Some applications of substructuring and domain decomposition techniques to radiation and scattering of time-harmonic electromagnetic waves

Abstract

After a quick review of the domain decomposition methods, and, more particularly on their application to large size problems relative to radiation and scattering of time-harmonic waves, we describe two contributions of the authors in this context. The two contributions are related to the scattering of a time-harmonic electromagnetic wave by a large perfectly conducting structure including a deep cavity. The first contribution is a substructuring technique. It is used to increase the speed of the convergence of the iterative algorithm in a Multi-Level Fast Multipol Method (MLFMM) solution. Numerical experiments illustrate the effectiveness of the approach since the number of iterations of the underlying Krylov iterative method remains almost constant while increasing a characteristic length in the problem. The second contribution proposes an adaptation of the overlapping domain decomposition techniques for a boundary integral formulation. It is used here to perform a hybridization of an exact formulation, used at the opening of the cavity, with an asymptotic high-frequency method employed for the rest of the exterior boundary of the structure. Numerical results demonstrate the reliability and the efficiency of the method.

Convergence bounds of GMRES with Schwarz' preconditioner for the scattering problem

“…The Schwarz' preconditioner, as formulated in (11), has been used in earlier works for the computation of the Helmholtz problem in bounded domains (see [29,30]) and for the scattering waves (see [10,13,14]). …”

Coupling of a non‐overlapping domain decomposition method for a nodal finite element method with a boundary element method

“…Also, the solution of the BEM can be performed using an adapted non-overlapping DDM (see [32]) so that each iteration is carried out by mainly doing one iteration for the FEM and another one for the BEM. In this manner, only small-size systems have to be solved at each iteration.…”

Implementation of spectral basis functions in BEM/FEM/GSM Domain Decomposition Methods devoted to scattering and radiation applications

“…Many references are available in the literature [1][2][3][4][5][6][7][8]. They can be used as a process of distributing data from a computational domain among the machines of a cluster of PCs, or for the separation of the physical domain into regions which occurs when using different and adapted solving methods, and finally as a process of subdividing the solution of large linear system into smaller problems whose solutions can be used as a pre-conditioner or a solver.…”

“…These functions f n and g n are extended by zero on all of the other surfaces of the domain. Finally the electromagnetic field decomposition (11) is introduced in relation (9) and then in relation (8), yielding to the following decomposition of the currents on the equipment interface: 9) is the electric current induced on Γ by the nth magnetic current K n = f n × n belonging to Γ and extended by zero on the other surfaces of the scatterer. In the same way, J source solution of Eq.…”