On a Low-Frequency and Contrast Stabilized Full-Wave Volume Integral Equation Solver for Lossy Media

TL;DR

This paper introduces a regularized volume integral equation method for modeling high-contrast, lossy dielectric media across a broad frequency spectrum, improving stability and accuracy especially at low frequencies.

Contribution

The paper proposes a novel regularized D-VIE formulation using volume quasi-Helmholtz projectors to enhance stability and accuracy for high-permittivity objects at low frequencies.

Findings

Method remains stable and accurate at low frequencies.

Numerical results confirm improved convergence and precision.

Applicable to biological tissue modeling and electromagnetic applications.

Abstract

In this paper we present a new regularized electric flux volume integral equation (D-VIE) for modeling high-contrast conductive dielectric objects in a broad frequency range. This new formulation is particularly suitable for modeling biological tissues at low frequencies, as it is required by brain epileptogenic area imaging, but also at higher ones, as it is required by several applications including, but not limited to, transcranial magnetic and deep brain stimulation (TMS and DBS, respectively). When modeling inhomogeneous objects with high complex permittivities at low frequencies, the traditional D-VIE is ill-conditioned and suffers from numerical instabilities that result in slower convergence and in less accurate solutions. In this work we address these shortcomings by leveraging a new set of volume quasi-Helmholtz projectors. Their scaling by the material permittivity matrix allows for the re-balancing of the equation when applied to inhomogeneous scatterers and thereby makes the proposed method accurate and stable even for high complex permittivity objects until arbitrarily low frequencies. Numerical results, canonical and realistic, corroborate the theory and confirm the stability and the accuracy of this new method both in the quasi-static regime and at higher frequencies.