A multi-mesh, preconditioned conjugate gradient solver for eigenvalue problems in finite element models

Abstract

A multi-mesh, preconditioned conjugate gradient solver is proposed to solve large finite element eigenvalue problems in engineering applications. A generalized eigenvalue problem is solved by a conjugate gradient iteration method with a preconditioner matrix which is a partially factorized stiffness matrix. Initial trial eigenvectors for the proposed solver are obtained by interpolation using the eigenvectors obtained from a coarser mesh with a much smaller number of degrees of freedom. The employment of these trial eigenpairs was found to significantly increase the rate of convergence of the solver and also to prevent slow convergence/convergence failure in problems with closely spaced eigenvalues and repeated eigenvalues. Hence, the proposed solver presents a significant performance improvement over the existing preconditioned conjugate gradient method. Finite element eigenvalue problems for plates and shells are evaluated in this study and the proposed methods are shown to provide savings in both memory and computational time for large size problems. In the examples conducted, an eigenvalue problem for a square plate with about 50,000 degrees of freedom was solved two times faster than the direct (Lanczos) method, along with a memory storage requirement that is about 12 times smaller. However, very thin plate examples (length to thickness ratio > 150) show slow convergence and sometimes convergence failure. The convergence failure for very thin plates can be controlled by allowing more fill-ins in the incomplete Cholesky factorization of the preconditioner matrix.