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Mixed Finite Elements, Compatibility Conditions, and Applications pp 195–232Cite as

Finite Elements for the Reissner–Mindlin Plate

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References

  1. S. M. Alessandrini, D. N. Arnold, R. S. Falk, and A. L. Madureira. Derivation and justification of plate models by variational methods. In Plates and Shells (Quebec 1996), pp. 1–20. American Mathematical Society, Providence, RI, 1999.

    Google Scholar 

  2. D. N. Arnold. Innovative finite element methods for plates. Math. Appl. Comput., 10(2):77–88, 1991.

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  4. D. N. Arnold, D. Boffi, and R. S. Falk. Remarks on quadrilateral Reissner-Mindlin plate elements. In Proceedings of the Fifth World Congress on Computational Mechanics (WCCM V). Vienna University of Technology, Austria, 2002.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. D. N. Arnold, D. Boffi, R. S. Falk, and L. Gastaldi. Finite element approximation on quadrilateral meshes. Commun. Numer. Methods Eng., 17(11):805–812, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  7. D. N. Arnold and F. Brezzi. 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 

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

    Article  MathSciNet  Google Scholar 

  9. D. N. Arnold, F. Brezzi, and L. D. Marini. A family of discontinuous Galerkin finite elements for the Reissner–Mindlin plate. J. Sci. Comput., 22/23:25–45, 2005.

    Article  MathSciNet  Google Scholar 

  10. D. N. Arnold and R. S. Falk. Edge effects in the Reissner–Mindlin plate theory. In Analytic and Computational Models of Shells, pp. 71–90. A.S.M.E., New York, 1989.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  14. D. N. Arnold and R. S. Falk. Analysis of a linear–linear finite element for the Reissner–Mindlin plate model. Math. Models Methods Appl. Sci., 7(2):217–238, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  15. F. Auricchio and C. Lovadina. Analysis of kinematic linked interpolation methods for Reissner–Mindlin plate problems. Comput. Methods Appl. Mech. Eng., 190(18–19):2465–2482, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  16. F. Auricchio and R. L. Taylor. A triangular thick plate element with an exact thin limit. Finite Elem. Anal. Des., 19:57–68, 1995.

    Article  MATH  Google Scholar 

  17. K.-J. Bathe and F. Brezzi. On the convergence of a four-node plate bending element based on Mindlin–Reissner plate theory and a mixed interpolation. In The Mathematics of Finite Elements and Applications, V (Uxbridge, 1984), pp. 491–503. Academic Press, London, 1985.

    Google Scholar 

  18. K.-J. Bathe and F. Brezzi. A simplified analysis of two plate bending elements – the MITC4 and MITC9 elements. In Numerical Techniques for Engineering Analysis and Design, vol. 1. Martinus Nijhoff, Amsterdam, 1987.

    Google Scholar 

  19. K.-J. Bathe and F. Brezzi and S. W. Cho. The MITC7 and MITC9 plate bending elements. In Comput. Struct, 32:797–841, 1989.

    Article  MATH  Google Scholar 

  20. K.-J. Bathe and E. N. Dvorkin. A four-node plate bending element based on Mindlin/Reissner plate theory and mixed interpolation. Int. J. Numer. Methods Eng., 21:367–383, 1985.

    Article  MATH  Google Scholar 

  21. K.-J. Bathe and E. N. Dvorkin. A formulation of general shell elements—the use of mixed interpolation of tensorial components. Int. J. Numer. Methods Eng., 22:697–722, 1986.

    Article  MATH  Google Scholar 

  22. J. H. Bramble and T. Sun. A negative-norm least squares method for Reissner–Mindlin plates. Math. Comp., 67(223):901–916, 1998.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  24. F. Brezzi and M. Fortin. Numerical approximation of Mindlin–Reissner plates. Math. Comput., 47(175):151–158, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  25. F. Brezzi, M. Fortin, and R. Stenberg. 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 

  26. F. Brezzi and L. D. Marini. A nonconforming element for the Reissner-Mindlin plate. Comput. & Struct., 81(8–11):515–522, 2003. In honour of Klaus-Jürgen Bathe.

    Article  MathSciNet  Google Scholar 

  27. C. Chinosi, C. Lovadina, and L. D. Marini. Nonconforming finite elements for Reissner-Mindlin plates. In Applied and Industrial Mathematics in Italy, Ser. Adv. Math. Appl. Sci., vol. 69, pp. 213–224. World Sci. Publ., Hackensack, NJ, 2005.

