A fast compound direct iterative algorithm for solving transient line contact elastohydrodynamic lubrication problems

Abstract

A fast compound direct iterative algorithm for solving transient line contact elastohydrodynamic lubrication (EHL) problems is presented. First, by introducing a special matrix splitting iteration method into the traditional compound direct iterative method, the full matrices for the linear systems of equations are transformed into sparse banded ones with any half-bandwidth; then, an extended Thomas method which can solve banded linear systems with any half-bandwidth is derived to accelerate the computing speed. Through the above two steps, the computational complexity of each iteration is reduced approximately from O\left(N^{\mathrm{3}}/\mathrm{3}\right) to O\left({\beta }^{\mathrm{2}}N\right), where N is the total number of nodes, and \beta is the half-bandwidth. Two kinds of numerical results of transient EHL line contact problems under sinusoidal excitation or pure normal approach process are obtained. The results demonstrate that the new algorithm increases computing speed several times more than the traditional compound direct iterative method with the same numerical precision. Also the results show that the new algorithm can get the best computing speed and robustness when the ratio, half-bandwidth to total number of nodes, is about 7.5%–10.0% in moderate load cases.

Keywords

elastohydrodynamic lubrication / transient / line contact / matrix splitting iteration method / the Thomas method

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Higher Education Press and Springer-Verlag Berlin Heidelberg