A study of the numerical robustness of single-layer method with Fourier basis for multiple obstacle scattering in homogeneous media

Hélène Barucq 1 Juliette Chabassier 1 Ha Pham 1 Sébastien Tordeux 1
1 Magique 3D - Advanced 3D Numerical Modeling in Geophysics
LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau], Inria Bordeaux - Sud-Ouest
Abstract : We investigate efficient numerical methods for the problem of multiple-scattering of obstacles in homogeneous media. This is a first step towards the more general problem in strongly inhomogeneous media. The inhomegeneity for the multiple-scattering problem is caused by the presence of obstacles. For formations composed of a small number of medium-sized obstacles, satisfactory results can be obtained with optimized softwares based on standard discretization technique such as Finite Element Method (FEM). However, constraint by its need for meshing, a FEM loses its robustness, as the number of obstacles increases, or when their size decreases. As an alternative, we work with a Galerkin Integral Equation method, which we call Fourier Series - Single Layer (FS-SL) method, which describes the scattered wave as a superposition of single layer potentials and uses truncation of Fourier series to discretize the continuous problem. We describe in details the systems generated by the method, accompanied by a well-posedness study, for penetrable and impenetrable obstacles, the later involving Dirichlet, Neuman and Impedance boundary conditions. To study the numerical performance of the method, we limit ourselves to the case of disc - shaped obstacles. We first compare the results of our mesh-free method with Montjoie (a FE-based software) to validate the robustness for problems with a large number of small obstacles. We then investigate then efficiency of different solver types in the resolution of the dense linear system generated by FS-SL method. The study is done for Direct Solvers (Mumps, Lapack and Scalapack) and iterative GMRES-type Solvers with various preconditioners. We show that the optimal choice depends on the distance between obstacles, their size and number.
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Hélène Barucq, Juliette Chabassier, Ha Pham, Sébastien Tordeux. A study of the numerical robustness of single-layer method with Fourier basis for multiple obstacle scattering in homogeneous media. [Research Report] RR-8988, Inria Bordeaux Sud-Ouest. 2016. ⟨hal-01408904⟩

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