A new numerical scheme for non uniform homogenized problems: Application to the non linear Reynolds compressible equation

A new numerical approach is proposed to alleviate the computational cost of solving non-linear non-uniform homogenized problems, The article details t he application of the proposed approach to lubrication problems with roughn ess effects. The method is based on a two-parameter Taylor expansion of the implicit dependence of the homogenized coefficients on the average pressur e and on the local value of the air gap thickness. A fourth-order Taylor ex pansion provides an approximation that is accurate enough to be used in the global problem solution instead of the exact dependence, without introduci ng significant errors. In this way, when solving the global problem, the so lution of local problems is simply replaced by the evaluation of a polynomi al. Moreover, the method leads naturally to Newton-Raphson nonlinear iterat ions, that further reduce the cost. The overall efficiency of the numerical methodology makes it feasible to ap ply rigorous homogenization techniques in the analysis of compressible flui d contact considering roughness effects. Previous work makes use of an heur istic averaging technique. Numerical comparison proves that homogenization- based methods are superior when the roughness is strongly anisotropic and n ot aligned with the flow direction.