A Three-field Augmented Lagrangian Formulation of Unilateral Contact Problems with Cohesive Forces

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 decompositioncoordination approach and a nonsmooth Newton’s method. Finally, numerical results illustrating the theoretical analysis are presented. Mathematics Subject Classification. 65N30, 65K10, 74S05, 74M15, 74R99. Received December 23, 2008. Revised July 27, 2009. Published online January 27, 2010.