Skip to main content
SpringerLink
Log in

Stabilised finite element methods for a bending moment formulation of the Reissner-Mindlin plate model

  • Published:
Calcolo Aims and scope Submit manuscript

Abstract

This work presents new stabilised finite element methods for a bending moments formulation of the Reissner-Mindlin plate model. The introduction of the bending moment as an extra unknown leads to a new weak formulation, where the symmetry of this variable is imposed strongly in the space. This weak problem is proved to be well-posed, and stabilised Galerkin schemes for its discretisation are presented and analysed. The finite element methods are such that the bending moment tensor is sought in a finite element space constituted of piecewise linear continuos and symmetric tensors. Optimal error estimates are proved, and these findings are illustrated by representative numerical experiments.

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

Similar content being viewed by others

References

  1. Amara, M., Capatina-Papaghiuc, D., Chatti, A.: New locking-free mixed method for the Reissner- Mindlin thin plate model. SIAM J. Numer. Anal. 40, 1561–1582 (2002)

    Article  MathSciNet  Google Scholar 

  2. Arnold, D.N., Falk, R.S.: A uniformly accurate finite element method for the Reissner- Mindlin plate. SIAM J. Numer. Anal. 26(6), 1276–1290 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  3. Behrens, E., Guzman, J.: A new family of mixed methods for the Reissner-Mindling plate model based on a system of first-order equations. J. Sci. Comput. 49, 137–166 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  4. Beirao, L., da Veiga, D., Mora Rodríguez, R.: Numerical analysis of a locking-free mixed finite element method for a bending moment formulation of Reissner-Mindlin plate model. Numer. Methods Partial Differ. Equ. 29, 40–63 (2013)

    Article  MATH  Google Scholar 

  5. Bochev, P., Dohrmann, C., Gunzburger, M.: Stabilisation of low-order mixed finite elements for the Stokes equations. SIAM J. Numer. Anal. 44(1), 82–101 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Braess, D., Blömer, C.: A multigrid method for a parameter dependent problem in solid mechanics. Numer. Math. 57, 747–761 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  7. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  8. Brezzi, F., Fortin, M., Stenberg, R.: Error analysis of mixed-interpolated elements for Reissner-Mindlin plates. Math. Models Methods Appl. Sci. 1(2), 125–151 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  9. Brezzi, F., Fortin, M.: A minimal stabilisation procedure for mixed finite element methods. Numer. Math. 89(2001), 457–492 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  10. Burman, E.: A unified analysis for conforming and nonconforming stabilised finite element methods using interior penalty. SIAM J. Numer. Anal. 43(5), 2012–2033 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. Carstensen, C., Xie, X., Yu, G., Zhou, T.: A priori and a posterior analysis for a locking-free low order quadrilateral hybrid finite element for Reissner-Mindlin plates. Comput. Methods Appl. Mech. Eng. 200, 1161–1175 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. Durán, R., Liberman, E.: On mixed finite elements methods for the Reissner-Mindlin plate model. Math. Comput. 58, 561–573 (1992)

    Article  MATH  Google Scholar 

  13. Ern, A., Guermond, J.L.: Theory and Practice of Finite Element. Springer, Berlin (2004)

    Book  Google Scholar 

  14. Falk, R.: Finite Elements for the Reissner-Mindlin problem. In: Mixed Finite Elements, Compatibility Conditions, and Applications. Lecture Notes in Mathematics, vol. 1939, pp. 195–232, Springer, Berlin (2008)

  15. Girault, V., Raviart, P.A.: Finite element methods for Navier-Stokes equations: theory and algorithms. Springer series in computational mathematics. Springer, Berlin, Heidelberg, New York, Tokyo (1986)

  16. Hansbo, P., Larson, M.: Locking free quadrilateral continuous/discontinuous finite element methods for the Reissner-Mindlin plate. Comput. Methods Appl. Mech. Eng. 269, 381–393 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  17. Hughes, T.J.R., Franca, L.P.: A mixed finite element formulation for Reissner-Mindlin plate theory: uniform convergence of all higher-order spaces. Comput. Methods Appl. Mech. Eng. 67, 223–240 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kouhia, R.: On stabilised finite element methods for the Reissner-Mindlin plate model. Int. J. Numer. Methods Eng. 74(8), 1314–1328 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  19. Lovadina, C.: Analysis of a mixed finite element method for the Reissner-Mindlin plate problems. Comput. Methods Appl. Mech. Eng. 163, 71–85 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  20. Oswald, P.: On a BPX-preconditioner for P1 elements. Tech. Report, FSU Jena (1991)

    Google Scholar 

  21. Schöberl, J., Stenberg, R.: Multigrid methods for a stabilised Reissner-Mindlin plate formulation. SIAM J. Numer. Anal. 47(4), 2735–2751 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  22. Stenberg, R.: A new finite element formulation for the plate bending problem. In: Asymptotic methods for elastic structures (Lisbon, 1993), pp. 209–221, de Gruyter, Berlin (1995)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow,  G1 1XH, Scotland

    Gabriel R. Barrenechea & Andreas Wachtel

  2. Departamento de Matemática y Física Aplicadas, Universidad Católica de la Santísima Concepción, Casilla 297, Concepción, Chile

    Tomás P. Barrios

Authors
  1. Gabriel R. Barrenechea
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tomás P. Barrios
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Andreas Wachtel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gabriel R. Barrenechea.

Additional information

This research was partially supported by the Leverhulme Trust through the Research Grant Project No. RPG-2012-483. The work of TB was also partially funded by Dirección de Investigación de Universidad Católica de la Santísima Concepción, Chile.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Barrenechea, G.R., Barrios, T.P. & Wachtel, A. Stabilised finite element methods for a bending moment formulation of the Reissner-Mindlin plate model. Calcolo 52, 343–369 (2015). https://doi.org/10.1007/s10092-014-0120-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10092-014-0120-1

Keywords

Mathematics Subject Classification

Search

Navigation