Skip to main content
SpringerLink
Log in

A priori error analysis for a finite element approximation of dynamic viscoelasticity problems involving a fractional order integro-differential constitutive law

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

We consider a fractional order viscoelasticity problem modelled by a power-law type stress relaxation function. This viscoelastic problem is a Volterra integral equation of the second kind with a weakly singular kernel where the convolution integral corresponds to fractional order differentiation/integration. We use a spatial finite element method and a finite difference scheme in time. Due to the weak singularity, fractional order integration in time is managed approximately by linear interpolation so that we can formulate a fully discrete problem. In this paper, we present a stability bound as well as a priori error estimates. Furthermore, we carry out numerical experiments with varying regularity of exact solutions at the end.

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. Hunter, S.C.: Mechanics of continuous media. Ellis Horwood series in mathematics and its applications. E. Horwood, Australia (1976). 9780853120421, 82021328

    MATH  Google Scholar 

  2. Shaw, S., Whiteman, J.R.: Some partial differential Volterra equation problems arising in viscoelasticity. In: Proceedings of Equadiff, vol. 9, pp 183–200 (1998)

  3. Rivière, B., Shaw, S., Whiteman, J.R.: Discontinuous Galerkin finite element methods for dynamic linear solid viscoelasticity problems. Numer. Meth. Partial Diff. Eq. 23(5), 1149–1166 (2007)

    Article  MathSciNet  Google Scholar 

  4. Findley, W.N., F.A. Davis: Creep and relaxation of nonlinear viscoelastic materials. Courier Corporation, Massachusetts (2013)

  5. Drozdov, A.D.: Viscoelastic structures: mechanics of growth and aging. Academic Press, Cambridge, Massachusetts (1998)

    MATH  Google Scholar 

  6. Golden, J.M., Graham, G.A.: Boundary value problems in linear viscoelasticity. Springer Science & Business Media, Berlin, Germany (2013)

  7. Riviére, B., Shaw, S., Wheeler, M.F., Whiteman, J.R.: Discontinuous Galerkin finite element methods for linear elasticity and quasistatic linear viscoelasticity. Numer Math 95(2), 347–376 (2003)

    Article  MathSciNet  Google Scholar 

  8. Shaw, S., Whiteman, J.: Numerical solution of linear quasistatic hereditary viscoelasticity problems ii: a posteriori estimates. BICOM Technical Report 98-3, see www. brunel. ac. uk/˜ icsrbicm, Tech. Rep. (1998)

  9. Shaw, S., Whiteman, J.: Numerical solution of linear quasistatic hereditary viscoelasticity problems i: a priori estimates. Recall 11(4) (1999)

  10. Jang, Y., Shaw, S.: Finite element approximation and analysis of viscoelastic wave propagation with internal variable formulations. arXiv:http://arxiv.org/abs/2001.04745 (2020)

  11. Nutting, P.: A new general law of deformation. J. Franklin Ins. 191(5), 679–685 (1921)

    Article  Google Scholar 

  12. Torvik, P.J., Bagley, R.L.: On the appearance of the fractional derivative in the behavior of real materials. J Appl Mech 51(2), 294–298 (1984)

    Article  Google Scholar 

  13. Koeller, R.: Applications of fractional calculus to the theory of viscoelasticity (1984)

  14. Li, C., Chen, A., Ye, J.: Numerical approaches to fractional calculus and fractional ordinary differential equation. J. Comput. Phys. 230(9), 3352–3368 (2011)

    Article  MathSciNet  Google Scholar 

  15. McLean, W., Thomée, V.: Numerical solution of an evolution equation with a positive-type memory term. The ANZIAM Journal 35(1), 23–70 (1993)

    MathSciNet  MATH  Google Scholar 

  16. McLean, W., Thomée, V.: Numerical solution via Laplace transforms of a fractional order evolution equation. J. Integral. Equ. Appl. 57–94 (2010)

  17. McLean, W., Thomée, V.: Maximum-norm error analysis of a numerical solution via laplace transformation and quadrature of a fractional-order evolution equation. IMA J. Numer. Anal. 30(1), 208–230 (2010)

