensitiesZeros", "[monad]", Types) {
-    using DensityList = typename TestType::DensityList;
-
-    TestType grid({2, 3}, {0.5, 1.5});
-
-    grid.setDensitiesZeros();
-
-    const DensityList expected(grid.numElements(), 0.0);
-
-    REQUIRE_THAT(grid.densities(), Catch::Matchers::Approx(expected).margin(NUMERICAL_ZERO));
-}
-
-TEMPLATE_LIST_TEST_CASE("monad::Grid2d: Test setDensitiesOnes", "[monad]", Types) {
-    using DensityList = typename TestType::DensityList;
-
-    TestType grid({2, 3}, {0.5, 1.5});
-
-    grid.setDensitiesOnes();
-
-    const DensityList expected(grid.numElements(), 1.0);
-
-    REQUIRE_THAT(grid.densities(), Catch::Matchers::Approx(expected).margin(NUMERICAL_ZERO));
-}
-
-TEMPLATE_LIST_TEST_CASE("monad::Grid2d: Test setDensitiesRandom", "[monad]", Types) {
-    using DensityList = typename TestType::DensityList;
-
-    TestType grid({2, 3}, {0.5, 1.5});
-
-    grid.setDensitiesRandom(1234);
-
-    const DensityList expected {
-        0.49766366630595177,
-        0.81783844288953345,
-        0.61211189358442286,
-        0.77135991905475021,
-        0.86066977348643747,
-        0.15063696570521984
-    };
-
-    REQUIRE_THAT(grid.densities(), Catch::Matchers::Approx(expected).margin(NUMERICAL_ZERO));
-}
-
-TEMPLATE_LIST_TEST_CASE("monad::Grid2d: Test setDensitiesFunction", "[monad]", Types) {
-    using Point = typename TestType::Point;
-
-    TestType grid({2, 3}, {0.5, 1.5});
-
-    SECTION("No errors") {
-        // 0.1x+0.2y
-        auto f = [](const Point &p) -> double {
-            return 0.1 * p(0) + 0.2 * p(1);
-        };
-
-        grid.setDensitiesFunction(f);
-
-        // The average of a linear function over a convex region equals its value at the centroid
-        const auto nodes = grid.elementNodes(1);
-        const Point centroid = nodes.colwise().mean();
-        const double expected = f(centroid);
-
-        REQUIRE_THAT(grid.getDensity(1), Catch::Matchers::WithinAbs(expected, NUMERICAL_ZERO));
-    }
-
-    SECTION("Bad function") {
-        // exp(x+y)
-        auto f = [](const Point &point) -> double {
-            const double x = point(0);
-            const double y = point(1);
-
-            return std::exp(x + y);
-        };
-
-        REQUIRE_THROWS_AS(grid.setDensitiesFunction(f), std::invalid_argument);
-    }
-}
-
-TEMPLATE_LIST_TEST_CASE("monad::Grid2d: Test setDensitiesFile", "[monad]", Types) {
-    using DensityList = typename TestType::DensityList;
-
-    TestType grid({2, 3}, {0.5, 1.5});
-
-    const auto folder = std::filesystem::path(__FILE__).parent_path() / "csv";
-
-    SECTION("No errors") {
-        const auto file = folder / "good.csv";
-
-        grid.setDensitiesFile(file.string());
-
-        const DensityList expected {
-            0.1,
-            0.2,
-            0.3,
-            0.4,
-            0.5,
-            0.6
-        };
-
-        REQUIRE_THAT(grid.densities(), Catch::Matchers::Approx(expected).margin(NUMERICAL_ZERO));
-    }
-
-    SECTION("Invalid csv data") {
-        // data < 0
-        auto file = folder / "bad1.csv";
-
-        REQUIRE_THROWS_AS(grid.setDensitiesFile(file.string()), std::runtime_error);
-
-        // data > 1
-        file = folder / "bad2.csv";
-
-        REQUIRE_THROWS_AS(grid.setDensitiesFile(file.string()), std::runtime_error);
-
-        // not enough data
-        file = folder / "bad3.csv";
-
-        REQUIRE_THROWS_AS(grid.setDensitiesFile(file.string()), std::runtime_error);
-
-        // too much data
-        file = folder / "bad4.csv";
-
-        REQUIRE_THROWS_AS(grid.setDensitiesFile(file.string()), std::runtime_error);
-
-        // no data
-        file = folder / "bad5.csv";
-
-        REQUIRE_THROWS_AS(grid.setDensitiesFile(file.string()), std::runtime_error);
-
-        // non-numeric data
-        file = folder / "bad6.csv";
-
-        REQUIRE_THROWS_AS(grid.setDensitiesFile(file.string()), std::runtime_error);
-    }
-}
-
-TEMPLATE_LIST_TEST_CASE("monad::Grid2d: Test translate", "[monad]", Types) {
-    TestType grid({2, 3}, {0.5, 1.5});
-
-    // Index=1 → Point=(1,0)
-    grid.setDensity(1, 0.5);
-
-    // Point=(0,2)
-    grid.translate({1, 2});
-
-    REQUIRE_THAT(grid.getDensity(4), Catch::Matchers::WithinAbs(0.5, NUMERICAL_ZERO));
-}
-
-TEST_CASE("monad::Grid2d: Test elementNodes", "[monad]") {
-    SECTION("Quad4Grid") {
-        const Quad4Grid grid({2, 3}, {0.5, 1.5});
-
-        SECTION("No errors") {
-            const Quad4Grid::NodesMatrix expected {
-                {0.25, 0.0},
-                {0.5, 0.0},
-                {0.5, 0.5},
-                {0.25, 0.5}
-            };
-
-            REQUIRE(grid.elementNodes(1).isApprox(expected, NUMERICAL_ZERO));
-        }
-
-        SECTION("Invalid index") {
-            REQUIRE_THROWS_AS(grid.elementNodes(grid.numElements()), std::out_of_range);
-        }
-    }
-
-    SECTION("Quad8Grid") {
-        const Quad8Grid grid({2, 3}, {0.5, 1.5});
-
-        SECTION("No errors") {
-            const Quad8Grid::NodesMatrix expected {
-                {0.25, 0.0},
-                {0.5, 0.0},
-                {0.5, 0.5},
-                {0.25, 0.5},
-                {0.375, 0.0},
-                {0.5, 0.25},
-                {0.375, 0.5},
-                {0.25, 0.25}
-            };
-
-            REQUIRE(grid.elementNodes(1).isApprox(expected, NUMERICAL_ZERO));
-        }
-
-        SECTION("Invalid index") {
-            REQUIRE_THROWS_AS(grid.elementNodes(grid.numElements()), std::out_of_range);
-        }
-    }
-}
-
-TEMPLATE_LIST_TEST_CASE("monad::Grid2d: Test measure/area", "[monad]", Types) {
-    const TestType grid({2, 3}, {0.5, 1.5});
-
-    REQUIRE_THAT(grid.measure(), Catch::Matchers::WithinAbs(0.75, NUMERICAL_ZERO));
-    REQUIRE_THAT(grid.area(), Catch::Matchers::WithinAbs(0.75, NUMERICAL_ZERO));
-}
-
-TEMPLATE_LIST_TEST_CASE("monad::Grid2d: Test operator==", "[monad]", Types) {
-    const TestType grid1({2, 3}, {0.5, 1.5});
-    const TestType grid2({2, 3}, {0.5, 1.5});
-
-    REQUIRE(grid1 == grid2);
-}
-
-TEMPLATE_LIST_TEST_CASE("monad::Grid2d: Test operator!=", "[monad]", Types) {
-    const TestType grid1({2, 3}, {0.5, 1.5});
-
-    SECTION("Different resolution") {
-        const TestType grid2({3, 3}, {0.5, 1.5});
-        const TestType grid3({2, 4}, {0.5, 1.5});
-
-        REQUIRE(grid1 != grid2);
-        REQUIRE(grid1 != grid3);
-    }
-
-    SECTION("Different size") {
-        const TestType grid2({2, 3}, {0.6, 1.5});
-        const TestType grid3({2, 3}, {0.5, 1.6});
-
-        REQUIRE(grid1 != grid2);
-        REQUIRE(grid1 != grid3);
-    }
-}
diff --git a/tests/grid/hex20_topology_test.cpp b/tests/grid/hex20_topology_test.cpp
new file mode 100644
index 0000000..8bc4539
--- /dev/null
+++ b/tests/grid/hex20_topology_test.cpp
@@ -0,0 +1,35 @@
+#include <catch2/catch_test_macros.hpp>
+#include "monad/grid/hex20_topology.hpp"
+#include "monad/detail/constants.hpp"
+
+using namespace monad;
+using namespace monad::grid;
+
+TEST_CASE("monad::grid::Hex20Topology: test numNodes", "[monad]") {
+    REQUIRE(Hex20Topology::numNodes({2, 3, 4}) == 193);
+}
+
+TEST_CASE("monad::grid::Hex20Topology: test numPeriodicNodes", "[monad]") {
+    REQUIRE(Hex20Topology::numPeriodicNodes({2, 3, 4}) == 96);
+}
+
+TEST_CASE("monad::grid::Hex20Topology: test node", "[monad]") {
+    const Hex20Topology::Point expected(0.25, 0.0, 0.0);
+    const Hex20Topology::Point actual = Hex20Topology::node(1, {2, 3, 4}, {0.5, 1.5, 2.0});
+
+    REQUIRE(actual.isApprox(expected, NUMERICAL_ZERO));
+}
+
+TEST_CASE("monad::grid::Hex20Topology: test element", "[monad]") {
+    const Hex20Topology::ElementList expected{1, 2, 5, 4, 13, 14, 17, 16, 61, 102, 63, 101, 69, 111, 71, 110, 146, 147, 150, 149};
+    const Hex20Topology::ElementList actual = Hex20Topology::element(1, {2, 3, 4});
+
+    REQUIRE(actual == expected);
+}
+
+TEST_CASE("monad::grid::Hex20Topology: test periodicElement", "[monad]") {
+    const Hex20Topology::ElementList expected{1, 0, 2, 3, 7, 6, 8, 9, 25, 48, 27, 49, 31, 54, 33, 55, 73, 72, 74, 75};
+    const Hex20Topology::ElementList actual = Hex20Topology::periodicElement(1, {2, 3, 4});
+
+    REQUIRE(actual == expected);
+}
diff --git a/tests/grid/hex8_topology_test.cpp b/tests/grid/hex8_topology_test.cpp
new file mode 100644
index 0000000..1aa046b
--- /dev/null
+++ b/tests/grid/hex8_topology_test.cpp
@@ -0,0 +1,35 @@
+#include <catch2/catch_test_macros.hpp>
+#include "monad/grid/hex8_topology.hpp"
+#include "monad/detail/constants.hpp"
+
+using namespace monad;
+using namespace monad::grid;
+
+TEST_CASE("monad::grid::Hex8Topology: test numNodes", "[monad]") {
+    REQUIRE(Hex8Topology::numNodes({2, 3, 4}) == 60);
+}
+
+TEST_CASE("monad::grid::Hex8Topology: test numPeriodicNodes", "[monad]") {
+    REQUIRE(Hex8Topology::numPeriodicNodes({2, 3, 4}) == 24);
+}
+
+TEST_CASE("monad::grid::Hex8Topology: test node", "[monad]") {
+    const Hex8Topology::Point expected(0.25, 0.0, 0.0);
+    const Hex8Topology::Point actual = Hex8Topology::node(1, {2, 3, 4}, {0.5, 1.5, 2.0});
+
+    REQUIRE(actual.isApprox(expected, NUMERICAL_ZERO));
+}
+
+TEST_CASE("monad::grid::Hex8Topology: test element", "[monad]") {
+    const Hex8Topology::ElementList expected{1, 2, 5, 4, 13, 14, 17, 16};
+    const Hex8Topology::ElementList actual = Hex8Topology::element(1, {2, 3, 4});
+
+    REQUIRE(actual == expected);
+}
+
+TEST_CASE("monad::grid::Hex8Topology: test periodicElement", "[monad]") {
+    const Hex8Topology::ElementList expected{1, 0, 2, 3, 7, 6, 8, 9};
+    const Hex8Topology::ElementList actual = Hex8Topology::periodicElement(1, {2, 3, 4});
+
+    REQUIRE(actual == expected);
+}
diff --git a/tests/grid/quad4_topology_test.cpp b/tests/grid/quad4_topology_test.cpp
new file mode 100644
index 0000000..1e103b4
--- /dev/null
+++ b/tests/grid/quad4_topology_test.cpp
@@ -0,0 +1,35 @@
+#include <catch2/catch_test_macros.hpp>
+#include "monad/grid/quad4_topology.hpp"
+#include "monad/detail/constants.hpp"
+
+using namespace monad;
+using namespace monad::grid;
+
+TEST_CASE("monad::grid::Quad4Topology: test numNodes", "[monad]") {
+    REQUIRE(Quad4Topology::numNodes({2, 3}) == 12);
+}
+
+TEST_CASE("monad::grid::Quad4Topology: test numPeriodicNodes", "[monad]") {
+    REQUIRE(Quad4Topology::numPeriodicNodes({2, 3}) == 6);
+}
+
+TEST_CASE("monad::grid::Quad4Topology: test node", "[monad]") {
+    const Quad4Topology::Point expected(0.25, 0.0);
+    const Quad4Topology::Point actual = Quad4Topology::node(1, {2, 3}, {0.5, 1.5});
+
+    REQUIRE(actual.isApprox(expected, NUMERICAL_ZERO));
+}
+
+TEST_CASE("monad::grid::Quad4Topology: test element", "[monad]") {
+    const Quad4Topology::ElementList expected{1, 2, 5, 4};
+    const Quad4Topology::ElementList actual = Quad4Topology::element(1, {2, 3});
+
+    REQUIRE(actual == expected);
+}
+
+TEST_CASE("monad::grid::Quad4Topology: test periodicElement", "[monad]") {
+    const Quad4Topology::ElementList expected{1, 0, 2, 3};
+    const Quad4Topology::ElementList actual = Quad4Topology::periodicElement(1, {2, 3});
+
+    REQUIRE(actual == expected);
+}
diff --git a/tests/grid/quad8_topology_test.cpp b/tests/grid/quad8_topology_test.cpp
new file mode 100644
index 0000000..1d13bc7
--- /dev/null
+++ b/tests/grid/quad8_topology_test.cpp
@@ -0,0 +1,35 @@
+#include <catch2/catch_test_macros.hpp>
+#include "monad/grid/quad8_topology.hpp"
+#include "monad/detail/constants.hpp"
+
+using namespace monad;
+using namespace monad::grid;
+
+TEST_CASE("monad::grid::Quad8Topology: test numNodes", "[monad]") {
+    REQUIRE(Quad8Topology::numNodes({2, 3}) == 29);
+}
+
+TEST_CASE("monad::grid::Quad8Topology: test numPeriodicNodes", "[monad]") {
+    REQUIRE(Quad8Topology::numPeriodicNodes({2, 3}) == 18);
+}
+
+TEST_CASE("monad::grid::Quad8Topology: test node", "[monad]") {
+    const Quad8Topology::Point expected(0.25, 0.0);
+    const Quad8Topology::Point actual = Quad8Topology::node(1, {2, 3}, {0.5, 1.5});
+
+    REQUIRE(actual.isApprox(expected, NUMERICAL_ZERO));
+}
+
+TEST_CASE("monad::grid::Quad8Topology: test element", "[monad]") {
+    const Quad8Topology::ElementList expected{1, 2, 5, 4, 13, 22, 15, 21};
+    const Quad8Topology::ElementList actual = Quad8Topology::element(1, {2, 3});
+
+    REQUIRE(actual == expected);
+}
+
+TEST_CASE("monad::grid::Quad8Topology: test periodicElement", "[monad]") {
+    const Quad8Topology::ElementList expected{1, 0, 2, 3, 7, 12, 9, 13};
+    const Quad8Topology::ElementList actual = Quad8Topology::periodicElement(1, {2, 3});
+
+    REQUIRE(actual == expected);
+}
diff --git a/tests/grid/grid_3d_test.cpp b/tests/grid/structured_grid_test.cpp
similarity index 60%
rename from tests/grid/grid_3d_test.cpp
rename to tests/grid/structured_grid_test.cpp
index c0d3e27..479474d 100644
--- a/tests/grid/grid_3d_test.cpp
+++ b/tests/grid/structured_grid_test.cpp
@@ -1,140 +1,245 @@
 #include <tuple>
 #include <stdexcept>
-#include <cmath>
 #include <cstddef>
+#include <cmath>
 #include <catch2/catch_test_macros.hpp>
 #include <catch2/catch_template_test_macros.hpp>
-#include <catch2/matchers/catch_matchers_vector.hpp>
 #include <catch2/matchers/catch_matchers_floating_point.hpp>
-#include "monad/grid/hex8_grid.hpp"
-#include "monad/grid/hex20_grid.hpp"
+#include "monad/grid/grid_aliases.hpp"
 #include "monad/detail/constants.hpp"
-#include "test_common.hpp"
 
 using namespace monad;
-using namespace monad::testing;
 
