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ADVANCES IN COMPUTATIONAL MECHANICS
Edited by: M. Papadrakakis and B.H.V. Topping
Gradient Eigenanalysis on Nested Finite Elements
L. Bergamaschi, G. Gambolati, G. Pini and M. Putti
Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Universitá degli Studi di Padova, Padova, Italy
L. Bergamaschi, G. Gambolati, G. Pini, M. Putti, "Gradient Eigenanalysis on Nested Finite Elements", in M. Papadrakakis, B.H.V. Topping, (Editors), "Advances in Computational Mechanics", Civil-Comp Press, Edinburgh, UK, pp 225-238, 1994. doi:10.4203/ccp.26.8.1
The efficient computation of the leftmost eigenpairs of the generalized symmetric eigenproblem A x = lamda B x by a deflation accelerated conjugate gradient (DACG) method may be enhanced by an improved estimate of the initial eigenvectors obtained with a multigrid (MG) type approach. The DACG algorithm essentially optimizes the Rayleigh quotient in subspaces of decreasing size B-orthogonal to the eigenvectors previously computed by a preconditioned conjugate gradient (CG) scheme. The DACG 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 various preconditioners taken from the family of the incomplete Cholesky decompositions of A. The initial rate may be ameliorated by providing an initial guess calculated on nested finite element (FE) grids of growing resolution.
The overall algorithm has been applied to structural eigenproblems defined over four nested FE grids. The results for the computation of the 40 smallest eigenpairs indicate that the asymptotic convergence is very much dependent on the actual eigenvalue distribution and may be substantially improved by the use of appropriate and relatively inexpensive preconditioners. The nested iterations (NI) may lead to a marked reduction of the initial iterations on the finest grid level where the solution is finally required. NI decreases the CPU time by a factor of 2.5. The performance of the NI-DACG method is very promising and emphasizes the potential of this new approach in the partial solution of symmetric positive definite eigenproblems of large and very large size.
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