A Matrix Decomposition Method for Orthotropic Elasticity Problems

Abstract

The construction of an efficient numerical scheme for three-dimensional elasticity problems depends not only on understanding the nature of the physical problem involved, but also on exploiting special properties associated with its discretized system and incorporating these properties into the numerical algorithm. In this paper an efficient and parallelizable decomposition method is presented, referred to as the SAS domain decomposition method, for orthotropic elasticity problems with symmetrical domain and boundary conditions. Mathematically, this approach exploits important properties possessed by the special class of matrices A that satisfy the relation $A=PAP$, where P is some symmetrical signed permutation matrix. These matrices can be decomposed, via orthogonal transformations, into disjoint submatrices. Physically, the method takes advantage of the symmetry of a given problem and decomposes the whole domain of the original problem into independent subdomains. This method has potential for reducing the bandwidth of the stiffness (mass) matrix and lends itself to parallelism on three levels. Therefore, it is useful for sequential, vector, and multiprocessor computers.