# Three-Dimensional Hyperbolic Mesh Generation Method Based on the Neural Network
## 3 Results
### Terrain Solids
Figure 8: Mesh results of terrain: (**a**) volume diagonal skewness one; (**b**) volume diagonal skewness two; (**c**) orthogonality of the outer surface; (**d**) orthogonality of the terrain surface; (**e**) skewness of the outer surface; (**f**) skewness of the terrain surface. Figure 7: Mesh results of the surface with a central protrusion: (**a**) diagonal skewness one of the mesh; (**b**) diagonal skewness two of the mesh; (**c**) orthogonality distribution of the outer surface; (**d**) orthogonality distribution of the initial surface; (**e**) skewness distribution of the outer surface; (**f**) skewness distribution of the initial surface. To preliminarily evaluate the quality of the generated mesh, we continue to use the criteria provided by Equations (16) and (17) to assess the mesh quality of the initial surface and the outer surface. The orthogonality and skewness of the initial surface are shown in Figures 8d and 8f, respectively, while Figure 8c,e depict the orthogonality and skewness of the outer surface. For the volume mesh, we continue to use diagonal skewness to evaluate mesh quality. Unlike the test model, the terrain model lacks symmetry. The two diagonal skewness distributions of the volume mesh are different, as shown in Figure 8a,b. Based on Figure 8, it can be observed that the neural network-based method can effectively constrain the boundaries along the advancing direction as well as the outer boundary. Meanwhile, good mesh quality on the initial terrain surface can be observed from Figure 8d,f, with orthogonality values all greater than 0.75 and skewness values all less than 0.22. However, Figure 8c,e reveal that the mesh quality on the outer boundary surface is somewhat inferior to the initial surface, indicating that even though the outer boundary is a plane, its mesh distribution is still non-uniform. Furthermore, as shown in Figure 8e,f, the diagonal skewness distribution of the volume mesh also indicates the volume mesh generated by the neural network is of good quality. To illustrate the characteristics of the mesh generated by the neural network, we also generated a solid mesh for the same terrain surface using a traditional method. However, due to the inability of the traditional method to control the outer boundary, we can only constrain the boundaries along the advancing direction using constant plane boundary conditions, while ensuring the same number of mesh layers and a similar model size. To make a preliminary quantitative comparison between the neural network method and the traditional method, we compared the diagonal skewness and the mesh smoothness in different directions. Given that a smaller diagonal skewness value indicates better mesh quality, we compared the maximum values of diagonal skewness and the minimum values of smoothness in different directions. For convenience, we define 1 minus the diagonal distortion, denoted as 1-DS, so that higher values of all parameters correspond to better mesh quality. The result is shown in Figure 9.  
As shown in Figure 9, the diagonal skewness of the meshes generated by the neural network method is superior to the traditional methods. In the I, J direction, the smoothness of the meshes generated by the two methods is close, while in the K direction, the smoothness of the meshes generated by the neural network method is better than the traditional method. Similar to the three-dimensional surface model, we also selected equiangular skewness and maximum deviation angle to illustrate the difference between the neural network method and the traditional method. Given the significant differences in the number of grids across different ranges, we compared the logarithm of the element number plus one. The mesh distribution within different ranges is shown in Figure 10.  
Figure 9: Quantitatively compare the extreme values of diagonal skewness and smoothness for the meshes generated by different methods. As shown in Figure 10, the mesh generated by the neural network exhibits better quality compared to the traditional method.