In this paper, several active set methods based on classical problems arising in Contact Mechanics are analyzed, namely unilateral/bilateral contact associated with Tresca’s/Coulomb’s law of friction in small and large deformation. The purpose of this work is to extend an Inexact Primal–Dual Active Set (IPDAS) method already used in Hueber et al. (2008) to the formalism of dynamics and hyper-elasticity. This method permits to solve the unilateral problem with Coulomb’s law of friction by taking into account an alternative for the latter based on the approximation of the Coulomb’s law by a succession of states of Tresca friction in which the friction threshold is fixed at each fixed point iteration. The mechanical formulation in the hyper-elasticity framework is first presented, next, we establish weak formulations of the different cases of frictional contact problems and we give the finite element approximation of the problems. Then, we detail the numerical treatment within the framework of the primal–dual active set strategy for different frictional contact conditions. We finally provide some numerical experiments to bring into light the efficiency of the IPDAS method and to carry out a comparison with the augmented Lagrangian method by considering representative contact problems in both small and large deformation cases.

本文分析了基于接触力学中经典问题的几种主动集方法，即与小变形和大变形中的Tresca / Coulomb定律相关的单边/双边接触。这项工作的目的是扩展已经在Hueber等人中使用的不精确的原始对偶有效集（IPDAS）方法。（2008年）的动力学和超弹性形式主义。该方法允许通过考虑一系列的Tresca摩擦状态来近似库仑定律，从而解决库仑摩擦定律的单边问题，其中在每个定点迭代中，摩擦阈值都固定。首先介绍超弹性框架中的机械公式，然后，我们针对摩擦接触问题的不同情况建立了弱公式，并给出了问题的有限元近似。然后，我们在不同摩擦接触条件下的原始-对偶主动集策略框架内详细说明了数值处理。最后，我们提供了一些数值实验，以揭示IPDAS方法的效率，并通过考虑小变形和大变形情况下的代表性接触问题，与增强拉格朗日方法进行比较。