A three-field augmented lagrangian formulation of unilateral contact problems with cohesive forces

Doyen, David; Ern, Alexandre; Piperno, Serge

ESAIM: Modélisation mathématique et analyse numérique, Tome 44 (2010) no. 2, pp. 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.

[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

[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. | EuDML | Zbl

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

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

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

[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

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

[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

[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

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

[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

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

[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

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

[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

[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

[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

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

[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

[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

[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

[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

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

[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

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

[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

[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

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

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