A Boundary Integral Formulation and a Topological Energy-Based Method for an Inverse 3D Multiple Scattering Problem with Sound-Soft, Sound-Hard, Penetrable, and Absorbing Objects

Abstract

Abstract In this paper, we study numerical methods for simulating acoustic scattering by multiple three-dimensional objects of different nature (penetrable, sound-soft, sound-hard and absorbing targets) simultaneously present in the background media. We derive and analyze a boundary integral system of equations that arises when the solution of the problem is represented via single-layer potentials. We give abstract necessary and sufficient conditions for convergence of Petrov–Galerkin discretizations and show that spectral methods satisfy these conditions. Superalgebraic convergence order of the discrete method for smooth objects is illustrated in some test cases. After that, we tackle the inverse problem of finding the shape of objects of different unknown nature from measurements of the total field at a set of receptors. We propose a numerical algorithm based on the computation of the topological energy of a weighted multifrequency least squares cost functional and present some numerical examples to illustrate its capabilities.