Contact with friction using the augmented Lagrangian Method: a conditional constrained minimization problem

Abstract

This work presents a formulation of the contact with friction between elastic bodies. This is a non linear problem due to unilateral constraints (inter-penetration of bodies) and friction. The solution of this problem can be found using optimization concepts, modelling the problem as a constrained minimization problem. The Finite Element Method is used to construct approximation spaces. The minimization problem has the total potential energy of the elastic bodies as the objective function, the non-inter-penetration conditions are represented by inequality constraints, and equality constraints are used to deal with the friction. Due to the presence of two friction conditions (stick and slip), specific equality constraints are present or not according to the current condition. Since the Coulomb friction condition depends on the normal and tangential contact stresses related to the constraints of the problem, it is devised a conditional dependent constrained minimization problem. An Augmented Lagrangian Method for constrained minimization is employed to solve this problem. This method, when applied to a contact problem, presents Lagrange Multipliers which have the physical meaning of contact forces. This fact allows to check the friction condition at each iteration. These concepts make possible to devise a computational scheme which lead to good numerical results

Influence of the Main Contact Parameters in Finite Element Analysis of Elastic Bodies in Contact

“…Another approach to the contact problem is shown by [8], where minimal principles without an explicit connection with variational methods are applied directly in the contact problem. The concept of conditionally dependent constraints is used to deal with sticking or sliding situations.…”

Contact stiffness estimation in ANSYS using simplified models and artificial neural networks

“…In the case of FEM, the use of constrained minimization methods such as the Augmented Lagrangian Method is widely used to model the contact problem. In this case, the constraints of the minimization problem have physical equivalence, as non-penetration conditions (inequalities) or adherence (equality) [20,22].…”