A Posteriori Error Control at Numerical Solution of Plate Bending Problem

Abstract:

The classical approach to a posteriori error control considered in this paper bears on the counter variational principles of Lagrange and Castigliano. Its efficient implementation for problems of mechanics of solids assumes obtaining equilibrated stresses/resultants which, at the same time, are sufficiently close to the exact values. Besides, it is important that computation of the error bound with the use of such stresses/resultants would be cheap in respect to the arithmetic work. Following these guide lines, we expand the preceding results for elliptic partial differential equations and theory elasticity equations upon the problem of thin plate bending. We obtain guaranteed a posteriori error bounds of simple forms for solutions by the finite element method and discuss the algorithms of linear complexity for their computation. The approach of the paper also allows to improve some known general a posteriori estimates by means of arbitrary not equilibrated stress fields.

• Periodical: Applied Mechanics and Materials (Volumes 725-726)
• Pages: 674-680
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.725-726.674
• Online since: January 2015
• Authors: V.G. Korneev*
• Keywords: Accuracy of Approximations, Boundary Value Problems, Mechanics of Continuum, Posteriory Error Bounds, Theory Elasticity

References

[1] Ladev`eze, P., Leguillon, D. Error estimate procedure in the finite element method and applications (1983) SIAM J. Numer. Anal., 20, p.485–509.

DOI: 10.1137/0720033

[2] Ainsworth, M., Oden, J. T. A posteriori estimation in finite element analysis (2000) John Wiley & Sons, Inc., 240 p.

[3] Luce, R., Wohlmuth, B. A local a posteriori error estimator based on equilibrated fluxes (2004) SIAM J. Num. Anal., 42, pp.1394-1414.

DOI: 10.1137/s0036142903433790

[4] Vejchodsky, T. Local a posteriori error estimator based on the hypercircle method. In: Proc. of the European Congress on Computational Method in Applied Mechanics and Engrg. ECCOMAS, Yavaskyla, Finland, (2004).

[5] P. Neittaanmaki, P., Repin, S. I. Reliable methods for computer simulation Error control and a posteriori estimates (2004) Elsevier, 305 p.

[6] M. Ainsworth, M., L., Demkowicz, L., Kim, C-W. Analysis of the equilibrated residual method for а posterioti estimation on meshes with hanging nodes (2007) Computer Meth. Appl. Math. Engrg., 196 , 37-40, pp.3493-3507.

DOI: 10.1016/j.cma.2006.10.020

[7] Braess, D., Schoberl, J. Equilibrated residual error estimator for Maxswell's equations (2008) Math. Comp., 77, pp.651-672.

DOI: 10.1090/s0025-5718-07-02080-7

[8] Dorfler, W. A convergent adaptive algorithm for Poisson's equation (1996) SIAM J. Numer. Anal., 33, 3, pp.1106-1124.

DOI: 10.1137/0733054

[9] Dorfler, W., Rumpf, M. An adaptive strategy for elliptic problems including a posteriori controlled boundary approximation algorithm (1998) Mathematics of computation, 67, 224, pp.1361-1382.

DOI: 10.1090/s0025-5718-98-00993-4

[10] Cochez-Dhondt, S., Nicaise, S. A posteriori error estimators based on equilibrated fluxes (2010) Computational methods in applied mathematics, 10, 1, p.49–68.

DOI: 10.2478/cmam-2010-0002

[11] Rabus, R. A natural adaptive nonconforming FEM of quasi-optimal complexity (2010) Computational methods in applied mathematics, 10, 3, pp.315-325.

DOI: 10.2478/cmam-2010-0018

[12] Anufriev, I. E., Korneev, V. G., Kostylev, V. S. Exactly equilibrated fields, can they be efficiently used for a posteriori error estimation? (2006).

[13] Korneev, V. G. Prostyye algoritmy vychisleniya aposteriornyh ocenok chislennyh reshenii ellipticheskih yravnenii [Simple algorithms for computation of a posteriori bounds for numerical solutions of elliptic equations] (2011).

[15] Verfurt, R. A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques (1996) Wiley, 127 p.

[16] P. Neittaanmaki, P., Repin, S. I. A posteriori error estimates for boundary-value problems related to the biharmonic operator (2001) East-West J. Numer. Math., 2, p.157–178.

DOI: 10.1515/jnma.2001.157

[17] Adjerid, S. A posteriori error estimates for fourth-order elliptic problems (2002) Comput. Methods Appl. Mech. Eng., 191, p.2539–2559.

DOI: 10.1016/s0045-7825(01)00412-1

[18] Liu, K. A Gradient Recovery-based a Posteriori Error Estimators for the Ciarlet - Raviart Formulation of the Second Biharmonic Equations (2007) Applied Mathematical Sciences, 1, 21, p.997 – 1007.

[19] Beirao da Veiga, Niiranen, J., Stenberg, R. L. A posteriori error estimates for the Morley plate bending element (2007) Numer. Math., 106, p.165–179.

DOI: 10.1007/s00211-007-0066-1

[20] Hansbo, P., Larson, M. G. A posteriori error estimates for continuous/discontinuous Galerkin approximations of the Kirchhoff–Love plate (2008) Preprint 2008: 10, Chalmers University of Technology, Geoteborg, 19 p.

DOI: 10.1007/s00466-015-1204-8

[21] Georgoulis, E. H., Houston, P., Virtanen, J. An a posteriori error indicator for discontinuous Galerkin approximations of fourth-order elliptic problems (2011) IMA Journal of Numerical Analysis, 31, p.281–298.

DOI: 10.1093/imanum/drp023

[22] Korneev, V. G. Numerical solution in stresses of problems of shell theory using oblique-angled meshes (1981) USSR Computational Mathematics and Mathematical Physics, 21, 2, pp.184-194.

DOI: 10.1016/0041-5553(81)90018-5

[23] Mikhlin, S.G. Variatsionnye methody v mathematicheskoi fizike [Variational methods in mathematical physics] (1970) Moskva , Nauka, 512 p. (rus).

[24] Abovskii, N. P., Andreyev, N. P., Deruga, A. P. Variacionnye principy v teorii uprugosti i teorii obolochek [Variational principles in theory elasticity and shell theory] (1978) Moskva, Nauka, 288 p. (rus).

[25] Korneev, V. G. Supershodimost' reshenii metoda konechnyh elementov v setochnyh normah [Super-convergence of finite element method solutions in mesh norms] (1982).