-using Types = std::tuple<Hex8Grid, Hex20Grid>;
+using Types = std::tuple<Quad4Grid, Quad8Grid, Hex8Grid, Hex20Grid>;
+
+TEMPLATE_LIST_TEST_CASE("monad::grid::StructuredGrid: Test initalization", "[monad]", Types) {
+    using Resolution = typename TestType::Resolution;
+    using Size = typename TestType::Size;
+
+    Resolution resolution;
+    resolution.fill(2);
+
+    Size size;
+    size.fill(0.4);
 
-TEMPLATE_LIST_TEST_CASE("monad::Grid3d: Test initalization", "[monad]", Types) {
     SECTION("Invalid resolution") {
-        REQUIRE_THROWS_AS(TestType({0, 3, 4}, {0.5, 1.5, 2.0}), std::invalid_argument);
-        REQUIRE_THROWS_AS(TestType({2, 0, 4}, {0.5, 1.5, 2.0}), std::invalid_argument);
-        REQUIRE_THROWS_AS(TestType({2, 3, 0}, {0.5, 1.5, 2.0}), std::invalid_argument);
+        for (std::size_t i = 0; i < TestType::Dim; ++i) {
+            Resolution resolutionInvalid = resolution;
+            resolutionInvalid[i] = 0;
+
+            REQUIRE_THROWS_AS(TestType(resolutionInvalid, size), std::invalid_argument);
+        }
     }
 
     SECTION("Invalid size") {
-        REQUIRE_THROWS_AS(TestType({2, 3, 4}, {0.0, 1.5, 2.0}), std::invalid_argument);
-        REQUIRE_THROWS_AS(TestType({2, 3, 4}, {-0.5, 1.5, 2.0}), std::invalid_argument);
-        REQUIRE_THROWS_AS(TestType({2, 3, 4}, {0.5, 0.0, 2.0}), std::invalid_argument);
-        REQUIRE_THROWS_AS(TestType({2, 3, 4}, {0.5, -1.5, 2.0}), std::invalid_argument);
-        REQUIRE_THROWS_AS(TestType({2, 3, 4}, {0.5, 1.5, 0.0}), std::invalid_argument);
-        REQUIRE_THROWS_AS(TestType({2, 3, 4}, {0.5, 1.5, -2.0}), std::invalid_argument);
+        for (std::size_t i = 0; i < TestType::Dim; ++i) {
+            Size sizeInvalid = size;
+            sizeInvalid[i] = 0.0;
+
+            REQUIRE_THROWS_AS(TestType(resolution, sizeInvalid), std::invalid_argument);
+
+            sizeInvalid[i] = -1.0;
+
+            REQUIRE_THROWS_AS(TestType(resolution, sizeInvalid), std::invalid_argument);
+        }
     }
 }
 
-TEMPLATE_LIST_TEST_CASE("monad::Grid3d: Test resolution", "[monad]", Types) {
+TEMPLATE_LIST_TEST_CASE("monad::grid::StructuredGrid: Test resolution", "[monad]", Types) {
     using Resolution = typename TestType::Resolution;
+    using Size = typename TestType::Size;
+
+    Resolution resolution;
+    resolution.fill(2);
 
-    const TestType grid({2, 3, 4}, {0.5, 1.5, 2.0});
+    Size size;
+    size.fill(0.4);
 
-    const Resolution expected{2, 3, 4};
+    const TestType grid(resolution, size);
 
-    REQUIRE(grid.resolution() == expected);
+    REQUIRE(grid.resolution() == resolution);
 }
 
-TEMPLATE_LIST_TEST_CASE("monad::Grid3d: Test size", "[monad]", Types) {
+TEMPLATE_LIST_TEST_CASE("monad::grid::StructuredGrid: Test size", "[monad]", Types) {
+    using Resolution = typename TestType::Resolution;
     using Size = typename TestType::Size;
 
-    const TestType grid({2, 3, 4}, {0.5, 1.5, 2.0});
+    Resolution resolution;
+    resolution.fill(2);
+
+    Size size;
+    size.fill(0.4);
 
-    const Size expected{0.5, 1.5, 2.0};
+    const TestType grid(resolution, size);
 
-    REQUIRE(grid.size() == expected);
+    REQUIRE(grid.size() == size);
 }
 
-TEMPLATE_LIST_TEST_CASE("monad::Grid3d: Test densities", "[monad]", Types) {
-    using DensityList = typename TestType::DensityList;
+TEMPLATE_LIST_TEST_CASE("monad::grid::StructuredGrid: Test numElements", "[monad]", Types) {
+    using Resolution = typename TestType::Resolution;
+    using Size = typename TestType::Size;
+
+    Resolution resolution;
+    resolution.fill(2);
+
+    Size size;
+    size.fill(0.4);
 
-    const TestType grid({2, 3, 4}, {0.5, 1.5, 2.0});
+    const TestType grid(resolution, size);
 
-    const DensityList expected(24, 0.0);
+    const std::size_t expected = static_cast<std::size_t>(std::pow(2.0, TestType::Dim));
+    const std::size_t actual = grid.numElements();
 
-    REQUIRE_THAT(grid.densities(), Catch::Matchers::Approx(expected).margin(NUMERICAL_ZERO));
+    REQUIRE(actual == expected);
 }
 
-TEMPLATE_LIST_TEST_CASE("monad::Grid3d: Test numElements", "[monad]", Types) {
-    const TestType grid({2, 3, 4}, {0.5, 1.5, 2.0});
+TEMPLATE_LIST_TEST_CASE("monad::grid::StructuredGrid: Test numNodes", "[monad]", Types) {
+    using Resolution = typename TestType::Resolution;
+    using Size = typename TestType::Size;
+    using Topology = typename TestType::Topology;
 
-    REQUIRE(grid.numElements() == 24);
+    Resolution resolution;
+    resolution.fill(2);
+
+    Size size;
+    size.fill(0.4);
+
+    const TestType grid(resolution, size);
+
+    const std::size_t expected = Topology::numNodes(resolution);
+    const std::size_t actual = grid.numNodes();
+
+    REQUIRE(actual == expected);
 }
 
-TEST_CASE("monad::Grid3d: Test numNodes", "[monad]") {
-    SECTION("Hex8Grid") {
-        const Hex8Grid grid({2, 3, 4}, {0.5, 1.5, 2.0});
+TEMPLATE_LIST_TEST_CASE("monad::grid::StructuredGrid: Test numPeriodicNodes", "[monad]", Types) {
+    using Resolution = typename TestType::Resolution;
+    using Size = typename TestType::Size;
+    using Topology = typename TestType::Topology;
 
-        REQUIRE(grid.numNodes() == 60);
-    }
+    Resolution resolution;
+    resolution.fill(2);
 
-    SECTION("Hex20Grid") {
-        const Hex20Grid grid({2, 3, 4}, {0.5, 1.5, 2.0});
+    Size size;
+    size.fill(0.4);
 
-        REQUIRE(grid.numNodes() == 193);
-    }
+    const TestType grid(resolution, size);
+
+    const std::size_t expected = Topology::numPeriodicNodes(resolution);
+    const std::size_t actual = grid.numPeriodicNodes();
+
+    REQUIRE(actual == expected);
 }
 
-TEST_CASE("monad::Grid3d: Test numPeriodicNodes", "[monad]") {
-    SECTION("Hex8Grid") {
-        const Hex8Grid grid({2, 3, 4}, {0.5, 1.5, 2.0});
+TEMPLATE_LIST_TEST_CASE("monad::grid::StructuredGrid: Test node", "[monad]", Types) {
+    using Resolution = typename TestType::Resolution;
+    using Size = typename TestType::Size;
+    using Topology = typename TestType::Topology;
+    using Point = typename TestType::Point;
 
-        REQUIRE(grid.numPeriodicNodes() == 24);
-    }
+    Resolution resolution;
+    resolution.fill(2);
 
-    SECTION("Hex20Grid") {
-        const Hex20Grid grid({2, 3, 4}, {0.5, 1.5, 2.0});
+    Size size;
+    size.fill(0.4);
 
-        REQUIRE(grid.numPeriodicNodes() == 96);
+    const TestType grid(resolution, size);
+
+    SECTION("No errors") {
+        const Point expected = Topology::node(1, resolution, size);
+        const Point actual = grid.node(1);
+
+        REQUIRE(actual.isApprox(expected, NUMERICAL_ZERO));
+    }
+
+    SECTION("Invalid index") {
+        const std::size_t badIndex = grid.numNodes();
+    
+        REQUIRE_THROWS_AS(grid.node(badIndex), std::out_of_range);
     }
 }
 
-TEST_CASE("monad::Grid3d: Test node", "[monad]") {
-    SECTION("Hex8Grid") {
-        const Hex8Grid grid({2, 3, 4}, {0.5, 1.5, 2.0});
+TEST_CASE("monad::grid::StructuredGrid: Test nodes", "[monad]") {
+    SECTION("Quad4Grid") {
+        using NodesList = typename Quad4Grid::NodesList;
 
-        SECTION("No errors") {
-            const Hex8Grid::Point expected(0.25, 0.0, 0.0);
+        const Quad4Grid grid({2, 3}, {0.5, 1.5});
 
-            REQUIRE(grid.node(1).isApprox(expected, NUMERICAL_ZERO));
-        }
+        const NodesList expected {
+            {0.0, 0.0},
+            {0.25, 0.0},
+            {0.5, 0.0},
+            {0.0, 0.5},
+            {0.25, 0.5},
+            {0.5, 0.5},
+            {0.0, 1.0},
+            {0.25, 1.0},
+            {0.5, 1.0},
+            {0.0, 1.5},
+            {0.25, 1.5},
+            {0.5, 1.5}
+        };
 
-        SECTION("Invalid index") {
-            REQUIRE_THROWS_AS(grid.node(grid.numNodes()), std::out_of_range);
+        const auto actual = grid.nodes();
+
+        REQUIRE(actual.size() == expected.size());
+
+        for (std::size_t i = 0; i < actual.size(); ++i) {
+            REQUIRE(actual[i].isApprox(expected[i], NUMERICAL_ZERO));
         }
     }
 
-    SECTION("Hex20Grid") {
-        const Hex20Grid grid({2, 3, 4}, {0.5, 1.5, 2.0});
+    SECTION("Quad8Grid") {
+        using NodesList = typename Quad8Grid::NodesList;
 
-        SECTION("No errors") {
-            const Hex20Grid::Point expected(0.25, 0.0, 0.0);
+        const Quad8Grid grid({2, 3}, {0.5, 1.5});
 
-            REQUIRE(grid.node(1).isApprox(expected, NUMERICAL_ZERO));
-        }
+        const NodesList expected {
+            {0.0, 0.0},
+            {0.25, 0.0},
+            {0.5, 0.0},
+            {0.0, 0.5},
+            {0.25, 0.5},
+            {0.5, 0.5},
+            {0.0, 1.0},
+            {0.25, 1.0},
+            {0.5, 1.0},
+            {0.0, 1.5},
+            {0.25, 1.5},
+            {0.5, 1.5},
+            {0.125, 0.0},
+            {0.375, 0.0},
+            {0.125, 0.5},
+            {0.375, 0.5},
+            {0.125, 1.0},
+            {0.375, 1.0},
+            {0.125, 1.5},
+            {0.375, 1.5},
+            {0.0, 0.25},
+            {0.25, 0.25},
+            {0.5, 0.25},
+            {0.0, 0.75},
+            {0.25, 0.75},
+            {0.5, 0.75},
+            {0.0, 1.25},
+            {0.25, 1.25},
+            {0.5, 1.25}
+        };
 
-        SECTION("Invalid index") {
-            REQUIRE_THROWS_AS(grid.node(grid.numNodes()), std::out_of_range);
+        const auto actual = grid.nodes();
+
+        REQUIRE(actual.size() == expected.size());
+
+        for (std::size_t i = 0; i < actual.size(); ++i) {
+            REQUIRE(actual[i].isApprox(expected[i], NUMERICAL_ZERO));
         }
     }
-}
 
-TEST_CASE("monad::Grid3d: Test nodes", "[monad]") {
     SECTION("Hex8Grid") {
         using NodesList = typename Hex8Grid::NodesList;
 
         const Hex8Grid grid({2, 3, 4}, {0.5, 1.5, 2.0});
 
-        const auto actual = grid.nodes();
-
         const NodesList expected {
             {0.0, 0.0, 0.0},
             {0.25, 0.0, 0.0},
@@ -198,6 +303,8 @@ TEST_CASE("monad::Grid3d: Test nodes", "[monad]") {
             {0.5, 1.5, 2.0}
         };
 
+        const auto actual = grid.nodes();
+
         REQUIRE(actual.size() == expected.size());
 
         for (std::size_t i = 0; i < actual.size(); ++i) {
@@ -210,8 +317,6 @@ TEST_CASE("monad::Grid3d: Test nodes", "[monad]") {
 
         const Hex20Grid grid({2, 3, 4}, {0.5, 1.5, 2.0});
 
-        const auto actual = grid.nodes();
-
         const NodesList expected {
             {0.0, 0.0, 0.0},
             {0.25, 0.0, 0.0},
@@ -408,6 +513,8 @@ TEST_CASE("monad::Grid3d: Test nodes", "[monad]") {
             {0.5, 1.5, 1.75}
         };
 
+        const auto actual = grid.nodes();
+
         REQUIRE(actual.size() == expected.size());
 
         for (std::size_t i = 0; i < actual.size(); ++i) {
@@ -416,37 +523,65 @@ TEST_CASE("monad::Grid3d: Test nodes", "[monad]") {
     }
 }
 
-TEST_CASE("monad::Grid3d: Test element", "[monad]") {
-    SECTION("Hex8Grid") {
-        const Hex8Grid grid({2, 3, 4}, {0.5, 1.5, 2.0});
+TEMPLATE_LIST_TEST_CASE("monad::grid::StructuredGrid: Test element", "[monad]", Types) {
+    using Resolution = typename TestType::Resolution;
+    using Size = typename TestType::Size;
+    using Topology = typename TestType::Topology;
+    using ElementList = typename TestType::ElementList;
 
-        SECTION("No errors") {
-            const Hex8Grid::ElementList expected{1, 2, 5, 4, 13, 14, 17, 16};
+    Resolution resolution;
+    resolution.fill(2);
 
-            REQUIRE(grid.element(1) == expected);
-        }
+    Size size;
+    size.fill(0.4);
 
-        SECTION("Invalid index") {
-            REQUIRE_THROWS_AS(grid.element(grid.numElements()), std::out_of_range);
-        }
-    }
+    const TestType grid(resolution, size);
 
-    SECTION("Hex20Grid") {
-        const Hex20Grid grid({2, 3, 4}, {0.5, 1.5, 2.0});
+    SECTION("No errors") {
+        const ElementList expected = Topology::element(1, resolution);
+        const ElementList actual = grid.element(1);
 
