The boundary element method (BEM) is widely used in fast numerical solvers for concentrated elastic contact problems arising from the wheel-rail contact in the railway industry. In this paper we extend the range of applicability of BEM by computing the influence coefficients (ICs) numerically. These ICs represent the Green's function of the problem, i.e. the surface deformation due to unit loads. They are not analytically available when the half-space is invalid, for instance in conformal contact. An elastic model is proposed to compute these ICs numerically, by the finite element method (FEM). We present a detailed investigation to find proper strategies of FEM meshing and element types, considering accuracy and computational cost. Moreover, the effects of computed ICs to contact solutions are examined for a Cattaneo shift contact problem. The work in this paper provides a guidance to study fast solvers for the conformal contact.

Rail transportation plays an important role in our everyday life, and there is fast development and modernization in the railway industry to meet the growing demand for swifter, safer and more comfortable trains. At the same time, the security of train operation and the maintenance of rails have to be considered. A lot of research on these issues has been carried out, among which the study of the contact between a train's wheel and the rail is particularly significant. The contact problem considers two elastic bodies. When they are pressed together, a contact area is formed where the two body surfaces coincide with each other. Moreover, an elastic field of stress, strain and displacement arises in each body. These stresses consist of normal stress (pressure) and frictional stress (traction) acting in the tangential direction. When solving the so-called normal contact problem, we search for the contact area and the pressure on it. The tangential contact problem is studied when the two bodies are brought into relative motion. If the relative velocity of the two surfaces is small, a creeping motion may be observed which is largely caused by the elastic deformation at the contact region. In those parts of the contact area where the tangential stress is small, the surfaces of the two bodies stick to each other. Otherwise, local relative sliding may occur. The research question is to find the adhesion and slip areas,and the tangential tractions.