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
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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.
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G. Duvaut and J. L. Lions, Inequalities in mechanics and physics, Springer-Verlag, Berlin, 1976
J. J. Kalker, Variational principles of contact elastostatics, J. Inst. Maths. Applics. 20, 199–219 (1977)
I. Hlavacek and J. Lovisek, A finite-element analysis for the Signorini problem in plane elastostatics, Aplikace Matematiky, 22, 215–228 (1977)
N. Kikuchi and J. T. Oden, Contact problems in elasticity, SIAM, Philadelphia, 1981
R. Glowinski, J. L. Lions and R. Tremolieres, Analyse numérique des Inéquations variationnelles, 2 vols., Dunod, Paris, 1976
I. Paczelt, Solution of elastic contact problems by the finite-element displacement method. Acta Techica Acad. Sci. Hungaricae, 82, 354–375 (1976)
N. Kikuchi and Y. J. Song, Contact problems involving forces and movements for incompressible linearly elastic materials, Int. J. Engng Sci. 18, 357–377 (1980)
S. K. Chan and I. S. Tuba, A finite-element method for contact problems, Int. J. Mech. Sci. 13, 615–639 (1971)
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)
R. Courant, K. Friedrichs and H. Lewy, On the partial difference equations of mathematical physics, IBM Journal 11, 215–234 (1967)
R. Courant, Variational methods for the solutions of problems of equilibrium and vibrations. Bull. Amer. Math. Soc. 49, 1–23 (1943)
W. I. Zangwill, Nonlinear programming via penalty functions, Management Science 13, 344–358 (1967)
J. L. Lions, Quelques methodes de resolution des problèmes aux limites Nonlinéaires, Dunod, Paris, 1969
J. P. Aubin, Approximation of elliptic boundary value problems, Wiley-Interscience, New York, 1972
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
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
M. Okabe and N. Nikuchi, An application of penalty methods to a two-body contact problem, Proc. Third EMD Speciality Conf., ASCE, 1979
J. Necas, Les méthodes directes et théorie des équations elliptiques, Masson, 1967
L. A. Garlin, Theory of elastic contact problems, Moscow, 1953. Japanese translation by T. Sato, Tokyo, 1956
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
I. Babuska and A. K. Aziz, The mathematical foundations of the finite-element methods with applications to partial differential equations, Academic Press, New York, 1972
R. S. Falk, Error estimates for the approximation of a class of variational inequalities, Math. Computation 28, 963–971 (1974)
P. G. Ciarlet, The finite-element method for elliptic problems, North-Holland, Amsterdam, 1978
W. Goldsmith, Impact, Edward Arnold, London, 1960
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Article copyright:
© Copyright 1981
American Mathematical Society
