# Mapping Based Quality Metrics for Mesh Deformation Algorithms Using Radial Basis Functions
## 3 Mapping-Based Metrics
### Skewness Metrics  
Skewness metrics is to analyse distortions of the deformed elements compared to the initial elements. According to Equation (22), denote \(f_{skew}\) as the mapping-based skewness metric of angle change, and:  
\[f_{skew}=\sin\theta \tag{25}\]  
\[\theta=\arccos(\frac{1+2E_{12}}{\sqrt{1+2E_{11}}\sqrt{1+2E_{22}}}) \tag{26}\]  
where \(E_{11},E_{12},E_{22}\) are items in a green strain tensor \(E\).  
\(f_{skew}=1\) means that the deformed element has equal angles with the origin reference one.  
\(f_{skew}\leq 0\) means that the deformed element is degenerate.  
In a traditional method [6; 22], mesh-based skewness metrics, which could be also called shape metrics, could be expressed as:  
\[\vec{f}_{skew}=\frac{\sqrt{3}v^{(k)}}{a_{1,1}^{(k)}+a_{2,2}^{(k)}-a_{1,2}^{(k)}} \tag{27}\]  
The definition of variables in the expression and calculation process can be found in Reference [30]. The evaluation requires coordinates of mesh elements and determinant calculation, which brings a computational burden.  
Based on the above description, the quality metrics of the generalized TPS mesh deformation algorithm used in this paper can be evaluated. The specific steps are as follows:  
**Step 1.** Calculate the deformation of mesh nodes over the computational domain based on the RBFs method with Equation (6).  
**Step 2.** Calculate the displacement gradient tensor \(H\) according to Equation (11), based on a central difference method. With the adjacent three mesh nodes \(\alpha,\beta,\eta\), a difference item \(\frac{\partial u_{i}^{\beta}}{\partial x_{j}^{\beta}}\)\((i,j=1,2,3)\) of the central node \(\beta\) in a displacement gradient tensor \(H\) can be calculated as:  
\[\frac{\partial u_{i}^{\beta}}{\partial X_{j}^{\beta}}=\frac{u_{i}^{\alpha}-2u _{i}^{\beta}+u_{i}^{\eta}}{\Delta X_{j}^{2}}\ (i,j=1,2,3) \tag{28}\]  
\[\Delta X_{j}=\frac{\left|X_{j}^{\alpha}-X_{j}^{\beta}\right|+\left|X_{j}^{ \eta}-X_{j}^{\beta}\right|}{2} \tag{29}\]  
where \(u_{i}^{\alpha},u_{i}^{\beta},u_{i}^{\eta}\)\((i=1,2,3)\) are displacement components of the mesh node \(\alpha,\beta,\eta\) and \(X_{j}^{\alpha},X_{j}^{\beta},X_{j}^{\eta}\) are coordinates of the initial mesh of node \(\alpha,\beta,\eta\).  
**Step 3.** Calculate the green strain tensor \(E\) according to Equation (17).  
\[E=\frac{1}{2}(H^{T}H+H^{T}+H) \tag{30}\]  
**Step 4.** Calculate the mapping-based size metrics of the mesh deformation method according to Equation (23).  
**Step 5**. Calculate the mapping-based skewness metrics of the mesh deformation method according to Equations (25) and (26).  
It should be noted that, in step 2, quality evaluation of mesh nodes on a fluid boundary is not valuable because the deformation of the fluid boundary is not clear. A difference item of boundary nodes can be equal to adjacent nodes in the fluid field inside. The mapping-based metrics only require the coordinates of initial mesh nodes and displacement of the mesh nodes as a description of step 1, which can be easily obtained from a mesh deformation process. Therefore, it can be regarded as an efficient method and can reveal the characteristic of the mesh deformation algorithm itself without information of deformed mesh.