Analysis of some low order quadrilateral Reissner-Mindlin plate elements

Ming, Pingbing; Shi, Zhong-ci

doi:10.1090/S0025-5718-06-01833-3

Analysis of some low order quadrilateral Reissner-Mindlin plate elements
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by Pingbing Ming and Zhong-ci Shi;

Math. Comp. 75 (2006), 1043-1065

DOI: https://doi.org/10.1090/S0025-5718-06-01833-3

Published electronically: May 1, 2006

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Abstract:

Four quadrilateral elements for the Reissner-Mindlin plate model are considered. The elements are the stabilized MITC4 element of Lyly, Stenberg and Vihinen (1933), the MIN4 element of Tessler and Hughes (1983), the Q4BL element of Zienkiewicz et al. (1993) and the FMIN4 element of Kikuchi and Ishii (1999). For all elements except the Q4BL element, a unifying variational formulation is introduced, and optimal H$^1$ and L$^2$ error bounds uniform in the plate thickness are proven. Moreover, we propose a modified Q4BL element and show that it admits the optimal H$^1$ and L$^2$ error bounds uniform in the plate thickness. In particular, we study the convergence behavior of all elements regarding the mesh distortion.

References

Douglas N. Arnold, Daniele Boffi, and Richard S. Falk, Approximation by quadrilateral finite elements, Math. Comp. 71 (2002), no. 239, 909–922. MR 1898739, DOI 10.1090/S0025-5718-02-01439-4
Douglas N. Arnold, Daniele Boffi, and Richard S. Falk, Quadrilateral $H(\textrm {div})$ finite elements, SIAM J. Numer. Anal. 42 (2005), no. 6, 2429–2451. MR 2139400, DOI 10.1137/S0036142903431924
Douglas N. Arnold, Daniele Boffi, Richard S. Falk, and Lucia Gastaldi, Finite element approximation on quadrilateral meshes, Comm. Numer. Methods Engrg. 17 (2001), no. 11, 805–812. MR 1872639, DOI 10.1002/cnm.450
Douglas N. Arnold and Franco Brezzi, Some new elements for the Reissner-Mindlin plate model, Boundary value problems for partial differential equations and applications, RMA Res. Notes Appl. Math., vol. 29, Masson, Paris, 1993, pp. 287–292. MR 1260452
Douglas N. Arnold and Richard S. Falk, A uniformly accurate finite element method for the Reissner-Mindlin plate, SIAM J. Numer. Anal. 26 (1989), no. 6, 1276–1290. MR 1025088, DOI 10.1137/0726074
Douglas N. Arnold and Richard S. Falk, Asymptotic analysis of the boundary layer for the Reissner-Mindlin plate model, SIAM J. Math. Anal. 27 (1996), no. 2, 486–514. MR 1377485, DOI 10.1137/S0036141093245276
Ferdinando Auricchio and Carlo Lovadina, Partial selective reduced integration schemes and kinematically linked interpolations for plate bending problems, Math. Models Methods Appl. Sci. 9 (1999), no. 5, 693–722. MR 1697391, DOI 10.1142/S021820259900035X
Rezak Ayad, Gouri Dhatt, and Jean Louis Batoz, A new hybrid-mixed variational approach for Reissner-Mindlin plates. The MiSP model, Internat. J. Numer. Methods Engrg. 42 (1998), no. 7, 1149–1179. MR 1633100, DOI 10.1002/(SICI)1097-0207(19980815)42:7<1149::AID-NME391>3.0.CO;2-2
R. Ayad and A. Rigolot, An improved four-node hybrid-mixed element based upon Mindlin’s plate theory, Internat. J. Numer. Methods Engrg. 55 (2002), 705–731.
Klaus-Jürgen Bathe, Finite-Elemente-Methoden, Springer-Verlag, Berlin, 1986 (German). Matrizen und lineare Algebra, die Methode der finiten Elemente, Lösung von Gleichgewichtsbedingungen und Bewegungsgleichungen. [Matrices and linear algebra, the finite-element method, solution of equilibrium conditions and equations of motion]; Translated from the English and with a foreword by Peter Zimmermann. MR 851177, DOI 10.1007/978-3-642-96905-8
K.-J. Bathe and E. Dvorkin, A four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation, Internat. J. Numer. Methods Engrg. 21 (1985), 367–383.
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. MR 1278258, DOI 10.1007/978-1-4757-4338-8
F. Brezzi and M. Fortin, Numerical approximation of Mindlin-Reissner plates, Math. Comp. 47 (1986), no. 175, 151–158. MR 842127, DOI 10.1090/S0025-5718-1986-0842127-7
Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
D. Chapelle and R. Stenberg, An optimal low-order locking-free finite element method for Reissner-Mindlin plates, Math. Models Methods Appl. Sci. 8 (1998), no. 3, 407–430. MR 1624871, DOI 10.1142/S0218202598000172
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 520174
