Elastohydrodynamic Lubrication

Variable
timesteps

Variable timestepping

The convential approach in EHL calculations has always been to use fixed time steps when doing transient calculations. These have normally been such that the non-dimensional timestep size is the non-dimensional mesh spacing. In the work of my PhD thesis the time step was allowed to vary based on a local error test. The error test chosen was the Shampine and Gordon one (Computer Solution of Ordinary Differential Equations: the Initial Value Problem, 1975). This error test required two or three extra multigrid cycles per timestep, but in cases where there was a large change of operating behaviours - especially including non-linear effects, then the extra work could make significant savings in the overall computational cost.

This is seen very clearly in the example of reversal of lubricant entrainment. Initially the lubricant is travelling at 5cm/s but this velocity decreases linearly to -5cm/s at 0.2s. This mean that momentarily at 0.1s the lubricant is stationary. In reality this situation occurs all the time with cams and followers and a disc of highly viscous fluid forms at the point of reversal in the centre of the contact. This then moves towards the new outflow before the original profile in restored. It is this disc which is the non-linear portion of the process.

Numerical results are shown in this movie of reversal (MPEG format). It can be noticed how more timesteps are taken around the point of reversal than in the initial and later linear phases. For more detailed analysis of this consider the three graphs shown. The first (right) shows the central and minimum film thicknesses over the course of the solve, the second (below left) shows the time step size, and the third (below right) the stepsize change ratio, from the local error test. Film thickness graph
Central and minimum film thickness

Timestep size graph Stepsize change ratio graph
Timestep size Step size change ratio

This work has been published, along with many further examples in my PhD thesis and also in the Proceedings of the Leeds-Lyon Symposium on Tribology, 2000.

Introduction - Variable Timestepping - Mesh Adaptation - PSEs - Parallelism - The Grid
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