Constrained problem involving mesh deformation

dolfinx
1

Hi,

I have a problem exemplified by the following equations:

\nabla.\varepsilon_\mathrm{r}\nabla u = 0,

subject to the following constraint

\intop \mathbf{dr}\varepsilon_\mathrm{r}|\nabla u|^2 = C,

where u is the unknown field and \varepsilon_\mathrm{r} is a domain dependent material parameter.

The problem is to deform the geometry/mesh so as to realize the given constraint. I already know the mesh deformation field V_s as a solution to Laplace equation.

My question is, can I define a variational form where the material parameter \varepsilon_\mathrm{r} is expressed as a function of deformed field V_s rather than the original mesh? If true, I believe I should be able to define it as a constrained nonlinear problem. I would appreciate if knowledgeable people could give me pointers or refer me to relevant examples.

Thanks a lot

2

Could you elaborate on your problem a bit more? I struggle to understand some details.

  • What is the \mathbf{dr} operator?
  • You write “The problem is to deform the geometry…” but also “I already know the mesh deformation”, so is the deformation known or unknown?

Jumping the gun:

Yes, certainly. Similar constructs show up all the time for Lagrangian and ALE formulations. This work comes to mind: mae-207-fea-for-coupled-problems/fsi/fitted-fsi-example.py at master · david-kamensky/mae-207-fea-for-coupled-problems · GitHub In lines 169 to 175 you’ll see that the differential operators are written for the deformed grid, given the underformed grid and the deformation field.

3

Thank you very much for your feedback and links. I agree the post is ambiguous but let me try to clarify by answering your questions:

  1. The operator \mathbf{dr} is over the entire volume/geometry.
  2. I know how the geometry (i.e. the mesh) is deformed according to \mathbf{V}_s which is a solution of Laplace equation. But the magnitude of deformation (a scalar s) is unknown and needs to be determined by the constraint equation. I hope I do not misuse the notations by relating the deformed space \mathbf{x}_s in terms of the original space \mathbf{x}:
\mathbf{x}_s = \mathbf{x} + s\mathbf{V}_\mathrm{s}(\mathbf{x}).

Of course one can solve this problem using the optimization approach. But my problem involves only a single parameter for which optimization appears to be overkill and I wanted to try the idea described in the question.