Locking-Free Finite Elements for Unilateral Crack Problems in Elasticity

Z. Belhachmi; J.-M. Sac-Epée; S. Tahir

Mathematical Modelling of Natural Phenomena (2009)

  • Volume: 4, Issue: 1, page 1-20
  • ISSN: 0973-5348

Abstract

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We consider mixed and hybrid variational formulations to the linearized elasticity system in domains with cracks. Inequality type conditions are prescribed at the crack faces which results in unilateral contact problems. The variational formulations are extended to the whole domain including the cracks which yields, for each problem, a smooth domain formulation. Mixed finite element methods such as PEERS or BDM methods are designed to avoid locking for nearly incompressible materials in plane elasticity. We study and implement discretizations based on such mixed finite element methods for the smooth domain formulations to the unilateral crack problems. We obtain convergence rates and optimal error estimates and we present some numerical experiments in agreement with the theoretical results.

How to cite

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Belhachmi, Z., Sac-Epée, J.-M., and Tahir, S.. "Locking-Free Finite Elements for Unilateral Crack Problems in Elasticity." Mathematical Modelling of Natural Phenomena 4.1 (2009): 1-20. <http://eudml.org/doc/222437>.

@article{Belhachmi2009,
abstract = { We consider mixed and hybrid variational formulations to the linearized elasticity system in domains with cracks. Inequality type conditions are prescribed at the crack faces which results in unilateral contact problems. The variational formulations are extended to the whole domain including the cracks which yields, for each problem, a smooth domain formulation. Mixed finite element methods such as PEERS or BDM methods are designed to avoid locking for nearly incompressible materials in plane elasticity. We study and implement discretizations based on such mixed finite element methods for the smooth domain formulations to the unilateral crack problems. We obtain convergence rates and optimal error estimates and we present some numerical experiments in agreement with the theoretical results.},
author = {Belhachmi, Z., Sac-Epée, J.-M., Tahir, S.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {crack problems; variational inequalities; smooth domain method; mixed finite elements; a priori estimates; mixed finite elements},
language = {eng},
month = {1},
number = {1},
pages = {1-20},
publisher = {EDP Sciences},
title = {Locking-Free Finite Elements for Unilateral Crack Problems in Elasticity},
url = {http://eudml.org/doc/222437},
volume = {4},
year = {2009},
}

TY - JOUR
AU - Belhachmi, Z.
AU - Sac-Epée, J.-M.
AU - Tahir, S.
TI - Locking-Free Finite Elements for Unilateral Crack Problems in Elasticity
JO - Mathematical Modelling of Natural Phenomena
DA - 2009/1//
PB - EDP Sciences
VL - 4
IS - 1
SP - 1
EP - 20
AB - We consider mixed and hybrid variational formulations to the linearized elasticity system in domains with cracks. Inequality type conditions are prescribed at the crack faces which results in unilateral contact problems. The variational formulations are extended to the whole domain including the cracks which yields, for each problem, a smooth domain formulation. Mixed finite element methods such as PEERS or BDM methods are designed to avoid locking for nearly incompressible materials in plane elasticity. We study and implement discretizations based on such mixed finite element methods for the smooth domain formulations to the unilateral crack problems. We obtain convergence rates and optimal error estimates and we present some numerical experiments in agreement with the theoretical results.
LA - eng
KW - crack problems; variational inequalities; smooth domain method; mixed finite elements; a priori estimates; mixed finite elements
UR - http://eudml.org/doc/222437
ER -

