A stabilized Lagrange multiplier method for the enriched finite-element approximation of contact problems of cracked elastic bodies

Saber Amdouni; Patrick Hild; Vanessa Lleras; Maher Moakher; Yves Renard

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

  • Volume: 46, Issue: 4, page 813-839
  • ISSN: 0764-583X

Abstract

top
The purpose of this paper is to provide a priori error estimates on the approximation of contact conditions in the framework of the eXtended Finite-Element Method (XFEM) for two dimensional elastic bodies. This method allows to perform finite-element computations on cracked domains by using meshes of the non-cracked domain. We consider a stabilized Lagrange multiplier method whose particularity is that no discrete inf-sup condition is needed in the convergence analysis. The contact condition is prescribed on the crack with a discrete multiplier which is the trace on the crack of a finite-element method on the non-cracked domain, avoiding the definition of a specific mesh of the crack. Additionally, we present numerical experiments which confirm the efficiency of the proposed method.

How to cite

top

Amdouni, Saber, et al. "A stabilized Lagrange multiplier method for the enriched finite-element approximation of contact problems of cracked elastic bodies." ESAIM: Mathematical Modelling and Numerical Analysis 46.4 (2012): 813-839. <http://eudml.org/doc/277840>.

@article{Amdouni2012,
abstract = {The purpose of this paper is to provide a priori error estimates on the approximation of contact conditions in the framework of the eXtended Finite-Element Method (XFEM) for two dimensional elastic bodies. This method allows to perform finite-element computations on cracked domains by using meshes of the non-cracked domain. We consider a stabilized Lagrange multiplier method whose particularity is that no discrete inf-sup condition is needed in the convergence analysis. The contact condition is prescribed on the crack with a discrete multiplier which is the trace on the crack of a finite-element method on the non-cracked domain, avoiding the definition of a specific mesh of the crack. Additionally, we present numerical experiments which confirm the efficiency of the proposed method.},
author = {Amdouni, Saber, Hild, Patrick, Lleras, Vanessa, Moakher, Maher, Renard, Yves},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Extended finite element method (Xfem); crack; unilateral contact; Signorini’s problem; extended finite element method (XFEM); Signorini's problem},
language = {eng},
month = {2},
number = {4},
pages = {813-839},
publisher = {EDP Sciences},
title = {A stabilized Lagrange multiplier method for the enriched finite-element approximation of contact problems of cracked elastic bodies},
url = {http://eudml.org/doc/277840},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Amdouni, Saber
AU - Hild, Patrick
AU - Lleras, Vanessa
AU - Moakher, Maher
AU - Renard, Yves
TI - A stabilized Lagrange multiplier method for the enriched finite-element approximation of contact problems of cracked elastic bodies
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2012/2//
PB - EDP Sciences
VL - 46
IS - 4
SP - 813
EP - 839
AB - The purpose of this paper is to provide a priori error estimates on the approximation of contact conditions in the framework of the eXtended Finite-Element Method (XFEM) for two dimensional elastic bodies. This method allows to perform finite-element computations on cracked domains by using meshes of the non-cracked domain. We consider a stabilized Lagrange multiplier method whose particularity is that no discrete inf-sup condition is needed in the convergence analysis. The contact condition is prescribed on the crack with a discrete multiplier which is the trace on the crack of a finite-element method on the non-cracked domain, avoiding the definition of a specific mesh of the crack. Additionally, we present numerical experiments which confirm the efficiency of the proposed method.
LA - eng
KW - Extended finite element method (Xfem); crack; unilateral contact; Signorini’s problem; extended finite element method (XFEM); Signorini's problem
UR - http://eudml.org/doc/277840
ER -

