Skip to main content
SpringerLink
Log in

A virtual element method for contact

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

The problem of contact between two elastic bodies is addressed computationally using the virtual element method (VEM). The use of the VEM allows the use of non-matching meshes for the two bodies, and hence obviates the need for node-to-node contact on the candidate contact interfaces. The contact constraint is imposed using either a Lagrange multiplier or penalty formulation. A number of numerical examples illustrate the robustness and accuracy of the algorithm.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  1. Wriggers P (2006) Computational contact mechanics, 2nd edn. Springer, Berlin

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

  3. Hallquist JO (1979) NIKE2d: an implicit, finite-deformation, finite element code for analysing the static and dynamic response of two-dimensional solids. Technical Report UCRL–52678, Lawrence Livermore National Laboratory, University of California

  4. Wriggers P, Simo J (1985) A note on tangent stiffnesses for fully nonlinear contact problems. Commun Appl Numer Methods 1:199–203

    Article  MATH  Google Scholar 

  5. Hallquist JO, Goudreau GL, Benson DJ (1985) Sliding interfaces with contact-impact in large-scale Lagrange computations. Comput Methods Appl Mech Eng 51:107–137

    Article  MathSciNet  MATH  Google Scholar 

  6. Pietrzak G, Curnier A (1997) Continuum mechanics modeling and augmented Lagrange formulation of multibody, Large deformation frictional contact problems. In: Owen DR, Hinton E, Onate E (eds) Proceedings of COMPLAS 5. CIMNE, Barcelona, pp 878–883

  7. Padmanabhan V, Laursen T (2001) A framework for development of surface smoothing procedures in large deformation frictional contact analysis. Finite Elem Anal Des 37:173–198

    Article  MATH  Google Scholar 

  8. Wriggers P, Krstulovic-Opara L, Korelc J (2001) Smooth \({C}^1\)- interpolations for two-dimensional frictional contact problems. Int J Numer Methods Eng 51:1469–1495

    Article  MathSciNet  MATH  Google Scholar 

  9. Krstulovic-Opara L, Wriggers P, Korelc J (2002) A \({C}^1\)-continuous formulation for 3D finite deformation frictional contact. Comput Mech 29:27–42

    Article  MathSciNet  MATH  Google Scholar 

  10. Wohlmuth BI (2000) A mortar finite element method using dual spaces for the Lagrange multiplier. SIAM J Numer Anal 38:989–1012

    Article  MathSciNet  MATH  Google Scholar 

  11. Wohlmuth B, Krause R (2004) Monotone methods on non-matching grids for non linear contact problems. SISC 25:324–347

    Article  MathSciNet  Google Scholar 

  12. Puso MA (2004) A 3D mortar method for solid mechanics. Int J Numer Methods Eng 59(3):315–336

    Article  MATH  Google Scholar 

  13. 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  MATH  Google Scholar 

  14. Flemisch B, Puso MA, Wohlmuth B (2005) A new dual mortar method for curved interfaces: 2D elasticity. Int J Numer Methods Eng 63:813–832

    Article  MathSciNet  MATH  Google Scholar 

  15. Fischer KA, Wriggers P (2005) Frictionless 2D contact formulations for finite deformations based on the mortar method. Comput Mech 36:226–244

    Article  MATH  Google Scholar 

  16. Fischer KA, Wriggers P (2006) Mortar based frictional contact formulation for higher order interpolations using the moving friction cone. Comput Methods Appl Mech Eng 195:5020–5036

    Article  MathSciNet  MATH  Google Scholar 

  17. Tur M, Fuenmayor F, Wriggers P (2009) A mortar-based frictional contact formulation for large deformations using Lagrange multipliers. Comput Methods Appl Mech Eng 198:2860–2873

    Article  MathSciNet  MATH  Google Scholar 

  18. Popp A, Gitterle M, Gee M, Wall WA (2010) A dual mortar approach for 3D finite deformation contact with consistent linearization. Int J Numer Methods Eng 83(11):1428–1465

    Article  MathSciNet  MATH  Google Scholar 

  19. Temizer I, Wriggers P, Hughes TJR (2011) Contact treatment in isogeometric analysis with NURBS. Comput Methods Appl Mech Eng 200:1100–1112

