Numerical methods in lubrication modelling

Over the past two decades, many numerical schemes have been developed
to solve elasto-hydrodynamic lubrication problems. New schemes are continuously
being sought with the aim of improving efficiency and robustness. The two main
issues of concern when solving these problems are large computational costs and
numerical instabilities. The multigrid method, first used by Lubrecht et al. [68]
when solving these problems, has proved to be very successful in dealing with the
issue of computational costs. Venner [97] took this work further and developed
a relaxation scheme which dealt with the issue of instability. However, Venner’s
scheme is not only difficult to understand because it is not presented in its entirety
but also difficult to implement due its complexity. Hence, a new easy to understand
and simple relaxation scheme will be developed and employed in this work, [74], to
solve elasto-hydrodynamic lubrication problems.
The aim of this work is to present an efficient, robust and general purpose
numerical solver for isothermal (Newtonian) elasto-hydrodynamic lubrication circular contact problems. The solver will be based on the FDMG Multigrid Software
[92] and the new relaxation scheme. Elasto-hydrodynamic lubrication problems are
very important in engineering applications and there is a need for general purpose
solvers for industrial applications.
The multigrid solver will be used to solve both steady-state and time dependent (transient) problems. A wide range of steady-state problems will be
solved and the obtained solutions will be compared with those obtained using other
numerical methods, [75]. Up to this date, transient problems are constantly being
solved using fixed time step methods where the step sizes are chosen arbitrarily.
We will present solutions to both fixed and variable time step methods. The governing equations of transient problems will be written as a system of differential
algebraic equations and methods from this area will be employed in variable-step
time integration and convergence testing, [90].