# Mapping Based Quality Metrics for Mesh Deformation Algorithms Using Radial Basis Functions
## 4 Results and Discussions
### Test Case 1: Translation and Rotation of Structured Square Mesh  
In the first test cases, a standard test case where the initial mesh elements are all square with a side length \(C\) has been produced. As shown in Figure 1(a), the computational domain has dimension \(80C\times 40C\) and the rectangle at the center of the domain has dimension \(4C\times 2C\). The rectangle is rotated 45 degrees around the center. The deformed mesh based on the RBFs' method using generalized TPS is shown in Figure 1(b).  
The mapping-based metrics proposed in this paper are used to evaluate the mesh deformation algorithm. Figure 3 shows the skewness metric and size metric over computational domain. It can be seen that the blue area is of the minimal value of quality metrics, which means the elements undergo the greatest compression and distortion over this domain.  
The accuracy of mapping-based metrics of mesh deformation algorithm is checked using a specific element with a minimal metrics value in the blue area. Figure 4 has marked the edge of a specific element with a red line. Calculating the angle change and area ratio of a specific element, and compass between mapping-based metrics, mesh-based metrics, and a geometrical method can be completed.  
Figure 2: Mesh of test case 1.  
With a geometrical method, the angles of deformed mesh and area ratio from the initial mesh to deformed mesh can be calculated directly based on geometrical relationships. As quadrilateral elements, the geometrical configuration of a specific element is shown in Figure 5. The element has four nodes denoted as \(k,k+1,k+2,k+3\) and \(\theta_{i}\) is the angle between two sides joined at the \(i\)th node. Calculation results of the angle change have been presented in Table 3. It can be seen that calculation results by the mapping-based metric have a very small error compared to a geometrical calculation directly, which is less than 1.5%. The area ratio calculation results of mapping-based metrics is 0.5810. It is very close to the result of the geometrical method, which has a value of 0.5803.  
Figure 6 shows the calculation results of traditional mesh-based metric described in Sections 3.2 and 3.3 in this test case. The metrics distribution is nearly the same with that of mapping-based metrics presented in Figure 3. This is because every initial cell is a square, and the sides of elements are parallel to the base vectors of the initial configuration.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Node** & **Coordinate X** & **Coordinate Y** & **Angle by** & **Angle by** & **Error** \\  & & & & **Metric/Deg** & **Geometry/Deg** & \\ \hline \(k\) & 1.73 & 5.04 & 133.05 & 133.67 & 0.46\% \\ \(k+1\) & 2.19 & 4.86 & 48.73 & 48.14 & 1.23\% \\ \(k+2\) & 2.05 & 5.24 & 130.04 & 129  
Therefore, the changed angle between the two unit base vectors equals to the changed angle between the two sides of the deformed mesh element. Table 4 shows the comparison between two metrics and average metrics value that has also been presented. The average value of the metric over all the elements indicates the average quality of the mesh. The higher the average quality of the mesh, the more stable, accurate, and efficient the computation will be. The minimum value of the metric over all the elements indicates the quality of the cell with the lowest quality. This value is required to be larger than zero. Otherwise, the mesh will contain degenerate elements. Degenerate elements have a very negative influence on the stability and accuracy of numerical computations [22]. All the average and minimal quality metrics of size and skew calculated by mapping-based metrics and traditional mesh-based metrics methods are closed, resulting in small error values of less than 0.22%.