GRADIENT EIGENANALYSIS ON NESTED FINITE-ELEMENTS

The efficient computation of the leftmost eigenpairs of the generalize d symmetric eigenproblem Ax = lambda Bx by a deflation accelerated con jugate gradient (DACG) method may be enhanced by an improved estimate of the initial eigenvectors obtained with a multigrid (MG)-type approa ch. The DACG algorithm essentially optimizes the Rayleigh quotient in subspaces of decreasing size B-orthogonal to the eigenvectors previous ly computed by a preconditioned conjugate gradient (CG) scheme. The DA CG asymptotic rate of convergence may be shown to be controlled by the relative separation of the eigenvalue being currently sought and the next higher one and can be effectively accelerated by the use of vario us preconditioners taken from the family of the incomplete Cholesky de compositions of A. The initial rate may be ameliorated by providing an initial guess calculated on nested finite element (FE) grids of growi ng resolution. The overall algorithm has been applied to structural ei genproblems defined over four nested FE grids. The results for the com putation of the 40 smallest eigenpairs indicate that the asymptotic co nvergence is very much dependent on the actual eigenvalue distribution and may be substantially improved by the use of appropriate and relat ively inexpensive preconditioners. The nested iterations (NI) may lead to a marked reduction of the initial iterations on the finest grid le vel where the solution is finally required. NI decreases the CPU time by a factor of 25. The performance of the NI-DACG method is very promi sing and emphasizes the potential of this new approach in the partial solution of symmetric positive definite eigenproblems of large and ver y large size. Copyright (C) 1996 Civil-Comp Limited and Elsevier Scien ce Limited