# On finite element uniqueness studies for Coulombs frictional contact model

International Journal of Applied Mathematics and Computer Science (2002)

- Volume: 12, Issue: 1, page 41-50
- ISSN: 1641-876X

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topHild, Patrick. "On finite element uniqueness studies for Coulombs frictional contact model." International Journal of Applied Mathematics and Computer Science 12.1 (2002): 41-50. <http://eudml.org/doc/207567>.

@article{Hild2002,

abstract = {We are interested in the finite element approximation of Coulomb's frictional unilateral contact problem in linear elasticity. Using a mixed finite element method and an appropriate regularization, it becomes possible to prove existence and uniqueness when the friction coefficient is less than Cε^\{2\}|log(h)|^\{-1\}, where h and ε denote the discretization and regularization parameters, respectively. This bound converging very slowly towards 0 when h decreases (in comparison with the already known results of the non-regularized case) suggests a minor dependence of the mesh size on the uniqueness conditions, at least for practical engineering computations. Then we study the solutions of a simple finite element example in the non-regularized case. It can be shown that one, multiple or an infinity of solutions may occur and that, for a given loading, the number of solutions may eventually decrease when the friction coefficient increases.},

author = {Hild, Patrick},

journal = {International Journal of Applied Mathematics and Computer Science},

keywords = {Coulomb's friction law; non-uniqueness example; mesh-size dependent uniqueness conditions; finite elements; non-uniqueness; linear elasticity; mixed finite element method; regularization},

language = {eng},

number = {1},

pages = {41-50},

title = {On finite element uniqueness studies for Coulombs frictional contact model},

url = {http://eudml.org/doc/207567},

volume = {12},

year = {2002},

}

TY - JOUR

AU - Hild, Patrick

TI - On finite element uniqueness studies for Coulombs frictional contact model

JO - International Journal of Applied Mathematics and Computer Science

PY - 2002

VL - 12

IS - 1

SP - 41

EP - 50

AB - We are interested in the finite element approximation of Coulomb's frictional unilateral contact problem in linear elasticity. Using a mixed finite element method and an appropriate regularization, it becomes possible to prove existence and uniqueness when the friction coefficient is less than Cε^{2}|log(h)|^{-1}, where h and ε denote the discretization and regularization parameters, respectively. This bound converging very slowly towards 0 when h decreases (in comparison with the already known results of the non-regularized case) suggests a minor dependence of the mesh size on the uniqueness conditions, at least for practical engineering computations. Then we study the solutions of a simple finite element example in the non-regularized case. It can be shown that one, multiple or an infinity of solutions may occur and that, for a given loading, the number of solutions may eventually decrease when the friction coefficient increases.

LA - eng

KW - Coulomb's friction law; non-uniqueness example; mesh-size dependent uniqueness conditions; finite elements; non-uniqueness; linear elasticity; mixed finite element method; regularization

UR - http://eudml.org/doc/207567

ER -

## References

top- Adams R.A. (1975): Sobolev Spaces. - New-York: Academic Press. Zbl0314.46030
- Alart P. (1993): Critères d'injectivite et de surjectivite pourcertaines applications de R^n dans lui même: application à la mecanique du contact. - Math. Model. Numer. Anal., Vol. 27, No. 2, pp. 203-222.
- Ben Belgacem F. (2000): Numerical simulation of some variational inequalities arisen fromunilateral contact problems by the finite element method. - SIAM J. Numer. Anal., Vol. 37, No. 4, pp. 1198-1216. Zbl0974.74055
- Ciarlet P.G. (1991): The finite element method for elliptic problems, In: Handbook of Numerical Analysis, Volume II (P.G. Ciarlet and J.L. Lions, Eds.). - Amsterdam: North Holland, pp.17-352.
- Coorevits P., Hild P., Lhalouani K. and Sassi T. (2002): Mixed finite element methods for unilateral problems: convergence analysis and numerical studies. - Internal report of Laboratoire de Mathematiques de l'Universite de Savoie n^o 00-01c, to appear in Mathematics of Computation (published online May 21, 2001). Zbl1013.74062
- Duvaut G. and Lions J.-L. (1972): Les Inequations en Mecanique et en Physique. - Paris: Dunod. Zbl0298.73001
- Eck C. and Jarušek J. (1998): Existence results for the static contact problem with Coulomb friction. - Math. Mod. Meth. Appl. Sci., Vol. 8, No. 3, pp. 445-468. Zbl0907.73052
- Haslinger J. (1983): Approximation of the Signorini problem with friction, obeying the Coulomb law. - Math. Meth. Appl. Sci., Vol. 5, No. 3, pp. 422-437. Zbl0525.73130
- Haslinger J. (1984): Least square method for solving contact problems with friction obeying Coulomb's law. - Apl. Mat., Vol. 29, No. 3, pp. 212-224. Zbl0557.73100
- Haslinger J., Hlavaček I. and Nečas J. (1996): Numerical methods for unilateral problems in solid mechanics, In: Handbook of Numerical Analysis, Vol. IV (P.G. Ciarlet and J.L. Lions, Eds.). - Amsterdam: North Holland, pp. 313-485. Zbl0873.73079
- Hassani R., Hild P. and Ionescu I. (2001): On non-uniqueness of the elastic equilibrium with Coulomb friction: A spectral approach. - Internal report of Laboratoire de Mathematiques de l'Universite de Savoie no. 01-04c. Submitted.
- Janovsky V. (1981): Catastrophic features of Coulomb friction model, In: The Mathematics of Finite Elements and Aplications (J.R. Whiteman, Ed.). - London: Academic Press, pp.259-264.
- Jarušek J. (1983): Contact problems with bounded friction. Coercive case. - Czechoslovak. Math. J., Vol. 33, No. 2, pp. 237-261. Zbl0519.73095
- Kato Y. (1987): Signorini's problem with friction in linear elasticity. - Japan J. Appl. Math., Vol. 4, No. 2, pp. 237-268. Zbl0627.73098
- Klarbring A. (1990): Examples of non-uniqueness and non-existence of solutions to quasistatic contact problems with friction. - Ing. Archiv, Vol. 60, pp. 529-541.
- Nečas J., Jarušek J. and Haslinger J. (1980): On the solution of the variational inequality to the Signorini problem with small friction. - Boll. Unione Mat. Ital., Vol. 17-B(5), No. 2, pp. 796-811. Zbl0445.49011

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