A Linear Complexity Direct Volume Integral Equation Solver for Full-Wave 3-D Circuit Extraction in Inhomogeneous Materials

Abstract

An -matrix based linear complexity direct matrix solution is developed for the volume integral equation (VIE) based broadband full-wave extraction of general 3-D circuits. Such circuits are in general electrically small or moderate, but contain arbitrarily shaped lossy conductors immersed in inhomogeneous dielectrics with ports located anywhere in the physical layout of the circuit. In the proposed direct solver, we first develop a well-conditioned VIE formulation without sacrificing the rigor and the advantages of the prevailing formulations. This formulation facilitates a robust direct solution of good accuracy even with a rank-1 representation. We then overcome the numerical challenge of solving the resultant highly unstructured system matrix mixed with both square and rectangular dense and sparse matrices by developing a fast linear complexity direct solution. This direct solution is capable of inverting dense matrices involving over 2 million unknowns in less than 1 h on a single CPU core running at 3 GHz. Numerical simulations of large-scale 3-D circuits and comparisons with state-of-the-art linear complexity iterative VIE solvers have demonstrated the accuracy, efficiency, and linear complexity of the proposed direct VIE solver.

A Unified View of Computational Electromagnetics

“…A paper by Sankaran [68] presents quite well the advantages and disadvantages of various methods with an extensive discussion of flexibility, accuracy, and computational loads. To deal with computational expenses, many fast algorithms have been developed; they accelerate solution processes for modeling electrically large and complex problems [69]; they include multilevel fast multipole algorithm (MLFMA) [70]- [72], fast direct solution methods [73], [74], graphics processing unit (GPU) accelerated methods [75], [76], discontinuous Galerkin (DG) methods [57], [77], domain decomposition (DD) methods [78], and so on [79], [80]. Note that GPU acceleration is not exclusively a matter of technology but also requires a modification of the computational procedure and implementation scheme to run on specialized hardware.…”

A Butterfly-Accelerated Volume Integral Equation Solver for Broad Permittivity and Large-Scale Electromagnetic Analysis

“…Existing fast direct solvers for the VIE include Huygens' equivalence principle-based algorithms [26], [27] yielding an O(N 2 ) complexity, and low-rank-based algorithms such as H matrices [28], [29], H 2 matrices [30]- [32], and skeletonization [20], [33], [34]. These low-rank-based algorithms can attain O(N ) or O(N log N ) complexities for static or low-frequency scattering problems, but tend to become less efficient as the electrical size of the scatterer increases.…”

Fast direct solution of 3‐D volume integral equations by skeletonization for dynamic electromagnetic wave problems

“…Suppose that N is the number of unknowns, these methods work well when N is small but become infeasible when N is large because of the complexity of O(N 3 ). A variety of efficient direct solvers for linear systems have been developed [26][27][28][29][30][31][32][33][34][35][36] (and the references therein). These solvers partially or completely side-step the challenges related to the convergence issue of iterative solvers.…”