## 2 Numerical Method and Model Development
### Spatial Discretization and Adaptive Algorithm  
#### 2.3.1 Spatial Discretization  
In contrast to the point generation for a curved line boundary in a 2D space, that in a 3D space is considerably more complex. In addition, owing to the meshless property of the algorithm, the generated points over the curved surface lack connecting relationships and thus lack the structure required for the traditional ray method [39]. To address these challenges, as shown in Figure 1, this work firstly applies a mapping method [40] to discretize the curved surfaces into points. Consequently, an improved method of the vector dot product is employed to evaluate the attributes of background Cartesian points. Finally, attribute-based background grids are used to pre-mark the computational domain and thus accelerate the spatial discretization. The process of generating the discrete points using the mapping method is shown in Figure 2. It comprises the following four steps. First, the boundary curves of the surface are extracted and then discretized into a series of points. Second, the discrete points of the boundary curves are mapped onto an \(x_{1}\)-\(x_{2}\) 2D plane to form a boundary of a 2D domain. Third, the meshless point generation algorithm for the 2D domain is applied to generate a series of discrete points within the domain. Fourth, all the discrete points on the \(x_{1}\)-\(x_{2}\) plane are mapped back onto the 3D surface to obtain the meshless discrete point set. Figure 1: Flowchart of the spatial discretization of a 3D space. The coordinates of the background points in the 3D domain are calculated as  
\[ \begin{array}{l}x_{i,j,k}=x_{start}+(i-1)\Delta x,(1\leq i\leq N_{x})\\ y_{i,j,k}=y_{start}+(j-1)\Delta y,(1\leq j\leq N_{y})\\ z_{i,j,k}=z_{start}+(k-1)\Delta z,(1\leq k\leq N_{z})\end{array}  \tag{9}\]  
where the subscript _start_ represents the background points at the start; \(\Delta x\), \(\Delta y\), and \(\Delta z\) represent the spacings between two neighboring background points along the \(x\), the \(y\), and the \(z\) directions, respectively; \(N_{x}\), \(N_{y}\), and \(N_{z}\) represent the total number of rows of all background points generated along the \(x\), the \(y\), and the \(z\) directions, respectively; and the subscripts \(i\), \(j\), and \(k\) denote the orders of the rows with certain background points along the \(x\), the \(y\), and the \(z\) directions, respectively. After generating the background points, the spatial relationship between the background points and the boundary points must be determined. A vector dot product method is introduced in this work to determine the attribute of the background points and thereby accelerate the generation of the discrete points. Considering a computational domain partially composed of a hemispherical surface, as shown in Figure 3, as an example, it is defined that the right side and the left side of the hemispherical surface are the outside and the inside of the computational domain, respectively. Point A is a point on the surface, whose normal vector is \(\overrightarrow{n}\), and point B beside the surface is a background Cartesian point to be evaluated. Point A is strictly selected by following two principles. First, point A is inside the 3D background points where point B is located. Second, point A is the closest surface boundary point to point B.  
Figure 3: Schematic of the vector dot product method used for determining the spatial relationship between boundary point A and background Cartesian point B. \(\overrightarrow{n}\) is the normal vector of the surface. Figure 2: Generation of discrete points using the mapping method. Considering the vector pointing from point A to point B as AB, the attribute of the background Cartesian point B can be determined as  
\[ \begin{split}\overrightarrow{\text{AB}}\cdot\overrightarrow{n}>0, \quad\text{In}\\ \overrightarrow{\text{AB}}\cdot\overrightarrow{n}<0,\quad\text{Out }\end{split}