Compute derivatives in FEniCSX

1

Hi all! I am doing the sensitivity analysis and I have several questions about FEniCSX.

Question 1

Take the classical demo Hyperelasticity — FEniCSx tutorial (jorgensd.github.io) as an example. Now we assume the elasticity parameter mu is a cell-wise constant instead of a constant on the whole mesh.

S = dolfinx.FunctionSpace(mesh, ('DG', 0))
mu = dolfinx.Function(S)

And we can set some arbitrary values to mu

import numpy as np
NElem = mesh.topology.index_map(mesh.topology.dim).size_local # number of elements
with mu.vector.localForm() as mu_loc:
    mu_loc.setValues(np.arange(NElem).tolist(), np.arange(NElem).tolist())

How can I output the values of mu ? In this case it should be np.arange(NElem).tolist().

Question 2

Now we want to compute the derivative of F with respect to the parameter mu of each element

F = ufl.inner(ufl.grad(v), P)*dx - ufl.inner(v, B)*dx - ufl.inner(v, T)*ds(2) 
dFdmu1 = ufl.diff(F, mu1)
dFdmu2 = ufl.diff(F, mu2)
dFdmu3 = ufl.diff(F, mu3)
...

This process should be equivalent to compute cell-wise integrals as (since mu1, mu2, and mu3 are independent variables):

dFdmu1 = ufl.diff(F, mu) # take the integral on element 1
dFdmu2 = ufl.diff(F, mu) # take the integral on element 2
dFdmu3 = ufl.diff(F, mu) # take the integral on element 3
...

So is it possible to compute these derivatives conveniently in FEniCSX? I think it is doable with for-loops, but it will be very slow if we have a larger mesh.

Question 3
Finally, I want to assemble these vectors:

R1 = dolfinx.fem.assemble_vector(dFdmu1) # vector 1
R2 = dolfinx.fem.assemble_vector(dFdmu2) # vector 2
R3 = dolfinx.fem.assemble_vector(dFdmu2) # vector 3

Is there some way to assemble multiple vectors simultaneously to improve computational efficiency? It might be something like

dolfinx.fem.assemble_vector([dFdmu1, dFdmu2, dFdmu3])

and then we will obtain a matrix containing the assembled vectors.

Thanks!

2

As mu is a dolfinx.Function, you can print the values using mu.vector.array

For your other questions, you need to provide minimal working code examples, as you have not defined mu1-mu3, and I do not understand what you are trying to do.

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