A Stable, Accurate and Well-Conditioned Time-Domain PMCHWT Formulation

TL;DR

The paper tackles instability and ill-conditioning in time-domain PMCHWT for dielectric scattering by introducing a Calderón-based dense-mesh preconditioner and a loop-star, quasi-Helmholtz regularization with temporal differentiation/integration rescaling. A right-left rescaling strategy paired with a carefully designed diagonal preconditioner yields a well-conditioned, stable marching-on-in-time formulation, effective even for highly non-smooth, multiply-connected geometries. Numerical results demonstrate late-time stability, robust conditioning under mesh refinement and large timesteps, and accurate far-field predictions at low frequencies, outperforming classical TD-PMCHWT and FD-PMCHWT approaches. The method enables efficient iterative-solver solutions and broad applicability to transient dielectric scattering problems, with potential extensions to higher-order discretizations and convolution-based time integration.

Abstract

This paper introduces a new boundary element formulation for transient electromagnetic scattering by homogeneous dielectric objects based on the time-domain PMCHWT equation. To address dense-mesh breakdown, a multiplicative Calderon preconditioner utilizing a modified static electric field integral operator is employed. Large-timestep breakdown and late-time instability are simultaneously resolved by rescaling the Helmholtz components leveraging the quasi-Helmholtz projectors and using temporal differentiation and integration as rescaling operators. This rescaling also balances the loop and star components at large timesteps, improving solution accuracy. The resulting discrete system is solved using a marching-on-in-time scheme and iterative solvers. Numerical experiments for simply- and multiply-connected dielectric scatterers, including highly non-smooth geometries, corroborate the accuracy, stability, and efficiency of the proposed approach.

Figures (5)

• Figure 1: Scatterer geometries used in numerical experiments. From left to right: a smooth and simply-connected sphere of radius $1\mathrm{m}$; a smooth and multiply-connected torus of 2 radii $0.75\mathrm{m}$ and $0.25\mathrm{m}$; and a highly non-smooth star-based pyramid of height $0.5\mathrm{m}$, with 24 vertices of the base lie on two concentric circles of radius $1\mathrm{m}$ and $0.3\mathrm{m}$.
• Figure 2: Intensity of the electric current density $\bm{j}$ on the unit sphere, the toroidal surface, and the star-based pyramid (from left to right). Different quadrature strategies are employed to discretize the classical TD-PMCHWT formulation, characterized by the number of quadrature points $N_q$ = 4, 13, 78, and 400 appearing on the legends. Numerical solutions to the TD-PMCHWT equation are severely unstable at late times and highly sensitive to quadrature errors. In contrast, the solution to the proposed qHP TD-PMCHWT formulation is stable even when the integrals are computed with a very low precision. In all three cases, only $N_q$ = 4 quadrature points are used when discretizing the qHP TD-PMCHWT.
• Figure 3: Spectra of the companion matrices for the TD-PMCHWT (discretized using $N_q = 78$ quadrature points) and the proposed qHP TD-PMCHWT ($N_q = 4$), on the unit sphere, the toroidal surface, and the star-based pyramid (from left to right). The TD-PMCHWT's spectra exhibit eigenvalues clustered around $1+0\iota$ that cause late-time instability to numerical solution. In contrast, the qHP TD-PMCHWT does not support any eigenvalue around $1+0\iota$, rendering its solution stable. In all cases, the eigenvalues near $-1+0\iota$ are unproblematic as they reside strictly inside the unit circle, and correspond to the oscillation of the solution below the machine precision VVV+2013.
• Figure 4: (Left) Condition number of the TD-PMCHWT and qHP TD-PMCHWT systems for scattering by the unit sphere, the torus, and the star-based pyramid as a function of the timestep $\Delta t$. (Middle) Condition number of the TD-PMCHWT and qHP TD-PMCHWT systems for scattering by the sphere and the torus as a function of the mesh size $h$. (Right) Convergence history for the GMRES solution of the star-based pyramid scattering problem. The classical TD-PMCHWT formulation suffers from dense-mesh breakdown and large-timestep breakdown, whereas the proposed qHP TD-PMCHWT formulation is immune to both of them.
• Figure 5: Scattered electric far field of the incident plane wave with the frequency $10^{-5} \mathrm{Hz}$, computed using the Mie series, the qHP TD-PMCHWT and the FD-PMCHWT. The far field computed from the qHP TD-PMCHWT solution is accurate, whereas the one computed from the FD-PMCHWT is completely wrong due to the loss of solution accuracy at low frequencies.