Franz Chouly, Patrick Hild, Yves Renard:
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 (see for example (Rockafellar 1974, 1976)). The augmented Lagrangian has since quite widely become established for the approximation and resolution of contact problems in small and large strains, mainly following the research of (Curnier and Alart 1988 ; Alart and Curnier 1991 ; Simo and Laursen 1992). The method by Nitsche (1971) was originally proposed to allow a Dirichlet- type boundary condition to be weakly taken into account, precisely avoiding the use of Lagrange multipliers. Only recently has it been extended to contact conditions with or without friction in (Chouly and Hild 2013a ; Annavarapu et al. 2014 ; Chouly 2014 ; Chouly et al. 2015). The close connection between Nitsche and Lagrangian methods is however quite clear and it is the objective of this chapter to shed some light on this relationship. This is achieved namely by looking into the mechanisms underlying these methods, and also by way of presenting some recent developments within the framework of small and large elastic strains. Section 2 first presents the continuous problem of frictional contact between two elastic solids, within the framework of small strains. Section 3 is dedi- cated to finite element approximation within the framework of small strains, where mathematical analysis of numerical methods is possible. Section 4 fi- nally presents the extension of the methods described in previous sections to the regime of large elastic transformations, as well as numerical results related to this context.