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A three-field augmented Lagrangian formulationof unilateral contact problems with cohesive forces

Published online by Cambridge University Press:  27 January 2010

David Doyen ,
Alexandre Ern  and
Serge Piperno
David Doyen
Affiliation:
EDF R&D, 1 avenue du Général de Gaulle, 92141 Clamart Cedex, France. david.doyen@edf.fr
Alexandre Ern
Affiliation:
Université Paris-Est, CERMICS, École des Ponts, 77455 Marne-la-Vallée Cedex 2, France. ern@cermics.enpc.fr; piperno@cermics.enpc.fr
Serge Piperno
Affiliation:
Université Paris-Est, CERMICS, École des Ponts, 77455 Marne-la-Vallée Cedex 2, France. ern@cermics.enpc.fr; piperno@cermics.enpc.fr
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Abstract

We investigate unilateral contact problems with cohesive forces, leading tothe constrained minimization of a possibly nonconvex functional. Weanalyze the mathematical structure of the minimization problem. The problem is reformulated in terms of a three-field augmentedLagrangian, and sufficient conditions for the existence of a localsaddle-point are derived. Then, we derive and analyze mixed finiteelement approximations to the stationarity conditions of the three-fieldaugmented Lagrangian. The finite element spaces for the bulk displacement andthe Lagrange multiplier must satisfy a discrete inf-sup condition, whilediscontinuous finite element spaces spanned by nodal basis functions areconsidered for the unilateral contact variable so as to use collocationmethods. Two iterative algorithms are presented and analyzed, namely anUzawa-type method within a decomposition-coordination approach and anonsmooth Newton's method. Finally, numerical results illustrating thetheoretical analysis are presented.

Keywords

Unilateral contactcohesive forcesaugmented Lagrangianmixed finite elementsdecomposition-coordination methodNewton's method
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 2 , March 2010 , pp. 323 - 346
Copyright
© EDP Sciences, SMAI, 2010

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References

Alart, P. and Curnier, A., A mixed formulation for frictional contact problems prone to Newton like solution methods. Comput. Methods Appl. Mech. Engrg. 92 (1991) 353–375. CrossRef
Bathe, K.J. and Brezzi, F., 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.
D.P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods. Athena Scientific (1982).
D.P. Bertsekas, Nonlinear Programming. Athena Scientific (1999).
Bourdin, B., Francfort, G.A. and Marigo, J.-J., The variational approach to fracture. J. Elasticity 91 (2008) 5–148. CrossRef
Champaney, L., Cognard, J.-Y. and Ladevèze, P., Modular analysis of assemblages of three-dimensional structures with unilateral contact conditions. Comput. Struct. 73 (1999) 249–266. CrossRef
Chen, Z., On the augmented Lagrangian approach to Signorini elastic contact problem. Numer. Math. 88 (2001) 641–659. CrossRef
P.G. Ciarlet, Mathematical elasticity, Vol. I: Three-dimensional elasticity, Studies in Mathematics and its Applications 20. North-Holland Publishing Co., Amsterdam (1988).
F.H. Clarke, Optimization and nonsmooth analysis, Classics in Applied Mathematics 5. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, USA, second edition (1990).
Z. Denkowski, S. Migórski and N.S. Papageorgiou, An introduction to nonlinear analysis: applications. Kluwer Academic Publishers, Boston, USA (2003).
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).
A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Applied Mathematical Sciences 159. Springer-Verlag, New York, USA (2004).
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).
M. Frémond, Contact with adhesion, in Topics in nonsmooth mechanics, Birkhäuser, Basel, Switzerland (1988) 157–185.
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).
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.
Hauret, P. and Le Tallec, P., A discontinuous stabilized mortar method for general 3d elastic problems. Comput. Methods Appl. Mech. Engrg. 196 (2007) 4881–4900. CrossRef
Hild, P. and Laborde, P., Quadratic finite element methods for unilateral contact problems. Appl. Numer. Math. 41 (2002) 401–421. CrossRef
Hüeber, S. and Wohlmuth, B.I., An optimal a priori error estimate for nonlinear multibody contact problems. SIAM J. Numer. Anal. 43 (2005) 156–173 (electronic). CrossRef
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).
Kinderlehrer, D., Remarks about Signorini's problem in linear elasticity. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981) 605–645.
Kunisch, K. and Stadler, G., Generalized Newton methods for the 2D-Signorini contact problem with friction in function space. ESAIM: M2AN 39 (2005) 827–854. CrossRef
P. Ladevèze, Nonlinear Computational Structural Mechanics – New Approaches and Non-Incremental Methods of Calculation. Springer-Verlag (1999).
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).
Lorentz, E., A mixed interface finite element for cohesive zone models. Comput. Methods Appl. Mech. Engrg. 198 (2008) 302–317. CrossRef
Marcus, M. and Mizel, V.J., Every superposition operator mapping one Sobolev space into another is continuous. J. Funct. Anal. 33 (1979) 217–229. CrossRef
Moussaoui, M. and Khodja, K., Régularité des solutions d'un problème mêlé Dirichlet-Signorini dans un domaine polygonal plan. Commun. Partial Differ. Equ. 17 (1992) 805–826. CrossRef
Qi, L. and Sun, J., A nonsmooth version of Newton's method. Math. Program. 58 (1993) 353–367. CrossRef
Slimane, L., Bendali, A. and Laborde, P., Mixed formulations for a class of variational inequalities. ESAIM: M2AN 38 (2004) 177–201. CrossRef