Ricardo G. Durán, Erwin Hernández, Luis Hervella-Nieto, Elsa Liberman, and Rodolfo Rodríguez, Error estimates for low-order isoparametric quadrilateral finite elements for plates, SIAM J. Numer. Anal. 41 (2003), no. 5, 1751–1772. MR 2035005, DOI 10.1137/S0036142902409410
I. Fried and S. K. Yang, Triangular, nine-degrees-of-freedoms, C$^{\,0}$ plate bending element of quadratic accuracy, Quart. Appl. Math. 31 (1973), 303–312.
V. Girault and P.-A. Raviart, An analysis of a mixed finite element method for the Navier-Stokes equations, Numer. Math. 33 (1979), no. 3, 235–271. MR 553589, DOI 10.1007/BF01398643
Thomas J. R. Hughes, The finite element method, Prentice Hall, Inc., Englewood Cliffs, NJ, 1987. Linear static and dynamic finite element analysis; With the collaboration of Robert M. Ferencz and Arthur M. Raefsky. MR 1008473
F. Kikuchi and K. Ishii, An improved $4$-node quadrilateral plate bending element of the Reissner-Mindlin type, Comput. Mech. 23 (1999), 240–249.
A. M. Lakhany, I. Marek, and J. R. Whiteman, Superconvergence results on mildly structured triangulations, Comput. Methods Appl. Mech. Engrg. 189 (2000), no. 1, 1–75. MR 1779678, DOI 10.1016/S0045-7825(99)00281-9
C. Lovadina, A new class of mixed finite element methods for Reissner-Mindlin plates, SIAM J. Numer. Anal. 33 (1996), no. 6, 2457–2467. MR 1427474, DOI 10.1137/S0036142994265061
Zi Qiang Yang and Shi Bo Zhou, Constellation graphical method and classification, J. Numer. Methods Comput. Appl. 5 (1984), no. 1, 54–57 (Chinese, with English summary). MR 751364
Mikko Lyly and Rolf Stenberg, The stabilized MITC plate bending elements, Computational mechanics (Buenos Aires, 1998) Centro Internac. Métodos Numér. Ing., Barcelona, 1998, pp. CD-ROM file. MR 1839065
M. Lyly and R. Stenberg, Stabilized finite element methods for Reissner-Mindlin plates, preprint, 2001.
Mikko Lyly, Rolf Stenberg, and Teemu Vihinen, A stable bilinear element for the Reissner-Mindlin plate model, Comput. Methods Appl. Mech. Engrg. 110 (1993), no. 3-4, 343–357. MR 1256325, DOI 10.1016/0045-7825(93)90214-I
Pingbing Ming and Zhong-Ci Shi, Nonconforming rotated $\scr Q_1$ element for Reissner-Mindlin plate, Math. Models Methods Appl. Sci. 11 (2001), no. 8, 1311–1342. MR 1859825, DOI 10.1142/S0218202501001343
Pingbing Ming and Zhong-ci Shi, Optimal $\scr L^2$ error bounds for MITC3 type element, Numer. Math. 91 (2002), no. 1, 77–91. MR 1896088, DOI 10.1007/s002110100334
P. B. Ming and Z.-C. Shi, Optimal L$^2$ error estimate for stabilized MITC4 element, preprint, 2001.
Pingbing Ming and Zhong-Ci Shi, Quadrilateral mesh revisited, Comput. Methods Appl. Mech. Engrg. 191 (2002), no. 49-50, 5671–5682. MR 1941167, DOI 10.1016/S0045-7825(02)00471-1
Pingbing Ming and Zhong-Ci Shi, Two nonconforming quadrilateral elements for the Reissner-Mindlin plate, Math. Models Methods Appl. Sci. 15 (2005), no. 10, 1503–1517. MR 2168943, DOI 10.1142/S0218202505000868
Juhani Pitkäranta, Analysis of some low-order finite element schemes for Mindlin-Reissner and Kirchhoff plates, Numer. Math. 53 (1988), no. 1-2, 237–254. MR 946378, DOI 10.1007/BF01395887
Zhong Ci Shi, A convergence condition for the quadrilateral Wilson element, Numer. Math. 44 (1984), no. 3, 349–361. MR 757491, DOI 10.1007/BF01405567
A. Tessler and T. J. R. Hughes, An improved treatment of transverse shear in the Mindlin type four-node quadrilateral element, Comput. Methods Appl. Mech. Engrg. 39 (1983), 311–335.
J. M. Thomas, Sur l’analyse numérique des méthods d’éléments finis hybrides et mixtes, Thése d’Etat, Université Pierre et Marie Curie, Paris, 1977.
Z. Xu, A simple and efficient triangular finite element for plate bending, Acta Mech. Sin. 2 (1986), 185–192.
Z. Xu, O. C. Zienkiewicz and L. F. Zhang, Linked interpolation for Reissner-Mindlin plate elements, III, an alternative quadrilateral element, Internat. J. Numer. Methods Engrg. 37 (1994), 1437–1443.
O. C. Zienkiewicz, Z. Xu, L. F. Zeng, A. Samuelsson and N. E. Wiberg, Linked interpolation for Reissner-Mindlin plate elements, I: A simple quadrilateral, Internat. J. Numer. Methods. Engrg. 36 (1993), 3043–3056.

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Analysis of some low order quadrilateral Reissner-Mindlin plate elements
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Abstract: