Cholesky decomposition with fixing nodes to stable computation of a generalized inverse of the stiffness matrix of a floating structure
Tomáš Brzobohatý, Zdeněk Dostál, Tomáš Kozubek, Petr Kovář, Alexandros Markopoulos · International Journal for Numerical Methods in Engineering · 2011
Abstract The direct methods for the solution of systems of linear equations with a symmetric positive‐semidefinite (SPS) matrixAusually comprise the Cholesky decomposition of a nonsingular diagonal blockA𝒥𝒥 ofAand 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 blockA𝒥𝒥 ofA, then decomposesA𝒥𝒥 by the Cholesky decomposition, and finally evaluates a generalized inverse of the Schur complementSofA𝒥𝒥. The Schur complementSis typically very small, so the generalized inverse can be effectively evaluated by the singular value decomposition (SVD). If the rank ofAor a lower bound on the nonzero eigenvalues ofAare known, then the SVD can be implemented without any ‘epsilon’. Moreover, if the kernel ofAis 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.