Abstract
In this paper, we explore the possibility of solving 3D Maxwell’s equations in the presence of a nonlinear and/or inhomogeneous material response. We propose using a hybrid approach, which combines a boundary integral representation with a domain-based method. This hybrid approach has previously been successfully applied to 1D linear and nonlinear transient wave scattering problems. The basic idea of the approach is to propagate Maxwell’s equations inside the scattering objects forward in time using a domain-based method, while a boundary integral representation of the electromagnetic field is used to supply the domain-based method with the Required surface values. Thus, no grids outside the scattering objects are needed, and this greatly reduces the computational cost and complexity.
Highlights
Boundary Element method (BEM), as a tool for solving scattering problems, has several attractive features
The surface localization of boundary integral equations means that the boundary discretization, which leads to the BEM equations, can be optimized with respect to the geometry of each surface separately when there are more than one scattering object, which there usually is
For domain-based methods like Finite Element method (FEM) [3,4,5] and Finite Difference Time Domain method (FDTD), such an optimization can only be achieved by using non-uniform or multiple grids tailored to the geometrical shape of the objects
Summary
Boundary Element method (BEM), as a tool for solving scattering problems, has several attractive features. BEM, is exceptionally well suited for modeling scattering problems where the sources are slow on relevant timescales In this setting the boundary integral equations are derived in the spectral domain and the discretized equations defining BEM only needs to be solved for the small set of discrete frequencies that are required for an accurate representation of the time dependent source. For a domain-based method like FDTD, near-stationary sources are the worst possible case since FDTD is, as the name indicate, a time domain method and slow sources mean that the Maxwell’s equations have to be solved for a long interval of time, which is costly in terms of computational resources Given all these attractive features of BEM for solving scattering problems in electromagnetics, it is somewhat surprising that in a popularity contest, FEM and FDTD beat BEM hands down [6].
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