**Authors:** Saad Omar

**Source:** FERMAT, Volume 11, Communications 11, Sep_Oct, 2015

**Abstract:** Among existing computational electromagnetic methods, volume integral equation (VIE) based methods have unique
advantages in modeling both open-region problems and complicated geometry and materials. However, to accentuate the unique advantages
of the VIE-based methods, two major obstacles must be overcome: one is the generality of the VIE formulation; the other being the high computational
cost of a VIE-solver.

Traditional VIE-based formulations developed for solving wave-related problems are not amenable for solving circuit problems, while existing circuit-based VIE-formulations involve simplifications and approximations that are invalid for wave-related problems. In this work, we develop a new first-principles-based VIE-formulation that bridges the gap between wave- and circuit-based electromagnetic analysis, using which the analysis and design of circuits exposed to external electromagnetic fields is made possible in a full electromagnetic spectrum.

The linear system of equations resulting from a VIE-based analysis is not only dense but also large involving volume unknowns in a 3-D computational domain. To address this computational challenge, we overcame the related numerical issues to develop an H2- matrix based linear complexity direct VIE solver for large-scale circuit parameter extraction, which is capable of solving millions of VIE unknowns using modest computational resources on a single CPU core. Lastly, our newly developed minuscule cost SVD-mimicking 2-matrix recompression schemes have made it possible, for the first time, to achieve linear complexity iterative and fast direct solvers for general large-scale electrodynamic scattering problems.

**Index Terms:** Fast Direct Inverse, Linear complexity direct VIE solvers, Volume Integral Equation, Direct Circuit Solver,
Fast electrodynamic solvers, Simultaneous circuit radiation formulations.

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