An Improved Fixed-Point Algorithm to Solve the Lubrication Problem with Cavitation

Abstract

The most widely adopted model to treat thin film (lubrication) problems with cavitation is the Elrod-Adams model. An efficient numerical solution is by no means trivial, as the classical NewtonRaphson method is not suitable for this problem. The best approach so far is to solve it with a Jacobi or Gauss-Seidel type algorithm, as done by Ausas (R. Ausas et al, Journal of Tribology, 131(3):031702 (2009)). However, it becomes computationally expensive for large problems, due to the elliptic problem being solved for the pressure. Multigrid methods arise like a natural choice to accelerate results, yet their implementation for the Elrod-Adams model is not straightforward. A mass-conserving multigrid algorithm was presented by Checo (Doctoral dissertation, Universidade de São Paulo (2016)), where it was shown that care must be taken when transferring the solution between multigrid meshes due to the cavitation boundary shift between them. Here we propose a fixed-point method to solve stationary flow in porous media, which outperforms our previous algorithm.