-        SECTION("No errors") {
-            const Hex20Grid::ElementList expected{1, 2, 5, 4, 13, 14, 17, 16, 61, 102, 63, 101, 69, 111, 71, 110, 146, 147, 150, 149};
+        REQUIRE(actual == expected);
+    }
 
-            REQUIRE(grid.element(1) == expected);
-        }
+    SECTION("Invalid index") {
+        const std::size_t badIndex = grid.numElements();
 
-        SECTION("Invalid index") {
-            REQUIRE_THROWS_AS(grid.element(grid.numElements()), std::out_of_range);
-        }
+        REQUIRE_THROWS_AS(grid.element(badIndex), std::out_of_range);
     }
 }
 
-TEST_CASE("monad::Grid3d: Test elements", "[monad]") {
+TEST_CASE("monad::grid::StructuredGrid: Test elements", "[monad]") {
+    SECTION("Quad4Grid") {
+        const Quad4Grid grid({2, 3}, {0.5, 1.5});
+
+        const Quad4Grid::ElementsList expected {
+            {0, 1, 4, 3},
+            {1, 2, 5, 4},
+            {3, 4, 7, 6},
+            {4, 5, 8, 7},
+            {6, 7, 10, 9},
+            {7, 8, 11, 10}
+        };
+
+        REQUIRE(grid.elements() == expected);
+    }
+
+    SECTION("Quad8Grid") {
+        const Quad8Grid grid({2, 3}, {0.5, 1.5});
+
+        const Quad8Grid::ElementsList expected {
+            {0, 1, 4, 3, 12, 21, 14, 20},
+            {1, 2, 5, 4, 13, 22, 15, 21},
+            {3, 4, 7, 6, 14, 24, 16, 23},
+            {4, 5, 8, 7, 15, 25, 17, 24},
+            {6, 7, 10, 9, 16, 27, 18, 26},
+            {7, 8, 11, 10, 17, 28, 19, 27}
+        };
+
+        REQUIRE(grid.elements() == expected);
+    }
+
     SECTION("Hex8Grid") {
         const Hex8Grid grid({2, 3, 4}, {0.5, 1.5, 2.0});
 
@@ -514,37 +649,65 @@ TEST_CASE("monad::Grid3d: Test elements", "[monad]") {
     }
 }
 
-TEST_CASE("monad::Grid3d: Test periodicElement", "[monad]") {
-    SECTION("Hex8Grid") {
-        const Hex8Grid grid({2, 3, 4}, {0.5, 1.5, 2.0});
+TEMPLATE_LIST_TEST_CASE("monad::grid::StructuredGrid: Test periodicElement", "[monad]", Types) {
+    using Resolution = typename TestType::Resolution;
+    using Size = typename TestType::Size;
+    using Topology = typename TestType::Topology;
+    using ElementList = typename TestType::ElementList;
 
-        SECTION("No errors") {
-            const Hex8Grid::ElementList expected{1, 0, 2, 3, 7, 6, 8, 9};
+    Resolution resolution;
+    resolution.fill(2);
 
-            REQUIRE(grid.periodicElement(1) == expected);
-        }
+    Size size;
+    size.fill(0.4);
 
-        SECTION("Invalid index") {
-            REQUIRE_THROWS_AS(grid.periodicElement(grid.numElements()), std::out_of_range);
-        }
+    const TestType grid(resolution, size);
+
+    SECTION("No errors") {
+        const ElementList expected = Topology::periodicElement(1, resolution);
+        const ElementList actual = grid.periodicElement(1);
+
+        REQUIRE(actual == expected);
     }
 
-    SECTION("Hex20Grid") {
-        const Hex20Grid grid({2, 3, 4}, {0.5, 1.5, 2.0});
+    SECTION("Invalid index") {
+        const std::size_t badIndex = grid.numElements();
+        
+        REQUIRE_THROWS_AS(grid.element(badIndex), std::out_of_range);
+    }
+}
 
-        SECTION("No errors") {
-            const Hex20Grid::ElementList expected{1, 0, 2, 3, 7, 6, 8, 9, 25, 48, 27, 49, 31, 54, 33, 55, 73, 72, 74, 75};
+TEST_CASE("monad::grid::StructuredGrid: Test periodicElements", "[monad]") {
+    SECTION("Quad4Grid") {
+        const Quad4Grid grid({2, 3}, {0.5, 1.5});
+
+        const Quad4Grid::ElementsList expected {
+            {0, 1, 3, 2},
+            {1, 0, 2, 3},
+            {2, 3, 5, 4},
+            {3, 2, 4, 5},
+            {4, 5, 1, 0},
+            {5, 4, 0, 1}
+        };
 
-            REQUIRE(grid.periodicElement(1) == expected);
-        }
+        REQUIRE(grid.periodicElements() == expected);
+    }
 
-        SECTION("Invalid index") {
-            REQUIRE_THROWS_AS(grid.periodicElement(grid.numElements()), std::out_of_range);
-        }
+    SECTION("Quad8Grid") {
+        const Quad8Grid grid({2, 3}, {0.5, 1.5});
+
+        const Quad8Grid::ElementsList expected {
+            {0, 1, 3, 2, 6, 13, 8, 12},
+            {1, 0, 2, 3, 7, 12, 9, 13},
+            {2, 3, 5, 4, 8, 15, 10, 14},
+            {3, 2, 4, 5, 9, 14, 11, 15},
+            {4, 5, 1, 0, 10, 17, 6, 16},
+            {5, 4, 0, 1, 11, 16, 7, 17}
+        };
+
+        REQUIRE(grid.periodicElements() == expected);
     }
-}
 
-TEST_CASE("monad::Grid3d: Test periodicElements", "[monad]") {
     SECTION("Hex8Grid") {
         const Hex8Grid grid({2, 3, 4}, {0.5, 1.5, 2.0});
 
@@ -612,184 +775,57 @@ TEST_CASE("monad::Grid3d: Test periodicElements", "[monad]") {
     }
 }
 
-TEMPLATE_LIST_TEST_CASE("monad::Grid3d: Test getDensity/setDensity", "[monad]", Types) {
-    TestType grid({2, 3, 4}, {0.5, 1.5, 2.0});
-
-    SECTION("No errors") {
-        REQUIRE_THAT(grid.getDensity(1), Catch::Matchers::WithinAbs(0.0, NUMERICAL_ZERO));
-
-        grid.setDensity(1, 0.1);
-
-        REQUIRE_THAT(grid.getDensity(1), Catch::Matchers::WithinAbs(0.1, NUMERICAL_ZERO));
-    }
-
-    SECTION("Invalid index") {
-        REQUIRE_THROWS_AS(grid.getDensity(grid.numElements()), std::out_of_range);
-
-        REQUIRE_THROWS_AS(grid.setDensity(grid.numElements(), 0.1), std::out_of_range);
-    }
-
-    SECTION("Invalid density") {
-        REQUIRE_THROWS_AS(grid.setDensity(1, -0.1), std::invalid_argument);
-        REQUIRE_THROWS_AS(grid.setDensity(1, 1.1), std::invalid_argument);
-    }
-}
-
-TEMPLATE_LIST_TEST_CASE("monad::Grid3d: Test setDensities", "[monad]", Types) {
-    using DensityList = typename TestType::DensityList;
-
-    TestType grid({2, 3, 4}, {0.5, 1.5, 2.0});
-
-    SECTION("No errors") {
-        const DensityList densities(grid.numElements(), 0.5);
-
-        grid.setDensities(densities);
-
-        REQUIRE_THAT(grid.densities(), Catch::Matchers::Approx(densities).margin(NUMERICAL_ZERO));
-    }
-
-    SECTION("Invalid size") {
-        const DensityList densities(grid.numElements() + 1, 0.5);
-
-        REQUIRE_THROWS_AS(grid.setDensities(densities), std::invalid_argument);
-    }
-
-    SECTION("Invalid density") {
-        DensityList densities(grid.numElements(), 0.5);
-        densities[1] *= -1;
-
-        REQUIRE_THROWS_AS(grid.setDensities(densities), std::invalid_argument);
-    }
-}
-
-TEMPLATE_LIST_TEST_CASE("monad::Grid3d: Test setDensitiesConstant", "[monad]", Types) {
-    using DensityList = typename TestType::DensityList;
+TEST_CASE("monad::grid::StructuredGrid: Test elementNodes", "[monad]") {
+    SECTION("Quad4Grid") {
+        const Quad4Grid grid({2, 3}, {0.5, 1.5});
 
-    TestType grid({2, 3, 4}, {0.5, 1.5, 2.0});
+        SECTION("No errors") {
+            const Quad4Grid::NodesMatrix expected {
+                {0.25, 0.0},
+                {0.5, 0.0},
+                {0.5, 0.5},
+                {0.25, 0.5}
+            };
 
-    SECTION("No errors") {
-        grid.setDensitiesConstant(0.5);
+            const auto actual = grid.elementNodes(1);
 
-        const DensityList expected(grid.numElements(), 0.5);
+            REQUIRE(actual.isApprox(expected, NUMERICAL_ZERO));
+        }
 
-        REQUIRE_THAT(grid.densities(), Catch::Matchers::Approx(expected).margin(NUMERICAL_ZERO));
-    }
+        SECTION("Invalid index") {
+            const std::size_t badIndex = grid.numElements();
 
-    SECTION("Invalid density") {
-        REQUIRE_THROWS_AS(grid.setDensitiesConstant(-0.1), std::invalid_argument);
-        REQUIRE_THROWS_AS(grid.setDensitiesConstant(1.1), std::invalid_argument);
+            REQUIRE_THROWS_AS(grid.elementNodes(badIndex), std::out_of_range);
+        }
     }
-}
-
-TEMPLATE_LIST_TEST_CASE("monad::Grid3d: Test setDensitiesZeros", "[monad]", Types) {
-    using DensityList = typename TestType::DensityList;
 
-    TestType grid({2, 3, 4}, {0.5, 1.5, 2.0});
+    SECTION("Quad8Grid") {
+        const Quad8Grid grid({2, 3}, {0.5, 1.5});
 
-    grid.setDensitiesZeros();
-
-    const DensityList expected(grid.numElements(), 0.0);
-
-    REQUIRE_THAT(grid.densities(), Catch::Matchers::Approx(expected).margin(NUMERICAL_ZERO));
-}
-
-TEMPLATE_LIST_TEST_CASE("monad::Grid3d: Test setDensitiesOnes", "[monad]", Types) {
-    using DensityList = typename TestType::DensityList;
-
-    TestType grid({2, 3, 4}, {0.5, 1.5, 2.0});
-
-    grid.setDensitiesOnes();
-
-    const DensityList expected(grid.numElements(), 1.0);
-
-    REQUIRE_THAT(grid.densities(), Catch::Matchers::Approx(expected).margin(NUMERICAL_ZERO));
-}
-
-TEMPLATE_LIST_TEST_CASE("monad::Grid3d: Test setDensitiesRandom", "[monad]", Types) {
-    using DensityList = typename TestType::DensityList;
-
-    TestType grid({2, 3, 4}, {0.5, 1.5, 2.0});
-
-    grid.setDensitiesRandom(1234);
-
-    const DensityList expected {
-        0.49766366630595177,
-        0.81783844288953345,
-        0.61211189358442286,
-        0.77135991905475021,
-        0.86066977348643747,
-        0.15063696570521984,
-        0.19851876010104652,
-        0.8151629340839277,
-        0.15881535340057207,
-        0.11613783027882053,
-        0.01290753305838846,
-        0.48683344491314973,
-        0.33101542682467566,
-        0.80263957374559314,
-        0.09825193844042972,
-        0.05599344953922467,
-        0.44266265526020054,
-        0.02214390901237923,
-        0.29072854825594158,
-        0.24639444143572856,
-        0.73828676607573551,
-        0.88922613337994483,
-        0.98713930325475108,
-        0.1174433832915556
-    };
-
-    REQUIRE_THAT(grid.densities(), Catch::Matchers::Approx(expected).margin(NUMERICAL_ZERO));
-}
-
-TEMPLATE_LIST_TEST_CASE("monad::Grid3d: Test setDensitiesFunction", "[monad]", Types) {
-    using Point = typename TestType::Point;
-
-    TestType grid({2, 3, 4}, {0.5, 1.5, 2.0});
-
-    SECTION("No errors") {
-        // 0.1x+0.2y+0.3z
-        auto f = [](const Point &p) -> double {
-            return 0.1 * p(0) + 0.2 * p(1) + 0.3 * p(2);
-        };
-
-        grid.setDensitiesFunction(f);
-
-        // The average of a linear function over a convex region equals its value at the centroid
-        const auto nodes = grid.elementNodes(1);
-        const Point centroid = nodes.colwise().mean();
-        const double expected = f(centroid);
+        SECTION("No errors") {
+            const Quad8Grid::NodesMatrix expected {
+                {0.25, 0.0},
+                {0.5, 0.0},
+                {0.5, 0.5},
+                {0.25, 0.5},
+                {0.375, 0.0},
+                {0.5, 0.25},
+                {0.375, 0.5},
+                {0.25, 0.25}
+            };
 
-        REQUIRE_THAT(grid.getDensity(1), Catch::Matchers::WithinAbs(expected, NUMERICAL_ZERO));
-    }
+            const auto actual = grid.elementNodes(1);
 
-    SECTION("Bad function") {
-        // exp(x+y+z)
-        auto f = [](const Point &point) -> double {
-            const double x = point(0);
-            const double y = point(1);
-            const double z = point(2);
+            REQUIRE(actual.isApprox(expected, NUMERICAL_ZERO));
+        }
 
-            return std::exp(x + y + z);
-        };
+        SECTION("Invalid index") {
+            const std::size_t badIndex = grid.numElements();
 
-        REQUIRE_THROWS_AS(grid.setDensitiesFunction(f), std::invalid_argument);
+            REQUIRE_THROWS_AS(grid.elementNodes(badIndex), std::out_of_range);
+        }
     }
-}
-
-TEMPLATE_LIST_TEST_CASE("monad::Grid3d: Test translate", "[monad]", Types) {
-    TestType grid({2, 3, 4}, {0.5, 1.5, 2.0});
-
-    // Index=1 → Point=(1,0,0)
-    grid.setDensity(1, 0.5);
-
-    // Point=(0,2,3)
-    grid.translate({1, 2, 3});
 
-    REQUIRE_THAT(grid.getDensity(22), Catch::Matchers::WithinAbs(0.5, NUMERICAL_ZERO));
-}
-
-TEST_CASE("monad::Grid3d: Test elementNodes", "[monad]") {
     SECTION("Hex8Grid") {
         const Hex8Grid grid({2, 3, 4}, {0.5, 1.5, 2.0});
 
@@ -805,11 +841,15 @@ TEST_CASE("monad::Grid3d: Test elementNodes", "[monad]") {
                 {0.25, 0.5, 0.5}
             };
 
-            REQUIRE(grid.elementNodes(1).isApprox(expected, NUMERICAL_ZERO));
+            const auto actual = grid.elementNodes(1);
+
+            REQUIRE(actual.isApprox(expected, NUMERICAL_ZERO));
         }
 
         SECTION("Invalid index") {
-            REQUIRE_THROWS_AS(grid.elementNodes(grid.numElements()), std::out_of_range);
+            const std::size_t badIndex = grid.numElements();
+
+            REQUIRE_THROWS_AS(grid.elementNodes(badIndex), std::out_of_range);
         }
     }
 
@@ -840,49 +880,91 @@ TEST_CASE("monad::Grid3d: Test elementNodes", "[monad]") {
                 {0.25, 0.5, 0.25}
             };
 
-            REQUIRE(grid.elementNodes(1).isApprox(expected, NUMERICAL_ZERO));
+            const auto actual = grid.elementNodes(1);
+
+            REQUIRE(actual.isApprox(expected, NUMERICAL_ZERO));
         }
 
         SECTION("Invalid index") {
-            REQUIRE_THROWS_AS(grid.elementNodes(grid.numElements()), std::out_of_range);
+            const std::size_t badIndex = grid.numElements();
+
+            REQUIRE_THROWS_AS(grid.elementNodes(badIndex), std::out_of_range);
         }
     }
 }
 
