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A new numerical scheme for non uniform homogenized problems: application to the nonlinear Reynolds compressible equation. (English) Zbl 1112.76433

Summary: A new numerical approach is proposed to alleviate the computational cost of solving nonlinear 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.

MSC:

76M25 Other numerical methods (fluid mechanics) (MSC2010)
76M50 Homogenization applied to problems in fluid mechanics
76D08 Lubrication theory
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
76N15 Gas dynamics (general theory)
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