Performances of non-dissipative structure-dependent integration methods

  • Chang, Shuenn-Yih (Department of Civil Engineering, National Taipei University of Technology)
  • Received : 2017.07.23
  • Accepted : 2017.10.07
  • Published : 2018.01.10


Three structure-dependent integration methods with no numerical dissipation have been successfully developed for time integration. Although these three integration methods generally have the same numerical properties, such as unconditional stability, second-order accuracy, explicit formulation, no overshoot and no numerical damping, there still exist some different numerical properties. It is found that TLM can only have unconditional stability for linear elastic and stiffness softening systems for zero viscous damping while for nonzero viscous damping it only has unconditional stability for linear elastic systems. Whereas, both CEM and CRM can have unconditional stability for linear elastic and stiffness softening systems for both zero and nonzero viscous damping. However, the most significantly different property among the three integration methods is a weak instability. In fact, both CRM and TLM have a weak instability, which will lead to an adverse overshoot or even a numerical instability in the high frequency responses to nonzero initial conditions. Whereas, CEM possesses no such an adverse weak instability. As a result, the performance of CEM is much better than for CRM and TLM. Notice that a weak instability property of CRM and TLM might severely limit its practical applications.


Supported by : National Science Council


  1. Alamatian, J. (2013), "New implicit higher order time integration for dynamic analysis", Struct. Eng. Mech., 48(5), 711-736.
  2. Belytschko, T. and Hughes, T.J.R. (1983), Computational Methods for Transient Analysis, Elsevier Science Publishers B.V., North-Holland, Amsterdam.
  3. Chang, S.Y. (2002), "Explicit pseudodynamic algorithm with unconditional stability", J. Eng. Mech., ASCE, 128(9), 935-947.
  4. Chang, S.Y. (2006), "Accurate representation of external force in time history analysis", J. Eng. Mech., ASCE, 132(1), 34-45.
  5. Chang, S.Y. (2007), "Improved explicit method for structural dynamics", J. Eng. Mech., ASCE, 133(7), 748-760.
  6. Chang, S.Y. (2009), "An explicit method with improved stability property", Int. J. Numer. Meth. Eng., 77(8), 1100-1120.
  7. Chang, S.Y. (2010), "A new family of explicit method for linear structural dynamics", Comput. Struct., 88(11-12), 755-772.
  8. Chang, S.Y. (2014a), "Family of structure-dependent explicit methods for structural dynamics", J. Eng. Mech., ASCE, 140(6), 06014005.
  9. Chang, S.Y. (2014b), "Numerical dissipation for explicit, unconditionally stable time integration methods", Earthq. Struct., 7(2), 157-176.
  10. Chang, S.Y. (2014c), "A family of non-iterative integration methods with desired numerical dissipation", Int. J. Numer. Meth. Eng., 100(1), 62-86.
  11. Chang, S.Y. (2015a), "Dissipative, non-iterative integration algorithms with unconditional stability for mildly nonlinear structural dynamics", Nonlin. Dyn., 79(2), 1625-1649.
  12. Chang, S.Y. (2015b), "A general technique to improve stability property for a structure-dependent integration method", Int. J. Numer. Meth. Eng., 101(9), 653-669.
  13. Chang, S.Y., Wu, T.H. and Tran, N.C. (2015), "A family of dissipative structure-dependent integration methods", Struct. Eng. Mech., 55(4), 815-837.
  14. Chang, S.Y., Wu, T.H. and Tran, N.C. (2016), "Improved formulation for a structure-dependent integration method", Struct. Eng. Mech., 60(1), 149-162.
  15. Chen, C. and Ricles, J.M. (2008), "Development of direct integration algorithms for structural dynamics using discrete control theory", J. Eng. Mech., ASCE, 134(8), 676-683.
  16. Chung, J. and Hulbert, G.M. (1993), "A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-${\alpha}$ method", J. Appl. Mech., 60(6), 371-375.
  17. Civalek, O. (2007), "Nonlinear dynamic response of MDOF systems by the method of harmonic differential quadrature (HDQ)", Struct. Eng. Mech., 25(2), 201-217.
  18. Fung, T.C. (2001), Solving initial value problems by differential quadrature method-Part 2: second-and higher-order equations. Int. J. Numer. Meth. Eng., 50, 1429-1454.<1429::AID-NME79>3.0.CO;2-A
  19. Fung, T.C. (2002), "Stability and accuracy of differential quadrature method in solving dynamic problems", Comput. Meth. Appl. Mech. Eng., 191, 1311-1331.
  20. Gao, Q., Wu, F., Zhang, H.W., Zhong, W.X., Howson, W.P. and Williams, F.W. (2012), "A fast precise integration method for structural dynamics problems", Struct. Eng. Mech., 43(1), 1-13.
  21. Goudreau, G.L. and Taylor, R.L. (1972), "Evaluation of numerical integration methods in elasto-dynamics", Comput. Meth. Appl. Mech. Eng., 2, 69-97.
  22. Gui, Y., Wang, J.T., Jin, F., Chen, C. and Zhou, M.X. (2014), "Development of a family of explicit algorithms for structural dynamics with unconditional stability", Nonlin. Dyn., 77(4), 1157-1170.
  23. Hadianfard, M.A. (2012), "Using integrated displacement method to time-history analysis of steel frames with nonlinear flexible connections", Struct. Eng. Mech., 41(5), 675-689.
  24. Hilber, H.M. and Hughes, T.J.R. (1978), "Collocation, dissipation, and 'overshoot' for time integration schemes in structural dynamics", Earthq. Eng. Struct. Dyn., 6, 99-118.
  25. Hilber, H.M., Hughes, T.J.R. and Taylor, R.L. (1977), "Improved numerical dissipation for time integration algorithms in structural dynamics", Earthq. Eng. Struct. Dyn., 5, 283-292.
  26. Kolay, C. and Ricles, J. (2016), "Assessment of explicit and semi-explicit classes of model-based algorithms for direct integration in structural dynamics", Int. J. Numer. Meth. Eng., 107, 49-73.
  27. Kolay, C. and Ricles, J.M. (2014), "Development of a family of unconditionally stable explicit direct integration algorithms with controllable numerical energy dissipation", Earthq. Eng. Struct. Dyn., 43, 1361-1380.
  28. Krenk, S. (2008), "Extended state-space time integration with high-frequency energy dissipation", Int. J. Numer. Meth. Eng., 73, 1767-1787.
  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.
  30. Newmark, N.M. (1959), "A method of computation for structural dynamics", J. Eng. Mech. Div., ASCE, 85, 67-94.
  31. Penzien, J. (2004), "Dynamic analysis of structure/foundation systems", Struct. Eng. Mech., 17(3), 281-290.
  32. Su, C. and Xu, R. (2014), "Random vibration analysis of structures by a time-domain explicit formulation method", Struct. Eng. Mech., 52(2), 239-260.
  33. Tang, Y. and Lou, M.L. (2017), "New unconditionally stable explicit integration algorithm for real-time hybrid testing", J. Eng. Mech., 143(7), 04017029-1-15
  34. Wilson, E.L. (1968), "A computer program for the dynamic stress analysis of underground structures", SESM Report No.68-1, Division of Structural Engineering Structural Mechanics, University of California, Berkeley, USA.
  35. Wood, W.L., Bossak, M. and Zienkiewicz, O.C. (1981), "An alpha modification of Newmark's method", Int. J. Numer. Meth. Eng., 15, 1562-1566.