A stabilized time-domain combined field integral equation using the quasi-Helmholtz projectors

TL;DR

The paper tackles instability-laden time-domain boundary integral equations for PEC scattering by introducing a stabilized time-domain combined field integral equation (TD-CFIE). It leverages quasi-Helmholtz projectors to separate solenoidal and irrotational components, applies Yukawa-type Calderón preconditioning, and linearly combines stabilized TD-EFIE and TD-MFIE into a Calderón-like TD-CFIE, all discretized in time via a marching-on-in-time scheme. This approach yields a well-conditioned, non-resonant, and non-dc-unstable solver that remains accurate for large time steps and fine spatial meshes on both simply- and multiply-connected geometries, with near-linear complexity when accelerated by fast time-domain methods. While near-dc instabilities can appear for multiply-connected surfaces under refinement, the overall method significantly enhances robustness and practicality of TD boundary integral solvers for PEC scattering.

Abstract

This paper introduces a time-domain combined field integral equation for electromagnetic scattering by a perfect electric conductor. The new equation is obtained by leveraging the quasi-Helmholtz projectors, which separate both the unknown and the source fields into solenoidal and irrotational components. These two components are then appropriately rescaled to cure the solution from a loss of accuracy occurring when the time step is large. Yukawa-type integral operators of a purely imaginary wave number are also used as a Calderon preconditioner to eliminate the ill-conditioning of matrix systems. The stabilized time-domain electric and magnetic field integral equations are linearly combined in a Calderon-like fashion, then temporally discretized using an appropriate pair of trial functions, resulting in a marching-on-in-time linear system. The novel formulation is immune to spurious resonances, dense discretization breakdown, large-time step breakdown and dc instabilities stemming from non-trivial kernels. Numerical results for both simply-connected and multiply-connected scatterers corroborate the theoretical analysis.

Figures (11)

• Figure 1: Rao-Wilton-Glisson (RWG) function $\bm{f}_n(\bm{r})$ associated with the edge $e_n$. The function $\bm{f}_n(\bm{r})$ is defined on two adjacent triangles $c^-_n$ and $c^+_n$ sharing the common edge $e_n$. The bold arrow indicates the direction of $\bm{f}_n(\bm{r})$ across $e_n$.
• Figure 2: Buffa-Christiansen (BC) function $\bm{g}_n(\bm{r})$ associated with the edge $e_n$. The function $\bm{g}_n(\bm{r})$ is a linear combination of RWG functions defined on the barycentric refinement of the original mesh. The gray area indicates the support of $\bm{g}_n(\bm{r})$, while the bold arrow indicates the direction of $\bm{g}_n(\bm{r})$ along $e_n$.
• Figure 3: The continuous piecewise-linear function $h_0$.
• Figure 4: The continuously differentiable piecewise-quadratic function $q_0$.
• Figure 5: Triangular mesh of two surface geometries used in numerical experiments. Left: a sphere of radius $1\mathrm{m}$. Right: a torus of two radii $3\mathrm{m}$ and $1\mathrm{m}$.

Theorems & Definitions (3)

• Remark 2.1
• Remark 3.1
• Remark 4.1