A mortar algorithm combined with an Augmented Lagrangian approach for treatment of frictional contact problems

Contact mechanics is present in a wide range of mechanical engineering applications, and numerous works have been dedicated to the numerical solution of contact problems. Mathematically, the frictional contact problem can be seen as defined by a variational inequality. The solution corresponds to the minimum of the total energy of the system, subjected to inequalities constraints associated to the normal and tangential components of the traction and distance vectors, respectively. Nonlinear contact mechanics can be related to nonlinear optimization problems formulated by using the method of Lagrange multipliers, resulting in a saddle point system to be solved at each iteration. The method of Lagrange multipliers is very popular in contact mechanics because it overcomes the ill-conditioning inconvenience of the penalty methods. However, the size of the global matrix increases due to the additional unknowns (the Lagrange multipliers) and zero entries appear on the main diagonal of the stiffness matrix. These drawbacks can be avoided by using an augmented Lagrangian method, which consists in a combination of both the penalty and the Lagrange multipliers techniques.
In this work, a mixed penalty-duality formulation based on an augmented Lagrangian approach for treating the contact and the friction inequality constraints is presented. The augmented Lagrangian approach allows to regularize the non differentiable contact terms, giving a C1 differentiable saddle-point function. The relative displacement of the contacting bodies is described in the framework of the Finite Element Method (FEM) using the mortar method, which gives a smooth representation of the contact forces across the bodies interface.