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

David Doyen (1) , Alexandre Ern (2) , Serge Piperno (2)

1 EDF - EDF
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

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 mini- mization 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 La- grangian. 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 algo- rithms 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.

Domaines: Analyse numérique [math.NA]

David Doyen, Alexandre Ern, Serge Piperno. A three-field augmented Lagrangian formulation of unilateral contact problems with cohesive forces. ESAIM: Mathematical Modelling and Numerical Analysis, 2010, 44 (2), pp.323-346. ⟨10.1051/m2an/2010004⟩. ⟨hal-00349836⟩