Reducing sparse nonlinear eigenproblems by automated multi-level substructuring
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Computers and Structures
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This paper is concerned with solving large-scale eigenvalue problems by algebraic substructuring. Algebraic substructuring refers to the process of applying matrix reordering and partitioning algorithms to divide a large sparse matrix into smaller submatrices from which a subset of spectral components are extracted and combined to form approximate solutions to the original problem. Through an algebraic analysis, we identify critical conditions under which a simple version of algebraic substructuring works well. This particular version of substructuring is identical to the component mode synthesis (CMS) method (see [R. R. Craig and M. C. C. Bampton, {\em Coupling of substructures for dynamic analysis}, AIAA J., 6 (1968), pp. 1313--1319] and [W. C. Hurty, {\em Vibrations of structure systems by component-mode synthesis}, J. Engrg. Mech., 86 (1960), pp. 51--69]) when the matrix reordering is based on a geometric partitioning of the computational domain. We observe an interesting connection between the accuracy of an approximate eigenpair obtained through substructuring and the distribution of the components of eigenvectors of a canonical matrix pencil congruent to the original problem. A priori error bounds for the smallest eigenpair approximation are developed. This development leads to a simple heuristic for choosing spectral components (modes) from each substructure. The effectiveness of such a heuristic is demonstrated with numerical examples. We show that algebraic substructuring can be effectively used to solve a generalized eigenvalue problem arising from the finite element analysis of an accelerator structure.