Time Integration Method with High Accuracy and Efficiency for Structural Dynamic Analysis
Publication: Journal of Engineering Mechanics
Volume 145, Issue 3
Abstract
In this study, a novel composite time integration method is proposed for more accurately and efficiently solving typical structural dynamic problems. In this method, the second-order accuracy is ensured for dynamic problems. First, the stability, accuracy properties, local truncation error, and global error are analyzed and compared with available state-of-the-art methods in the literature. Then, three sets of parameters are recommended and discussed, and optimization of these parameters results in a high accuracy and efficiency of the proposed method. Finally, three classical examples with high-frequency vibrations, where a large ratio of the time step size to the period is adopted, are presented to demonstrate the accuracy, efficiency, and applicability of the proposed method.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The first author appreciates the financial support of the China Scholarship Council and the Independent Exploration and Innovation Project for Ph.D. students at Central South University (2016zzts072). The work described in this paper was also supported by a key basic research project (Project 973) of P.R. China under Contract 2015CB057701. Louisiana State University provided the high-performance computing resources. All the findings reported here are those of the authors and do not necessarily represent those of the sponsors.
References
Balah, M., and H. N. Al-Ghamedy. 2005. “Energy-momentum conserving algorithm for nonlinear dynamics of laminated shells based on a third-order shear deformation theory.” J. Eng. Mech. 131 (1): 12–22. https://doi.org/10.1061/(ASCE)0733-9399(2005)131:1(12).
Bathe, K. J. 1996. Finite element procedures. Upper Saddle River, NJ: Prentice-Hall.
Bathe, K. J. 2007. “Conserving energy and momentum in nonlinear dynamics: A simple implicit time integration scheme.” Comput. Struct. 85 (7–8): 437–445. https://doi.org/10.1016/j.compstruc.2006.09.004.
Bathe, K. J., and M. M. I. Baig. 2005. “On a composite implicit time integration procedure for nonlinear dynamics.” Comput. Struct. 83 (31–32): 2513–2524. https://doi.org/10.1016/j.compstruc.2005.08.001.
Bathe, K. J., and G. Noh. 2012. “Insight into an implicit time integration scheme for structural dynamics.” Comput. Struct. 98–99: 1–6. https://doi.org/10.1016/j.compstruc.2012.01.009.
Bathe, K. J., and E. L. Wilson. 1973. “Stability and accuracy analysis of direct integration methods.” Earthquake Eng. Struct. 1 (3): 283–291. https://doi.org/10.1002/eqe.4290010308.
Béjar, L. A., and K. T. Danielson. 2016. “Critical time-step estimation for explicit integration of dynamic higher-order finite-element formulations.” J. Eng. Mech. 142 (7): 04016043. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001082.
Chang, S. Y. 2002. “Explicit pseudodynamic algorithm with unconditional stability.” J. Eng. Mech. 128 (9): 935–947. https://doi.org/10.1061/(ASCE)0733-9399(2002)128:9(935).
Chang, S. Y. 2003. “Nonlinear error propagation analysis for explicit pseudodynamic algorithm.” J. Eng. Mech. 129 (8): 841–850. https://doi.org/10.1061/(ASCE)0733-9399(2003)129:8(841).
Chen, C., and J. M. Ricles. 2008. “Development of direct integration algorithms for structural dynamics using discrete control theory.” J. Eng. Mech. 134 (8): 676–683. https://doi.org/10.1061/(ASCE)0733-9399(2008)134:8(676).
Chung, J. T., and J. M. Lee. 1994. “A new family of explicit time integration methods for linear and non-linear structural dynamics.” Int. J. Numer. Methods Eng. 37 (23): 3961–3976. https://doi.org/10.1002/nme.1620372303.
Erlieher, S., L. Bonvaentura, and O. S. Bursi. 2002. “The analysis of the Generalized-a method for non-linear dynamic problems.” Comput. Mech. 28 (2): 83–104. https://doi.org/10.1007/s00466-001-0273-z.
Fu, C. C. 1970. “A method for the numerical integration of the equation of motions arising from a finite element analysis.” J. Appl. Mech. ASME 37 (3): 599–605. https://doi.org/10.1115/1.3408586.
Gottlieb, D., and S. A. Orszag. 1993. Numerical analysis of spectral methods: Theory and applications. Philadelphia: Capital City Press.
Guddati, M. N., and B. Yue. 2004. “Modified integration rules for reducing dispersion error in finite element methods.” Comput. Methods Appl. Mech. Eng. 193 (3–5): 275–287. https://doi.org/10.1016/j.cma.2003.09.010.
Ham, S., and K. J. Bathe. 2012. “A finite element method enriched for wave propagation problems.” Comput. Struct. 94–95: 1–12. https://doi.org/10.1016/j.compstruc.2012.01.001.
Hilber, H. M., and T. J. R. Hughes. 1978. “Collocation, dissipation and [overshoot] for time integration schemes in structural dynamics.” Earthquake Eng. Struct. Dyn. 6 (1): 99–117. https://doi.org/10.1002/eqe.4290060111.
