[firedrake] Index notation with higher-order derivatives

[Christian Jacobs]

I'm trying to express this in UFL:
http://amcg.ese.ic.ac.uk/~ctj10/images/equations.png

For the case of k = 2, I can write
F = inner(w[0], grad(grad(s0[0,0])[0])[0] + grad(grad(s0[0,1])[1])[1])*dx +
inner(w[1], grad(grad(s0[1,0])[0])[0] + grad(grad(s0[1,1])[1])[1])*dx
where w is a TestFunction in a VectorFunctionSpace and s0 (sigma) is a
Function in a TensorFunctionSpace.

I'd like to generalise this for an arbitrary value of k. I've tried

Aij = s0[i,j]
k = 2
for c in range(0, k):
   Aij = Aij.dx(j)
A = as_vector(Aij, i)
print A
F = inner(w, A)*dx

For k = 1, { A | A_{i_0} = sum_{i_1} (grad(StressOld[i_0, i_1]))[i_1]  }
For k = 2, { A | A_{i_0} = (grad(sum_{i_1} (grad(StressOld[i_0, i_1]))[i_1]
))[i_1] }
For k = 3, { A | A_{i_0} = sum_{i_1} (grad((grad(sum_{i_1}
(grad(StressOld[i_0, i_1]))[i_1] ))[i_1]))[i_1]

The cases k = 1 and k = 3 compile fine, but with k = 2 I'm getting the
following error message:
ufl.log.UFLException: Can only integrate scalar expressions. The integrand
is a tensor expression with value rank 0 and free indices (1,).

Using inner(w[i], A) doesn't get rid of the "free indices" issue. Any ideas
as to what's going wrong here?

The error can be reproduced with:

from firedrake import *
mesh = UnitSquareMesh(10, 10)
tfs = TensorFunctionSpace(mesh, "CG", 1)
vfs = VectorFunctionSpace(mesh, "CG", 1)
w = TestFunction(vfs)
s0 = Function(tfs).interpolate(Expression([[1,2], [3,4]]))
Aij = s0[i,j]
k = 2
for c in range(0, k):
   Aij = Aij.dx(j)
A = as_vector(Aij, i)
print A
F = inner(w, A)*dx
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