A three-field augmented lagrangian formulation of unilateral contact problems with cohesive forces
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 2, p. 323-346

We investigate unilateral contact problems with cohesive forces, leading to the constrained minimization of a possibly nonconvex functional. We analyze the mathematical structure of the minimization problem. The problem is reformulated in terms of a three-field augmented lagrangian, and sufficient conditions for the existence of a local saddle-point are derived. Then, we derive and analyze mixed finite element approximations to the stationarity conditions of the three-field augmented lagrangian. The finite element spaces for the bulk displacement and the Lagrange multiplier must satisfy a discrete inf-sup condition, while discontinuous finite element spaces spanned by nodal basis functions are considered for the unilateral contact variable so as to use collocation methods. Two iterative algorithms are presented and analyzed, namely an Uzawa-type method within a decomposition-coordination approach and a nonsmooth Newton's method. Finally, numerical results illustrating the theoretical analysis are presented.

DOI : https://doi.org/10.1051/m2an/2010004
Classification:  65N30,  65K10,  74S05,  74M15,  74R99
Keywords: unilateral contact, cohesive forces, augmented lagrangian, mixed finite elements, decomposition-coordination method, Newton's method
@article{M2AN_2010__44_2_323_0,
author = {Doyen, David and Ern, Alexandre and Piperno, Serge},
title = {A three-field augmented lagrangian formulation of unilateral contact problems with cohesive forces},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {44},
number = {2},
year = {2010},
pages = {323-346},
doi = {10.1051/m2an/2010004},
zbl = {1192.74355},
mrnumber = {2655952},
language = {en},
url = {http://www.numdam.org/item/M2AN_2010__44_2_323_0}
}

Doyen, David; Ern, Alexandre; Piperno, Serge. A three-field augmented lagrangian formulation of unilateral contact problems with cohesive forces. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 2, pp. 323-346. doi : 10.1051/m2an/2010004. http://www.numdam.org/item/M2AN_2010__44_2_323_0/

[1] P. Alart and A. Curnier, A mixed formulation for frictional contact problems prone to Newton like solution methods. Comput. Methods Appl. Mech. Engrg. 92 (1991) 353-375. | Zbl 0825.76353

[2] K.J. Bathe and F. Brezzi, Stability of finite element mixed interpolations for contact problems. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 12 (2001) 167-183. | Zbl 1097.74054

[3] D.P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods. Athena Scientific (1982). | Zbl 0572.90067

[4] D.P. Bertsekas, Nonlinear Programming. Athena Scientific (1999). | Zbl 1015.90077

[5] B. Bourdin, G.A. Francfort and J.-J. Marigo, The variational approach to fracture. J. Elasticity 91 (2008) 5-148. | Zbl 1176.74018

[6] L. Champaney, J.-Y. Cognard and P. Ladevèze, Modular analysis of assemblages of three-dimensional structures with unilateral contact conditions. Comput. Struct. 73 (1999) 249-266. | Zbl 1049.74562

[7] Z. Chen, On the augmented Lagrangian approach to Signorini elastic contact problem. Numer. Math. 88 (2001) 641-659. | Zbl 1047.74054

[8] P.G. Ciarlet, Mathematical elasticity, Vol. I: Three-dimensional elasticity, Studies in Mathematics and its Applications 20. North-Holland Publishing Co., Amsterdam (1988). | Zbl 0648.73014

[9] F.H. Clarke, Optimization and nonsmooth analysis, Classics in Applied Mathematics 5. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, USA, second edition (1990). | Zbl 0696.49002

[10] Z. Denkowski, S. Migórski and N.S. Papageorgiou, An introduction to nonlinear analysis: applications. Kluwer Academic Publishers, Boston, USA (2003). | Zbl 1040.46001

[11] I. Ekeland and R. Témam, Convex analysis and variational problems, Classics in Applied Mathematics. 28. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, USA (1999). | Zbl 0939.49002

[12] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Applied Mathematical Sciences 159. Springer-Verlag, New York, USA (2004). | Zbl 1059.65103

[13] M. Fortin and R. Glowinski, Augmented Lagrangian methods: Applications to the numerical solution of boundary value problems, Studies in Mathematics and its Applications 15. North-Holland Publishing Co., Amsterdam (1983). | Zbl 0525.65045

[14] M. Frémond, Contact with adhesion, in Topics in nonsmooth mechanics, Birkhäuser, Basel, Switzerland (1988) 157-185. | Zbl 0656.73051

[15] R. Glowinski and P. Le Tallec, Augmented Lagrangian and operator-splitting methods in nonlinear mechanics, SIAM Studies in Applied Mathematics 9. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, USA (1989). | Zbl 0698.73001

[16] J. Haslinger, I. Hlaváček and J. Nečas, Numerical methods for unilateral problems in solid mechanics, in Handbook of numerical analysis IV, Amsterdam, North-Holland (1996) 313-485. | Zbl 0873.73079

[17] P. Hauret and P. Le Tallec, A discontinuous stabilized mortar method for general 3d elastic problems. Comput. Methods Appl. Mech. Engrg. 196 (2007) 4881-4900. | Zbl 1173.74424

[18] P. Hild and P. Laborde, Quadratic finite element methods for unilateral contact problems. Appl. Numer. Math. 41 (2002) 401-421. | Zbl 1062.74050

[19] S. Hüeber and B.I. Wohlmuth, An optimal a priori error estimate for nonlinear multibody contact problems. SIAM J. Numer. Anal. 43 (2005) 156-173 (electronic). | Zbl 1083.74047

[20] N. Kikuchi and J.T. Oden, Contact problems in elasticity: a study of variational inequalities and finite element methods, SIAM Studies in Applied Mathematics 8. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, USA (1988). | Zbl 0685.73002

[21] D. Kinderlehrer, Remarks about Signorini's problem in linear elasticity. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981) 605-645. | Numdam | Zbl 0482.73017

[22] K. Kunisch and G. Stadler, Generalized Newton methods for the 2D-Signorini contact problem with friction in function space. ESAIM: M2AN 39 (2005) 827-854. | Numdam | Zbl pre02213941

[23] P. Ladevèze, Nonlinear Computational Structural Mechanics - New Approaches and Non-Incremental Methods of Calculation. Springer-Verlag (1999). | Zbl 0912.73003

[24] J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications I, Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York, USA (1972). | Zbl 0223.35039

[25] E. Lorentz, A mixed interface finite element for cohesive zone models. Comput. Methods Appl. Mech. Engrg. 198 (2008) 302-317. | Zbl 1194.74438

[26] M. Marcus and V.J. Mizel, Every superposition operator mapping one Sobolev space into another is continuous. J. Funct. Anal. 33 (1979) 217-229. | Zbl 0418.46024

[27] M. Moussaoui and K. Khodja, Régularité des solutions d'un problème mêlé Dirichlet-Signorini dans un domaine polygonal plan. Commun. Partial Differ. Equ. 17 (1992) 805-826. | Zbl 0806.35049

[28] L. Qi and J. Sun, A nonsmooth version of Newton's method. Math. Program. 58 (1993) 353-367. | Zbl 0780.90090

[29] L. Slimane, A. Bendali and P. Laborde, Mixed formulations for a class of variational inequalities. ESAIM: M2AN 38 (2004) 177-201. | Numdam | Zbl 1100.65059