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http://dx.doi.org/10.12989/sem.2003.16.2.219

More reliable responses for time integration analyses  

Soroushian, A. (Civil Engineering Department, Faculty of Engineering, University of Tehran)
Farjoodi, J. (Civil Engineering Department, Faculty of Engineering, University of Tehran)
Publication Information
Structural Engineering and Mechanics / v.16, no.2, 2003 , pp. 219-240 More about this Journal
Abstract
One of the most versatile approaches for analyzing the dynamic behavior of structural systems is direct time integration of semi-discrete equations of motion. However responses computed by time integration are generally inexact and hence the corresponding errors would rather be studied in advance. In spite of the various error estimation formulations that exist in the literature, it is accepted practice to repeat the analyses with smaller time steps, followed by a comparison between the results. In this paper, after a review of this simple method and disregarding the round-off errors, a more efficient, reliable and yet simple method for estimating errors and enhancing the accuracy is proposed. The main objectives of this research are more realistic error estimation based on the concept of convergence, approximately controlling the reliability by comparing the actual rate of convergence with the integration method's order of accuracy, and enhancement of reliability by applying Richardson's extrapolation. Starting from the errors at specific time instants, the study is then generalized to cases in which the errors should be estimated and decreased at specific events e.g. peak responses. Numerical study illustrates the efficacy of the proposed method.
Keywords
direct time integration; error estimation; reliable responses; rate of convergence; Richardson's extrapolation; round-off error; computational cost;
Citations & Related Records

