Integral Equation Methods for Electrostatics, Acoustics, and Electromagnetics in Smoothly Varying, Anisotropic Media

Abstract

We present a collection of well-conditioned integral equation methods for the solution of electrostatic, acoustic or electromagnetic scattering problems involving anisotropic, inhomogeneous media. In the electromagnetic case, our approach involves a minor modification of a classical formulation. In the electrostatic or acoustic setting, we introduce a new vector partial differential equation, from which the desired solution is easily obtained. It is the vector equation for which we derive a well-conditioned integral equation. In addition to providing a unified framework for these solvers, we illustrate their performance using iterative solution methods coupled with the FFT-based technique of [1] to discretize and apply the relevant integral operators.

An FFT-accelerated direct solver for electromagnetic scattering from penetrable axisymmetric objects

“…It will also be useful to define the wavenumber k = ω √ µ. In the most general case, the material parameters are allowed to be spatially dependent tensors [38]. There are two canonical boundary value problems in classical electromagnetics, that of scattering from perfect electric conductors (PECs) and scattering in non-conducting (dielectric, or penetrable) materials with piecewise constant material properties.…”

Optimization of 2D Heterogeneous Lenses via BFGS and Volume Integral Equation Method

“…In general we can write the solution as a sum of two terms: E = E inc +E scat , H = H inc +H scat the incoming field and the scattered field where the incoming field E inc , H inc verifies the free space Maxwell's equations and the scattered field E scat , H scat verifies the radiation condition at infinity. This problem can be reformulated by a volume integral equation (see [4]). In this paper we will consider z-invariant geometries with TE waves and µ(x) = µ 0 .…”

“…In all cases we need a fast and efficient convolution with the free-space Green's function algorithm. In this paper we apply the recently developed method (see [3] [4]) to analyze homogeneous and inhomogeneous lens problems.…”

A fast rapidly convergent method for approximation of convolutions with applications to wave scattering and some other problems

“…On the other hand, most solvers involving the Helmholtz kernel are devoted to the numerical solution of the Lippmann-Schwinger integral equation [13, page 216]. While we do not intend to review all such work here, some recent contributions in this direction can be found in [1,2,5,6,10,14,15,17,21,22,24,25] and references therein. To the best of our knowledge, none of the methods cited above are designed to handle the more general class of weakly singular kernels.…”