Stability of finite element mixed interpolations for contact problems
Bathe, Klaus Jürgen ; Brezzi, Franco
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 12 (2001), p. 167-183 / Harvested from Biblioteca Digitale Italiana di Matematica
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We consider the formulation of contact problems using a Lagrange multiplier to enforce the contact no-penetration constraint. The finite element discretization of the formulation must satisfy stability conditions which include an inf-sup condition. To identify which finite element interpolations in the contact constraint lead to stable (and optimal) numerical solutions we focus on the finite element discretization and solution of a «simple» model problem. While a simple problem to avoid the need for technicalities, the analysis of the finite element discretizations to solve the problem gives valuable insight and allows quite general conclusions on the use of different interpolation schemes.

Si considera il problema del contatto senza penetrazione di due corpi elastici, usando la tecnica dei moltiplicatori di Lagrange per il trattamento del vincolo unilaterale. La discretizzazione con elementi finiti di tale problema deve soddisfare opportune condizioni di stabilità, che includono una condizione di inf-sup. Per identificare la tipologia degli elementi finiti che possono portare a schemi discretizzati stabili (ed ottimali) ci concentriamo sulla discretizzazione di un problema modello «semplice». Tale scelta permette di evitare un certo numero di tecnicismi, pur fornendo valide indicazioni sulle scelte da operare in contesti molto più generali.

Publié le : 2001-09-01
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     author = {Klaus J\"urgen Bathe and Franco Brezzi},
     title = {Stability of finite element mixed interpolations for contact problems},
     journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
     volume = {12},
     year = {2001},
     pages = {167-183},
     zbl = {1097.74054},
     mrnumber = {1898458},
     language = {en},
     url = {http://dml.mathdoc.fr/item/RLIN_2001_9_12_3_167_0}
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Bathe, Klaus Jürgen; Brezzi, Franco. Stability of finite element mixed interpolations for contact problems. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 12 (2001) pp. 167-183. http://gdmltest.u-ga.fr/item/RLIN_2001_9_12_3_167_0/

[1] El-Abbasi, N. - Bathe, K.J., Stability and Patch Test Performance of Contact Discretizations. Submitted to Computers & Structures.

[2] Baiocchi, C. - Buttazzo, G. - Gastaldi, F. - Tomarelli, F., General Existence Theorems for Unilateral Problems in Continuum Mechanics. Arch. Rational Mech. Anal., 100, 1988, 149-189. | MR 913962 | Zbl 0646.73011

[3] Baiocchi, C. - Capelo, A., Variational and Quasivariational Inequalities. J. Wiley and Sons, Chichester 1984. | MR 745619 | Zbl 0551.49007

[4] Bathe, K.J., Finite Element Procedures. Prentice Hall, 1996. | Zbl 0994.74001

[5] Bathe, K.J., The Inf-Sup Condition and its Evaluation for Mixed Finite Element Methods. Computers & Structures, 79, 2001, 243-252 and 971. | MR 1813187

[6] K.J. Bathe (Ed.), Computational Fluid and Solid Mechanics. Elsevier, 2001. | Zbl 0985.74004

[7] Brezis, H., Problèmes unilateraux. J. Math. Pures Appl., 51, 1972, 1-168. | MR 428137 | Zbl 0237.35001

[8] Brezzi, F. - Bathe, K.J., A Discourse on the Stability Conditions for Mixed Finite Element Formulations. Computer Methods in Applied Mechanics and Engineering, 82, 1990, 27-57. | MR 1077650 | Zbl 0736.73062

[9] Brezzi, F. - Fortin, M., Mixed and Hybrid Finite Element Methods. Springer-Verlag, 1991. | MR 1115205 | Zbl 0788.73002

[10] Brezzi, F. - Hager, W.W. - Raviart, P.-A., Error Estimates for the Finite Element Solution of Variational Inequalities. Part II Mixed Methods. Numer. Math., 31, 1978, 1-16. | MR 508584 | Zbl 0427.65077

[11] Ciarlet, Ph.G., The Finite Element Method for Elliptic Problems. North Holland, 1978. | MR 520174 | Zbl 0511.65078

[12] Duvaut, G. - Lions, J.L., Inequalities in Mechanics and Physics. Springer-Verlag, 1976. | MR 521262 | Zbl 0331.35002

[13] Fichera, G., Unilateral Constraints in Elasticity. In: Actes du Congrès International des Mathématiciens (Nice, Paris 1970), 3, Gauthier-Villars, septembre 1971, vol. 3, 79-84. | MR 426557 | Zbl 0254.73021

[14] Fortin, M., An Analysis of Convergence of the Mixed Finite Element Method. RAIRO Analyse Numérique, 11, 1997, 341-354. | MR 464543 | Zbl 0373.65055

[15] Kikuchi, N. - Oden, J.T., Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM Studies in Applied mathematics, 8, 1988. | MR 961258 | Zbl 0685.73002

[16] Kinderlehrer, D. - Stampacchia, G., An Introduction to Variational Inequalities and Their Applications. Academic Press, 1980. | MR 567696 | Zbl 0457.35001

[17] Lions, J.L. - Magenes, E., Nonhomogeneous Boundary Value Problems and Applications. Vol. 1, Springer-Verlag, 1972. | Zbl 0223.35039

[18] Stenberg, R., A Family of Mixed Finite Elements for the Elasticity Problem. Numer. Math., 53, 1988, 513-538. | MR 954768 | Zbl 0632.73063

[19] Widlund, O., An Extension Theorem for Finite Element Spaces with three Applications. In: W. Hackbush - K. Witsch (eds.), Numerical Techniques in Continuum Mechanics. Vieweg & Sohn, 1987. | Zbl 0615.65114