[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.12989/sem.2008.28.5.549

Nonlinear dynamic analysis by Dynamic Relaxation method

Rezaiee-Pajand, M. (Department of Civil Engineering, Ferdowsi University)
Alamatian, J. (Department of Civil Engineering, Ferdowsi University)

Publication Information

Structural Engineering and Mechanics / v.28, no.5, 2008 , pp. 549-570 More about this Journal

Abstract

Numerical integration is an efficient approach for nonlinear dynamic analysis. In this paper, general category of the implicit integration errors will be discussed. In order to decrease the errors, Dynamic Relaxation method with modified time step (MFT) will be used. This procedure leads to an alternative algorithm which is very general and can be utilized with any implicit integration scheme. For numerical verification of the proposed technique, some single and multi degrees of freedom nonlinear dynamic systems will be analyzed. Moreover, results are compared with both exact and other available solutions. Suitable accuracy, high efficiency, simplicity, vector operations and automatic procedures are the main merits of the new algorithm in solving nonlinear dynamic problems.

Keywords

Modified Dynamic Relaxation; implicit time integration; nonlinear dynamic analysis;

Citations & Related Records

Times Cited By Web Of Science : 10 (Related Records In Web of Science)
Times Cited By SCOPUS : 9

Reference
Cited By SCOPUS

1	Frankel, S.P. (1950), "Convergence rates of iterative treatments of partial differential equations", Math. Tables Aids Comp., 4, 65-75 DOI
2	Hoff, C. and Taylor, R.L. (1990), "Higher derivative explicit one step methods for non-linear dynamic problems. Part I: Design and theory", Int. J. Numer. Meth. Eng., 29, 275-290 DOI ScienceOn
3	Hulbert, G.M. (1994), "A unified set of single-step asymptotic annihilation algorithms for structural dynamics", Comput. Meth. Appl. Mech. Eng., 113, 1-9 DOI ScienceOn
4	Modak, S. and Sotelino, E. (2002), "The generalized method for structural dynamic applications", Adv. Eng. Software, 33, 565-575 DOI ScienceOn
5	Papadrakakis, M. (1881), "A method for the automatic evaluation of the dynamic relaxation parameters", Comput. Meth. Appl. Mech. Eng., 25, 35-48 DOI ScienceOn
6	Paz, M. (1978), Structural Dynamics: Theory and Computation, John Willy & Sons DOI ScienceOn
7	Rao, A.R.M. (2005), "MPI-based parallel finite element approach for implicit nonlinear dynamic analysis employing sparse PCG solvers", Adv. Eng. Software, 36, 181-198 DOI ScienceOn
8	Rao, A.R.M., Rao, T.V.S.R.A. and Dattaguru, B. (2003), "A new parallel overlapped domain decomposition method for nonlinear dynamic finite element analysis", Comput. Struct., 81, 2441-2454 DOI ScienceOn
9	Tamma, K.K., Zhou, X. and Sha, D. (2001), "A theory of development and design of generalized integration operators for computational structural dynamics", Int. J. Numer. Meth. Eng., 60, 1619-1664 DOI ScienceOn
10	Zhai, W.M. (1996), "Two simple fast integration methods for large-scale dynamic problems in engineering", Int. J. Numer. Meth. Eng., 39, 4199-4214 DOI ScienceOn
11	Zienkiewicz, O.C., Wood, W.L. and Taylor, R.L. (1984), "A unified set of single step algorithms Part 1: General formulation and applications", Int. J. Numer. Meth. Eng., 20, 1529-1552 DOI ScienceOn
12	Hoff, C. and Taylor, R.L. (1990), "Higher derivative explicit one step methods for non-linear dynamic problems. Part II: Practical calculations and comparison with other higher order methods", Int. J. Numer. Meth. Eng., 29, 291-301 DOI
13	Chung, J. and Hulbert, G. (1993), "A time integration method for structural dynamics with improved numerical dissipation: The generalized-method", J. Appl. Mech., 30, 371-384
14	Felippa, C.A. (1999), Nonlinear Finite Element Methods. ASEN 5017, Course Material, www.colorado.edu/courses.d/nfemd/; Spring
15	Fung, T.C. (1998), "Complex-time step newmark methods with controllable numerical dissipation", Int. J. Numer. Meth. Eng., 41, 65-93
16	Chen, C.N. (2000), "Efficient and reliable solutions of static and dynamic nonlinear structural mechanics problems by an integration numerical approach using DQFEM and direct time integration with accelerated equilibrium iteration schemes", Appl. Math. Model., 24, 637-655 DOI ScienceOn
17	Clough, R.W. and Penzien, J. (1993) Dynamics of Structures, McGraw Hill: New York
18	Anvoner, S. (1970), Solution of Problems in Mechanics of Machines 1. Pitman Paperbacks: London
