CONTINUOUS FINITE ELEMENT METHODS FOR REISSNER-MINDLIN PLATE PROBLEM

doi:10.1016/S0252-9602(18)30760-4

Abstract

Abstract:

On triangle or quadrilateral meshes, two finite element methods are proposed for solving the Reissner-Mindlin plate problem either by augmenting the Galerkin formulation or modifying the plate-thickness. In these methods, the transverse displacement is approximated by conforming (bi)linear macroelements or (bi)quadratic elements, and the rotation by conforming (bi)linear elements. The shear stress can be locally computed from transverse displacement and rotation. Uniform in plate thickness, optimal error bounds are obtained for the transverse displacement, rotation, and shear stress in their natural norms. Numerical results are presented to illustrate the theoretical results.

Key words: Reissner-Mindlin plate, continuous element, triangle element, quadrilateral element, finite element method, uniform convergence

Huoyuan DUAN, Junhua MA. CONTINUOUS FINITE ELEMENT METHODS FOR REISSNER-MINDLIN PLATE PROBLEM[J].Acta mathematica scientia,Series B, 2018, 38(2): 450-470.

Trendmd

References

[1] Arnold D N, Falk R S. The boundary layer for the Reissner-Mindlin plate model. SIAM J Math Anal, 1990, 21:281-317
[2] Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods. New-York:Springer-Verlag, 1991
[3] Hu J, Shi Z C. Analysis of nonconforming rotated Q1 element for the Reissner-Mindlin plate problem//Industrial and Applied Mathematics in China, Ser Contemp Appl Math CAM, 10. Beijing:Higher Ed Press, 2009:101-111
[4] Hu J, Shi Z C. Analysis for quadrilateral MITC elements for the Reissner-Mindlin plate problem. Math Comp, 2009, 78:673-711
[5] Hu J, Shi Z C. Two lower order nonconforming quadrilateral elements for the Reissner-Mindlin plate. Sci China Ser A, 2008, 51:2097-2114
[6] Hu J, Shi Z C. Error analysis of quadrilateral Wilson element for Reissner-Mindlin plate. Comput Methods Appl Mech Engrg, 2008, 197:464-475
[7] Hu J, Shi Z C. Two lower order nonconforming rectangular elements for the Reissner-Mindlin plate. Math Comp, 2007, 76:1771-1786
[8] Hu J, Shi Z C. On the convergence of Weissman-Taylor element for Reissner-Mindlin plate. Int J Numer Anal Model, 2004, 1:65-73
[9] Hu J, Ming P B, Shi Z C. Nonconforming quadrilateral rotated Q1 element for Reissner-Mindlin plate. J Comput Math, 2003, 21:25-32
[10] Ming P B, Shi Z C. Optimal L2 error bounds for MITC3 type element. Numer Math, 2002, 91:77-91
[11] Ming P B, Shi Z C. Convergence analysis for quadrilateral rotated Q1 elements//Kananaskis A B. Scientific Computing and Applications, Adv Comput Theory Pract, 7. New-York:Nova Sci Publ, 2001:115-124
[12] Ming P B, Shi Z C. Nonconforming rotated Q1 element for Reissner-Mindlin plate. Math Models Methods Appl Sci, 2001, 11:1311-1342
[13] Ming P B, Shi Z C. Some low order quadrilateral Reissner-Mindlin plate elements//Recent Advances in Scientific Computing and Partial Differential Equations (Hong Kong, 2002), Contemp Math, 330. Providence, RI:Amer Math Soc, 2003:155-168
[14] Ming P B, Shi Z C. Two nonconforming quadrilateral elements for the Reissner-Mindlin plate. Math Models Methods Appl Sci, 2005,15:1503-1517
[15] Ming P B, Shi Z C. Analysis of some low order quadrilateral Reissner-Mindlin plate elements. Math Comp, 2006, 75:1043-1065
[16] Guo Y H, Xie X P. Uniform analysis of a stabilized hybrid finite element method for Reissner-Mindlin plates. Sci China Math, 2013, 56:1727-1742
[17] Carstensen C, Xie X P, Yu G Z, Zhou T X. A priori and a posteriori analysis for a locking-free low order quadrilateral hybrid finite element for Reissner-Mindlin plates. Comput Methods Appl Mech Engrg, 2001,200:1161-1175
[18] Yu G Z, Xie X P, Zhang X. Parameter extension for combined hybrid finite element methods and application to plate bending problems. Appl Math Comput, 2010,216:3265-3274
[19] Guo G H, Xie X P. On choices of stress modes for lower order quadrilateral Reissner-Mindlin plate elements. Numer Math J Chin Univ (Engl Ser), 2006, 15:120-126
[20] Xie X P. From energy improvement to accuracy enhancement:improvement of plate bending elements by the combined hybrid method. J Comput Math, 2004,22:581-592
[21] Zhou T X, Xie X P. A combined hybrid finite element method for plate bending problems. J Comput Math, 2003,21:347-356
[22] Duan H Y, Liang G P. A locking-free Reissner-Mindlin quadrilateral element. Math Comp, 2004, 73:1655-1671
[23] Duan H Y. A finite element method for plate problem. Math Comp, 2014, 83:701-733
[24] Duan H Y, Liang G P. Analysis of some stabilized low-order mixed finite element methods for ReissnerMindlin plates. Comput Methods Appl Mech Engrg, 2001,191:157-179
[25] Duan H Y, Liang G P. An improved Reissner-Mindlin triangular element. Comput Methods Appl Mech Engrg, 2002, 191:2223-2234
[26] Duan H Y, Liang G P. Mixed and nonconforming finite element approximations of Reissner-Mindlin plates. Comput Methods Appl Mech Engrg, 2003, 192:5265-5281
[27] Arnold D N, Brezzi F, Falk R S, Marini L D. Locking-Free Reissner-Mindlin elements without reduced integration. Comput Methods Appl Mech Engrg, 2007, 196:3660-3671
[28] Arnold D N, Falk R S. A uniformly accurate finite element method for the Reissner-Mindlin plate. SIAM J Numer Anal, 1989, 26:1276-1290
[29] Behrens E M, Guzmán J. A new family of mixed methods for the Reissner-Mindlin plate model based on a system of first-order equations. J Sci Comput, 2011, 49:137-166
[30] Bosing P R, Madureira A L, Mozolevski I. A new interior penalty discontinuous Galerkin method for the Reissner-Mindlin model. Math Models Methods Appl Sci, 2010, 20:1343-1361
[31] Zhou T X. The partial projection method in the finite element discretization of the Reissner-Mindlin plate. J Comput Math, 1995, 13:172-191
[32] Schöberl J, Stenberg R. Multigrid methods for a stabilized Reissner-Mindlin plate formulation. SIAM J Numer Anal, 2009, 47:2735-2751
[33] Destuynder P. Mathematical Analysis of Thin Shell Problems. Paris:Masson, 1991
[34] Ciarlet P G. The Finite Element Method for Elliptic Problems. Amsterdam:North-Holland, 1978
[35] Clément P. Approximation by finite elements using local regularization. RAIRO Anal Numer, 1975, 8:77-83
[36] Girault V, Raviart P A. Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Berlin:Springer-Verlag, 1986
[37] Duan H Y, Liang G P. Nonconforming elements in least-squares mixed finite element methods. Math Comp, 2004, 73:1-18