Project 09: Nonmatching Grid Control with Overlapping Meshes - NMOPT

Goal¶

Use nonmatching meshes to control a PDE by prescribing values on an overlapping control grid that is independent of the state mesh.

This project is inspired by deal.II step-60.

Mathematical Formulation¶

Let $\Omega$ be the physical domain and let $\Omega_c$ be an overlapping control domain with its own mesh. The state is defined on $\Omega$ , while the control $u$ is defined on $\Omega_c$ .

Let

\Pi:U_h(\Omega_c)\to L^2(\Omega)

(1)

be the transfer operator from the nonmatching control mesh to the state mesh. For example, $\Pi u$ may be the restriction or interpolation of the control onto the overlap.

Solve

\min_{(y,u)} \frac12\|y-y_d\|_{L^2(\Omega)}^2 + \frac\alpha2\|u-u_{\mathrm{ref}}\|_{L^2(\Omega_c)}^2

(2)

subject to

\begin{cases} -\Delta y=\Pi u+f & \text{in }\Omega,\\ y=0 & \text{on }\partial\Omega. \end{cases}

(3)

The weak state equation is

\int_\Omega \nabla y\cdot\nabla v\,dx = \int_\Omega (\Pi u)v\,dx + \int_\Omega f v\,dx.

(4)

The adjoint equation is

\int_\Omega \nabla p\cdot\nabla v\,dx = \int_\Omega (y-y_d)v\,dx.

(5)

The control equation is

\alpha(u-u_{\mathrm{ref}}) - \Pi^*p=0 \qquad\text{in }U_h(\Omega_c),

(6)

where $\Pi^*$ is the adjoint transfer from the state mesh to the control mesh.

Implementation Hints¶

Build two triangulations and two DoFHandlers.
Assemble coupling matrices by quadrature over intersections.
Verify adjointness:

(\Pi u,p)_{L^2(\Omega)} = (u,\Pi^*p)_{L^2(\Omega_c)}.

(7)

Then implement reduced gradient.

Relevant Examples¶

deal.II: step-60 for nonmatching grids and overlapping mesh coupling.
Course code: codes/dealii/execs/laplacian.cc, codes/dealii/execs/unconstrained_poisson_optimization.cc.
Course lectures: lecture06.md, lecture07.md.

Deliverables¶

Coupling matrix between nonmatching meshes.
Adjoint consistency test.
Reduced-gradient optimization.
Optional box constraints on the control mesh.