# Three-Dimensional Hyperbolic Mesh Generation Method Based on the Neural Network
## 3 Results
### Terrain Solids  
Before starting the three-dimensional solid mesh generation of the terrain model, a standard surface with a central protrusion on a flat plane was chosen as our test model because three-dimensional solids are more complex than three-dimensional surfaces. By generating the mesh for the test model and evaluating the convergence process of the neural network, we can verify the effectiveness of the proposed method and the robustness of the grid generation process. In flow field calculations, a square computational domain is typically used for general terrain surfaces. Therefore, we select the constraint surfaces along the advancing direction and the outer surfaces as planes to ultimately generate a rectangular mesh result. On the selected test model, we assigned \(\xi,\eta,\zeta\) values to each point. Based on loss function 15, we use a neural network with four hidden layers, each containing 64 nodes, to obtain the solution of Equation (3) in the region enclosed by all surfaces. Before the computation, we appropriately scaled the model and set the value of the Jacobian matrix determinant to 0.01. Once the computation begins, the loss function converges stably to 1 \(\times\) 10\({}^{-7}\), indicating the effectiveness of the method and the robustness of the solution process. Considering that the model is symmetrical in both directions perpendicular to the advancing direction, we use diagonal skewness to preliminarily assess the quality of the generated grid. This criterion is defined as one minus the ratio of the two diagonals on the same plane of the hexahedron, the larger the value, the more distorted the mesh. The resulting mesh and quality distribution are shown in Figure 7.  
From Figure 7a,b, it can be seen that the boundaries along the advancing direction and the outer boundaries are well constrained. The maximum skewness value of the generated grid is less than 0.03, indicating a high-quality hyperbolic mesh. Moreover, the mesh distribution is relatively uniform. According to expressions 16 and 17, Figure 7c-f show the calculated skewness and orthogonality of the initial and outer boundaries. It can be observed that the grid quality on all surfaces is quite good. However, unlike conventionally generated grids, the grid distribution on the outer boundary is not uniform. The non-uniformity of the outer boundary can better satisfy the orthogonality constraints in Equation (3). Notably, due to the small spatial range along the advancing direction, this grid was obtained through one training process of the neural network. After confirming the feasibility of three-dimensional grid generation, we selected a different terrain surface to generate its mesh. The constraint surface along the advancing direction and outer surfaces are also planes. The setting allows us to verify boundary constraints under a general model while closely approximating grids generated by the traditional hyperbolic method for comparison. After appropriately scaling the terrain surface, we chose a neural network with five hidden layers to solve Equation (3), the specified Jacobian determinant value remaining at 0.01. After the neural network convergence, the hyperbolic mesh was generated, as shown in Figure 8. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Method** & \multicolumn{3}{c}{**Equiangular Skewness**} & \multicolumn{3}{c}{**Deviation Angle**} \\ \hline  & **Min** & **Max** & **Average** & **Min** & **Max** & **Average** \\ \hline Neural-based & 0.889 & 0.999 & 0.980 & 0.036 & 9.971 & 1.762 \\ Interpolation & 0.824 & 0.999 & 0.985 & 0.000 & 15.850 & 1.395 \\ Projection & 0.774 & 0.999 & 0.984 & 0.002 & 20.306 & 1.457 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Extreme and average values of equiangular skewness and deviation angle for different methods.