Evaluation of discretization and integration methods for the analysis of finite hydrodynamic bearings with surface texturing

Abstract

Efficient numerical methods are essential in the analysis of finite hydrodynamic bearings with surface texturing. This is especially evident in optimization and parametric studies where the discretization and integration methods are used to solve the governing two-dimensional Reynolds equation multiple times. Performance comparison studies of the methods are thus required to select the method that is most suitable for a particular bearing geometry. In this work, we conduct a comprehensive and systematic comparison of typical implementations of the finite difference, finite volume, finite element and spectral element discretization methods together with the Newton-Cotes formula and Gauss quadratures for hydrodynamic bearings governed by the two-dimensional Reynolds equation. The methods were evaluated by comparing the approximation errors in the calculation of the maximum pressure, load capacity, coefficient of friction, and minimum film thickness for parallel and convergent bearings textured with elliptical grooves or trapezoidal dimples. The number of degrees of freedom required by the methods to achieve the error cut-off values of 5%, 1% and 0.1% were calculated. Our results demonstrate that the spectral element method uses up to 72 times fewer degrees of freedom than the other methods for the same cut-off values. Also, our study revealed that the shape of the groove/dimple and the bearing convergence ratio can have a significant effect on the approximation errors of the numerical methods used. Specifically, for piecewise-linear texture features (e.g. trapezoidal dimples) and positive convergence ratio it is easier, for the methods, to accurately approximate the solution. In such cases, the finite volume and finite element methods are reasonable choices and provide a good trade-off between the ease of implementation and approximation errors. The worse performance was observed for the finite difference and thus this method is not recommended when the computational efficiency and the accuracy of results are of importance.Our results demonstrate that the spectral element method uses up to 72 times fewer degrees of freedom than the other methods for the same cut-off values. Also, our study revealed that the shape of the groove/dimple and the bearing convergence ratio can have a significant effect on the approximation errors of the numerical methods used. Specifically, for piecewise-linear texture features (e.g. trapezoidal dimples) and positive convergence ratio it is easier, for the methods, to accurately approximate the solution. In such cases, the finite volume and finite element methods are reasonable choices and provide a good trade-off between the ease of implementation and approximation errors. The worse performance was observed for the finite difference and thus this method is not recommended when the computational efficiency and the accuracy of results are of importance.