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

• Published in: IEEE transactions on antennas and propagation Vol. 71; no. 3; pp. 2571 - 2586
• Main Authors: Henry, Clement, Merlini, Adrien, Rahmouni, Lyes, Andriulli, Francesco P.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.03.2023; The Institute of Electrical and Electronics Engineers, Inc. (IEEE); Institute of Electrical and Electronics Engineers
• Subjects: Bioelectromagnetism; Biological models (mathematics); Brain; Brain modeling; Complex permittivity; Computational Physics; Electric breakdown; Electric flux; Frequency ranges; high-contrast (HC) conductive media; Integral equations; Lossy media; Low frequencies; Mathematical models; Numerical models; Permittivity; Physics; preconditioning; Projectors; Solid modeling; Tissues; Volume integral equations; volume integral equations (VIEs)
• ISSN: 0018-926X; 1558-2221
• DOI: 10.1109/TAP.2022.3161390

In this article, 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, deep brain stimulation (DBS). 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 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 rebalancing 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.