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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.

In this paper we compute closed-form expressions for the topological derivative for three-dimensional time-harmonic electromagnetic waves for perfect conductors (Dirichlet condition), electromagnetic cavities (Neumann condition), absorbing obstacles (impedance condition), and dielectric inclusions (transmission conditions). The proofs are based on the computation of shape derivatives followed by asymptotic expansions using Mie series when infinitesimal spheres are considered. An exhaustive gallery of numerical experiments is presented, which demonstrate that the topological derivative is a very powerful tool for the detection of multiple electromagnetic scatterers without a priori information about their number, size, shape, or location. Numerical examples include highly demanding configurations where only a few incident directions and a few observation points (for near-field data) or a few far-field observation directions (for far-field data) are considered.

In this work we propose and analyze numerical methods for the approximation of the solution of Helmholtz transmission problems in two or three dimensions. This kind of problems arises in many applications related to scattering of acoustic, thermal and electromagnetic waves. Formulations based on boundary integral methods are powerful tools to deal with transmission problems in unbounded media. Different formulations using boundary integral equations can be found in the literature. We propose here new symmetric formulations based on a paper by Martin Costabel and Ernst P. Stephan (1985), that uses the Calderon projector for the interior and exterior problems to develop closed expressions for the interior and exterior Dirichlet-to-Neumann operators. These operators are then matched to obtain an integral system that is equivalent to the Helmholtz transmission problem and uses Cauchy data on the transmission boundary as unknowns. We show how to simplify the aspect and analysis of the method by employing an additional mortar unknown with respect to the ones used in the original paper, writing it in an appropriate way to devise Krylov type iterations based on the separate Dirichlet-to-Neumann operators.

In this paper we survey computational techniques based on boundary integral formulations for the simulation of thermal waves. Time-harmonic solutions to diffusion problems appear in many physical situations of interest and give rise to many interesting problems related to material characterization, parameter assessment or detection of defects. We review the main direct, indirect and mixed integral numerical methods for a model of scattering of thermal waves by many obstacles and discuss how multiple scattering techniques and other physical tools can be understood as iterative methods or used as preconditioners. We also deal with some transient problems that can be solved with boundary element methods using the Laplace transform and with coupled finite and boundary element schemes for non-homogeneous obstacles

In this paper we propose a hybrid between direct and indirect boundary integral methods to solve a transmission problem for the Helmholtz equation in Lipschitz and smooth domains. We present an exhaustive abstract study of the numerical approximation of the resulting system of boundary integral equations by means of Galerkin methods. Some particular examples of convergent schemes in the smooth case in two dimensions are given. Finally, we extend the results to a thermal scattering problem in a half plane with several obstacles and provide numerical results that illustrate the accuracy of our methods depending on the regularity of the interface.