    Chapter  Google Scholar 

  28. M. Crouzeix and P.-A. Raviart. Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, 7(R-3):33–75, 1973.

    MathSciNet  Google Scholar 

  29. R. Durán. The inf–sup condition and error estimates for the Arnold–Falk plate bending element. Numer. Math., 59(8):769–778, 1991.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  31. R. Durán, E. Hernández, L. Hervella-Nieto, E. Liberman, and R. Rodríguez. Error estimates for low-order isoparametric quadrilateral finite elements for plates. SIAM J. Numer. Anal., 41(5):1751–1772 (electronic), 2003.

    Article  MATH  MathSciNet  Google Scholar 

  32. R. Durán, L. Hervella-Nieto, E. Liberman, R. Rodríguez, and J. Solomin. Approximation of the vibration modes of a plate by Reissner–Mindlin equations. Math. Comp., 68(228):1447–1463, 1999.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  34. R. Durán and E. Liberman. On the convergence of a triangular mixed finite element method for Reissner–Mindlin plates. Math. Models Methods Appl. Sci., 6(3):339–352, 1996.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  36. L. P. Franca and R. Stenberg. 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 

  37. T. J. R. Hughes and L. P. Franca. A mixed finite element formulation for Reissner–Mindlin plate theory: uniform convergence of all higher-order spaces. Comput. Methods Appl. Mech. Eng., 67(2):223–240, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  38. T. J. R. Hughes and T. E. Tezuyar. Finite elements based upon Mindlin plate theory with particular reference to the four node blinear isoparamtric element. J. Appl. Mech. Eng., 48:587–598, 1981.

    Article  MATH  Google Scholar 

  39. C. Lovadina. Analysis of a mixed finite element method for the Reissner–Mindlin plate problems. Comput. Methods Appl. Mech. Eng., 163(1–4):71–85, 1998.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  41. M. Lyly. On the connection between some linear triangular Reissner–Mindlin plate bending elements. Numer. Math., 85(1):77–107, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  42. M. Lyly, R. Stenberg, and T. Vihinen. A stable bilinear element for the Reissner–Mindlin plate model. Comput. Methods Appl. Mech. Eng., 110(3–4):343–357, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  43. E. Oñate, F. Zarate, and F. Flores. A simple triangular element for thick and thin plate and shell analysis. Int. J. Numer. Methods Eng., 37:2569–2582, 1994.

    Article  MATH  Google Scholar 

  44. P. Peisker and D. Braess. Uniform convergence of mixed interpolated elements for Reissner–Mindlin plates. RAIRO Modél. Math. Anal. Numér., 26(5):557–574, 1992.

    MATH  MathSciNet  Google Scholar 

  45. I. Perugia and T. Scapolla. Optimal rectangular MITC finite elements for Reissner–Mindlin plates. Numer. Methods Partial Differen. Equations, 13(5):575–585, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  46. J. Pitkäranta. Analysis of some low-order finite element schemes for Mindlin–Reissner and Kirchhoff plates. Numer. Math., 53(1–2):237–254, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  47. R. Stenberg. 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.

    Google Scholar 

  48. R. Stenberg and M. Suri. An hp error analysis of MITC plate elements. SIAM J. Numer. Anal., 34(2):544–568, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  49. R. L. Taylor and F. Auricchio. Linked interpolation for Reissner–Mindlin plate elements: Part II – a simple triangle. Int. J. Num. Meths. Engrg., 50:71–101, 1985.

    Google Scholar 

  50. A. Tessler and T. J. R. Hughes. A three-node mindlin plate element with improved transverse shear. Comput. Methods Appl. Mech. Eng., 50:71–101, 1985.

    Article  MATH  Google Scholar 

  51. Z. Xu. A thick-thin triangular plate element. Int. J. Numer. Methods Eng., 33:963–973, 1992.

    Article  Google Scholar 

  52. O. C. Zienkiewicz and D. Lefebvre. A robust triangular plate bending element of the Reissner–Mindlin plate. Int. J. Numer. Methods Eng., 26:1169–1184, 1998.

    Article  Google Scholar 

  53. O. C. Zienkiewicz, R. L. Taylor, P. Papadopoulos, and E. Oñate. Plate bending elements with discrete constraints: New triangular elements. Comput. & Struct., 35:505–522, 1990.

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, The State University of New Jersey, 110 Frelinghuysen Rd, 08854-801, Piscataway, NJ, USA

    Richard S. Falk

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  1. Richard S. Falk
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Editors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Brescia, Via Valotti 9, 25133, Brescia, Italy

    Lucia Gastaldi

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Falk, R.S. (2008). Finite Elements for the Reissner–Mindlin Plate. In: Boffi, D., Gastaldi, L. (eds) Mixed Finite Elements, Compatibility Conditions, and Applications. Lecture Notes in Mathematics, vol 1939. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78319-0_5

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