An H-LU Based Direct Finite Element Solver Accelerated by Nested Dissection for Large-scale Modeling of ICs and Packages

Abstract

In this work, we prove that for the sparse matrix resulting from a flnite-element- based analysis of electrodynamic problems, its inverse has a data-sparse H-matrix approximation with error well controlled. Based on this proof, we develop a fast direct flnite element solver. In this direct solver, the H-matrix-based LU factorization is developed, which is further accelerated by nested dissection. We show that the proposed direct solver has an O(kNlogN) memory complexity and O(k 2 Nlog 2 N) time complexity, where k is a small number that is adaptively determined based on accuracy requirements, and N is the number of unknowns. A comparison with the state-of-the-art direct flnite element solver that employs the most advanced sparse matrix solution has shown a clear advantage of the proposed solver. Applications to large-scale package modeling involving millions of unknowns have demonstrated the accuracy and almost linear complexity of the proposed direct solver. In addition, the proposed method is applicable to arbitrarily-shaped three-dimensional structures and arbitrary inhomogeneity.