Cholesky decomposition with fixing nodes to stable computation of a generalized inverse of the stiffness matrix of a floating structure

Abstract

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[MATHEMATICAL SCRIPT CAPITAL J][MATHEMATICAL SCRIPT CAPITAL J] 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[MATHEMATICAL SCRIPT CAPITAL J][MATHEMATICAL SCRIPT CAPITAL J] of A, then decomposes A[MATHEMATICAL SCRIPT CAPITAL J][MATHEMATICAL SCRIPT CAPITAL J] by the Cholesky decomposition, and finally evaluates a generalized inverse of the Schur complement S of A[MATHEMATICAL SCRIPT CAPITAL J][MATHEMATICAL SCRIPT CAPITAL J]. 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.