The effect of near-zone preconditioning on electromagnetic integral equations of first and second kind

Oliver Wiedenmann ,Thomas F Eibert

doi:10.5194/ars-11-61-2013

Abstract

Abstract. The linear equation systems which arise from the discretization of surface integral equations are conveniently solved with iterative methods because of the possibility to employ fast integral methods like the Multilevel Fast Multipole Method. However, especially integral equations of the first kind often lead to very ill-conditioned systems, which require the usage of effective preconditioners. In this paper, the regularization property of near-zone preconditioning operators on the Electric Field Integral Equation is demonstrated and investigated for problems of different size. Furthermore, comparisons are drawn to second-kind integral equations such as the Combined Field Integral Equation.

Highlights

Boundary integral equations are widely used for the solution of electromagnetic scattering and radiation problems, mainly because they require only the discretization of the surfaces of the involved objects and the radiation condition is imposed implicitly
The Magnetic Field Integral Equation (MFIE) shows better convergence behavior, since it is an integral equation of the second kind with a compact integral operator, but it can only be formulated for objects with closed surfaces and shows worse solution accuracy due to the difficulty of discretizing the identity operator accurately
We investigate the convergence characteristics of near-zone preconditioned Electric Field Integral Equation (EFIE) formulations and second-kind integral equations such as Combined Field Integral Equation (CFIE) for problems of different size

Summary

Introduction

Boundary integral equations are widely used for the solution of electromagnetic scattering and radiation problems, mainly because they require only the discretization of the surfaces of the involved objects and the radiation condition is imposed implicitly. The factorization of the near-field matrix leads to a regularization of the first-kind integral equation which is similar to a transformation into a second-kind integral equation in terms of convergence behavior Another way of doing this is the use of Calderon-based preconditioners. The discretization of the outer integral operator cannot be performed with the commonly employed Rao-WiltonGlisson (RWG) basis functions Instead it requires the usage of Buffa-Christiansen basis functions (Buffa and Christiansen, 2007) or other basis functions with similar properties and in particular the same disadvantages. It shall be shown that a factorization of the near-zone matrix improves the convergence properties of the EFIE for small and medium-sized problems considerably so that it behaves like a second kind integral equation. The regularization property of a ”perfect” near-zone preconditioner is investigated and explained in theory

Integral equation formulation

Numerical Results

10-2 CFIE - LU preconditioner

Conclusions