A Deterministic-Solution Based Fast Eigenvalue Solver With Guaranteed Convergence for Finite-Element Based 3-D Electromagnetic Analysis

摘要

A fast solution to both the quadratic eigenvalue problem and the generalized eigenvalue problem is developed for the finite-element based analysis of general 3-D electromagnetic problems. Given an arbitrary frequency band of interest, denoting the number of physically important eigenvalues for this band by $k$, the proposed eigenvalue solution is capable of solving a significantly reduced eigenvalue problem of $O (k)$ to find a complete set of eigenvalues and eigenvectors that are physically important for the given frequency band. In addition to bypassing the need of solving a large-scale eigenvalue problem of $O (N)$, with $N$ being the system matrix size, the reduced eigenvalue problem is constructed from $O (k)$ solutions to a deterministic problem. As a result, the methods that have been developed to solve deterministic problems and their fast solvers can all be readily leveraged to solve eigenvalue problems. Moreover, the proposed fast eigenvalue solution has guaranteed convergence and controlled accuracy, which is theoretically proved in this paper. The solution is applicable to general 3-D problems where the structures are arbitrary, materials are inhomogeneous, and both dielectrics and conductors can be lossy. Applications to microwave devices, package structures, and on-chip integrated circuits have demonstrated the accuracy, efficiency, and convergence of the proposed fast eigenvalue solution.