Lagrangian and Nitsche Methods for Frictional Contact

Augmented Lagrangians and Lagrangians are constrained optimization tools that very early have naturally been applied to contact problems with deformable solids. This chapter presents the continuous problem of frictional contact between two elastic solids, within the framework of small strains. It focuses on finite element approximation within the framework of small strains, where mathematical analysis of numerical methods is possible. The use of Lagrange multipliers leads to transforming a constrained minimization problem into a problem with simple constraints on multipliers. The chapter also presents a stabilized method that allows more flexibility in choosing the finite element spaces for the displacement fields and the multiplier. The concern for the numerical integration on the contact boundary can be included in the practical aspects for the implementation of the methods, whether for the augmented Lagrangian or the Nitsche method. This is common to all mortar-type methods, where an integration of quantities between two incompatible meshes is necessary.