Adaptive finite element analysis of fractal interfaces in contact problems

Abstract

Structures involving interfaces with fractal geometry are referred here as a sequence of classical interfaces problems, which result from the consideration of the fractal interfaces as the unique “fixed point” or the “deterministic attractor” of a given Iterated Function System (IFS). On the interface, unilateral contact conditions are assumed to hold. The approximations of the fractal interfaces are combined with a penalty regularization based on the minimization of the potential energy, after some appropriate transformations are performed. For this type of contact problems there often result singular points on the interfaces which lead to possible stress concentrations. Further-more, the convergence of finite element solution under a sufficient discretization can not be determined from the outset. An adaptive finite element strategy appears to be suitable for such kind of contact problems in that it possesses the properties of adjusting automatically the mesh sizes both in the interior of the bodies and on the contact zone. In this spirit, both the goals of exactly determining the real contact areas, and of enhancing the accuracy of finite element solution (meanwhile consuming reasonable computational costs) may be achieved. The error estimator based on the residual stress analysis is discussed. Numerical examples illustrate the validity and effectiveness of the method proposed in this paper.