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A sign preserving mixed finite element approximation for contact problems

Patrick Hild

International Journal of Applied Mathematics and Computer Science (2011)

  • Volume: 21, Issue: 3, page 487-498
  • ISSN: 1641-876X

Abstract

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This paper is concerned with the frictionless unilateral contact problem (i.e., a Signorini problem with the elasticity operator). We consider a mixed finite element method in which the unknowns are the displacement field and the contact pressure. The particularity of the method is that it furnishes a normal displacement field and a contact pressure satisfying the sign conditions of the continuous problem. The a priori error analysis of the method is closely linked with the study of a specific positivity preserving operator of averaging type which differs from the one of Chen and Nochetto. We show that this method is convergent and satisfies the same a priori error estimates as the standard approach in which the approximated contact pressure satisfies only a weak sign condition. Finally we perform some computations to illustrate and compare the sign preserving method with the standard approach.

How to cite

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Patrick Hild. "A sign preserving mixed finite element approximation for contact problems." International Journal of Applied Mathematics and Computer Science 21.3 (2011): 487-498. <http://eudml.org/doc/208063>.

@article{PatrickHild2011,
abstract = {This paper is concerned with the frictionless unilateral contact problem (i.e., a Signorini problem with the elasticity operator). We consider a mixed finite element method in which the unknowns are the displacement field and the contact pressure. The particularity of the method is that it furnishes a normal displacement field and a contact pressure satisfying the sign conditions of the continuous problem. The a priori error analysis of the method is closely linked with the study of a specific positivity preserving operator of averaging type which differs from the one of Chen and Nochetto. We show that this method is convergent and satisfies the same a priori error estimates as the standard approach in which the approximated contact pressure satisfies only a weak sign condition. Finally we perform some computations to illustrate and compare the sign preserving method with the standard approach.},
author = {Patrick Hild},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {variational inequality; positive operator; averaging operator; contact problem; Signorini problem; mixed finite element method},
language = {eng},
number = {3},
pages = {487-498},
title = {A sign preserving mixed finite element approximation for contact problems},
url = {http://eudml.org/doc/208063},
volume = {21},
year = {2011},
}

TY - JOUR
AU - Patrick Hild
TI - A sign preserving mixed finite element approximation for contact problems
JO - International Journal of Applied Mathematics and Computer Science
PY - 2011
VL - 21
IS - 3
SP - 487
EP - 498
AB - This paper is concerned with the frictionless unilateral contact problem (i.e., a Signorini problem with the elasticity operator). We consider a mixed finite element method in which the unknowns are the displacement field and the contact pressure. The particularity of the method is that it furnishes a normal displacement field and a contact pressure satisfying the sign conditions of the continuous problem. The a priori error analysis of the method is closely linked with the study of a specific positivity preserving operator of averaging type which differs from the one of Chen and Nochetto. We show that this method is convergent and satisfies the same a priori error estimates as the standard approach in which the approximated contact pressure satisfies only a weak sign condition. Finally we perform some computations to illustrate and compare the sign preserving method with the standard approach.
LA - eng
KW - variational inequality; positive operator; averaging operator; contact problem; Signorini problem; mixed finite element method
UR - http://eudml.org/doc/208063
ER -

