Single-level Fast Multipole Method for frictionless rough contact problem
Résumé
Surface roughness plays an important role in contact mechanics, especially when soft materials (as e.g. rubber) are involved at the contact interface. In the case of tyre/road contact, the computational complexity is high due to the time variation of contact spots between the tyre tread and the road surface during rolling and the relatively large contact area. Furthermore, the contact operates at multiple scales due to the multi-asperity nature of real road surfaces. In order to reduce the computational complexity, a single-level Fast Multipole Method (FMM) is developed to accelerate the conventional method, to solve rough contact problems. This work relies mainly on Boussinesq’s contact theory, which is based on frictionless contact problems in static conditions at the surface of an elastic half-space. The influence function of Boussinesq contact theory is approximate by solid harmonics to enable the single-level Fast Multipole Method to accelerate the contact pressure calculation at the micro-scale. Therefore, a non-adaptive quad-tree, to divide the studied surface into equal cells is used. Due to the single-level approach the subdivision of the quad-tree stops at level 2, wherefore the far-field operation of the FMM is neglected. In a first step, the Multipole Expansion operation of the Fast Multipole Method, is validated. In fact, no contact problem is solved during this step. Instead a given analytical contact pressure is applied to verify the computed result of displacement and to compare it to the analytical result, for a single axisymmetrical asperity (cylinder, cone or paraboloid). Then, the single-level Fast Multipole Method is first applied to a surface composed of 25 sphere shaped asperities (50 mm x 50 mm) and afterwards to a real road surface (50 mm x 50 mm). The computed results for the contact pressure, its accuracy and overall CPU time are compared to a reference method. Future work will emphasise the development of a multi-level Fast Multipole Method and its comparison to the reference method in regards to a surface with axisymmetrical asperities and real road surfaces.