    Article  MathSciNet  Google Scholar 

  18. Linz, P.: Theoretical numerical analysis: an introduction to advanced techniques. Courier Corporation, Massachusetts, United States (2001)

    MATH  Google Scholar 

  19. Oldham, K., Spanier, J.: The fractional calculus theory and applications of differentiation and integration to arbitrary order, vol. 111. Elsevier, Amsterdam, Netherlands (1974)

  20. Malinowska, A.B., Torres, D.F.: Introduction to the fractional calculus of variations. World Scientific Publishing Company, Singapore (2012)

    Book  Google Scholar 

  21. Miller, K.S., Ross, B.: An introduction to the fractional calculus and fractional differential equations. Wiley-Interscience, Hoboken, United States (1993)

    MATH  Google Scholar 

  22. Brenner, S., Scott, R.: The mathematical theory of finite element methods, vol. 15. Springer Science & Business Media, Berlin, Germany (2007)

  23. Wheeler, M.F.: A priori L2 error estimates for Galerkin approximations to parabolic partial differential equations. SIAM J. Numer. Anal. 10(4), 723–759 (1973)

    Article  MathSciNet  Google Scholar 

  24. Ciarlet, P.G.: On Korn’s inequality. Chinese Annals. Math., Series B 31(5), 607–618 (2010)

    Article  MathSciNet  Google Scholar 

  25. Horgan, C.O., Payne, L.E.: On inequalities of Korn, Friedrichs and Babuška-Aziz. Arch Ration Mech An 82(2), 165–179 (1983)

    Article  Google Scholar 

  26. Nitsche, J.A.: On Korn’s second inequality. RAIRO. Anal Numér. 15(3), 237–248 (1981)

    Article  MathSciNet  Google Scholar 

  27. Li, J., Huang, Y., Lin, Y.: Developing finite element methods for Maxwell’s equations in a Cole–Cole dispersive medium. SIAM J. Sci. Comput. 33 (6), 3153–3174 (2011)

    Article  MathSciNet  Google Scholar 

  28. Warburton, T., Hesthaven, J.: On the constants in hp-finite element trace inverse inequalities. Comput. Meth. Appl. Mech. Eng. 192(25), 2765 – 2773 (2003). [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0045782503002949

    Article  MathSciNet  Google Scholar 

  29. Rivière, B.: Discontinuous Galerkin methods for solving elliptic and parabolic equations: theory and implementation. SIAM (2008)

  30. Ozisik, S., Riviere, B., Warburton, T.: On the constants in inverse inequalities in L2. Rice University, Tech. Rep. (2010)

  31. Thomée, V.: Galerkin finite element methods for parabolic problems, vol. 1054. Springer, Berlin, Germany (1984)

  32. Grenander, U., Szegö, G.: Toeplitz forms and their applications. University of California Press, California, United States (1958)

    Book  Google Scholar 

  33. Lopez-Marcos, J.: A difference scheme for a nonlinear partial integrodifferential equation. SIAM J. Numer. Anal. 27(1), 20–31 (1990)

    Article  MathSciNet  Google Scholar 

  34. Dauge, M.: Elliptic boundary value problems on corner domains, volume 1341 of lecture notes in mathematics (1988)

  35. Grisvard, P.: Elliptic problems in nonsmooth domains. SIAM (2011)

Download references

Funding

Jang gratefully acknowledges the supported of a scholarship from Brunel University London.

Author information

Author notes

  1. Yongseok Jang

    Present address: CERFACS, 42 Avenue Gaspard Coriolis, 31057, Toulouse, France

Authors and Affiliations

  1. Department of Mathematics, Brunel University London, Uxbridge, UB8 3PH, UK

    Yongseok Jang & Simon Shaw

Authors
  1. Yongseok Jang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Simon Shaw
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yongseok Jang.

Additional information

Communicated by: Bangti Jin

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jang, Y., Shaw, S. A priori error analysis for a finite element approximation of dynamic viscoelasticity problems involving a fractional order integro-differential constitutive law. Adv Comput Math 47, 46 (2021). https://doi.org/10.1007/s10444-021-09857-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10444-021-09857-8

Keywords

Mathematics Subject Classification 2010

Search

Navigation