Skip to main content
SpringerLink
Log in

A New Family of Mixed Methods for the Reissner-Mindlin Plate Model Based on a System of First-Order Equations

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

The mixed method for the biharmonic problem introduced in (Behrens and Guzmán, SIAM J. Numer. Anal., 2010) is extended to the Reissner-Mindlin plate model. The Reissner-Mindlin problem is written as a system of first order equations and all the resulting variables are approximated. However, the hybrid form of the method allows one to eliminate all the variables and have a final system only involving the Lagrange multipliers that approximate the transverse displacement and rotation at the edges of the triangulation. Mixed finite element spaces for elasticity with weakly imposed symmetry are used to approximate the bending moment matrix. Optimal estimates independent of the plate thickness are proved for the transverse displacement, rotations and bending moments. A post-processing technique is provided for the displacement and rotations variables and we show numerically that they converge faster than the original approximations.

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. Amara, M., Capatina-Papaghiuc, D., Chatti, A.: New locking-free mixed method for the Reissner-Mindlin thin plate model. SIAM J. Numer. Anal. 40(4), 1561–1582 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Amara, M., Thomas, J.M.: Equilibrium finite elements for the linear elastic problem. Numer. Math. 33(4), 367–383 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Modél. Math. Anal. Numér. 19(1), 7–32 (1985)

    MathSciNet  MATH  Google Scholar 

  4. Arnold, D.N., Brezzi, F.: Some new elements for the Reissner-Mindlin plate model. In: Boundary Value Problems for Partial Differential Equations and Applications. RMA Res. Notes Appl. Math., vol. 29, pp. 287–292. Masson, Paris (1993)

    Google Scholar 

  5. Arnold, D.N., Brezzi, F., Falk, R., Marini, D.L.: Locking-free Reissner-Mindlin elements without reduced integration. Comput. Methods Appl. Mech. Eng. 196(37–40), 3660–3671 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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  MathSciNet  MATH  Google Scholar 

  7. Arnold, D.N., Falk, R.S.: Edge effects in the Reissner-Mindlin plate theory. In: Noor, A.K., Belytschhko, T., Simo, J.C. (eds.) Analytic and Computational Models of Shells, pp. 71–90. ASME, New York (1989)

    Google Scholar 

  8. Arnold, D.N., Falk, R.S.: The boundary layer for the Reissner-Mindlin plate model. SIAM J. Math. Anal. 21(2), 281–312 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  9. Arnold, D.N., Falk, R.S.: Asymptotic analysis of the boundary layer for the Reissner-Mindlin plate model. SIAM J. Math. Anal. 27(2), 486–514 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Arnold, D.N., Falk, R.S., Winther, R.: Differential complexes and stability of finite element methods, II: the elasticity complex. In: Compatible Spatial Discretizations. IMA Vol. Math. Appl., vol. 142, pp. 47–67. Springer, New York (2006)

    Chapter  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Arruchio, F., Taylor, R.L.: A triangular thick plate element with an exact thin limit. Finite Elem. Anal. Des. 19, 57–68 (1995)

    Article  Google Scholar 

  13. Brezzi, F., Douglas, J., Marini, D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math.

  14. Behrens, E.M., Guzmán, J.: A mixed method for the biharmonic problem based on a system of first-order equations. SIAM J. Numer. Anal. (2010, to appear)

  15. Brezzi, F., Bathe, K.-J., Fortin, M.: Mixed-interpolated elements for Reissner-Mindlin plates. Int. J. Numer. Methods Eng. 28(8), 1787–1801 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  16. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15. Springer, New York (1991)

    Book  MATH  Google Scholar 

  17. Brezzi, F., Marini, L.D.: A nonconforming element for the Reissner-Mindlin plate. Comput. Struct. 81, 515–522 (2003)

    Article  MathSciNet  Google Scholar 

  18. Christiansen, S.H., Winther, R.: Smoothed projections in finite element exterior calculus. Math. Comput. 77(262), 813–829 (2008)

    MathSciNet  MATH  Google Scholar 

  19. Cockburn, B., Dong, B., Guzmán, J.: A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems. Math. Comput. 77(264), 1887–1916 (2008)

    Article  MATH  Google Scholar 

  20. Cockburn, B., Dong, B.: An analysis of the minimal dissipation local discontinuous Galerkin method for convection-diffusion problems. J. Sci. Comput. 32(2), 233–262 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chinosi, C., Lovadina, C., Marini, L.D.: Nonconforming locking-free finite elements for Reissner-Mindlin plates. Comput. Methods Appl. Mech. Eng. 195(25–28), 3448–3460 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Cockburn, B., Gopalakrishnan, J.: A characterization of hybridized mixed methods for second order elliptic problems. SIAM J. Numer. Anal. 42(1), 283–301 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  24. Durán, R., Ghioldi, A., Wolanski, N.: A finite element method for the Mindlin-Reissner plate model. SIAM J. Numer. Anal. 28(4), 1004–1014 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  25. Durán, R., Liberman, E.: On mixed finite element methods for the Reissner-Mindlin plate model. Math. Comput. 58(198), 561–573 (1992)

    MATH  Google Scholar 

  26. 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)

    Chapter  Google Scholar 

  27. Falk, R.S., Tu, T.: Locking-free finite elements for the Reissner-Mindlin plate. Math. Comput. 69(231), 911–928 (2000)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Franca, L.P., Stenberg, R.: A modification of a low-order Reissner-Mindlin plate bending element. In: The Mathematics of Finite Elements and Applications, VII, Uxbridge, 1990, pp. 425–436. Academic Press, London (1991)

    Google Scholar 

  30. Gopalarkrishnan, J., Guzmán, J.: A second elasticity element using the matrix bubble with tightened stress symmetry. Preprint

  31. Guzmán, J.: A unified analysis of several mixed methods for elasticity with weak stress symmetry. J. Sci. Comput. 44(2), 156–169 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numer. 11, 237–339 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lovadina, C.: A low-order nonconforming finite element for Reissner-Mindlin plates. SIAM J. Numer. Anal. 42(6), 2688–2705 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  34. Nédélec, J.-C.: Mixed finite elements in R 3. Numer. Math. 35(3), 315–341 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  35. Raviart, P.-A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. In: Mathematical Aspects of Finite Element Methods, Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975. Lecture Notes in Mathematics, vol. 606, pp. 292–315. Springer, Berlin (1977)

    Chapter  Google Scholar 

  36. Schöberl, J.: A posteriori error estimates for Maxwell equations. Math. Comput. 77(262), 633–649 (2008)

    MATH  Google Scholar 

  37. Schöberl, J.: Commuting quasi-interpolation operators for mixed finite elements. Preprint ISC-01-10-MATH, Institute for Scientific Computing, Texas A&M University (2001)

  38. Scott, R.L., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Ingeniería Civil, Universidad Católica de la Santísima Concepción, Casilla 297, Concepción, Chile

    Edwin M. Behrens

  2. Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA

    J. Guzmán

Authors
  1. Edwin M. Behrens
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Guzmán
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Guzmán.

Additional information

The first author was partially supported by CONICYT-Chile through the FONDECYT Grant 11070085, and by the Dirección de Investigación of the Universidad Católica de la Santísima Concepción. The second author was partially supported by the National Science Foundation (grant DMS-0914596).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Behrens, E.M., Guzmán, J. A New Family of Mixed Methods for the Reissner-Mindlin Plate Model Based on a System of First-Order Equations. J Sci Comput 49, 137–166 (2011). https://doi.org/10.1007/s10915-010-9451-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-010-9451-5

Keywords

Search

Navigation