# An Animated Visualization Method for Large-Scale Unstructured Unsteady Flow
## 3 Methods
### Unsteady Animation Visualization Method
#### 3.2.3 Mesh Simplification  
Non-Three-Dimensional Cells  
Zero-dimensional (0D) cells consist only of vertex and multi-vertex cells, which are made up of two or more vertices with no connection between them. As shown in Figure 7, a vertex cell can be saved directly to the surface output cell, while a multi-vertex cell is divided into multiple vertex cells and then the variable attributes of the corresponding cells are copied and processed as vertex cells. Figure 7: Simplification method of multi-vertex, polyline, and quadratic curve. The dots in the figure represent the points of the cell. The numbers record the index of the cell after subdividing. There are three different varieties of 1D cells: line, polyline, and quadratic curve. A polyline is a cell that consists of multiple lines. The method of processing is comparable to the 0D cell. As output cells, the polyline is divided into several separate lines during mesh simplification. A quadratic curve is the simplest nonlinear cell, defined by its beginning point, ending point, and inflection point. When rendering nonlinear cells, computer graphics libraries must typically subdivide nonlinear cells into linear cells before converting them into linear primitives for rendering. Consequently, it can be directly subdivided into linear cells when simplifying the mesh, without both affecting the final visual effect and requiring repetitive cell subdivision operations during rendering. As shown in Figure 7, a quadratic curve cell can be viewed as a polyline composed of two lines, after which a straightforward linear cell division is performed. Two-dimensional cells consist of triangle, quadratic triangle, quadrilateral, quadratic quadrilateral, etc. Even though there are more cell types than 0D and 1D, the processing principle is still to subdivide nonlinear cells into linear cells and then divide the linear cells into simple linear cells. Using a quadratic quadrilateral cell as an example, first add a midpoint, then use the midpoint as a common vertex and form four linear quadrilateral cells with the original eight vertices, as depicted in Figure 8. This completes the simplification of a nonlinear 2D cell. Three-Dimensional Cells  
The processing of 3D cells is the most complicated part of mesh simplification, which requires analyzing all 2D surfaces of each 3D cell and only retaining the 2D surfaces that constitute the flow field surface. The judgment rules are as follows: let the set of 3D cells in the flow field be \(V\), the set of surfaces that constitute the flow field surface be \(S\), and the vertices of a 2D surface \(f\) be \(P\), then for this 2D surface, if there exists a unique \(v\in V\) such that \(P\subseteq v\), that is, all vertices in \(P\) are contained in the unique 3D cell \(v\), then the surface \(f\) is the external surface of the flow field. On the contrary, if there is no such unique 3D cell, then the surface \(f\) is the internal surface of the flow field. Let the set of vertices of 3D cell \(v\) be \(V_{v}\). Then, for a 2D surface \(f=p_{1},p_{2},\ldots,p_{k}|k\in N,\ k\geq 3\), where \(p_{i}(i\in N^{*})\) denotes the \(i\)th vertex of \(f\), the judgment process can be described as follows:  
\[\forall f,P\subseteq V_{v},v\in V,f=\{p_{1},p_{2},\ldots,p_{k}\ |\ k\in N,k\geq 3\}, \exists!v\in V\cdot s.t.P\subseteq V_{v}\Longrightarrow f\in S \tag{1}\]  
For each 2D surface of a 3D cell, the connectivity of the 3D cell where it is located can be analyzed to determine whether the surface belongs to the external surface of the flow field. First, obtain all the vertices that make up the 2D surface and then traverse all the cells connected to the current cell. If there exists any adjacent cell that contains these vertices, it means that it is shared by at least two 3D cells, so it is not an external surface. Otherwise, it can be processed as an external surface of the flow field according to the 2D cell.