A Locking-Free and Reduction-Free Conforming Finite Element Method for the Reissner-Mindlin Plate on Rectangular Meshes

Abstract

A family of conforming finite elements are designed on rectangular grids for solving the Reissner-Mindlin plate equation. The rotation is approximated by $C^1-Q_{k+1}$ in one direction and $C^0-Q_k$ in the other direction finite elements. The displacement is approximated by $C^1-Q_{k+1,k+1}$. The method is locking-free without using any projection/reduction operator. Theoretical proof and numerical confirmation are presented.

Keywords

Locking-free / Reissner-Mindlin equation / Finite element / Rectangular mesh / 65N15 / 65N30 / 65M60 / 76M10

References

[1] Arnold DN, Brezzi F, Falk RS, Marini LD. Locking-free Reissner-Mindlin elements without reduced integration. Comput. Methods Appl. Mech. Eng.. 2007, 196(37/38/39/40): 3660-3671

[2] Arnold DN, Brezzi F, Marini D. A family of discontinuous Galerkin finite elements for the Reissner-Mindlin plate. J. Sci. Comput.. 2005, 22: 25-45

[3] Arnold DN, Falk RS. A uniformly accurate finite element method for the Reissner-Mindlin plate. SIAM J. Numer. Anal.. 1989, 26: 1276-1290

[4] Bathe, K.-J., Brezzi, F.: On the convergence of a four-node plate bending element based on Mindlin-Reissner plate theory and a mixed interpolation. In: The Mathematics of Finite Elements and Applications. V (Uxbridge, 1984), pp. 491–503. Academic Press, London (1985)

[5] Bathe, K.-J., Brezzi, F.: A simplified analysis of two plate bending elements—the MITC4 and MITC9 elements. In: Numerical Techniques for Engineering Analysis and Design, vol. 1. Martinus Nijhoff, Amsterdam (1987)

[6] Bathe K-J, Brezzi F, Cho SW. The MITC7 and MITC9 plate bending elements. Comput. Struct.. 1989, 32: 797-841

[7] Bathe K-J, Dvorkin EN. A four-node plate bending element based on Mindlin/Reissner plate theory and mixed interpolation. Int. J. Numer. Methods Eng.. 1985, 21: 367-383

[8] Bathe K-J, Dvorkin EN. A formulation of general shell elements—the use of mixed interpolation of tensorial components. Int. J. Numer. Methods Eng.. 1986, 22: 697-722

[9] Beirao da Veiga L, Buffa A, Lovadina C, Martinelli M, Sangalli G. An isogeometric method for the Reissner-Mindlin plate bending problem. Comput. Methods Appl. Mech. Eng.. 2012, 209(212): 45-53

[10] Brezzi F, Fortin M, Stenberg R. Error analysis of mixed-interpolated elements for Reissner-Mindlin plates. Math. Models Methods Appl. Sci.. 1991, 1(2): 125-151

[11] Durán R, Hernández E, Hervella-Nieto L, Liberman E, Rodríguez R. Error estimates for low-order isoparametric quadrilateral finite elements for plates. SIAM J. Numer. Anal.. 2003, 41: 1751-1772

[12] Durán R, Hervella-Nieto L, Liberman E, Rodríguez R, Solomin J. Approximation of the vibration modes of a plate by Reissner-Mindlin equations. Math. Comp.. 1999, 68(228): 1447-1463

[13] Durán R, Liberman E. On mixed finite element methods for the Reissner-Mindlin plate model. Math. Comp.. 1992, 58(198): 561-573

[14] Durán R, Liberman E. On the convergence of a triangular mixed finite element method for Reissner-Mindlin plates. Math. Models Methods Appl. Sci.. 1996, 6: 339-352

[15] Falk RS, Tu T. Locking-free finite elements for the Reissner-Mindlin plate. Math. Comp.. 2000, 69(231): 911-928

[16] Hansbo P, Heintz D, Larson M. A finite element method with discontinuous rotations for the Mindlin-Reissner plate model. Comput. Methods Appl. Mech. Eng.. 2011, 200: 638-648

