Numerical dissipation for explicit, unconditionally stable time integration methods

  • Chang, Shuenn-Yih (Department of Civil Engineering, National Taipei University of Technology)
  • Received : 2013.12.29
  • Accepted : 2014.03.24
  • Published : 2014.08.29


Although the family methods with unconditional stability and numerical dissipation have been developed for structural dynamics they all are implicit methods and thus an iterative procedure is generally involved for each time step. In this work, a new family method is proposed. It involves no nonlinear iterations in addition to unconditional stability and favorable numerical dissipation, which can be continuously controlled. In particular, it can have a zero damping ratio. The most important improvement of this family method is that it involves no nonlinear iterations for each time step and thus it can save many computationally efforts when compared to the currently available dissipative implicit integration methods.


Supported by : National Science Council


  1. Bathe, K.J. (1986), Finite Element Procedure in Engineering Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., USA.
  2. Bathe, K.J. and Wilson, E.L. (1973), "Stability and accuracy analysis of direct integration methods", Earthq. Eng. Struct. Dyn., 1(3), 283-291.
  3. Belytschko, T. and Schoeberle, D.F. (1975), "On the unconditional stability of an implicit algorithm for nonlinear structural dynamics", J. Appl. Mech., 42(4), 865-869.
  4. Belytschko, T. and Hughes, T.J.R. (1983), Computational Methods for Transient Analysis, Elsevier Science Publishers B.V., North-Holland.
  5. Chang, S.Y. (1997), "Improved numerical dissipation for explicit methods in pseudodynamic tests", Earthq. Eng. Struct. Dyn, 26(9), 917-929.<917::AID-EQE685>3.0.CO;2-9
  6. Chang, S.Y. (2000), "The $\gamma$-function pseudodynamic algorithm", J. Earthq. Eng., 4(3), 303-320.
  7. Chang, S.Y. (2002), "Explicit pseudodynamic algorithm with unconditional stability", J. Eng. Mech., ASCE, 128(9), 935-947.
  8. Chang, S.Y. (2007), "Improved explicit method for structural dynamics", J. Eng. Mech., ASCE, 133(7), 748-760.
  9. Chang, S.Y. (2009), "An explicit method with improved stability property", Int. J. Numer. Method Eng., 77(8), 1100-1120.
  10. Chang, S.Y. (2010), "A new family of explicit method for linear structural dynamics", Comput. Struct., 88(11-12), 755-772.
  11. 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.
  12. Dobbs, M.W. (1974), "Comments on 'stability and accuracy analysis of direct integration methods by Bathe and Wilson", Earthq. Eng. Struct. Dyn., 2, 295-299.
  13. Goudreau, G.L. and Taylor, R.L. (1972), "Evaluation of numerical integration methods in elasto-dynamics", Comput. Method. Appl. Mech. Eng., 2(1), 69-97.
  14. 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(3), 283-292.
  15. 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(1), 99-118.
  16. Hughes, T.J.R. (1987), The Finite Element Method, Prentice-Hall, Inc., Englewood Cliffs, NJ, USA.
  17. Krieg, R.D. (1973), "Unconditional stability in numerical time integration methods", J. Appl. Mech., 40(2), 417-421.
  18. Lambert, J.D. (1973), Computational Methods in Ordinary Differential Equations, John Wiley, London, UK.
  19. Lax, P.D. and Richmyer, R.D. (1956), "Survey of the stability of linear difference equations", Commun. Pure Appl. Math., 9(2), 267-293.
  20. Newmark, N.M. (1959), "A method of computation for structural dynamics", J. Eng. Mech. Div., ASCE, 85(3), 67-94.
  21. Simo, J.C., Tarnow, N. and Wong, K.K. (1992), "Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics", Comput. Method. Appl. Mech. Eng., 100(1), 63-116.
  22. Gonzalez, O. and Simo, J.C. (1996), "On the stability of symplectic and energy-momentum algorithms for non-linear Hamiltonian systems with symmetry", Comput. Method. Appl. Mech. Eng., 134(3-4), 197-222.
  23. Wood, W.L., Bossak, M. and Zienkiewicz, O.C. (1981), "An alpha modification of Newmark's method", Int. J. Numer. Method. Eng., 15(10), 1562-1566.
  24. Zhou, X. and Tamma, K.K. (2004), "Design, analysis and synthesis of generalized single step single solve and optimal algorithms for structural dynamics", Int. J. Numer. Method. Eng., 59(5), 597-668.
  25. Zhou, X. and Tamma, K.K. (2006), "Algorithms by design with illustrations to solid and structural mechanics/ dynamics", Int. J. Numer. Method. Eng., 66(11), 1841-1870.
  26. Zienkiewicz, O.C. (1977), The Finite Element Method, (3rd Edition), McGraw-Hill Book Co. Ltd., UK.

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