References

top
  1. R. Adams, Sobolev spaces. Academic Press, New York (1975).  Zbl0314.46030
  2. P. Alart and A. Curnier, A mixed formulation for frictional contact problems prone to Newton like solution methods. Comput. Methods Appl. Mech. Eng.92 (1991) 353–375.  Zbl0825.76353
  3. H.J. Barbosa and T. Hughes, The finite element method with Lagrange multipliers on the boundary : circumventing the Babuška-Brezzi condition. Comput. Methods Appl. Mech. Eng.85 (1991) 109–128.  Zbl0764.73077
  4. H.J. Barbosa and T. Hughes, Boundary Lagrange multipliers in finite element methods : error analysis in natural norms. Numer. Math.62 (1992) 1–15.  Zbl0765.65102
  5. H.J. Barbosa and T. Hughes, Circumventing the Babuška-Brezzi condition in mixed finite element approximations of elliptic variational inequalities. Comput. Methods Appl. Mech. Eng.97 (1992) 193–210.  Zbl0768.65033
  6. R. Becker, P. Hansbo and R. Stenberg, A finite element method for domain decomposition with non-matching grids. ESAIM : M2AN37 (2003) 209–225.  Zbl1047.65099
  7. F. Ben Belgacem, Numerical simulation of some variational inequalities arisen from unilateral contact problems by the finite element method. SIAM J. Numer. Anal.37 (2000) 1198–1216.  Zbl0974.74055
  8. F. Ben Belgacem and Y. Renard, Hybrid finite element methods for the Signorini problem. Math. Comp.72 (2003) 1117–1145.  Zbl1023.74043
  9. H. Ben Dhia and M. Zarroug, Hybrid frictional contact particles in elements. Revue Européenne des Éléments Finis9 (2002) 417–430.  Zbl1120.74814
  10. S. Bordas and M. Duflot, Derivative recovery and a posteriori error estimate for extended finite elements. Comput. Methods Appl. Mech. Eng.196 (2007) 3381–3399.  Zbl1173.74401
  11. S. Bordas and M. Duflot, A posteriori error estimation for extended finite elements by an extended global recovery. Int. J. Numer. Methods Eng.76 (2008) 1123–1138.  Zbl1195.74171
  12. S. Bordas, M. Duflot and P. Le, A simple error estimator for extended finite elements. Commun. Numer. Methods Eng.24 (2008) 961–971.  Zbl1156.65093
  13. E. Chahine, P. Laborde and Y. Renard, Crack-tip enrichment in the XFEM method using a cut-off function. Int. J. Numer. Methods Eng.75 (2008) 629–646.  Zbl1195.74167
  14. P. Ciarlet, The finite element method for elliptic problems, in Handbook of Numerical Analysis. Part 1, edited by P. Ciarlet and J. Lions, North Holland II (1991) 17–352.  
  15. J. Dolbow, N. Moës and T. Belytschko, An extended finite element method for modelling crack growth with frictional contact. Int. J. Numer. Methods Eng.46 (1999) 131–150.  Zbl0955.74066
  16. S. Géniaut, Approche XFEM pour la fissuration sous contact des structures industrielles. Thèse, École Centrale Nantes (2006).  
  17. S. Géniaut, P. Massin and N. Moës, A stable 3D contact formulation for cracks using XFEM. Revue Européenne de Mécanique Numérique, Calculs avec Méthodes sans Maillage16 (2007) 259–275.  Zbl1208.74096
  18. P. Grisvard, Elliptic problems in nonsmooth domains. Pitman (1985).  Zbl0695.35060
  19. P. Hansbo, C. Lovadina, I. Perugia and G. Sangalli, A Lagrange multiplier method for the finite element solution of elliptic interface problems using nonmatching meshes. Numer. Math.100 (2005) 91–115.  Zbl1066.65125
  20. J. Haslinger and Y. Renard, A new fictitious domain approach inspired by the extended finite element method. SIAM J. Numer. Anal. 47 (2009) 1474–1499.  Zbl1205.65322