Times Cited By Web Of Science : 5  (Related Records In Web of Science)
Times Cited By SCOPUS : 7
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1 Bernal, D. (1991), "Locating events in step-by-step integration of Eqs. of motion", J. Struct. Eng., ASCE, 117(2),530-545.   DOI
2 Cardona, A. and Geradin, M. (1989), "Time integration of the Eqs. of motion in mechanism analysis", Comput.Struct., 33(3), 801-820.   DOI   ScienceOn
3 Farjoodi, J. and Soroushian, A. (2002), "Shortcomings in numerical dynamic analysis of nonlinear systems",Report No. 614/2/696, University of Tehran, Tehran, Iran. (In Persian)
4 Henrici, P. (1962), Discrete Variable Methods in Ordinary Differential Eqs., John Wiley and Sons, USA.
5 Hughes, T.J.R. (1987), The Finite Element Method: Linear Static and Dynamic Finite Element Analysis,Prentice-Hall, USA.
6 Jacob, B.P. and Ebecken, N.F.F. (1994), "An optimized implementation of the Newmark/Newton-RaphsonAlgorithm for the time integration of nonlinear problems", Commun. Numer. Methods Eng., 10(12), 983-992.   DOI   ScienceOn
7 Kuhl, D. and Crisfield, M.A. (1999), "Energy conserving and decaying algorithms in nonlinear structuraldynamics", Int. J. Numer. Methods Eng., 45(5), 569-599.   DOI   ScienceOn
8 Low, K.H. (1991), "Convergence of the numerical methods for problems of structural dynamics", J. Sound Vib.,150(2), 342-349.   DOI   ScienceOn
9 Monro, D.M. (1985), Fortran 77, Edward Arnold, UK.
10 Rashidi, S. and Saadeghvaziri, M.A. (1997), "Seismic modeling of multi-span simply supported bridges usingadina", Comput. Struct., 64(5/6), 1025-1039.   DOI   ScienceOn
11 Schueller, G.I. and Pradlwarter, H.J. (1999), "On the stochastic response of nonlinear FE models", Arch. Appl.Mech., 69(9-10), 765-784.   DOI
12 Zeng, L.F., Wiberg, N-E., Li, X.D. and Xie, Y.M. (1992), "A posteriori local error estimation and adaptive timesteppingfor Newmark integration in dynamic analysis", Earthq. Eng. Struct. Dyn., 21(7), 555-571.   DOI
13 Chopra, A.K. (1995), Dynamics of Structures: Theory and Application to Earthquake Engineering, Prentice-Hall,USA.
14 Zienkiewicz, O.C., Borroomand, B. and Zhu, J.Z. (1999), "Recovery procedures in error estimation andadaptivity in linear problems", Comput. Methods Appl. Mech. Eng., 176(1-4), 111-125.   DOI   ScienceOn
15 Gupta, A.K. (1992), Response Spectrum Method: In Seismic Analysis and Design of Structures, CRC, USA.
16 Bathe, K.J. (1996), Finite Element Procedures; 2nd edn, Prentice-Hall, USA.
17 Ralston, A. and Rabinowitz, P. (1978), A First Course in Numerical Analysis; 2nd edn, McGraw-Hill, Japan.
18 Ruge, P.A. (1999), "A priori local error estimation with adaptive time-stepping", Commun. Numer. MethodsEng., 15(7), 479-491.   DOI
19 Wood, W.L. (1990), Practical Time-Stepping Schemes, Oxford, USA.
20 Newmark, N.M. (1959), "A method for computation for structural dynamics", J. Eng. Mech., ASCE, 85(3), 67-94.
21 Farjoodi, J. and Soroushian, A. (2000), "More accuracy in step-by-step analysis of nonlinear dynamic systems",Proc. of '5 Int. Conf. on Civil Eng., Iran, May. (In Persian)
22 Fung, T.C. (1997), "Third order time-step integration methods with controllable numerical dissipation", Commun.Numer. Methods Eng., 13(4), 307-315.   DOI   ScienceOn
23 Kardestuncer, H. (1987), Finite Element Handbook, McGraw-Hill, USA.
24 Xie, Y.M. and Steven, G.P. (1994), "Instability, chaos, and growth and decay of energy of time-stepping schemesfor nonlinear dynamic Eqs.", Commun. Numer. Methods Eng., 10(5), 393-401.   DOI   ScienceOn
25 Lambert, J.D. (1983), Computational Methods in Ordinary Differential Eqs., John Wiley and Sons, UK.
26 Choi, C-K. and Chung, H.J. (1996), "Adaptive time stepping for various direct time integration methods",Comput. Struct., 60(6), 923-944.   DOI   ScienceOn
27 Clough, R.W. and Penzien, J. (1993), Dynamics of Structures, 2nd edition, McGraw-Hill, USA.
28 Farjoodi, J. and Soroushian, A. (2001), "Robust convergence for the dynamic analysis of MDOF elastoplasticsystems", Proc. of the SEMC2001 Conf., South-Africa, April.
29 Xie, Y.M. (1996), "An assessment of time integration schemes for nonlinear dynamic Eqs.", J. Sound Vib.,192(1), 321-331.   DOI   ScienceOn
30 Kim, S.J., Cho, J.Y. and Kim, W.D. (1999), "From the trapezoidal rule to higher order accurate andunconditionally stable time-integration methods for structural dynamic", Comput. Methods Appl. Mech. Eng.,149(1), 73-88.   DOI   ScienceOn
31 Belytschko, T. and Hughes, T.J.R. (1983), Computational Methods for Transient Analysis, Elsevier: USA.
32 Bismarck-Nasr, M-N. and De Oliveira, A.M. (1991), "On enhancement of accuracy in direct integration dynamicresponse problems", Earthq. Eng. Struct. Dyn., 20(7), 699-703.   DOI
33 Mahin, S.A. and Lin, J. (1983), "Construction of inelastic response spectra for single degree-of-freedomsystems", UCB/EERC Report No. 83/17, University of California, Berkeley.
34 Penry, S.N. and Wood, W.L. (1985), "Comparison of some single-step methods for the numerical solution of thestructural dynamic Eqs.", Int. J. Numer. Methods Eng., 21(11), 1941-1955.   DOI   ScienceOn
35 Nau, J.M. (1993), "Computation of inelastic spectra", J. Eng. Mech., ASCE, 109(1), 279-288.
36 Zienkiewicz, O.C. and Xie, Y.M. (1991), "Simple error estimator and adaptive time stepping procedure fordynamic analysis", Earthq. Eng. Struct. Dyn., 20(9), 871-887.   DOI
37 Golley, B.W. (1998), "A weighted residual development of a time-stepping algorithm for structural dynamicsusing two general weight functions", Int. J. Numer. Methods Eng., 42(1), 93-103.   DOI   ScienceOn