Abstract
In this paper, we analyze the convergence of the Schwarz method for contact problems with Tresca friction formulated in stress variables. In this dual variable, the problem is written as a variational inequality in the space \(H_\mathrm{div}(\Omega )\), \(\Omega \) being the domain of the problem. The method is introduced as a subspace correction algorithm. In this case, the global convergence and the error estimation of the method are already proved in the literature under some assumptions. However, the checking of these hypotheses in the space \(H_\mathrm{div}(\Omega )\) cannot be proved easily, as for the space \(H^1(\Omega )\). The main result of this paper is to prove that these hypotheses are verified for this particular variational inequality. As in the case of the classical problems formulated in primal variables, the error estimate we obtain depends on the overlapping parameter of the domain decomposition.
Similar content being viewed by others
References
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
Alart P, Lebon F (1995) Solution of frictional contact problems using ILU and coarse/fine preconditioners. Comput Mech 16:98–105
Badea L (2003) Convergence rate of a multiplicative Schwarz method for strongly nonlinear inequalities. In: Barbu V et al (eds) Analysis and optimization of differential systems. Kluwer Academic Publishers, Boston, pp 31–42 (also available from http://www.imar.ro/lbadea)
Badea L (2006) Convergence rate of a Schwarz multilevel method for the constrained minimization of non-quadratic functionals. SIAM J Numer Anal 44(2):449–477
Badea L, Krause R (2012) One- and two-level Schwarz methods for inequalities of the second kind and their application to frictional contact. Numer Math 120(4):573–599
Bisegna P, Lebon F, Maceri F (2001) D-PANA : a convergent block-relaxation solution method for the discretized dual formulation of the Signorini–Coulomb contact problem. C R Math Acad Sci Paris 333:1053–1058
Bisegna P, Lebon F, Maceri F (2004) Relaxation procedures for solving Signorini–Coulomb contact problems. Adv Eng Softw 35:595–600
Cocou M, Pratt E, Raous M (1996) Formulation and approximation of quasistatic frictional contact. Int J Eng Sci 34:783–798
Cocu M (1984) Existence of solutions of Signorini problems with friction. Int J Eng Sci 22:567–575
Duvaut G, Lions JL (1972) Les inéquations en mécanique et en physique. Dunod, Paris
Girault V, Raviart P-A (1986) Finite element methods for Navier–Stokes equations. In: Springer series in computational mechanics, vol 5. Springer, New York
Kuss F, Lebon F (2009) Stress based finite element methods for solving contact problems: comparisons between various solution methods. Adv Eng Softw 40:697–706
Kuss F, Lebon F (2011) Error estimation and mesh adaptation for SignoriniCoulomb problems using E-FEM. Comput Struct 89:1148–1154
Lebon F (2003) Contact problems with friction: models and simulations. Simul Model Pract Theory 11:449–463
Licht C, Pratt E, Raous M (1991) International series of numerical mathematics, vol 101. Remarks on a numerical method for unilateral contact including friction. Birkhauser, Basel, pp 129–143
Raous M, Chabrand P, Lebon F (1988) Numerical methods for frictional contact problems and applications. J Theor Appl Mech 7:111–128
Telega JJ (1991) Quasi-static Signorini’s contact problem with friction and duality. Int Ser Numer Math 101:199–214
Toselli A, Widlund O (2005) Domain decomposition methods—algorithms and theory. Springer, New York
Wriggers P (2002) Computational contact mechanics. Wiley, New York
Acknowledgments
The authors acknowledge the support of the Laboratoire Européen Associé CNRS Franco-Roumain de Mathématiques et Modélisation, LEA Math-Mode, for this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Hector Ramirez.
Rights and permissions
About this article
Cite this article
Badea, L., Lebon, F. Schwarz method for dual contact problems. Comp. Appl. Math. 36, 719–731 (2017). https://doi.org/10.1007/s40314-015-0255-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40314-015-0255-y
Keywords
- Contact problems
- Dual formulation
- Domain decomposition methods
- Schwarz method
- Subspace correction methods
- Variational inequalities