Modelling and numerical simulation of contact and lubrication

This thesis considers the simulation of a steel forming process including the lubricant between tool and workpiece. The model equations, that describe deformation, contact and hydrodynamic flow, are derived from fundamental physical laws. For the contact-simulation the two models of Signorini- and z-contact are introduced. The 3-dimensional Stokes problem for a thin fluid film is condensed to the 2-dimensional SubStokes model (for velocity and pressure) and further reduced to the Reynolds model, which is an equation for the pressure only. All fluid models are extended to variational inequations to consider cavitation. Basic mathematical concepts for the numerical treatment with the finite element method and the corresponding analysis are presented. Error estimation is done separate for the modelling error and discretisation error. An algorithm for model adaptivity is given and model error estimates are derived out of the physical models for z- and Signorini-contact aswell as SubStokes- and Reynolds-fluids.
Error estimates for the discretisation error of elliptic problems - deformation and Reynolds-flow - are presented. The SubStokes problem is stabilized to apply linear finite elements. To estimate the discretisation error of the SubStokes inequation a Lagrangian multiplier is introduced. An algorithm for a simultanous refinement of model and grid is given. Finally the presented methods and estimates are validated by the application to various prototype examples. The results allow to compare the different models (z- and Signorini-contact, Reynolds- and SubStokes-fluid).