ObstacleDesign via RegularizedComplementarity

[thartland]

This PR adds the capability to solve a design problem subject to an obstacle problem.

The obstacle problem is, formally

$$ \min_{u}\mathcal{E}_{\text{obstacle}}(u:\theta) \text{ s.t. } g(u:\theta):=M_{\text{lump}}(u-\theta)\geq 0 $$

The optimality conditions, with pressure $p$ Lagrange multiplier associated to the inequality constraint and regularized Fisher-Burmeister complementarity, are then constraints of an upper-level design problem

$$ \min_{u,p,\theta}E_{\text{design}}(u,p,\theta) \text{ s.t. regularized obstacle optimality constraints are satisfied} $$

The example problem uses a discretization of the design objective functional

$$ E_{\text{design}}[u,p,\theta]:=\frac{\gamma_{0}}{2}\int_{\Omega}\theta^{2}\mathrm{d}V+\frac{\gamma_{1}}{2}\int_{\Omega}|\nabla\theta|^{2}\mathrm{d}V+\frac{\gamma_{2}}{2}\int_{\Omega}p^{2}\mathrm{d}V+\frac{\gamma_{3}}{2}\int_{\Omega}|\nabla p|^{2}\mathrm{d}V $$

with scalars $\gamma_{0},\gamma_{1},\gamma_{2},\gamma_{3}\geq 0$.

This PR updates smiths submodule ContinuationSolvers which contains some of the necessary backend solver technology. The smith obstacle design problem example uses a SmithObstacleDesignProblem deriving from ContinuationSolvers ObstacleDesignProblem to then utilize a smith::FunctionalObjective and smith::FunctionalWeakForm's for design objective function, gradient, and Hessian callbacks. Furthermore, the solver uses appropriate mass weighted norms so that the nonlinear solver iterations do not show dramatic dependence on the degree of mesh refinement.

Here is the design fields I obtain when running with serial refinement 2. For context the upper surface is the displacement field, the bottom surface is the optimal obstacle, and the scalar field plotted in color over each surface is the pressure field.

[Copilot]

Pull request overview

Adds a new obstacle-design example that formulates an obstacle problem via regularized complementarity and solves the resulting MPEC using ContinuationSolvers, integrating Smith functional objectives/weak forms and enabling mass-weighted norms for mesh-refinement robustness.

Changes:

• Introduces obstacle_design_example.cpp demonstrating an obstacle design workflow built on MPECSolver.
• Adds CMake plumbing to build/install the new example and register it as an example test.
• Hooks the new obstacle_design example directory into the top-level examples build.

Reviewed changes

Copilot reviewed 4 out of 4 changed files in this pull request and generated 6 comments.

File	Description
examples/obstacle_design/obstacle_design_example.cpp	New obstacle design example and a Smith-backed ObstacleDesignProblem adapter for objective/derivative callbacks.
examples/obstacle_design/CMakeLists.txt	Builds/installs the new example and adds an MPI example test when continuation support is enabled.
examples/CMakeLists.txt	Adds the obstacle_design subdirectory to the examples build.

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[Copilot]

Pull request overview

Copilot reviewed 4 out of 4 changed files in this pull request and generated 2 comments.

[Copilot]

Pull request overview

Copilot reviewed 4 out of 4 changed files in this pull request and generated 2 comments.

[Copilot]

Pull request overview

Copilot reviewed 4 out of 4 changed files in this pull request and generated 3 comments.