    Article  MathSciNet  MATH  Google Scholar 

  20. de Lorenzis L, Temizer I, Wriggers P, Zavarise G (2011) A large deformation frictional contact formulation using NURBS-based isogeometric analysis. Int J Numer Methods Eng 87:1278–1300

    MathSciNet  MATH  Google Scholar 

  21. Temizer I, Wriggers P, Hughes TJR (2012) Three-dimensional mortar-based frictional contact treatment in isogeometric analysis with NURBS. Comput Methods Appl Mech Eng 209–211(2012):115–128

    Article  MathSciNet  MATH  Google Scholar 

  22. de Lorenzis L, Wriggers P, Zavarise G (2012) Isogeometric analysis of 3D large deformation contact problems with the augmented Lagrangian formulation. Comput Mech 49:1–20

    Article  MathSciNet  MATH  Google Scholar 

  23. Sukumar N (2004) Construction of polygonal interpolants: a maximum entropy approach. Int J Numer Methods Eng 61:21592181

    MathSciNet  MATH  Google Scholar 

  24. Sukumar N, Malsch EA (2006) Recent advances in the construction of polygonal finite element interpolants. Arch Comput Methods Eng 13:129163

    Article  MathSciNet  MATH  Google Scholar 

  25. Biabanaki S, Khoei A (2012) A polygonal finite element method for modeling arbitrary interfaces in large deformation problems. Comput Mech 50:1933

    Article  MathSciNet  MATH  Google Scholar 

  26. Biabanaki S, Khoei A, Wriggers P (2014) Polygonal finite element methods for contact-impact problems on non-conformal meshes. Comput Methods Appl Mech Eng 269:198221

    Article  MathSciNet  MATH  Google Scholar 

  27. Chi H, Talischi C, Lopez-Pamies O, Paulino GH (2015) Polygonal finite elements for finite elasticity. Int J Numer Methods Eng 101(4):305–328

    Article  MathSciNet  Google Scholar 

  28. Beirão da Veiga L, Brezzi F, Cangiani A, Manzini G, Marini L, Russo A (2013) Basic principles of virtual element methods. Math Models Methods Appl Sci 23(01):199–214

    Article  MathSciNet  MATH  Google Scholar 

  29. Beirão da Veiga L, Brezzi F, Marini LD, Russo A (2014) The Hitchhiker’s guide to the virtual element method. Math Models Methods Appl Sci 24(08):1541–1573

    Article  MathSciNet  MATH  Google Scholar 

  30. Beirão Da Veiga L, Brezzi F, Marini L (2013) Virtual elements for linear elasticity problems. SIAM J Numer Anal 51(2):794–812

    Article  MathSciNet  MATH  Google Scholar 

  31. Gain AL, Talischi C, Paulino GH (2014) On the virtual element method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes. Comput Methods Appl Mech Eng 282:132–160

    Article  MathSciNet  Google Scholar 

  32. Bathe K-J, El-Abbasi N (2001) Stability and patch test performance of contact discretizations and a new solution algorithm. Comput Struct 79(16):1473–1486

    Article  Google Scholar 

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

    Book  MATH  Google Scholar 

Download references

Acknowledgments

Special care had to be taken to obtain the smootheness of the displacement and contact tractions at the contact interface. In this context the authors would like to thank Prof. Franco Brezzi for pointing out a method for a special stabilization that enhanced the results considerably, see Sect. 3.2. The third author acknowledges the generous support of the Alexander von Humboldt Foundation, through a Georg Forster Research Award.

Author information

Authors and Affiliations

  1. Institute of Continuum Mechanics, Leibniz Universität Hannover, Appelstr. 11, 30167, Hannover, Germany

    P. Wriggers & W. T. Rust

  2. Centre for Research in Computational and Applied Mechanics, University of Cape Town, Rondebosch, Cape Town, 7701, South Africa

    B. D. Reddy

Authors
  1. P. Wriggers
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. W. T. Rust
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. D. Reddy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to W. T. Rust.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wriggers, P., Rust, W.T. & Reddy, B.D. A virtual element method for contact. Comput Mech 58, 1039–1050 (2016). https://doi.org/10.1007/s00466-016-1331-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-016-1331-x

Keywords

Search

Navigation