Mixed finite element approximation of 3D contact problems with given friction: Error analysis and numerical realization

Jaroslav Haslinger; Taoufik Sassi

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 38, Issue: 3, page 563-578
  • ISSN: 0764-583X

Abstract

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This contribution deals with a mixed variational formulation of 3D contact problems with the simplest model involving friction. This formulation is based on a dualization of the set of admissible displacements and the regularization of the non-differentiable term. Displacements are approximated by piecewise linear elements while the respective dual variables by piecewise constant functions on a dual partition of the contact zone. The rate of convergence is established provided that the solution is smooth enough. The numerical realization of such problems will be discussed and results of a model example will be shown.

How to cite

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Haslinger, Jaroslav, and Sassi, Taoufik. "Mixed finite element approximation of 3D contact problems with given friction: Error analysis and numerical realization ." ESAIM: Mathematical Modelling and Numerical Analysis 38.3 (2010): 563-578. <http://eudml.org/doc/194228>.

@article{Haslinger2010,
abstract = { This contribution deals with a mixed variational formulation of 3D contact problems with the simplest model involving friction. This formulation is based on a dualization of the set of admissible displacements and the regularization of the non-differentiable term. Displacements are approximated by piecewise linear elements while the respective dual variables by piecewise constant functions on a dual partition of the contact zone. The rate of convergence is established provided that the solution is smooth enough. The numerical realization of such problems will be discussed and results of a model example will be shown. },
author = {Haslinger, Jaroslav, Sassi, Taoufik},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Mixed finite element methods; unilateral contact problems with friction; a priori error estimates.; dualization; rate of convergence},
language = {eng},
month = {3},
number = {3},
pages = {563-578},
publisher = {EDP Sciences},
title = {Mixed finite element approximation of 3D contact problems with given friction: Error analysis and numerical realization },
url = {http://eudml.org/doc/194228},
volume = {38},
year = {2010},
}

TY - JOUR
AU - Haslinger, Jaroslav
AU - Sassi, Taoufik
TI - Mixed finite element approximation of 3D contact problems with given friction: Error analysis and numerical realization
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 38
IS - 3
SP - 563
EP - 578
AB - This contribution deals with a mixed variational formulation of 3D contact problems with the simplest model involving friction. This formulation is based on a dualization of the set of admissible displacements and the regularization of the non-differentiable term. Displacements are approximated by piecewise linear elements while the respective dual variables by piecewise constant functions on a dual partition of the contact zone. The rate of convergence is established provided that the solution is smooth enough. The numerical realization of such problems will be discussed and results of a model example will be shown.
LA - eng
KW - Mixed finite element methods; unilateral contact problems with friction; a priori error estimates.; dualization; rate of convergence
UR - http://eudml.org/doc/194228
ER -

References

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  1. R.A. Adams, Sobolev Spaces. Academic Press (1975).  
  2. G. Amontons, Sur l'origine de la résistance dans les machines. Mémoires de l'Académie Royale (1699) 206–222.  
  3. L. Baillet and T. Sassi, Méthodes d'éléments finis avec hybridisation frontière pour les problèmes de contact avec frottement. C.R. Acad. Sci. Paris, Ser. I334 (2002) 917–922.  Zbl1073.74047
  4. G. Bayada, M. Chambat, K. Lhalouani and T. Sassi, Éléments finis avec joints pour des problèmes de contact avec frottement de Coulomb non local. C.R. Acad. Sci. Paris, Ser. I325 (1997) 1323–1328.  Zbl0898.73059
  5. P.-G. Ciarlet, The finite element method for elliptic problems, Handbook of Numerical Analysis, P.G. Ciarlet and J.L. Lions Eds., Vol. 2, Part 1, North-Holland (1991) 17–352.  
  6. C.A. Coulomb, Théorie des machines simples. Mémoire de Mathématique et de Physique de l'Académie Royale10 (1785) 145–173.  
  7. Z. Dostál, Box constrained quadratic programming with proportioning and projections. SIAM J. Opt.7 (1997) 871–887.  Zbl0912.65052
  8. G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique. Dunod, Paris (1972).  Zbl0298.73001
  9. I. Ekeland and R. Temam, Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976).  Zbl0322.90046
  10. R. Glowinski, Numerical methods for nonlinear variational problems. Springer, New York (1984).  Zbl0536.65054
  11. P. Grisvard, Elliptic Problems in Nonsmooth Domains. Monogr. Studies Math., Pitman 24 (1985).  Zbl0695.35060
  12. J. Haslinger and I. Hlaváček, Approximation of the Signorini problem with friction by mixed finite element method, J. Math. Anal. Appl.86 (1982) 99–122.  Zbl0486.73099
  13. J. Haslinger and P.D. Panagiolopoulas, Approximation of contact problems with friction by reciprocal variational formulations. Proc. Roy. Soc. Edinburgh98A (1984) 365–383.  
  14. J. Haslinger, I. Hlaváček and J. Nečas, Numerical methods for unilateral problems in solid mechanics, Handbook of Numerical Analysis, P.G. Ciarlet and J.L. Lions Eds., Vol. 4, Part 2, North-Holland (1996) 313–485.  Zbl0873.73079
  15. J. Haslinger, R. Kučera and Z. Dostál, An algorithm for numerical realization of 3D contact problems with Coulomb friction. J. Comput. Appl. Math.164-165 (2004) 387–408.  Zbl1107.74328
  16. P. Hild, À propos d'approximation par éléments finis optimale pour les problèmes de contact unilatéral. C.R. Acad. Sci. Paris, Ser. I326 (1998) 1233–1236.  Zbl0914.73060
  17. N. Kikuchi and J.T. Oden, Contact problems in elasticity: a study of variational inequalities and finite element methods. SIAM, Philadelphia (1988).  Zbl0685.73002
  18. D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications. Academic Press (1980).  Zbl0457.35001
  19. K. Lhalouani and T. Sassi, Nonconforming mixed variational formulation and domain decomposition for unilateral problems. East-West J. Numer. Math.7 (1999) 23–30.  Zbl0923.73061

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