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

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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.

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References

  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: 353–375

    Article  MathSciNet  Google Scholar 

  2. Belhachmi Z, Ben Belgacem F (2003) Quadratic finite element approximation of the Signorini problem. Math Comput 72: 83–104

    MathSciNet  Google Scholar 

  3. Chen Z, Nochetto R (2000) Residual type a posteriori error estimates for elliptic obstacle problems. Numer Math 84: 527–548

    Article  MathSciNet  Google Scholar 

  4. Eck C, Jarusek J, Krbec M (2005) Unilateral contact problems: variational methods and existence theorems. Chapman & Hall/CRC, Boca Raton

    Book  Google Scholar 

  5. Fischer-Cripps A (2000) Introduction to contact mechanics, mechanical engineering series. Springer, New York

    Google Scholar 

  6. Flemisch B, Melenk J, Wohlmuth BI (2005) Mortar methods with curved interfaces. Appl Numer Math 54: 339–361

    Article  MathSciNet  Google Scholar 

  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: 543–571

    MathSciNet  Google Scholar 

  8. Hager C, Wohlmuth BI (2010) Semismooth Newton methods for variational problems with inequality constraints. GAMM Mitteilungen 33: 8–24

    Article  MathSciNet  Google Scholar 

  9. Han W, Sofonea M (2002) Quasistatic contact problems in viscoelasticity and viscoplasticity, Studies in Advanced Mathematics, American Mathematical Society. International Press, Somerville

    Google Scholar 

  10. Hauret P, Le Tallec P (2007) A discontinuous stabilized mortar method for general 3d elastic problems. Comput Methods Appl Mech Eng 196: 4881–4900

    Article  MathSciNet  Google Scholar 

  11. Hertz H (1882) Über die Berührung fester elastischer Körper. J Reine Angew Math 92: 156–171

    Google Scholar 

  12. Hild P, Laborde P (2002) Quadratic finite element methods for unilateral contact problems. Appl Numer Math 41: 410–421

    Article  MathSciNet  Google Scholar 

  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: 555–576

    Article  MathSciNet  Google Scholar 

  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: 572–596

    Article  MathSciNet  Google Scholar 

  15. Hüeber S, Wohlmuth BI (2005) A primal-dual active set strategy for non-linear multibody contact problems. Comput Methods Appl Mech Eng 194: 3147–3166

    Article  Google Scholar 

  16. Johnson K (1985) Contact mechanics. Cambridge University Press, Cambridge

    Google Scholar 

  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), Philadelphia

  18. Laursen TA (2002) Computational contact and impact mechanics. Springer, Berlin

    Google Scholar 

  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: 19–37

    Article  MathSciNet  Google Scholar 

  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: 805–826

    Article  MathSciNet  Google Scholar 

  21. Nochetto R, Wahlbin L (2002) Positivity preserving finite element approximation. Math Comput 71: 1405–1419

    MathSciNet  Google Scholar 

  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: 1354–1391

    Article  MathSciNet  Google Scholar 

  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: 1428–1465

    Article  MathSciNet  Google Scholar 

  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 München

  25. Puso MA, Laursen TA (2002) A 3D contact smoothing method using Gregory patches. Int J Numer Methods Eng 54: 1161–1194

    Article  MathSciNet  Google Scholar 

  26. Puso MA, Laursen TA (2004) A mortar segment-to-segment contact method for large deformation solid mechanics. Comput Methods Appl Mech Eng 193: 601–629

    Article  MathSciNet  Google Scholar 

  27. Puso MA, Laursen TA (2004) A mortar segment-to-segment frictional contact method for large deformations. Comput Methods Appl Mech Eng 193: 4891–4913

    Article  MathSciNet  Google Scholar 

  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: 555–566

    Article  MathSciNet  Google Scholar 

  29. Timoshenko SP, Goodier JN (1970) Theory of elasticity. McGraw-Hill, New York

    Google Scholar 

  30. Wall WA, Gee MW (2010) Baci—a multiphysics simulation environment. Tech. report, Technische Universität München

  31. Wohlmuth BI (2011) Variationally consistent discretization schemes and numerical algorithms for contact problems. Acta Numerica, pp 569–734

  32. Wriggers P (2002) Computational contact mechanics. Wiley, New York

    Google Scholar 

  33. Wriggers, P, Nackenhorst, U (eds) (2007) Computational methods in contact mechanics, vol 3 of IUTAM Bookseries. Springer, Berlin

    Google Scholar 

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Authors and Affiliations

  1. Institute for Numerical Mathematics, Technische Universität München, Boltzmannstrasse 3, 85748, Garching, Germany

    B. I. Wohlmuth

  2. Institute for Computational Mechanics, Technische Universität München, Boltzmannstrasse 15, 85748, Garching, Germany

    A. Popp & W. A. Wall

  3. Mechanics and High Performance Computing Group, Technische Universität München, Boltzmannstrasse 15, 85748, Garching, Germany

    M. W. Gee

Authors
  1. B. I. Wohlmuth
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  2. A. Popp
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  3. M. W. Gee
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  4. W. A. Wall
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Correspondence to A. Popp.

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Wohlmuth, B.I., Popp, A., Gee, M.W. et al. An abstract framework for a priori estimates for contact problems in 3D with quadratic finite elements. Comput Mech 49, 735–747 (2012). https://doi.org/10.1007/s00466-012-0704-z

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  • DOI: https://doi.org/10.1007/s00466-012-0704-z

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