# Mapping Based Quality Metrics for Mesh Deformation Algorithms Using Radial Basis Functions
## 4 Results and Discussions
### Test Case 3: Transform of Airfoil with a Viscous Layer  
The mesh deformation algorithm performed very well for rigid airfoil with large rotation, as shown in Section 4.2. However, it is not clear how it works when the airfoil is changed. In this test case, an airfoil transform problem with a C-Type structure mesh are investigated. The initial airfoil is NACA0012, which is symmetrical and the target airfoil is NACA4412, which is asymmetric, as shown in Figure 14. Additionally, 100 nodes on the airfoil and 84 nodes on the boundary are selected as the center node. Figure 15 gives the deformed mesh over the calculation domain. Due to the small displacement of the nodes, it is almost impossible to detect the distortion of elements at a certain distance from the airfoil. Deformed mesh near the training edge and the leading edge of the airfoil has been given in Figure 15. As can be seen, the mesh near the airfoil maintains good orthogonality. Mapping-based metrics are used to illustrate the ability of the deformation. Figure 16 shows the calculation results of mapping-based metrics. The minimum value of skewness metrics of deformation is from the area around the leading edges, which is larger than 0.955. The area around the training edge experiences a relative heavy area change. However, the size metrics value is large than 0.85. Figure 14: Airfoil change in test case 3. Figure 15: Deformed mesh in test case 3. For this case, the mesh far away from the airfoil is slightly deformed, and the quality of elements in a viscous layer should be paid more attention, especially the grid orthogonality. Figure 17 shows the skewness metrics and size metrics of each deformed mesh near the airfoil. The minimal value of skewness metrics is larger than 0.91 and size metrics is larger than 0.86. The deformed elements maintain good orthogonality and the area are also well preserved. Table 6 shows the comparison between two metrics and average metrics value has also been presented. 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 0.35%  
Compared with a geometrical method, mapping-based metrics show high accuracy for an angle change and area change of a specific element. Meanwhile, when it is used to give the quality of deformed mesh, mapping-based metrics do not need the information of deform mesh. The method proposed here is efficient to calculate the quality of the mesh deformation process, especially for the dynamic cases to check if there are degradable problems. Meanwhile, the method has a high potential to be used in commercial software, which only needs the initial mesh coordinates and deformation vectors with uncomplicated calculations. The display of the results is consistent with existing software processing. Although the method in this paper is deduced by generalized TPS, the approach is consequently suitable for any other RBFs. Finally, it should be noted that, because three 2-dimensional test cases are used, the calculation time and computing resource consumption is not different enough between a mesh-based metrics method and a new method proposed. In the next research, the mapping-based metrics method will be extended to a more complex problem in a 3-dimensional space, and the improvement of efficiency is expected. Conceptualization, S.J. and C.X. Methodology, C.X., C.A. and Y.L. Software, S.J. Validation, C.X., S.J. and C.A. Writing--original draft preparation, Y.L. and C.A. Writing--review and editing, S.J. and C.A. Visualization, C.X. and C.A. Project administration, C.Y. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Key Research and Development Program (2016 YFB0200703). Not applicable. The data presented in this study are available on request from the corresponding author. The data are not publicly available due to restriction of research funding. The authors declare no conflict of interest.