Computational Mechanics

, Volume 51, Issue 6, pp 949–959 | Cite as

Improvement of stability conditions, accuracy and uniqueness of penalty approach in contact modeling

Open Access
Original Paper


The main objective of this paper is to improve stability conditions, uniqueness and convergence of numerical analysis of metal forming processes with contact constraints enforced by the penalty method. A commonly known drawback of this approach is the choice of penalty factor values. When assumed too low, they result in inaccurate fulfillment of the constraints while when assumed too high, they lead to ill-conditioning of the equations system which affects stability and uniqueness of the solution. The proposed modification of the penalty algorithm consists in adaptive estimation of the penalty factor values for the particular system of finite element equations and for the assumed allowed inaccuracy in fulfillment of the contact constraints. The algorithm is tested on realistic examples of sheet metal forming. The finite element code based on flow approach formulation (for rigid-plastic and rigid-viscoplastic material model) has been used.


Contact modeling Penalty approach Metal forming Deep drawing 



This study was partially financed by European Regional Development Fund within the framework of the Innovative Economy Programme, Project Number POIG.01.03.01-14-209/09.

Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.


  1. 1.
    Wriggers P (2002) Computational contact mechanics. Chichester, WileyGoogle Scholar
  2. 2.
    Oñate E, Zienkiewicz O (1983) A viscous shell formulation for the analysis of thin sheet metal forming. Int J Mech Sci 25: 305–335MATHCrossRefGoogle Scholar
  3. 3.
    Agelet de Saracibar C (1990) Finite element analysis of sheet metal forming processes. PhD thesis, Universitat Politecnica de Catalunya (in Spanish)Google Scholar
  4. 4.
    Oñate E, de Saracibar CA (1990) Analysis of sheet metal forming problems usig a selective bending-membrane formulations. Int J Num Meth Eng 30Google Scholar
  5. 5.
    Sosnowski W, Bednarek T, Kowalczyk P (2010) Stability and uniqueness of flow approach algorithm in sheet metal forming simulations. Comput Methods Mater Sci 10(1): 30–36Google Scholar
  6. 6.
    Nour-Omid B, Wriggers P (1987) A note on the optimum choice for penalty parameters. Commun Appl Numer Methods 3: 581–585MATHCrossRefGoogle Scholar
  7. 7.
    Barlam D, Zahavi E (1999) The reliability of solutions in contact problems. Comput Struct 70: 35–45MATHCrossRefGoogle Scholar
  8. 8.
    Pantano A, Averill RC (2002) A penalty-based finite element interface technology. Comput Struct 80: 1725–1748CrossRefGoogle Scholar
  9. 9.
    Mik M-S, Choi D-H (2000) A new penalty parameter update rule in the augmented lagrange multiplier method for dynamic response optimization. KSME Int J 14: 1122–1130Google Scholar
  10. 10.
    Zavarise G, Wriggers P, Stein E, Schrefler B (1992) Real contact mechanisms and finite element formulation—a coupled thermomechanical approach. Int J Numer Methods Eng 35: 767–785MATHCrossRefGoogle Scholar
  11. 11.
    Zavarise G, Wriggers P, Stein E, Schrefler B (1992) A numerical model for thermodynamical contact based on microscopic interface laws. Mech Res Commun 19: 173–182MATHCrossRefGoogle Scholar
  12. 12.
    Paggi M, Barber J (2011) Contact conductance of rough surfaces composed of modified rmd patches. Int J Heat Mass Transf 4: 4664–4672CrossRefGoogle Scholar
  13. 13.
    Zavarise G, Lorenzis LD, Taylor RL (2012) A non-consistent start-up procedure for contact problems with large load-step. Comp Meth Appl Mech Eng 205(208): 91–109CrossRefGoogle Scholar
  14. 14.
    Luenberger D (1984) Linear and nonlinear programming, 2nd edn. Chichester, Addison WesleyGoogle Scholar
  15. 15.
    Perzyna P (1966) Fundamental problems in viscoplasticity. Adv Appl Mech 9: 243–377CrossRefGoogle Scholar
  16. 16.
    Oñate E, Agelet de Saracibar C (1992) Numerical modelling of sheet metal forming problems. In: Hartley P, Pillinger I, Sturgess C (eds) Numerical modelling of material deformation processes: research, development and applications. Springer, New YorkGoogle Scholar
  17. 17.
    Batoz J, Bathe K, Wo L (1980) A study of three node triangular plate bending element. Int J Num Methods Eng 15: 1771–1812MATHCrossRefGoogle Scholar
  18. 18.
    Woo D (1968) On the complete solution of the deep drawing problem. Int J Mech Sci 10: 83–94CrossRefGoogle Scholar

Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.Institute of Fundamental Technological ResearchPolish Academy of SciencesWarsawPoland
  2. 2.Kazimierz Wielki UniversityBydgoszczPoland

Personalised recommendations