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PML-based boundary integral equation method for electromagnetic scattering problems in a layered-medium

Gang Bao, Wangtao Lu, Tao Yin, Lu Zhang

Abstract

This paper proposes a new boundary integral equation (BIE) methodology based on the perfectly matched layer (PML) truncation technique for solving the electromagnetic scattering problems in a multi-layered medium. Instead of using the original PML stretched fields, artificial fields which are also equivalent to the solutions in the physical region are introduced. This significantly simplifies the study of the proposed methodology to derive the PML problem. Then some PML transformed layer potentials and the associated boundary integral operators (BIOs) are defined and the corresponding jump relations are shown. Under the assumption that the fields vanish on the PML boundary, the solution representations, as well as the related BIEs and regularization of the hyper-singular operators, in terms of the current density functions on the truncated interface, are derived. Numerical experiments are presented to demonstrate the efficiency and accuracy of the method.

PML-based boundary integral equation method for electromagnetic scattering problems in a layered-medium

Abstract

This paper proposes a new boundary integral equation (BIE) methodology based on the perfectly matched layer (PML) truncation technique for solving the electromagnetic scattering problems in a multi-layered medium. Instead of using the original PML stretched fields, artificial fields which are also equivalent to the solutions in the physical region are introduced. This significantly simplifies the study of the proposed methodology to derive the PML problem. Then some PML transformed layer potentials and the associated boundary integral operators (BIOs) are defined and the corresponding jump relations are shown. Under the assumption that the fields vanish on the PML boundary, the solution representations, as well as the related BIEs and regularization of the hyper-singular operators, in terms of the current density functions on the truncated interface, are derived. Numerical experiments are presented to demonstrate the efficiency and accuracy of the method.
Paper Structure (13 sections, 7 theorems, 115 equations, 13 figures)

This paper contains 13 sections, 7 theorems, 115 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Two-layered medium problems
  4. Classical layer potentials and BIOs
  5. The truncated PML problems and BIEs
  6. The PML stretching
  7. The truncated PML problems
  8. BIEs
  9. Regularization of the hyper-singular operators
  10. Multi-layered medium problems
  11. Numerical experiments
  12. Implementation strategy
  13. Numerical examples

Key Result

Theorem 2.2

Let $\bm\varphi\in C^1(\Gamma)^3$ be a tangential field. It holds that and the following jump relations hold:

Figures (13)

  • Figure 1: Geometry description of the problem under consideration: scattering by a planar layered medium (a) or a defect on a penetrable layer (b). $\Pi$ and $\Gamma$ denote the interface between the two-layered medium.
  • Figure 2: (a) The PML-truncated domain; (b) The vertical view of PML truncation on the bottom infinite surface.
  • Figure 3: Geometry description of a planar layered medium (a) and a locally perturbed multi-layered medium (b) for the case $N=3$.
  • Figure 4: Example 1. Absolute values of the numerical solution to the BIEs (\ref{['PMLPECBIE']})-(\ref{['PMLPMCBIE']}) on $\Gamma_*^b$ as well as the exact values. (a)(b): PEC problem; (c)(d): PMC problem.
  • Figure 5: Example 1. Numerical errors $\epsilon_{\infty}$ for the PEC and PMC problems of scattering by a spherical obstacle on the half-space.
  • ...and 8 more figures

Theorems & Definitions (17)

  • Remark 2.1
  • Theorem 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 7 more