[QUESTION] Gradients wrt. to vertex position in FEM simulation

[andrinr]

Can we compute the gradients with respect to the mesh vertex position in a finite element simulation. If yes are there any examples that do so?

[shi-eric]

Are you asking about warp.fem specifically?

[andrinr]

yes

[gdaviet]

Hi @andrinr, I think the short answer is yes -- I am not sure we have an existing example that does exactly that, but happy to provide one. I would need a bit more context to answer more precisely. Do you mean:

1. Computing the gradient of a fem.integrate or fem.interpolate call with respect to the vertices of the underlying geometry? If so, that is possible using the usual backward (adjoint) differentiation: capture the integrate call in a wp.Tape and make sure the underlying geometry vertex position array has requires_grad=True, e.g, if geo is a fem.Tetmesh or fem.Hexmesh, geo.positions.requires_grad=True. Note that bilinear forms (sparse matrices) are not currently differentiable, though that is something I want to fix
2. Computing the gradient of a field or expression with respect to a space degrees of freedom (which will coincide with the mesh vertices for linear interpolation). In the current dev version of warp that is also possible in a forward differentiation way (i.e. computing the full gradient rather than the adjoint w.r.t some loss), by passing a "trial" field to fem.interpolate. The result will be a sparse matrix containing the gradient of the interpolated function with respect to trial field degrees of freedom at each quadrature epoint

[andrinr]

Hi @gdaviet :) I thinkk I am mostly interested in the first case. If you could come up with an example, that would be wonderful. If it helps, I have some ideas:

• Computing the gradients of the boundary nodes of a (2d) cylinder in a RANS airflow simulation. The objective function would be the mean of the RANS field. If that works, a shape optimization could be tried.
• The same thing as above, but instead of coming up with a RANS sim, it could be a simple adapation of the 2D NS example. The objective would be minimzing the Velocity Fluctuations (Turbulent Kinetic Energy - TKE).

Thanks for any help, much appreciated!

[gdaviet]

Cool, I'll try to setup a shape optimization example, I agree that it would be good to have one. Probably with a linear model to keep it simple