-TEMPLATE_LIST_TEST_CASE("monad::Grid3d: Test measure/volume", "[monad]", Types) {
-    const TestType grid({2, 3, 4}, {0.5, 1.5, 2.0});
+TEMPLATE_LIST_TEST_CASE("monad::grid::StructuredGrid: Test measure/area/volume", "[monad]", Types) {
+    using Resolution = typename TestType::Resolution;
+    using Size = typename TestType::Size;
 
-    REQUIRE_THAT(grid.measure(), Catch::Matchers::WithinAbs(1.5, NUMERICAL_ZERO));
-    REQUIRE_THAT(grid.volume(), Catch::Matchers::WithinAbs(1.5, NUMERICAL_ZERO));
+    Resolution resolution;
+    resolution.fill(2);
+
+    Size size;
+    size.fill(0.4);
+
+    const TestType grid(resolution, size);
+
+    const double expected = std::pow(0.4, TestType::Dim);
+    const double actual = grid.measure();
+
+    REQUIRE_THAT(actual, Catch::Matchers::WithinAbs(expected, NUMERICAL_ZERO));
+
+    if constexpr (TestType::Dim == 2) {
+        REQUIRE_THAT(grid.area(), Catch::Matchers::WithinAbs(expected, NUMERICAL_ZERO));
+    }
+    else {
+        REQUIRE_THAT(grid.volume(), Catch::Matchers::WithinAbs(expected, NUMERICAL_ZERO));
+    }
 }
 
-TEMPLATE_LIST_TEST_CASE("monad::Grid3d: Test operator==", "[monad]", Types) {
-    const TestType grid1({2, 3, 4}, {0.5, 1.5, 2.0});
-    const TestType grid2({2, 3, 4}, {0.5, 1.5, 2.0});
+TEMPLATE_LIST_TEST_CASE("monad::grid::StructuredGrid: Test operator==", "[monad]", Types) {
+    using Resolution = typename TestType::Resolution;
+    using Size = typename TestType::Size;
+
+    Resolution resolution;
+    resolution.fill(2);
+
+    Size size;
+    size.fill(0.4);
+
+    const TestType grid1(resolution, size);
+    const TestType grid2(resolution, size);
 
     REQUIRE(grid1 == grid2);
 }
 
-TEMPLATE_LIST_TEST_CASE("monad::Grid3d: Test operator!=", "[monad]", Types) {
-    const TestType grid1({2, 3, 4}, {0.5, 1.5, 2.0});
+TEMPLATE_LIST_TEST_CASE("monad::grid::StructuredGrid: Test operator!=", "[monad]", Types) {
+    using Resolution = typename TestType::Resolution;
+    using Size = typename TestType::Size;
+
+    Resolution resolution;
+    resolution.fill(2);
+
+    Size size;
+    size.fill(0.4);
+
+    const TestType grid1(resolution, size);
 
     SECTION("Different resolution") {
-        const TestType grid2({3, 3, 4}, {0.5, 1.5, 2.0});
-        const TestType grid3({2, 4, 4}, {0.5, 1.5, 2.0});
-        const TestType grid4({2, 3, 5}, {0.5, 1.5, 2.0});
+        for (std::size_t i = 0; i < TestType::Dim; ++i) {
+            Resolution resolutionDifferent = resolution;
+            resolutionDifferent[i] += 1;
+
+            const TestType grid2(resolutionDifferent, size);
 
-        REQUIRE(grid1 != grid2);
-        REQUIRE(grid1 != grid3);
-        REQUIRE(grid1 != grid4);
+            REQUIRE(grid1 != grid2);
+        }
     }
 
     SECTION("Different size") {
-        const TestType grid2({2, 3, 4}, {0.6, 1.5, 2.0});
-        const TestType grid3({2, 3, 4}, {0.5, 1.6, 2.0});
-        const TestType grid4({2, 3, 4}, {0.5, 1.5, 2.1});
+        for (std::size_t i = 0; i < TestType::Dim; ++i) {
+            Size sizeDifferent = size;
+            sizeDifferent[i] += 0.1;
+
+            const TestType grid2(resolution, sizeDifferent);
 
-        REQUIRE(grid1 != grid2);
-        REQUIRE(grid1 != grid3);
-        REQUIRE(grid1 != grid4);
+            REQUIRE(grid1 != grid2);
+        }
     }
 }
diff --git a/tests/io/gmsh/write_gmsh_densities_test.cpp b/tests/io/gmsh/write_gmsh_densities_test.cpp
index ba42ea0..db855f9 100644
--- a/tests/io/gmsh/write_gmsh_densities_test.cpp
+++ b/tests/io/gmsh/write_gmsh_densities_test.cpp
@@ -2,33 +2,26 @@
 #include <sstream>
 #include <string>
 #include <catch2/catch_template_test_macros.hpp>
-#include "monad/grid/quad4_grid.hpp"
-#include "monad/grid/quad8_grid.hpp"
-#include "monad/grid/hex8_grid.hpp"
-#include "monad/grid/hex20_grid.hpp"
+#include "monad/field/field_aliases.hpp"
 #include "monad/io/gmsh/write_gmsh_densities.hpp"
 
 using namespace monad;
 using namespace monad::io::gmsh;
 
-using Types = std::tuple<Quad4Grid, Quad8Grid, Hex8Grid, Hex20Grid>;
+using Types = std::tuple<DensityField2d, DensityField3d>;
 
 TEMPLATE_LIST_TEST_CASE("monad::io::gmsh: Test writeGmshDensities", "[monad]", Types) {
     using Resolution = typename TestType::Resolution;
-    using Size = typename TestType::Size;
 
     Resolution resolution;
     resolution.fill(1);
 
-    Size size;
-    size.fill(0.5);
-
-    TestType grid(resolution, size);
-    grid.setDensitiesRandom(1234);
+    TestType densityField(resolution);
+    densityField.setRandom(1234);
 
     std::ostringstream oss;
 
-    writeGmshDensities(oss, grid);
+    writeGmshDensities(oss, densityField);
 
     const std::string expected = "$ElementData\n"
                                  "1\n"
@@ -38,7 +31,7 @@ TEMPLATE_LIST_TEST_CASE("monad::io::gmsh: Test writeGmshDensities", "[monad]", T
                                  "0\n"
                                  "1\n"
                                  "1\n"
-                                 "1 0.497664\n"
+                                 "1 0.191519\n"
                                  "$EndElementData";
 
     REQUIRE(oss.str() == expected);
diff --git a/tests/io/gmsh/write_gmsh_elements_test.cpp b/tests/io/gmsh/write_gmsh_elements_test.cpp
index 0ca2c3c..4c5b71f 100644
--- a/tests/io/gmsh/write_gmsh_elements_test.cpp
+++ b/tests/io/gmsh/write_gmsh_elements_test.cpp
@@ -1,10 +1,7 @@
 #include <sstream>
 #include <string>
 #include <catch2/catch_test_macros.hpp>
-#include "monad/grid/quad4_grid.hpp"
-#include "monad/grid/quad8_grid.hpp"
-#include "monad/grid/hex8_grid.hpp"
-#include "monad/grid/hex20_grid.hpp"
+#include "monad/grid/grid_aliases.hpp"
 #include "monad/io/gmsh/write_gmsh_elements.hpp"
 
 using namespace monad;
diff --git a/tests/io/gmsh/write_gmsh_nodal_data_test.cpp b/tests/io/gmsh/write_gmsh_nodal_field_test.cpp
similarity index 100%
rename from tests/io/gmsh/write_gmsh_nodal_data_test.cpp
rename to tests/io/gmsh/write_gmsh_nodal_field_test.cpp
diff --git a/tests/io/gmsh/write_gmsh_nodes_test.cpp b/tests/io/gmsh/write_gmsh_nodes_test.cpp
index ea1f894..9f9ac4d 100644
--- a/tests/io/gmsh/write_gmsh_nodes_test.cpp
+++ b/tests/io/gmsh/write_gmsh_nodes_test.cpp
@@ -1,10 +1,7 @@
 #include <sstream>
 #include <string>
 #include <catch2/catch_test_macros.hpp>
-#include "monad/grid/quad4_grid.hpp"
-#include "monad/grid/quad8_grid.hpp"
-#include "monad/grid/hex8_grid.hpp"
-#include "monad/grid/hex20_grid.hpp"
+#include "monad/grid/grid_aliases.hpp"
 #include "monad/io/gmsh/write_gmsh_nodes.hpp"
 
 using namespace monad;
diff --git a/tests/io/save_grid_and_density_field_test.cpp b/tests/io/save_grid_and_density_field_test.cpp
new file mode 100644
index 0000000..59c8516
--- /dev/null
+++ b/tests/io/save_grid_and_density_field_test.cpp
@@ -0,0 +1,55 @@
+#include <tuple>
+#include <filesystem>
+#include <stdexcept>
+#include <catch2/catch_template_test_macros.hpp>
+#include "monad/grid/grid_aliases.hpp"
+#include "monad/field/density_field.hpp"
+#include "monad/io/save_grid_and_density_field.hpp"
+
+using namespace monad;
+using namespace monad::field;
+
+using Types = std::tuple<Quad4Grid, Quad8Grid, Hex8Grid, Hex20Grid>;
+
+TEMPLATE_LIST_TEST_CASE("monad: Test saveGridAndDensityField", "[monad]", Types) {
+    using Resolution = typename TestType::Resolution;
+    using Size = typename TestType::Size;
+    
+    Resolution resolution;
+    resolution.fill(2);
+
+    Size size;
+    size.fill(0.5);
+
+    const TestType grid(resolution, size);
+    const DensityField<TestType::Dim> densityField(resolution);
+
+    const std::filesystem::path folder = std::filesystem::path(__FILE__).parent_path();
+    std::filesystem::path file = folder / "output.msh";
+
+    SECTION("No errors") {
+        saveGridAndDensityField(grid, densityField, file.string());
+
+        REQUIRE(std::filesystem::is_regular_file(file));
+
+        std::filesystem::remove(file);
+    }
+
+    SECTION("Invalid field size") {
+        resolution[0] += 1;
+
+        REQUIRE_THROWS_AS(saveGridAndDensityField(grid, DensityField<TestType::Dim>(resolution), file.string()), std::invalid_argument);
+    }
+
+    SECTION("Invalid file extension") {
+        file = folder / "output.csv";
+
+        REQUIRE_THROWS_AS(saveGridAndDensityField(grid, densityField, file.string()), std::invalid_argument);
+    }
+
+    SECTION("Invalid file path") {
+        file = folder / "NonExistentDir" / "output.msh";
+
+        REQUIRE_THROWS_AS(saveGridAndDensityField(grid, densityField, file.string()), std::runtime_error);
+    }
+}
diff --git a/tests/io/save_grid_and_field_test.cpp b/tests/io/save_grid_and_field_test.cpp
deleted file mode 100644
index b8d149c..0000000
--- a/tests/io/save_grid_and_field_test.cpp
+++ /dev/null
@@ -1,63 +0,0 @@
-#include <tuple>
-#include <filesystem>
-#include <stdexcept>
-#include <catch2/catch_template_test_macros.hpp>
-#include "monad/grid/quad4_grid.hpp"
-#include "monad/grid/quad8_grid.hpp"
-#include "monad/grid/hex8_grid.hpp"
-#include "monad/grid/hex20_grid.hpp"
-#include "monad/io/save_grid_and_field.hpp"
-
-using namespace monad;
-
-using Types = std::tuple<Quad4Grid, Quad8Grid, Hex8Grid, Hex20Grid>;
-
-TEMPLATE_LIST_TEST_CASE("monad: Test saveGridAndField", "[monad]", Types) {
-    using Resolution = typename TestType::Resolution;
-    using Size = typename TestType::Size;
-    
-    Resolution resolution;
-    resolution.fill(2);
-
-    Size size;
-    size.fill(0.5);
-
-    TestType grid(resolution, size);
-    grid.setDensitiesRandom();
-    const int numNodes = static_cast<int>(grid.numNodes());
-
-    const std::filesystem::path folder = std::filesystem::path(__FILE__).parent_path();
-    std::filesystem::path file = folder / "output.msh";
-
-    SECTION("No errors") {
-        const Eigen::MatrixX2d field = Eigen::MatrixX2d::Random(numNodes, 2);
-
-        saveGridAndField(grid, field, file.string());
-
-        REQUIRE(std::filesystem::is_regular_file(file));
-
-        std::filesystem::remove(file);
-    }
-
-    SECTION("Invalid field size") {
-        const Eigen::MatrixX4d field = Eigen::MatrixX4d::Random(numNodes, 4);
-
-        REQUIRE_THROWS_AS(saveGridAndField(grid, field, file.string()), std::invalid_argument);
-    }
-
-    SECTION("Invalid file extension") {
-        const Eigen::MatrixX2d field = Eigen::MatrixX2d::Random(numNodes, 2);
-
-        file = folder / "output.csv";
-
-        REQUIRE_THROWS_AS(saveGridAndField(grid, field, file.string()), std::invalid_argument);
-    }
-
-    SECTION("Invalid file path") {
-        const Eigen::MatrixX2d field = Eigen::MatrixX2d::Random(numNodes, 2);
-
-        file = folder / "NonExistentDir" / "output.msh";
-
-        REQUIRE_THROWS_AS(saveGridAndField(grid, field, file.string()), std::runtime_error);
-    }
-}
diff --git a/tests/io/save_grid_and_nodal_field_test.cpp b/tests/io/save_grid_and_nodal_field_test.cpp
new file mode 100644
index 0000000..2ef0cde
--- /dev/null
+++ b/tests/io/save_grid_and_nodal_field_test.cpp
@@ -0,0 +1,81 @@
+#include <tuple>
+#include <filesystem>
+#include <stdexcept>
+#include <catch2/catch_template_test_macros.hpp>
+#include "monad/grid/grid_aliases.hpp"
+#include "monad/io/save_grid_and_nodal_field.hpp"
+
+using namespace monad;
+
+using Types = std::tuple<Quad4Grid, Quad8Grid, Hex8Grid, Hex20Grid>;
+
+TEMPLATE_LIST_TEST_CASE("monad: Test saveGridAndNodalField", "[monad]", Types) {
+    using Resolution = typename TestType::Resolution;
+    using Size = typename TestType::Size;
+    
+    Resolution resolution;
+    resolution.fill(2);
+
+    Size size;
+    size.fill(0.5);
+
+    const TestType grid(resolution, size);
+    const int numNodes = static_cast<int>(grid.numNodes());
+
+    const std::filesystem::path folder = std::filesystem::path(__FILE__).parent_path();
+    std::filesystem::path file = folder / "output.msh";
+
+    SECTION("No errors") {
+        SECTION("Scalar field") {
+            const Eigen::VectorXd field = Eigen::VectorXd::Random(numNodes);
+
+            saveGridAndNodalField(grid, field, file.string());
+
+            REQUIRE(std::filesystem::is_regular_file(file));
+
+            std::filesystem::remove(file);
+        }
+
+        SECTION("2D Vector field") {
+            const Eigen::MatrixX2d field = Eigen::MatrixX2d::Random(numNodes, 2);
+
+            saveGridAndNodalField(grid, field, file.string());
+
+            REQUIRE(std::filesystem::is_regular_file(file));
+
+            std::filesystem::remove(file);
+        }
+
+        SECTION("3D Vector field") {
+            const Eigen::MatrixX3d field = Eigen::MatrixX3d::Random(numNodes, 3);
+
+            saveGridAndNodalField(grid, field, file.string());
+
+            REQUIRE(std::filesystem::is_regular_file(file));
+
+            std::filesystem::remove(file);
+        }
+    }
+
+    SECTION("Invalid field size") {
+        const Eigen::VectorXd field = Eigen::VectorXd::Random(numNodes - 1);
+
+        REQUIRE_THROWS_AS(saveGridAndNodalField(grid, field, file.string()), std::invalid_argument);
+    }
+
+    SECTION("Invalid file extension") {
+        const Eigen::VectorXd field = Eigen::VectorXd::Random(numNodes);
+
+        file = folder / "output.csv";
+
+        REQUIRE_THROWS_AS(saveGridAndNodalField(grid, field, file.string()), std::invalid_argument);
+    }
+
+    SECTION("Invalid file path") {
+        const Eigen::VectorXd field = Eigen::VectorXd::Random(numNodes);
+
+        file = folder / "NonExistentDir" / "output.msh";
+
+        REQUIRE_THROWS_AS(saveGridAndNodalField(grid, field, file.string()), std::runtime_error);
+    }
+}
diff --git a/tests/io/save_grid_test.cpp b/tests/io/save_grid_test.cpp
index dc5f319..80ce26d 100644
--- a/tests/io/save_grid_test.cpp
+++ b/tests/io/save_grid_test.cpp
@@ -2,10 +2,7 @@
 #include <filesystem>
 #include <stdexcept>
 #include <catch2/catch_template_test_macros.hpp>
-#include "monad/grid/quad4_grid.hpp"
-#include "monad/grid/quad8_grid.hpp"
-#include "monad/grid/hex8_grid.hpp"
-#include "monad/grid/hex20_grid.hpp"
+#include "monad/grid/grid_aliases.hpp"
 #include "monad/io/save_grid.hpp"
 
 using namespace monad;
@@ -23,13 +20,12 @@ TEMPLATE_LIST_TEST_CASE("monad: Test saveGrid", "[monad]", Types) {
     size.fill(0.5);
 
