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Cholesky decomposition with fixing nodes to stable computation of a generalized inverse of the stiffness matrix of a floating structure
Author(s) -
Brzobohatý T.,
Dostál Z.,
Kozubek T.,
Kovář P.,
Markopoulos A.
Publication year - 2011
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.3187
Subject(s) - cholesky decomposition , schur complement , singular value decomposition , mathematics , positive definite matrix , schur decomposition , domain decomposition methods , invertible matrix , inverse , eigendecomposition of a matrix , eigenvalues and eigenvectors , block matrix , matrix (chemical analysis) , rank (graph theory) , pure mathematics , algorithm , combinatorics , finite element method , geometry , quantum mechanics , composite material , physics , materials science , thermodynamics
The direct methods for the solution of systems of linear equations with a symmetric positive‐semidefinite (SPS) matrix A usually comprise the Cholesky decomposition of a nonsingular diagonal block A of A and effective evaluation of the action of a generalized inverse of the corresponding Schur complement. In this note we deal with both problems, paying special attention to the stiffness matrices of floating structures without mechanisms. We present a procedure which first identifies a well‐conditioned positive‐definite diagonal block A of A , then decomposes A by the Cholesky decomposition, and finally evaluates a generalized inverse of the Schur complement S of A . The Schur complement S is typically very small, so the generalized inverse can be effectively evaluated by the singular value decomposition (SVD). If the rank of A or a lower bound on the nonzero eigenvalues of A are known, then the SVD can be implemented without any ‘epsilon’. Moreover, if the kernel of A is known, then the SVD can be replaced by effective regularization. The results of numerical experiments show that the proposed method is useful for effective implementation of the FETI‐based domain decomposition methods. Copyright © 2011 John Wiley & Sons, Ltd.