Mixed FEM of higher-order for time-dependent contact problems

In this paper mixed finite element methods of higher-order for time-dependent contact problems are discussed. The mixed methods are based on resolving the contact conditions by the introduction of Lagrange multipliers. Dynamic Signorini problems with and without friction are considered involving thermomechanical and rolling contact. Rothe’s method is used to provide a suitable time and space discretization. To discretize in time, a stabilized Newmark method is applied as an adequate time stepping scheme. The space discretization relies on finite elements of higher-order. In each time step the resulting problems are solved by Uzawa‘s method or, alternatively, by methods of quadratic programming via a suitable formulation in terms of the Lagrange multipliers. Numerical results are presented towards an application in production engineering. The results illustrate the performance of the presented techniques for a variety of problem formulations.