Hilber, H. M., T. J. R. Hughes, and R. L. Taylor. 1977. “Improved numerical dissipation for time integration algorithms in structural dynamics.” Earthquake Eng. Struct. Dyn. 5 (3): 283–292. https://doi.org/10.1002/eqe.4290050306.
Idesman, A. V., M. Schmidt, and J. R. Foley. 2011. “Accurate finite element modeling of linear elastodynamics problems with the reduced dispersion error.” Comput. Mech. 47 (5): 555–572. https://doi.org/10.1007/s00466-010-0564-3.
Keierleber, C. W., and B. T. Rosson. 2005. “Higher-order implicit dynamic time integration method.” J. Struct. Eng. 131 (8): 1267–1276. https://doi.org/10.1061/(ASCE)0733-9445(2005)131:8(1267).
Khamlichi, A., L. Elbakkali, and A. Limam. 2001. “Postbuckling of elastic beams considering higher order strain terms.” J. Eng. Mech. 127 (4): 372–378. https://doi.org/10.1061/(ASCE)0733-9399(2001)127:4(372).
Kim, J., and K. J. Bathe. 2013. “The finite element method enriched by interpolation covers.” Comput. Struct. 116: 35–49. https://doi.org/10.1016/j.compstruc.2012.10.001.
Kinmark, I. P. E., and W. G. Gray. 1986. “Fourth-order accurate one-step integration methods with large imaginary stability limits.” Numer. Methods Partial Differ. Equations 2 (1): 63–70.
Kolay, C., and J. M. Ricles. 2014. “Development of a family of unconditionally stable explicit direct integration algorithms with controllable numerical energy dissipation.” Earthquake Eng. Struct. 43 (9): 1361–1380. https://doi.org/10.1002/eqe.2401.
Kujawski, J., and R. H. Gallagher. 1984. “A family of higher order explicit algorithms for transient dynamic analysis.” Trans. Soc. Comp. Simul. 1 (2): 155–166.
Miranda, I., R. M. Ferencz, and T. J. R. Hughes. 1989. “An improved implicit-explicit time integration method for structural dynamics.” Earthquake Eng. Struct. 18 (5): 643–653. https://doi.org/10.1002/eqe.4290180505.
Newmark, N. M. 1959. “A method of computation for structural dynamics.” J. Eng. Mech. 85 (3): 67–94.
Noh, G., and K. J. Bathe. 2013. “An explicit time integration scheme for the analysis of wave propagations.” Comput. Struct. 129: 178–193. https://doi.org/10.1016/j.compstruc.2013.06.007.
Nsiampa, N., J. P. Ponthot, and L. Noels. 2008. “Comparative study of numerical explicit schemes for impact problems.” Int. J. Impact Eng. 35 (12): 1688–1694. https://doi.org/10.1016/j.ijimpeng.2008.07.003.
Rezaiee-Pajand, M., and J. Alamatian. 2008. “Implicit higher-order accuracy method for numerical integration in dynamic analysis.” J. Struct. Eng. 134 (6): 973–985. https://doi.org/10.1061/(ASCE)0733-9445(2008)134:6(973).
Rio, G., A. Soive, and V. Grolleau. 2005. “Comparative study of numerical explicit time integration algorithms.” Adv. Eng. Software 36 (4): 252–265. https://doi.org/10.1016/j.advengsoft.2004.10.011.
Rostami, S., S. Shojaee, and H. Saffari. 2013. “An explicit time integration method for structural dynamics using cubic B-spline polynomial functions.” Scientia Iranica 20 (1): 23–33. https://doi.org/10.1016/j.scient.2012.12.003.
Wen, W. B., K. L. Jian, and S. M. Luo. 2014. “An explicit time integration method for structural dynamics using septuple B-spline functions.” Int. J. Numer. Methods Eng. 97 (9): 629–657. https://doi.org/10.1002/nme.4599.
Wen, W. B., K. Wei, H. S. Lei, S. Y. Duan, and D. N. Fang. 2017. “A novel sub-step composite implicit time integration scheme for structural dynamics.” Comput. Struct. 182: 176–186. https://doi.org/10.1016/j.compstruc.2016.11.018.
Wilson, E. L. 1973. “Nonlinear dynamic analysis of complex structures.” Earthquake Eng. Struct. 1 (3): 241–252. https://doi.org/10.1002/eqe.4290010305.
Wood, W. L. 1990. Practical time-stepping schemes. Oxford, UK: Clarendon Press.
Yue, B., and M. N. Guddati. 2005. “Dispersion-reducing finite elements for transient acoustics.” J. Acoust. Soc. Am. 118 (4): 2132–2141. https://doi.org/10.1121/1.2011149.
Zhai, W. M. 1996. “Two simple fast integration methods for large-scale dynamic problems in engineering.” Int. J. Numer. Methods Eng. 39 (24): 4199–4214. https://doi.org/10.1002/(SICI)1097-0207(19961230)39:24%3C4199::AID-NME39%3E.0.CO;2-Y.
Information & Authors
Information
Published In
Copyright
©2019 American Society of Civil Engineers.
History
Received: Feb 6, 2018
Accepted: Aug 24, 2018
Published online: Jan 11, 2019
Published in print: Mar 1, 2019
Discussion open until: Jun 11, 2019
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.