Finite Element Analysis in Concurrent Processing: Computational Issues

The purpose of this research is to investigate the potential application of new methods for solving large-scale static structural problems on concurrent computers. It is well known that traditional single-processor computational speed will be limited by inherent physical limits. The only path to achieve higher computational speeds lies through concurrent processing. Traditional factorization solution methods for sparse matrices are ill suited for concurrent processing because the null entries get filled, leading to high communication and memory requirements. The research reported herein investigates alternatives to factorization that promise a greater potential to achieve high concurrent computing efficiency. Two methods, and their variants, based on direct energy minimization are studied: a) minimization of the strain energy using the displacement method formulation; b) constrained minimization of the complementary strain energy using the force method formulation. Initial results indicated that in the context of the direct energy minimization the displacement formulation experienced convergence and accuracy difficulties while the force formulation showed promising potential. I. Introduction HIS report summarizes the results of a research to investigate the potential of alternate methods for solving large-scale structural problems on future computers. The finite element method has proven to be a widely used technique to perform structural analysis. The method allows continuum problems of solid mechanics to be approximated by a finite number of unknowns. Using variational principles of mechanics, static problems reduce to the solution of a system of linear equations (or a sequence of linear equations for non-linear problems). The need for detailed analyses of complex structural systems (aerospace, automotive, and civil engineering) will soon lead to structural models with 10 to 100 million degrees of freedom. Solution of such large problems in a reasonable time will require computing at the processing rate far exceeding the capacity of today’s single processor CPU (Central Processing Unit)-based computers. CPU speeds will eventually reach inherent physical limits as processor miniaturization descends to the molecular and atomic scales. Recent computer technology advances (large-scale multi-processors, clusters or grids of commodity CPU’s, and reconfigurable computers such as Field Programmable Gate Arrays) are being developed to improve aggregate computing speeds. Although widely different in their details, these technologies share the common theme of concurrent processing. However, engineers cannot use the new computing technology until they have applications software developed that exploits the technology to its full potential. Traditional methods (algorithms) cannot realize the full potential of concurrent processing because they are rooted in the sequential thinking of the human brain. The method commonly used in the Finite Element Analysis (FEA) solves the load-deflection equations, [K] {U} = {P}, for displacements U