A fast, accurate Choleski method for the solution of symmetric systems of linear equations is presented. This direct method is based on a variable-band storage scheme and takes advantage of column heights to reduce the number of operations in the Choleski factorization. The method employs parallel computation in the outermost DO-loop and vector computation via the "loop unrolling" technique in the innermost DO-loop. The method avoids computations with zeros outside the column heights, and as an option, zeros inside the band. The close relationship between Choleski and Gauss elimination methods is examined. The minor changes required to convert the Choleski code to a Gauss code to solve non-positive-definite symmetric systems of equations are identified. The results for two large-scale structural analyses performed on supercomputers, demonstrate the accuracy and speed of the method. Nomenclature e a error norm for solution residuals e s strain energy error norm {f} load vector hpm har...

Engineers are constantly faced with solving problems of increasing complexity and detail. They frequently rely upon numerical methods to solve these problems, and their insatiable appetite for improved performance from computing hardware has reached a point where the computational requirements exceed reasonable expectations of the performance of Von-Neumann (serial) computers. Multiple Instruction stream Multiple Data stream (MIMD) computers have been devel-oped to overcome the performance limitations of serial computers. The hardware architec-tures of MIMD computers vary considerably and are much more sophisticated than serial computers. Developing large scale software for a variety of MIMD computers is difficult and expensive. There is a need to provide tools that facilitate programming these machines. The first part of this report examines the issues that must be considered to develop those tools. The two main areas of concern were architecture independence and data man-agement. Architecture independent software facilitates software portability and improves the longevity and utility of the software product. It provides some form of insurance for the

This paper presents parallel 3-D finite element analysis for distributed memory multiprocessors. Traditionally, finite element analysis has been performed on sequential computers. Current research in high performance finite element analysis shows considerable promise for fast, efficient implementation on MIMD and SIMD computers [Farhat 87a], [Johnsson 90]. The finite element system of equations can be solved by a direct method, or by an iterative method [Carter 89]. Although iterative methods are attractive when a good initial estimate of the solution is available, direct methods are generally preferred because once the coefficient matrix (global stiffness matrix) is factored, solution to multiple load cases and sensitivity analysis becomes considerably simpler. Several researchers have used the substructuring method for parallel finite element analysis on hypercubes [Adams 84], [Farhat 87b], [Carey 86]. It is described only briefly here; refer to [Adams 84] for details. In this method, the structural domain is decomposed into several subdomains (substructures), one for each processor. Nodes that are common to more than one subdomain form the `interface'. The resulting global stiffness matrix has an `arrowhead' pattern, with diagonal blocks corresponding to each substructure and the interface, and off-diagonal blocks coupling each substructure with the interface. Each processor assembles the element stiffness matrices for its substructure. Each processor then factorizes independently, one diagonal block and updates the interface block. All processors, in parallel, solve a system of equations corresponding to the interface block. Subsequently, the processors recover the displacement solution for their substructure, independently. While the substructuring method appears t...

This research is directed toward the numerical analysis of large, three dimensional, nonlinear dynamic problems in structural and solid mechanics. Such problems include those exhibiting large deformations, displacements, or rotations, those requiring finite strain plasticity material models that couple geometric and material nonlinearities, and those demanding detailed geometric modeling. A finite element code was developed, designed around the 3D isoparametric family of elements, and using a Total Lagrangian formulation and implicit integration of the global equations of motion. The research was conducted using the Alliant FXl8 and Convex C240 supercomputers. The research focuses on four main areas: Development of element computation algorithms that exploit the inherent opportunities for concurrency and vectorization present in the finite element method; Comparison of the preconditioned conjugate gradient method to a representative direct solver; Investigation of various nonlinear solution algorithms, such as modified Newton-Raphson, secant-Newton, and nonlinear preconditioned conjugate gradient; and,