Multiscale Orthogonal Finite-Element Reduction-Recovery method for transient analysis of integrated circuits and package problems

Driven by the continued advance of integrated circuit (IC) technology, large-scale electromagnetic (EM) modeling and simulation problems have been encountered across the design of on-chip circuits, package, and die-package interfaces. To give a few examples, full-chip post-layout performance verification that includes transmission line and full-wave effects, system-level signal and noise integrity analysis, and the characterization of broadband die-package interaction for clean power delivery. Two major difficulties exist for the analysis of the aforementioned large-scale problems. One is the large problem size. The modeling of a die, a package, and a combined die-package system results in numerical problems of ultra-large scale, requiring billions of parameters to describe them accurately. The other is the multiscale nature of the problem. An electromagnetic simulator is required to span scale ranges of at least 10000:1 to analyze a combined die-package system. Recently, a time domain Orthogonal Finite-Element Reduction-Recovery method (OrFE-RR) was developed to handle the large problem size associated with the design of very large-scale integrated circuits [1]. In this method, a set of orthogonal prism vector bases are developed. With this set of bases, the original ultralarge scale 3-D system of order TV is rigorously reduced to a 2-D single-layered system of order Mwith negligible computational cost, where Mis much less than N. The reduced 2D system is diagonal, and hence can be solved readily. After obtaining the solution of the reduced system, the rest of the solutions can be recovered in linear complexity. This method entails no theoretical approximation. It applies to any arbitrarily-shaped multilayer structure involving inhomogeneous materials. Numerical experiments have demonstrated its accuracy and capacity in simulating very large-scale on-chip, package, and die-package interaction problems.