Fast algorithms for frequency-domain finite element based analysis of integrated circuits and packages
Abstract
In the full-wave electromagnetics-based analysis of integrated circuit (IC) and package structures, large problem sizes are often encountered. In this thesis, fast techniques in the frequency domain finite element method are developed to overcome the challenge of simulating large-scale problems. The techniques we develop fall into two main categories: reducing the number of unknowns, for which we develop efficient layered finite element methods; and reducing the order of the numerical system, for which we develop minimal order models and fast algorithms for generating the minimal order models. We reduce the number of unknowns by developing efficient layered finite element techniques. The system matrix of the original 3D problem of dimension O(N) is reduced to that of 2D layers by a fast reduction algorithm that has an optimal complexity. The system matrix of 2D layers is then reduced to that of a single layer of O(M). The reduced system is solved by a fast iterative solution algorithm that can converge in a small number of iterations. The solution to the original system can be recovered from that of the reduced system if desired. The procedure is rigorous without making thereotical approximations. It is applicable to arbitrarily shaped 3-D structures in non-uniform materials. We then propose a minimal order model for the frequency-domain finite element based analysis of integrated circuits and packages. Starting from a system matrix of O(N) resulting from a finite element based analysis of lossless or lossy electromagnetic problems, we find a minimal order model of O(k) for a required accuracy. We also develop an efficient algorithm to generate such a minimal order model in fast CPU time. The proposed minimal order model can be used to perform an analytical fast frequency sweep from zero to high frequencies. It can also be used for fast right hand side sweep, model order reduced time-domain solution, fast synthesis of the integrated circuits, etc. To facilitate a fast modal analysis, we develop a solution based fast quadratic eigenvalue solver. Given an arbitrary frequency band, the proposed solver is capable of solving a significantly reduced eigenvalue problem to find a complete set of the eigenvalues and eigenvectors that are physically important for the given frequency band. The reduced system is constructed from a set of solutions to a deterministic problem in the given frequency band. The effectiveness and efficiency of this method are theoretically and numerically validated. This work provides efficient algorithms for fast frequency-domain finite element based analysis of integrated circuit and package problems.
Degree
Ph.D.
Advisors
Jiao, Purdue University.
Subject Area
Electrical engineering
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