Skip to main content
SpringerLink
Log in

A symplectic finite element method based on Galerkin discretization for solving linear systems

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

We propose a novel symplectic finite element method to solve the structural dynamic responses of linear elastic systems. For the dynamic responses of continuous medium structures, the traditional numerical algorithm is the dissipative algorithm and cannot maintain long-term energy conservation. Thus, a symplectic finite element method with energy conservation is constructed in this paper. A linear elastic system can be discretized into multiple elements, and a Hamiltonian system of each element can be constructed. The single element is discretized by the Galerkin method, and then the Hamiltonian system is constructed into the Birkhoffian system. Finally, all the elements are combined to obtain the vibration equation of the continuous system and solved by the symplectic difference scheme. Through the numerical experiments of the vibration response of the Bernoulli-Euler beam and composite plate, it is found that the vibration response solution and energy obtained with the algorithm are superior to those of the Runge-Kutta algorithm. The results show that the symplectic finite element method can keep energy conservation for a long time and has higher stability in solving the dynamic responses of linear elastic 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

Similar content being viewed by others

References

  1. GONZALEZ, O. Exact energy and momentum conserving algorithms for general models in nonlinear elasticity. Computer Methods in Applied Mechanics and Engineering, 190(13–14), 1763–1783 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. GONZALEZ, O. Time integration and discrete Hamiltonian systems. Journal of Nonlinear Science, 6(5), 449–467 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. GONZALEZ, O. and SIMO, J. C. On the stability of symplectic and energy-momentum algorithms for non-linear Hamiltonian systems with symmetry. Computer Methods in Applied Mechanics and Engineering, 134(3–4), 197–222 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. KRENK, S. Energy conservation in Newmark based time integration algorithms. Computer Methods in Applied Mechanics and Engineering, 195(44–47), 6110–6124 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. SIMO, J. C. and TARNOW, N. The discrete energy-momentum method-conserving algorithms for nonlinear elastodynamics. Zeitschrift für Angewandte Mathematik und Physik, 43(5), 757–792 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  6. SIMO, J. C., TARNOW, N., and DOBLARE, M. Non-linear dynamics of three-dimensional rods: exact energy and momentum conserving algorithms. International Journal for Numerical Methods in Engineering, 38(9), 1431–1473 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. SIMO, J. C., TARNOW, N., and WONG, K. K. Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics. Computer Methods in Applied Mechanics and Engineering, 100(1), 63–116 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  8. FENG, K. On difference schemes and symplectic geometry. Proceedings of the 1984 Beijing Symposium on Differential Geometry and Differential Equations, Science Press, Beijing, 42–58 (1985)

    Google Scholar 

  9. KANG, F., WU, H., and QIN, M. Symplectic difference schemes for linear Hamiltonian canonical systems. Journal of Computational Mathematics, 8(4), 371–380 (1990)

    MathSciNet  MATH  Google Scholar 

  10. KHANG, A., HOWSMON, D. P., LEJEUNE, E., and SACKS, M. S. Multi-scale Modeling of the Heart Valve Interstitial Cell, Multi-Scale Extracellular Matrix Mechanics and Mechanobiology, Springer International Publishing AG, Cham, 21–53 (2020)

    Google Scholar 

  11. SANZ-SERNA, J. M. Runge-Kutta schemes for Hamiltonian systems. BIT Numerical Mathematics, 28(4), 877–883 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  12. LASAGNI, F. M. Canonical Runge-Kutta methods. Zeitschrift für Angewandte Mathematik und Physik, 39(6), 952–953 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  13. SURIS, Y. B. On the conservation of the symplectic structure in the numerical solution of Hamiltonian systems. Numerical Solution of Ordinary Differential Equations, 232, 148–160 (1988)

