# Three-Dimensional Hyperbolic Mesh Generation Method Based on the Neural Network
## 3 Results
### Terrain Surface
For the smoothness in the I-direction, the difference between the neural network method and the other methods is less than 1%. To conduct a more detailed quantitative analysis for comparing the differences in meshes generated by different methods, we use the equiangular skewness and the maxi  
Figure 4: Comparison of mesh quality generated by different methods. (**a**) Neural network; (**b**) linear interpolation; (**c**) projection. Figure 5: The extreme value of orthogonality, skewness, and smoothness concerning the mesh generated by different methods. mum deviation angle to compare the grid quality distribution. The equivalent skewness is defined as:  
\[1-\text{max}[\frac{\theta_{\text{max}}-\theta_{e}}{180-\theta_{e}},\frac{\theta_{ e}-\theta_{\text{min}}}{\theta_{e}}], \tag{18}\]  
where \(\theta_{\text{max}}\) and \(\theta_{\text{min}}\) are the largest and smallest angles in a grid, and \(\theta_{e}\) is the angle for an equiangular grid (e.g., 60 degrees for a triangle and 90 degrees for a square). The range of equiangular skewness is 0-1, and the larger value indicates a better mesh quality. It is commonly used in practical flow field computations to evaluate the success or failure of the calculation. The maximum deviation angle is defined as the maximum deviation of the internal angle from 90 degrees for each grid. The smaller the deviation, the closer the element is to a regular hexahedron, leading to a more accurate element integration in flow field computations. So, the larger the skewness, the better the mesh quality, and the larger the deviation angle, the worse the mesh quality. Based on these two mesh quality criteria, we have statistically analyzed the number of meshes within different quality intervals generated by different methods. The results are shown in Figure 6.  
As we can see from Figure 6, for skewness, the grid quality generated by the neural network is mainly distributed in the range of 0.95-1, and even the poorest quality grid is greater than 0.85. In contrast, the grid quality generated by the projection method is similar to linear interpolation in terms of distribution across each range. The difference is that the minimum skewness value generated by the projection method is smaller than the interpolation method. For the deviation angle, the grid generated by the neural network has a significant increase in the range of 0-5 degrees and contains no grids with deviation angles greater than 15 degrees. The grids generated by the projection and interpolation methods have similar distributions across different ranges, except that the maximum deviation angle in the projection method is greater than the interpolation method. Under the criteria of equiangular skewness and deviation angle, Table 1 presents the extreme and average values of mesh quality generated by different methods. It can be seen that, in terms of the minimum skewness and maximum deviation angle, the mesh quality generated by the neural network is superior to the other two methods. However, in terms of average values, the other methods outperform the neural network method. The reason is that, in the surface model, flat areas occupy a large proportion. As observed in the previous comparison in Figure 4, the latter two methods perform well in flat regions. In contrast, the neural network method shows a more uniform distribution overall. However, when it comes to the flow field computation, the grids with the poorest quality have the most significant impact, thus giving the neural network method a considerable advantage. \[\max\Bigl{\{}1-\Bigl{|}\overset{\rightarrow}{t_{i}}\cdot\overset{\rightarrow} {t_{j}}\Bigr{|}\Bigr{\}}. \tag{16}\]0 Histogram of the mesh quality distribution of the terrain surface generated by different methods.