In many technical and engineering applications, numerical simulation is becoming more and more important for the design of products or the optimization of industrial production lines. However, the simulation of complex processes like the forming of sheet metal or the rolling of a car tire is a very challenging task, as nonlinear elastic or elastoplastic material behaviour needs to be combined with frictional contact and dynamic effects. In addition, these processes often feature a small mobile contact zone which needs to be resolved very accurately to get a good picture of the evolution of the contact stress. In order to be able to perform an accurate simulation of such intricate systems, there is a huge demand for a robust numerical scheme that combines a suitable multiscale discretization of the geometry with an efficient solution algorithm capable of dealing with the material and contact nonlinearities. The aim of this work is to design such an algorithm by combining domain decomposition methods with a suitable space discretization and efficient solvers for the resulting nonlinear systems.