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Boundary-field formulation for transient electromagnetic scattering by dielectric scatterers and coated conductors

George C. Hsiao, Tonatiuh Sánchez-Vizuet, Wolfgang L. Wendland

TL;DR

This work develops a boundary-field formulation for transient electromagnetic scattering by dielectric objects and coated conductors. By transferring the time-domain problem to the Laplace domain, representing the exterior field with electromagnetic layer potentials, and coupling to an interior transmission problem, the authors derive a nonlocal boundary-field system and prove its existence, uniqueness, and stability via a Calderón-projector framework. They then translate the Laplace-domain estimates to the time domain, obtaining regularity and energy bounds that underpin Convolution Quadrature-based time discretization and error control. The approach naturally extends to coated-conductor configurations and provides a rigorous foundation for stable, nonlocal boundary-integral methods for time-domain Maxwell scattering.

Abstract

We examine the transient scattered and transmitted fields generated when an incident electromagnetic wave impinges on a dielectric scatterer or a coated conductor embedded in an infinite space. By applying a boundary-field equation method, we reformulate the problem in the Laplace domain using the electric field equation inside the scatterer and a system of boundary integral equations for the scattered electric field in free space. To analyze this nonlocal boundary problem, we replace it by an equivalent boundary value problem. Existence, uniqueness and stability of the weak solution to the equivalent BVP are established in appropriate function spaces in terms of the Laplace transformed variable. The stability bounds are translated into time-domain estimates which determine the regularity of the solution in terms of the regularity of the problem data. These estimates can be easily converted into error estimates for a numerical discretization on the convolution quadrature for time evolution.

Boundary-field formulation for transient electromagnetic scattering by dielectric scatterers and coated conductors

TL;DR

This work develops a boundary-field formulation for transient electromagnetic scattering by dielectric objects and coated conductors. By transferring the time-domain problem to the Laplace domain, representing the exterior field with electromagnetic layer potentials, and coupling to an interior transmission problem, the authors derive a nonlocal boundary-field system and prove its existence, uniqueness, and stability via a Calderón-projector framework. They then translate the Laplace-domain estimates to the time domain, obtaining regularity and energy bounds that underpin Convolution Quadrature-based time discretization and error control. The approach naturally extends to coated-conductor configurations and provides a rigorous foundation for stable, nonlocal boundary-integral methods for time-domain Maxwell scattering.

Abstract

We examine the transient scattered and transmitted fields generated when an incident electromagnetic wave impinges on a dielectric scatterer or a coated conductor embedded in an infinite space. By applying a boundary-field equation method, we reformulate the problem in the Laplace domain using the electric field equation inside the scatterer and a system of boundary integral equations for the scattered electric field in free space. To analyze this nonlocal boundary problem, we replace it by an equivalent boundary value problem. Existence, uniqueness and stability of the weak solution to the equivalent BVP are established in appropriate function spaces in terms of the Laplace transformed variable. The stability bounds are translated into time-domain estimates which determine the regularity of the solution in terms of the regularity of the problem data. These estimates can be easily converted into error estimates for a numerical discretization on the convolution quadrature for time evolution.
Paper Structure (20 sections, 10 theorems, 115 equations, 2 figures)

This paper contains 20 sections, 10 theorems, 115 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The scattering problem
  3. Formulation of the problem
  4. Governing equations in the time domain
  5. Preliminaries and notation
  6. Sobolev spaces for the curl
  7. Causal tempered distributions for Maxwell's problem
  8. Laplace transform
  9. Notational remarks
  10. Governing equations in the Laplace domain
  11. Operator equations in $\Omega_-$
  12. Electromagnetic layer potentials and boundary integral operators
  13. Reduction to BIEs in $\Omega_+$
  14. Boundary-field formulation
  15. An equivalent BVP
  16. ...and 5 more sections

Key Result

Proposition 4.1

Given $({\bf m, j} ) \in {\bf H}_{\perp}^{-1/2}({\rm curl}_{\Gamma}, \Gamma) \times {\bf H}_{||}^{-1/2} ({\rm div}_{\Gamma} , \Gamma)$, the function ${\bf E}^{scat} \in {\bf H}({ \bf curl}, \mathbb{R}^3\setminus \Gamma) \cap {\bf H}({ \bf curl}^2, \mathbb{R}^3\setminus \Gamma )$ represented fo is the unique distributional solution of the transmission problem Moreover, the solution ${\bf E}^

Figures (2)

  • Figure 1: A schematic depiction of the problem geometry.
  • Figure 2: A conducting material, contained in the region $\Omega_c$ with boundary $\Gamma_c$, is enclosed by a dielectric in the sorrounding region $\Omega_-$. The boundary of $\Omega_-$ has two disjoint components: the vacuum-dielectric interface $\Gamma$ and the dielectric-conductor interface $\Gamma_c$. The unit vector $\boldsymbol{n}$ anchored on $\Gamma_d$ and is exterior to $\Omega_-$.

Theorems & Definitions (17)

  • Proposition 4.1
  • proof
  • Proposition 5.1
  • proof
  • Proposition 5.2
  • proof
  • Proposition 6.1
  • proof
  • Theorem 6.1
  • proof
  • ...and 7 more