Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Finite-element convergence for contact problems in plane linear elastostatics

Author: Joachim Gwinner
Journal: Quart. Appl. Math. 50 (1992), 11-25
MSC: Primary 65N30; Secondary 73C99, 73T05, 73V05
DOI: https://doi.org/10.1090/qam/1146620
MathSciNet review: MR1146620
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper presents a convergence analysis for the finite-element approximation of unilateral problems in plane linear elastostatics. We consider in particular the deformation of a body unilaterally supported by a frictionless rigid foundation, solely subjected to body forces and surface tractions without being fixed along some part of its boundary, and establish convergence of piecewise polynomial finite-element approximations for mechanically definite problems without imposing any regularity assumption. Moreover we study the discretization of the contact problem with given friction along the rigid foundation.

References [Enhancements On Off] (What's this?)

  • [1] Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR 0520174
  • [2] G. Duvaut and J.-L. Lions, Inequalities in mechanics and physics, Springer-Verlag, Berlin-New York, 1976. Translated from the French by C. W. John; Grundlehren der Mathematischen Wissenschaften, 219. MR 0521262
  • [3] H. Engels, Numerical quadrature and cubature, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980. Computational Mathematics and Applications. MR 587486
  • [4] G. Fichera, Boundary value problems of elasticity with unilateral constraints, Handbuch der Physik--Encyclopedia of Physics, Band VI a/2 Festkörpermechanik II, Springer, Berlin, 1972, pp. 391-424
  • [5] R. Glowinski, Lectures on numerical methods for nonlinear variational problems, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 65, Tata Institute of Fundamental Research, Bombay; sh Springer-Verlag, Berlin-New York, 1980. Notes by M. G. Vijayasundaram and M. Adimurthi. MR 597520
  • [6] Roland Glowinski, Numerical methods for nonlinear variational problems, Springer Series in Computational Physics, Springer-Verlag, New York, 1984. MR 737005
  • [7] Roland Glowinski, Jacques-Louis Lions, and Raymond Trémolières, Numerical analysis of variational inequalities, Studies in Mathematics and its Applications, vol. 8, North-Holland Publishing Co., Amsterdam-New York, 1981. Translated from the French. MR 635927
  • [8] J. Gwinner, Convergence and Error Analysis for Variational Inequalities and Unilateral Boundary Value Problems, Habilitationsschrift, TH Darmstadt, 1989
  • [9] I. Hlaváček, J. Haslinger, J. Nečas, and J. Lovíšek, Solution of variational inequalities in mechanics, Applied Mathematical Sciences, vol. 66, Springer-Verlag, New York, 1988. Translated from the Slovak by J. Jarník. MR 952855
  • [10] Ivan Hlaváček and Ján Lovíšek, A finite element analysis for the Signorini problem in plane elastostatics, Apl. Mat. 22 (1977), no. 3, 215–228 (English, with Czech and loose Russian summaries). MR 0446014
  • [11] H. G. Jeggle, Nichtlineare Funktionalanalysis, Teubner, Stuttgart, 1978
  • [12] N. Kikuchi and J. T. Oden, Contact problems in elasticity: a study of variational inequalities and finite element methods, SIAM Studies in Applied Mathematics, vol. 8, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. MR 961258
  • [13] Noboru Kikuchi and Young Joon Song, Penalty/finite-element approximations of a class of unilateral problems in linear elasticity, Quart. Appl. Math. 39 (1981/82), no. 1, 1–22. MR 613950, https://doi.org/10.1090/S0033-569X-1981-0613950-3
  • [14] J. Nečas, Les Méthodes Directes en Théorie des Équations Élliptiques, Academia, Prague, and Masson, Paris, 1967
  • [15] J. Nečas and I. Hlavaček, Mathematical Theory of Elastic and Elastoplastic Bodies: Introduction, Elsevier, Amsterdam, 1981
  • [16] P. D. Panagiotopoulos, Inequality problems in mechanics and applications, Birkhäuser Boston, Inc., Boston, MA, 1985. Convex and nonconvex energy functions. MR 896909
  • [17] A. Signorini, Sopra alcune questioni di elastostatica, Atti Soc. Ital. Progr. Sci., 1933
  • [18] Guido Stampacchia, Variational inequalities, Theory and Applications of Monotone Operators (Proc. NATO Advanced Study Inst., Venice, 1968) Edizioni “Oderisi”, Gubbio, 1969, pp. 101–192. MR 0425699
  • [19] Tran Van Bon, Finite element analysis of primal and dual formulations of semi-coercive elliptic problems with nonhomogeneous obstacles on the boundary, Apl. Mat. 33, 1-21 (1988)
  • [20] R. Zurmühl, Praktische Mathematik, Springer, Berlin, 1957/65

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 65N30, 73C99, 73T05, 73V05

Retrieve articles in all journals with MSC: 65N30, 73C99, 73T05, 73V05

Additional Information

DOI: https://doi.org/10.1090/qam/1146620
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society