Skip to main content
Log in

Mixed finite elements for elasticity on quadrilateral meshes

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

Abstract

We present stable mixed finite elements for planar linear elasticity on general quadrilateral meshes. The symmetry of the stress tensor is imposed weakly and so there are three primary variables, the stress tensor, the displacement vector field, and the scalar rotation. We develop and analyze a stable family of methods, indexed by an integer r ≥ 2 and with rate of convergence in the L 2 norm of order r for all the variables. The methods use Raviart–Thomas elements for the stress, piecewise tensor product polynomials for the displacement, and piecewise polynomials for the rotation. We also present a simple first order element, not belonging to this family. It uses the lowest order BDM elements for the stress, and piecewise constants for the displacement and rotation, and achieves first order convergence for all three variables.

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

Access this article

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. Amara, M., Thomas, J.M.: Equilibrium finite elements for the linear elastic problem. Numer. Math. 33, 367–383 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  2. Arnold, D.N., Awanou, G.: Rectangular mixed finite elements for elasticity. Math. Models Methods Appl. Sci. 15(9), 1417–1429 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. Arnold, D.N., Boffi, D., Falk, R.S.: Approximation by quadrilateral finite elements. Math. Comp. 71(239), 909–922 (2002). electronic

    Article  MATH  MathSciNet  Google Scholar 

  4. Arnold, D.N., Boffi, D., Falk, R.S.: Quadrilateral H(div) finite elements. SIAM J. Numer. Anal. 42(6), 2429–2451 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Arnold, D.N., Brezzi, F., Douglas, J. Jr.: PEERS: a new mixed finite element for plane elasticity. Jpn. J. Appl. Math. 1(2), 347–367 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  6. Arnold, D.N., Falk, R.S., Winther, R.: Differential complexes and stability of finite element methods II: The elasticity complex. In: Arnold, D., Bochev, P., Lehoucq, R., Nicolaides, R., Shaskov, M. (eds.) Compatible Spatial Discretizations, IMA Vol. Math. Appl., vol. 142, pp 47–68. Springer, Berlin (2006)

    Chapter  Google Scholar 

  7. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer 15, 1–155 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Arnold, D.N., Falk, R.S., Winther, R.: Mixed finite element methods for linear elasticity with weakly imposed symmetry. Math. Comput. 76, 1699–1723 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Arnold, D.N., Winther, R.: Mixed finite elements for elasticity. Numer. Math. 92, 401–419 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. Awanou, G.: Rectangular mixed elements for elasticity with weakly imposed symmetry condition. Adv. Comput. Math. 38(2), 351–367 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  11. Boffi, D., Brezzi, F., Fortin, M.: Reduced symmetry elements in linear elasticity. Commun. Pure Appl. Anal. 8(1), 95–121 (2009)

    MATH  MathSciNet  Google Scholar 

  12. Boffi, D., Brezzi, F., Fortin, M.: Mixed finite element methods and applications, Springer Series in Computational Mathematics, vol. 44. Springer, Heidelberg (2013). doi:10.1007/978-3-642-36519-5

    Book  Google Scholar 

  13. Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8, 129–151 (1974)

    MATH  MathSciNet  Google Scholar 

  14. Cockburn, B., Gopalakrishnan, J., Guzmán, J.: A new elasticity element made for enforcing weak stress symmetry. Math. Comp. 79(271), 1331–1349 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Falk, R.S.: Finite element methods for linear elasticity. In: Boffi, D., Gastaldi, L. (eds.) Mixed finite elements, compatibility conditions, and applications, Lecture Notes in Mathematics, vol. 1939. Springer-Verlag, Berlin (2008). Lectures given at the C.I.M.E. Summer School held in Cetraro, June 26–July 1, 2006

    Google Scholar 

  16. Farhloul, M., Fortin, M.: Dual hybrid methods for the elasticity and the Stokes problems: a unified approach. Numer. Math. 76, 419–440 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  17. Fortin, M.: Old and new finite elements for incompressible flows. Internat. J. Numer. Methods Fluids 1(4), 347–364 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  18. Girault, V., Raviart, P.A.: Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5. Springer, Berlin (1986)

    Book  Google Scholar 

  19. Gopalakrishnan, J., Guzmán, J.: A second elasticity element using the matrix bubble. IMA J. Numer. Anal. 32(1), 352–372 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  20. Johnson, C., Mercier, B.: Some equilibrium finite element methods for two-dimensional elasticity problems. Numer. Math. 30(1), 103–116 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  21. Morley, M.E.: A family of mixed finite elements for linear elasticity. Numer. Math. 55(6), 633–666 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  22. Stenberg, R.: On the construction of optimal mixed finite element methods for the linear elasticity problem. Numer. Math. 48(4), 447–462 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  23. Stenberg, R.: A family of mixed finite elements for the elasticity problem. Numer. Math. 53(5), 513–538 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  24. Stenberg, R.: Two low-order mixed methods for the elasticity problem. In: The mathematics of finite elements and applications, VI (Uxbridge, 1987), pp. 271–280. Academic Press, London (1988)

    Google Scholar 

  25. Fraeijs de Veubeke, B.M.: Displacement and equilibrium models in the finite element method. In: Zienkiewicz, O.C., Holister, G.S. (eds.) Stress Analysis, pp. 145–197. Wiley, New York (1965)

    Google Scholar 

  26. Watwood, V.B. Jr., Hartz, B.J.: An equilibrium stress field model for finite element solution of two-dimensional elastostatic problems. Internat. J. Solids Structures 4, 857–873 (1968)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Gerard Awanou.

Additional information

Communicated by: Jinchao Xu

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Arnold, D.N., Awanou, G. & Qiu, W. Mixed finite elements for elasticity on quadrilateral meshes. Adv Comput Math 41, 553–572 (2015). https://doi.org/10.1007/s10444-014-9376-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-014-9376-x

Keywords

Mathematics Subject classification

Navigation