Advertisement

Computational Mechanics

, Volume 49, Issue 6, pp 735–747 | Cite as

An abstract framework for a priori estimates for contact problems in 3D with quadratic finite elements

  • B. I. Wohlmuth
  • A. PoppEmail author
  • M. W. Gee
  • W. A. Wall
Original Paper

Abstract

In this paper, a variationally consistent contact formulation is considered and we provide an abstract framework for the a priori error analysis in the special case of frictionless contact and small deformations. Special emphasis is put on quadratic mortar finite element methods. It is shown that under quite weak assumptions on the Lagrange multiplier space \({\mathcal{O} (h^{t-1}), 2 < t < \frac52}\) , a priori results in the H 1-norm for the error in the displacement and in the H −1/2-norm for the error in the surface traction can be established provided that the solution is regular enough. We discuss several choices of Lagrange multipliers ranging from the standard lowest order conforming finite elements to locally defined biorthogonal basis functions. The crucial property for the analysis is that the basis functions have a local positive mean value. Numerical results are exemplarily presented for one particular choice of biorthogonal (i.e. dual) basis functions and also comprise the case of finite deformation contact.

Keywords

Mortar finite element methods Lagrange multipliers Contact problems A priori error analysis 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alart P, Curnier A (1991) A mixed formulation for frictional contact problems prone to Newton like solution methods. Comput Methods Appl Mech Eng 92: 353–375MathSciNetCrossRefGoogle Scholar
  2. 2.
    Belhachmi Z, Ben Belgacem F (2003) Quadratic finite element approximation of the Signorini problem. Math Comput 72: 83–104MathSciNetGoogle Scholar
  3. 3.
    Chen Z, Nochetto R (2000) Residual type a posteriori error estimates for elliptic obstacle problems. Numer Math 84: 527–548MathSciNetCrossRefGoogle Scholar
  4. 4.
    Eck C, Jarusek J, Krbec M (2005) Unilateral contact problems: variational methods and existence theorems. Chapman & Hall/CRC, Boca RatonCrossRefGoogle Scholar
  5. 5.
    Fischer-Cripps A (2000) Introduction to contact mechanics, mechanical engineering series. Springer, New YorkGoogle Scholar
  6. 6.
    Flemisch B, Melenk J, Wohlmuth BI (2005) Mortar methods with curved interfaces. Appl Numer Math 54: 339–361MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gitterle M, Popp A, Gee MW, Wall WA (2010) Finite deformation frictional mortar contact using a semi-smooth Newton method with consistent linearization. Int J Numer Methods Eng 84: 543–571MathSciNetGoogle Scholar
  8. 8.
    Hager C, Wohlmuth BI (2010) Semismooth Newton methods for variational problems with inequality constraints. GAMM Mitteilungen 33: 8–24MathSciNetCrossRefGoogle Scholar
  9. 9.
    Han W, Sofonea M (2002) Quasistatic contact problems in viscoelasticity and viscoplasticity, Studies in Advanced Mathematics, American Mathematical Society. International Press, SomervilleGoogle Scholar
  10. 10.
    Hauret P, Le Tallec P (2007) A discontinuous stabilized mortar method for general 3d elastic problems. Comput Methods Appl Mech Eng 196: 4881–4900MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hertz H (1882) Über die Berührung fester elastischer Körper. J Reine Angew Math 92: 156–171Google Scholar
  12. 12.
    Hild P, Laborde P (2002) Quadratic finite element methods for unilateral contact problems. Appl Numer Math 41: 410–421MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hüeber S, Mair M, Wohlmuth BI (2005) A priori error estimates and an inexact primal-dual active set strategy for linear and quadratic finite elements applied to multibody contact problems. Appl Numer Math 54: 555–576MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hüeber S, Stadler G, Wohlmuth BI (2008) A primal-dual active set algorithm for three-dimensional contact problems with Coulomb friction. SIAM J Sci Comput 30: 572–596MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hüeber S, Wohlmuth BI (2005) A primal-dual active set strategy for non-linear multibody contact problems. Comput Methods Appl Mech Eng 194: 3147–3166CrossRefGoogle Scholar
