Preconditioning Matrix for Fast Hierarchical Method in 3-D Capacitance Extraction of IC interconnect

Abstract

In 3-D capacitance extraction by indirect boundary element method (BEM), the dense potential coefficient matrix arising from the integral equation formulation is usually represented implicitly and concisely. This is a significant reduction compared to the O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) storage that would be required to store potential coefficient matrix directly. Further, the multiplication of matrix-vector product can be accomplished in time proportional to the size of the representation. However, the conventional preconditioning techniques for an explicit matrix are not applicable since no coefficient matrix is actually constructed. In this work, a preconditioning matrix for the fast hierarchical method (FHM) is proposed by carefully examining inherent properties of the hierarchical data structure in the FHM. This preconditioning matrix significantly enhances convergence of the iterative solution yet preserves the implicit nature of the matrix operations in the FHM. Numerical experiments included confirm: this preconditioned FHM reduces iteration times dramatically, and thereby the computational time is reduced.