Comptes Rendus
Numerical Analysis
Estimates of the modeling error for the Kirchhoff–Love plate model
[Estimation de l'erreur de modélisation pour le modèle de plaque de Kirchhoff–Love]
Comptes Rendus. Mathématique, Volume 348 (2010) no. 17-18, pp. 1039-1043.

Dans ce travail on considère le modèle de Kirchhoff–Love pour approcher les problèmes de plaques minces sous certaines conditions. Nous présentons des majorants d'erreur calculables. La borne de l'erreur relative converge comme O(h1/2) en terme du paramètre d'épaisseur h pourvu que la KL solution ait un peu plus de régularité.

In this Note we consider the Kirchhoff–Love model for approximating problems in linear elasticity on thin plates under certain hypotheses. We will present computable error majorants for the arising modelling error. The majorant for the relative error converges with a rate O(h1/2) in the thickness parameter h provided that the KL solution possesses extra reqularity.

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DOI : 10.1016/j.crma.2010.09.004

Sergey Repin 1 ; Stefan A. Sauter 2

1 St. Petersburg Department of V.A. Steklov Institute of Mathematics, Fontanka 27, 191 023 St. Petersburg, Russia
2 Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
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Sergey Repin; Stefan A. Sauter. Estimates of the modeling error for the Kirchhoff–Love plate model. Comptes Rendus. Mathématique, Volume 348 (2010) no. 17-18, pp. 1039-1043. doi : 10.1016/j.crma.2010.09.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.09.004/

[1] A.L. Alessandrini; D.N. Arnold; R.S. Falk; A.L. Madureira Derivation and justification of plate models by variational methods (M. Fortin, ed.), Plates and Shells, Quebec, 1996, CRM Proceedings and Lecture Notes, vol. 21, American Mathematical Society, Providence, RI, 1996, pp. 1-20

[2] D. Braess, S. Sauter, C. Schwab, On the justification of plate models, Journal of Elasticity, in press.

[3] P.G. Ciarlet Mathematical Elasticity, vol. II: Theory of Plates, North-Holland, Amsterdam, 1997

[4] P.G. Ciarlet; P. Destuynder A justification of a nonlinear model in plate theory, Comput. Methods Appl. Mech. Engrg., Volume 17/18 (1979), pp. 227-258

[5] M. Dauge; I. Gruais Asymptotics of arbitrary order in thin elastic plates and optimal estimates for the Kirchhoff–Love model, Asymptot. Anal., Volume 13 (1996), pp. 167-197

[6] P. Destuynder Comparaison entre les modeles tridimensionnels et bidimensionnels de plaque en élasticité, RAIRO Anal. Numer., Volume 15 (1981), pp. 331-369

[7] G. Kirchhoff Über das Gleichgewicht und die Bewegung einer elastischen Scheibe, J. Reine Angew. Math., Volume 40 (1850), pp. 51-58

[8] D. Morgenstern Herleitung der Plattentheorie aus der dreidimensionalen Elastizitätstheorie, Arch. Ration. Mech. Anal., Volume 4 (1959), pp. 145-152

[9] J.C. Paumier; A. Raoult Asymptotic consistency of the polynomial approximation in the linearized plate theory. Application to the Reissner–Mindlin model, Lyon, 1995 (ESAIM Proc.), Volume vol. 2, Soc. Math. Appl. Indust., Paris (1997), pp. 203-213

[10] A. Raoult Construction d'un modèle d'évolution de plaques avec terme d'inertie de rotation, Ann. Mat. Pura Appl., Volume CXXXIX (1985), pp. 361-400

[11] S. Repin Estimates for errors in two-dimensional models of elasticity theory, J. Math. Sci. (N. Y.), Volume 106 (2001), pp. 3027-3041

[12] S. Repin A Posteriori Estimates for Partial Differential Equations, Walter de Gruyter, Berlin, 2008

[13] S. Repin, S. Sauter, Computable estimates of the modeling error related to Kirchhoff–Love plate model, Preprint Series, No. 11-2009, Inst. für Mathematik, University of Zurich, 2009.

[14] B.A. Shoikhet On asymptotically exact equations of thin plates of complex structure, J. Appl. Math. Mech., Volume 37 (1974) no. 1973, pp. 867-877

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