Improving and expanding solid material model tests

[aabrown100-git]

Problem

I recently added solid material model tests (#225, #229). These test whether the implementations for the 2nd PK stress S and material elasticity tensor CC in get_pk2cc() are correct by comparing them against finite difference approximations of the same quantities (S_FD, CC_FD). In these comparisons, we must specify a tolerance, that is, check that S = S_FD and CC = CC_FD within a user-defined tolerance. It can be difficult to determine what is an appropriate tolerance. A more reliable method is to calculate the order of convergence of S - S_FD and CC - CC_FD. If everything is correct, then the order of convergence will match the order of the finite difference scheme.

Solution

Modify solid material model tests to check the order of convergence of S - S_FD and CC - CC_FD. This can be done by computing S_FD for several finite difference perturbations $$h$$. For each $$h$$, we can compute the error, for example for S

$$e(h) = ||S - S_{FD}(h)||$$

where we can use the Frobenius norm

Then, the order of convergence can be calculated by considering $$log(e)$$ vs. $$log(h)$$ and computing the slope of the line that best fits this data.

I have implemented this in this branch.

Additional context

No response

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