Skip to main content
SpringerLink
Log in

A novel class of highly efficient and accurate time-integrators in nonlinear computational mechanics

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

A new class of time-integrators is presented for strongly nonlinear dynamical systems. These algorithms are far superior to the currently common time integrators in computational efficiency and accuracy. These three algorithms are based on a local variational iteration method applied over a finite interval of time. By using Chebyshev polynomials as trial functions and Dirac–Delta functions as the test functions over the finite time interval, the three algorithms are developed into three different discrete time-integrators through the collocation method. These time integrators are labeled as Chebyshev local iterative collocation methods. Through examples of the forced Duffing oscillator, the Lorenz system, and the multiple coupled Duffing equations (which arise as semi-discrete equations for beams, plates and shells undergoing large deformations), it is shown that the new algorithms are far superior to the 4th order Runge–Kutta and ODE45 of MATLAB, in predicting the chaotic responses of strongly nonlinear dynamical systems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Adomian G (1988) A review of the decomposition method in applied mathematics. J Math Anal Appl 135:501–544

    Article  MathSciNet  MATH  Google Scholar 

  2. Atluri SN (2005) Methods of computer modeling in engineering & the sciences, vol I. Tech Science Press, Forsyth

    Google Scholar 

  3. Bai X, Junkins JL (2011) Modified Chebyshev–Picard iteration methods for solution of boundary value problems. Journal of Astronaut Sci 58(4):615–642

    Article  Google Scholar 

  4. Cheung YK, Chen SH, Lau SL (1991) A modified Lindstedt–Poincare method for certain strongly nonlinear oscillators. Int J Nonlinear Mech 26(3/4):367–378

    Article  MathSciNet  MATH  Google Scholar 

  5. Dai HH, Schnoor M, Alturi SN (2012) S simple collocation scheme for obtaining the periodic solutions of the Duffing equation, and its equivalence to the high dimensional harmonic balance method: subharmonic oscillations. Comput Model Eng Sci 84(5):459–497

    MathSciNet  MATH  Google Scholar 

  6. Dong L, Alotaibi A, Mohiuddine SA, Atluri SN (2014) Computational methods in engineering: a variety of primal & mixed methods, with global & local interpolations, for well-posed or ill-posed BCs. Comput Model Eng Sci 99(1):1–85

    MathSciNet  MATH  Google Scholar 

  7. Elgohary TA, Dong L, Junkins JL, Atluri SN (2014) A simple, fast, and accurate time-integrator for strongly nonlinear dynamical systems. Comput Model Eng Sci 100(3):249–275

    MathSciNet  MATH  Google Scholar 

  8. Fehlberg E (1969) Low-order classical Runge-Kutta fomulae with stepsize control and their application to some heat transfer problems. Technical report, NASA

  9. Fukushima T (1997) Picard iteration method, Chebyshev polynomial approximation, and global numerical integration of dynamical motions. Astron J 113(5):1909–1914

    Article  Google Scholar 

  10. Har J, Tamma K (2012) Advances in computational dynamics of particles, materials and structures. Wiley, West Sussex

    Book  MATH  Google Scholar 

  11. He J (1999) Variational iteration method: a kind of non-linear analytical technique: some examples. Int J Nonlinear Mech 34:699–708

    Article  MATH  Google Scholar 

  12. He J (2006) Homotopy perturbation method for solving boundary value problems. Phys Lett A 350(1–2):87–88

    Article  MathSciNet  MATH  Google Scholar 

  13. Hilber HM, Hughes TJ, Taylor RL (1977) Improved numerical dissipation for time integration algorithms in structural dynamics. Earthq Eng Struct Dyn 5(3):283–292

    Article  Google Scholar 

  14. Inokuti M, Sekine H, Mura T (1978) General use of the Lagrange multiplier in nonlinear mathematical physics. In: Nemat-Nasser S (ed) Variational method in the mechanics of solids. Pergamon Press, Oxford, pp 156–162

    Google Scholar 

  15. Jang M, Chen C, Liy Y (2000) On solving the initial-value problems using the differential transformation method. Appl Math Comput 115:145–160

    MathSciNet  MATH  Google Scholar 

  16. Liu CS, Yeih W, Kuo CL, Atluri SN (2009) A scalar homotopy method for solving an over/under determined system of non-linear algebraic equations. Comput Model Eng Sci 53(1):47–71

    MathSciNet  MATH  Google Scholar 

  17. Liu CS, Atluri SN (2011) An iterative algorithm for solving a system of nonlinear algebraic equations, \({ F}({ x})={ 0}\), using the system of ODEs woth an optimum \(\alpha \) in \(\dot{{ x}}=\lambda [\alpha { F}+(1-\alpha ){ B}^{T}{ F}]\); \(B_{ij} ={\partial F_i }/{\partial x_j }\). Comput Model Eng Sci 73(4):395–431

    MathSciNet  Google Scholar 

  18. Liu L, Dowell EH, Hall K (2007) A novel harmonic balance analysis for the van der pol oscillator. Int J Nonlinear Mech 42(1):2–12

    Article  MathSciNet  MATH  Google Scholar 

  19. Newmark NM (1959) A method of computation for structural dynamics. J Eng Mech Div 85(3):67–94

    Google Scholar 

  20. Thomas JP, Hall KC, Dowell EH (2003) A harmonic balance approach for modeling nonlinear aeroelastic behavior of wings in transonic viscous flow. AIAA Paper No. 1924, pp 1–6

  21. Wang X, Atluri SN (2016) A unification of the concepts of the variational iteration, adomian decomposition and picard iteration methods; and a local variational iteration method. Comput Model Eng Sci 111(6):567–585

    Google Scholar 

Download references

Acknowledgements

The authors thank Texas Tech University for its support. The first author also gratefully acknowledges the guidance from Professor Atluri, and the support from China Scholarship Council and Northwestern Polytechnical University.

Author information

Authors and Affiliations

  1. National Key Laboratory of Aerospace Flight Dynamics, Northwestern Polytechnical University, Xi’an, 710072, Shaanxi, China

    Xuechuan Wang

  2. Center for Advanced Research in the Engineering Sciences, Texas Tech University, Lubbock, TX, 79409, USA

    Xuechuan Wang & Satya N. Atluri

Authors
  1. Xuechuan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Satya N. Atluri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xuechuan Wang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, X., Atluri, S.N. A novel class of highly efficient and accurate time-integrators in nonlinear computational mechanics. Comput Mech 59, 861–876 (2017). https://doi.org/10.1007/s00466-017-1377-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-017-1377-4

Keywords

Search

Navigation