# Three-Dimensional Hyperbolic Mesh Generation Method Based on the Neural Network
## 2 Methodology  
Similar to all structured mesh generation methods, hyperbolic mesh generation also seeks to find the mapping relationship between the irregular physical domain and the regular computational domain through different algorithms. Figure 1 provides a schematic diagram of the mapping relationship, divided into three-dimensional surfaces and three-dimensional solids, which are arranged in rows. As we can see, in physical space, the coordinates are denoted as \(x,y,z\), while in computational space, the coordinates are \(\xi,\eta,\xi\). Since the computational space is regular, its range can be represented as:  
\[[0,\xi_{\max}]\times[0,\eta_{\max}]\times[0,\xi_{\max}]. \tag{1}\]  
For the mesh generation of the three-dimensional model, traditional hyperbolic mesh generation solves the governing equations on computational space using numerical methods, and then the mesh coordinates are obtained layer by layer. As depicted in Figure 1, it is the process from right to left; we can view it as solving the coordinates distribution of the physical domain on the computational domain. The mapping relationship is represented by the equation at the arrow from right to left in Figure 1. For three-dimensional surfaces and solids, it is denoted as \(\Re\) and \(\Re^{\prime}\), respectively. It is worth noting that, unlike three-dimensional solids, a three-dimensional surface is essentially a two-dimensional manifold and can be represented by a parametric surface that includes only two parameters. Therefore, its mapping in computational space is a plane and does not involve the parameter \(\zeta\), or \(\zeta_{\max}\) equal to zero. Traditional methods solve the mesh coordinates of the physical domain on the computational domain, leading to issues related to solution stability, grid smoothing, and boundary control, as described in the introduction. Therefore, this paper proposes using neural networks to directly solve the coordinates of the computational domain on the  
Figure 1: Mapping relationship diagram of three-dimensional surface and three-dimensional solid: (**a**) physical domain of three-dimensional surface; (**b**) computational domain of three-dimensional surface; (**c**) physical domain of three-dimensional solid; (**d**) computational domain of three-dimensional solid. physical domain; it is a process from left to right, as shown in Figure 1. In this scheme, the well-trained neural network represents the inverse mapping relation of the \(\Re\) or \(\Re^{\prime}\), as shown in the arrow from left to right in Figure 1, denoted as \(\Re^{-1}\) and \(\Re^{\prime-1}\). The advantage of this approach is that physical boundary conditions can be directly incorporated into the loss function, enabling constraints on general boundary conditions. Considering the governing equation of the three-dimensional model in the traditional hyperbolic mesh generation scheme [13; 15], when we treat a three-dimensional surface as a three-dimensional solid with unit thickness, it becomes apparent that the governing equations for three-dimensional surfaces and three-dimensional solids are essentially the same; i.e., two equations govern the orthogonality of the grid, while the final determinant controls the size of the grids. Therefore, to conduct the inverse process mentioned above, we transform the governing equations for surface and solid into a form defined in the physical domain. Thus, the governing equations for a three-dimensional surface are:  
\[ \begin{array}{l}\
abla_{\xi}\cdot\
abla_{\eta}=0\\ \vec{n}\cdot\
abla_{\eta}=0\\ \vec{n}\cdot(\
abla_{\xi}\times\
abla_{\eta})=J\end{array} , \tag{2}\]  
the governing equations for a three-dimensional solid are:  
\[ \begin{array}{l}\
abla_{\xi}\cdot\
abla_{\zeta}=0\\ \
abla_{\eta}\cdot\
abla_{\zeta}=0\\ \
abla_{\zeta}\cdot(\
abla_{\xi}\times\
abla_{\eta})=J^{\prime}\end{array} , \tag{3}\]  
where the \(\vec{n}\) represents the unit normal vector of each point on the surface. Both \(J\) and \(J^{\prime}\) are Jacobian determinants whose values need to be specified manually.