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Residual a posteriori error estimators for contact problems in elasticity

Published online by Cambridge University Press:  23 October 2007

Patrick Hild  and
Serge Nicaise
Patrick Hild
Affiliation:
Université de Franche-Comté, Laboratoire de Mathématiques de Besançon, CNRS UMR 6623, 16 route de Gray, 25030 Besançon, France. [email protected]
Serge Nicaise
Affiliation:
Université de Valenciennes et du Hainaut Cambrésis, MACS, ISTV, 59313 Valenciennes Cedex 9, France. [email protected]
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Abstract

This paper is concerned with the unilateral contact problem in linear elasticity. We define two a posteriori error estimators of residual type to evaluate the accuracy of the mixed finite element approximation of the contact problem.Upper and lower bounds of the discretization error are proved forboth estimators and several computations are performed toillustrate the theoretical results.

Keywords

Mixed finite element methoda posteriori error estimatesresidualsunilateral contact.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 5 , September 2007 , pp. 897 - 923
Copyright
© EDP Sciences, SMAI, 2007

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References

R.A. Adams, Sobolev spaces. Academic Press (1975).
K. Atkinson and W. Han, Theoretical numerical analysis: a functional analysis framework, in Texts in Applied Mathematics 39, Springer, New-York (2001); (second edition 2005).
Ben Belgacem, F., Numerical simulation of some variational inequalities arisen from unilateral contact problems by the finite element method. SIAM J. Numer. Anal. 37 (2000) 1198–1216. CrossRef
Ben Belgacem, F. and Renard, Y., Hybrid finite element methods for the Signorini problem. Math. Comp. 72 (2003) 1117–1145. CrossRef
Bostan, V., Han, W. and Reddy, B.D., A posteriori error estimation and adaptive solution of elliptic variational inequalities of the second kind. Appl. Numer. Math. 52 (2005) 13–38. CrossRef
Braess, D., A posteriori error estimators for obstacle problems - another look. Numer. Math. 101 (2005) 415–421. CrossRef
Carstensen, C., Scherf, O. and Wriggers, P., Adaptive finite elements for elastic bodies in contact. SIAM J. Sci. Comput. 20 (1999) 1605–1626. CrossRef
Chen, Z. and Nochetto, R.H., Residual type a posteriori error estimates for elliptic obstacle problems. Numer. Math. 84 (2000) 527–548. CrossRef
P.G. Ciarlet, The finite element method for elliptic problems, in Handbook of Numerical Analysis, Volume II, Part 1, P.G. Ciarlet and J.-L. Lions Eds., North Holland (1991) 17–352.
Coorevits, P., Hild, P. and Pelle, J.-P., A posteriori error estimation for unilateral contact with matching and nonmatching meshes. Comput. Methods Appl. Mech. Engrg. 186 (2000) 65–83. CrossRef
Coorevits, P., Hild, P. and Hjiaj, M., A posteriori error control of finite element approximations for Coulomb's frictional contact. SIAM J. Sci. Comput. 23 (2001) 976–999. CrossRef
Coorevits, P., Hild, P., Lhalouani, K. and Sassi, T., Mixed finite element methods for unilateral problems: convergence analysis and numerical studies. Math. Comp. 71 (2002) 1–25. CrossRef
G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique. Dunod (1972).
Eck, C. and Wendland, W., A residual-based error estimator for BEM-discretizations of contact problems. Numer. Math. 95 (2003) 253–282. CrossRef
Fichera, G., Problemi elastici con vincoli unilaterali il problema di Signorini con ambigue condizioni al contorno. Mem. Accad. Naz. Lincei. 8 (1964) 91–140.
G. Fichera, Existence theorems in elasticity, in Handbuch der Physik, Band VIa/2, Springer (1972) 347–389.
R. Glowinski, Lectures on numerical methods for nonlinear variational problems, in Lectures on Mathematics and Physics 65, Notes by M. G. Vijayasundaram and M. Adimurthi, Tata Institute of Fundamental Research, Bombay; Springer-Verlag, Berlin-New York (1980).
W. Han and M. Sofonea, Quasistatic contact problems in viscoelasticity and viscoplasticity. American Mathematical Society (2002).
J. Haslinger, I. Hlaváček and J. Nečas, Numerical methods for unilateral problems in solid mechanics, in Handbook of Numerical Analysis, Volume IV, Part 2, P.G. Ciarlet and J.-L. Lions Eds., North Holland (1996) 313–485.
P. Hild, A priori error analysis of a sign preserving mixed finite element method for contact problems. Preprint 2006/33 of the Laboratoire de Mathématiques de Besançon, submitted.
Hild, P. and Nicaise, S., A posteriori error estimations of residual type for Signorini's problem. Numer. Math. 101 (2005) 523–549. CrossRef
J.-B. Hiriart-Urruty and C. Lemaréchal, Convex analysis and minimization algorithms I. Springer (1993).
Hüeber, S. and Wohlmuth, B., An optimal error estimate for nonlinear contact problems. SIAM J. Numer. Anal. 43 (2005) 156–173. CrossRef
N. Kikuchi and J.T. Oden, Contact problems in elasticity. SIAM (1988).
T. Laursen, Computational contact and impact mechanics. Springer (2002).
Lee, C.Y. and Oden, J.T., A posteriori error estimation of h-p finite element approximations of frictional contact problems. Comput. Methods Appl. Mech. Engrg. 113 (1994) 11–45. CrossRef
Veeser, A., Efficient and reliable a posteriori error estimators for elliptic obstacle problems. SIAM J. Numer. Anal. 39 (2001) 146–167. CrossRef
R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley and Teubner (1996).
Verfürth, R., A review of a posteriori error estimation techniques for elasticity problems. Comput. Methods Appl. Mech. Engrg. 176 (1999) 419–440. CrossRef
P. Wriggers, Computational Contact Mechanics. Wiley (2002).
Wriggers, P. and Scherf, O., Different a posteriori error estimators and indicators for contact problems. Mathl. Comput. Modelling 28 (1998) 437–447. CrossRef