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A three-field augmented Lagrangian formulation of 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 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.

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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