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                                                                                             to ‘tweak’ the numerical parameters of
                                                                                             the multigrid method in order to get

FIGURE 2. Typical calculated oil film thickness (red curves) and pressure (blue curves) for  èEXAMINING POINT
an elastohydrodynamically lubricated line contact. The simulations were performed for an     CONTACTS IN FUTURE WORK
oil temperature of 373 K and a contact entrainment speed of 10 m/s, which is typical for
gear contacts. The results are from Shell’s proprietary solver.                              The Shell team is also interested in
                                                                                             lubricated ‘point contacts,’ such as rolling
	 The Shell team has found that this          use COMSOL for more difficult analyses,”       element bearings, as there are many
approach has converged reasonably well        says Taylor. “One of the fundamental           billions of these in machines across the
for pressures less than 1 GPa. For higher     advantages of COMSOL is that we can            world. Some of them are lubricated with
pressures from higher loads, it becomes       develop models without any lines of code       grease, a viscoelastic material, rather than
necessary to dampen oscillations in the       whatsoever. This makes maintenance and         a liquid lubricant. “Whereas the Reynolds’
calculated pressure distribution. Taylor      modification of models much simpler.           lubrication equation cannot be used for
explains: “Small diffusive terms can be       It is also much easier when we need to         viscoelastic fluids without significant
added to the Reynolds’ equation, or even      consider real lubricated contacts. These       modification, multiphysics simulation
better, SUPG (Streamline Upwind Petrov        are rough, at the micron scale, and so a       could solve the same lubricated contact
Galerkin) methods can be used to add          typical 3 mm contact would need to be          with the full Navier-Stokes equations,
residual based stabilizing terms to the       described by approximately 1,000 nodes,        using the correct viscoelastic properties
original equation. This approach then         if we model it accurately. This would          of the grease,” Taylor observes. “As most
enables successful simulation of contacts     take up too much computer time and             lubricated contacts are subject to time
for pressures up to 4 GPa.”                   memory for the direct method used in           varying loads, the ability to predict the
	 Over the years the Shell team has           the proprietary solver. While we could use     time dependence of oil film thickness
conducted many simulations using a            multigrid methods, it is often necessary       and friction is essential. We have in
proprietary direct solver, which also uses                                                   fact already used COMSOL to look at
adaptive meshing. The software, written                                                      such effects for gear teeth, which are
in a mixture of FORTRAN and C, consists                                                      effectively line contacts.”
of approximately 20,000 lines of code                                                        	 The accurate modeling of grease-
(although this includes the solver code                                                      lubricated rolling element-bearing
as well). “This solver works very well                                                       contacts is now an active area of current
for smooth surfaces; a small number of                                                       research for many companies, including
discretization nodes; ‘simple’ rheology,                                                     Shell, and Taylor expects to be making
with no viscoelasticity; and it assumes                                                      even more use of COMSOL Multiphysics. v
Reynolds’ equation holds.” The results
shown in Figure 2 are from this solver.
	 However, the proprietary solver code is
not so straightforward to modify and it
sometimes poses a steep learning curve for
newcomers. Members of the Shell team are
increasingly turning to
COMSOL Multiphysics® software for
complex lubrication problems (see Figure 3).

èMODELING AT THE MICRON                       FIGURE 3. Calculation of pressure distribution in lubricated line contact, using COMSOL®
SCALE                                         software. The conditions are the same as those used in Figure 2.

Over the past three years the team has
been using both systems to confirm
that they obtain the same results from
COMSOL Multiphysics as they do from
their own solver. “It is still easy to use
the solver for some problems, but we

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