A locking-free Reissner-Mindlin quadrilateral element

Duan, Huo-Yuan; Liang, Guo-Ping

doi:10.1090/S0025-5718-03-01619-3

A locking-free Reissner-Mindlin quadrilateral element
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by Huo-Yuan Duan and Guo-Ping Liang PDF

Math. Comp. 73 (2004), 1655-1671 Request permission

Abstract:

On arbitrary regular quadrilaterals, a new finite element method for the Reissner-Mindlin plate is proposed, where both transverse displacement and rotation are approximated by isoparametric bilinear elements, with local bubbles enriching rotation, and a local reduction operator is applied to the shear energy term. This new method gives optimal error bounds, uniform in the thickness of the plate, for both transverse displacement and rotation with respect to $H^1$ and $L^2$ norms.

References

K.J. Bathe and E.N. Dvorkin, A four-node plate bending element based on Mindlin-Reissner theory and a mixed interpolation, Internat. J. Numer. Methods Engrg., 21(1985), pp. 367-383.
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, The mathematics of finite elements and applications, V (Uxbridge, 1984) Academic Press, London, 1985, pp. 491–503. MR 811058
Ricardo Durán and Elsa Liberman, On mixed finite element methods for the Reissner-Mindlin plate model, Math. Comp. 58 (1992), no. 198, 561–573. MR 1106965, DOI 10.1090/S0025-5718-1992-1106965-0
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
Richard S. Falk and Tong Tu, Locking-free finite elements for the Reissner-Mindlin plate, Math. Comp. 69 (2000), no. 231, 911–928. MR 1665950, DOI 10.1090/S0025-5718-99-01165-5
Tian Xiao Zhou, The partial projection method in the finite element discretization of the Reissner-Mindlin plate model, J. Comput. Math. 13 (1995), no. 2, 172–191. MR 1330561
Franco Brezzi, Michel Fortin, and Rolf Stenberg, Error analysis of mixed-interpolated elements for Reissner-Mindlin plates, Math. Models Methods Appl. Sci. 1 (1991), no. 2, 125–151. MR 1115287, DOI 10.1142/S0218202591000083
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
Douglas N. Arnold and Richard S. Falk, The boundary layer for the Reissner-Mindlin plate model, SIAM J. Math. Anal. 21 (1990), no. 2, 281–312. MR 1038893, DOI 10.1137/0521016
Huo-Yuan Duan and Guo-Ping Liang, Analysis of some stabilized low-order mixed finite element methods for Reissner-Mindlin plates, Comput. Methods Appl. Mech. Engrg. 191 (2001), no. 3-5, 157–179. MR 1868502, DOI 10.1016/S0045-7825(01)00228-6
Rolf Stenberg, A new finite element formulation for the plate bending problem, Asymptotic methods for elastic structures (Lisbon, 1993) de Gruyter, Berlin, 1995, pp. 209–221. MR 1333214
Juhani Pitkäranta and Manil Suri, Design principles and error analysis for reduced-shear plate-bending finite elements, Numer. Math. 75 (1996), no. 2, 223–266. MR 1421988, DOI 10.1007/s002110050238
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
P. Destuynder, Mathematical Analysis of Thin Shell Problems, Masson, Paris, 1991.
P. Peisker and D. Braess, Uniform convergence of mixed interpolated elements for Reissner-Mindlin plates, RAIRO Modél. Math. Anal. Numér. 26 (1992), no. 5, 557–574 (English, with English and French summaries). MR 1177387, DOI 10.1051/m2an/1992260505571
F. Auricchio and C. Lovadina, Analysis of kinematic linked interpolation methods for Reissner-Mindlin plate problems, Comput. Methods Appl. Mech. Engrg. 190 (2001), no. 18-19, 2465–2482. MR 1815650, DOI 10.1016/S0045-7825(00)00247-4
Ricardo Durán, Adriana Ghioldi, and Noemí Wolanski, A finite element method for the Mindlin-Reissner plate model, SIAM J. Numer. Anal. 28 (1991), no. 4, 1004–1014. MR 1111450, DOI 10.1137/0728053
