A linear complexity ℌ<sup>2</sup>-matrix based direct volume integral solver for broadband 3-D circuit extraction in inhomogeneous materials

Abstract

An ℌ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -matrix based linear-complexity direct matrix solution for the volume integral equation (VIE) is developed for the broadband full-wave extraction of general 3-D circuits, containing arbitrarily shaped lossy conductors immersed in inhomogeneous dielectrics, with ports located anywhere in the physical layout of the circuit. A well-conditioned VIE formulation, without sacrificing any advantage of the conventional formulations, is developed to facilitate a robust direct solution of controlled accuracy even with a rank-1 representation. The resultant highly irregular system matrix composed of both square and rectangular dense and sparse matrices is solved by a fast linear-complexity direct solution, which is capable of inverting dense matrices involving over 2 millions of unknowns in less than 1 hour on a single CPU running at 3 GHz. Numerical simulations of large-scale 3-D circuits and comparisons with linear-complexity iterative solvers have demonstrated the accuracy and efficiency of the proposed direct solver.