     TestType grid(resolution, size);
-    grid.setDensitiesRandom();
 
     const std::filesystem::path folder = std::filesystem::path(__FILE__).parent_path();
     std::filesystem::path file = folder / "output.msh";
 
     SECTION("No errors") {
-        saveGrid(grid, file.string(), true);
+        saveGrid(grid, file.string());
 
         REQUIRE(std::filesystem::is_regular_file(file));
 
@@ -39,12 +35,12 @@ TEMPLATE_LIST_TEST_CASE("monad: Test saveGrid", "[monad]", Types) {
     SECTION("Invalid file extension") {
         file = folder / "output.csv";
 
-        REQUIRE_THROWS_AS(saveGrid(grid, file.string(), true), std::invalid_argument);
+        REQUIRE_THROWS_AS(saveGrid(grid, file.string()), std::invalid_argument);
     }
 
     SECTION("Invalid file path") {
         file = folder / "NonExistentDir" / "output.msh";
 
-        REQUIRE_THROWS_AS(saveGrid(grid, file.string(), true), std::runtime_error);
+        REQUIRE_THROWS_AS(saveGrid(grid, file.string()), std::runtime_error);
     }
 }
diff --git a/tests/material/bounds_test.cpp b/tests/material/bounds_test.cpp
new file mode 100644
index 0000000..c77a3e1
--- /dev/null
+++ b/tests/material/bounds_test.cpp
@@ -0,0 +1,44 @@
+#include <stdexcept>
+#include <catch2/catch_template_test_macros.hpp>
+#include "monad/material/transport/linear_transport_material.hpp"
+#include "monad/field/density_field.hpp"
+#include "monad/material/bounds.hpp"
+#include "monad/detail/constants.hpp"
+
+using namespace monad;
+using namespace monad::material;
+using namespace monad::field;
+
+using Types = std::tuple<LinearTransportMaterial<2>, LinearTransportMaterial<3>>;
+
+TEMPLATE_LIST_TEST_CASE("monad: Test voigtBound/reussBound", "[monad]", Types) {
+    using Resolution = typename DensityField<TestType::Dim>::Resolution;
+
+    const TestType material(2.1);
+
+    Resolution resolution;
+    resolution.fill(2);
+
+    DensityField<TestType::Dim> densityField(resolution);
+
+    SECTION("Solid material: reuss=voigt=K") {
+        const auto K = material.materialTensor();
+
+        densityField.setOnes();
+
+        const auto voigt = voigtBound(material, densityField);
+        const auto reuss = reussBound(material, densityField);
+
+        REQUIRE(voigt.isApprox(K, NUMERICAL_ZERO));
+        REQUIRE(reuss.isApprox(K, NUMERICAL_ZERO));
+    }
+
+    SECTION("Random material: reuss≤voigt") {
+        densityField.setRandom(1234);
+        
+        const auto voigt = voigtBound(material, densityField);
+        const auto reuss = reussBound(material, densityField);    
+
+        REQUIRE(reuss.trace() <= voigt.trace());
+    }
+}
diff --git a/tests/material/mechanical/linear_elastic_material_2d_test.cpp b/tests/material/mechanical/linear_elastic_material_2d_test.cpp
index ab824f0..dd466dc 100644
--- a/tests/material/mechanical/linear_elastic_material_2d_test.cpp
+++ b/tests/material/mechanical/linear_elastic_material_2d_test.cpp
@@ -1,7 +1,6 @@
 #include <stdexcept>
 #include <catch2/catch_test_macros.hpp>
 #include "monad/material/mechanical/linear_elastic_material_2d.hpp"
-#include "monad/grid/quad4_grid.hpp"
 #include "monad/detail/constants.hpp"
 
 using namespace monad;
@@ -54,34 +53,6 @@ TEST_CASE("monad::LinearElasticMaterial2d: Test materialTensor", "[monad]") {
     REQUIRE(material.materialTensor().isApprox(C, NUMERICAL_ZERO));
 }
 
-TEST_CASE("monad::LinearElasticMaterial2d: Test voigt/reuss", "[monad]") {        
-    const LinearElasticMaterial2d material(1.0, 0.3, LinearElasticMaterial2d::PlaneCondition::PlaneStress);
-
-    // Just using to get density values (does not matter 2D vs. 3D)
-    Quad4Grid grid({3, 3}, {1.0, 1.0});
-
-    SECTION("Solid material: reuss=voigt=C") {
-        const auto C = material.materialTensor();
-
-        grid.setDensitiesOnes();
-        
-        const auto voigt = material.voigt(grid);
-        const auto reuss = material.reuss(grid);
-        
-        REQUIRE(voigt.trace() == reuss.trace());
-        REQUIRE(voigt.trace() == C.trace());
-    }
-
-    SECTION("Random material: reuss≤voigt") {
-        grid.setDensitiesRandom(1234);
-        
-        const auto voigt = material.voigt(grid);
-        const auto reuss = material.reuss(grid);        
-
-        REQUIRE(reuss.trace() <= voigt.trace());
-    }
-}
-
 TEST_CASE("monad::LinearElasticMaterial2d: Test operator==", "[monad]") {
     LinearElasticMaterial2d material1(1.0, 0.3, LinearElasticMaterial2d::PlaneCondition::PlaneStress);
     LinearElasticMaterial2d material2(1.0, 0.3, LinearElasticMaterial2d::PlaneCondition::PlaneStress);
diff --git a/tests/material/mechanical/linear_elastic_material_3d_test.cpp b/tests/material/mechanical/linear_elastic_material_3d_test.cpp
index 7536679..3fd3def 100644
--- a/tests/material/mechanical/linear_elastic_material_3d_test.cpp
+++ b/tests/material/mechanical/linear_elastic_material_3d_test.cpp
@@ -1,7 +1,6 @@
 #include <stdexcept>
 #include <catch2/catch_test_macros.hpp>
 #include "monad/material/mechanical/linear_elastic_material_3d.hpp"
-#include "monad/grid/quad4_grid.hpp"
 #include "monad/detail/constants.hpp"
 
 using namespace monad;
@@ -47,34 +46,6 @@ TEST_CASE("monad::LinearElasticMaterial3d: Test materialTensor", "[monad]") {
     REQUIRE(material.materialTensor().isApprox(C, NUMERICAL_ZERO));
 }
 
