Time-domain orthogonal finite-element reduction-recovery (OrFE-RR) method for fast and accurate broadband simulation of die-package interaction

The co-simulation of a die and a package results in numerical problems of ultra-large scale, requiring billions of parameters to describe them accurately. In this paper, an orthogonal finite-element reduction-recovery method is proposed to overcome the large problem sizes encountered in the simulation of an integrated die-package system. In this method, a set of orthogonal prism vector basis functions are developed to render the surface-unknown-based matrix in each layer diagonal. An arbitrary 3D multilayered system such as a combined package and die is then reduced to a single-layer system with negligible computational cost. The reduced single-layer system is diagonal due to the orthogonal bases, and hence can be solved readily. From the solution of the reduced single-layer system, the solution in other layers is recovered in linear complexity. The method entails no approximation. It applies to arbitrarily-shaped multilayer structures and arbitrary inhomogeneity. In addition, it permits nonlinear device modeling and broadband simulation within one run. Numerical and experimental results have demonstrated its accuracy and high capacity in simulating on-chip, package, and die-package interface problems.