References

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  1. J. Alberty, C. Carstensen, S. A. Funken, R. Klose. Matlab Implementation of the Finite Element Method in Elasticity. Berichtsreihe des Mathematischen Seminars Kiel, 00-21 (2000).  
  2. D. N. Arnold, F. Brezzi, J. Douglas. PEERS: A new finite element for plane elasticity Japan J. Appl. Math., No. 1 (1984), 347–367.  
  3. Z. Belhachmi, F. Ben Belgacem. Quadratic finite element for Signorini problem. Math. Comp., 72 (2003), No. 241, 83–104.  
  4. Z. Belhachmi, J.M. Sac-Epée, J. Sokolowski. Mixed finite element methods for a smooth domain formulation of a crack problem. SIAM J. Numer. Anal., 43 (2005), No. 3, 1295–1320.  
  5. F. Ben Belgacem. Numerical simulation of some variational inequalities arisen from unilateral contact problems by finite element method. Siam J. Numer. Anal, 37 (2000),No. 4, 1198–1216.  
  6. F. Ben Belgacem, P. Hild, P. Laborde. Extension of the mortar finite element method to a variational inequality modelling unilateral contact. Math. Models Methods Appl. Sci., 9 (1999), No. 2, 287–303.  
  7. F. Ben Belgacem, Y. Renard. Hybrid finite element methods for the Signorini problem. Math. Comput., 72 (2003), No. 243, 1117–1145.  
  8. C. Bernardi, V. Girault. A local regularization operator for triangular and quadrilateral finite elements. SIAM. J. Numer. Anal., 35 (1998), No. 5, 1893–1916.  
  9. D. Braess, O. Klaas, R. Niekamp, E. Stein, F. Wobschal. Error Indicators For Mixed Finite Elements in 2-dimensional Linear Elasticity. Comput. Methods. Appl. Mech. Engrg., 127 (1995), No. 1-4, 345–356.  
  10. F. Brezzi, J. Douglas Jr, L.D. Marini. Two families of mixed finite elements for second order elliptic problems. Numer. Math., 47 (1985), No. 2, 217–235.  
  11. F. Brezzi, M. Fortin. Mixed and hybrid finite element methods. Springer Verlag, New York, Springer Series in Computational Mathematics, 15, 1991.  
  12. C. Carstensen, G. Dolzmann, S.A. Funken, D.S. Helm. Locking-free adaptive mixed finite element in linear elasticity. Comput. Methods. Appl.Mech. Engrg., 190 (2000), No. 13-14, 1701–1718.  
  13. P.G. Ciarlet. Basic Error Estimates for Elliptic Problems. In the Handbook of Numerical Analysis, Vol II, P.G. Ciarlet & J.-L. Lions eds, North-Holland, (1991), 17–351.  
  14. P. Coorevits, P. Hild, K. Lhalouani, T. Sassi. Mixed finite element methods for unilateral problems: convergence analysis and numerical studies. Math. Comp., 71, (2001), No. 237, 1–25.  
  15. G. Duvaut, J.-L. Lions. Les inéquations en mécanique et en physique. Dunod, 1972.  
  16. V. Girault, P.-A. Raviart. Finite element methods for the Navier-Stokes equations, Theory and algorithms. Springer-Verlag 1986.  
  17. R. Glowinski. Lectures on numerical methods for nonlinear variational problems. Springer, Berlin, 1980.  
  18. J. Haslinger, I. Hlaváček. Contact between Elastic Bodies -2.Finite Element Analysis, Aplikace Matematiky, 26 (1981), 263–290.  
  19. J. Haslinger, I. Hlaváček, J. Nečas. Numerical Methods for Unilateral Problems in Solid Mechanics, in the Handbook of Numerical Analysis, Vol IV, Part 2, P.G. Ciarlet & J.-L. Lions eds, North-Holland, 1996.  
  20. F. Hecht, O. Pironneau. FreeFem++, www.freefem.org  
  21. P. Hild, Y. Renard. An error estimates for the Signorini problem with Coulomb friction approximated by finite elements. Siam J. Numer. Anal., 45 (2007), No. 5, 2012–2031.  
  22. S. Hüeber, B.I. Wohlmuth. An optimal a priori error estimates for nonlinear multibody contact problems. SIAM J. Numer. Anal., 43 (2005), No. 1, 156–173  
  23. A.M. Khludnev, J. Sokolowski. Smooth domain method for crack problems. Quarterly of Applied Mathematics., 62 (2004), No. 3, 401–422.  
  24. N. Kikuchi, J. Oden. Contact problems in elasticity: A study of variational inequalities and finite element methods. SIAM, 1988.  
  25. D. Kinderlehrer, G. Stamppachia. An introduction to variational inequalities and their applications, Academic Press, 1980.  
  26. K. Lhalouani, T. Sassi. Nonconforming mixed variational formulation and domain decomposition for unilateral problems. East-West J. Numer. Math., 7 (1999), No. 1, 23–30.  
  27. L. Slimane, A. Bendali, P. Laborde. Mixed formulations for a class of variational inequalities. M2AN, 38 (2004), 1, 177–201.  
  28. R. Stenberg. A family of mixed finite elements for the elasticity problem. Numer. Math., 53 (1988), 5, 513–538.  
  29. S. Tahir. Méthodes d'approximation par éléments finis et analyse a posteriori d'inéquations variationnelles modélisant des problèmes de fissures unilatérales en élasticité linéaire. Ph.D. Thesis, University of Metz, France (2006).  
  30. S. Tahir, Z. Belhachmi. Mixed finite elements discretizations of some variational inequalities arising in elasticity problems in domains with cracks. Electron. J. Diff. Eqns., Conference 11 (2004), 33–40.  
  31. Z.-H. Zhong. Finite Element Procedures for Contact-Impact Problems. Oxford. University. Press, Oxford 1993.  

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