A geometrical-based contact algorithm using a barrier method

Most methods employed in the numerical solution of contact problems in finite element simulations rely on equality-based optimization methods. Typically, a gap function which is non-differentiable at the point of contact is used in these kind of approaches. The gap function can be seen as the Macaulay bracket of some distance function, where the latter is differentiable at the point of contact. In this article, we propose to use the distance function directly instead of using the gap function. This will give rise to a formulation involving inequality constraints. This approach eliminates the artificially introduced non-differentiability. To this end we propose a barrier algorithm as the method of choice to solve the problem. The method originates in optimization literature, where convergence proofs for the method are available