Advertisement

Numerische Mathematik

, Volume 139, Issue 3, pp 593–631 | Cite as

An unbiased Nitsche’s approximation of the frictional contact between two elastic structures

  • Franz Chouly
  • Rabii MlikaEmail author
  • Yves Renard
Article

Abstract

Most of the numerical methods dedicated to the contact problem involving two elastic bodies are based on the master/slave paradigm. It results in important detection difficulties in the case of self-contact and multi-body contact, where it may be impractical, if not impossible, to a priori nominate a master surface and a slave one. In this work we introduce an unbiased finite element method for the finite element approximation of frictional contact between two elastic bodies in the small deformation framework. In the proposed method the two bodies expected to come into contact are treated in the same way (no master and slave surfaces). The key ingredient is a Nitsche-based formulation of contact conditions, as in Chouly et al. (Math Comput 84:1089–1112, 2015). We carry out the numerical analysis of the method, and prove its well-posedness and optimal convergence in the \(H^1\)-norm. Numerical experiments are performed to illustrate the theoretical results and the performance of the method.

Mathematics Subject Classification

74M10 65N30 74M15 

Notes

Acknowledgements

We would like to sincerely thank the company “Manufacture Française des Pneumatiques Michelin” for the financial and technical support. We thank, as well, Région Franche-Comté for partial funding (Convention Région 2015C-4991 “Modèles mathématiques et méthodes numériques pour l’élasticité non-linéaire”).

References

  1. 1.
    Becker, R., Hansbo, P., Stenberg, R.: A finite element method for domain decomposition with non-matching grids. Math. Model. Numer. Anal. 37, 209–225 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ben Belgacem, F., Hild, P., Laborde, P.: Extension of the mortar finite element method to a variational inequality modeling unilateral contact. Math. Models Methods Appl. Sci. 09, 287–303 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brenner, S.-C., Scott, L.-R.: The Mathematical Theory of Finite Element Methods, vol. 15. Springer, New York (2007). Texts in Applied MathematicszbMATHGoogle Scholar
  4. 4.
    Brezis, H.: Équations et inéquations non linéaires dans les espaces vectoriels en dualité. Ann. Inst. Fourier (Grenoble) 18, 115–175 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chouly, F.: An adaptation of Nitsche’s method to the Tresca friction problem. J. Math. Anal. Appl. 411, 329–339 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chouly, F., Hild, P.: Nitsche-based method for unilateral contact problems: numerical analysis. SIAM J. Numer. Anal. 51, 1295–1307 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chouly, F., Hild, P.: On convergence of the penalty method for unilateral contact problems. App. Numer. Math. 65, 27–40 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chouly, F., Hild, P., Renard, Y.: Symmetric and non-symmetric variants of Nitsche’s method for contact problems in elasticity: theory and numerical experiments. Math. Comput. 84, 1089–1112 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dupont, T., Scott, R.: Polynomial approximation of functions in Sobolev spaces. Math. Comput. 34, 441–463 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Applied Mathematical Sciences, vol. 159. Springer, New York (2004)CrossRefzbMATHGoogle Scholar
  11. 11.
    Fabre, M., Pousin, J., Renard, Y.: A fictitious domain method for frictionless contact problems in elasticity using Nitsche’s method. SMAI J. Comp. Math. 2, 19–50 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Fritz, A., Hüeber, S., Wohlmuth, B.: A comparison of mortar and Nitsche techniques for linear elasticity. Calcolo 41, 115–137 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hansbo, A., Hansbo, P.: A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Comput. Methods Appl. Mech. Eng. 193, 3523–3540 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Haslinger, J., Hlaváĉek, I., Neĉas, J.: Numerical methods for unilateral problems in solid mechanics. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis. Elsevier, London (1996)Google Scholar
  15. 15.
    Heintz, P., Hansbo, P.: Stabilized lagrange multiplier methods for bilateral elastic contact with friction. Comput. Methods Appl. Mech. Eng. 195, 4323–4333 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hild, P., Renard, Y.: A stabilized lagrange multiplier method for the finite element approximation of contact problems in elastostatics. Numer. Math. 115, 101–129 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1988)CrossRefzbMATHGoogle Scholar
  18. 18.
    Laursen, T.: Formulation and treatment of frictional contact problems using finite elements. PhD thesis, Stanford Univ., CA. (1992)Google Scholar
  19. 19.
    Laursen, T.: Computational Contact and Impact Mechanics. Springer, Berlin (2002)zbMATHGoogle Scholar
  20. 20.
    Laursen, T., Simo, J.: A continuum-based finite element formulation for the implicit solution of multibody, large deformation frictional contact problems. Int. J. Numer. Meth. Eng. 36, 451–3485 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    McDevitt, T.W., Laursen, T.A.: A mortar-finite element formulation for frictional contact problems. Int. J. Numer. Methods Eng. 48, 1525–1547 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Moussaoui, M., Khodja, K.: Régularité des solutions d’un problème mêlé Dirichlet-Signorini dans un domaine polygonal plan. Commun. Partial Differ. Equ. 17, 805–826 (1992)CrossRefzbMATHGoogle Scholar
  23. 23.
    Nitsche, J.: Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Semin. Univ. Hamb. 36, 9–15 (1971)CrossRefzbMATHGoogle Scholar
  24. 24.
    Popp, A., Wohlmuth, B.I., Gee, M.W., Wall, W.A.: Dual quadratic mortar finite element methods for 3D finite deformation contact. SIAM J. Sci. Comput. 34, B421–B446 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Renard, Y.: Generalized Newton’s methods for the approximation and resolution of frictional contact problems in elasticity. Comp. Methods Appl. Mech. Eng. 256, 38–55 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Sauer, R.A., DeLorenzis, L.: An unbiased computational contact formulation for 3D friction. Int. J. Numer. Meth. Eng. 101, 251–280 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Stenberg, R.: On some techniques for approximating boundary conditions in the finite element method. J. Comput. Appl. Math. 63, 139–148 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Wohlmuth, B.: Variationally consistent discretization schemes and numerical algorithms for contact problems. Acta Numer. 20, 569–734 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Wohlmuth, B.I., Krause, R.H.: Monotone multigrid methods on nonmatching grids for nonlinear multibody contact problems. SIAM J. Sci. Comput. 25, 324–347 (2003)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques de Besançon - UMR CNRS 6623Université de Franche ComtéBesançon CedexFrance
  2. 2.CNRS, INSA-Lyon, LaMCoS UMR5259Université de LyonVilleurbanneFrance
  3. 3.CNRS, INSA-Lyon, ICJ UMR5208, LaMCoS UMR5259Université de LyonVilleurbanneFrance

Personalised recommendations