Writing the variational form

General
1

Tangent matrix for my problem is given by,

Kij = −∫∇N_{i} C^{n+1} ∇N_{j} dV

C is a forth order tensor(shape(3,3,3,3))
∇N is a second order tensor.

And I tried writing the variational form as

a_Newton = inner(gradv(v), inner(consistent_tang(u), gradv(u_))) * dxm

v and u_ are my test and trial functions and u is the displacement from the last time step.
it shows the error

Shapes do not match.

Why am I unable to write the action of a 4 the order tensor on a second order tensor in fenics.

Thanks for reading the question
Anupama

2

You should read up on the definition of inner, and how it differs wrt dot:
https://fenics.readthedocs.io/projects/ufl/en/latest/manual/form_language.html?highlight=Inner#inner
and Form language — Unified Form Language (UFL) 2021.1.0 documentation

1 Like
3

Thank you very much for the reply. I understood I needed to use dot product. But still it is making problems. I tried writing the weak form as below,

a1,a2,a3,a4 = indices(4)
a_Newton = inner(gradv(v), as_tensor(consistent_tang(u)[a1,a2,a3,a4]* gradv(u_)[a3,a4],(a1,a2))) * dxm

It is now giving me a new error.

RecursionError: maximum recursion depth exceeded

I am still a beginner in Fenics. Is there anything important that I am missing? Is there any other way to write this product without using indices?