weak forms for hyperelasticity

@bhaveshshrimali (from #438):

Thanks! I was starting to take a look at implementing the hyperelasticity demo. The most natural examples to follow seem to be linear elasticity, nonlinear poisson and laplace with inhomogeneous bcs.

What would be the best place to look for functions like log, inv, det (like in UFL) when writing the weak forms? For instance the stress would have an expression that looks like the following in UFL:

def firstPKStress(u):
    F = Identity(len(u)) + grad(u) 
    J = det(F)
    return mu * F - mu * inv(F).T + J * (J-1) * inv(F).T

I can see the helper functions eye and grad, which should help in defining F as eye(1, u.shape[0]) + grad(u), but how about the determinant (det) and inverse (inv) ?

[kinnala]

I’m closing this discussion for now since the hyperelasticity example is merged. Let’s create a new issue if something pops up.

[kinnala]

And just to have an idea, there the bulk of the time is indeed taken by the solve

It’s not clear to me that this timing properly separates assembly time and linear solve time.

Looking at the source code of solving.py it seems that _solve_varproblem includes both: https://bitbucket.org/fenics-project/dolfin/src/946dbd3e268dc20c64778eb5b734941ca5c343e5/python/dolfin/fem/solving.py#lines-276

I assume linear solve time is more or less equal in both cases because no magical differences there. Your previous profiling indicates that in scikit-fem linear solve takes a total of 15 seconds. Assuming that linear solve time is more or less equal, the assembly of the Jacobian takes approximately a total of 10 seconds in FEniCS and 70 seconds in scikit-fem.

It’s actually surprisingly good, I would have expected a larger difference given the amount of optimizations done automatically by FEniCS.