A Multigrid Approach in the Numerical Problem of Tangential Contact

Summary

This work presents the numerical simulation of rolling bodies in three dimensions with multigrid-methods and boundary discretization. The use of different scales during the calculation overcomes known problems, regarding convergence velocity and required time for summations. The final objective is the study of roughness in dry, tangential contact. Discretization up to 300 × 300 points are shown.

Geometrical assumptions and formulation of boundary conditions follow the pioneering work in numerics of contact by Kalker [1]. He developed different algorithms for various simplifications, and most of them are based on maximizing complementary energy. In this case there are proofs for existence and uniqueness of solutions. A serious disadvantage is a relationship of cubic order between number of unknowns and calculation time. This problem makes it impossible to handle fine resolutions, even with fast computers. A set of multigrid methods, introduced by Venner et. al. for contact with separating film of liquid [2] is used to solve this problem.

Contact areas between rough surfaces can be structured finely, therefore, the method of grid-transformation must be chosen carefully. The common geometry, used in literature, leads to an incorrect mapping of areas, thus existing methods are reformulated with suitable geometry. The problems appeared also in normal contact, were the adapted methods have been applied successfully [3].

In contrast to finite elements, the discretization of the boundary method can be restricted to the potential contact area. That way discretizations up to 300 × 300 points are accessible, which is sufficient for study of disturbed surfaces. The influence coefficients, that appear in the boundary approach, can be saved in reduced form under assumption of smooth macro-geometries.