Refined solutions of thermal lubrication problems generally require fine mesh and many iteration steps. To resolve these difficulties, Elrod and Brewe proposed an efficient algorithm based on the use of a Lobatto point version of the Gauss quadrature, which is typically twice as fast as the other quadrature methods. The original Lobatto algorithm was only applied for incompressible hydrodynamic lubrication. This paper presents a Lobatto point quadrature algorithm which is applicable for thermal elastohydrodynamic lubrication (EHL) problems where both density and viscosity of the lubricant are taken to be temperature and pressure dependent and the transverse velocity term in the energy equation is obtained from the continuity equation. Within this approach, the unknown temperature across the film is written in a series of Legendre polynomials. Regardless of the order of the series expansions, the thermal Reynolds equation can explicitly contain only the information from the first three Legendre polynomials, i.e., data from up to a second-order polynomial. Use of the Lobatto point calculation method has resulted in accuracy without the use of a larger number of grid points.
Lobatto Point Quadrature for Thermal Lubrication Problems Involving Compressible Lubricants. EHL Applications
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Moraru, L., and Keith, T. G. (October 19, 2006). "Lobatto Point Quadrature for Thermal Lubrication Problems Involving Compressible Lubricants. EHL Applications." ASME. J. Tribol. January 2007; 129(1): 194–198. https://doi.org/10.1115/1.2404965
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