19	Bathe, K.J. and Baig, M.M.I. (2005), "On a composite implicit time integration procedure for nonlinear dynamics", Comput. Struct., 83, 2513-2524 DOI ScienceOn
20	Lei, Z. and Qui, C. (2000), "A stochastic variational formulation for nonlinear dynamic analysis of structure", Comput. Meth. Appl. Mech. Eng., 190, 597-608 DOI ScienceOn
21	Mickens, R.E. (2005), "A numerical integration technique for conservative oscillators combining non-standard finite differences methods with a Hamilton's principle", J. Sound Vib., 285, 477-482 DOI ScienceOn
22	Kim, S.J., Cho, J.Y. and Kim, W.D. (1997), "From the trapezoidal rule to higher order accurate and unconditionally stable time-integration method for structural dynamics", Comput. Meth. Appl. Mech. Eng., 149, 73-88 DOI ScienceOn
23	Lee, I.J., Oh, I.K., Lee, I. and Rhiu, J.J. (2003), "Nonlinear static and dynamic instability of complete spherical shells using mixed finite element method", Int. J. Nonlinear Mech., 38, 923-934 DOI ScienceOn
24	Kadkhodayan, M., Alamatian, J. and Turvey, G.J. (2007), "A new fictitious time for the Dynamic Relaxation (DXDR) method", Int. J. Numer. Meth. Eng. DOI ScienceOn
25	Katona, M. and Zienkiewicz, O.C. (1985), "A unified set of single step algorithms Part 3: The beta-m method, A generalization of the Newmark scheme", Int. J. Numer. Meth. Eng., 21, 1345-1359 DOI ScienceOn
26	Kim, S.E., Huu, N.C. and Lee, D.H. (2005), "Second order inelastic dynamic analysis of 3-D steel frames", Int. J. Solids Struct. DOI ScienceOn
27	Hulbert, G. and Chung, J. (1996), "Explicit time integration algorithm for structural dynamics with optimal numerical dissipation", Comput. Meth. Appl. Mech. Eng., 137, 175-188 DOI
28	Zhang, L.C., Kadkhodayan, M. and Mai, Y.W. (1994), "Development of the maDR method", Comput. Struct., 52, 1-8 DOI ScienceOn
29	Undewood, P. (1983), Computational Method for Transient Analysis Chapter 5: Dynamic Relaxation, McGraw Hill, Amsterdam
30	Zhang, L.C. and Yu, T.X. (1989), "Modified adaptive dynamic relaxation method and its application to elasticplastic bending and wrinkling of circular plates", Comput. Struct., 33, 609-614 DOI ScienceOn
31	Wood, W.L. (1984), "A unified set of single step algorithms Part 2: Theory", Int. J. Numer. Meth. Eng., 20, 2303-2309 DOI ScienceOn
32	Soares, D. and Mansur, W.J. (2005), "A frequency-domain FEM approach based on implicit Green's functions for non-linear dynamic analysis", Int. J. Solids Struct., 42(23), 6003-6014 DOI ScienceOn
33	Soroushian, A., Wriggeres, P. and Farjoodi, J. (2005), "On practical integration of semi-discritized nonlinear equation of motions, Part I: Reasons for probable instability and improper convergence", J. Sound Vib., 284, 705-731 DOI ScienceOn
34	Qu, W.L., Chen, Z.H. and Xu, Y.L. (2001), "Dynamic analysis of wind-excited truss tower with friction dampers", Comput. Struct., 79, 2817-2831 DOI ScienceOn
35	Qu, W.L., Chen, Z.H. and Xu, Y.L. (2001), "Dynamic analysis of wind-excited truss tower with friction dampers", Comput. Struct., 79, 2817-2831 DOI ScienceOn
36	Rao, A.R.M. (2002), "A parallel mixed time integration for nonlinear dynamic analysis", Adv. Eng. Software, 33, 261-271
37	Ozkul, T.A. (2004), "A finite element formulation for dynamic analysis of shells of general shape by using the Wilson- ${\theta}$ method", Thin Wall. Struct., 42, 497-503 DOI ScienceOn
38	Penry, S.N. and Wood, W.L. (1985), "Comparison of some single-step methods for the numerical solution of the structural dynamic equation", Int. J. Numer. Meth. Eng., 21, 1941-1955 DOI ScienceOn

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1	/ Structural Engineering and Mechanics
2	/ Journal of Structural Engineering
3	/ Computers and Structures
4	/ Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science
5	/ International Journal of Computational Methods
6	/ Composite Structures
7	/ International Journal of Engineering, Transactions A: Basics
8	/ Computers and Structures
9	/ Computers and Structures

1	The dynamic relaxation method using new formulation for fictitious mass and damping / [Rezaiee-Pajand, M.;Alamatian, J.;] / Structural Engineering and Mechanics
2	Estimating the Region of Attraction via collocation for autonomous nonlinear systems / [Rezaiee-Pajand, M.;Moghaddasie, B.;] / Structural Engineering and Mechanics
3	New implicit higher order time integration for dynamic analysis / [Alamatian, Javad;] / Structural Engineering and Mechanics