Indication on Notation
5. Summary and Discussion
The objective of this work is to get a fast and precise solver for the EHD con- tact problem by applying model order reduction techniques. As a starting point, the EHD point and line contact problem are formulated as one equation system. Therefore, the Reynolds equation is discretized using finite differences. The elas- ticity equation on the basis of the halfspace theory and the load balance equations are discretized by numerical integration using the midpoint rule. The film rupture problem is solved using the Penalty method.
The EHD contact problem is a nonlinear system with nonlinear parameter de- pendency. For the first time, the complete nonlinear EHD contact problem is reduced using model order reduction techniques. Due to the nonlinearity, all lin- ear model order reduction techniques such as the Krylov-based approaches drop out. Within this work, the POD method is used as reduction procedure. It relies on the extraction of the most important information given by a precomputed set of solutions of the full nonlinear system. Furthermore, a system approximation is applied to the reduced sytem by replacing the reduced system function and its Jacobian by less complex surrogates. Thus, the resulting reduced system with system approximation (RNSA) contains no large-scaled operations anymore.
Within this work, the stationary as well as the transient EHD contact problem is investigated. A system is referred to as quasi-stationary, if the transient behavior of the components are much faster than the change of the system itself. Thus, the problem can be formulated as a chronological sequence of stationary problems. Here, the aim of this work is to develop a method for a fully automated generation of compact models for a specified parameter space. The idea is to use an adaptive snapshot selection algorithm which is based on a random sampling and refinement of the parameter space. Since the evaluation is executed on the reduced framework, a fast error measure is required. Furthermore, a new method to determine a distance measure is presented. It relates the distances between two solutions in parameter and state space. On the one hand, the distance measure allows an a- priori goal-oriented snapshot selection, in order to get a reduced system in advance. On the other hand, it acts as an estimator for the selection of the best starting solution. The algorithm generates a reduced system, which provides a reliable, highly accurate and very fast approximation of the full system within the specified
94 5. Summary and Discussion parameter space. The speed-up factor is about 50 for a line contact problem and about 6000 for a point contact problem with an overall loss in accuracy of less than 1 %.
Furthermore, the stationary EHD point contact problem is extended to consider Non-Newtonian effects by using the generalized Reynolds equation. Here, the im- plicitly given Non-Newtonian EHD contact problem is solved for the first time as a fully coupled system of equations using the Newton-Raphson scheme. Thereby, the gain in convergence rate predominates the higher costs in film thickness inte- gration. Thus, a faster solution can be achieved. Furthermore, in comparison to the full system, the reduced system is about 900 times faster even though a slower processor is used.
For the transient EHD contact problem a new formulation is introduced which relates the computational area to the current contact size. Here, two descriptions are developed. The first is a formulation in an ALE coordinate system, treating the non-constant parameter as explicit time functions. As a side effect the convective term includes an artificial flow due to the grid movement requiring a spatially adjusted discretization in order to maintain the upwind stabilization. The second description uses Eulerian coordinates on a grid which is adjusted for every time step referred to as remeshing. Here, required solutions of former time steps have to be projected onto the new grid. The adjustment of the computational area shows clear improvements in performance for systems with excitations of large amplitude.
Additionally, an alternative approach, TPWL, is applied and adapted for the EHD contact problem for the first time. Here, a new selection procedure for the operating points is presented, leading to a moderate reduction of the necessary number of operating points at a lower offline computational effort. In comparison to the RNSA method, the TPWL model provides a faster solution whose quality is decent but not as good as the Newton based one. In this connection, the perfor- mance of the TPWL model is strongly dependent on the complexity of weigthing. It is determined by the extent of input variance and parameter ranges or more general, on the total number of operating points.
Since the time reduction for the simulation is very effective and due to their abil- ity to take into account the parameterization of their boundary conditions (load, velocity, radius of curvature,. . . ), the proposed reduced models could be embedded into more global models of entire mechanics as subroutine focusing on critical con- tact points. Thus, the full nonlinear dynamical behavior of an EHD contact could be introduced into more general system simulations. Another application lies in their use within optimization loops to tune mechanical components with respect to
5. Summary and Discussion 95
their operating conditions (geometry, material, eventually lubricant properties,. . . ) keeping the total complexity of the underlying physical problem.
6. Outlook
Within this work, smooth isothermal Newtonian EHD line and point contacts are considered. However, the observance of more complex physics within the simu- lation is gaining more and more attention. The introduction of Non-Newtonian effects into the MOR model has been a first step towards a realistic friction mod- eling of contacts including sliding. A further step could be the consideration of temperature effects by introducing the energy balance into the equation system. Therewith, the friction prediction would become more accurate for a wider range of operating conditions such as higher surface velocities.
Another contemporary tribological research field is the examination of the in- fluence of rough surfaces within an EHD contact. In general, surface roughness features can be introduced into a parameterized reduced model whilst the surface topography can be described by a small number of parameters. Here, the arising question is: how good global basis functions can aproximate the resulting scattered solution space of pressure and deformation. A promising alternative to introduce surface roughness effects are the use of flow factors, where the solution remains its smooth form.
Not only surface and temperature effects acting especially within the contact area are of interest, but also effects outside of the contact as e.g. the availability of fluid at the inlet zone. In general the Reynolds equation assumes a fully flooded gap which is not fullfilled for an EHD contact with starved lubrication. In order to handle a starved lubricated problem, the Jakobsson-Floberg-Olsson (JFO) cavita- tion model can be used leading to the form of a complementarity problem similar to the one introduced within this work.
Having a more numerically based point of view, it might be noted that a fi- nite difference discretization scheme is applied within this work using a spatially equidistant grid. Nevertheless, the reduction procedure is not restricted to a special discretization scheme. Alternatively, a discretization with finite elements should be applicable, too. Therewith, an adapted discretization can be arranged including local refinements within the contact zone and a coarsening outside of the contact. Thus, the computational area can be chosen larger having a fine grid at the critical areas at the same time. Furthermore, the formulation of the elasticity with finite
98 6. Outlook elements enables the consideration of solid bodies which cannot be formulated as a halfspace.
Finally, the weightings of the TPWL model have a strong influence on the ac- curacy of the TPWL solution. However, within this work no problem specific weighting procedure is used but only the standard one. The performance of the TPWL model might be improvable by applying an EHD problem adjusted weight- ing by e.g. considering not only the state but also the inputs for the determination of the weighting functions.