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Local Multiple Traces Formulation for Heterogeneous Electromagnetic Scattering: Implementation and Preconditioning

Paul Escapil-Inchauspé, Carlos Jerez-Hanckes

TL;DR

This paper extends the local MTF to three-dimensional EM scattering in heterogeneous media, introducing a skeleton-mesh framework that assigns independent boundary unknowns per subdomain and weakly enforces Calderón identities and inter-subdomain transmissions. It delivers a practical, scalable EM solver implemented in Bempp-cl, augmented by block-OSRC preconditioning to accelerate GMRES without heavy barycentric refinements. The numerical experiments demonstrate robustness to an increasing number of subdomains, triple points, and complex geometries, with favorable scaling in solver time and memory when using the OSRC preconditioner. Overall, the work provides a viable pathway to large-scale, preconditioned EM scattering computations in heterogeneous domains, and it outlines future directions including FMM acceleration and parallelization. The approach is validated on spheres and concentric cuboids, highlighting practical impact for industrial EM applications and open-source accessibility.

Abstract

We consider the three-dimensional time-harmonic electromagnetic (EM) wave scattering transmission problem involving heterogeneous scatterers. The fields are approximated using the local multiple traces formulation (MTF), originally introduced for acoustic scattering. This scheme assigns independent boundary unknowns to each subdomain and weakly enforces Calderón identities along with interface transmission conditions. As a result, the MTF effectively handles shared points or edges among multiple subdomains, while supporting various preconditioning and parallelization strategies. Nevertheless, implementing standard solvers presents significant challenges, particularly in managing the degrees of freedom associated with subdomains and their interfaces. To address these difficulties, we propose a novel framework that suitably defines approximation spaces and enables the efficient exchange of normal vectors across subdomain boundaries. This framework leverages the skeleton mesh, representing the union of all interfaces, as the computational backbone, and constitutes the first scalable implementation of the EM MTF. Furthermore, we conduct several numerical experiments, exploring the effects of increasing subdomains and block On-Surface-Raditation-Condition (OSRC) preconditioning, to validate our approach and provide insights for future developments.

Local Multiple Traces Formulation for Heterogeneous Electromagnetic Scattering: Implementation and Preconditioning

TL;DR

This paper extends the local MTF to three-dimensional EM scattering in heterogeneous media, introducing a skeleton-mesh framework that assigns independent boundary unknowns per subdomain and weakly enforces Calderón identities and inter-subdomain transmissions. It delivers a practical, scalable EM solver implemented in Bempp-cl, augmented by block-OSRC preconditioning to accelerate GMRES without heavy barycentric refinements. The numerical experiments demonstrate robustness to an increasing number of subdomains, triple points, and complex geometries, with favorable scaling in solver time and memory when using the OSRC preconditioner. Overall, the work provides a viable pathway to large-scale, preconditioned EM scattering computations in heterogeneous domains, and it outlines future directions including FMM acceleration and parallelization. The approach is validated on spheres and concentric cuboids, highlighting practical impact for industrial EM applications and open-source accessibility.

Abstract

We consider the three-dimensional time-harmonic electromagnetic (EM) wave scattering transmission problem involving heterogeneous scatterers. The fields are approximated using the local multiple traces formulation (MTF), originally introduced for acoustic scattering. This scheme assigns independent boundary unknowns to each subdomain and weakly enforces Calderón identities along with interface transmission conditions. As a result, the MTF effectively handles shared points or edges among multiple subdomains, while supporting various preconditioning and parallelization strategies. Nevertheless, implementing standard solvers presents significant challenges, particularly in managing the degrees of freedom associated with subdomains and their interfaces. To address these difficulties, we propose a novel framework that suitably defines approximation spaces and enables the efficient exchange of normal vectors across subdomain boundaries. This framework leverages the skeleton mesh, representing the union of all interfaces, as the computational backbone, and constitutes the first scalable implementation of the EM MTF. Furthermore, we conduct several numerical experiments, exploring the effects of increasing subdomains and block On-Surface-Raditation-Condition (OSRC) preconditioning, to validate our approach and provide insights for future developments.

Paper Structure

This paper contains 21 sections, 1 theorem, 63 equations, 21 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Geometry and parameters
  4. Functional spaces and traces
  5. Heterogenous EM wave transmission problem
  6. Boundary integral potentials and operators
  7. Local multiple traces formulation
  8. Continuous setting
  9. Discretization setting
  10. Practical implementation of the MTF
  11. Motivation
  12. Top-down approach for space definitions
  13. Implementation in Bempp-cl
  14. Preconditioning
  15. Numerical experiments
  16. ...and 6 more sections

Key Result

Proposition 1

Let ${\bf u}$ and ${\bf u}^\textup{inc}$ be defined as in eqn:boundary_conditions_traces. For $i \in \{ 0,\ldots,M\}$, there holds that

Figures (21)

  • Figure 1: Example of an admissible partition with $M=4$. Junction points are represented in red.
  • Figure 2: $\texttt{MTF(2)}$ for Case A (top) and Case B (bottom): Squared amplitude in plane $z=0$ for the scattered field $|{\bf E}^\text{sc}|^2$ (left) and total field $|{\bf E}|^2$ (right) for a precision of $r=10$ points per wavelength.
  • Figure 3: (a)
  • Figure 4: Mesh for the half-sphere and half-cube for precisions of $r=10$ and $r=20$ points per wavelength.
  • Figure 5: MTF for $M=2$. Convergence of GMRES is shown for the half-sphere (top) and the half-cube (bottom) in Case A (left) and Case B (right). We compare unpreconditioned GMRES (black) with block-OSRC preconditioned GMRES (blue). Convergence results are provided for $N_0$ (solid lines) and $N_1>N_0$ (dashed lines), enabling the study of $h$-dependence, and demonstrating the robust performance of the block-OSRC preconditioned scheme.
  • ...and 16 more figures

Theorems & Definitions (9)

  • Remark 1
  • Proposition 1: Calderón identities
  • proof
  • Remark 2
  • Remark 3
  • Remark 4: PMCHWT
  • Remark 5: Geometries and subdomains
  • Remark 6: Conventions for meshing $\Gamma_{ij}$
  • Remark 7