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A Stabilized Lagrange Multiplier Method for the Finite ElementApproximation of Frictional Contact Problems in Elastostatics

Published online by Cambridge University Press:  27 January 2009

V. Lleras
V. Lleras*
Affiliation:
Laboratoire de mathématiques, Université de Besançon, CNRS UMR 6623, 16 route de Gray, 25030 Besançon Cedex, France
*
[email protected]
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Abstract

In this work we consider a stabilized Lagrange multiplier method in order toapproximate the Coulomb frictional contact model in linear elastostatics. Theparticularity of the method is that no discrete inf-sup condition is needed. Westudy the existence and the uniqueness of solution of the discrete problem.

Keywords

unilateral contactCoulomb's friction lawfinite elementsmixedmethodstabilization
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 4 , Issue 1: Modelling and numerical methods in contact mechanics , 2009 , pp. 163 - 182
Copyright
© EDP Sciences, 2009

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References

R.A. Adams. Sobolev spaces, Academic Press, 1975.
Alart, P., Curnier, A.. Generalisation of Newton type methods to contact problems with friction, J. Mecan. Theor. Appl., 7 (1988), 67–82.
Babuška, I.. The finite element method with Lagrange multipliers, Numer. Math., 20 (1973), 179–192. CrossRef
Barbosa, H.J.C., Hughes, T.J.R.. The finite element method with Lagrange multipliers on the boundary: circumventing the Babuška-Brezzi condition, Comput. Methods Appl. Mech. Engrg., 85 (1991), 109–128. CrossRef
Barbosa, H.J.C., Hughes, T.J.R.. Boundary Lagrange multipliers in finite element methods: error analysis in natural norms, Numer. Math., 62 (1992), 1–15. CrossRef
Barbosa, H.J.C., Hughes, T.J.R.. Circumventing the Babuška-Brezzi condition in mixed finite element approximations of elliptic variational inequalities, Comput. Methods Appl. Mech. Engrg., 97 (1992), 193–210. CrossRef
Becker, R., Hansbo, P., Stenberg, R.. A finite element method for domain decomposition with non-matching grids, Math. Model. Numer. Anal., 37 (2003), 209–225. CrossRef
Belhachmi, Z., Ben Belgacem, F.. Quadratic finite element approximation of the Signorini problem, Math. Comp., 72 (2003), 83–104. CrossRef
Belhachmi, Z., Sac-Epée, J.M., Sokolowski, J.. Mixed finite element methods for smooth domain formulation of crack problems, SIAM J. Numer. Anal., 43 (2005), 1295–1320. CrossRef
Ben Belgacem, F., Renard, Y.. Hybrid finite element methods for the Signorini problem, Math. Comp., 72 (2003), 1117–1145. CrossRef
S.C. Brenner, L.R. Scott. The mathematical theory of finite element methods, Springer-Verlag, 2002.
F. Brezzi. On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers, Rev. Franç. Automatique Inform. Rech. Opér., Sér. Rouge Anal. Numér., 8 (1974), 129–151.
F. Brezzi, M. Fortin. Mixed and hybrid finite element methods, Springer, 1991.
Chen, Z.. On the augmented Lagrangian approach to Signorini elastic contact problem, Numer. Math., 88 (2001), 641–659. CrossRef
P.G. Ciarlet. The finite element method for elliptic problems, in Handbook of Numerical Analysis, eds. P.G. Ciarlet and J.L. Lions, North Holland, 2 (1991), 17–352.
Cocou, M., Roccou, R.. Numerical analysis of quasistatic unilateral contact problems with local friction, SIAM J. Numer. Anal., 39 (2001), 1324–1342.
Coorevits, P., Hild, P., 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., Sassi, T.. Mixed finite element methods for unilateral problems: convergence analysis and numerical studies, Math. Comp., 71 (2002), 1–25. CrossRef
G. Duvaut. Problèmes unilatéraux en mécanique des milieux continus, in Actes du congrès international des mathématiciens (Nice 1970), Gauthier-Villars, 3 (1971), 71–77.
G. Duvaut, J.L. Lions. Les inéquations en mécanique et en physique, Dunod, 1972.
Eck, C., Jarušek, J.. Existence results for the static contact problem with Coulomb friction, Math. Models Meth. Appl. Sci., 8 (1998), 445–468. CrossRef
