# Mapping Based Quality Metrics for Mesh Deformation Algorithms Using Radial Basis Functions
## 3 Mapping-Based Metrics
### Mapping Relationships  
In order to describe the motion and deformation of an element, the initial configuration \(\Omega\) is selected as the reference configuration and \(\omega\) as the current configuration deformed. A nonsingular single-valued mapping \(M:\Omega\rightarrow\omega\) is established, as shown in Figure 1. \(OR_{1}R_{2}R_{3}\) is set as the global coordinate system. The position of the point \(P\) in the reference configuration is \(\mathbf{X}(X_{1},X_{2},X_{3})\) and its image point \(P^{\prime}\) with the deformation process in the current configuration is \(\mathbf{x}(x_{1},x_{2},x_{3})\). \(X_{i}(i=1,2,3)\) and \(x_{i}(i=1,2,3)\) are coordinates in \(OR_{1}R_{2}R_{3}\). The mapping of the point can be written as:  
\[x_{i}=x_{i}(X_{1},X_{2},X_{3})\ (i=1,\ 2,\ 3) \tag{7}\]  
As shown in Figure 1, the position of two adjacent points in the reference configuration is given by \(\mathbf{X}\) and \(\mathbf{X}+d\mathbf{X}\). The distance vector between two adjacent points is \(d\mathbf{X}\). In the current configuration, the distance vector between two adjacent points after mapping is \(d\mathbf{x}\). The transformation between \(d\mathbf{X}\) and \(d\mathbf{x}\) is investigated as:  
\[d\mathbf{x}=\mathbf{x}(\mathbf{X}+d\mathbf{X})-\mathbf{x}(\mathbf{X})=\frac{\partial\mathbf{x}}{\partial \mathbf{X}}d\mathbf{X}=Fd\mathbf{X} \tag{8}\]  
where \(F\) is the deformation gradient tensor. \begin{table}
\begin{tabular}{c c c} \hline
**No.** & **Name** & \(\mathbf{f}(\mathbf{x})\) \\ \hline
1 & \(\text{CPC}^{0}\) & \((1-\theta)^{2}\) \\
2 & \(\text{CPC}^{2}\) & \((1-\theta)^{4}(4\theta+1)\) \\
3 & \(\text{CPC}^{4}\) & \((1-\theta)^{6}(35/3-\theta^{2}+6\theta+1)\) \\
4 & \(\text{CPC}^{6}\) & \((1-\theta)^{8}(32\theta^{3}+25\theta^{2}+8\theta+1)\) \\
5 & \(\text{CTPSC}^{0}\) & \((1-\theta)^{5}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Radial basis functions (RBFs) with compact functions (\(\theta=\frac{r}{r}\), \(x\) is the Euclidean distance and \(r\) is the support radius). \begin{table}
\begin{tabular}{c c c} \hline
**No.** & **Name** & \(\mathbf{f}(\mathbf{x})\) \\ \hline
1 & Thin plate spline (TPS) & \(x^{2}\log(x)\) \\
2 & Multi-quadric bi-harmonics (MQB) & \(\sqrt{a^{2}+x^{2}}\) \\
3 & Inverse multi-quadric bi-harmonics (IMQB) & \(\sqrt{1/(a^{2}+x^{2})}\) \\
4 & Quadric bi-harmonics (QB) & \(1+x^{2}\) \\
5 & Gaussian & \(e^{-x^{2}}\) \\ \hline \end{tabular}
\end{table} 
Table 2: RBFs with global functions. Assuming the displacement vector of point is \(\mathbf{u}=\mathbf{x}-\mathbf{X}\), then it easily follows from Equation (8) that:  
\[d\mathbf{x}=\frac{\partial(\mathbf{X}+\mathbf{u})}{\partial\mathbf{X}}d\mathbf{X}=(\mathbf{I}+\frac{ \partial\mathbf{u}}{\partial\mathbf{X}})d\mathbf{X} \tag{9}\]  
Given a displacement gradient tensor \(\mathbf{H}=\mathbf{F}-\mathbf{I}\), the formulation of distance can be expressed as:  
\[dx_{i}=(\delta_{ij}+H_{ij})dX_{j} \tag{10}\]  
\[\mathbf{H}=\frac{\partial\mathbf{u}}{\partial\mathbf{X}}=\left( \begin{array}{ccc}\frac{ \partial u_{1}}{\partial X_{1}}&\frac{\partial u_{1}}{\partial X_{2}}&\frac{ \partial u_{1}}{\partial X_{3}}\\ \frac{\partial u_{2}}{\partial X_{1}}&\frac{\partial u_{2}}{\partial X_{2}}& \frac{\partial u_{2}}{\partial X_{3}}\\ \frac{\partial u_{3}}{\partial X_{1}}&\frac{\partial u_{3}}{\partial X_{2}}& \frac{\partial u_{3}}{\partial X_{3}}\end{array} \right) \tag{11}\]  
It is assumed that \(\left[\mathbf{E}^{1},\mathbf{E}^{2},\mathbf{E}^{3}\right]\) are the unit base vectors of the reference configuration. Any vector \(\mathbf{V}\) in the reference can be expressed as \(\mathbf{V}=V_{i}\mathbf{E}^{i}\)\((i=1,2,3)\). Images of unit base vectors in the current configuration is \(\left[\mathbf{e}^{1},\mathbf{e}^{2},\mathbf{e}^{3}\right]\) and the images of vector \(\mathbf{V}\) can be expressed as \(\mathbf{v}=v_{i}\mathbf{E}^{i}\)\((i=1,2,3)\). According to continuum mechanics [34], the relationships between coordinates of two vectors in the reference coordinates are:  
\[\left[ \begin{array}{c}v_{1}\\ v_{2}\\ v_{3}\end{array} \right]=\left[