# A Local Adaptive Mesh Refinement for JFO Cavitation Model on Cartesian Meshes
## 2 AMR Algorithm
### Neighbor Fixing
The AMR method provides almost the same accuracy results while the number of cells is reduced to around 50% of the uniform mesh. The computational efficiency is shown to be effectively improved. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Bearings**} & \multirow{2}{*}{**Refinement Level**} & **Number of Tagged Cells for** & **Number of Tagged Cells for** & **Number of Cells Actually** & **Total Number of Cells before** & **Refinement Ratio (A/B)** \\  & & **for Pressure** & **Density Ratio** & **defined (A)** & **Refinement (B)** & **Ratio (A/B)** \\ \hline Case 1: journal & 1 & 30 & 40 & 240 & 265 & 91\% \\ bearing & 2 & 90 & 40 & 672 & 1024 & 65\% \\  & 3 & 300 & 180 & 1664 & 4096 & 41\% \\ \hline Case 2: thrust & 1 & 30 & 40 & 252 & 265 & 95\% \\ bearing & 2 & 150 & 60 & 776 & 1024 & 76\% \\  & 3 & 350 & 180 & 1960 & 4096 & 48\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Refinement strategy for the two cases. Figure 9: Final refined mesh for journal bearing: (**a**) pressure distribution; (**b**) density ratio distribution. The second case is the dimple-enhanced seal-like thrust bearing investigated by [PERSON] and [PERSON] [22]. The bearing surface was textured with duplicated micro circular holes. The variation of the hole height results in a divergence and convergence clearance region. Hence, oil film rupture and reformation simultaneously occur around the hole. The bearing was operated in the condition of \(r_{0}\)= 750 \(\mu\)m, \(h_{g}\) = 10 \(\mu\)m, \(L\) = 3 mm, \(h_{0}\) = 4 \(\mu\)m, \(\mu\) = 0.0035 Pa-s, \(U\) = 1.45 m/s, \(p_{c}\) = 0.9 \(\times\) 10\({}^{5}\) Pa (a), \(p_{a}\) = 1.0 \(\times\) 10\({}^{5}\) Pa (a). The refinement strategy for the bearing is listed in Table 3. The refinement ratios were 91%, 76%, and 48% for the three-time refinements. They are higher than those of case 1. A more computational cost was indicated to be paid. The solution of the three-times refined mesh is compared with the solution of the corresponding uniform mesh and the solution provided by [PERSON] and [PERSON] [22], as shown in Figures 11 and 12. It can be seen that the AMR solution is close to the solution of the uniform mesh (128 \(\times\) 128 cells) with a small difference compared with the solution provided by [PERSON] and [PERSON] [22]. The difference is mainly reflected by the value of maximum pressure. [PERSON] algorithm is sensitive to the number of grids. Insufficient number of grids often leads to high maximum pressure. The difference is acceptable under the current refinement strategy. The number of the final refined mesh is 9220, which is 56% of the uniform mesh. The AMR method provides almost the same accuracy solution while the computational cost is saved. Figure 11: Final refined mesh for thrust bearing: (**a**) pressure distribution; (**b**) density ratio distribution. Figure 10: Comparison of pressure distribution between different results for \(y\) = 50 mm. ## 6 Conclusions

A quadtree-based AMR procedure was developed for bearing cavitation simulation. The universal Reynolds equation was numerically solved on the nonuniform mesh by [PERSON] cavitation algorithm. The pressure and density ratio were both taken as the indicators of the error estimation with the Richardson extrapolation method. The investigated cases showed that the AMR method provided nearly the same accuracy results compared with the uniform mesh, while the number of mesh was reduced to 50-60% of the number of the uniform mesh. The computational efficiency was effectively improved. The AMR method is suggested to be a potential tool for bearing cavitation simulation. Conceptualization, W.X. ; investigation, K.L. and Z.G. ; software, M.Z. and J.Y. ; writing--original draft preparation, W.X. writing--review and editing, W.X. All authors have read and agreed to the published version of the manuscript. This research was funded by the project YKJ201814 supported by Science Foundation of Nanjing Institute of Technology. Not applicable.