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 0520174
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
Alexander Iosilevich, Klaus-Jürgen Bathe, and Franco Brezzi, Numerical inf-sup analysis of MITC plate bending elements, Plates and shells (Québec, QC, 1996) CRM Proc. Lecture Notes, vol. 21, Amer. Math. Soc., Providence, RI, 1999, pp. 225–242. MR 1696482, DOI 10.1090/crmp/021/15
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
M. Lyly and R. Stenberg, Stabilized MITC plate bending elements, in: M. Papadrakakis and B.H.V. Topping, eds., Advances in Finite Element Techniques, CIVIL-COMP Ltd, Edinburgh, Scotland, 1994, pp. 11-16.
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
Shmuel L. Weissman and Robert L. Taylor, Resultant fields for mixed plate bending elements, Comput. Methods Appl. Mech. Engrg. 79 (1990), no. 3, 321–355. MR 1050152, DOI 10.1016/0045-7825(90)90067-V
O.C. Zienkiewicz, Z. Xu, L.F. Zeng, A. Samuelsson and N.E. Wiberg, Linked interpolation for Reissner-Mindlin plate elements: Part I—A simple quadrilateral; Part III—An alternative quadrilateral, Int. J. Numer. Methods Engrg., 36(1993), pp. 3043-3056; 37(1994), pp. 1437-1443.
R. L. Taylor and F. Auricchio, Linked interpolation for Reissner-Mindlin plate elements: Part II-a simple triangle, Int. J. Numer Methods Engrg., 36(1993), pp. 3057-3066.
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
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
Franco Brezzi, Klaus-Jürgen Bathe, and Michel Fortin, Mixed-interpolated elements for Reissner-Mindlin plates, Internat. J. Numer. Methods Engrg. 28 (1989), no. 8, 1787–1801. MR 1008138, DOI 10.1002/nme.1620280806
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
J. Pitkäranta and R. Stenberg, Error bounds for the approximation of Stokes problem with bilinear/constant elements on irregular quadrilateral meshes, in: J. R. Whiteman, ed., The Mathematics of Finite Elements and Applications, V. MAFELAP, Academic Press, New-York, 1985, pp. 325-334.
K. J. Bathe and F. Brezzi, A simplified analysis of two plate bending elements-the MITC4 and MITC9 elements, in: G.N. Pande and J. Middleton, eds., NUMETA 87, Vol. 1, Numerical Techniques for Engineering Analysis and Design, Martinus Nijhoff, Amsterdam, 1987.
H.-Y Duan and G.-P. Liang, Nonconforming elements in least-squares mixed finite element methods, Math. Comp. 73(2004), pp. 1-18.
Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383, DOI 10.1007/978-3-642-61623-5
M. Lyly and R. Stenberg, Stabilized finite element methods for Reissner-Mindlin plates (in: the web page of the second author)
J. P. Wang and T. Mathew, Mixed finite element methods over quadrilaterals, in: Proceedings of 3rd Internat. Conference on Advances in Numerical Methods and Applications, J.T. Dimov, B. Sendov, and P. Vassilevski, eds., World Scientific, River Edge, NJ, 1994, pp. 203-214.
R. G. 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 (1999), no. 228, 1447–1463. MR 1648387, DOI 10.1090/S0025-5718-99-01094-7
C. Lovadina, Analysis of a mixed finite element method for the Reissner-Mindlin plate problems, Comput. Methods Appl. Mech. Engrg. 163 (1998), no. 1-4, 71–85. MR 1653723, DOI 10.1016/S0045-7825(98)00003-6
Ricardo G. Durán and Elsa Liberman, On the convergence of a triangular mixed finite element method for Reissner-Mindlin plates, Math. Models Methods Appl. Sci. 6 (1996), no. 3, 339–352. MR 1388710, DOI 10.1142/S0218202596000110
H.-Y. Duan, Studies On Mixed Finite Element Methods, Ph.D. thesis, Institute of Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, March, 2002.