## 4 Basic Interpolation Rule and Interpolation Functions
### Moving Kriging and Moving Least Squares Interpolation Methods  
In this section, two interpolation methods are introduced to build an interpolation matrix: the moving Kriging and moving least squares interpolation functions. Both methods satisfy the normalization condition.  
The moving Kriging (MK) interpolation method is based on the work of [PERSON] [27]. The moving Kriging interpolation \(u(x)\) is defined by  
\[\mathbf{u}(x)=p^{T}(x)\mathbf{SU}+r(x)^{T}R^{-1}(I-P\mathbf{S})\mathbf{U} \tag{65}\]  
\[\mathbf{U}=\begin{bmatrix}\mathbf{u}(x_{1})\\ \vdots\\ \mathbf{u}(x_{n})\end{bmatrix} \tag{66}\]  
where \(\mathbf{P}\), \(\mathbf{R}\), \(r(x)\) and \(\mathbf{S}\) are given by Equations (67)-(70), respectively.  
\[\mathbf{P}=\begin{bmatrix}\mathbf{p}(x_{1})&\cdots&\mathbf{p}(x_{i})&\cdots&\mathbf{p}(x_{n}) \end{bmatrix} \tag{67}\]  
\[\mathbf{R}=\begin{bmatrix}1&R\begin{pmatrix}\mathbf{x}_{1},\mathbf{x}_{2} \end{pmatrix}&\cdots&R\begin{pmatrix}\mathbf{x}_{1},\mathbf{x}_{n}\end{pmatrix}\\ R\begin{pmatrix}\mathbf{x}_{2},\mathbf{x}_{1}\end{pmatrix}&1&\cdots&R\begin{pmatrix} \mathbf{x}_{2},\mathbf{x}_{n}\end{pmatrix}\\ \vdots&\vdots&\ddots&\vdots\\ R\begin{pmatrix}\mathbf{x}_{n},\mathbf{x}_{1}\end{pmatrix}&R\begin{pmatrix}\mathbf{x}_{n}, \mathbf{x}_{2}\end{pmatrix}&\cdots&1\end{bmatrix} \tag{68}\]  
\[\mathbf{r}(x)=\begin{bmatrix}R(\mathbf{x}_{1},\mathbf{x})&\cdots&R(\mathbf{x}_{i^{\prime}},\bm {x})&\cdots&R(\mathbf{x}_{n},\mathbf{x})\end{bmatrix} \tag{69}\]  
\[\mathbf{S}=\begin{pmatrix}P^{T}\mathbf{R}^{-1}\mathbf{P}\end{pmatrix}^{-1}P^{T}\mathbf{R}^{-1} \tag{70}\]  
\(R\begin{pmatrix}x_{i},x_{j}\end{pmatrix}\) is the correlation function between any two of the n nodes, \(x_{i}\) and \(x_{j}\). \(p(x)\) is a polynomial basis given by Equation (71), and m denotes the number of terms in the basis.  
\[\mathbf{p}(\mathbf{x})^{T}=\begin{bmatrix}1&x&y&x^{2}&y^{2}&xy\end{bmatrix} \tag{71}\]  
We can introduce the notation  
\[\mathbf{B}=\mathbf{R}^{-1}(\mathbf{I}-\mathbf{P}\mathbf{S}) \tag{72}\]  
Then, Equation (65) can be rewritten as:  
\[\mathbf{u}(x)=\begin{bmatrix}p^{T}(x)\mathbf{S}+r(x)^{T}\mathbf{B}\end{bmatrix}\mathbf{U}= \sum\limits_{i=1}^{n_{c}}\phi_{i}(x)u_{i}=\varphi(x)\mathbf{U} \tag{73}\]  
where  
\[\phi_{i}(x)=\sum\limits_{j=1}^{m}p_{j}(x)\mathbf{S}_{ji}+\sum\limits_{k=1}^{n_{c}} r_{k}(x)^{T}\mathbf{B}_{ki} \tag{74}\]  
According to [26], the shape function \(\phi_{i}(x)\) has the characteristic that \(\sum\limits_{i}^{n}\phi_{i}=1\), which satisfies Equation (64). So, the MK interpolation method satisfies the normalization condition and could be used for the assembly of two substructures.  
Similar to the MK approximation, the moving least squares interpolation \(u(x)\) can be defined by  
\[\mathbf{u}(x)=p^{T}(x)\mathbf{A}(x)^{-1}\mathbf{B}(\mathbf{x})\mathbf{U}=\Phi(\mathbf{x})\mathbf{U} \tag{75}\]  
where \(\mathbf{A}(\mathbf{x})\) and \(\mathbf{B}(\mathbf{x})\) can be defined by  
\[\mathbf{A}(\mathbf{x})=\sum\limits_{i=1}^{n}\mathbf{R}(\mathbf{x},\mathbf{x}_{i})\mathbf{p}(\mathbf{x}_{i })\mathbf{p}^{T}(\mathbf{x}_{i}) \tag{76}\]  
\[\mathbf{B}(\mathbf{x})=\begin{bmatrix}R(\mathbf{x},\mathbf{x}_{1})\mathbf{p}(\mathbf{x}_{1})&\cdots&R (\mathbf{x},\mathbf{x}_{n})\mathbf{p}(\mathbf{x}_{n})\end{bmatrix} \tag{77}\]According to [26], the MLS interpolation method also satisfies the normalization condition and could be used for the assembly of two substructures.