An error estimate for the signorini problem with Coulomb friction approximated by finite elements

SIAM Journal on Numerical Analysis

Volumn 45, Issue 5, 2007, Pages 2012-2031

  Hild, Patrick ^a   Renard, Yves ^b  

^a UNIVERSITÉ DE FRANCHE COMTÉ   (France)

^b CNRS   (France)

Author keywords

A priori error estimate; Coulomb friction; Finite elements; Unilateral contact; Uniqueness of solution

Indexed keywords

FRICTION; TRIBOLOGY;

A PRIORI ERROR ESTIMATE; COULOMB FRICTION; FINITE ELEMENTS; UNILATERAL CONTACT; UNIQUENESS OF SOLUTION;

FINITE ELEMENT METHOD;

EID: 55549102664     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/050645439     Document Type: Article

References (43)

1. R.A. ADAMS, Sobolev Spaces, Academic Press, New York, 1975.
2. L.-E. ANDERSSON, Existence results for quasistatic contact problems with Coulomb friction, Appl. Math. Optim., 42 (2000), pp. 169-202.
3. F. BEN BELGACEM, The mortar finite element method with Lagrange multipliers, Numer. Math., 84 (1999), pp. 173-197.
4. F. BEN BELGACEM, Numerical simulation of some variational inequalities arisen from unilateral contact problems by the finite element methods, SIAM J. Numer. Anal., 37 (2000), pp. 1198-1216.
5. F. BEN BELGACEM, P. HILD, AND P. LABORDE, Extension of the mortar finite element method to a variational inequality modeling unilateral contact, Math. Models Methods Appl. Sci., 9 (1999), pp. 287-303.
6. F. BEN BELGACEM AND Y. RENARD, Hybrid finite element methods for the Signorini problem, Math. Comp., 72 (2003), pp. 1117-1145.
7. C. BERNARDI, Y. MADAY. AND A.T. PATERA, A new nonconforming approach to domain decomposition: The mortar element method, in Collège de France seminar, H. Brezis and J.-L. Lions, eds., Pitman, Boston, 1994, pp. 13-51.
8. Z. CHEN AND R.H. NOCHETTO, Residual type a posteriori error estimates for elliptic obstacle problems, NUMER. MATH., 84 (2000), PP. 527-548.
9. 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, Amsterdam, 1991, pp. 17-352.
10. P. CLÉMENT, Approximation by finite elements functions using local regularization, RAIRO Anal. Numer., 9 (1975), pp. 77-84.
11. M. COCU, Existence of solutions of Signorini problems with friction, Internat. J. Engrg. Sci., 22 (1984), pp. 567-575.
12. P. COOREVITS, P. HILD, K. LHALOUANI, AND T. SASSI, Mixed finite element methods for unilateral problems: Convergence analysis and numerical studies, Math. Comp., 71 (2002), pp. 1-25.
13. 2 projection on finite element function spaces, Math. Comp., 48 (1987), pp. 521-532.
14. L. DEMKOVICZ AND J.T. ODEN, On some existence and uniqueness results in contact problems with nonlocal friction, Nonlinear Anal., 6 (1982), pp. 1075-1093.
15. G. DIJVAUT, Problèmes unilatéraux en mécanique des milieux continus, in Actes du Congrès International des Mathématiciens (Nice 1970), Tome 3, Gauthier-Villars, Paris, 1971, pp. 71-77.
16. G. DUVAUT, Equilibre d'un solide élastique avec contact unilatéral et frottement de Coulomb, C. R. Acad. Sci. Sér. I Math., 290 (1980), pp. 263-265.
17. G. DUVAUT AND J.L. LIONS, Les inéquations en mécanique et en physique, Dunod, Paris, 1972.
18. C. ECK AND J. JARUŠEK, Existence results for the static contact problem with Coulomb friction, Math. Models Methods Appl. Sci., 8 (1998), pp. 445-468.
19. C. ECK, J. JARUŠEK, AND M. KRBEC, Unilateral Contact Problems: Variational Methods and Existence Theorems, Pure Appl. Math. 270, CRC Press, Boca Raton, FL, 2005.
