Mathematical Problems in Engineering
Volume 7 (2001), Issue 4, Pages 355-378
A new numerical scheme for non uniform homogenized problems: Application to the non linear Reynolds compressible equation
1Instituto Balseiro and Centro Atómico Bariloche, Bariloche 8400, Argentina
2M.M.C.S, CNRS-UMR 5585, Insa de Lyon, Bat 401, Villeurbanne Cedex 69621, France
Received 1 May 2000
Copyright © 2001 Gustavo C. Buscaglia and Mohammed Jai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A new numerical approach is proposed to alleviate the computational cost of solving non-linear non-uniform homogenized problems. The article details the application of the proposed approach to lubrication problems with roughness effects. The method is based on a two-parameter Taylor expansion of the implicit dependence of the homogenized coefficients on the average pressure and on the local value of the air gap thickness. A fourth-order Taylor expansion provides an approximation that is accurate enough to be used in the global problem solution instead of the exact dependence, without introducing significant errors. In this way, when solving the global problem, the solution of local problems is simply replaced by the evaluation of a polynomial. Moreover, the method leads naturally to Newton-Raphson nonlinear iterations, that further reduce the cost.
The overall efficiency of the numerical methodology makes it feasible to apply rigorous homogenization techniques in the analysis of compressible fluid contact considering roughness effects. Previous work makes use of an heuristic averaging technique. Numerical comparison proves that homogenization-based methods are superior when the roughness is strongly anisotropic and not aligned with the flow direction.