Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

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)
  • Received : 2002.12.11
  • Accepted : 2003.06.16
  • Published : 2003.08.25
⟨ Previous Next ⟩

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

References

  1. Bathe, K.J. (1996), Finite Element Procedures; 2nd edn, Prentice-Hall, USA.
  2. Belytschko, T. and Hughes, T.J.R. (1983), Computational Methods for Transient Analysis, Elsevier: USA.
  3. Bernal, D. (1991), "Locating events in step-by-step integration of Eqs. of motion", J. Struct. Eng., ASCE, 117(2),530-545. https://doi.org/10.1061/(ASCE)0733-9445(1991)117:2(530)
  4. 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. https://doi.org/10.1002/eqe.4290200708
  5. Cardona, A. and Geradin, M. (1989), "Time integration of the Eqs. of motion in mechanism analysis", Comput.Struct., 33(3), 801-820. https://doi.org/10.1016/0045-7949(89)90255-1
  6. Choi, C-K. and Chung, H.J. (1996), "Adaptive time stepping for various direct time integration methods",Comput. Struct., 60(6), 923-944. https://doi.org/10.1016/0045-7949(95)00452-1
  7. Chopra, A.K. (1995), Dynamics of Structures: Theory and Application to Earthquake Engineering, Prentice-Hall,USA.
  8. Clough, R.W. and Penzien, J. (1993), Dynamics of Structures, 2nd edition, McGraw-Hill, USA.
  9. 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)
  10. Farjoodi, J. and Soroushian, A. (2001), "Robust convergence for the dynamic analysis of MDOF elastoplasticsystems", Proc. of the SEMC2001 Conf., South-Africa, April.
  11. 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)
  12. Fung, T.C. (1997), "Third order time-step integration methods with controllable numerical dissipation", Commun.Numer. Methods Eng., 13(4), 307-315. https://doi.org/10.1002/(SICI)1099-0887(199704)13:4<307::AID-CNM64>3.0.CO;2-2
  13. 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. https://doi.org/10.1002/(SICI)1097-0207(19980515)42:1<93::AID-NME353>3.0.CO;2-W
  14. Gupta, A.K. (1992), Response Spectrum Method: In Seismic Analysis and Design of Structures, CRC, USA.
  15. Henrici, P. (1962), Discrete Variable Methods in Ordinary Differential Eqs., John Wiley and Sons, USA.
  16. Hughes, T.J.R. (1987), The Finite Element Method: Linear Static and Dynamic Finite Element Analysis,Prentice-Hall, USA.
  17. 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. https://doi.org/10.1002/cnm.1640101204
  18. Kardestuncer, H. (1987), Finite Element Handbook, McGraw-Hill, USA.
  19. 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. https://doi.org/10.1016/S0045-7825(97)00061-3
  20. Kuhl, D. and Crisfield, M.A. (1999), "Energy conserving and decaying algorithms in nonlinear structuraldynamics", Int. J. Numer. Methods Eng., 45(5), 569-599. https://doi.org/10.1002/(SICI)1097-0207(19990620)45:5<569::AID-NME595>3.0.CO;2-A
  21. Lambert, J.D. (1983), Computational Methods in Ordinary Differential Eqs., John Wiley and Sons, UK.
  22. Low, K.H. (1991), "Convergence of the numerical methods for problems of structural dynamics", J. Sound Vib.,150(2), 342-349. https://doi.org/10.1016/0022-460X(91)90628-W
  23. 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.
  24. Monro, D.M. (1985), Fortran 77, Edward Arnold, UK.
  25. Nau, J.M. (1993), "Computation of inelastic spectra", J. Eng. Mech., ASCE, 109(1), 279-288.
  26. Newmark, N.M. (1959), "A method for computation for structural dynamics", J. Eng. Mech., ASCE, 85(3), 67-94.
  27. 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. https://doi.org/10.1002/nme.1620211102
  28. Ralston, A. and Rabinowitz, P. (1978), A First Course in Numerical Analysis; 2nd edn, McGraw-Hill, Japan.
  29. Rashidi, S. and Saadeghvaziri, M.A. (1997), "Seismic modeling of multi-span simply supported bridges usingadina", Comput. Struct., 64(5/6), 1025-1039. https://doi.org/10.1016/S0045-7949(97)00016-3
  30. Ruge, P.A. (1999), "A priori local error estimation with adaptive time-stepping", Commun. Numer. MethodsEng., 15(7), 479-491. https://doi.org/10.1002/(SICI)1099-0887(199907)15:7<479::AID-CNM262>3.0.CO;2-7
  31. Schueller, G.I. and Pradlwarter, H.J. (1999), "On the stochastic response of nonlinear FE models", Arch. Appl.Mech., 69(9-10), 765-784. https://doi.org/10.1007/s004190050255
  32. Wood, W.L. (1990), Practical Time-Stepping Schemes, Oxford, USA.
  33. Xie, Y.M. (1996), "An assessment of time integration schemes for nonlinear dynamic Eqs.", J. Sound Vib.,192(1), 321-331. https://doi.org/10.1006/jsvi.1996.0190
  34. 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. https://doi.org/10.1002/cnm.1640100505
  35. 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. https://doi.org/10.1002/eqe.4290210701
  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. https://doi.org/10.1002/eqe.4290200907
  37. 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. https://doi.org/10.1016/S0045-7825(98)00332-6

Cited by

  1. On practical integration of semi-discretized nonlinear equations of motion. Part 1: reasons for probable instability and improper convergence vol.284, pp.3-5, 2005, https://doi.org/10.1016/j.jsv.2004.07.008
  2. A technique for time integration analysis with steps larger than the excitation steps vol.24, pp.12, 2008, https://doi.org/10.1002/cnm.1097
  3. Asymptotic upper bounds for the errors of Richardson extrapolation with practical application in approximate computations vol.80, pp.5, 2009, https://doi.org/10.1002/nme.2642
  4. Practical Integration of Semidiscretized Nonlinear Equations of Motion: Proper Convergence for Systems with Piecewise Linear Behavior vol.139, pp.2, 2013, https://doi.org/10.1061/(ASCE)EM.1943-7889.0000434
  5. A unified starting procedure for the Houbolt method vol.24, pp.1, 2008, https://doi.org/10.1002/cnm.949
  6. Selection of time step for pseudodynamic testing vol.10, pp.3, 2011, https://doi.org/10.1007/s11803-011-0079-8
  7. Efficient Static Analysis of Assemblies of Beam-Columns Subjected to Continuous Loadings Available as Digitized Records vol.4, pp.None, 2003, https://doi.org/10.3389/fbuil.2018.00083