A remedy for a family of dissipative, non-iterative structure-dependent integration methods

  • Chang, Shuenn-Yih (Department of Civil Engineering, National Taipei University of Technology) ;
  • Wu, Tsui-Huang (Department of Civil Engineering, National Taipei University of Technology)
  • Received : 2016.08.22
  • Accepted : 2018.01.23
  • Published : 2018.01.25


A family of the structure-dependent methods seems very promising for time integration since it can simultaneously have desired numerical properties, such as unconditional stability, second-order accuracy, explicit formulation and numerical dissipation. However, an unusual overshoot, which is essentially different from that found by Goudreau and Taylor in the transient response, has been experienced in the steady-state response of a high frequency mode. The root cause of this unusual overshoot is analytically explored and then a remedy is successfully developed to eliminate it. As a result, an improved formulation of this family method can be achieved.


  1. Bathe, K.J. and Wilson, E.L. (1973), "Stability and accuracy analysis of direct integration methods", Earthq. Eng. Struct. Dyn., 1, 283-291.
  2. Bayat, M., Bayat, M. and Pakar, I. (2015), "Analytical study of nonlinear vibration of oscillators with damping", Earthq. Struct., 9(1), 221-232.
  3. Belytschko, T. and Hughes, T.J.R. (1983), Computational Methods for Transient Analysis, Elsevier Science Publishers B.V., North-Holland.
  4. Chang, S.Y. (2002), "Explicit pseudodynamic algorithm with unconditional stability", J. Eng. Mech., ASCE, 128(9), 935-947.
  5. Chang, S.Y. (2006), "Accurate representation of external force in time history analysis", J. Eng. Mech., ASCE, 132(1), 34-45.
  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), "A family of non-iterative integration methods with desired numerical dissipation", Int. J. Numer. Meth. Eng., 100(1), 62-86.
  9. Chang, S.Y. (2014b), "Numerical dissipation for explicit, unconditionally stable time integration methods", Earthq. Struct., 7(2), 157-176.
  10. Chang, S.Y. (2015), "Dissipative, non-iterative integration algorithms with unconditional stability for mildly nonlinear structural dynamics", Nonlin. Dyn., 79(2), 1625-1649.
  11. Chang, S.Y. (2016), "A virtual parameter to improve stability properties for an integration method", Earthq. Struct., 11(2), 297-313.
  12. 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.
  13. 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.
  14. Fattah, M.Y., Hamoo, M.J. and Dawood, S.H. (2015), "Dynamic response of a lined tunnel with transmitting boundaries", Earthq. Struct., 8(1), 275-304.
  15. 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.
  16. Goudreau, G.L. and Taylor, R.L. (1972), "Evaluation of numerical integration methods in elasto- dynamics", Comput. Meth. Appl. Mech. Eng., 2, 69-97.
  17. 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.
  18. 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.
  19. Kaveh, A., Aghakouchak, A.A. and Zakian, P. (2015), "Reduced record method for efficient time history dynamic analysis and optimal design", Earthq. Struct., 8(3), 639-663.
  20. Rezaiee-Pajand, M. and Alamatian, J. (2008), "Implicit higherorder accuracy method for numerical integration in dynamic analysis", J. Struct. Eng., 134 (6), 973-985.
  21. Rezaiee-Pajand, M. and Karimi-Rad, M. (2017), "A family of second-order fully explicit time integration schemes", Comput. Appl. Math., 1-24.
  22. Rezaiee-Pajand, M., Sarafrazi, S.R. and Hashemian, M. (2011), "Improving stability domains of the implicit higher order accuracy method", Int. J. Numer. Meth. Eng., 88 (9), 880-896.
  23. Rezaiee-Pajanda, M. and Hashemian, M. (2016), "Time integration method based on discrete transfer function", Int. J. Struct. Stab. Dyn., 16(5), 1550009.
  24. Rezaiee-Pajanda, M., Hashemian, M. and Bohlulyb, A. (2017), "A novel time integration formulation for nonlinear dynamic analysis", Aerosp. Sci. Technol., 69, 625-635.
  25. Romero, A., Galvin, P. and Dominguez, J. (2012) "A time domain analysis of train induced vibrations", Earthq. Struct., 3(3), 297-313.
  26. Su, C., Huang, H., Ma, H. and Xu, R. (2014), "Efficient MCS for random vibration of hysteretic systems by an explicit iteration approach", Earthq. Struct., 7(2), 119-139.
  27. 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.
  28. 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.