[17] Hu J, Huang Y, Zhang S. The lowest order differentiable finite element on rectangular grids. SIAM Num. Anal.. 2011, 49(4): 1350-1368

[18] Hu J, Jiang L, Shi Z. New a posteriori error estimate and quasi-optimal convergence of the adaptive nonconforming Wilson element. J. Comput. Appl. Math.. 2014, 265: 173-186

[19] Hu J, Ming P, Shi Z. Nonconforming quadrilateral rotated Q1 element for Reissner-Mindlin plate. J. Comput. Math.. 2003, 21: 25-32

[20] Hu J, Shi Z-C. On the convergence of Weissman-Taylor element for Reissner-Mindlin plate. Int. J. Numer. Anal. Model.. 2004, 1(1): 65-73

[21] Hu J, Shi Z-C. Two lower order nonconforming rectangular elements for the Reissner-Mindlin plate. Math. Comp.. 2007, 76(260): 1771-1786

[22] Hu J, Shi ZC. Error analysis of quadrilateral Wilson element for Reissner-Mindlin plate. Comput. Methods Appl. Mech. Eng.. 2008, 197(6/7/8): 464-475

[23] Hu J, Shi ZC. Two lower order nonconforming quadrilateral elements for the Reissner-Mindlin plate. Sci. China Ser. A. 2008, 51(11): 2097-2114

[24] Hu J, Shi ZC. Analysis for quadrilateral MITC elements for the Reissner-Mindlin plate problem. Math. Comp.. 2009, 78(266): 673-711

[25] Hu J, Shi ZC. A new a posteriori error estimate for the Morley element. Numer. Math.. 2009, 112(1): 25-40

[26] Hu, J., Shi, Z.-C.: Analysis of nonconforming rotated Q1 element for the Reissner-Mindlin plate problem. Industrial and Applied Mathematics in China, 101–111, Ser. Contemp. Appl. Math. CAM, 10, Higher Ed. Press, Beijing (2009)

[27] Hu J, Shi ZC. The best L² norm error estimate of lower order finite element methods for the fourth order problem. J. Comput. Math.. 2012, 30(5): 449-460

[28] Hu J, Shi ZC. A lower bound of the L² norm error estimate for the Adini element of the biharmonic equation. SIAM J. Numer. Anal.. 2013, 51(5): 2651-2659

[29] Hu J, Shi ZC. Constrained nonconforming rotated Q1 element methods for the Reissner-Mindlin plate problem. Math. Numer. Sin.. 2016, 38(3): 325-340

[30] Hu J, Shi ZC, Xu J. Convergence and optimality of the adaptive Morley element method. Numer. Math.. 2012, 121(4): 731-752

[31] Hu J, Zhang S. The minimal conforming $H^k$ finite element spaces on $R^n$ rectangular grids. Math. Comp.. 2015, 84(292): 563-579

[32] Huang J, Guo L, Shi ZC. Vibration analysis of Kirchhoff plates by the Morley element method. J. Comput. Appl. Math.. 2008, 213(1): 14-34

[33] Huang Y, Zhang S. A lowest order divergence-free finite element on rectangular grids. Front. Math. China. 2011, 6(2): 253-270

[34] Huang Y, Zhang S. Supercloseness of the divergence-free finite element solutions on rectangular grids. Commun. Math. Stat.. 2013, 1: 143-162

[35] Hughes TJR, Tezuyar TE. Finite elements based upon Mindlin plate theory with particular reference to the four node blinear isoparamtric element. J. Appl. Mech. Eng.. 1981, 48: 587-598

[36] Lai J, Huang J, Shi ZC. A lumped mass finite element method for vibration analysis of elastic plate-plate structures. Sci. China Math.. 2010, 53(6): 1453-1474

[37] Li H, Ming P, Shi ZC. The quadratic Specht triangle. J. Comput. Math.. 2020, 38(1): 103-124