    MATH  Google Scholar 

  14. BRIDGES, and THOMAS, J. Multi-symplectic structures and wave propagation. Mathematical Proceedings of the Cambridge Philosophical Society, 121(1), 147–190 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  15. REICH, S. Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations. Journal of Computational Physics, 157(2), 473–499 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. YAN, X., VEGT, J., and BOKHOVE, O. Discontinuous Hamiltonian finite element method for linear hyperbolic systems. Journal of Scientific Computing, 35, 241–265 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. XU, Y., VAN DER VEGT, J. J. W., and BOKHOVE, O. Discontinuous Hamiltonian finite element method for linear hyperbolic systems. Journal of Scientific Computing, 35(2–3), 241–265 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. KIRBY, R. C. and KIEU, T. T. Symplectic-mixed finite element approximation of linear acoustic wave equations. Numerische Mathematik, 130(2), 257–291 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. SÁNCHEZ, M., CIUCA, C., NGUYEN, N. C., PERAIRE, J., and COCKBURN, B. Symplectic Hamiltonian HDG methods for wave propagation phenomena. Journal of Computational Physics, 350, 951–973 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. APPELO, D. and HAGSTROM, T. An energy-based discontinuous Galerkin discretization of the elastic wave equation in second order form. Computer Methods in Applied Mechanics & Engineering, 338, 362–391 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  21. QIU, Z. P. and JIANG, N. A symplectic conservative perturbation series expansion method for linear Hamiltonian systems with perturbations and its applications. Advances in Applied Mathematics and Mechanics, 13(6), 1535–1557 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  22. QIU, Z. and XIA, H. Symplectic perturbation series methodology for non-conservative linear Hamiltonian system with damping. Acta Mechanica Sinica, 37, 983–996 (2021)

    Article  MathSciNet  Google Scholar 

  23. QIU, Z. P. and JIANG, N. Comparative study and application of random and interval inhomogeneous linear Hamiltonian systems (in Chinese). Chinese Journal of Theoretical and Applied Mechanics, 52, 60–72 (2020)

    Google Scholar 

  24. QIU, Z. P. and JIANG, N. Random and interval uncertainty symplectic preserving algorithms for linear Birkhoff equations and their comparative studies (in Chinese). Chinese Science: Physics, Mechanics, Astronomy, 50(8), 084611 (2020)

    Google Scholar 

  25. ZHOU, S., MA, Y. C., SUN, Z., and HU, X. F. A novel super symplectic analytical singular element for crack propagation along a bimaterial interface. Theoretical and Applied Fracture Mechanics, 122, 103565 (2022)

    Article  Google Scholar 

  26. SU, B., SHEN, W., LANG, C., and LI, H. Order-corrected symplectic finite element method for elastic wave modelling. Exploration Geophysics, 52, 321–334 (2020)

    Article  Google Scholar 

  27. SUN, Z., HU, G., NIE, X., and SUN, J. An analytical symplectic method for buckling of ring-stiffened graphene platelet-reinforced composite cylindrical shells subjected to hydrostatic pressure. Journal of Marine Science & Engineering, 10(12), 1834 (2022)

    Article  Google Scholar 

  28. LAI, A., FU, G., and LIU, P. A symplectic analytical approach for torsional buckling of cylindrical shells with asymmetric local defects. International Journal of Structural Stability and Dynamics, 23, 2350093 (2022)

    Article  MathSciNet  Google Scholar 

  29. CHEN, G. and ZHU, G. Symplectic algorithms for stable manifolds in control theory. IEEE Transactions on Automatic Control, 67(6), 3105–3111 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  30. ZHOU, Z., FAN, J., LIM, C. W., RONG, D., and XU, X. A size-dependent coupled symplectic and finite element method for steady-state forced vibration of built-up nanobeam systems. International Journal of Structural Stability & Dynamics, 19(7), 1950081 (2019)

    Article  MathSciNet  Google Scholar 

  31. SHIRAZI, H. M. S. and KOUROSH, H. A new approach to analytical solution of cantilever beam vibration with nonlinear boundary condition. Journal of Computational and Nonlinear Dynamics, 7(3), 034502 (2012)

    Article  Google Scholar 

  32. SHEN, Y. and GIBBS, B. M. An approximate solution for the bending vibrations of a combination of rectangular thin plates. Journal of Sound and Vibration, 105(1), 73–90 (1986)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors wish to express their many thanks to the reviewers for their useful and constructive comments.

Author information

Authors and Affiliations

  1. Institute of Solid Mechanics, School of Aeronautic Science and Engineering, Beihang University, Beijing, 100083, China

    Zhiping Qiu, Zhao Wang & Bo Zhu

Authors
  1. Zhiping Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhao Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bo Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhiping Qiu.

Additional information

Conflict of interest

The authors declare no conflict of interest.

Project supported by the National Natural Science Foundation of China (Nos. 12132001 and 52192632)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qiu, Z., Wang, Z. & Zhu, B. A symplectic finite element method based on Galerkin discretization for solving linear systems. Appl. Math. Mech.-Engl. Ed. 44, 1305–1316 (2023). https://doi.org/10.1007/s10483-023-3012-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-023-3012-5

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

Search

Navigation