Modern integrated circuits (IC) and package design has scaled into the deep submicron regime and the nanometer regime. Fast and broadband frequency-domain electromagnetic analysis has become increasingly important. The large problem size encountered in the analysis of ICs and packages is a major challenge especially for a finite element method (FEM) based electromagnetic analysis.

To reduce the computational cost for large-scale electromagnetic analysis, model order reduction (MOR) methods have been developed to preprocess the huge linear system into reduced order models. However, in order to meet the modeling and simulation challenges arising from the IC and package design, existing MOR methods still have to overcome the following shortcomings. First, many existing MOR methods lack a closed-form error bound. Given an accuracy requirement, the model generated from existing methods may not be compact enough. Second, most of the existing reduced order models depend on frequency and right hand side. They lose efficiency when analyzing frequency-dependent problems with a large number of right hand sides. Last but not least, many existing MOR methods suffer from low frequency breakdown problem. Additional models have to be built if low frequency solutions, including DC solution, are required.

This paper proposes a minimal order model for any prescribed accuracy for the finite element based solution of general 3-D problems having arbitrary lossless/lossy structures and inhomogeneous materials. This model entails no theoretical approximations. It is frequency and right hand side independent, and hence can be employed for both fast frequency and right hand side sweep. Moreover, the model does not suffer from low-frequency breakdown and is accurate from zero to high frequencies. To facilitate the application of such a minimal order model, we have also developed an efficient algorithm to generate this model. Numerical experiments have demonstrated the accuracy and efficiency of the proposed method. In addition to frequency-domain analysis, the proposed model can also be used for fast time-domain analysis.

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