Numerical analysis of the generalized von Kármán equations

Philippe G. Ciarlet; Liliana Gratie; Srinivasan Kesavan

doi:10.1016/j.crma.2005.09.031

Numerical Analysis

Numerical analysis of the generalized von Kármán equations

Presented by: Alain Bensoussan

Philippe G. Ciarlet ¹; Liliana Gratie ²; Srinivasan Kesavan ³

¹ Department of Mathematics, City University of Hong Kong, 83, Tat Chee Avenue, Kowloon, Hong Kong
² Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, 83, Tat Chee Avenue, Kowloon, Hong Kong
³ The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai-600113, India

Comptes Rendus. Mathématique, Volume 341 (2005) no. 11, pp. 695-699.

Abstract
Résumé

The ‘generalized von Kármán equations’ constitute a mathematical model for a nonlinearly elastic plate subjected to boundary conditions ‘of von Kármán type’ only on a portion of its lateral face, the remaining portion being free. We establish here the convergence of a conforming finite element approximation to these equations. The proof relies in particular on a compactness method due to J.-L. Lions and on Brouwer's fixed point theorem.

Les « équations de von Kármán généralisées » constituent un modèle mathématique d'une plaque non linéairement élastique soumise à des conditions aux limites « de von Kármán » sur une partie seulement de sa face latérale, la partie restante étant libre. On établit ici la convergence de la solution approchés de ces équations, obtenue par une méthode d'élements finis conformes. La démonstration repose en particulier sur une méthode de compacité due à J.-L. Lions et sur le théorème du point fixe de Brouwer.

Received: 2005-09-05
Accepted: 2005-09-14
Published online: 2005-11-02

DOI: 10.1016/j.crma.2005.09.031

Author's affiliations:

Philippe G. Ciarlet ¹; Liliana Gratie ²; Srinivasan Kesavan ³

¹ Department of Mathematics, City University of Hong Kong, 83, Tat Chee Avenue, Kowloon, Hong Kong
² Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, 83, Tat Chee Avenue, Kowloon, Hong Kong
³ The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai-600113, India

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     author = {Philippe G. Ciarlet and Liliana Gratie and Srinivasan Kesavan},
     title = {Numerical analysis of the generalized von {K\'arm\'an} equations},
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     pages = {695--699},
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     doi = {10.1016/j.crma.2005.09.031},
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Philippe G. Ciarlet; Liliana Gratie; Srinivasan Kesavan. Numerical analysis of the generalized von Kármán equations. Comptes Rendus. Mathématique, Volume 341 (2005) no. 11, pp. 695-699. doi : 10.1016/j.crma.2005.09.031. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.09.031/

References
Cited by

[1] P.G. Ciarlet The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978

[2] P.G. Ciarlet A justification of the von Kármán equations, Arch. Rational Mech. Anal., Volume 73 (1980), pp. 349-389

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

[4] P.G. Ciarlet; L. Gratie Generalized von Kármán equations, J. Math. Pures Appl., Volume 80 (2001), pp. 263-279

[5] P.G. Ciarlet, L. Gratie, S. Kesavan, On the generalized von Kármán equations and their approximation, in preparation

[6] P.G. Ciarlet; L. Gratie; N. Sabu An existence theorem for generalized von Kármán equations, J. Elasticity, Volume 62 (2001), pp. 239-248

[7] P.G. Ciarlet; P. Rabier Les équations de von Kármán, Lecture Notes in Math., vol. 826, Springer-Verlag, Berlin, 1980

[8] R.B. Kellogg; T.Y. Li; J. Yorke A method of continuation for calculating a Brouwer fixed point (S. Karamardian, ed.), Fixed Points: Algorithms and Applications, Academic Press, New York, 1977, pp. 133-147

[9] S. Kesavan La méthode de Kikuchi appliquée aux équations de von Kármán, Numer. Math., Volume 32 (1979), pp. 209-232

[10] J.L. Lions Quelques méthodes de résolution de problèmes aux limites non linéaires, Dunod, Paris, 1969

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