The combined approximations (CA) method is an effective reanalysis approach providing high quality results. The CA method is suitable for a wide range of structural optimization problems including linear reanalysis, nonlinear reanalysis and eigenvalue reanalysis. However, with increasing complexity and scale of engineering problems, the efficiency of the CA method might not be guaranteed. A major bottleneck of the CA is how to obtain reduced basis vectors efficiently. Therefore, a modified CA method, based on approximation of the inverse matrix, is suggested. Based on the symmetric successive over-relaxation (SSOR) and compressed sparse row (CSR), the efficiency of CA method is shown to be much improved and corresponding storage space markedly reduced. In order to further improve the efficiency, the suggested strategy is implemented on a graphic processing unit (GPU) platform. To verify the performance of the suggested method, several case studies are undertaken. Compared with the popular serial CA method, the results demonstrate that the suggested GPU-based CA method is an order of magnitude faster for the same level of accuracy.
Issue Section:
Research Papers
References
1.
Kirsch
, U.
, 2000
, “Combined Approximations—A General Reanalysis Approach for Structural Optimization
,” Struct. Optim.
, 20
, pp. 97
–106
.10.1007/s0015800501412.
Kirsch
, U.
, and Papalambros
, P. Y.
, 2001
, “Exact and Accurate Solutions in the Approximate Reanalysis of Structures
,” AIAA J.
, 39
, pp. 2198
–2205
.10.2514/2.12193.
Kirsch
, U.
, 1999
, “Efficient-Accurate Reanalysis for Structural Optimization
,” AIAA J.
, 37
, pp. 1663
–1669
.10.2514/2.6494.
Kirsch
, U.
, 1991
, “Reduced Basis Approximations of Structural Displacements for Optimal Design
,” AIAA J.
, 29
, pp. 1751
–1758
10.2514/3.107995.
Kirsch
, U.
, 1993
, Structural Optimizations, Fundamentals and Applications
, Springer
, Berlin, Heidelberg, New York
.6.
Kirsch
, U.
, 1993
, “Approximate Reanalysis for Topological Optimization
,” Struct. Optim.
, 6
, pp. 143
–150
.10.1007/BF017435057.
Leu
, L.-J.
, and Huang
, C.-W.
, 1998
, “A Reduced Basis Method for Geometric Nonlinear Analysis of Structures
,” IASS J.
, 39
, pp. 71
–75
.8.
Aktas
, A.
, and Moses
, F.
, 1998
, “Reduced Basis Eigenvalue Solutions or Damaged Structures
,” Mech. Struct. Mach.
, 26
, pp. 63
–79
.10.1080/089054598089454209.
Chen
, S. H.
, and Yang
, X. W.
, 2000
, “Extended Kirsch Combined Method for Eigenvalue Reanalysis
,” AIAA J.
, 38
, pp. 927
–930
.10.2514/2.104910.
Chen
, S. H.
, and Yang
, Z. J.
, 2004
, “A Universal Method for Structural Static Reanalysis of Topological Modifications
,” Int. J. Numer. Methods Eng.
, 61
, pp. 673
–686
.10.1002/nme.108411.
Kirsch
, U.
, and Bogomolni
, M.
, 2004
, “Procedures for Approximate Eigenproblem Reanalysis of Structures
,” Int. J. Numer. Methods Eng.
, 60
, pp. 1969
–198
.10.1002/nme.103212.
Kirsch
, U.
, Bogomolni
, M.
, and Sheinman
, I.
, 2007
, “Efficient Dynamic Reanalysis of Structures
,” ASCE J. Struct. Eng.
, 133
, pp. 440
–448
.10.1061/(ASCE)0733-9445(2007)133:3(440)13.
Kirsch
, U.
, Bogomolni
, M.
, and Sheinman
, I.
, 2006
, “Nonlinear Dynamic Reanalysis for Structural Optimization
,” Comput. Methods Appl. Mech. Eng.
, 195
, pp. 4420
–4432
.10.1016/j.cma.2005.09.01314.
Kirsch
, U.
, 2008
, Reanalysis of Structures
, Springer
, Dordrecht
.15.
Kirsch
, U.
, 2010
, “Reanalysis and Sensitivity Reanalysis by Combined Approximations
,” Struct. Multidiscip. Optim.
, 40
, pp. 1
–15
.10.1007/s00158-009-0369-116.
Raz
, R.
, 2003
, “On the Complexity of Matrix Product
,” SIAM J. Comput.
, 32
(5
), pp. 1356
–1369
.10.1137/S009753970240214717.
Unterkircher
, A.
, and Reissner
, J.
