Nonlinear Material Law from Table

Question

Hello,

I would like to use the NonlinearVariationalProblem with a nonlinear material which depends on the solution (e.g. in Maxwell equations the material gets saturated if the magnetic field is high enough). The material law is given as a chi(grad(u)) table and is interpolated between the given nodes (linear, spline, ...). I would like to do something like this:

a = chi(grad(u)) * inner(grad(u), grad(v)) * dx 

Is there a way to utilize the automatic differentiation feature and the nonlinear variational solver for this? or is it necessary to explicitly implement the newton method?

thanks
Florian

Answer 1

To enjoy the differentiation and Newton method chi would need to be implemented using UFL. I suppose that a table, you mention, would result in piece-wise expression which can be implemented using UFL Conditional. (Differentiation of Conditional is correct assuming that the expression is continuous.)

But I warn you that a deeply-nested combination of Conditionals (implementing a long table) may not scale well with the size of the table.

Implementing the table using DOLFIN Expression (possibly C++ expression for performance), treating the term using fixed-point, may be useful and efficient.

EDIT: I can imagine, the latter approach could be improved by the derivative of chi dchi provided by the code providing chi.

u = Function(V)
chi, dchi = compute_material_data(u) # returns Expressions
v = TestFunction(V)
F = chi*inner(grad(u), grad(v))*dx - rhs
dF  = derivative(F, u) + dchi*inner(grad(u), grad(v))*dx # or using UFL replace
solve(F == 0, bcs=bcs, J=dF)

Note this can really work as a proper Newton method if Expressions chi and dchi have a reference to u and update their values dynamically within each Newton step.

Comments

Added a comment how derivative of chi could be specified manually if available.