Keywords: Maxwell's equations, perfectly matched layer formalism, non-convex unbounded domain, finite volume time domain method, finite difference time domain method, discontinuous Galerkin time domain method.

Nowadays, one important problem with electromagnetic simulations remains in the need for more and more memory and computing time to solve real configurations. Indeed, the structures studied by the industry are always larger with more detail to obtain the most accurate simulated response for the given problem. To take into account these industrial constraints, research has been done on different numerical schemes to reduce the costs in terms of computing time and memory storage. In particular, one approach to this research consists of reducing as much as possible the computational domain necessary to the simulation.

In this context, many papers about the perfectly matched layer (PML) formalism have been produced. Generally, this formalism is described as a set of boundary layers with a distribution of absorbing materials located around a convex boundary [1]. In the most case, the boundary is defined by a Cartesian box. Theorical and numerical studies have shown the numerous advantages of the PML formalism, particularly its simplicity and its efficiency. This method also permits the reduction of the computational domain but it is not optimal. Actually, for a non convex structure, like an aircraft for example, by using this approach, we have still a lot of cells in the mesh of the domain which do not define the structure. Recently, some researchers have proposed a PML formalism adapted to the non-convex boundary, which allows us to avoid this problem. A theoretical study of this new model has been done and its advantage in the frequency domain has been numerically established [2], also in the temporal domain with a dispersive formulation [3,4]. In the time domain, the advantage obtained by the "non convex PML formalism" is not so obvious. Indeed, to have an efficient absorption of the wave, we need to reduce the C.F.L. condition, which reduces also the gain of the method, in terms of the computing time.

In our contribution, we present the "non convex PML formalism" for Maxwell's equations and study numerically its performace in terms of gain in computing time and memory storage by comparison with a classical "convex PML formalism". This study proposes different numerical schemes such as finite difference, finite volume and three-dimensional discontinuous Galerkin time domain methods.

1

J.P. Bérenger, "Three-Dimensional Perfectly Matched Layer for the Absorption of Electromagnetic Waves", Journal of Computational Physics, 127(2), 363-379, 1996. doi:10.1006/jcph.1996.0181

2

S.D. Gedney, "An Anisantropic PML Absorbing Media for the FDTD Simulation of Fields in Lossy and Dispersive Media", Electromagnetics, 16(4), 399-415, 1996. doi:10.1080/02726349608908487

3

S. Laurens, P.A. Mazet, "Perfectly Matched Layers for Star-Shaped Domains", WAVES 2008, 2009.

4

P.A. Mazet, L. Segui, B. Dah, "Sur l'existence et l'unicité des solutions pour le système de Maxwell harmonique en présence de couches de Bérenger", Compte-rendu de l'Académie des Sciences, 2001. doi:10.1016/S0764-4442(01)02077-8

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