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[Johan Hake <johan.hake@xxxxxxxxx>]

Dominique!

Take a look in the hyperelastisity demo, where the energy is differentiated 
wrt the displacement twice:

  http://www.fenics.org/doc/demos/pde/hyperelasticity/python/documentation.html

Johan

On Friday September 3 2010 09:28:17 Dominique wrote:
> New question #123984 on DOLFIN:
> https://answers.launchpad.net/dolfin/+question/123984
> 
> Hello,
> 
> I am trying to use derivative() twice to compute the second-derivatives of
> a variational form with respect to the unknown. Here is an example
> program:
> 
> ---
> from dolfin import *
> import numpy as np
> 
> mesh = UnitSquare(5,5)
> V = FunctionSpace(mesh, 'CG', 1)
> u = Function(V)
> v = TestFunction(V)
> a = dot(grad(v), (1+u**2) * grad(u))*dx   # Nonlinear variational form
> 
> # First derivative of a.
> du = TrialFunction(V)
> da = derivative(a, u, du)
> 
> # Second derivative of a.
> d2u = TrialFunction(V)
> d2a = derivative(da, u, d2u)
> 
> # Evaluate a somewhere.
> u.vector()[:] = np.random.random(V.dim())
> a_val = assemble(a, mesh=mesh) ; print 'a(u) = ', a_val
> 
> # Evaluate da somewhere.
> da_val = assemble(da, mesh=mesh) ; print 'da(u) = ', da_val
> 
> # Evaluate d2a somewhere.
> d2a_val = assemble(d2a, mesh=mesh) ; print 'd2a(u) = ', d2a_val
> ---
> 
> The first two 'print' commands output:
> 
> a(u) =  <PETScVector of size 36>
> da(u) =  <PETScMatrix of size 36 x 36>
> 
> but the last assemble() command raises an exception:
> 
> "ufl.log.UFLException: Found product of basis functions, forms must be
> linear in each basis function argument"
> 
> What am I doing wrong here? Shouldn't I obtain a tensor of order 3 by
> differentiating da?
> 
> Thanks in advance!
> Dominique