A high order spectral volume method for elastohydrodynamic lubrication problems: Formulation and application of an implicit p-multigrid algorithm for line contact problems

摘要

A high-order spectral volume (SV) method is implemented for solving the steady-state, isothermal, line-contact elastohydrodynamic lubrication problem. The standard procedure involves solving the Reynolds equation, the film thickness equation and the force balance equation. The nonlinear, integral and implicit nature of the film thickness equation creates a complex pressure updating procedure. In this paper, the Reynolds equation is modified so as to automatically result in a pressure update during time marching. The convective flux is handled using an approximate Riemann flux. Two viscous flux formulations were attempted: a. The classical local discontinuous Galerkin (LDG) formulation and b. The recently formulated LDG2 formulation. An implicit pre-conditioned Lower Upper Symmetrical Gauss Seidel (LU-SGS) p-multigrid method developed for the spectral volume Navier Stokes (NS) solver by Kannan and Wang is adopted here for time marching. A convergence speedup of more than two orders is obtained with the implicit p-multigrid method, when compared to the traditional Runge-Kutta explicit method. The entire formulation is compact and hence can be easily parallelized. A comparison between results obtained using the SV formulation and those from an existing low order finite difference simulation confirmed that the SV delivers extremely accurate solutions in spite of using relatively small number of degrees of freedom (DOF). The LDG2 formulation delivers slightly more accurate results than the traditional LDG approach. In general, the numerical results are very promising and indicate that the approach has a great potential for more complicated elastohydrodynamic problems such as the transient line contact problem and the point contact problem.