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

CAVALIERI FEDERICO J.; BRÜLS OLIVIER; CARDONA ALBERTO

Zagreb

Congreso; Multibody Dynamics 2013, ECCOMAS Thematic Conference; 2013

University of Zagreb, Faculty of Mechanical Engineering and Naval Architecture

Contact mechanic is present in a wide range of several mechanical engineering applications, and numerous works have been dedicated to the numerical solution of contact problems. Mathematically, the contact problem can be interpreted as a physical system defined by a variation inequality. Solving the frictional contact problems, corresponds to minimize the total energy of the system, subjected to inequalities constrains associated the normal and tangential components of the traction and distance vector, respectively. Nonlinear contact mechanics can be related to nonlinear optimization problems formulated by using the method of Lagrange multipliers, which results 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, there are an increasing of the global matrix size due to additional unknowns: the Lagrange multipliers and zero entries on the main diagonal of the stiffness matrix. These drawbacks can be avoided by using an augmented Lagrangian method. It is a combination of both the penalty and the Lagrange multipliers techniques. In this work, a mixed penalty-duality formulation from an augmented Lagrangian approach for treating the contact and the friction inequality constrains, is presented. The augmented Lagrangian approach allows to regularize the non differentiable contact terms and gives a C1 differentiable saddle-point functional. The relative displacement of two 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.