Compurers & Smrfures Vol. 63. No. 3, pp. 413-428. 1991 0 1997 Elwier Science Ltd PII: s0045-7949(%)00343 Printed in Great Britain. All rights rewwd 004s7949197 $17.00 + 0.00 NONLINEAR FINITE ELEMENT ANALYSIS OF BEAMS AND ARCHES USING PARALLEL PROCESSORS L. N. B. Gummadi and A. N. Palazotto Department of Aeronautics and Astronautics, Air Force Institute of Technology, WPAFB, OH 45433, U.S.A. (Received 17 January 1996) Abstract-A nonlinear finite element formulation for beams and arches is presented for implementation on a multiple instruction multiple data (MIMD) parallel machine. A twelve degree of freedom arch element is developed using Green strains and 2nd Piola Kirchhoff stresses. Concepts like loop splitting and expression splitting are used to improve the load balancing capability among the processors. Using this developed element, beams and arches that are undergoing large displacements and large rotations are analyzed, and efficiency of the program is compared with a sequential implementation. 0 1997 Elsevier Science Ltd. All rights reserved INTRODUCTION Ever since the serial computers based on the architecture by John Von Neumann [l] was devel- oped, their speed has increased steadily to match the needs of ever growing engineering applications in science and engineering. However during the 197Os, it became evident that the maximum possible speed limits of computers were approached due to the fundamental physical limitations imposed by the speed of light. A viable alternative to improve the speed of a computer in solving an engineering problem is in the use of a group of computers. This led to the development of the concept of parallel processing. In a serial computer, one processor is used for all the co-mputation involved in solving a problem. In a parallel computer, many processors are available for simultaneous processing and the effectiveness becomes possible through the sharing of the computational effort among themselves thereby improving the speed of the computer. The concept of parallel processing has made tremendous impact on many areas of computer application such as biosphere modeling, pollution modeling, semi con- ductor material modeling, ocean modeling, computer tomography, analysis of protein structures, vehicle design and dynamics etc [2]. A parallel computer is useless unless an efficient parallel algorithm is developed. An algorithm that is developed for a serial computer can be highly inefficient in a parallel computing environment as that algorithm will make use of only one processor at a time keeping the other available processors idle. Development of a parallel algorithm for a given problem is dependent on the control mechanism of the processors (SIMD, MIMD etc.), memory sharing mechanism of the processors (shared memory or distributed memory) and the arrangement of the processors (hypercube, mesh, ring, BBN butterfly etc.) [3]. This dependency on the architecture poses problems of portability with the parallel computers. A program written for one parallel computer may require extensive modification for a different parallel program unlike a serial computer (as a matter of fact, all serial computers fall into one sub category of parallel computers with a single instruction single data (SISD) control mechanism incorporating a single memory unit architecture). The concept of parallel processing has been successfully used in the past for various structural. analysis applications such as static analysis, dynamic analysis and optimization. The review of progress in these applications can be found in Refs [4-6]. In a typical linear static finite element analysis, the most time consuming operation is the solution of linear simultaneous equations. Hence, the majority of the efforts in the parallelization of a linear static analysis were focused on developing and implementing various equation solver algorithms on different parallel machines of different architecture. Some such efforts include linear static and transient finite element analysis on BBN butterfly parallel processor by Henno et al. [7] using parallel Gaussian elimin- ation algorithm, LDL decomposition algorithm for linear and nonlinear static and dynamic analysis on local memory MIMD machines by Farhat et al. [8,9], and column oriented Cholesky algorithm by Goehlich et al. [lo] on shared memory MIMD machines etc. Some other techniques that are implemented for the parallel computational environ- ment include global local analysis by Sun and 413
NONLINEAR FINITE ELEMENT ANALYSIS OF BEAMS AND ARCHES USING PARALLEL PROCESSORS
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