Preconditioned Conjugate Gradient Method for Static Reanalysis with Modifications of Supports
Publication: Journal of Engineering Mechanics
Volume 141, Issue 2
Abstract
This paper focuses on the problem of static reanalysis with modifications of supports. An efficient approximate reanalysis method is proposed. The definition of an augmented stiffness matrix is first introduced, and the preconditioned conjugate gradient method is then used. A preconditioner is constructed by fully using the Cholesky factorization of the initial stiffness matrix. The method preserves the ease of implementation, and the accuracy of the approximate solutions can be adaptively monitored. Numerical examples show that high-quality results can be provided by the proposed method and that the convergence is fast.
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Acknowledgments
This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 11072085 and 51005096) and the Fundamental Research Funds for the Central Universities.
References
Abu Kassim, A. M., and Topping, B. H. V. (1987). “Static reanalysis: A review.” J. Struct. Eng., 1029–1045.
Akgün, M. A., Garcelon, J. H., and Haftka, R. T. (2001). “Fast exact linear and non-linear structural reanalysis and the Sherman–Morrison–Woodbury formulas.” Int. J. Numer. Methods Eng., 50(7), 1587–1606.
Amir, O., and Sigmund, O. (2011). “On reducing computational effort in topology optimization: How far can we go?” Struct. Multidisciplin. Optim., 44(1), 25–29.
Amir, O., Sigmund, O., Lazarov, B. S., and Schevenels, M. (2012). “Efficient reanalysis techniques for robust topology optimization.” Comput. Methods Appl. Mech. Eng., 245–246, 217–231.
Compaq Visual Fortran 6.5 [Computer software]. Khost, TX, Compaq Computer Corporation.
Demmel, J. W. (1997). Applied numerical linear algebra, SIAM Publications, Philadelphia.
Friswell, M. I. (2006). “Efficient placement of rigid supports using finite element models.” Commun. Numer. Methods Eng., 22(3), 205–213.
Golub, G. H., and Van Loan, C. F. (1996). Matrix computations, 3rd Ed., Johns Hopkins University Press, Baltimore.
Kirsch, U. (2003). “Design-oriented analysis of structures—Unified approach.” J. Eng. Mech., 264–272.
Kirsch, U. (2008). Reanalysis of structures, Springer, Dordrecht, Netherlands.
Kirsch, U., Kocvara, M., and Zowe, J. (2002). “Accurate reanalysis of structures by a preconditioned conjugate gradient method.” Int. J. Numer. Methods Eng., 55(2), 233–251.
Kirsch, U., and Papalambros, P. Y. (2001). “Structural reanalysis for topological modifications—A unified approach.” Struct. Multidisciplin. Optim., 21(5), 333–344.
Li, Z. G., Lim, C. W., and Wu, B. S. (2008). “A comparison of several reanalysis methods for structural layout modifications with added degrees of freedom.” Struct. Multidisciplin. Optim., 36(4), 403–410.
Li, Z. G., and Wu, B. S. (2007). “A preconditioned conjugate gradient approach to structural reanalysis for general layout modifications.” Int. J. Numer. Methods Eng., 70(5), 505–522.
Liu, H. F., Wu, B. S., and Li, Z. G. (2012a). “An efficient approach to structural static reanalysis with added support constraints.” Struct. Eng. Mech., 43(3), 273–285.
Liu, H. F., Wu, B. S., and Li, Z. G. (2014). “Method of updating the Cholesky factorization for structural reanalysis with added degrees of freedom.” J. Eng. Mech., 384–392.
Liu, H. F., Wu, B. S., Lim, C. W., and Li, Z. G. (2012b). “An approach for structural static reanalysis with unchanged number of degrees of freedom.” Struct. Multidisciplin. Optim., 45(5), 681–692.
Mróz, Z., and Rozvany, G. I. N. (1975). “Optimal design structures with variable support conditions.” J. Optim. Theory Appl., 15(1), 85–101.
Nair, P. B. (2002). “Equivalence between the combined approximations technique and Krylov subspace methods.” AIAA J., 40(5), 1021–1023.
Olhoff, N., and Taylor, J. E. (1983). “On structural optimization.” J. Appl. Mech., 50(4b), 1139–1151.
Pais, M. J., Yeralan, S. N., Davis, T. A., and Kim, N. H. (2012). “An exact reanalysis algorithm using incremental Cholesky factorization and its application to crack growth modeling.” Int. J. Numer. Methods Eng., 91(12), 1358–1364.
Takezawa, A., Nishiwaki, S., Izui, K., and Yoshimura, M. (2006). “Structural optimization using function-oriented elements to support conceptual designs.” J. Mech. Des., 128(4), 689–700.
Tanskanen, P. (2006). “A multiobjective and fixed elements based modification of the evolutionary structural optimization method.” Comput. Methods Appl. Mech. Eng., 196(1–3), 76–90.
Terdalkar, S. S., and Rencis, J. J. (2006). “Graphically driven interactive finite element stress reanalysis for machine elements in the early design stage.” Finite Elem. Anal. Des., 42(10), 884–899.
Turan, A., and Mugan, A. (2013). “Structural and sensitivity reanalyses based on singular value decomposition.” Struct. Multidisciplin. Optim., 48(2), 327–337.
Van der Vorst, H. A. (2003). Iterative Krylov methods for large linear systems, Cambridge University Press, Cambridge, U.K.
Voormeeren, S., and Rixen, D. (2013). “Updating component reduction bases of static and vibration modes using preconditioned iterative techniques.” Comput. Methods Appl. Mech. Eng., 253, 39–59.
Wang, B. P., and Chen, J. L. (1996). “Application of genetic algorithm for the support location optimization of beams.” Comput. Struct., 58(4), 797–800.
Wang, D., Jiang, J. S., and Zhang, W. H. (2004). “Optimization of support positions to maximize the fundamental frequency of structures.” Int. J. Numer. Methods Eng., 61(10), 1584–1602.
Wang, H., Li, E. Y., and Li, G. Y. (2013). “A parallel reanalysis method based on approximate inverse matrix for complex engineering problems.” J. Mech. Des., 135(8), 081001.
Wu, B. S., and Li, Z. G. (2005). “Reanalysis of structural modifications due to removal of degrees of freedom.” Acta Mech., 180(1–4), 61–71.
Wu, B. S., and Li, Z. G. (2006). “Static reanalysis of structures with added degrees of freedom.” Commun. Numer. Methods Eng., 22(4), 269–281.
Wu, B. S., Xu, Z. H., and Li, Z. G. (2008). “A note on imposing displacement boundary conditions in finite element analysis.” Commun. Numer. Methods Eng., 24(9), 777–784.
Zhu, J. H., and Zhang, W. H. (2010). “Integrated layout design of supports and structures.” Comput. Methods Appl. Mech. Eng., 199(9–12), 557–569.
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Published In
Journal of Engineering Mechanics
Volume 141 • Issue 2 • February 2015
Copyright
© 2014 American Society of Civil Engineers.
History
Received: Feb 26, 2014
Accepted: Jun 5, 2014
Published online: Jun 25, 2014
Published in print: Feb 1, 2015
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