# A Local Adaptive Mesh Refinement for JFO Cavitation Model on Cartesian Meshes
## 2 AMR Algorithm
### Neighbor Fixing
It affects not only the accuracy of the solution but also the computational cost. An appropriate refinement strategy can realize a more efficient solution. Although the use of variable gradient as a refinement indicator is more natural, the paper adopted a posteriori error estimation to refine the areas where the discretization error was large. As mentioned above, the Richardson extrapolation method is a simple way to find the area where it needs to be refined. Pressure and density ratio are the main flow variables of a bearing film. If an individual parameter (pressure or density ratio) is used as error estimation, this is inappropriate in [PERSON] cavitation algorithm since the pressure is always constant cavitation pressure in the cavitation region, and the density ratio is always constant 1 in the liquid region. The refinement on the whole bearing film is impossible. Significant discretization error always exists in the cavitation region or liquid region. Thus, pressure and density ratio are both taken as the variables to estimate error. Two cases were studied below to show the AMR results. The first case is the cavitated journal bearing studied by [PERSON] et al. [8; 9] and [PERSON] [10]. The bearing is a finite type, with \(L/D=4/3\) (\(D=100\) mm), operating in the condition of \(\varepsilon=0.60\), \(n=459\) r/min, \(p_{c}=28\) kPa (a), and \(\mu=0.0127\) Pa-s. The solution of the three-time refined mesh is compared with the solution of the corresponding uniform mesh and the experimental data. The initial mesh is taken as a very coarse mesh with \(16\times 16\) cells. This mesh is generated by recursively refining the whole mesh four times. In the first refinement, 30 cells in the liquid region and 40 cells in the cavitation region were tagged to be refined. The tagged cells were the cells of the coarsen mesh with \(8\times 8\) cells in the implementation of the Richardson extrapolation method. After the refinement operation, the number of the cells refined was 240. It was lower than the expected number of grids, 280, as calculated by 70 \(\times\) 4. This is because some of the tagged cells with large errors were common in the cavitation and liquid regions. In the subsequent refinements, 90 and 300 cells in the liquid region and 40 and 180 cells in the cavitation region were tagged to be refined for the second and third refinements, respectively. It is noted that the number of cells tagged was empirically determined according to the initial numerical tests. Although the refinement strategy is empirical, it still shows the law that the ratio of the tagged number to the total number reduces with the increase in total cell numbers. As shown in Table 3, the refinement ratios are 91%, 65%, and 41% in descending order. The refinement strategy indicates that the discretization error of the coarse mesh should be reduced sufficiently. This is because the coarse mesh has the largest discretization error, where the discretization error is proportional to the power \(p\) of the grid spacing \(h\). Figure 9 shows the pressure and density ratio for the final refined mesh. It can be seen that the refined meshes tagged by the pressure-based error estimation were basically distributed in the region with large pressure gradients, and the refined meshes tagged by density ratio-based error estimation were basically distributed in the region of cavitation interfaces where the density ratio gradients were also large. This behavior indicated that the discretization error was concentrated in the area with large variable gradients. Figure 10 compares the pressure distribution for the AMR results, uniform mesh results, and experimental results provided by [PERSON] and [PERSON] [12] and [PERSON] [13]. The pressure obtained by the AMR method is nearly consistent with that obtained by the uniform mesh (128 \(\times\) 128 cells), and there is a slight difference in the experimental results. The number of the final refined mesh is 7984, while the number of the uniform mesh is 16,384.