Multi-Trace Müller Boundary Integral Equation for Electromagnetic Scattering by Composite Objects

TL;DR

This work develops a global multi-trace Müller boundary integral equation for electromagnetic scattering by composite dielectric objects, yielding a fully second-kind system for junctions and dense meshes with $N$ components. It relies on the Stratton–Chu representation in the background region (extinction property) to regularize off-diagonal blocks, enabling a conforming mixed discretization using RWG and BC functions. Numerical experiments demonstrate accurate field traces and robust conditioning, with shorter solution times than MT-PMCHWT and Calderón-preconditioned variants, despite higher assembly cost. The framework offers a scalable, stable approach for multi-component scattering and points to promising extensions to time-domain problems.

Abstract

This paper introduces a boundary integral equation for time-harmonic electromagnetic scattering by composite dielectric objects. The formulation extends the classical Müller equation to composite structures through the global multi-trace method. The key ingredient enabling this extension is the use of the Stratton-Chu representation in complementary region, also known as the extinction property, which augments the off-diagonal blocks of the interior representation operator. The resulting block system is composed entirely of second-kind operators. A Petrov-Galerkin (mixed) discretization using Rao-Wilton-Glisson trial functions and Buffa-Christiansen test functions is employed, yielding linear systems that remain well conditioned on dense meshes and at low frequencies without the need for additional stabilization. This reduces computational costs associated with matrix-vector multiplications and iterative solving. Numerical experiments demonstrate the accuracy of the method in computing field traces and derived quantities.

Figures (7)

• Figure 1: Left: A composite object consisting of several components $\Omega_k, k = 1, 2, \ldots, N,$ immersed in the background $\Omega_0$. This configuration usually gives rise to junctions that are curves where three or more regions meet (dashed red circles). Right: Illustration of the global multi-trace approach, in which a conceptual gap filled by the background medium is inserted between adjacent components (dashed blue lines).
• Figure 2: Geometries used in numerical experiments. From left to right: a two-component unit sphere in which one component occupies three quadrants and the other occupies the remaining quadrant; a configuration of two fused square-section tori featuring a fully occluded cavity; and a domain constructed by stacking multiple unit cubes (illustrated here using three cubes).
• Figure 3: Magnitude of the electric surface current density on the boundaries of the three geometries, excited by a plane wave with wavenumber $\kappa_0 = 6 \, \mathrm{m^{-1}}$, computed using the global multi-trace Müller equation. For visualization purposes, the subdomains of the sphere (leftmost) and the cubes (rightmost) are slightly separated. The solutions exhibit continuity across the interfaces between adjacent materials.
• Figure 4: Electric far field scattered by the unit sphere for incident plane waves with wavenumbers $\kappa_0 = 6 \, \mathrm{m^{-1}}$(left) and $\kappa_0 = 0.06 \, \mathrm{m^{-1}}$(right). The far fields computed using the MT-Müller formulation show excellent agreement with those obtained from the Mie series.
• Figure 5: Real part of the $y$-component of the electric field reconstructed from the Cauchy data on the interfaces of the dual-torus domain using the Stratton--Chu representation potentials, evaluated on a grid at $y = 0.5 \mathrm{m}$. From top to bottom and left to right, the subplots show the fields reconstructed using the potentials associated with the background region, with the two subdomains, and finally the total electric field. The surface currents fulfill the extinction property and therefore constitute a Maxwell solution. The total field also exhibits the continuity across the material interfaces.