Solving a free boundary problem with non-constant coefficients

The present article is concerned with the numerical solution of a free boundary problem for an elliptic state equation with non-constant coefficients. We maximize the Dirichlet energy functional over all domains of fixed volume. The domain under consideration is represented by a level set function which is driven by the objective’s shape gradient. The state is computed by the finite element method where the underlying triangulation is constructed by means of a marching cubes algorithm. We show that the combination of these tools lead to an efficient solver for general shape optimization problems.