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Discretization error for the discrete Kirchhoff plate finite element approximation. (English) Zbl 1296.65154
Summary: We provide in this work the discretization error estimates that can guide an adaptive mesh refinement for the Discrete Kirchhoff plate finite elements. The proposed developments are built upon the concept of error estimates for classical elasticity and adapted to suit the Kirchhoff plate finite elements. We give a detailed illustration of the proposed procedures for the Discrete Kirchhoff triangular plate element, along with several different possibilities for constructing the enhancement of test space needed for error estimates. The first novelty concerns the consistent displacement field in terms of the third order polynomial for the Discrete Kirchhoff triangle, whereas the second novelty is the use of the Argyris triangle with fifth order polynomials for constructing the enhanced test for error estimates. We compare the latter against several alternatives that can be used for Kirchhoff plates. The results of numerical examples are given to illustrate the effectiveness of proposed discretization error estimates.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74S05 Finite element methods applied to problems in solid mechanics
74K20 Plates
Software:
Gmsh
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[1] Ainsworth, M.; Oden, J. T., A unified approach to a posteriori error estimation using element residual methods, Numer. Math., 65, 1, 23-50, (1993) · Zbl 0797.65080
[2] Ainsworth, M.; Oden, J. T., A posteriori error estimation in finite element analysis, (2000), Wiley · Zbl 1008.65076
[3] S.M. Allesandrini, D.N. Arnold, R.S. Falk, A.L. Madureira, Derivation and justification of plate models by variational methods, in: M. Fortin (Ed.), CRM/AMS Proceedings ‘Plates and Shells’, 1999, pp. 1-21. · Zbl 0958.74033
[4] Argyris, J. H.; Fried, I.; Scharpf, D. W., The TUBA family of plate elements for the matrix displacement method, Aeronaut. J. R. Aeronaut. Soc., 72, 701-709, (1968)
[5] Babuska, I.; Osborn, J.; Pitkaranta, J., Analysis of mixed methods using mesh dependent norms, Math. Comput., 35, 152, 1039-1062, (1980) · Zbl 0472.65083
[6] Babuska, I.; Rheinboldt, W. C., Error estimates for adaptive finite element computations, SIAM J. Numer. Anal., 15, 736-754, (1978) · Zbl 0398.65069
[7] Batoz, J. L., An explicit formulation for an efficient triangular plate-bending element, Int. J. Numer. Methods Eng., 18, 1077-1089, (1982) · Zbl 0487.73087
[8] Batoz, J. L.; Bathe, K. J.; Ho, L. W., A study of three-node triangular plate bending elements, Int. J. Numer. Methods Eng., 15, 1771-1812, (1980) · Zbl 0463.73071
[9] R. Becher, R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods, in: A. Iserles (Ed.), Acta Numerica, Cambridege University Press, 2001, pp. 1-102. · Zbl 1105.65349
[10] Benoit, C.; Coorevits, P.; Pelle, J. P., Error estimation for plate structures: application using the DKT element, Eng. Comput., 16, 5, 584-600, (1999) · Zbl 0986.74066
[11] Bernadou, M., Finite element method for the shell problem, (1996), Wiley
[12] Bohinc, U.; Ibrahimbegovic, A.; Brank, B., Model adaptivity for finite element analysis of thin or thick plates based on equilibrated boundary stress resultants, Eng. Comput., 26, 1/2, 69-99, (2009) · Zbl 1257.74147
[13] Braess, D.; Sauter, S.; Schwab, C., On the justification of plate models, J. Elast., 103, 53-71, (2011) · Zbl 1237.74127
[14] Brank, B., On boundary layer in the Mindlin plate model: levy plates, Thin-Walled Struct., 46, 5, 451-465, (2008)
[15] Brank, B.; Korelc, J.; Ibrahimbegovic, A., Nonlinear shell problem formulation accounting for through-the-thickness stretching and its finite element implementation, Comput. Struct., 80, 699-717, (2002)
[16] F. Brezzi, Nonstandard finite elements for fourth order elliptic problems, in: R. Glowinski, E.Y. Rodin, O.C. Zienkiewicz (Eds.), ‘Energy methods in FE Analysis, 1979, pp. 193-211 (Chapter 10).