  21. J. Haslinger, I. Hlaváček and J. Nečas, Numerical methods for unilateral problems in solid mechanics, in Handbook of Numerical Analysis. Part 2, edited by P. Ciarlet and J.-L. Lions, North Holland IV (1996) 313–485.  Zbl0873.73079
  22. P. Heintz and P. Hansbo, Stabilized Lagrange multiplier methods for bilateral elastic contact with friction. Comput. Methods Appl. Mech. Eng.195 (2006) 4323–4333.  Zbl1123.74045
  23. P. Hild, Numerical implementation of two nonconforming finite element methods for unilateral contact. Comput. Methods Appl. Mech. Eng.184 (2000) 99–123.  Zbl1009.74062
  24. P. Hild and Y. Renard, An error estimate for the Signorini problem with Coulomb friction approximated by finite elements. SIAM J. Numer. Anal.45 (2007) 2012–2031.  Zbl1146.74050
  25. P. Hild and Y. Renard, A stabilized Lagrange multiplier method for the finite element approximation of contact problems in elastostatics. Numer. Math.15 (2010) 101–129.  Zbl1194.74408
  26. P. Hild, V. Lleras and Y. Renard, A residual error estimator for the XFEM approximation of the elasticity problem. Submitted.  Zbl1169.65107
  27. S. Hüeber, B.I. Wohlmuth, An optimal a priori error estimate for nonlinear multibody contact problems. SIAM J. Numer. Anal.43 (2005) 156–173.  Zbl1083.74047
  28. 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) 163–192.  Zbl1089.74046
  29. A. Khoei and M. Nikbakht, Contact friction modeling with the extended finite element method (XFEM). J. Mater. Proc. Technol.177 (2006) 58–62.  
  30. A. Khoei and M. Nikbakht, An enriched finite element algorithm for numerical computation of contact friction problems. Int. J. Mech. Sci.49 (2007) 183–199.  
  31. N. Kikuchi and J. Oden, Contact problems in elasticity. SIAM, Philadelphia (1988).  Zbl0685.73002
  32. P. Laborde and Y. Renard, Fixed point strategies for elastostatic frictional contact problems. Math. Methods Appl. Sci.31 (2008) 415–441.  Zbl1132.74032
  33. N. Moës, J. Dolbow and T. Belytschko, A finite element method for cracked growth without remeshing. Int. J. Numer. Methods Eng.46 (1999) 131–150.  Zbl0955.74066
  34. M. Moussaoui and K. Khodja, Regularité des solutions d’un problème mêlé Dirichlet-Signorini dans un domaine polygonal plan. Commun. Partial Differential Equations17 (1992) 805–826.  Zbl0806.35049
  35. S. Nicaise, Y. Renard and E. Chahine, Optimal convergence analysis for the extended finite element method. Int. J. Numer. Methods Eng.86 (2011) 528–548.  Zbl1216.74029
  36. E. Pierres, M.-C. Baietto and A. Gravouil, A two-scale extended finite element method for modeling 3D crack growth with interfacial contact. Comput. Methods Appl. Mech. Eng.199 (2010) 1165–1177.  Zbl1227.74088
  37. J. Pommier and Y. Renard, Getfem++, an open source generic C++ library for finite element methods. Available on : , December, 23rd (2011).  URIhttp://download.gna.org/getfem/html/homepage/userdoc/index.html
  38. J.J. Rodenas, O.A. Gonzales-Estrada and J.E. Tarancon, A recovery-type error estimator for the extended finite element method based on singular plus smooth stress field splitting. Int. J. Numer. Methods Eng.76 (2008) 545–571.  Zbl1195.74194
  39. R. Stenberg, On some techniques for approximating boundary conditions in the finite element method. J. Comput. Appl. Math.63 (1995) 139–148.  Zbl0856.65130
  40. G. Strang and G. Fix, An analysis of the finite element method. Prentice-Hall, Englewood Cliffs (1973).  Zbl0356.65096

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.