-TEST_CASE("monad::LinearElasticMaterial3d: Test voigt/reuss", "[monad]") {        
-    const LinearElasticMaterial3d material(1.0, 0.3);
-
-    // Just using to get density values (does not matter 2D vs. 3D)
-    Quad4Grid grid({3, 3}, {1.0, 1.0});
-
-    SECTION("Solid material: reuss=voigt=C") {
-        const auto C = material.materialTensor();
-
-        grid.setDensitiesOnes();
-        
-        const auto voigt = material.voigt(grid);
-        const auto reuss = material.reuss(grid);
-
-        REQUIRE(voigt.trace() == reuss.trace());
-        REQUIRE(voigt.trace() == C.trace());
-    }
-
-    SECTION("Random material: reuss≤voigt") {
-        grid.setDensitiesRandom(1234);
-        
-        const auto voigt = material.voigt(grid);
-        const auto reuss = material.reuss(grid);        
-
-        REQUIRE(reuss.trace() <= voigt.trace());
-    }
-}
-
 TEST_CASE("monad::LinearElasticMaterial3d: Test operator==", "[monad]") {
     LinearElasticMaterial3d material1(1.0, 0.3);
     LinearElasticMaterial3d material2(1.0, 0.3);
diff --git a/tests/material/multiphysics/linear_piezoelectric_material_2d_test.cpp b/tests/material/multiphysics/linear_piezoelectric_material_2d_test.cpp
deleted file mode 100644
index 4ca5965..0000000
--- a/tests/material/multiphysics/linear_piezoelectric_material_2d_test.cpp
+++ /dev/null
@@ -1,166 +0,0 @@
-#include <stdexcept>
-#include <catch2/catch_test_macros.hpp>
-#include "monad/material/mechanical/linear_elastic_material_2d.hpp"
-#include "monad/material/transport/linear_transport_material_aliases.hpp"
-#include "monad/material/multiphysics/linear_piezoelectric_material.hpp"
-#include "monad/detail/constants.hpp"
-
-using namespace monad;
-
-TEST_CASE("monad::LinearPiezoelectricMaterial2d: Test initalization", "[monad]") {
-    using CouplingTensor = typename LinearPiezoelectricMaterial2d::CouplingTensor;
-
-    SECTION("Invalid piezoelectric tensor") {
-        const LinearElasticMaterial2d elasticMaterial(1.0, 0.3, monad::LinearElasticMaterial2d::PlaneCondition::PlaneStress);
-        const LinearDielectricMaterial2d dielectricMaterial(2.1);
-
-        const CouplingTensor d {
-            {10.0, 0.0, 0.0},
-            {0.0, 10.0, 10.0}
-        };
-
-        REQUIRE_THROWS_AS(LinearPiezoelectricMaterial2d(elasticMaterial, dielectricMaterial, d), std::invalid_argument);
-    }
-}
-
-TEST_CASE("monad::LinearPiezoelectricMaterial2d: Test elasticMaterial", "[monad]") {
-    using CouplingTensor = typename LinearPiezoelectricMaterial2d::CouplingTensor;
-
-    const LinearElasticMaterial2d elasticMaterial(1.0, 0.3, monad::LinearElasticMaterial2d::PlaneCondition::PlaneStress);
-    const LinearDielectricMaterial2d dielectricMaterial(2.1);
-
-    const CouplingTensor d {
-        {0.01, 0.0, 0.0},
-        {0.0, 0.01, 0.01}
-    };
-
-    const LinearPiezoelectricMaterial2d material(elasticMaterial, dielectricMaterial, d);
-
-    REQUIRE(material.elasticMaterial() == elasticMaterial);
-}
-
-TEST_CASE("monad::LinearPiezoelectricMaterial2d: Test dielectricMaterial", "[monad]") {
-    using CouplingTensor = typename LinearPiezoelectricMaterial2d::CouplingTensor;
-
-    const LinearElasticMaterial2d elasticMaterial(1.0, 0.3, monad::LinearElasticMaterial2d::PlaneCondition::PlaneStress);
-    const LinearDielectricMaterial2d dielectricMaterial(2.1);
-
-    const CouplingTensor d {
-        {0.01, 0.0, 0.0},
-        {0.0, 0.01, 0.01}
-    };
-
-    const LinearPiezoelectricMaterial2d material(elasticMaterial, dielectricMaterial, d);
-
-    REQUIRE(material.dielectricMaterial() == dielectricMaterial);
-}
-
-TEST_CASE("monad::LinearPiezoelectricMaterial2d: Test couplingTensor", "[monad]") {
-    using CouplingTensor = typename LinearPiezoelectricMaterial2d::CouplingTensor;
-
-    const LinearElasticMaterial2d elasticMaterial(1.0, 0.3, monad::LinearElasticMaterial2d::PlaneCondition::PlaneStress);
-    const LinearDielectricMaterial2d dielectricMaterial(2.1);
-
-    const CouplingTensor d {
-        {0.01, 0.0, 0.0},
-        {0.0, 0.01, 0.01}
-    };
-
-    const LinearPiezoelectricMaterial2d material(elasticMaterial, dielectricMaterial, d);
-
-    REQUIRE(material.couplingTensor().isApprox(d, NUMERICAL_ZERO));
-}
-
-TEST_CASE("monad::LinearPiezoelectricMaterial2d: Test materialTensor", "[monad]") {
-    using CouplingTensor = typename LinearPiezoelectricMaterial2d::CouplingTensor;
-    using MaterialTensor = typename LinearPiezoelectricMaterial2d::MaterialTensor;
-
-    const LinearElasticMaterial2d elasticMaterial(1.0, 0.3, monad::LinearElasticMaterial2d::PlaneCondition::PlaneStress);
-    const LinearDielectricMaterial2d dielectricMaterial(2.1);
-
-    const CouplingTensor d {
-        {0.01, 0.0, 0.0},
-        {0.0, 0.01, 0.01}
-    };
-
-    const LinearPiezoelectricMaterial2d material(elasticMaterial, dielectricMaterial, d);
-
-    MaterialTensor expected;
-    expected << elasticMaterial.materialTensor(), -d.transpose(),
-                -d, -dielectricMaterial.materialTensor();
-
-    REQUIRE(material.materialTensor().isApprox(expected, NUMERICAL_ZERO));
-}
-
-TEST_CASE("monad::LinearPiezoelectricMaterial2d: Test operator==", "[monad]") {
-    using CouplingTensor = typename LinearPiezoelectricMaterial2d::CouplingTensor;
-
-    const LinearElasticMaterial2d elasticMaterial(1.0, 0.3, monad::LinearElasticMaterial2d::PlaneCondition::PlaneStress);
-    const LinearDielectricMaterial2d dielectricMaterial(2.1);
-
-    const CouplingTensor d {
-        {0.01, 0.0, 0.0},
-        {0.0, 0.01, 0.01}
-    };
-
-    const LinearPiezoelectricMaterial2d material1(elasticMaterial, dielectricMaterial, d);
-    const LinearPiezoelectricMaterial2d material2(elasticMaterial, dielectricMaterial, d);
-
-    REQUIRE(material1 == material2);
-}
-
-TEST_CASE("monad::LinearPiezoelectricMaterial2d: Test operator!=", "[monad]") {
-    using CouplingTensor = typename LinearPiezoelectricMaterial2d::CouplingTensor;
-
-    SECTION("Different elastic materials") {
-        const LinearElasticMaterial2d elasticMaterial1(1.0, 0.3, monad::LinearElasticMaterial2d::PlaneCondition::PlaneStress);
-        const LinearElasticMaterial2d elasticMaterial2(1.1, 0.3, monad::LinearElasticMaterial2d::PlaneCondition::PlaneStress);
-        const LinearDielectricMaterial2d dielectricMaterial(2.1);
-
-        const CouplingTensor d {
-            {0.01, 0.0, 0.0},
-            {0.0, 0.01, 0.01}
-        };
-
-        LinearPiezoelectricMaterial2d material1(elasticMaterial1, dielectricMaterial, d);
-        LinearPiezoelectricMaterial2d material2(elasticMaterial2, dielectricMaterial, d);
-
-        REQUIRE(material1 != material2);
-    }
-
-    SECTION("Different electric materials") {
-        const LinearElasticMaterial2d elasticMaterial(1.0, 0.3, monad::LinearElasticMaterial2d::PlaneCondition::PlaneStress);
-        const LinearDielectricMaterial2d dielectricMaterial1(2.1);
-        const LinearDielectricMaterial2d dielectricMaterial2(2.2);
-
-        const CouplingTensor d {
-            {0.01, 0.0, 0.0},
-            {0.0, 0.01, 0.01}
-        };
-
-        const LinearPiezoelectricMaterial2d material1(elasticMaterial, dielectricMaterial1, d);
-        const LinearPiezoelectricMaterial2d material2(elasticMaterial, dielectricMaterial2, d);
-
-        REQUIRE(material1 != material2);
-    }
-
-    SECTION("Different piezoelectric tensors") {
-        const LinearElasticMaterial2d elasticMaterial(1.0, 0.3, monad::LinearElasticMaterial2d::PlaneCondition::PlaneStress);
-        const LinearDielectricMaterial2d dielectricMaterial(2.1);
-
-        const CouplingTensor d1 {
-            {0.01, 0.0, 0.0},
-            {0.0, 0.01, 0.01}
-        };
-
-        const CouplingTensor d2 {
-            {0.02, 0.0, 0.0},
-            {0.0, 0.01, 0.01}
-        };
-
-        const LinearPiezoelectricMaterial2d material1(elasticMaterial, dielectricMaterial, d1);
-        const LinearPiezoelectricMaterial2d material2(elasticMaterial, dielectricMaterial, d2);
-
-        REQUIRE(material1 != material2);
-    }
-}
diff --git a/tests/material/multiphysics/linear_piezoelectric_material_3d_test.cpp b/tests/material/multiphysics/linear_piezoelectric_material_3d_test.cpp
deleted file mode 100644
index c35c629..0000000
--- a/tests/material/multiphysics/linear_piezoelectric_material_3d_test.cpp
+++ /dev/null
@@ -1,177 +0,0 @@
-#include <stdexcept>
-#include <catch2/catch_test_macros.hpp>
-#include "monad/material/mechanical/linear_elastic_material_3d.hpp"
-#include "monad/material/transport/linear_transport_material_aliases.hpp"
-#include "monad/material/multiphysics/linear_piezoelectric_material.hpp"
-#include "monad/detail/constants.hpp"
-
-using namespace monad;
-
-TEST_CASE("monad::LinearPiezoelectricMaterial3d: Test initalization", "[monad]") {
-    using CouplingTensor = typename LinearPiezoelectricMaterial3d::CouplingTensor;
-
-    SECTION("Invalid piezoelectric tensor") {
-        const LinearElasticMaterial3d elasticMaterial(1.0, 0.3);
-        const LinearDielectricMaterial3d dielectricMaterial(2.1);
-
-        const CouplingTensor d {
-            {0.0, 0.0, 0.0, 10.0, 0.0, 0.0},
-            {0.0, 0.0, 10.0, 0.0, 0.0, 0.0},
-            {10.0, 10.0, 10.0, 0.0, 0.0, 0.0}
-        };
-
-        REQUIRE_THROWS_AS(LinearPiezoelectricMaterial3d(elasticMaterial, dielectricMaterial, d), std::invalid_argument);
-    }
-}
-
-TEST_CASE("monad::LinearPiezoelectricMaterial3d: Test elasticMaterial", "[monad]") {
-    using CouplingTensor = typename LinearPiezoelectricMaterial3d::CouplingTensor;
-
-    const LinearElasticMaterial3d elasticMaterial(1.0, 0.3);
-    const LinearDielectricMaterial3d dielectricMaterial(2.1);
-
-    const CouplingTensor d {
-        {0.0, 0.0, 0.0, 0.01, 0.0, 0.0},
-        {0.0, 0.0, 0.01, 0.0, 0.0, 0.0},
-        {0.01, 0.01, 0.01, 0.0, 0.0, 0.0}
-    };
-
-    const LinearPiezoelectricMaterial3d material(elasticMaterial, dielectricMaterial, d);
-
-    REQUIRE(material.elasticMaterial() == elasticMaterial);
-}
-
-TEST_CASE("monad::LinearPiezoelectricMaterial3d: Test dielectricMaterial", "[monad]") {
-    using CouplingTensor = typename LinearPiezoelectricMaterial3d::CouplingTensor;
-
-    const LinearElasticMaterial3d elasticMaterial(1.0, 0.3);
-    const LinearDielectricMaterial3d dielectricMaterial(2.1);
-
-    const CouplingTensor d {
-        {0.0, 0.0, 0.0, 0.01, 0.0, 0.0},
-        {0.0, 0.0, 0.01, 0.0, 0.0, 0.0},
-        {0.01, 0.01, 0.01, 0.0, 0.0, 0.0}
-    };
-
-    const LinearPiezoelectricMaterial3d material(elasticMaterial, dielectricMaterial, d);
-
-    REQUIRE(material.dielectricMaterial() == dielectricMaterial);
-}
-
-TEST_CASE("monad::LinearPiezoelectricMaterial3d: Test couplingTensor", "[monad]") {
-    using CouplingTensor = typename LinearPiezoelectricMaterial3d::CouplingTensor;
-
-    const LinearElasticMaterial3d elasticMaterial(1.0, 0.3);
-    const LinearDielectricMaterial3d dielectricMaterial(2.1);
-
-    const CouplingTensor d {
-        {0.0, 0.0, 0.0, 0.01, 0.0, 0.0},
-        {0.0, 0.0, 0.01, 0.0, 0.0, 0.0},
-        {0.01, 0.01, 0.01, 0.0, 0.0, 0.0}
-    };
-
-    const LinearPiezoelectricMaterial3d material(elasticMaterial, dielectricMaterial, d);
-
-    REQUIRE(material.couplingTensor().isApprox(d, NUMERICAL_ZERO));
-}
-
-TEST_CASE("monad::LinearPiezoelectricMaterial3d: Test materialTensor", "[monad]") {
-    using CouplingTensor = typename LinearPiezoelectricMaterial3d::CouplingTensor;
-    using MaterialTensor = typename LinearPiezoelectricMaterial3d::MaterialTensor;
-
-    const LinearElasticMaterial3d elasticMaterial(1.0, 0.3);
-    const LinearDielectricMaterial3d dielectricMaterial(2.1);
-
-    const CouplingTensor d {
-        {0.0, 0.0, 0.0, 0.01, 0.0, 0.0},
-        {0.0, 0.0, 0.01, 0.0, 0.0, 0.0},
-        {0.01, 0.01, 0.01, 0.0, 0.0, 0.0}
-    };
-
-    const LinearPiezoelectricMaterial3d material(elasticMaterial, dielectricMaterial, d);
-
-    MaterialTensor expected;
-    expected << elasticMaterial.materialTensor(), -d.transpose(),
-                -d, -dielectricMaterial.materialTensor();
-
-    REQUIRE(material.materialTensor().isApprox(expected, NUMERICAL_ZERO));
-}
-
-TEST_CASE("monad::LinearPiezoelectricMaterial3d: Test operator==", "[monad]") {
-    using CouplingTensor = typename LinearPiezoelectricMaterial3d::CouplingTensor;
-
-    const LinearElasticMaterial3d elasticMaterial(1.0, 0.3);
-    const LinearDielectricMaterial3d dielectricMaterial(2.1);
-
-    const CouplingTensor d {
-        {0.0, 0.0, 0.0, 0.01, 0.0, 0.0},
-        {0.0, 0.0, 0.01, 0.0, 0.0, 0.0},
-        {0.01, 0.01, 0.01, 0.0, 0.0, 0.0}
-    };
-
-    const LinearPiezoelectricMaterial3d material1(elasticMaterial, dielectricMaterial, d);
-    const LinearPiezoelectricMaterial3d material2(elasticMaterial, dielectricMaterial, d);
-
-    REQUIRE(material1 == material2);
-}
-
-TEST_CASE("monad::LinearPiezoelectricMaterial3d: Test operator!=", "[monad]") {
-    using CouplingTensor = typename LinearPiezoelectricMaterial3d::CouplingTensor;
-
-    SECTION("Different elastic materials") {
-        const LinearElasticMaterial3d elasticMaterial1(1.0, 0.3);
-        const LinearElasticMaterial3d elasticMaterial2(1.1, 0.3);
-
-        const LinearDielectricMaterial3d dielectricMaterial(2.1);
-
-        const CouplingTensor d {
-            {0.0, 0.0, 0.0, 0.01, 0.0, 0.0},
-            {0.0, 0.0, 0.01, 0.0, 0.0, 0.0},
-            {0.01, 0.01, 0.01, 0.0, 0.0, 0.0}
-        };
-
-        const LinearPiezoelectricMaterial3d material1(elasticMaterial1, dielectricMaterial, d);
-        const LinearPiezoelectricMaterial3d material2(elasticMaterial2, dielectricMaterial, d);
-
-        REQUIRE(material1 != material2);
-    }
-
-    SECTION("Different electric materials") {
-        const LinearElasticMaterial3d elasticMaterial(1.0, 0.3);
-        const LinearDielectricMaterial3d dielectricMaterial1(2.1);
-        const LinearDielectricMaterial3d dielectricMaterial2(2.2);
-
-        const CouplingTensor d {
-            {0.0, 0.0, 0.0, 0.01, 0.0, 0.0},
-            {0.0, 0.0, 0.01, 0.0, 0.0, 0.0},
-            {0.01, 0.01, 0.01, 0.0, 0.0, 0.0}
-        };
-
-        const LinearPiezoelectricMaterial3d material1(elasticMaterial, dielectricMaterial1, d);
-        const LinearPiezoelectricMaterial3d material2(elasticMaterial, dielectricMaterial2, d);
-
-        REQUIRE(material1 != material2);
-    }
-
-    SECTION("Different piezoelectric tensors") {
-        const LinearElasticMaterial3d elasticMaterial(1.0, 0.3);
-        const LinearDielectricMaterial3d dielectricMaterial(2.1);
-
-        const CouplingTensor d1 {
-            {0.0, 0.0, 0.0, 0.01, 0.0, 0.0},
-            {0.0, 0.0, 0.01, 0.0, 0.0, 0.0},
-            {0.01, 0.01, 0.01, 0.0, 0.0, 0.0}
-        };
-
-        const CouplingTensor d2 {
-            {0.0, 0.0, 0.0, 0.02, 0.0, 0.0},
-            {0.0, 0.0, 0.01, 0.0, 0.0, 0.0},
-            {0.01, 0.01, 0.01, 0.0, 0.0, 0.0}
-        };
-
-        const LinearPiezoelectricMaterial3d material1(elasticMaterial, dielectricMaterial, d1);
-        const LinearPiezoelectricMaterial3d material2(elasticMaterial, dielectricMaterial, d2);
-
-        REQUIRE(material1 != material2);
-    }
-}
diff --git a/tests/material/multiphysics/linear_piezoelectric_material_test.cpp b/tests/material/multiphysics/linear_piezoelectric_material_test.cpp
new file mode 100644
index 0000000..ff69eea
--- /dev/null
+++ b/tests/material/multiphysics/linear_piezoelectric_material_test.cpp
@@ -0,0 +1,239 @@
+#include <stdexcept>
+#include <catch2/catch_template_test_macros.hpp>
+#include "monad/material/mechanical/linear_elastic_material_2d.hpp"
+#include "monad/material/material_aliases.hpp"
+#include "monad/detail/constants.hpp"
+
+using namespace monad;
+
+using Types = std::tuple<LinearPiezoelectricMaterial2d, LinearPiezoelectricMaterial3d>;
+
+TEMPLATE_LIST_TEST_CASE("monad::LinearPiezoelectricMaterial: Test initalization", "[monad]", Types) {
+    using StiffnessTensor = typename TestType::StiffnessTensor;
+    using PermittivityTensor = typename TestType::PermittivityTensor;
+    using CouplingTensor = typename TestType::CouplingTensor;
+
+    SECTION("Invalid stiffness tensor") {
+        StiffnessTensor c = StiffnessTensor::Random();
+        // Make PSD
+        c = c.transpose() * c;
+        c(0, 0) = 0.0;
+
+        PermittivityTensor epsilon = PermittivityTensor::Random();
+        // Make PSD
+        epsilon = epsilon.transpose() * epsilon;
+        // Make PD
+        epsilon += PermittivityTensor::Identity();
+
+        const CouplingTensor d = CouplingTensor::Constant(0.01);
+
+        REQUIRE_THROWS_AS(TestType(c, epsilon, d), std::invalid_argument);
+    }
+
+    SECTION("Invalid permittivity tensor") {
+        StiffnessTensor c = StiffnessTensor::Random();
+        // Make PSD
+        c = c.transpose() * c;
+        // Make PD
+        c += StiffnessTensor::Identity();
+
+        PermittivityTensor epsilon = PermittivityTensor::Random();
+        // Make PSD
+        epsilon = epsilon.transpose() * epsilon;
+        epsilon(0, 0) = 0.0;
+
+        const CouplingTensor d = CouplingTensor::Constant(0.01);
+
+        REQUIRE_THROWS_AS(TestType(c, epsilon, d), std::invalid_argument);
+    }
+
+    SECTION("Invalid piezoelectric coupling tensor") {
+        StiffnessTensor c = StiffnessTensor::Random();
+        // Make PSD
+        c = c.transpose() * c;
+        // Make PD
+        c += StiffnessTensor::Identity();
+
+        PermittivityTensor epsilon = PermittivityTensor::Random();
+        // Make PSD
+        epsilon = epsilon.transpose() * epsilon;
+        // Make PD
+        epsilon += PermittivityTensor::Identity();
+
+        const CouplingTensor d = CouplingTensor::Constant(10.0);
+
+        REQUIRE_THROWS_AS(TestType(c, epsilon, d), std::invalid_argument);
+    }
+}
+
+TEMPLATE_LIST_TEST_CASE("monad::LinearPiezoelectricMaterial: Test stiffnessTensor", "[monad]", Types) {
+    using StiffnessTensor = typename TestType::StiffnessTensor;
+    using PermittivityTensor = typename TestType::PermittivityTensor;
+    using CouplingTensor = typename TestType::CouplingTensor;
+
+    StiffnessTensor c = StiffnessTensor::Random();
+    // Make PSD
+    c = c.transpose() * c;
+    // Make PD
+    c += StiffnessTensor::Identity();
+
+    PermittivityTensor epsilon = PermittivityTensor::Random();
+    // Make PSD
+    epsilon = epsilon.transpose() * epsilon;
+    // Make PD
+    epsilon += PermittivityTensor::Identity();
+
+    const CouplingTensor d = CouplingTensor::Constant(0.01);
+
+    const TestType material(c, epsilon, d);
+
+    REQUIRE(material.stiffnessTensor().isApprox(c, NUMERICAL_ZERO));
+}
+
+TEMPLATE_LIST_TEST_CASE("monad::LinearPiezoelectricMaterial: Test permittivityTensor", "[monad]", Types) {
+    using StiffnessTensor = typename TestType::StiffnessTensor;
+    using PermittivityTensor = typename TestType::PermittivityTensor;
+    using CouplingTensor = typename TestType::CouplingTensor;
+
+    StiffnessTensor c = StiffnessTensor::Random();
+    // Make PSD
+    c = c.transpose() * c;
+    // Make PD
+    c += StiffnessTensor::Identity();
+
+    PermittivityTensor epsilon = PermittivityTensor::Random();
+    // Make PSD
+    epsilon = epsilon.transpose() * epsilon;
+    // Make PD
+    epsilon += PermittivityTensor::Identity();
+
+    const CouplingTensor d = CouplingTensor::Constant(0.01);
+
+    const TestType material(c, epsilon, d);
+
+    REQUIRE(material.permittivityTensor().isApprox(epsilon, NUMERICAL_ZERO));
+}
+
+TEMPLATE_LIST_TEST_CASE("monad::LinearPiezoelectricMaterial: Test couplingTensor", "[monad]", Types) {
+    using StiffnessTensor = typename TestType::StiffnessTensor;
+    using PermittivityTensor = typename TestType::PermittivityTensor;
+    using CouplingTensor = typename TestType::CouplingTensor;
+
+    StiffnessTensor c = StiffnessTensor::Random();
+    // Make PSD
+    c = c.transpose() * c;
+    // Make PD
+    c += StiffnessTensor::Identity();
+
+    PermittivityTensor epsilon = PermittivityTensor::Random();
+    // Make PSD
+    epsilon = epsilon.transpose() * epsilon;
+    // Make PD
+    epsilon += PermittivityTensor::Identity();
+
+    const CouplingTensor d = CouplingTensor::Constant(0.01);
+
+    const TestType material(c, epsilon, d);
+
+    REQUIRE(material.couplingTensor().isApprox(d, NUMERICAL_ZERO));
+}
+
+TEMPLATE_LIST_TEST_CASE("monad::LinearPiezoelectricMaterial: Test materialTensor", "[monad]", Types) {
+    using StiffnessTensor = typename TestType::StiffnessTensor;
+    using PermittivityTensor = typename TestType::PermittivityTensor;
+    using CouplingTensor = typename TestType::CouplingTensor;
+    using MaterialTensor = typename TestType::MaterialTensor;
+
+    StiffnessTensor c = StiffnessTensor::Random();
+    // Make PSD
+    c = c.transpose() * c;
+    // Make PD
+    c += StiffnessTensor::Identity();
+
+    PermittivityTensor epsilon = PermittivityTensor::Random();
+    // Make PSD
+    epsilon = epsilon.transpose() * epsilon;
+    // Make PD
+    epsilon += PermittivityTensor::Identity();
+
+    const CouplingTensor d = CouplingTensor::Constant(0.01);
+
+    const TestType material(c, epsilon, d);
+
+    MaterialTensor op;
+    op << c, -d.transpose(),
+          -d, -epsilon;
+
+    REQUIRE(material.materialTensor().isApprox(op, NUMERICAL_ZERO));
+}
+
+TEMPLATE_LIST_TEST_CASE("monad::LinearPiezoelectricMaterial: Test operator==", "[monad]", Types) {
+    using StiffnessTensor = typename TestType::StiffnessTensor;
+    using PermittivityTensor = typename TestType::PermittivityTensor;
+    using CouplingTensor = typename TestType::CouplingTensor;
+
+    StiffnessTensor c = StiffnessTensor::Random();
+    // Make PSD
+    c = c.transpose() * c;
+    // Make PD
+    c += StiffnessTensor::Identity();
+
+    PermittivityTensor epsilon = PermittivityTensor::Random();
+    // Make PSD
+    epsilon = epsilon.transpose() * epsilon;
+    // Make PD
+    epsilon += PermittivityTensor::Identity();
+
+    const CouplingTensor d = CouplingTensor::Constant(0.01);
+
+    const TestType material1(c, epsilon, d);
+    const TestType material2(c, epsilon, d);
+
+    REQUIRE(material1 == material2);
+}
+
+TEMPLATE_LIST_TEST_CASE("monad::LinearPiezoelectricMaterial: Test operator!=", "[monad]", Types) {
+    using StiffnessTensor = typename TestType::StiffnessTensor;
+    using PermittivityTensor = typename TestType::PermittivityTensor;
+    using CouplingTensor = typename TestType::CouplingTensor;
+
+    StiffnessTensor c = StiffnessTensor::Random();
+    // Make PSD
+    c = c.transpose() * c;
+    // Make PD
+    c += StiffnessTensor::Identity();
+
+    PermittivityTensor epsilon = PermittivityTensor::Random();
+    // Make PSD
+    epsilon = epsilon.transpose() * epsilon;
+    // Make PD
+    epsilon += PermittivityTensor::Identity();
+
+    CouplingTensor d = CouplingTensor::Constant(0.01);
+
+    const TestType material1(c, epsilon, d);
+
+    SECTION("Different stiffness tensor") {
+        c(1, 1) += 0.01;
+
+        const TestType material2(c, epsilon, d);
+
+        REQUIRE(material1 != material2);
+    }
+    
+    SECTION("Different permittivity tensor") {
+        epsilon(1, 1) += 0.01;
+
+        const TestType material2(c, epsilon, d);
+
+        REQUIRE(material1 != material2);
+    }
+
+    SECTION("Different piezoelectric coupling tensor") {
+        d(1, 1) += 0.01;
+
+        const TestType material2(c, epsilon, d);
+
+        REQUIRE(material1 != material2);
+    }
+}
diff --git a/tests/material/transport/linear_transport_material_test.cpp b/tests/material/transport/linear_transport_material_test.cpp
index 971dde3..b11458e 100644
--- a/tests/material/transport/linear_transport_material_test.cpp
+++ b/tests/material/transport/linear_transport_material_test.cpp
@@ -1,10 +1,10 @@
 #include <stdexcept>
 #include <catch2/catch_template_test_macros.hpp>
 #include "monad/material/transport/linear_transport_material.hpp"
-#include "monad/grid/quad4_grid.hpp"
 #include "monad/detail/constants.hpp"
 