References

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  1. Adams, R. (1975). Sobolev Spaces, Academic Press, New York, NY/London. Zbl0314.46030
  2. Belhachmi, Z., Sac-Epée, J.-M. and Sokolowski, J. (2005). Mixed finite element methods for smooth domain formulation of crack problems, SIAM Journal on Numerical Analysis 43(3): 1295-1320. Zbl1319.74027
  3. Ben Belgacem, F. and Brenner, S. (2001). Some nonstandard finite element estimates with applications to 3D Poisson and Signorini problems, Electronic Transactions on Numerical Analysis 12: 134-148. Zbl0981.65131
  4. Ben Belgacem, F., Hild, P. and Laborde, P. (1999). Extension of the mortar finite element method to a variational inequality modeling unilateral contact, Mathematical Models and Methods in the Applied Sciences 9(2): 287-303. Zbl0940.74056
  5. Ben Belgacem, F. and Renard, Y. (2003). Hybrid finite element methods for the Signorini problem, Mathematics of Computation 72(243): 1117-1145. Zbl1023.74043
  6. Bernardi, C. and Girault, V. (1998). A local regularisation operator for triangular and quadrilateral finite elements, SIAM Journal on Numerical Analysis 35(5): 1893-1916. Zbl0913.65007
  7. Brenner, S. and Scott, L. (2002). The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, NY. Zbl1012.65115
  8. Chen, Z. and Nochetto, R. (2000). Residual type a posteriori error estimates for elliptic obstacle problems, Numerische Mathematik 84(4): 527-548. Zbl0943.65075
  9. Ciarlet, P. (1991). The finite element method for elliptic problems, in P.G. Ciarlet and J.-L. Lions (Eds.), Handbook of Numerical Analysis, Vol. II, Part 1, North Holland, Amsterdam, pp. 17-352. 
  10. Clément, P. (1975). Approximation by finite element functions using local regularization, RAIRO Modélisation Mathématique et Analyse Numérique 2(R-2): 77-84. Zbl0368.65008
  11. Coorevits, P., Hild, P., Lhalouani, K. and Sassi, T. (2002). Mixed finite element methods for unilateral problems: Convergence analysis and numerical studies, Mathematics of Computation 71(237): 1-25. Zbl1013.74062
  12. Duvaut, G. and Lions, J.-L. (1972). Les inéquations en mécanique et en physique Dunod, Paris. Zbl0298.73001
  13. Eck, C., Jarušek, J. and Krbec, M. (2005). Unilateral Contact Problems. Variational Methods and Existence Theorems, CRC Press, Boca Raton, FL. Zbl1079.74003
  14. Fichera, G. (1964). Elastic problems with unilateral constraints, the problem of ambiguous boundary conditions, Memorie della Accademia Nazionale dei Lincei 8(7): 91-140, (in Italian). 
  15. Fichera, G. (1974). Existence theorems in linear and semilinear elasticity, Zeitschrift für Angewandte Mathematik und Mechanik 54(12): 24-36. 
  16. Grisvard, P. (1985). Elliptic Problems in Nonsmooth Domains, Pitman, Boston, MA. Zbl0695.35060
  17. Han, W. and Sofonea, M. (2002). Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, American Mathematical Society, Providence, RI. Zbl1013.74001
  18. Haslinger, J., Hlaváček, I. and Nečas, J. (1996). Numerical methods for unilateral problems in solid mechanics, in P. Ciarlet and J.-L. Lions (Eds.), Handbook of Numerical Analysis, Vol. IV, Part 2, North Holland, Amsterdam, pp. 313-485. Zbl0873.73079
  19. Hilbert, S. (1973). A mollifier useful for approximations in Sobolev spaces and some applications to approximating solutions of differential equations, Mathematics of Computation 27: 81-89. Zbl0257.65087
  20. Hild, P. (2000). Numerical implementation of two nonconforming finite element methods for unilateral contact, Computer Methods in Applied Mechanics and Engineering 184(1): 99-123. Zbl1009.74062
  21. Hild, P. (2002). On finite element uniqueness studies for Coulomb's frictional contact model, International Journal of Applied Mathematics and Computer Science 12(1): 41-50. Zbl1041.74070
  22. Hild, P. and Nicaise, S. (2007). Residual a posteriori error estimators for contact problems in elasticity, Mathematical Modelling and Numerical Analysis 41(5): 897-923. Zbl1140.74024
  23. Hiriart-Urruty, J.-B. and Lemaréchal, C. (1993). Convex Analysis and Minimization Algorithms I, Springer, Berlin. Zbl0795.49001
  24. Hüeber, S. and Wohlmuth, B. (2005a). An optimal error estimate for nonlinear contact problems, SIAM Journal on Numerical Analysis 43(1): 156-173. Zbl1083.74047
  25. Hüeber, S. and Wohlmuth, B. (2005b). A primal-dual active set strategy for non-linear multibody contact problems, Computer Methods in Applied Mechanics and Engineering 194(27-29): 3147-3166. Zbl1093.74056
  26. Khludnev, A. and Sokolowski, J. (2004). Smooth domain method for crack problems, Quarterly of Applied Mathematics 62(3): 401-422. Zbl1067.74056
  27. Kikuchi, N. and Oden, J. (1988). Contact Problems in Elasticity, SIAM, Philadelphia, PA. Zbl0685.73002
  28. Laursen, T. (2002). Computational Contact and Impact Mechanics, Springer, Berlin. Zbl0996.74003
  29. Nochetto, R. and Wahlbin, L. (2002). Positivity preserving finite element approximation, Mathematics of Computation 71(240): 1405-1419. Zbl1001.41011
  30. Scott, L. and Zhang, S. (1990). Finite element interpolation of nonsmooth functions satisfying boundary conditions, Mathematics of Computation 54(190): 483-493. Zbl0696.65007
  31. Strang, G. (1972). Approximation in the finite element method, Numerische Mathematik 19: 81-98. Zbl0221.65174
  32. Wohlmuth, B. and Krause, R. (2003). Monotone multigrid methods on nonmatching grids for nonlinear multibody contact problems, SIAM Journal on Scientific Computation 25(1): 324-347. Zbl1163.65334
  33. Wriggers, P. (2002). Computational Contact Mechanics, Wiley, Chichester. Zbl1104.74002

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