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

Abstract

SUMMARYThe 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 JJ 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 JJ of A, then decomposes A JJ by the Cholesky decomposition, and finally evaluates a generalized inverse of the Schur complement S of A JJ . 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.

Adaptive FEM with Domain Decomposition Method for Partitioned-Based Fluid–Structure Interaction

“…The Dirichlet boundary conditions are subdivided into each parallelized floating subdomains. The procedure allows for more flexible and modular parallel environment by treating the each individual subdomain stiffness matrices separately through the use of fixed node method [40]. This method simplifies implementation since any suitable library, for instance CHOLMOD library, can be integrated into the current code to solve for matrix inversion [41].…”

Scalable TFETI algorithm for the solution of multibody contact problems of elasticity

“…Let us note that the action of a generalized inverse that satisfies (14) may be evaluated at the cost comparable with that of Cholesky's decomposition applied to the regularized K (see [45][46][47]). It may be verified directly that if u solves (11), then there is a vector a such that…”

Solving Contact Mechanics Problems with PERMON

“…The key point is that kernels of subdomain stiffness matrices are known a priori, have the same dimension and can be formed without any computation from the mesh data. Furthermore, each local stiffness matrix can be regularized cheaply, and the inverse of the resulting nonsingular matrix is a pseudoinverse of the original singular one [6,4].…”