Shape of 2D mesh optimization

I have a FEniCSx code that solves coupled PDEs. The result of some postprocessing computations depend on the geometry of the mesh, which is 2D by the way. I plan to modify the mesh to study its impact on the solution.

I know about optimizers such as Optuna. What I do not know about is what are common approaches to this problem?

I am thinking to use Gmsh. I can fix 4 points that will not be changed (think of a square), I want to link 2 of those together via straight lines, and similarly with the remaining 2. I am left with a square or rectangle with 2 missing sides, and this is where I plan to modify the mesh. I thought about using Bsplines and possible patched BĂ©zier curves to complete the limits of the mesh. But I am not quite sure what parameters, other than the control points, I should pass to the optimizer. Maybe some score that comes out of an evaluation of the meshing? The less parameters or points, the higher the score.

Ideally I would like the mesh to be smooth (I need to compute a second spatial derivative).

Thanks for sharing any ideas.

This is a pretty open ended question as there are a lot of different optimization approaches. My answer is focused on the gradient based optimization case. Implementation is often a bit simpler for the gradient-free approach, but at the expense of significantly more expensive optimizations for large number of design variables.:

I would suggest looking at the legacy dolfin work done on dolfin-adjoint and it’s associated publications, although this software unfortunately hasn’t been updated to work with dolfinx (yet?).

Remeshing using gmsh between steps is likely going to introduce challenges for computing derivatives between steps (in a gradient based framework). A good place to start with the FEniCSx specific implementation might be this thread. I would also suggest looking at some of the work my colleagues Luca Scotzniovsky and @Ru_Xiang have done with a motor shape optimization problem where they solve a hyper-elasticity subproblem. The paper preprint is here and their work is up on github here.

A resource on Airfoil Shape Optimization. The authors spend a bit of time discussing some of the challenges of differentiable mesh deformation although they don’t solve a specific variational problem to deform the mesh.


Thanks and sorry for my low quality reply (on phone, in a rush). I didn’t know this was an active researxh domain, but I can understand because in general the solution to the PDE can drastically change for minor mesh adjustments. I don’t think my PDEs would be problematic in that regard but I am not 100 percent sure.

I am not worried about gradient method problems as I plan to use a random sampling and.then TPE (tree parzen estimator) of optuna. I would remesh from scratch every meshes rather than remeshing not from scratch. My geometry is 2D and doesn’t contain that many elements, so this part is fast compared to solving the PDE which takes several minutes (I have to use at least a degree 3 FE space).

This is definitely the benefit of gradient-free methods (easier implementation). Broadly speaking gradient based methods require something like 2-3 orders of magnitude less function evaluations, so if the cost of repeated function evaluations is preventing convergence, keep these approaches in the back of your mind.

A diagram might be helpful here, but I think the usage of b-splines or similar is the right approach. At the moment, I can’t think of other parameters that might be important here.

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