Parallel Time-Domain Finite-Element Simulator of Linear Speedup and Electromagnetic Accuracy for the Simulation of Die–Package Interaction

In this paper, we develop an almost embarrassingly parallel solution to an electromagnetic solver of linear complexity for overcoming the grand challenge of performing electromagnetically accurate co-simulation of die-package interaction. In this solution, through suitable basis functions and linear algebraic techniques, we directly and rigorously decompose the system matrix in a 3-D space to multiple matrices of 1-D sizes with negligible computational overhead. Each 1-D matrix is made tridiagonal and, hence, can be solved readily in linear complexity. We then achieve an almost embarrassingly parallel implementation of the fast electromagnetic solver with a low communication-to-computation ratio. Numerical experiments on a large-scale combined die-package system, involving more than 3.5 billion unknowns, have demonstrated superior performance of the proposed parallel transient simulator for simulating large-scale integrated circuits and package problems. In addition, the proposed solver is applicable to any arbitrarily shaped multilayer structure embedded in inhomogeneous materials.