胡俊¹, 石钟慈²

1. 北京大学数学科学学院, 北京 100871;
2. LSEC, 中国科学院数学与系统科学研究院计算数学与科学工程计算研究所, 北京 100190

收稿日期:2016-01-18出版日期:2016-08-15发布日期:2016-09-08

CONSTRAINED NONCONFORMING ROTATED Q₁ ELEMENT METHODS OF THE REISSNER–MINDLIN PLATE PROBLEM

Hu Jun¹, Shi Zhongci²

1. LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, China;
2. LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China

Received:2016-01-18Online:2016-08-15Published:2016-09-08







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本文利用带约束非协调旋转Q₁元逼近Reissner-Mindlin板问题中旋度的两个分量，并分别选择Wilson元、双线性元和带约束非协调旋转Q₁元逼近挠度，相应地选取不连续的矢量值分片线性函数空间、最低阶旋转Raviart-Thomas元空间和矢量值分片常数函数空间为离散的剪应力空间，在矩形网格上构造了三个板元.通过证明一个离散的Korn不等式，并借助MITC4元的解构造了旋度、挠度和剪应力一个具有某种特殊且关键的可交换性的插值，再利用Helmholtz分解分析相容性误差，我们证明了这三个矩形元在能量范数意义下与板厚无关的一致最优收敛性.数值算例验证了我们的理论结果.
MR(2010)主题分类:
65N30
65N15
35J25

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[1] Arnold D N and Falk R S. A uniformly accurate finite element method for the Reissner-Mindlin plate[J]. SIAM J. Numer. Anal., 1989, 26:1276-1290. [2] Arnold D N, Brezzi F, Falk R S and Donatella Marini L. Locking-free Reissner-Mindlin elements without reduced integration[J]. Comput. Methods Appl. Mech. Engrg., 2007, 196:3660-3671. [3] Arnold D N and Falk R S. The boundary layer for the Reissner-Mindlin plate model[J]. SIAM J. Math. Math. Anal., 1990, 21:281-312. [4] Arnold D N and Falk R S. Analysis of a linear-linear-element for the Reissner-Mindlin model[J]. M3AS, 1997, 7:486-514. [5] Bathe K J and Brezzi F. On the convergence of a four-node plate bending element based on Mindlin-Reissner plate theory and a mixed interpolation. in:J.R. whiteman. ed., MAFELAP V(London, 1985) 491-503. [6] Bathe K J, Brezzi F, Fortin M. Mixed-interpolated elements for Reissner-Mindlin plates, Internat[J]. J. Numer. Methods Engrg, 1989, 28:1787-1801. [7] Bathe K J, Brezzi F and Fortin M. 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. [8] Bathe K J and Dvorkin E. A four-node plate bending element based on Mindlin-Reissner plate theory and a mixed interpolation[J]. Internat. J. Numer. Methods Engrg., 1985, 21:367-383. [9] Braess D and Peisker P. Uniform convergence of mixed interpolated elements for Reissner-Mindlin plates[J]. M2AN, 1992, 26:557-574. [10] Brezzi F, Bathe K J and Fortin M. Mixed-interpolated elements for Reissner-Mindlin plates[J]. Internat. J. Numer. Methods Engrg., 1989, 28:1787-1801. [11] Brezzi F and Fortin M. Numerical approximation of Reissner-Mindlin plates, Math. Comp., 1986, 47:151-158. [12] Brezzi F and Fortin M. Mixed and Hybrid Finite Element Methods. Springer-Verlag 1991. [13] Brezzi F, Fortin M and Stenberg R. Error analysis of mixed-interpolated elements for the Reissner-Mindlin plate[J]. M3AS, 1991, 1:125-151. [14] Duran R. The inf-sup condition and error estimates for Arnold-Falk plate bending element[J]. Numer. Math., 1991, 58:769-778. [15] Duran R and Liberman E. On mixed finite element methodes for the Reissner-Mindlin plate model[J]. Math. Comp., 1992, 58:561-573. [16] Duran R and Hernández E, Hervella-Nieto L, Liberman E and Rodr?guez R. Error estimates for lower-order isoparametric quadrilateral finite elements for plates[J]. SIAM J. Numer. Anal., 2003, 41:1751-1772. [17] Falk R S. Finite Elements for the Reissner-Mindlin Plate, Mixed Finite Elements, Compatibility Conditions, and Applications Volume 1939(2008)of the series Lecture Notes in Mathematics, pp. 195-232, Springer-Verlag. [18] Gastald L and Nochetto R H. Quasi-optimal pointwise error estimates for the Reissner-Mindlin plate[J]. SIAM J. Numer. Anal., 1991, 28:363-377. [19] Girault V, Raviart P A. Finite Element Methods for Navier-Stokes Equations. Springer Verlag, 1986. [20] 胡俊. 弹性力学问题四边形Locking Free元[D]. 中国科学院数学与系统科学研究院博士学位论文, 2004. [21] 胡俊, 满红婴, 石钟慈. 带约束非协调旋转Q1元在Stokes和几乎不可压弹性力学问题的应用[J]. 计算数学, 2005, 27:311-324. [22] Hu J, Ming P B and Shi Z C. Nonconforming quadrilateral rotated Q1 element for Reissner-Mindlin plate[J]. J. Comp. Math., 2003, 21:25-32. [23] Hu J and Shi Z C. Constrained quadrilateral nonconforming rotated Q1 element[J]. J. Comp. Math., 2005, 23:561-586. [24] Hu J and Shi Z C. Error analysis of quadrilateral Wilson element for the Reissner-Mindlin plate[J]. Comput. Methods Appl. Mech. Engrg., 2008, 197:464-475. [25] 明平兵. 非协调元vs闭锁问题[D]. 中国科学院数学与系统科学研究院博士学位论文, 1999. [26] Ming P B and Shi Z C. Two nonconforming quadrilateral elements for the Reissner-Mindlin plate[J]. M3AS, 2005, 15:1503-1518. [27] Ming P B and Shi Z C. Nonconforming rotated Q1 finite element for Reissner-Mindlin plate[J]. M3AS, 2001, 11:1311-1342. [28] Onte E, Zorate F and Flores F. A simple triangular element for thick and thin plate and shell analysis[J]. Internat. J. Numer. Methods Engrg., 1994, 37:2569-2582. [29] Park C and Sheen D. P1-nonconforming quadrilateral finite element methods for second-order elliptic problems[J]. SIAM J. Numer. Anal., 2003, 41:624-640. [30] Pitkäranta J and Süri M. Design principles and error analysis for reduced-shear plate bending finite elements[J]. Numer. Math., 1996, 75:223-266. [31] Rannarcher R and Turek S. Simple nonconforming quadrilateral Stokes element[J]. Numer. Methods Partial Differential Equations, 1992, 8:97-111. [32] Shi Z C. A convergence condition for quadrilateral Wilson element[J]. Numer. Math., 1984, 44:349-361. [33] Raviart P A and Thomas J M. A mixed finite element method for second order elliptic problems, Proc.Sympos. Mathematical Aspects of the Finite Element Method(Rome, 1975)[J]. Lecture Notes in Math., 1977, 606:292-315, Springer-Verlag. [34] Zhang Z and Zhang S. Wilson element for the Reissner-Mindlin plate[J]. Comput. Methods Appl. Mech. Engrg., 1994, 113:55-65.

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