Algorithms and Experiments for Structural Mechanics of High Performance Architectures

摘要

Given a finite element model of a structure and a set of external loads, one of the primary objectives of elastic analysis and structural dynamics is to determine the internal forces, leading to the resulting stresses, strains and displacements. Considering the increasing demands upon the structural engineer to analyze larger and more complex structures, the need for multiprocessing and/or vectorization of the numerical schemes being used is substantial if time-to-solution reductions are desired. The simplest approach to structural mechanics computations is the formation and Cholesky factorization of the global stiffness matrix K. On the other hand, inherent ill-conditioning in K often requires extended precision computations. As a result, alternatives to the formation of K such as orthogonal factorization and preconditioned conjugate gradient techniques have gained in popularity. Presented here is a detailed study of the performance of some promising alternative algorithms to the stiffness equations approach on high performance architectures. Implementations of direct and iterative methods for the solution of structural mechanics problems have been made on the Alliant FX/8 and Cray X-MP systems. The direct methods include the classical Cholesky factorization of the stiffness matrices, the natural factor method of Argyris and a weighted least squares with iterative improvement method of Van Loan. The iterative methods include new preconditioned conjugate gradient methods for weighted least squares problems by Freund and for least squares with equality constraints by Barlow, Nichols, and Plemmons. (ERA citation 13:018107)