 using namespace monad;
+using namespace monad::material;
 
 using Types = std::tuple<LinearTransportMaterial<2>, LinearTransportMaterial<3>>;
 
@@ -45,34 +45,6 @@ TEMPLATE_LIST_TEST_CASE("monad::LinearTransportMaterial: Test materialTensor", "
     REQUIRE(material.materialTensor().isApprox(K, NUMERICAL_ZERO));
 }
 
-TEMPLATE_LIST_TEST_CASE("monad::LinearTransportMaterial: Test voigt/reuss", "[monad]", Types) {        
-    const TestType material(2.1);
-
-    // Just using to get density values (does not matter 2D vs. 3D)
-    Quad4Grid grid({3, 3}, {1.0, 1.0});
-
-    SECTION("Solid material: reuss=voigt=K") {
-        const auto K = material.materialTensor();
-
-        grid.setDensitiesOnes();
-
-        const auto voigt = material.voigt(grid);
-        const auto reuss = material.reuss(grid);
-
-        REQUIRE(voigt.trace() == reuss.trace());
-        REQUIRE(voigt.trace() == K.trace());
-    }
-
-    SECTION("Random material: reuss≤voigt") {
-        grid.setDensitiesRandom(1234);
-        
-        const auto voigt = material.voigt(grid);
-        const auto reuss = material.reuss(grid);        
-
-        REQUIRE(reuss.trace() <= voigt.trace());
-    }
-}
-
 TEMPLATE_LIST_TEST_CASE("monad::LinearTransportMaterial: Test operator==", "[monad]", Types) {
     const TestType material1(1.0);
     const TestType material2(1.0);
diff --git a/tests/solver/mechanical/linear_elastic_solver_test.cpp b/tests/solver/mechanical/linear_elastic_solver_test.cpp
index 9539859..78565f6 100644
--- a/tests/solver/mechanical/linear_elastic_solver_test.cpp
+++ b/tests/solver/mechanical/linear_elastic_solver_test.cpp
@@ -1,24 +1,41 @@
 #include <tuple>
 #include <stdexcept>
 #include <catch2/catch_template_test_macros.hpp>
-#include "monad/grid/quad4_grid.hpp"
-#include "monad/grid/quad8_grid.hpp"
-#include "monad/grid/hex8_grid.hpp"
-#include "monad/grid/hex20_grid.hpp"
-#include "monad/material/mechanical/linear_elastic_material.hpp"
-#include "monad/solver/mechanical/linear_elastic_solver.hpp"
+#include "monad/grid/grid_aliases.hpp"
+#include "monad/material/mechanical/linear_elastic_material_2d.hpp"
+#include "monad/material/mechanical/linear_elastic_material_3d.hpp"
+#include "monad/material/bounds.hpp"
+#include "monad/field/field_aliases.hpp"
+#include "monad/solver/solver_aliases.hpp"
 #include "monad/detail/eigen_utils.hpp"
 #include "monad/detail/constants.hpp"
 
 using namespace monad;
 using namespace monad::detail;
 
-using Types = std::tuple<Quad4Grid, Quad8Grid, Hex8Grid, Hex20Grid>;
-
-TEMPLATE_LIST_TEST_CASE("monad::LinearElasticSolver2d: Test solve", "[monad]", Types) {
-    using Resolution = typename TestType::Resolution;
-    using Size = typename TestType::Size;
-    using Material = LinearElasticMaterial<TestType::Dim>;
+template <class GridT, class MaterialT, class DensityFieldT, class SolverT>
+struct TypePair {
+    using Grid = GridT;
+    using Material = MaterialT;
+    using DensityField = DensityFieldT;
+    using Solver = SolverT;
+};
+
+using Types = std::tuple<
+    TypePair<Quad4Grid, LinearElasticMaterial2d, DensityField2d, LinearElasticSolver<Quad4Grid>>,
+    TypePair<Quad8Grid, LinearElasticMaterial2d, DensityField2d, LinearElasticSolver<Quad8Grid>>,
+    TypePair<Hex8Grid, LinearElasticMaterial3d, DensityField3d, LinearElasticSolver<Hex8Grid>>,
+    TypePair<Hex20Grid, LinearElasticMaterial3d, DensityField3d, LinearElasticSolver<Hex20Grid>>
+>;
+
+TEMPLATE_LIST_TEST_CASE("monad::LinearElasticSolver: Test solve", "[monad]", Types) {
+    using Grid = typename TestType::Grid;
+    using Material = typename TestType::Material;
+    using DensityField = typename TestType::DensityField;
+    using Solver = typename TestType::Solver;
+
+    using Resolution = typename Grid::Resolution;
+    using Size = typename Grid::Size;
     using StiffnessTensor = typename Material::MaterialTensor;
 
     Resolution resolution;
@@ -27,7 +44,7 @@ TEMPLATE_LIST_TEST_CASE("monad::LinearElasticSolver2d: Test solve", "[monad]", T
     Size size;
     size.fill(0.5);
 
-    TestType grid(resolution, size);
+    Grid grid(resolution, size);
     
     StiffnessTensor C = StiffnessTensor::Random();
     // Make PSD
@@ -37,23 +54,23 @@ TEMPLATE_LIST_TEST_CASE("monad::LinearElasticSolver2d: Test solve", "[monad]", T
 
     const Material material(C);
 
-    SECTION("Density=1 → CBar=material") {
-        grid.setDensitiesOnes();
-        const LinearElasticSolver solver(grid, material);
+    DensityField densityField(resolution);
 
-        const auto results = solver.solve();
-        const auto &CBar = results.CBar;
+    const Solver solver;
 
-        const auto expected = material.materialTensor();
+    SECTION("Density=1 → CBar=C") {
+        densityField.setOnes();
+
+        const auto results = solver.solve(grid, densityField, material);
+        const auto &CBar = results.CBar;
 
-        REQUIRE(CBar.isApprox(expected, NUMERICAL_ZERO));
+        REQUIRE(CBar.isApprox(C, NUMERICAL_ZERO));
     }
 
     SECTION("Density=0 → CBar=0") {
-        grid.setDensitiesZeros();
-        LinearElasticSolver solver(grid, material);
+        densityField.setZeros();
 
-        const auto results = solver.solve();
+        const auto results = solver.solve(grid, densityField, material);
         const auto &CBar = results.CBar;
 
         // Set zero threshold to one order of magnitude larger
@@ -61,18 +78,17 @@ TEMPLATE_LIST_TEST_CASE("monad::LinearElasticSolver2d: Test solve", "[monad]", T
     }
 
     SECTION("Random density") {
-        grid.setDensitiesRandom(1234);
-        const LinearElasticSolver solver(grid, material);
+        densityField.setRandom(1234);
 
-        auto results = solver.solve();
+        const auto results = solver.solve(grid, densityField, material);
         const auto &CBar = results.CBar;
 
         REQUIRE(isSymmetric(CBar));
         REQUIRE(isPD(CBar));
 
         SECTION("Voigt/Reuss bounds") {
-            const auto voigt = material.voigt(grid);
-            const auto reuss = material.reuss(grid);
+            const auto voigt = voigtBound(material, densityField);
+            const auto reuss = reussBound(material, densityField);
 
             REQUIRE(reuss.trace() <= CBar.trace());
             REQUIRE(CBar.trace() <= voigt.trace());
@@ -82,24 +98,22 @@ TEMPLATE_LIST_TEST_CASE("monad::LinearElasticSolver2d: Test solve", "[monad]", T
             Resolution shift;
             shift.fill(1);
 
-            grid.translate(shift);
+            densityField.translate(shift);
 
-            const LinearElasticSolver solver2(grid, material);
+            const auto resultsShift = solver.solve(grid, densityField, material);
 
-            results = solver.solve();
-            const auto &CBar2 = results.CBar;
+            const auto &CBarShift = resultsShift.CBar;
 
-            REQUIRE(CBar2.isApprox(CBar, NUMERICAL_ZERO));
+            REQUIRE(CBarShift.isApprox(CBar, NUMERICAL_ZERO));
         }
     }
 
     SECTION("Bad options") {
-        grid.setDensitiesRandom(1234);
-        const LinearElasticSolver solver(grid, material);
+        densityField.setRandom(1234);
 
         SolverOptions options;
         options.maxIterations = 1;
 
-        REQUIRE_THROWS_AS(solver.solve(options), std::runtime_error);
+        REQUIRE_THROWS_AS(solver.solve(grid, densityField, material, options), std::runtime_error);
     }
 }
diff --git a/tests/solver/multiphysics/linear_piezoelectric_solver_test.cpp b/tests/solver/multiphysics/linear_piezoelectric_solver_test.cpp
index 96d4b07..c7cfdfa 100644
--- a/tests/solver/multiphysics/linear_piezoelectric_solver_test.cpp
+++ b/tests/solver/multiphysics/linear_piezoelectric_solver_test.cpp
@@ -1,33 +1,43 @@
 #include <tuple>
 #include <stdexcept>
 #include <catch2/catch_template_test_macros.hpp>
-#include "monad/grid/quad4_grid.hpp"
-#include "monad/grid/quad8_grid.hpp"
-#include "monad/grid/hex8_grid.hpp"
-#include "monad/grid/hex20_grid.hpp"
-#include "monad/material/mechanical/linear_elastic_material.hpp"
-#include "monad/material/transport/linear_transport_material.hpp"
-#include "monad/material/multiphysics/linear_piezoelectric_material.hpp"
-#include "monad/solver/multiphysics/linear_piezoelectric_solver.hpp"
+#include "monad/grid/grid_aliases.hpp"
+#include "monad/material/material_aliases.hpp"
+#include "monad/material/bounds.hpp"
+#include "monad/field/field_aliases.hpp"
+#include "monad/solver/solver_aliases.hpp"
 #include "monad/detail/eigen_utils.hpp"
 #include "monad/detail/constants.hpp"
 
 using namespace monad;
 using namespace monad::detail;
 
-// using Types = std::tuple<Quad4Grid, Quad8Grid, Hex8Grid, Hex20Grid>;
-using Types = std::tuple<Hex20Grid>;
+template <class GridT, class MaterialT, class DensityFieldT, class SolverT>
+struct TypePair {
+    using Grid = GridT;
+    using Material = MaterialT;
+    using DensityField = DensityFieldT;
+    using Solver = SolverT;
+};
+
+using Types = std::tuple<
+    TypePair<Quad4Grid, LinearPiezoelectricMaterial2d, DensityField2d, LinearPiezoelectricSolver<Quad4Grid>>,
+    TypePair<Quad8Grid, LinearPiezoelectricMaterial2d, DensityField2d, LinearPiezoelectricSolver<Quad8Grid>>,
+    TypePair<Hex8Grid, LinearPiezoelectricMaterial3d, DensityField3d, LinearPiezoelectricSolver<Hex8Grid>>,
+    TypePair<Hex20Grid, LinearPiezoelectricMaterial3d, DensityField3d, LinearPiezoelectricSolver<Hex20Grid>>
+>;
 
