Keywords: automated multi-level sub-structuring, AMLS, eigenvalue, eigenvector, sparse matrix, Krylov subspace method, moment matching.

The automated multi-level sub-structuring (AMLS) method which was introduced by Bennighof [1] and co-workers quite recently is a multi-level extension of the component mode synthesis (CMS) method which was developed in the 1960s for solving eigenvalue problems arising in structural analysis. Recent studies in vibro-acoustic analysis of passenger car bodies where huge finite element models with more than six million degrees of freedom appear and several hundreds of eigenfrequencies and eigenmodes are needed have shown that AMLS is considerably faster than the shift-and-invert Lanczos algorithm commonly used in structural engineering for this sort of problems.

In AMLS the large finite element model of a structure is recursively divided into very many sub-structures on several levels based on the sparsity structure of the system matrices. Assuming that the interior degrees of freedom of sub-structures depend quasi-statically on the interface degrees of freedom, and modelling the deviation from quasi-static dependence in terms of a small number of selected sub-structure eigenmodes, the size of the finite element model is reduced substantially yet yielding satisfactory accuracy over a wide frequency range of interest.

It is important to note that the eigenapproximations produced with AMLS are usually less accurate than those computed by Lanczos type methods. Although the lack of accuracy can be tolerated in some application such as frequency response analysis, it may be a cause of concern in others.

The limited accuracy of AMLS results from the fact that the deviation from quasi-static dependence of sub-structures on their interfaces is compensated only by eigenmodes of the clamped sub-structures corresponding to eigenvalues which do not exceed a given cut-off frequency. This approach does not incorporate the coupling of sub-structures and their boundaries sufficiently.

An alternative selection rule of sub-structure eigenmodes which is based on the force applied by the sub-structure to its boundary was presented by Bai, Liao and Gao [2]. A further possibility to include the interaction of a sub-structure and its boundary is to replace eigenmodes by bases of block-Krylov subspaces which allow for the interaction of sub-structures and interfaces via the initial block. For the special symmetric eigenvalue problem this method was proposed by Bekas and Saad [3] (although motivated differently).

We discuss both alternative reduction methods and expound the problems of using the modifications in a multi-level environment. We report on numerical experiments for a multi-level implementation of the new AMLS versions. These demonstrate that the modified mode selection approach is promising. The AMLS method based on block-Krylov subspaces seems to be inferior to the standard AMLS algorithm, and more research is necessary to make it a competitive method for large scale problems.

1

J.K. Bennighof, R.B. Lehoucq, "An automated multilevel substructuring method for the eigenspace computation in linear elastodynamics", SIAM J. Sci. Comput., 25, 2084-2106, 2004. doi:10.1137/S1064827502400650

2

B.-S. Liao, Z. Bai, W. Gao, "The important modes of subsystems: A moment-matching approach", Internat. J. Numer. Meth. Engrg., 70, 1581-1597, 2007. doi:10.1002/nme.1940

3

C. Bekas, Y. Saad, "Computation of smallest eigenvalues using spectral Schur complements", SIAM J. Sci. Comput., 27, 458-481, 2005. doi:10.1137/040603528

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