Browse > Article
http://dx.doi.org/10.12989/eas.2017.13.3.279

A novel two sub-stepping implicit time integration algorithm for structural dynamics  

Yasamani, K. (Malek-ashtar University of Technology)
Mohammadzadeh, S. (College of Engineering, School of Civil Engineering, University of Tehran)
Publication Information
Earthquakes and Structures / v.13, no.3, 2017 , pp. 279-288 More about this Journal
Abstract
Having the ability to keep on yielding stable solutions in problems involving high potential of instability, composite time integration methods have become very popular among scientists. These methods try to split a time step into multiple sub-steps so that each sub-step can be solved using different time integration methods with different behaviors. This paper proposes a new composite time integration in which a time step is divided into two sub-steps; the first sub-step is solved using the well-known Newmark method and the second sub-step is solved using Simpson's Rule of integration. An unconditional stability region is determined for the constant parameters to be chosen from. Also accuracy analysis is perform on the proposed method and proved that minor period elongation as well as a reasonable amount of numerical dissipation is produced in the responses obtained by the proposed method. Finally, in order to provide a practical assessment of the method, several benchmark problems are solved using the proposed method.
Keywords
simpson rule; newmark method; composite time integration; unconditional stability; numerical damping; period elongation;
Citations & Related Records
Times Cited By KSCI : 3  (Citation Analysis)
연도 인용수 순위
1 Chang, S.Y. (2015), "Dissipative, noniterative integration algorithms with unconditional stability for mildly nonlinear structural dynamic problems", Nonlinear Dyn., 79(2), 1625-1649.   DOI
2 Chang, S.Y. (2016), "A virtual parameter to improve stability properties for an integration method", Earthq. Struct., 11(2), 297-313.   DOI
3 Chang, S.Y., and Liao, W.I. (2005), "An unconditionally stable explicit method for structural dynamics", J. Earthq. Eng., 9(3), 349-370.   DOI
4 Chang, S.Y. (2009), "Accurate integration of nonlinear systems using newmark explicit method", J. Mech., 25(3), 289-297.   DOI
5 Chang, S.Y. (2014), "A family of noniterative integration methods with desired numerical dissipation", Int. J. Numer. Meth. Eng., 100(1), 62-86.   DOI
6 Chopra, A. (2007), Dynamics of Structures: Theory and Applications to Earthquake Engineering, 3rd Edition, Prentice-Hall, Upper Saddle River, NJ.
7 Chung, J., and Hulbert, G. (1993), "A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-${\alpha}$ method", J. Appl. Mech., 60(2), 371-375.   DOI
8 Dokainish, M. and Subbaraj, K. (1989), "A survey of direct time-integration methods in computational structural dynamics-I. Explicit methods", Comput. Struct., 32(6), 1371-1386.   DOI
9 Dong, S. (2010), "BDF-like methods for nonlinear dynamic analysis", J. Comput. Phys., 229(8), 3019-3045.   DOI
10 Gautam, S.S. and Sauer, R.A. (2014), "A composite time integration scheme for dynamic adhesion and its application to gecko spatula peeling", Int. J. Comput. Meth. Eng., 11(5), 1350104.   DOI
11 Noh, G., and K.-J. Bathe (2013), "An explicit time integration scheme for the analysis of wave propagations", Comput. Struct.,129, 178-193.   DOI
12 Pezeshk, S. and Camp C.V. (1995), "An explicit time-intergration method for damped structural systems", Struct. Eng. Mech., 3(2), 145-162.   DOI
13 Pezeshk, S. and Camp C.V. (1995), "An explicit time integration technique for dynamic analysis", Int. J. Numer. Meth. Eng., 38(13), 2265-2281.   DOI
14 Rezaiee-Pajand, M. and Alamatian, J. (2008), "Numerical time integration for dynamic analysis using a new higher order predictor-corrector method", Eng. Comput., 25(6), 541-568.   DOI
15 Rezaiee-Pajand, M. and Hashemian M. (2016), "Time integration method based on discrete transfer function", Int. J. Struct. Stab. Dyn., 16(5), 1550009.   DOI
16 Scherer, P.O. (2017), Numerical Integration, in Computational Physics, Springer.
17 Soares, D. (2016), "An implicit family of time marching procedures with adaptive dissipation control", Appl. Math. Model., 40(4), 3325-3341.   DOI
18 Tornabene, F., Dimitri, R. and Viola E. (2016), "Transient dynamic response of generally-shaped arches based on a GDQ-timestepping method", Int. J. Mech. Sci., 114, 277-314   DOI
19 Verma, M., Rajasankar, J. and Iyer N.R. (2015), "Numerical assessment of step-by-step integration methods in the paradigm of real-time hybrid testing", Earthq. Struct., 8(6), 1325-1348.   DOI
20 Wen, W., Wei, K., Lei, H., Duan, S. and Fang, D. (2017), "A novel sub-step composite implicit time integration scheme for structural dynamics", Comput. Struct., 182, 176-186.   DOI
21 Gholampour, A.A. and Ghassemieh, M. (2013), "Nonlinear structural dynamics analysis using weighted residual integration", Mech. Adv. Mater. Struct., 20(3), 199-216.   DOI
22 Gholampour, A.A., Ghassemieh, M. and Razavi, H. (2011), "A time stepping method in analysis of nonlinear structural dynamics", Appl. Comput. Mech., 5(2), 143-150.