[17] Brink, U.; Stein, E., A posteriori error estimators in large strain elasticity using equilibrated Neumann problems, Comput. Methods Appl. Mech. Eng., 161, 77-101, (1998) · Zbl 0943.74062
[18] Ciarlet, P. G., Basic error estimates for elliptic problems, Handbook of Numerical Analysis, vol. II, (1991), Elsevier · Zbl 0875.65086
[19] R.W. Clough, C.A. Felippa, A refined quadrilateral element for analysis of plate bending, in: Proc. Conf. on Matrix Methods in Structural Mechanics, WPAFB, Ohio, 1965, pp. 399-440.
[20] R.W. Clough, J.L. Tocher, Finite element stiffness matrices for analysis of plate bending, in: Proc. Conf. on Matrix Methods in Structural Mechanics, WPAFB, Ohio, 1965, pp. 515-545.
[21] Da Veiga, L. B.; Niranen, J.; Stenberg, R., A posteriori error analysis for the Morley plate element with general boundary conditions, Numer. Math., 106, 2, 165-179, (2007) · Zbl 1110.74050
[22] Geuzaine, C.; Remacle, J.-F., Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities, Int. J. Numer. Methods Eng., 79, 11, 1309-1331, (2009) · Zbl 1176.74181
[23] Hu, J.; Shi, Z., A new a posteriori error estimate for the Morley element, Numer. Math., 112, 1, 25-40, (2009) · Zbl 1169.74646
[24] Hughes, T. J.R., The finite element method: linear static and dynamic finite element analysis, (2000), Dover publications · Zbl 1191.74002
[25] Ibrahimbegovic, A., Plate quadrilateral finite element with incompatible modes, Commun. Appl. Numer. Methods, 8, 497-504, (1992) · Zbl 0757.73056
[26] Ibrahimbegovic, A., Quadrilateral finite elements for analysis of thick and thin plates, Comput. Methods Appl. Mech. Eng., 110, 195-209, (1993) · Zbl 0845.73070
[27] Ibrahimbegovic, A., Nonlinear solid mechanics: theoretical formulations and finite element solution methods, (2009), Springer Berlin · Zbl 1168.74002
[28] Ibrahimbegovic, A.; Frey, F., Efficient implementation of stress resultant plasticity in analysis of Reissner-Mindlin plates, Int. J. Numer. Methods Eng., 36, 303-320, (1993) · Zbl 0825.73841
[29] Ladevèze, P.; Leguillon, D., Error estimate procedure in the finite element method and applications, SIAM J. Numer. Anal., 20, 3, 485-509, (1983) · Zbl 0582.65078
[30] Ladevèze, P.; Pelle, J. P., Mastering calculations in linear and nonlinear mechanics, (2005), Springer · Zbl 1077.74001
[31] Lee, K. H.; Lim, G. T.; Wang, C. M., Thick levy plates re-visited, Int. J. Solids Struct., 39, 127-144, (2002) · Zbl 1035.74037
[32] Lovadina, C.; Stenberg, R., A posteriori error analysis of the linked interpolation technique for plate bending problems, SIAM J. Numer. Anal., 43, 5, 2227, (2005) · Zbl 1098.74055
[33] Morley, L. S.D., Skew plates and structures, (1963), Pergamon press · Zbl 0124.17704
[34] Reddy, J. N., Mechanics of laminated composite plates, (1998), CRC press Theory and analysis
[35] Stein, E., Error controlled adaptive finite elements in solid mechanics, (2003), Wiley
[36] Stein, E.; Ohnimus, S., Anisotropic discretization - and model-error estimation in solid mechanics by local Neumann problems, Comput. Methods Appl. Mech. Eng., 176, 363-385, (1999) · Zbl 0954.74072
[37] R.L. Taylor, S. Govindjee, Solution of clamped rectangular plate problems, Report UCB/SEMM-2002/09, 2002. · Zbl 1087.74040
[38] Yunus, S. M.; Pawlak, T. P.; Wheeler, M. J., Application of the Zienkiewicz-Zhu error estimator for plate and shell analysis, Int. J. Numer. Methods Eng., 29, 6, 1281-1298, (1990) · Zbl 0713.73087
[39] Zienkiewicz, O. C.; Taylor, R. L., Finite element method, (2000), Elseiver · Zbl 0991.74002
[40] Zienkiewicz, O. C.; Zhu, J. Z., The superconvergent patch recovery and a posteriori error estimates. part 1: the recovery technique, Int. J. Numer. Methods Eng., 33, 1331-1364, (1992) · Zbl 0769.73084
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