generalizing linear elasticity to multiple materials

[gdmcbain]

As opposed to the mortar method of #305, I've been looking at 'monolithic' approaches to linear elastic problems (like say ‘structural vibration’ ex21) with multiple subdomains with different densities and Lamé parameters (as for heat transfer in ‘insulated wire’ ex17).

The approach of ex17 using subdomains worked fine, requiring only a minor modification of skfem.models.elasticity.linear_elasticity to:

@bilinear_form
def multielastic(u, du, v, dv, w):

    lame_lambda, lame_mu = w.w

    def C(T):
        return 2.0 * lame_mu * T + lame_lambda * eye(trace(T), T.shape[0])

    def Eps(dw):
        return 0.5 * (dw + transpose(dw))

    return ddot(C(Eps(du)), Eps(dv))

I think it should be possible to combine this into the one in skfem.models.elasticity so that it works whether w is a pair of scalars or, as here, a numpy.ndarray of shape (2, ib.nelems, len(ib.W)), formed with

lame_lambda_mu = np.stack([ib.zero_w() for _ in range(2)])
for subdomain, elements in m.subdomains.items():
    density[elements] = material[subdomain]['rho']
    for i, p in enumerate(lame_parameters(*map(
            material[subdomain].get, ['E', 'poisson']))):
        lame_lambda_mu[i, elements] = p