Topology optimization example

[jorgensd]

Implementation of the structural problem from Pasteur Labs (mosaic) using DOLFINx adjoint.
Heads up @dionhaefner :)

Instead of using ADAM I've opted for using an optimization solver that can use reduced hessian vector products as well.

Bug-fixes along the way

• Compute_adjoint split blocked elements when calling extract_blocks. This was not intentional and therefore we now ensure that blocks are not extracted for the un-blocked case. Test added to verify that this was an issue on main.
  - Nonlinear test locally exposed an issue with snes converging early if tolerance is too low.: This has been fixed properly on main with Mypy fixes #82

Summary of outputs:

[corner_load] one tape recompute (Jhat(rho)): 4.7843e-02 s
[corner_load] one adjoint solve (Jhat.derivative()): 6.0980e-02 s

[full_face_load] one tape recompute (Jhat(rho)): 5.1036e-02 s
[full_face_load] one adjoint solve (Jhat.derivative()): 6.2825e-02 s

[dionhaefner]

Cool stuff @jorgensd! (and CC @andrinr)

Please backport to Mosaic if you can, or tell us what + how to do it 🙏

[jorgensd]

Cool stuff @jorgensd! (and CC @andrinr)

Please backport to Mosaic if you can, or tell us what + how to do it 🙏

I'll clean up the demo a bit first, so it's clearer how things differ from legacy dolfin adjoint.
A general question is why did you choose the ADAM optimizer rather than another first order method like BFGS?

[dionhaefner]

We do use L-BFGS for most cases (example), and it's the optimizer we use to compare solvers in the paper.