Generalized impedance boundary conditions and shape derivatives for 3D Helmholtz problems

Abstract

This paper is concerned with the shape sensitivity analysis of the solution to the Helmholtz transmission problem for three-dimensional sound-soft or sound-hard obstacles coated by a thin layer. This problem can be asymptotically approached by exterior problems with an improved condition on the exterior boundary of the coated obstacle, called generalized impedance boundary condition (GIBC). Using a series expansion of the Laplacian operator in the neighborhood of the exterior boundary, we retrieve the first-order GIBCs characterizing the presence of an interior thin layer with a constant thickness. The first shape derivative of the solution to the original Helmholtz transmission problem solves a new thin layer transmission problem with non-vanishing jumps across the exterior and the interior boundary of the thin layer. We show that we can interchange the first-order differentiation with respect to the shape of the exterior boundary and the asymptotic approximation of the solution. Numerical experiments are presented to highlight the various theoretical results.