, 2005
, “Parallel Assembling and Equation Solving via Graph Algorithms With an Application to the FE Simulation of Metal Extrusion Processes
,” Comput. Struct.
, 83
(8–9
), pp. 627
–638
.10.1016/j.compstruc.2004.10.01218.
Farhat
, C.
, Crivelli
, L.
, and Roux
, F. X.
, 1994
, “A Transient FETI Methodology for Large-Scale Parallel Implicit Computations in Structural Mechanics
,” 37
(11
), pp. 1945
–1975
.19.
Komatitsch
, D.
, Micha
, D.
, and Erlebacher
, G.
, 2009
, “Porting a High-Order Finite-Element Earthquake Modeling Application to NVIDIA Graphics Cards Using CUDA
,” J. Parallel Distrib. Comput.
, 69
(5
), pp. 451
–460
.10.1016/j.jpdc.2009.01.00620.
Owens
, J. D.
, Houston
, M.
, Luebke
, D.
, Green
, S.
, Stone
, J. E.
, and Phillips
, J. C.
, 2008
, “GPU Computing
,” Proc. IEEE
, 5
, pp. 879
–899
.10.1109/JPROC.2008.91775721.
Cecka
, C.
, Lew
, A. J.
, and Darve
, E.
, 2011
, “Assembly of Finite Element Methods on Graphic Processors
,” Int. J. Numer. Methods Eng.
, 85
, pp. 640
–669
.10.1002/nme.298922.
Cai
, Y.
, Li
, G.
, Wang
, H.
, Zheng
, G.
, and Lin
, S.
, 2012
, “Development of Parallel Explicit Finite Element Sheet Forming Simulation System Based on GPU Architecture
,” Adv. Eng. Software
, 45
(1
), pp. 370
–379
.10.1016/j.advengsoft.2011.10.01423.
Joldes
, G. R.
, Wittek
, A.
, and Miller
, K.
, 2010
, “Real-Time Nonlinear Finite Element Computations on GPU—Application to Neurosurgical Simulation
,” Comput. Methods Appl. Mech. Eng.
, 199
(49–52
), pp. 3305
–3314
.10.1016/j.cma.2010.06.03724.
Krakiwsky
, S. E.
, Turner
, L. E.
, and Okoniewski
, M. M.
, 2004
, “Acceleration of Finite-Difference Time-Domain (FDTD) Using Graphics Processor Units(GPU)
,” Proceedings of the IEEE MTT-S International Microwave Symposium
, pp. 1033
–1036
.25.
Inman
, M. J.
, and Elsherbeni
, A. Z.
, 2005
, “Programming Video Cards for Computational Electromagnetics Applications
,” IEEE Antennas Propag. Mag.
, 47
, pp. 71
–78
.10.1109/MAP.2005.160873027.
Golub
, G. H.
, and Van Loan
, C. F.
, 1996
, Matrix Computations
, 3rd ed., The Johns Hopkins University Press
, Baltimore
.28.
Chen
, R. S.
, Yung
, E. K. N.
, Chan
, C. H.
, and Fang
, D. G.
, “Application of SSOR Preconditioned Conjugate Gradient Algorithm to Edge-FEM for 3-Dimensional Full Wave Electromagnetic Boundary Value Problems
,” IEEE Trans. Microwave Theory Tech.
, 50
(4
), pp. 1165
–1172
.10.1109/22.99342029.
Download CUDA
, http://developer.nvidia.com/cuda-downloads30.
Fletcher
, R.
, and Reeves
, C. M.
, 1964
, “Function Minimization by Conjugate Gradients
,” Comput. J.
, 7
, pp. 149
–154
.10.1093/comjnl/7.2.14931.
Langou
, J.
, Luszczek
, P.
, Kurzak
, J.
, Buttari
, A.
, and Dongarra
, J.
, 2006
, “Exploiting the Performance of 32 bit Floating Point Arithmetic in Obtaining 64 bit Accuracy (Revisiting Iterative Refinement for Linear Systems)
,” Proceedings of the ACM/IEEE Super Computing
.32.
Bell
, N.
, and Garland
, M.
, 2008
, “Efficient Sparse Matrix–Vector Multiplication on CUDA
,” Technical Report No. NVR-2008-04, NVIDIA.33.
Adams
, M. F.
, 2000
, “Parallel Multigrid Solvers for 3D Unstructured Finite Element Problems in Large Deformation Elasticity and Plasticity
,” Int. J. Numer. Methods Eng.
, 48
, pp. 1241
–62
.10.1002/(SICI)1097-0207(20000720)48:8<1241::AID-NME946>3.0.CO;2-RCopyright © 2013 by ASME
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