Acta Mechanica Sinica

, Volume 35, Issue 1, pp 112–128 | Cite as

A three-parameter single-step time integration method for structural dynamic analysis

  • Huimin Zhang
  • Yufeng XingEmail author
Research Paper


The existing three-parameter single-step time integration methods, such as the Generalized-\(\alpha \) method, improve numerical dissipation by modifying equilibrium equation at time points, which cause them to lose accuracy due to the interpolation of load vectors. Moreover, these three-parameter methods do not present an available formulation applied to a general second-order nonlinear differential equation. To solve these problems, this paper proposes an innovative three-parameter single-step method by introducing an additional variable into update equations. Although the present method is spectrally identical to the Generalized-\(\alpha \) method for undamped systems, it possesses higher accuracy since it strictly satisfies the equilibrium equation at time points, and can be readily used to solve nonlinear equations. By the analysis of accuracy, stability, numerical dissipation and dispersion, the optimal second-order implicit and explicit schemes are generated, which can maximize low-frequency accuracy when high-frequency dissipation is specified. To check the performance of the proposed method, several numerical experiments are conducted and the proposed method is compared with a few up-to-date methods.


Three-parameter Single-step Optimal scheme Higher accuracy 



This work was supported by the National Natural Science Foundation of China (11672019, 11372021, and 37686003).