[38] Lovadina C, Marini D. Nonconforming locking-free finite elements for Reissner-Mindlin plates. Comput. Methods Appl. Mech. Eng.. 2006, 195: 3448-3460

[39] Mao S, Nicaise S, Shi Z-C. Error estimates of Morley triangular element satisfying the maximal angle condition. Int. J. Numer. Anal. Model.. 2010, 7(4): 639-655

[40] Mao S, Shi Z. High accuracy analysis of two nonconforming plate elements. Numer. Math.. 2009, 111(3): 407-443

[41] Ming P, Shi Z-C. Nonconforming rotated Q1 element for Reissner-Mindlin plate. Math. Models Methods Appl. Sci.. 2001, 11: 1311-1342

[42] Ming P, Shi Z-C. Two nonconforming quadrilateral elements for the Reissner-Mindlin plate. Math. Models Methods Appl. Sci.. 2005, 15(10): 1503-1517

[43] Ming P, Shi Z-C. Analysis of some low order quadrilateral Reissner-Mindlin plate elements. Math. Comp.. 2006, 75(255): 1043-1065

[44] Perugia I, Scapolla T. Optimal rectangular MITC finite elements for Reissner-Mindlin plates. Numer. Methods Partial Differential Equations. 1997, 13(5): 575-585

[45] Scott LR, Zhang S. Higher-dimensional nonnested multigrid methods. Math. Comp.. 1992, 58(198): 457-466

[46] Shi ZC. On the convergence of the incomplete biquadratic nonconforming plate element. Math. Numer. Sinica. 1986, 8(1): 53-62

[47] Shi ZC. Convergence of the TRUNC plate element. Comput. Methods Appl. Mech. Eng.. 1987, 62(1): 71-88

[48] Shi ZC. The space of shape functions of an energy-orthogonal plate element. J. China Univ. Sci. Tech.. 1990, 20(2): 127-131

[49] Shi ZC. Error estimates for the Morley element. Math. Numer. Sinica. 1990, 12(2): 113-118

[50] Shi Z, Chen Q. An efficient rectangular plate element. Sci. China Ser. A. 2001, 44(2): 145-158

[51] Shi ZC, Xie Z-H. Substructure preconditioners for nonconforming plate elements. J. Comput. Math.. 1998, 16(4): 289-304

[52] Shi ZC, Xu X. A V-cycle multigrid method for TRUNC plate element. Comput. Methods Appl. Mech. Eng.. 2000, 188(1/2/3): 483-493

[53] Shi ZC, Xu X-J. The mortar element method for a nonlinear biharmonic equation. J. Comput. Math.. 2005, 23(5): 537-560

[54] Stenberg R, Suri M. An hp error analysis of MITC plate elements. SIAM J. Numer. Anal.. 1997, 34: 544-568

[55] Wang M, Shi ZC, Xu J. Some n-rectangle nonconforming elements for fourth order elliptic equations. J. Comput. Math.. 2007, 25(4): 408-420

[56] Ye X. A rectangular element for the Reissner-Mindlin plate. Numer. Methods Partial Differential Equations. 2000, 16(2): 184-193

[57] Ye X, Zhang S, Zhang Z. A locking-free weak Galerkin finite element method for Reissner-Mindlin plate on polygonal meshes. Comput. Math. Appl.. 2020, 80(5): 906-916

[58] Zhang S. A family of $Q_{k+1, k}\times Q_{k, k+1}$ divergence-free finite elements on rectangular grids. SIAM J. Numer. Anal.. 2009, 47: 2090-2107

[59] Zhang S. On the full $C_1$-$Q_k$ finite element spaces on rectangles and cuboids. Adv. Appl. Math. Mech.. 2010, 2: 701-721

[60] Zhang, S.: Robust Falk-Neilan finite elements for the Reissner-Mindlin plate. Commun. Appl. Math. Comput. 5, 1697–1712 (2023)

Funding

National Natural Science Foundation of China(12131005)

RIGHTS & PERMISSIONS

Shanghai University