23 Goudreau, G.L. and Taylor, R.L. (1972), "Evaluation of numerical integration methods in elastodynamics", Comput. Meth. Appl. Mech. Eng., 2(1), 69-97.   DOI
24 Howe, R. (1991), "A new family of real-time redictor-corrector integration algorithms", Simulation, 57(3), 177-186.   DOI
25 Kadapa, C., Dettmer, W. and Peric D. (2017), "On the advantages of using the first-order generalised-alpha scheme for structural dynamic problems", Comput. Struct.,193, 226-238.   DOI
26 Leontyev, V. (2010), "Direct time integration algorithm with controllable numerical dissipation for structural dynamics: two-step Lambda method", Appl. Numer. Math., 60(3), 277-292.   DOI
27 Lourderaj, U., Song, K., Windus, T.L., Zhuang, Y. and Hase, W.L. (2007), "Direct dynamics simulations using Hessian-based predictor-corrector integration algorithms", J. Chem. Phy., 126(4), 044105.   DOI
28 Matias Silva, W.T. and Mendes Bezerra, L. (2008), "Performance of composite implicit time integration scheme for nonlinear dynamic analysis", Math. Probl. Eng., 2008, 815029.
29 Mohammadzadeh, S., Ghassemieh, M. and Park Y. (2017), "Structure-dependent improved Wilson-${\theta}$ method with higher order of accuracy and controllable amplitude decay", Appl. Math. Model., 52, 417-436.   DOI
30 Newmark, N. M. (1959), "A method of computation for structural dynamics", J. Eng. Mech. Div., 85(3), 67-94.
31 Wen, W., Tao, Y., Duan, S., Yan, J., Wei, K. and Fang, D. (2017), "A comparative study of three composite implicit schemes on structural dynamic and wave propagation analysis", Comput. Struct., 190, 126-149.   DOI
32 Wilson, E.L. (1962), Dynamic Response by Step-by-Step Matrix Analysis, Labortorio Nacional de Engenharia Civil, Lisbon, Portugal, Lisbon, Portugal.
33 ZHAI, W. M. (1996), "Two simple fast integration methods for large-scale dynamic problems in engineering", Int. J. Numer. Meth. Eng., 39(24), 4199-4214.   DOI
34 Zhang, J., Liu, Y. and Liu D. (2017), "Accuracy of a composite implicit time integration scheme for structural dynamics", Int. J. Numer. Meth. Eng., 109(3), 368-406.   DOI
35 Zhang, L., Liu, T. and Li, Q. (2015), "A robust and efficient composite time integration algorithm for nonlinear structural dynamic analysis", Math. Probl. Eng., 2015, 907023.
36 Shrikhande, M. (2014), Finite Element Method and Computational Structural Dynamics, PHI Learning Pvt. Ltd.
37 Bathe, K.J. (2007), "Conserving energy and momentum in nonlinear dynamics: a simple implicit time integration scheme", Comput. Struct., 85(7), 437-445.   DOI
38 Bathe, K.J. and E.L. Wilson (1976), Numerical Methods in Finite Element Analysis, Prentice-Hall Englewood Cliffs, NJ.
39 Bathe, K.J. and Baig, M.M.I. (2005), "On a composite implicit time integration procedure for nonlinear dynamics", Comput. Struct., 83(31), 2513-2524.   DOI
40 Bathe, K.J. and Noh, G. (2012), "Insight into an implicit time integration scheme for structural dynamics", Comput. Struct., 98-99, 1-6.   DOI
41 Bathe, K.J. (2014), Finite Element Procedures, Prentice-Hall, NJ.
42 Belytschko, T. and Lu, Y. (1993), "Explicit multi-time step integration for first and second order finite element semidiscretizations", Comput. Meth. Appl. Mech. Eng., 3-4(108), 353-383.
43 Chandra, Y., Zhou, Y., Stanciulescu, I., Eason, T. and Spottswood S. (2015), "A robust composite time integration scheme for snap-through problems", Comput. Mech., 55(5), 1041-1056.   DOI
44 Chang, S.Y. (2002), "Explicit pseudodynamic algorithm with unconditional stability", J. Eng. Mech., 128(9), 935-947.   DOI
45 Chang, S.Y. (2007), "Improved explicit method for structural dynamics", J. Eng. Mech., 133(7), 748-760.   DOI
46 Chang, S.Y. (2014), "Numerical dissipation for explicit, unconditionally stable time integration methods", Earthq. Struct., 7(2), 159-178.   DOI