Abstract

We consider functionals depending on solutions of boundary value problem, They can be regarded as shape functionals, because they usually depend on the underlying domain. Their (directional) derivatives with respect to variations of the domain are shape-gradients and expressions for those can be derived by means of (i) domain transformations or (ii) Hadamard representation formulas.

The two approaches yield two different but equivalent formulas. Both rely on solutions of two boundary value problems (BVPs), but one involves integrating their traces on the boundary of the domain, while the other evaluates integrals in the volume. Usually, the two BVPs can only be solved approximately, for instance, by finite element methods. However, when used with finite element solutions, the equivalence of the two formulas breaks down. By means of a comprehensive convergence analysis, we establish that the volume based expression for the shape gradient generally offers better accuracy in a finite element setting. The results are confirmed by several numerical experiments.