Algorithms for eigenvalue problems in piezoelectric finite element analyses

Two algorithms for eigenvalue problems in piezoelectric finite element analyses are introduced. The first algorithm involves the use a Lanczos eigensolver, while the second algorithm uses a Rayleigh quotient iteration scheme. In both solution methods, schemes are implemented to reduce storage requirements and solution time. Also, both solution methods seek to preserve the sparsity structure of the stiffness matrix to realize major savings in memory. In the Lanczos solution method, the structural pattern of the consistent mass matrix is exploited to gain savings in both memory and solution time. In the Rayleigh quotient iteration method, an algorithm for generating good initial eigenpairs is employed to improve significantly its overall convergence rate, and convergence stability in the regions of closely spaced eigenvalues and repeated eigenvalues. The initial eigenvectors are obtained by interpolation from a coarse mesh. In order for this iterative method to be effective, an eigenvector of interest in the fine mesh must resemble an eigenvector in the coarse mesh. Hence, the method is effective for finding the set of eigenpairs in the low frequency range, and not in the high frequency range where the eigenvectors of the coarse mesh does not resemble well their counterparts in the fine mesh. Results of example problems are presented to show the savings in solution time and storage requirements of the proposed algorithms when compared with the conventional algorithm which uses static condensation to remove the electric potential degrees of freedom from the piezoelectric eigenvalue problem. The disadvantage of this conventional scheme is that the static condensation destroys the sparsity structure of the stiffness matrix, which leads to major increases in memory and solution time