# The Active Set Algorithm for Solving Frictionless Unilateral Contact Problems

• Georges Dumont
Chapter

## Abstract

In this paper is presented a method taking into account the frictionless unilateral contact phenomenon within a FEM program. This method is based on the active set algorithm for solving the constrained minimization problem associated with the unilateral contact model. Three main families of contact models are usually used: the first one is based on the penalty method regularizing the contact conditions. The second one (see for example: Bathe and Chaudary (1985), Klarbring (1986a) and Kalker (1990)), in which category falls our algorithm, is concerned with the treatment of the contact geometrical conditions by duality and requires the development of algorithms to detect the frontiers where contact may occur during the calculation steps. In this case, the variational equality turns into a variational inequality and thus involves constrained minimization methods to be solved. The third one, as proposed by Heegaard and Curnier (1993) and by Cescotto and Charlier (1993), often called mixed approach, is based either on the use of performing algorithms dedicated to the resolution of the problem, as the augmented lagrangian method, or on mixed variational formulations. Our approach lies on a node to node linearized contact modelization. The associated variational formulation leads equivalently to a constrained minimization problem, for further reference in Boot (1968), which is presented, with an unique solution characterized by the Kuhn-Tucker relations. The principle of the so called active set algorithm, which differs from the projected gradient algorithm, proposed by Rosen (1960), in the mean of updating the set of constraints, is then presented and discussed on a theoretical point of view, that is without any computer implementation considerations. This iterative algorithm ensures, at each step, the minimization in a convex set of the deformation energy associated to the structure and gives a condition to modify the set of active constraints in order to perform the next step. Eventually, the basis of an original demonstration of convergence in a finite number of steps is given. As a conclusion, an industrial study using the implementation of the active set algorithm in our FEM industrial software, Code Aster, is mentioned and the future investigations aiming at improving the behavior of this algorithm and extending it to the friction contact problem are presented.

## Keywords

Variational Inequality Contact Problem Augmented Lagrangian Method Constrain Minimization Problem Unilateral Contact
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