  16. 16.
    Johnson K (1985) Contact mechanics. Cambridge University Press, CambridgeGoogle Scholar
  17. 17.
    Kikuchi N, Oden JT (1988) Contact problems in elasticity: a study of variational inequalities and finite element methods. Society for Industrial and Applied Mathematics (SIAM), PhiladelphiaGoogle Scholar
  18. 18.
    Laursen TA (2002) Computational contact and impact mechanics. Springer, BerlinGoogle Scholar
  19. 19.
    Li J, Melenk J, Wohlmuth BI, Zou J (2010) Optimal a priori estimates for higher order finite elements for elliptic interface problems. Appl Numer Math 60: 19–37MathSciNetCrossRefGoogle Scholar
  20. 20.
    Moussaoui M, Khodja K (1992) Régularité des solutions d’un problème mêlé Dirichlet–Signorini dans un domaine polygonal plan. Commun Partial Differ Equ 17: 805–826MathSciNetCrossRefGoogle Scholar
  21. 21.
    Nochetto R, Wahlbin L (2002) Positivity preserving finite element approximation. Math Comput 71: 1405–1419MathSciNetGoogle Scholar
  22. 22.
    Popp A, Gee MW, Wall WA (2009) A finite deformation mortar contact formulation using a primal-dual active set strategy. Int J Numer Methods Eng 79: 1354–1391MathSciNetCrossRefGoogle Scholar
  23. 23.
    Popp A, Gitterle M, Gee MW, Wall WA (2010) A dual mortar approach for 3D finite deformation contact with consistent linearization. Int J Numer Methods Eng 83: 1428–1465MathSciNetCrossRefGoogle Scholar
  24. 24.
    Popp A, Wohlmuth BI, Gee MW, Wall WA (2011) Dual quadratic mortar finite element methods for 3D finite deformation contact. Tech. report, Technische Universität MünchenGoogle Scholar
  25. 25.
    Puso MA, Laursen TA (2002) A 3D contact smoothing method using Gregory patches. Int J Numer Methods Eng 54: 1161–1194MathSciNetCrossRefGoogle Scholar
  26. 26.
    Puso MA, Laursen TA (2004) A mortar segment-to-segment contact method for large deformation solid mechanics. Comput Methods Appl Mech Eng 193: 601–629MathSciNetCrossRefGoogle Scholar
  27. 27.
    Puso MA, Laursen TA (2004) A mortar segment-to-segment frictional contact method for large deformations. Comput Methods Appl Mech Eng 193: 4891–4913MathSciNetCrossRefGoogle Scholar
  28. 28.
    Puso MA, Laursen TA, Solberg J (2008) A segment-to-segment mortar contact method for quadratic elements and large deformations. Comput Methods Appl Mech Eng 197: 555–566MathSciNetCrossRefGoogle Scholar
  29. 29.
    Timoshenko SP, Goodier JN (1970) Theory of elasticity. McGraw-Hill, New YorkGoogle Scholar
  30. 30.
    Wall WA, Gee MW (2010) Baci—a multiphysics simulation environment. Tech. report, Technische Universität MünchenGoogle Scholar
  31. 31.
    Wohlmuth BI (2011) Variationally consistent discretization schemes and numerical algorithms for contact problems. Acta Numerica, pp 569–734Google Scholar
  32. 32.
    Wriggers P (2002) Computational contact mechanics. Wiley, New YorkGoogle Scholar
  33. 33.
    Wriggers, P, Nackenhorst, U (eds) (2007) Computational methods in contact mechanics, vol 3 of IUTAM Bookseries. Springer, BerlinGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • B. I. Wohlmuth
    • 1
  • A. Popp
    • 2
    Email author
  • M. W. Gee
    • 3
  • W. A. Wall
    • 2
  1. 1.Institute for Numerical MathematicsTechnische Universität MünchenGarchingGermany
  2. 2.Institute for Computational MechanicsTechnische Universität MünchenGarchingGermany
  3. 3.Mechanics and High Performance Computing GroupTechnische Universität MünchenGarchingGermany

Personalised recommendations