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Locking-free finite elements
for the Reissner-Mindlin plate


Authors: Richard S. Falk and Tong Tu
Journal: Math. Comp. 69 (2000), 911-928
MSC (1991): Primary 65N30, 73K10, 73K25
DOI: https://doi.org/10.1090/S0025-5718-99-01165-5
Published electronically: August 20, 1999
MathSciNet review: 1665950
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Abstract: Two new families of Reissner-Mindlin triangular finite elements are analyzed. One family, generalizing an element proposed by Zienkiewicz and Lefebvre, approximates (for $k\ge 1)$ the transverse displacement by continuous piecewise polynomials of degree $k+1$, the rotation by continuous piecewise polynomials of degree $k+1$ plus bubble functions of degree $k+3$, and projects the shear stress into the space of discontinuous piecewise polynomials of degree $k$. The second family is similar to the first, but uses degree $k$ rather than degree $k+1$ continuous piecewise polynomials to approximate the rotation. We prove that for $2\le s\le k+1$, the $L^2$ errors in the derivatives of the transverse displacement are bounded by $Ch^s$ and the $L^2$ errors in the rotation and its derivatives are bounded by $Ch^s\min(1,ht^{-1})$ and $Ch^{s-1}\min(1,ht^{-1})$, respectively, for the first family, and by $Ch^s$ and $Ch^{s-1}$, respectively, for the second family (with $C$ independent of the mesh size $h$ and plate thickness $t)$. These estimates are of optimal order for the second family, and so it is locking-free. For the first family, while the estimates for the derivatives of the transverse displacement are of optimal order, there is a deterioration of order $h$ in the approximation of the rotation and its derivatives for $t$ small, demonstrating locking of order $h^{-1}$. Numerical experiments using the lowest order elements of each family are presented to show their performance and the sharpness of the estimates. Additional experiments show the negative effects of eliminating the projection of the shear stress.


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  • 1. D. N. Arnold, Innovative finite element methods for plates, Mat. Apl. Comput. 10 (1991), 77-99. MR 93c:65136
  • 2. D. N. Arnold and R. S. Falk, A uniformly accurate finite element method for the Reissner-Mindlin plate, SIAM J. Numer. Anal. 26 (1989), 1276-1290. MR 91c:65068
  • 3. D. N. Arnold and R. S. Falk, The boundary layer for the Reissner-Mindlin plate model, SIAM J. Math. Anal. 21 (1990), 281-312. MR 91c:73053
  • 4. D. N. Arnold and R. S. Falk, Asymptotic analysis of the boundary layer for the Reissner-Mindlin plate model, SIAM J. Math. Anal. 27 (1996), 486-514. MR 97i:73064
  • 5. 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 (1997), 217-238. MR 98b:73034
  • 6. K. J. Bathe, F. Brezzi, and S. W. Cho, The MITC7 and MITC9 plate bending elements, Comput. & Structures 32 (1989), 797-841.
  • 7. K. J. Bathe, F. Brezzi and M. Fortin, Mixed-interpolated elements for Reissner-Mindlin plates, Internat. J. Numer. Meth. Engrg. 21 (1989), 1787-1801. MR 90g:73090
  • 8. J. H. Bramble and T. Sun, A negative-norm least squares method for Reissner-Mindlin plates, Math. Comp. 67 (1998), 901-916. MR 99d:73086
  • 9. F Brezzi and M. Fortin, Numerical approximation of Mindlin-Reissner plates, Math. Comp. 47 (1986), 151-158. MR 87g:73057
  • 10. F. Brezzi, M. Fortin, and R. Stenberg, Error analysis of mixed-interpolated elements for Reissner-Mindlin plates, Math. Models Methods Appl. Sci. 1 (1991), 125-151. MR 92e:73030
  • 11. R. Durán, A. Ghioldi, and N. Wolanski, A finite element method for Mindlin-Reissner plate model, SIAM J. Numer. Anal. 28 (1991), 1004-1014. MR 92f:73046
  • 12. R. Durán and E. Liberman, On mixed finite element methods for the Reissner-Mindlin plate model, Math. Comp. 58 (1992), 561-573. MR 92f:65135
  • 13. R. Durán and E. Liberman, On the convergence of a triangular mixed finite element method for Reissner-Mindlin plate, Math. Models Methods Appl. Sci. 6 (1996), 339-352. MR 97e:73064
  • 14. L. Franca and R. Stenberg, A modification of a low-order Reissner-Mindlin plate bending element, The Mathematics of Finite Elements and Applications VII (J. W. Whiteman, ed.), Academic Press, 1991, pp. 425-436. MR 92d:65007
  • 15. L. Franca, R. Stenberg and T. Vihinen, A nonconforming finite element method for the Reissner-Mindlin plate bending model, Proc. 13th IMACS World Conf. Computation and Applied Mathematics, (Vichnevetsky and Miller, eds.), Trinity College, Dublin, 4 (1991), 1907-1908.
  • 16. T. J. R. Hughes and L. Franca, A mixed finite element formulation for Reissner-Mindlin plate theory: Uniform convergence of all higher order spaces, Comp. Methods Appl. Mech. Engrg. 67 (1988), 223-240. MR 89g:73033
  • 17. E. Oñate, F. Zarate and F. Flores, A simple triangular element for thick and thin plate and shell analysis, Internat. J. Numer. Method. Engrg. 37 (1994), 2569-2582.
  • 18. P. Peisker and D. Braess, Uniform convergence of mixed interpolated elements for Reissner-Mindlin plates, RAIRO Modél. Math. Anal. Numér. 26 (1992), 557-574. MR 93j:73070
  • 19. J. Pitkäranta, Analysis of some low-order finite element schemes for Mindlin-Reissner and Kirchoff plates, Numer. Math. 53 (1988), 237-254. MR 89f:65126
  • 20. J. Pitkäranta and M. Suri, Design principles and error analysis for reduced-shear plate-bending finite elements, Numer. Math. 75 (1996), 223-266. MR 98c:73078
  • 21. R. Stenberg and T. Vihinen, Calculations with some linear elements for Reissner-Mindlin plates, Proc. European Conf. New Advances in Computational Structural Mechanics, Giens, France, (1991), 505-511.
  • 22. T. Tu, Performance of Reissner-Mindlin Elements, Ph.D. Thesis, Dept. Math., Rutgers University, 1998.
  • 23. O. C. Zienkiewicz and D. Lefebvre, A robust triangular plate bending element of the Reissner-Mindlin plate, Internat. J. Numer. Methods Engrg. 26 (1998), 1169-1184.
  • 24. O. C. Zienkiewicz, R. L. Taylor, P. Papadopoulos and E. Oñate, Plate bending elements with discrete constraints: New triangular elements, Comput. & Structures 35 (1990), 505-522.

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Additional Information

Richard S. Falk
Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854
Email: falk@math.rutgers.edu

Tong Tu
Affiliation: Bloomberg Princeton Index Group, 100 Business Park Drive, Skillman, New Jersey 08858
Email: tongtu@bloomberg.net

DOI: https://doi.org/10.1090/S0025-5718-99-01165-5
Keywords: Reissner-Mindlin plate, finite element, locking-free
Received by editor(s): August 14, 1998
Published electronically: August 20, 1999
Additional Notes: The first author was supported by NSF grant DMS-9704556
Article copyright: © Copyright 2000 American Mathematical Society

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