Analysis for quadrilateral MITC elements for the Reissner-Mindlin plate problem

Hu, Jun; Shi, Zhong-Ci

doi:10.1090/S0025-5718-08-02153-4

Analysis for quadrilateral MITC elements for the Reissner-Mindlin plate problem
HTML articles powered by AMS MathViewer

by Jun Hu and Zhong-Ci Shi;

Math. Comp. 78 (2009), 673-711

DOI: https://doi.org/10.1090/S0025-5718-08-02153-4

Published electronically: August 1, 2008

PDF | Request permission

Abstract:

The present paper is made up of two parts. In the first part, we study the mathematical stability and convergence of the quadrilateral MITC elements for the Reissner-Mindlin plate problem in an abstract setting. We generalize the Brezzi-Bathe-Fortin conditions to the quadrilateral MITC elements by weakening the second and fourth conditions. Under these conditions, we show the well-posedness of the discrete problem and establish an abstract error estimate in the energy norm. The conclusion of this part is sparsity in the mathematical research of the quadrilateral MITC elements in the sense that one only needs to check these five conditions.

In the second part, we extend four families of rectangular MITC elements of Stenberg and Süri to the quadrilateral meshes. We prove that these quadrilateral elements satisfy the generalized Brezzi-Bathe-Fortin conditions from the first part. We develop the h-p error estimates in both energy and $L^2$ norm for these quadrilateral elements. For the first three families of quadrilateral elements, the error estimates indicate that their convergent rates in both energy and $L^2$ norm depend on the mesh distortion parameter $\alpha$. We can get optimal error estimates for them provided that $\alpha =1$. In addition, we show the optimal convergence rates in energy norm uniformly in $\alpha$ for the fourth family of quadrilateral elements. Like their rectangular counterparts, these quadrilateral elements are locking-free.

References

Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 450957
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, 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
D. N. Arnold, D. Boffi and R. S. Falk. Remarks on quadrilateral Reissner-Mindlin plate elements, in Proceedings of Fifth World Congress on Computational Mechanics, H. A. Mang, F. G. Rammerstorfer and J. Eberhardsteiner, eds.
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
Ivo Babuška and Manil Suri, The $p$ and $h$-$p$ versions of the finite element method, basic principles and properties, SIAM Rev. 36 (1994), no. 4, 578–632. MR 1306924, DOI 10.1137/1036141
I. Babuška and Manil Suri, The $h$-$p$ version of the finite element method with quasi-uniform meshes, RAIRO Modél. Math. Anal. Numér. 21 (1987), no. 2, 199–238 (English, with French summary). MR 896241, DOI 10.1051/m2an/1987210201991
I. Babuška and Manil Suri, The optimal convergence rate of the $p$-version of the finite element method, SIAM J. Numer. Anal. 24 (1987), no. 4, 750–776. MR 899702, DOI 10.1137/0724049
K. J. Bathe, F. Brezzi and M. Fortin. A simplified analysis of two-plate elements: The MITC4 and MITC9 element, G. N. Pande and J. Middleton (eds.), Numeta 87 Vol.1, Numerical Techniques for Engineering Analysis and Design, Martinus Nijhoff, Amsterdam.
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), pp. 367–383.
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
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
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
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
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
J. Hu. Quadrilateral locking free elements in elasticity, Doctorate Dissertation, Institute of Computational Mathematics, CAS, 2004.
Jun Hu, Pingbing Ming, and Zhongci Shi, Nonconforming quadrilateral rotated $Q_1$ element for Reissner-Mindlin plate, J. Comput. Math. 21 (2003), no. 1, 25–32. Special issue dedicated to the 80th birthday of Professor Zhou Yulin. MR 1974269
J. Hu and Z. C. Shi, $h$-$p$ analysis for $\mathbf {H}(\operatorname {rot})$-Conforming Finite Elements Over Quadrilaterals, 2002, Preprint.
J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth. MR 350177
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
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, 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
Pingbing Ming and Zhongci Shi, Quadrilateral mesh, Chinese Ann. Math. Ser. B 23 (2002), no. 2, 235–252. Dedicated to the memory of Jacques-Louis Lions. MR 1924140, DOI 10.1142/S0252959902000237
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
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
Rolf Stenberg and Manil Suri, Mixed $hp$ finite element methods for problems in elasticity and Stokes flow, Numer. Math. 72 (1996), no. 3, 367–389. MR 1367655, DOI 10.1007/s002110050174
Rolf Stenberg and Manil Suri, An $hp$ error analysis of MITC plate elements, SIAM J. Numer. Anal. 34 (1997), no. 2, 544–568. MR 1442928, DOI 10.1137/S0036142994278486
Manil Suri, The $p$-version of the finite element method for elliptic equations of order $2l$, RAIRO Modél. Math. Anal. Numér. 24 (1990), no. 2, 265–304 (English, with French summary). MR 1052150, DOI 10.1051/m2an/1990240202651
Manil Suri, On the stability and convergence of higher-order mixed finite element methods for second-order elliptic problems, Math. Comp. 54 (1990), no. 189, 1–19. MR 990603, DOI 10.1090/S0025-5718-1990-0990603-X
Manil Suri, Ivo Babuška, and Christoph Schwab, Locking effects in the finite element approximation of plate models, Math. Comp. 64 (1995), no. 210, 461–482. MR 1277772, DOI 10.1090/S0025-5718-1995-1277772-6
P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Lecture Notes in Math., Vol. 606, Springer, Berlin-New York, 1977, pp. 292–315. MR 483555
J. P. Wang and T. Mathew. Mixed finite element methods over quadrilaterals, 1994, preprint.