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Nonconforming stabilized combined finite element method for Reissner-Mindlin plate. (English) Zbl 1160.74045

Summary: Based on combination of two variational principles, we develop a nonconforming stabilized finite element method for Reissner-Mindlin plates. The method is convergent when the finite element space is energy-compatible. Error estimates are derived. In particular, three finite element spaces are applied in the computation. Numerical results show that the method is insensitive to mesh distortion and has better performence than the MITC4 and DKQ methods. With properly chosen parameters, high accuracy can be obtained on coarse meshes.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74K20 Plates
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[1] Brezzi, F. Numerical approximation of Mindlin-Reissner plates. Math. Comput. 47(175), 151–158 (1986) · Zbl 0596.73058 · doi:10.1090/S0025-5718-1986-0842127-7
[2] Duran, R. and Liberman, E. On mixed finite element method for the Reissner-Mindlin plate model. Math. Comput. 58(198), 561–573 (1992) · Zbl 0763.73054
[3] Zhou, Tianxiao. The partial projection method in the finite element discretization of the Reissner-Mindlin plate model. J. Comput. Math. 13(2), 172–191 (1995) · Zbl 0826.73064
[4] Chinosi, C. Remarks on some mixed finite element schemes for Reissner-Mindlin plate model. Calcolo 39(2), 87–108 (2002) · Zbl 1072.65128 · doi:10.1007/s100920200006
[5] Lovadina, C. A low-order nonconforming finite element for Reissner-Mindlin plates. SIAM J. Numer. Anal. 42(6), 2688–2705 (2005) · Zbl 1080.74048 · doi:10.1137/040603474
[6] Chinosi, C., Lovadina, C., and Marini, L. D. Nonconforming locking-free finite elements for Reissner-Mindlin plates. Comput. Meth. Appl. Mech. Engng. 195(25–28), 3448–3460 (2006) · Zbl 1119.74047 · doi:10.1016/j.cma.2005.06.025
[7] Ming, Pingbing and Shi, Zhongci. Two nonconforming quadrilateral elements for the Reissner-Mindlin plate. Mathematical Models and Methods in Applied Science 15(10), 1503–1517 (2005) · Zbl 1096.74050 · doi:10.1142/S0218202505000868
[8] Luo, Kun. An accurate quadrilateral plate element based on energy optimization. Comm. Numer. Meth. Engng. 21(9), 487–498 (2005) · Zbl 1084.74057 · doi:10.1002/cnm.761
[9] Chen, Shaochun and Shi, Dongyang. General error estimates of nonconforming plate elements. Mathematica Numerica Sinica 22(1), 295–300 (2000) · Zbl 0997.74067
[10] Hu, Jun and Shi, Zhongci. Two lower order nonconforming rectangular elements for the Reissner-Mindlin plate. Math. Comput. 76(260), 1771–1786 (2007) · Zbl 1118.74050 · doi:10.1090/S0025-5718-07-01952-7
[11] Long, Yuqiu and Zhao, Junqing. A generalized conforming rectangular element for thick/thin plates (in Chinese). Engineering Mechanics 5(1), 1–8 (1988) · Zbl 0676.73045
[12] Long, Zhifei and Cen, Song. Finite Element Methods (in Chinese), China Waterpower Press, Beijing (2001) · Zbl 1047.74068
[13] Cen, Song, Long, Yuqiu, Yao, Zhenhan, and Chiew, Singping. Application of the quadrilateral area co-ordinate method: a new element for Mindlin-Reissner plate. Int. J. Numer. Meth. Engng. 66(1), 1–45 (2006) · Zbl 1110.74845 · doi:10.1002/nme.1533
[14] Taylor, R. L., Simo, J. C., Zienkiewicz, O. C., and Chan, A. H. C. The patch test–a condition for assessing FEM convergence. Int. J. Numer. Meth. Engng. 22(1), 39–62 (1986) · Zbl 0593.73072 · doi:10.1002/nme.1620220105
[15] Wu, Changchun and Bian, Xuehuang. Nonconforming Numerical Analysis and Combined Hybird Finite Element Methods, Science Press, Beijing (1997)
[16] Shi, Zhongci. A convergence condition for the quadrilateral Wilson element. Numer. Math. 44(3), 349–361 (1984) · Zbl 0581.65008 · doi:10.1007/BF01405567
[17] Zhang, Zhimin and Zhang, Shangyou. Derivative superconvergence of rectangular finite elements for the Reissner-Mindlin plate. Comput. Meth. Appl. Mech. Engng. 134(1), 1–16 (1995) · Zbl 0891.73071 · doi:10.1016/0045-7825(96)01025-0
[18] Bathe, K. J. and Dvorkin, E. A four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation. Int. J. Numer. Meth. Engng. 21(2), 367–383 (1985) · Zbl 0551.73072 · doi:10.1002/nme.1620210213
[19] Batoz, J. L. and Bentahar, M. Evaluation of a new quadrilateral thin plate bending element. Inter. J. Numer. Meth. Engng. 18(11), 1655–1677 (1982) · Zbl 0489.73080 · doi:10.1002/nme.1620181106
[20] Zhou, Tianxiao. Stabilized hubrid finite element methods based on the combination of saddle point principles of elasticity problems. Math. Comput. 72(244), 1655–1673 (2003) · Zbl 1081.74046 · doi:10.1090/S0025-5718-03-01473-X
[21] Feng, Minfu, Yang, Rongkui, and Xiong, Huaxin. Stabilized finite element methods for the Reissner-Mindlin plate. J. Numer. Math. 8(2), 125–132 (1999) · Zbl 1049.74776
[22] Brezzi, F. and Fortin, M. Mixed and Hybrid Finite Method, Springer-Verlag, New York (1991) · Zbl 0788.73002
[23] Feng, Minfu, Ming, Pingbing, and Yang, Rongkui. Absolute stable homotopy finite element methods for circular arch problem and asymptotic exactness posteriori error estimates. J. Comput. Math. 20(6), 653–672 (2002) · Zbl 1017.65087
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