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Maxwell à la Helmholtz: Electromagnetic scattering by 3D perfect electric conductors via Helmholtz integral operators

Juan Burbano-Gallegos, Carlos Pérez-Arancibia, Catalin Turc

TL;DR

The paper presents a novel class of indirect boundary integral equations for 3D PEC electromagnetic scattering that rely solely on Helmholtz integral operators, yielding frequency-robust, second-kind Fredholm formulations without spurious resonances. By proving a Maxwell–Helmholtz equivalence for both the electric and magnetic fields and addressing the zero-frequency limit, it develops Electric Field-Only and Magnetic Field-Only CFIEs that can be regularized via Calderón preconditioning. Numerical validation using high-order Density Interpolation Nyström methods in Inti.jl demonstrates accuracy across planewave and dipole manufactured solutions, including challenging low-frequency and multiply connected geometries, and shows favorable solver performance with FMM and H-matrix accelerations. The work offers a practical, scalable path to integrating Maxwell PEC scattering within the rich ecosystem of scalar Helmholtz solvers, with potential extensions to transmissions and inhomogeneous media. Overall, it delivers a rigorous, solver-friendly framework that enhances the robustness and efficiency of 3D PEC scattering computations.

Abstract

This paper introduces a novel class of indirect boundary integral equation (BIE) formulations for the solution of electromagnetic scattering problems involving smooth perfectly electric conductors (PECs) in three-dimensions. These combined-field-type BIE formulations rely exclusively on classical Helmholtz boundary operators, resulting in provably well-posed, frequency-robust, Fredholm second-kind BIEs. Notably, we prove that the proposed formulations are free from spurious resonances, while retaining the versatility of Helmholtz integral operators. The approach is based on the equivalence between the Maxwell PEC scattering problem and two independent vector Helmholtz boundary value problems for the electric and magnetic fields, with boundary conditions defined in terms of the Dirichlet and Neumann traces of the corresponding vector Helmholtz solutions. While certain aspects of this equivalence (for the electric field) have been previously exploited in the so-called field-only BIE formulations, we here rigorously establish and generalize the equivalence between Maxwell and Helmholtz problems for both fields. Finally, a variety of numerical examples highlights the robustness and accuracy of the proposed approach when combined with Density Interpolation-based Nyström methods and fast linear algebra solvers, implemented in the open-source Julia package Inti$.$jl.

Maxwell à la Helmholtz: Electromagnetic scattering by 3D perfect electric conductors via Helmholtz integral operators

TL;DR

The paper presents a novel class of indirect boundary integral equations for 3D PEC electromagnetic scattering that rely solely on Helmholtz integral operators, yielding frequency-robust, second-kind Fredholm formulations without spurious resonances. By proving a Maxwell–Helmholtz equivalence for both the electric and magnetic fields and addressing the zero-frequency limit, it develops Electric Field-Only and Magnetic Field-Only CFIEs that can be regularized via Calderón preconditioning. Numerical validation using high-order Density Interpolation Nyström methods in Inti.jl demonstrates accuracy across planewave and dipole manufactured solutions, including challenging low-frequency and multiply connected geometries, and shows favorable solver performance with FMM and H-matrix accelerations. The work offers a practical, scalable path to integrating Maxwell PEC scattering within the rich ecosystem of scalar Helmholtz solvers, with potential extensions to transmissions and inhomogeneous media. Overall, it delivers a rigorous, solver-friendly framework that enhances the robustness and efficiency of 3D PEC scattering computations.

Abstract

This paper introduces a novel class of indirect boundary integral equation (BIE) formulations for the solution of electromagnetic scattering problems involving smooth perfectly electric conductors (PECs) in three-dimensions. These combined-field-type BIE formulations rely exclusively on classical Helmholtz boundary operators, resulting in provably well-posed, frequency-robust, Fredholm second-kind BIEs. Notably, we prove that the proposed formulations are free from spurious resonances, while retaining the versatility of Helmholtz integral operators. The approach is based on the equivalence between the Maxwell PEC scattering problem and two independent vector Helmholtz boundary value problems for the electric and magnetic fields, with boundary conditions defined in terms of the Dirichlet and Neumann traces of the corresponding vector Helmholtz solutions. While certain aspects of this equivalence (for the electric field) have been previously exploited in the so-called field-only BIE formulations, we here rigorously establish and generalize the equivalence between Maxwell and Helmholtz problems for both fields. Finally, a variety of numerical examples highlights the robustness and accuracy of the proposed approach when combined with Density Interpolation-based Nyström methods and fast linear algebra solvers, implemented in the open-source Julia package Intijl.

Paper Structure

This paper contains 15 sections, 14 theorems, 185 equations, 5 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Scattering problem
  3. Maxwell’s equations as a vector Helmholtz boundary value problem
  4. The zero frequency limit
  5. Electromagnetic boundary integral equation formulations using Helmholtz integral operators
  6. Electric combined-field-only formulation
  7. Magnetic combined-field-only formulation
  8. Numerical examples
  9. Field corrections to achieve exact divergence-free conditions
  10. Error measures
  11. Planewave scattering by a PEC sphere
  12. Dipole fields as manufactured solutions
  13. Scattering by more general smooth surfaces
  14. Conclusions and future work
  15. Curvature operator and mean curvature

Key Result

Lemma 3.2

Let $U_e \subset \mathbb{R}^3 \setminus \overline{\Omega}$ be an open set such that $\Gamma \subset \overline{U_e}$, where $\Gamma = \partial \Omega$ is a closed, $C^{2,\alpha}$-smooth regular surface with unit normal vector $\nu \in C^{1,\alpha}(\Gamma, \mathbb{R}^3)$. Suppose $F \in C^{1,\beta}(\o where $\partial_\nu F \in C^{0,\beta}(\Gamma,\mathbb{C}^3)$, with $\mathscr{R} \in C^{0,\alpha}(\Ga

Figures (5)

  • Figure 1: Illustration of the domains and fields involved in the PEC scattering problem setup.
  • Figure 2: Error convergence plots at low frequencies. Left: Relative errors for various mesh sizes in the low-frequency regime, for the problem described in Table \ref{['tab:low_frequency']}, using the R-ECFOIE \ref{['eq:R-ECFOIE']} with parameter values $\xi = 1$ and $\eta = \pi$. Right: Corresponding relative errors obtained with the R-MCFOIE \ref{['eq:R-MCFOIE']}. While convergence is evident, a degradation in accuracy is observed.
  • Figure 3: Left: Magnitude of the surface trace of dipole fields used in the numerical experiments reported in Table \ref{['tab:low_frequency_torus']}. Right: Surface mesh used in the same experiments, colored according to the absolute error in the mean curvature computed using Inti.jl.
  • Figure 4: Electric fields resulting from planewave scattering by the bean- and flower-shaped PEC surfaces \ref{['eq:smooth_surfaces']}. Upper panels: Mean curvature on surface meshes. Middle and lower panels: Real part of the $x$-component of the total electric field on the $yz$-plane at $k=10^{-8}\pi$ (middle) and $k=5\pi$ (bottom).
  • Figure 5: Magnetic fields resulting from planewave scattering by the bean- and flower-shaped PEC surfaces \ref{['eq:smooth_surfaces']}. Upper panels: Gaussian curvature on surface meshes. Middle and lower panels: Real part of the $x$- component of the total magnetic field on the $yz$-plane at $k=10^{-8}\pi$ (middle) and $k=5\pi$ (bottom).

Theorems & Definitions (37)

  • Remark 3.1
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • proof
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • proof
  • Theorem 4.1
  • ...and 27 more