 TEMPLATE_LIST_TEST_CASE("monad::LinearPiezoelectricSolver: Test solve", "[monad]", Types) {
-    using Resolution = typename TestType::Resolution;
-    using Size = typename TestType::Size;
-    using Element = typename TestType::Element;
-    using MechanicalMaterial = LinearElasticMaterial<Element::Dim>;
-    using ElectricalMaterial = LinearTransportMaterial<Element::Dim>;
-    using Material = LinearPiezoelectricMaterial<MechanicalMaterial, ElectricalMaterial>;
-    using StiffnessTensor = typename MechanicalMaterial::MaterialTensor;
+    using Grid = typename TestType::Grid;
+    using Material = typename TestType::Material;
+    using DensityField = typename TestType::DensityField;
+    using Solver = typename TestType::Solver;
+
+    using Resolution = typename Grid::Resolution;
+    using Size = typename Grid::Size;
     using CouplingTensor = typename Material::CouplingTensor;
-    using MaterialTensor = typename Material::MaterialTensor;
+    using StiffnessTensor = typename Material::StiffnessTensor;
+    using PermittivityTensor = typename Material::PermittivityTensor;
 
     Resolution resolution;
     resolution.fill(2);
@@ -35,7 +45,7 @@ TEMPLATE_LIST_TEST_CASE("monad::LinearPiezoelectricSolver: Test solve", "[monad]
     Size size;
     size.fill(0.5);
 
-    TestType grid(resolution, size);
+    Grid grid(resolution, size);
 
     StiffnessTensor c = StiffnessTensor::Random();
     // Make PSD
@@ -43,36 +53,37 @@ TEMPLATE_LIST_TEST_CASE("monad::LinearPiezoelectricSolver: Test solve", "[monad]
     // Make PD
     c += StiffnessTensor::Identity();
 
-    const MechanicalMaterial elasticMaterial(c);
-    const ElectricalMaterial dielectricMaterial(2.1);
+    PermittivityTensor epsilon = PermittivityTensor::Random();
+    // Make PSD
+    epsilon = epsilon.transpose() * epsilon;
+    // Make PD
+    epsilon += PermittivityTensor::Identity();
+
+    const CouplingTensor d = CouplingTensor::Constant(0.01);
 
-    const CouplingTensor d = 0.1 * CouplingTensor::Random();
+    const Material material(c, epsilon, d);
 
-    const Material material(elasticMaterial, dielectricMaterial, d);
+    DensityField densityField(resolution);
+
+    const Solver solver;
 
     SECTION("Density=1 → homogenized=material") {
-        grid.setDensitiesOnes();
-        const LinearPiezoelectricSolver solver(grid, material);
+        densityField.setOnes();
 
-        const auto results = solver.solve();
+        const auto results = solver.solve(grid, densityField, material);
         const auto &cBar = results.cBar;
         const auto &epsilonBar = results.epsilonBar;
         const auto &dBar = results.dBar;
 
-        MaterialTensor op;
-        op << cBar, -dBar.transpose(),
-              -dBar, -epsilonBar;
-
-        const MaterialTensor expected = material.materialTensor();
-
-        REQUIRE(op.isApprox(expected, NUMERICAL_ZERO));
+        REQUIRE(cBar.isApprox(c, NUMERICAL_ZERO));
+        REQUIRE(epsilonBar.isApprox(epsilon, NUMERICAL_ZERO));
+        REQUIRE(dBar.isApprox(d, NUMERICAL_ZERO));
     }
 
-    SECTION("Density=0 → operator=0") {
-        grid.setDensitiesZeros();
-        const LinearPiezoelectricSolver solver(grid, material);
+    SECTION("Density=0 → CBar=0") {
+        densityField.setZeros();
 
-        const auto results = solver.solve();
+        const auto results = solver.solve(grid, densityField, material);
         const auto &cBar = results.cBar;
         const auto &epsilonBar = results.epsilonBar;
         const auto &dBar = results.dBar;
@@ -84,64 +95,42 @@ TEMPLATE_LIST_TEST_CASE("monad::LinearPiezoelectricSolver: Test solve", "[monad]
     }
 
     SECTION("Random density") {
-        grid.setDensitiesRandom(1234);
-        const LinearPiezoelectricSolver solver(grid, material);
+        densityField.setRandom(1234);
 
-        auto results = solver.solve();
-        const auto &cBar = results.cBar;
-        const auto &epsilonBar = results.epsilonBar;
-        const auto &dBar = results.dBar;
+        const auto results = solver.solve(grid, densityField, material);
+        const auto &opBar = results.opBar;
 
-        REQUIRE(isSymmetric(cBar));
-        REQUIRE(isPD(cBar));
-
-        REQUIRE(isSymmetric(epsilonBar));
-        REQUIRE(isPD(epsilonBar));
+        REQUIRE(isSymmetric(opBar));
+        REQUIRE(!isPD(opBar));
 
         SECTION("Voigt/Reuss bounds") {
-            SECTION("cBar") {
-                const auto voigt = elasticMaterial.voigt(grid);
-                const auto reuss = elasticMaterial.reuss(grid);
-
-                REQUIRE(reuss.trace() <= cBar.trace());
-                REQUIRE(cBar.trace() <= voigt.trace());
-            }
-
-            SECTION("epsilonBar") {
-                const auto voigt = dielectricMaterial.voigt(grid);
-                const auto reuss = dielectricMaterial.reuss(grid);
+            const auto voigt = voigtBound(material, densityField);
+            const auto reuss = reussBound(material, densityField);
 
-                REQUIRE(reuss.trace() <= epsilonBar.trace());
-                REQUIRE(epsilonBar.trace() <= voigt.trace());
-            }
+            REQUIRE(reuss.trace() <= opBar.trace());
+            REQUIRE(opBar.trace() <= voigt.trace());
         }
 
         SECTION("Translational invariance") {
             Resolution shift;
             shift.fill(1);
 
-            grid.translate(shift);
+            densityField.translate(shift);
 
-            const LinearPiezoelectricSolver solver2(grid, material);
+            const auto resultsShift = solver.solve(grid, densityField, material);
 
-            results = solver.solve();
-            const auto &cBar2 = results.cBar;
-            const auto &epsilonBar2 = results.epsilonBar;
-            const auto &dBar2 = results.dBar;
+            const auto &opBarShift = resultsShift.opBar;
 
-            REQUIRE(cBar2.isApprox(cBar, NUMERICAL_ZERO));
-            REQUIRE(epsilonBar2.isApprox(epsilonBar, NUMERICAL_ZERO));
-            REQUIRE(dBar2.isApprox(dBar, NUMERICAL_ZERO));
+            REQUIRE(opBarShift.isApprox(opBar, NUMERICAL_ZERO));
         }
     }
 
     SECTION("Bad options") {
-        grid.setDensitiesRandom(1234);
-        const LinearPiezoelectricSolver solver(grid, material);
+        densityField.setRandom(1234);
 
         SolverOptions options;
         options.maxIterations = 1;
 
-        REQUIRE_THROWS_AS(solver.solve(options), std::runtime_error);
+        REQUIRE_THROWS_AS(solver.solve(grid, densityField, material, options), std::runtime_error);
     }
 }
diff --git a/tests/solver/scalar/linear_scalar_diffusion_solver_test.cpp b/tests/solver/scalar/linear_scalar_diffusion_solver_test.cpp
index 7d8d8c8..81f3310 100644
--- a/tests/solver/scalar/linear_scalar_diffusion_solver_test.cpp
+++ b/tests/solver/scalar/linear_scalar_diffusion_solver_test.cpp
@@ -1,70 +1,81 @@
 #include <tuple>
 #include <stdexcept>
 #include <catch2/catch_template_test_macros.hpp>
-#include "monad/grid/quad4_grid.hpp"
-#include "monad/grid/quad8_grid.hpp"
-#include "monad/grid/hex8_grid.hpp"
-#include "monad/grid/hex20_grid.hpp"
-#include "monad/material/transport/linear_transport_material.hpp"
-#include "monad/solver/scalar/linear_scalar_diffusive_solver.hpp"
+#include "monad/grid/grid_aliases.hpp"
+#include "monad/material/material_aliases.hpp"
+#include "monad/material/bounds.hpp"
+#include "monad/field/field_aliases.hpp"
+#include "monad/solver/solver_aliases.hpp"
 #include "monad/detail/eigen_utils.hpp"
 #include "monad/detail/constants.hpp"
 
 using namespace monad;
 using namespace monad::detail;
-using namespace monad::fem::scalar;
 
-template <class GridT, GradientConvention C>
+template <class GridT, class MaterialT, class DensityFieldT, class SolverT>
 struct TypePair {
     using Grid = GridT;
-    static constexpr GradientConvention Convention = C;
+    using Material = MaterialT;
+    using DensityField = DensityFieldT;
+    using Solver = SolverT;
 };
 
+// Negative sign convention: linear dielectric material
+// Positive sign convention: linear mass diffusive material
 using Types = std::tuple<
-    TypePair<Quad4Grid, GradientConvention::Negative>,
-    TypePair<Quad4Grid, GradientConvention::Positive>,
-    TypePair<Quad8Grid, GradientConvention::Negative>,
-    TypePair<Quad8Grid, GradientConvention::Positive>,
-    TypePair<Hex8Grid, GradientConvention::Negative>,
-    TypePair<Hex8Grid, GradientConvention::Positive>,
-    TypePair<Hex20Grid, GradientConvention::Negative>,
-    TypePair<Hex20Grid, GradientConvention::Positive>
+    TypePair<Quad4Grid, LinearDielectricMaterial2d, DensityField2d, LinearDielectricSolver<Quad4Grid>>,
+    TypePair<Quad4Grid, LinearMassDiffusiveMaterial2d, DensityField2d, LinearMassDiffusiveSolver<Quad4Grid>>,
+    TypePair<Quad8Grid, LinearDielectricMaterial2d, DensityField2d, LinearDielectricSolver<Quad8Grid>>,
+    TypePair<Quad8Grid, LinearMassDiffusiveMaterial2d, DensityField2d, LinearMassDiffusiveSolver<Quad8Grid>>,
+    TypePair<Hex8Grid, LinearDielectricMaterial3d, DensityField3d, LinearDielectricSolver<Hex8Grid>>,
+    TypePair<Hex8Grid, LinearMassDiffusiveMaterial3d, DensityField3d, LinearMassDiffusiveSolver<Hex8Grid>>,
+    TypePair<Hex20Grid, LinearDielectricMaterial3d, DensityField3d, LinearDielectricSolver<Hex20Grid>>,
+    TypePair<Hex20Grid, LinearMassDiffusiveMaterial3d, DensityField3d, LinearMassDiffusiveSolver<Hex20Grid>>
 >;
 
 TEMPLATE_LIST_TEST_CASE("monad::Solver: Test solve", "[monad]", Types) {
     using Grid = typename TestType::Grid;
+    using Material = typename TestType::Material;
+    using DensityField = typename TestType::DensityField;
+    using Solver = typename TestType::Solver;
+
     using Resolution = typename Grid::Resolution;
     using Size = typename Grid::Size;
-    using Element = typename Grid::Element;
-    using Solver = LinearScalarDiffusiveSolver<Grid, Element, TestType::Convention>;
+    using MaterialTensor = typename Material::MaterialTensor;
 
     Resolution resolution;
-    resolution.fill(3);
-    
+    resolution.fill(2);
+
     Size size;
     size.fill(0.5);
 
     Grid grid(resolution, size);
 
-    const LinearTransportMaterial<Element::Dim> material(2.1);
+    MaterialTensor K = MaterialTensor::Random();
+    // Make PSD
+    K = K.transpose() * K;
+    // Make PD
+    K += MaterialTensor::Identity();
+
+    const Material material(K);
+
+    DensityField densityField(resolution);
+
+    const Solver solver;
 
     SECTION("Density=1 → KBar=material") {
-        grid.setDensitiesOnes();
-        const Solver solver(grid, material);
+        densityField.setOnes();
 
-        const auto results = solver.solve();
+        const auto results = solver.solve(grid, densityField, material);
         const auto &KBar = results.KBar;
 
-        const auto expected = material.materialTensor();
-
-        REQUIRE(KBar.isApprox(expected, NUMERICAL_ZERO));
+        REQUIRE(KBar.isApprox(K, NUMERICAL_ZERO));
     }
 
     SECTION("Density=0 → KBar=0") {
-        grid.setDensitiesZeros();
-        const Solver solver(grid, material);
+        densityField.setZeros();
 
-        const auto results = solver.solve();
+        const auto results = solver.solve(grid, densityField, material);
         const auto &KBar = results.KBar;
 
         // Set zero threshold to one order of magnitude larger
@@ -72,18 +83,17 @@ TEMPLATE_LIST_TEST_CASE("monad::Solver: Test solve", "[monad]", Types) {
     }
 
     SECTION("Random density") {
-        grid.setDensitiesRandom(1234);
-        const Solver solver(grid, material);
+        densityField.setRandom(1234);
 
-        auto results = solver.solve();
+        const auto results = solver.solve(grid, densityField, material);
         const auto &KBar = results.KBar;
 
         REQUIRE(isSymmetric(KBar));
         REQUIRE(isPD(KBar));
 
         SECTION("Voigt/Reuss bounds") {
-            const auto voigt = material.voigt(grid);
-            const auto reuss = material.reuss(grid);
+            const auto voigt = voigtBound(material, densityField);
+            const auto reuss = reussBound(material, densityField);
 
             REQUIRE(reuss.trace() <= KBar.trace());
             REQUIRE(KBar.trace() <= voigt.trace());
@@ -93,24 +103,22 @@ TEMPLATE_LIST_TEST_CASE("monad::Solver: Test solve", "[monad]", Types) {
             Resolution shift;
             shift.fill(1);
 
-            grid.translate(shift);
+            densityField.translate(shift);
 
-            const Solver solver2(grid, material);
+            const auto resultsShift = solver.solve(grid, densityField, material);
 
-            results = solver.solve();
-            const auto &KBar2 = results.KBar;
+            const auto &KBarShift = resultsShift.KBar;
 
-            REQUIRE(KBar2.isApprox(KBar, NUMERICAL_ZERO));
+            REQUIRE(KBarShift.isApprox(KBar, NUMERICAL_ZERO));
         }
     }
 
     SECTION("Bad options") {
-        grid.setDensitiesRandom(1234);
-        const Solver solver(grid, material);
+        densityField.setRandom(1234);
 
         SolverOptions options;
         options.maxIterations = 1;
 
-        REQUIRE_THROWS_AS(solver.solve(options), std::runtime_error);
+        REQUIRE_THROWS_AS(solver.solve(grid, densityField, material, options), std::runtime_error);
     }
 }
diff --git a/tests/solver/solver_options_test.cpp b/tests/solver/solver_options_test.cpp
index 1a5c3c7..6c32e38 100644
--- a/tests/solver/solver_options_test.cpp
+++ b/tests/solver/solver_options_test.cpp
@@ -4,41 +4,46 @@
 using namespace monad;
 
 TEST_CASE("monad::SolverOptions: Test defaults", "[monad]") {
-    const auto options = SolverOptions::defaults();
+    const SolverOptions options;
 
     REQUIRE(options.maxIterations == 1000);
     REQUIRE(options.tolerance == 1e-6);
     REQUIRE(options.fields == FieldSave::None);
+    REQUIRE(options.verbose == false);
 }
 
 TEST_CASE("monad::SolverOptions: Test operator==", "[monad]") {
-    const auto options1 = SolverOptions::defaults();
-    const auto options2 = SolverOptions::defaults();
+    const SolverOptions options1;
+    const SolverOptions options2;
 
     REQUIRE(options1 == options2);
 }
 
 TEST_CASE("monad::SolverOptions: Test operator!=", "[monad]") {
-    const auto options1 = SolverOptions::defaults();
+    const SolverOptions options1;
+    SolverOptions options2;
 
     SECTION("Different maxIterations") {
-        auto options2 = SolverOptions::defaults();
         options2.maxIterations += 1;
 
         REQUIRE(options1 != options2);
     }
 
     SECTION("Different tolerance") {
-        auto options2 = SolverOptions::defaults();
         options2.tolerance += 1e-5;
 
         REQUIRE(options1 != options2);
     }
 
     SECTION("Different fields") {
-        auto options2 = SolverOptions::defaults();
         options2.fields = FieldSave::Total;
 
         REQUIRE(options1 != options2);
     }
+
+    SECTION("Different verbose") {
+        options2.verbose = true;
+
+        REQUIRE(options1 != options2);
+    }
 }