Fast Solution of Transient Elastohydrodynamic Line Contact Problems Using the Trajectory Piecewise Linear Approach

Abstract

In order to obtain a fast solution scheme, the trajectory piecewise linear (TPWL) method is applied to the transient elastohydrodynamic (EHD) line contact problem for the first time. TPWL approximates the nonlinearity of a dynamical system by a weighted superposition of reduced linearized systems along specified trajectories. The method is compared to another reduced order model (ROM), based on Galerkin projection, Newton–Raphson scheme and an approximation of the nonlinear reduced system functions. The TPWL model provides further speed-up compared to the Newton–Raphson based method at a high accuracy.

An Exact and General Model Order Reduction Technique for the Finite Element Solution of Elastohydrodynamic Lubrication Problems

“…This shows the ability of the employed MOR technique to capture such unusual patterns. This is not the case with other MOR techniques found in the recent EHL literature [16][17][18][19] which require the specific incorporation of these unusual features into the reduced solution space for the reduced model to be able to capture them.…”

“…The order of reduction is significant since the size of the elastic problem is usually much larger than that of the hydrodynamic problem, given that it entails a higher space dimension. Yet, it is not as large as the reduction order obtained with MOR techniques introduced in recent years, such as the works by Habchi and Issa [16] for line contacts or Habchi [17] for circular contacts where in both cases the size of the elastic problem was reduced to less than 30DOFs or also Maier et al [18,19] who reduced both the elastic and hydrodynamic parts in line contacts under steady-state or transient operation to an overall size of less than 300DOFs. However, the advantages of the SCS algorithm are numerous over these MOR techniques.…”

“…This is not the case with other MOR techniques used so far in the numerical treatment of the EHL problem (e.g., Refs. [16][17][18][19]). It is also important to mention that, if needed, the deformations and stresses within the original solid domain may be recovered using Eq.…”

“…This being said, the excessive computational overhead associated with this offline phase is not considered prohibitive. Besides, when compared to other MOR techniques introduced in the recent EHL literature [16][17][18][19], these involve offline phases that are not only more computationally demanding but they also require a very high level of expertise in the generation of their corresponding reduced solution spaces. Next, the performance of the SCS model compared to that of the full model is investigated in terms of solution precision and computational times.…”

“…First, Habchi and Issa [16,17] introduced the "EHL-basis technique" which allowed reducing the size of the elastic problem to less than 30DOFs/unknowns. Maier et al [18,19] used similar MOR approaches, but extended the model reduction to the hydrodynamic problem too, reducing its size and giving rise to an even faster resolution procedure.…”