Finite Element Methods for Frictional and Frictionless Contact Problems

Resumen: Contact problems play a central role in various mechanical structures and are highly relevant in hydrostatics and thermostatics. These issues arise in everyday situations, such as the interaction of braking pads with wheels, tires with roads, and pistons with skirts, as well as in common industrial processes like metal forming and metal extrusion.

The primary objective of my talk is to explore finite element methods applied to both frictional and frictionless contact problems. This study focuses on contact problems characterized by various conditions, including the Signorini non-penetration condition and normal compliance condition, while modeling frictional force using Coulomb’s law of dry friction and Tresca friction laws. Variational inequalities, representing the variational formulations of contact problems, serve as powerful tools in the mathematical analysis of these models.

During the presentation, we will delve into both elliptic and parabolic variational inequalities. Regarding elliptic variational inequalities, our examination will include the Signorini contact problem and the frictional contact problem with normal compliance. In the domain of parabolic variational inequalities, I will provide an overview of the quasi-static contact problem.

The regularity of the solution plays a crucial role in determining the qualitative behaviour of the approximation. However, there are instances where the solution to variational inequalities exhibits irregularities due to factors such as the presence of free boundaries and reentrant corners within the domain. To address this, we aim to develop a rigorous theoretical framework to establish a posteriori error estimates and obtain convergence rates for some variational inequalities. These estimates quantify the accuracy of the numerical approximation and offer essential insights for evaluating the reliability of computed solutions. The theoretical findings will be substantiated by numerical investigations.