  1. 1.
    Newmark, N.M.: A method of computation for structural dynamics. Proc. ASCE 85, 67–94 (1959)Google Scholar
  2. 2.
    Wilson E.L.: A computer program for the dynamic stress analysis of underground structures. California Univ. Berkeley Structural Engineering Lab, No. SEL-68-1, 1968Google Scholar
  3. 3.
    Hilber, H.M., Hughes, T.J.R., Taylor, R.L.: Improved numerical dissipation for time integration algorithms in structural dynamics. Earthq. Eng. Struct. D 5, 283–292 (1977)CrossRefGoogle Scholar
  4. 4.
    Wood, W.L., Bossak, M., Zienkiewicz, O.C.: An alpha modification of Newmark’s method. Int. J. Numer. Methods Eng. 15, 1562–1566 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chung, J., Hulbert, G.: A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized \(\alpha \)-method. J. Appl. Mech. 32, 371–375 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Houbolt, J.C.: A recurrence matrix solution for the dynamic response of elastic aircraft. J. Aeronaut. Sci. 17, 540–550 (1950)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Park, K.C.: An improved stiffly stable method for direct integration of nonlinear structural dynamic equation. J. Appl. Mech. 42, 464–470 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Zhai, W.M.: Two simple fast integration methods for large-scale dynamic problems in engineering. Int. J. Numer. Methods Eng. 39, 4199–4214 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Rezaiee-Pajand, M., Alamatian, J.: Implicit higher order accuracy method for numerical integration in dynamic analysis. J. Struct. Eng. ASCE 134, 973–985 (2008)CrossRefzbMATHGoogle Scholar
  10. 10.
    Alamatian, J., Rezaiee-Pajand, M.: Numerical time integration for dynamic analysis using a new higher order predictor-corrector method. Eng. Comput. 25, 541–568 (2008)CrossRefzbMATHGoogle Scholar
  11. 11.
    Zienkiewicz, O.C., Wood, W.L., Hine, N.M., et al.: A unified set of single step algorithms. Part 1: general formulation and application. Int. J. Numer. Methods Eng. 20, 1529–1552 (1984)CrossRefzbMATHGoogle Scholar
  12. 12.
    Tamma, K.K., Sha, D., Zhou, X.: Time discretized operators. Part 1: towards the theoretical design of a new generation of a generalized family of unconditionally stable implicit and explicit representations of arbitrary order for computational dynamics. Comput. Methods Appl. Mech. Eng. 192, 257–290 (2003)CrossRefzbMATHGoogle Scholar
  13. 13.
    Sha, D., Zhou, X., Tamma, K.K.: Time discretized operators. Part 2: towards the theoretical design of a new generation of a generalized family of unconditionally stable implicit and explicit representations of arbitrary order for computational dynamics. Comput. Methods Appl. Mech. Eng. 192, 291–329 (2003)CrossRefzbMATHGoogle Scholar
  14. 14.
    Zhou, X., Tamma, K.K.: Design, analysis, and synthesis of generalized single step single solve and optimal algorithms for structural dynamics. Int. J. Numer. Methods Eng. 59, 597–668 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Bathe, K.J., Baig, M.M.I.: On a composite implicit time integration procedure for nonlinear dynamics. Comput. Struct. 83, 2513–2524 (2005)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Noh, G., Bathe, K.J.: An explicit time integration scheme for the analysis of wave propagations. Comput. Struct. 129, 178–193 (2013)CrossRefGoogle Scholar
  17. 17.
    Zhang, J., Liu, Y., Liu, D.: Accuracy of a composite implicit time integration scheme for structural dynamics. Int. J. Numer. Methods Eng. 109, 368–406 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Noh, G., Ham, S., Bathe, K.J.: Performance of an implicit time integration scheme in the analysis of wave propagations. Comput. Struct. 123, 93–105 (2013)CrossRefGoogle Scholar
  19. 19.
    Klarmann, S., Wagner, W.: Enhanced studies on a composite time integration scheme in linear and non-linear dynamics. Comput. Mech. 2015, 455–468 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Chandra, Y., Zhou, Y., Stanciulescu, I., et al.: A robust composite time integration scheme for snap-through problems. Comput. Mech. 55, 1041–1056 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Wen, W.B., Wei, K., Lei, H.S., et al.: A novel sub-step composite implicit time integration scheme for structural dynamics. Comput. Struct. 182, 176–186 (2017)CrossRefGoogle Scholar
  22. 22.
    Rezaiee-Pajand, M., Sarafrazi, S.R.: A mixed and multi-step higher-order implicit time integration family. J. Mech. Eng. Sci. 224, 2097–2108 (2010)CrossRefGoogle Scholar
  23. 23.
    Xing, Y., Guo, J.: Differential quadrature time element method for structural dynamics. Acta Mech. Sin. 28, 782–792 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Xing, Y., Qin, M., Guo, J.: A time finite element method based on the differential quadrature rule and Hamilton’s variational principle. Appl. Sci. 7, 138 (2017)CrossRefGoogle Scholar
  25. 25.
    Qin, M., Xing, Y., Guo, J.: An improved differential quadrature time element method. Appl. Sci. 7, 471 (2017)CrossRefGoogle Scholar
  26. 26.
    Wen, W.B., Luo, S.M., Jian, K.L.: A novel time integration method for structural dynamics utilizing uniform quintic B-spline functions. Arch. Appl. Mech. 85, 1743–1759 (2015)CrossRefGoogle Scholar
  27. 27.
    Shojaee, S., Rostami, S., Abbasi, A.: An unconditionally stable implicit time integration algorithm: modified quartic B-spline method. Comput. Struct. 153, 98–111 (2015)CrossRefGoogle Scholar
  28. 28.
    Tamma, K.K., Har, J., Zhou, X.M., et al.: An overview and recent advances in vector and scalar formalisms: space/Time discretization in computational dynamics-A unified approach. Arch. Comput. Methods E. 18, 119–283 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Shao, H.P., Cai, C.W.: The direct integration three-parameter optimal schemes for structural dynamics. In: Proceeding of the International Conference: Machine Dynamics and Engineering Applications, Xi’an, 1988, pp. 16–20Google Scholar
  30. 30.
    Lax, P.D., Richmyer, R.D.: Survey of the stability of linear limit difference equations. Commun. Pure Appl. Math. 9, 267–293 (1956)CrossRefGoogle Scholar
  31. 31.
    Bathe, K.J., Wilson, E.L.: Stability and accuracy analysis of direct direction methods. Earthq. Eng. Struct. D 1, 283–291 (1973)CrossRefGoogle Scholar
  32. 32.
    Dahlquist, G.: A special stability problem for linear multistep methods. BIT 3, 27–43 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Hulbert, G.M., Chung, J.: Explicit time integration algorithms for structural dynamics with optimal numerical dissipation. Comput. Methods Appl. Mech. Eng. J. 137, 175–188 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Gobat, J.I., Grosenbaugh, M.A.: Application of the generalized-\(\alpha \) method to the time integration of the cable dynamics equations. Comput. Methods Appl. Mech. Eng. J. 190, 4817–4829 (2001)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Solid MechanicsBeihang University (BUAA)BeijingChina

Personalised recommendations