20. P. GRISVARD, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.
21. W. HAN AND M. SOFONEA, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, American Mathematical Society, Providence, RI, 2002.
22. J. HASLINGER, Approximation of the Signorini problem with friction, Obeyinq the Coulomb law. Math. Methods Appl. Sci., 5 (1983), pp. 422-437.
23. J. HASLINGER AND I. HLAVÁČEK, CONTACT BETWEEN ELASTIC BODIES-2. FINITE ELEMENT ANALYSIS, APLIKACE MATEMATIKY, 26 (1981), PP. 263-290.
24. 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, Amsterdam, 1996, pp. 313-485.
25. R. HASSANI, P. HILD, I. IONESCU, AND N.-D. SAKKI, A mixed finite element method and solution multiplicity for Coulomb frictional contact, Comput. Methods Appl. Mech. Engrg., 192 (2003), pp. 4517-4531.
26. P. HILD, Non-unique slipping in the Coulomb friction model in two-dimensional linear elasticity, Quart. J. Mech. Appl. Math., 57 (2004), pp. 225-235.
27. P. HILD, Multiple solutions of stick and separation type in the Signorini model with Coulomb friction, Z. Angew. Math. Mech., 85 (2005), pp. 673-680.
28. P. HILD, A Priori Error Analysis of a Sign Preserving Mixed Finite Element Method for Contact Problems, Internal Report 2006-33 of the Laboratoire de Mathématiques de Besançon; Appl. Numer. Math., submitted.
29. S. HÜEBER AND B. I., WOHLMUTH, An optimal a priori error estimate for nonlinear multibody contact problems, SIAM J. Numer. Anal., 43 (2005), pp. 156-173.
30. H. KHENOUS, J. POMMIER, AND Y. RENARD, Hybrid discretization of the Signorini problem with Coulomb friction, Theoretical aspects and comparison of some numerical solvers, Appl. Numer. Math., 56 (2006), pp. 163-192.
31. N. KIKUCHI AND J.T. ODEN, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM Stud. Appl. Math. 8, SIAM, Philadelphia, 1988.
32. A. KLARBRING, A. MIKELÍC, AND M. SHILLOR, Frictional contact problems with normal compliance, Internat. J. Engrg. Sci., 26 (1988), pp. 811-832.
33. A. KLARBRING, A. MIKELÍC, AND M. SHILLOR, On friction problems with normal compliance, Nonlinear Anal., 13 (1989), pp. 935-955.
34. P. LABORDE AND Y. RENARD, Fixed points strategies for elastostatic frictional contact problems, Math. Methods Appl. Sci., to appear.
35. T. LAURSEN, Computational Contact and Impact Mechanics, Springer-Verlag, New York, 2002.
36. J.-L. LIONS AND E. MAGENES, Problèmes aux limites non homogènes, Dunod, Paris, 1968.
37. J.A.C. MARTINS AND J.T. ODEN, Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws, Nonlinear Anal., 11 (1987), pp. 407-428.
38. V.G. MAZ'YA AND T.O. SHAPOSHNIKOVA, Theory of Multipliers in Spaces of Differentiable Functions, Pitman, Boston, 1985.
39. J. NEČAS, J. JARUŠEK, AND J. HASLINGER, On the solution of the variational inequality to the Signorini problem with small friction, Boll. Unione Mat. Ital., 17 (1980), pp. 796-811.
40. J.T. ODEN AND J.A.C. MARTINS, Models and computational methods for dynamic friction phenomena, Comput. Methods. Appl. Mech. Engrg., 52 (1985), pp. 527-634.
41. Y. RENARD, A uniqueness criterion for the Signorini problem with. Coulomb friction, SIAM J. Math. Anal., 38 (2006), pp. 452-467.
42. R. ROCCA AND M. COCOU, Numerical analysis of quasi-static unilateral contact problems with local friction, SIAM J. Numer. Anal., 39 (2001), pp. 1324-1342.
43. P. WRIGGERS, Computational Contact Mechanics, Wiley, New York, 2002.