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

Abstract

An ${\cal H}^{2}$ -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.