C. Eck, J. Jarušek, M. Krbec. Unilateral contact problems: variational methods and existence theorems, Pure and Applied Mathematics, CRC Press, 270 (2005).
W. Han, M. Sofonea. Quasistatic contact problems in viscoelasticity and viscoplasticity, American Mathematical Society, International Press, 2002.
Hansbo, P., Lovadina, C., Perugia, I., Sangalli, G.. A Lagrange multiplier method for the finite element solution of elliptic interface problems using nonmatching meshes, Numer. Math., 100 (2005), 91–115. CrossRef
Haslinger, J.. Approximation of the Signorini problem with friction, obeying the Coulomb law, Math. Methods Appl. Sci., 5 (1983), 422–437. CrossRef
J. Haslinger, I. Hlaváček, J. Nečas. Numerical methods for unilateral problems in solid mechanics, in Handbook of Numerical Analysis, Eds. P.G. Ciarlet and J.-L. Lions, North Holland, 4 (1996), 313–485.
J. Haslinger, Y. Renard. A new fictitious domain approach inspired by the extended finite element method, submitted.
Hassani, R., Hild, P., Ionescu, I., Sakki, N.D.. A mixed finite element method and solution multiplicity for Coulomb frictional contact, Comput. Methods Appl. Mech. Engrg., 192 (2003), 4517–4531. CrossRef
Heintz, P., Hansbo, P.. Stabilized Lagrange multiplier methods for bilateral elastic contact with friction, Comput. Methods Appl. Mech. Engrg., 195 (2006), 4323–4333. CrossRef
Hild, P.. Non-unique slipping in the Coulomb friction model in two-dimensional linear elasticity, Q. Jl. Mech. Appl. Math., 57 (2004), 225–235. CrossRef
Hild, P.. Multiple solutions of stick and separation type in the Signorini model with Coulomb friction, Z. Angew. Math. Mech., 85 (2005), 673–680. CrossRef
Hild, P., Laborde, P.. Quadratic finite element methods for unilateral contact problems, Appl. Numer. Math., 41 (2002), 401–421. CrossRef
Hild, P., Renard, Y.. An error estimate for the Signorini problem with Coulomb friction approximated by the finite elements, SIAM J. Numer. Anal., 45 (2007), 2012–2031. CrossRef
P. Hild, Y. Renard. A stabilized Lagrange multiplier method for the finite element approximation of contact problems in elastostatics, submitted.
Jarušek, J.. Contact problems with bounded friction. Coercive case, Czechoslovak. Math. J., 33 (1983), 237–261.
N. Kikuchi, J.T. Oden. Contact problems in elasticity : a study of variational inequalities and finite element methods, SIAM, 1988.
D. Kinderlehrer, G. Stampacchia. An introduction to variational inequalities and their applications, Pure and Applied mathematics, Academic Press, New York-London, 1980.
T. Laursen. Computational contact and impact mechanics, Springer, 2002.
J.–L. Lions, E. Magenes. Problèmes aux limites non homogènes, Dunod, 1968.
V. Lleras. Thesis, in preparation.
V.G. Maz'ya, T.O. Shaposhnikova. Theory of multipliers in spaces of differentiable functions, Pitman, 1985.
Moës, N., Dolbow, J., Belytschko, T.. A finite element method for cracked growth without remeshing, Int. J. Numer. Meth. Engng., 46 (1999), 131–150. 3.0.CO;2-J>CrossRef
Nečas, J., Haslinger, J., Jarušek, J.. On the solution of the variational inequality to the Signorini problem with small friction, Bolletino U. M. I., 17 (1980), 796–811.
Nitsche, J.. Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Univ. Hamburg, 36 (1971), 9–15. CrossRef
Owen, D.R.J., Peric, D.. Computational model for 3D contact problems with friction based on the penalty method, Int. J. Num. Meth. Eng., 35 (1992), 1289–1309.
Renard, Y.. A uniqueness criterion for the Signorini problem with Coulomb friction, SIAM J. Math. Anal., 38 (2006), 452–467. CrossRef
M. Shillor, M. Sofonea, J.J. Telega. Models and analysis of quasistatic contact. Variational methods, Springer, 2004.
Stenberg, R.. On some techniques for approximating boundary conditions in the finite element method, J. Comput. Appl. Math., 63 (1995), 139–148. CrossRef
P. Wriggers. Computational Contact Mechanics, Wiley, 2002.