Shape derivatives for scattering problems

Abstract

In this paper we study shape derivatives of solutions of acoustic and elec- tromagnetic scattering problems in frequency domain from the perspective of differential forms following Hiptmair and Li’s work (2013 Ann. Mat. Pura Appl. 192 1077–98). Relying on variational formulations, we present a unified framework for the derivation of strong and weak forms of derivatives with respect to variations of the shape of an impenetrable (resp. penetrable) scatterer, when we impose Dirichlet, Neumann, or impedance (resp. transmission) conditions on its boundary (resp. interface). In 3D for degrees l = 0 and l = 1 of the forms we obtain known and new formulas for shape derivatives of solutions of Helmholtz and Maxwell equations. They can form the foundation for numerical approximation with finite elements or boundary elements.