A time discretization scheme based on Rothe's method for dynamical contact problems with friction

摘要

We present a new dissipative and contact-stabilized time discretization scheme for dynamic frictional contact problems, which is based on Rothe's method. Especially for the case of Coulomb friction the stability of the contact stresses is of crucial importance as they directly influence the frictional behavior. In our approach, we obtain an accurate description of the frictional stresses by a time-discretized friction law allowing for an implicit treatment of the contact forces in the framework of the Newmark scheme. Moreover, undesirable oscillations at the contact interface are removed by employing an additional L2-projection within the predictor step. Since the implicit treatment of the material behavior and the frictional response requires a (quasi-)variational inequality to be solved in each time step, we derive a non-smooth multiscale method, which allows for the efficient and robust solution of these highly nonlinear problems. The convergence of this multiscale method is proven for the case of Tresca friction. For the case of Coulomb friction, an inexact fixed point iteration in the normal stresses is used. We furthermore, consider the case of two-body contact. Here, the information transfer at the contact interface is realized by means of mortar methods, which provide a stable discretization of the relative displacements and the stresses at the contact boundary. Numerical results for the resulting fully discrete scheme in 3D are presented, showing the high accuracy of the proposed method.