Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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

   
 

 

Penalty/finite-element approximations of a class of unilateral problems in linear elasticity


Authors: Noboru Kikuchi and Young Joon Song
Journal: Quart. Appl. Math. 39 (1981), 1-22
MSC: Primary 73T05; Secondary 49D30, 65N30, 73K25
DOI: https://doi.org/10.1090/qam/613950
MathSciNet review: 613950
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The present paper is concerned with a development of a penalty/finite-element approximation of a class of unilateral problems in linear elasticity. A penalty method is applied to resolve the inequality constraint due to contact, and convergence with respect to the penalty parameter is discussed. Then finite-element approximations are introduced to the penalized formulation with a priori error estimates in terms of the penalty and mesh parameters. Several numerical examples are also given in the end of the paper.


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

  • [1] G. Fichera, Un teorema generale di semicontinuita per gli integrali multipli e sue applicazioni alla fisicamatematica, in Atti del convegno Lagrangiano, Torino, 1963, pp. 138-151
  • [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] J. J. Kalker, Variational principles of contact elastostatics, J. Inst. Math. Appl. 20 (1977), no. 2, 199–219. MR 0455732
  • [4] 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
  • [5] N. Kikuchi and J. T. Oden, Contact problems in elasticity, SIAM, Philadelphia, 1981
  • [6] R. Glowinski, J. L. Lions and R. Tremolieres, Analyse numérique des Inéquations variationnelles, 2 vols., Dunod, Paris, 1976
  • [7] I. Páczelt, Solution of elastic contact problems by the finite element displacement method, Acta Tech. Acad. Sci. Hungar. 82 (1976), no. 3-4, 353–375 (English, with German and Russian summaries). MR 0452064
  • [8] Noboru Kikuchi and Young Joon Song, Contact problems involving forces and moments for incompressible linearly elastic materials, Internat. J. Engrg. Sci. 18 (1980), no. 2, 357–377. Computational methods in nonlinear problems in mechanics and engineering science (Austin, Tex., 1979). MR 661276, https://doi.org/10.1016/0020-7225(80)90057-9
  • [9] S. K. Chan and I. S. Tuba, A finite-element method for contact problems, Int. J. Mech. Sci. 13, 615-639 (1971)
  • [10] T. J. R. Hughes, R. L. Taylor, L. Sackman, A. Curnier and W. Kanoknukulchai, A finite-element method for a class of contact-impact problems, Compt. Meth. Appl. Mech. Engng. 8, 249-276 (1976)
  • [11] R. Courant, K. Friedrichs, and H. Lewy, On the partial difference equations of mathematical physics, IBM J. Res. Develop. 11 (1967), 215–234. MR 0213764, https://doi.org/10.1147/rd.112.0215
  • [12] R. Courant, Variational methods for the solution of problems of equilibrium and vibrations, Bull. Amer. Math. Soc. 49 (1943), 1–23. MR 0007838, https://doi.org/10.1090/S0002-9904-1943-07818-4
  • [13] Willard I. Zangwill, Non-linear programming via penalty functions, Management Sci. 13 (1967), 344–358. MR 0252040, https://doi.org/10.1287/mnsc.13.5.344
  • [14] J. L. Lions, Quelques methodes de resolution des problèmes aux limites Nonlinéaires, Dunod, Paris, 1969
  • [15] Jean-Pierre Aubin, Approximation of elliptic boundary-value problems, Wiley-Interscience [A division of John Wiley & Sons, Inc.], New York-London-Sydney, 1972. Pure and Applied Mathematics, Vol. XXVI. MR 0478662
  • [16] T. Tsuta and S. Yamaji, Finite-element analysis of contact problem, in Theory and practice in finite-element structural analysis, Tokyo University Press, 177-194, 1973
  • [17] Y. Yamada, Y. Ezawa, I. Nishiguchi and M. Okabe, Handy incorporation of bond and singular elements in finite element solution routine, Trans. Fifth Int. Conf. on SMIRT, 1979
  • [18] M. Okabe and N. Nikuchi, An application of penalty methods to a two-body contact problem, Proc. Third EMD Speciality Conf., ASCE, 1979
  • [19] J. Necas, Les méthodes directes et théorie des équations elliptiques, Masson, 1967
  • [20] L. A. Garlin, Theory of elastic contact problems, Moscow, 1953. Japanese translation by T. Sato, Tokyo, 1956
  • [21] J. T. Oden, N. Nikuchi, and Y. J. Song, An analysis of exterior penalty methods and reduced integration for finite element approximations of contact problems in incompressible elasticity, TICOM Report, The University of Texas at Austin, 1980
  • [22] A. K. Aziz (ed.), The mathematical foundations of the finite element method with applications to partial differential equations, Academic Press, New York-London, 1972. MR 0347104
  • [23] Richard S. Falk, Error estimates for the approximation of a class of variational inequalities, Math. Comput. 28 (1974), 963–971. MR 0391502, https://doi.org/10.1090/S0025-5718-1974-0391502-8
  • [24] 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
  • [25] Werner Goldsmith, Impact:The theory and physical behaviour of colliding solids, Edward Arnold Publishers, Ltd., London, 1960. MR 0128163

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 73T05, 49D30, 65N30, 73K25

Retrieve articles in all journals with MSC: 73T05, 49D30, 65N30, 73K25


Additional Information

DOI: https://doi.org/10.1090/qam/613950
Article copyright: © Copyright 1981 American Mathematical Society


Brown University The Quarterly of Applied Mathematics
is distributed by the American Mathematical Society
for Brown University
Online ISSN 1552-4485; Print ISSN 0033-569X
© 2017 Brown University
Comments: qam-query@ams.org
AMS Website