Novel volume integral equations for broad frequency and material parameter electromagnetic analysis

Background. Determining electromagnetic phenomena inside highly heterogeneous media with regions of very large, very small, or negative permittivity has biological, geophysical, metamaterial, and plasma applications. Volume integral equations are the natural tool for such media, but when the permittivity of the object differs greatly from that of its surroundings, or when the frequency is low, standard formulations demand very fine meshes and produce ill-conditioned systems.

Objective. Develop volume integral equation solvers whose conditioning and accuracy are independent of material contrast, the sign of the permittivity, and frequency, so that one solver can serve as a general-purpose tool from bioelectromagnetics to plasma scattering.

Approach. The internally combined volume-surface integral equation judiciously combines volume and surface equivalence principles. When discretized with standard Galerkin methods it yields matrices whose condition numbers are unaffected by the maximum permittivity and its sign. More recently we introduced a volume integral equation for low-frequency electric field dosimetry of transcranial magnetic stimulation that is stable in the quasi-static regime, and we are developing high-contrast-stable potential-based volume integral equations for use with the potential-based solvers of the Consortium for Electromagnetic Technologies.

Main results. Unlike other methods, the internally combined volume-surface integral equation determines electromagnetic phenomena inside objects of arbitrarily high permittivity over a broad range of frequencies, making it the first general-purpose integral equation tool for electromagnetic analysis of medical applications. It is also stable for negative permittivity and was the first solver able to analyze the communication blackout that occurs when space vehicles travel through negative-permittivity plasma regions of the atmosphere.

Significance. A general-purpose electromagnetic solver lets medical researchers with little electromagnetics background analyze and improve their procedures, streamlining new technology development, and it aids the design of antennas whose signals penetrate re-entry plasma without blackout. The same conditioning ideas now feed our high-contrast, broadband solvers for layered media and chip packaging.

Publications.

S. S. Vaezi (G) and L. J. Gomez, "Novel Volume Integral Equation Approach for Low-Frequency E-Field Dosimetry of Transcranial Magnetic Stimulation," IEEE Transactions on Magnetics, vol. 60, no. 12, 2024. link

L. J. Gomez, A. C. Yucel, and E. Michielssen, "The ICVSIE: A General Purpose Integral Equation Method for Bio-Electromagnetic Analysis," IEEE Transactions on Biomedical Engineering, vol. 65, no. 3, pp. 565-574, 2018. link

A. C. Yucel, L. J. Gomez, and E. Michielssen, "Internally Combined Volume-Surface Integral Equation for EM Analysis of Inhomogeneous Negative Permittivity Plasma Scatterers," IEEE Transactions on Antennas and Propagation, vol. 66, no. 4, pp. 1903-1913, 2018. link

L. J. Gomez, A. C. Yucel, and E. Michielssen, "Internally Combined Volume-Surface Integral Equation for a 3D Electromagnetic Scattering Analysis of High Contrast Media," IEEE Antennas and Wireless Propagation Letters, vol. 16, pp. 1691-1694, 2017. link

L. J. Gomez, A. C. Yucel, and E. Michielssen, "Low-frequency stable internally combined volume-surface integral equation for high-contrast scatterers," IEEE Antennas and Wireless Propagation Letters, vol. 14, pp. 1423-1426, 2015. link

L. J. Gomez, A. C. Yucel, and E. Michielssen, "Volume-surface combined field integral equation for plasma scatterers," IEEE Antennas and Wireless Propagation Letters, vol. 14, pp. 1064-1067, 2015. link

M. Nekoie (G), S. S. Vaezi (G), and L. J. Gomez, "High-Contrast Stable Potential-Based Volume Integral Equation," International Conference on Electromagnetics in Advanced Applications, September 2026.

S. B. Sayed, L. J. Gomez, and A. C. Yucel, "A Multi-Region Internally Combined Volume Surface Integral Equation for EM Analysis of Inhomogeneous Negative and Positive Permittivity Scatterers," IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting, July 2020.