*4.3. Numerical Multi-Scale Modeling*

Finally, the third modeling approach which can be used to model multi-scale surface textures is based upon numerical multi-scale methods. It is important to emphasize that multi-scale modeling is not just about developing analytical models that use explicitly multi-physics coupling based upon the multi-scale expansion of the governing equations, it is also about developing numerical discretization schemes and algorithms. Numerical multi-scale methods can be divided into two main classes [143]. The first class are algorithms conceived to efficiently solve the details of the problem, including the small-scale behavior. Examples of these methods applied to solve tribological problems are the multi-grid [208–213] and adaptive mesh refinements [214–218] methods. In fact, these are linear scaling algorithms, which implies that their computational complexity scales linearly with the number of degrees of freedom necessary to represent the detailed micro-scale solution. The second class is denoted domain decomposition, which provides a platform on which multi-scale methods can be constructed. In this case, the computational domain is divided into sub-domains and a simulation strategy is adopted based upon solving the given problem on each sub-domain, thus making sure that the solutions on different sub-domains match [143]. An important subclass of numerical multi-scale methods are numerical discretization schemes (e.g., finite element, finite volume or finite difference methods), which modify the finite discretization space to consider explicitly the micro-scale features of the problem. In other words, the finite discretization space is adapted by including functions with the proper micro-scale characteristics or by creating multi-scale basis functions that relate the micro- to macro-scale simulations [219–223]. A more in-depth understanding of numerical multi-scale methods for micro-meso-macroscopic scales coupling can be found in [143,144,224,225].

The use of multi-scale domain decomposition methods and multi-scale numerical discretization schemes to solve lubrication-related problems is not widely explored in the literature and has received attention only recently. However, especially due to the inherent sub-domain strategy of these methods, they are convenient and potentially powerful to deal with multi-scale surface textures. For instance, Pei et al. [226] developed a new finite cell method for modeling surface textures in hydrodynamic lubrication. This technique uses a matrix transformation reduction strategy in which the computational domain is divided into a fine-scale domain with texture cells and a coarser-scale domain without texture cells, as can be seen in Figure 17. The proposed methodology was compared with several test cases involving FEM, CFD, and existing theoretical and experimental data, thus demonstrating that both computing time and storage were significantly reduced. Afterwards, the same authors extended their multi-scale method to lubrication problems with rough surfaces considering parallel computation to speed-up the overall numerical solution. The results showed that the method can be used to predict the average mixed lubrication effects on the global scale (coarser mesh) from deterministic calculations in the small-scale, and to accurately recover the deterministic small-scale effects from the global scale results [227]. The same multi-scale methodology was then applied to investigate the influence of surface texture on the lubrication performance of floating ring bearings including thermal effects. Nine different texture patterns were analyzed, and the results verified that textures significantly increased the side leakage and reduced the temperature rise [228].

A multi-scale approach combining a micro-deterministic mixed lubrication model for small-scales and a macro-scale model was proposed by Nyemeck et al. [229,230] to predict the hydrodynamic load carrying capacity with nominally parallel surfaces. In this model, the mass flow conservation is ensured at the boundaries of micro-cells through the calculation of pressure variations at the macro-cell boundaries obtained from a micro-deterministic model with mass-conservative cavitation and asperity contact.

The classical homogenization approach applied to model mixed lubrication was extended by Pérez-Ràfols et al. [231] to study small flows by coupling two scales with a stochastic element. The proposed stochastic element is established using a two-scale formulation based upon the framework of the heterogeneous multi-scale method.

Two important advantages of this model is that (i) the periodic repetition of the topography is not assumed as in conventional homogenization methods, which allows using much smaller micro-scale domains, and (ii) the prediction of more realistic flow patterns compared with conventional homogenization models for similar small-scale domain size is possible. The multi-scale framework proposed by Gao and Hewson [194] was extended by de Boer et al. [232] to 3D micro-scale simulations and more accurate lubricant behavior. Particularly, a two-scale method using a heterogeneous

multi-scale approach to study the EHL and micro-EHL effects in tilted-pad bearings were developed. The micro-scale problem was solved by CFD simulations including surface elastic deformation, and a method for the homogenization of the micro-scale results was proposed and coupled to the macro-scale via pressure gradient-mass flow rate relationship.

**Figure 17.** Overview of the finite cell method proposed by [227] for modeling surface textures in different lubrication regimes. (**a**) Illustration of the main domain decomposition steps and nomenclature of the proposed method.

More recently, Brunetière and Francisco [233] presented a multi-scale finite element method applied to the simulation of hydrodynamic lubrication of large rough contact surface. As illustrated in Figure 18a,b, the method is based on dividing the computational domain (macro-scale) into sub-domains (micro-scale) connected by a coarser mesh. The pressure distribution at the macro-scale is used as boundary conditions for the micro-scale problem, and then these boundary pressures are adjusted to guarantee the global mass flow conservation between contiguous sub-domains. A comparison between full deterministic and top-scale results with different values is shown in Figure 18c.

Finally, Costagliola et al. [234–237] proposed a spring-block modeling approach to investigate the fundamental mechanisms of dry friction between textured surfaces and how multi-scale surface textures influence static and dynamic friction. The model was used to show how the intricate surface geometry and local material properties on different length scales strongly affect the macroscopic friction force. Furthermore, it was also demonstrated how global friction properties can be tuned and optimized by designing composite surfaces with varying roughness features or local stiffness values.

**Figure 18.** Overview of the multi-scale finite element method proposed by [233] to simulate hydrodynamic lubrication of large rough contact surfaces, which can be extended to deal with multi-scale textures. (**a**) Multi-scale mesh. (**b**) Multi-scale solution procedure. (**c**) Comparison between full deterministic and top-scale pressure distribution with different values.
