Laplace transform to avoid numerical instabilities due to the effect of Peclet number in thermal problems

Abstract

It is well known that thermal analysis is important in tribology in both dry and lubricated cases. In lubricated contacts, the effect of temperature for high speeds and high loads governs the parameters that characterize the contact behaviour (frictional force, carrying capacity, flow, etc.). Thus the thermal analysis in the fluid and in the surrounding bodies is necessary to model the contact correctly. The finite element method (FEM) has proved in recent years to be one of the most general and useful procedures for prediction of the characteristics of lubricated contacts. However, where thermal effects are taken into account the method has limitations and can cause oscillations and inaccuracies to occur in solutions at high velocity (high Peclet number). For example, oscillations occur in the solutions for a 0.04m shaft running at 6r/min (Peclet number ≅ 9). Clearly, the range of velocity magnitudes that leads to good results with the FEM is unable to cover a broad range of industrial applications. Some remedies have already been suggested (smaller elements, hybrid formulation, the weighted residual method with a weighting function different from the interpolation function, etc.). An altogether different approach is to modify the initial equation.

It is well known that the oscillations and numerical instabilities arise from the fact that the thermal problem is not self-adjoint. The essential point, not usually stressed, concerns the symmetry of a linear operator (as for a matrix), which is not an absolute notion. It is relative to a bilinear form. If the given problem does not ensure this symmetry condition (as in the present case) an attempt may be made to transform the given problem into another that fulfils the symmetry conditions.

A method based on Gurtin's theorem is proposed here which allows a preliminary transformation of an equation into an integral-differential one and the convolution product of two functions. This method leads to a variational formulation of any linear initial value problems. Therefore, if the problem has a variational formulation the operator is symmetric and the stiffness matrix obtained from numerical discretization (here FEM) is also symmetric. When this is the case the effect of the Peclet number disappears. This method has been applied here to the thermal problem for slider and journal bearings and obtained very promising results in order to generalize this method to non-linear conditions in stationary thermal problems.