Transaction of JSCES, Paper No Parallel Finite Element Analysis in Large Scale Shell Structures using CGCG Solver

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1 Transaction of JSCES, Paper No. 257 * Parallel Finite Element Analysis in Large Scale Shell Structures using Solver Shohei HORIUCHI, Hirohisa NOGUCHI and Hiroshi KAWAI ) This paper presents a parallel finite element analysis of large scale shell structures where the iterative method is utilized as a linear equation solver. The method is one of the conjugated gradient methods with the coarse grid solver and has been developed for analysis of shell structures in our previous paper. As the method is based on the domain decomposition method, it can be adapted to the parallelization straightforwardly. As a numerical example, an analysis of pinched cylinder is conducted. Comparing with the results by the BDD method, the parallel solver shows competitive performance with regard to the computational time and the parallel efficiency and superior efficiency with regard to memory usage. Finally, a large scale shell problem with 1 million degrees of freedom is successfully solved by the present parallel solver. Key Words: Parallel Computing, Shell Element, Finite Element Method, Iterative Method, Domain Decomposition Method, Balancing Method 1. (1),(2) CG (3) BDD (4) (6) (7) PC PC Domain Decomposition Method (8) CPU * Manuscript received, December 1, 24; final version, February 21, 25; 8 16 Published, March 11, 25. Copyright 25 by the Japan Society for Computational Engineering and Science. PC 1

2 (1) 2.1 (11),(12) CG type A type B (8) BDD fine coarse coarse fine coarse coarse coarse (8) fine BDD BDD type A 3 (6) Fig.1 A coarse CG CG coarse fine Fig.1 coarse type B coarse LU BDD (4) FETI (5) CG BDD BDD BDD Neumann-Neumann (9) coarse coarse coarse BDD BDD fine BDD coarse CG sparse coarse BDD

3 Neumann-Neumann 1.x1-6 4 MITC ADVENTURE_Metis (15) PC 8 CPU Intel Xeon3.6GHzFSB 533MHz dual 2.5 Gbyte 512Mbyte 4 DDR coarse SDRAM PC Gigabit coarse Ethernet coarse 4.2 Pinched cylinder problem LU CPU Pinched cylinder problemfig.2 1/8 3.x1 6 MPa.3 2 BDD FETI Element P.E. Farhat FETI 2, FETI-DP model 2 1 P.E. Origin (13) START A A A Coarse Grid LU CG CG CG END Fig.1 Flowchart of method 4. (14) mpich Table1 model 1 1 Processor 16 Fig.3 Table1 Model mesh size model 1 Fig.4 2, DIAG16 P.E. 497sec 8 2 BDD Fig.5 Fig.5 BDD BDD BDD LDL T BDD 4.1 BDD model 12 1 P.E. BDD BDD BDD CG type A DIAG

4 BDD Fig.6 1 P.E. 2, 2 Fig.15 DIAG 16 P.E. P.E. 82% 1 P.E. 1, model 1 coarse 2 2, coarse Fig.16 model 12, 2 P.E. Fig.7Fig.8 2 P.E. coarse LDL T 1 2 PC z 3 1 model 2 Fig.9 BDD P thickness : 3 16 P.E. 8 Young's modulus : 3. x 1 6 Poisson's ratio :.3 1,sec DIAG Fig.2 Analysis model of pinched cylinder 14,sec P.E. 8 Mbyte Fig P.E.1,6 Fig.11 Fig (16) = Barrel vault roof problem 4 Barrel vault roof problem Fig.13 1/8 4.2 model DIAG 5 Fig.14 8 P.E. Elapsed time [sec] 8 2 P = 1 Fig.3 Domain decomposition (16 domains) y x 2 domains 2 domains BDD 2 domains BDD 2 domains Fig.4 Elapsed time (model 1) 6

5 Memory usage [Mbyte] Parallel efficiency Preconditioning time [sec] Preconditioning time [sec] Elapsed time [sec] domains 2 domains BDD 2 domains BDD 2 domains Fig.5 Memory usage (model 1) DIAG 2 domains 2 domains BDD 2 domains Fig.6 Parallel efficiency (model 1, ) process / node 1 process / node Fig.7 Elapsed time (model 1, ) 3 2 process / node 1 process / node 2 Fig.8 Preconditioning time (model 1, ) 3 8 P.E P.E Number of domains Fig.9 Elapsed time (model 2) Memory usage [Mbyte] Number of domains Fig.1 Memory usage of preconditioner (model 2, ) E E+ 1.E E E E E E E-7 Residual norms Residual norms 1-7 DIAG Number of iteration Fig.11 History of residual norms (model 2, 1,6 domains) 2. Displacement x log (Number of d.o.f.) Analysis Reference Fig.12 Displacement of pinched cylinder 5. 1 PC ADVENTURE (17) 9

6 Young s modulus: 3.x1 6 Poisson s ratio :. Thickness : 3. Weight density :.2833 Parallel efficiency Elapsed time [sec] Memory usage [Mbyte] free r = 3 symm. A diaphragm support symm. 3 free 4 diaphragm support x z y Fig.13 Analysis model of barrel vault roof problem domains 4 domains 8 domains Fig.14 Parallel efficiency 25 1 domains 2 4 domains 8 domains 15 1 (2),, I,, 24. (3) G. Yagawa and R. Shioya, Parallel finite elements on a massively parallel computer with domain decomposition, Comp. Sys. in Eng. 4, 1994, (4) J. Mandel, Balancing domain decomposition, Communications in Numerical Methods in Engineering, 9, 1993, (5) C. Farhat and F.X. Roux, Implicit parallel processing in structural mechanics, Comp. Mech. Advances, 2, 1994, pp (6),,,,,, A 653, 22, (7),,,, 3,, 2, 2, (8),,,, 25, Paper No.254. (9) Y.H. De Roeck and P. LeTallec, Analysis and test of a local domain decomposition preconditioner, in Fourth International Symposium on Domain Decomposition Methods, SIAM, Philadelphia, PA, (1) M. Benzi, R. Kouhia and M. Tuma, An assesment of some preconditioning techniques in shell problems, Communication in Numerical Methods in Engineering, 14, 1998, (11) E. E. Ovtchinnikov and L. S. Xanthis; Effective dimensional reduction algorithm for eigenvelue problems for thin elastic structures: a paradigm in three dimensions, Applied Mathematics, 97, 2, (12) E. Ramm, M. Gee and W.A. Wall, Parallel multilevel solution of nonlinear shell structure, WCCM VI, Beijing, China, Fig.15 Elapsed time (13) C. Farhat, M. Lesoinne, P. LeTallec, K. Pierson and D. 35 Rixen, FETI-DP: A dual-primal unifield FETI method Part I: 3 1 domains 4 domains A faster alternative to the two-level FETI method, 25 8 domains International Journal for Numerical Methods in Engineering, 2 5, 21, Fig.16 Total memory usage (1),, 1,, 21, Paper No (14) E.N. Dvorkin and K.-J. Bathe, A continuum mechanics based forur-node shell element for general nonlinear analysis, Eng. Comp., 1, 1984, (15) (16) ABAQUS Example Problems Manual, Hibbitt, Karlson & Sorensen. 5. 8, 1, 1998, 1.1.2