## 2 Materials and Methods
### Optimization Design Based on Radial Basis Interpolation of Surrogate Models  
#### 2.2.1 Basic Definition of the Radial Point Interpolation Method  
The radial point interpolation method (RPIM) combines the point interpolation method (PIM) and the radial base function (RBF), which have been widely used in meshless simulations and other fields. RPIM provides an efficient way to create accurate surrogate models for complex engineering problems by enabling the interpolation of scattered data points without requiring a structured grid. The method is particularly valuable for optimizing the flexible nozzle design as it can handle multiple design variables, non-linear relationships between inputs and outputs, and complex geometric constraints. This approach significantly reduces the computational cost compared to full numerical simulations while maintaining accuracy. The following equation describes RPIM as  
\[u^{h}(\mathbf{x})=\sum_{i=1}^{n}R_{i}(\mathbf{x})a_{i}=\mathbf{R}^{\mathbf{T} }(\mathbf{x})\mathbf{a}, \tag{10}\]  
where \(R_{i}(\mathbf{x})\) is the radial basis function. The RPIM interpolation with polynomial expansion is expressed as  
\[u(\mathbf{x})=\sum_{i=1}^{n}R_{i}(\mathbf{x})a_{i}+\sum_{j=1}^{m}P_{j}( \mathbf{x})b_{j}=\mathbf{R}^{\mathbf{T}}(\mathbf{x})\mathbf{a}+\mathbf{P}^{ \mathbf{T}}(\mathbf{x})\mathbf{b}, \tag{11}\]  
where \(P_{j}(x)\) is the polynomial basis term in the spatial coordinates \(x\); \(n\) and \(m\) denote the number of RBFs and the number of polynomial basis terms, respectively; \(a_{i}\) and \(b_{j}\) are the unknown coefficients; \(R_{i}(x)\) denotes the RBF with the variable \(r\); and \(r\) denotes the distance between the point \(x\) and the node \(x_{i}\). \[r=\sqrt{(x-x_{i})^{2}+(y-y_{i})^{2}}. \tag{12}\]  
To determine the unknown coefficients \(a_{i}\) and \(b_{j}\) in Equation (10), the \(a\) support domain containing \(n\) field nodes is created at point \(x\). The state function values of the \(n\)-sampled nodes are described in vector form as  
\[\mathbf{U_{S}}=\mathbf{R_{0}}\mathbf{a}+\mathbf{P_{m}}\mathbf{b}. \tag{13}\]  
In addition, m additional equations are as follows:  
\[\mathbf{P_{m}}^{\mathbf{T}}\mathbf{a}=\mathbf{0}. \tag{14}\]  
Combining Equations (13) and (14) yields  
\[\widetilde{\mathbf{U}}_{s}= \begin{bmatrix}\mathbf{U_{s}}\\ \mathbf{0}\end{bmatrix} = \begin{bmatrix}\mathbf{R_{0}}&\mathbf{P_{m}}\\ \mathbf{P_{m}^{T}}&\mathbf{0}\end{bmatrix}\begin{bmatrix}\mathbf{a}\\ \mathbf{b}\end{bmatrix} =\mathbf{Ga_{0}}, \tag{15}\]  
where  
\[\mathbf{G}= \begin{bmatrix}\mathbf{R_{0}}&\mathbf{P_{m}}\\ \mathbf{P_{m}}^{\mathbf{T}}&\mathbf{0}\end{bmatrix} . \tag{16}\]  
Equation (10) at point x is then expressed as  
\[u(\mathbf{X})=\widetilde{\mathbf{\Phi}}^{T}\widetilde{\mathbf{U}}_{s}=\Big{\{} \mathbf{R}^{\mathbf{T}}(\mathbf{x})\ \mathbf{P}^{\mathbf{T}}(\mathbf{x})\Big{\}}\mathbf{G}^{-1}\widetilde{\mathbf{ U}}_{s}, \tag{17}\]where \(\widetilde{\mathbf{\Phi}}^{T}\) is the node displacement corresponding to the first n components of the shape function derived from RPIM:  
\[\widetilde{\mathbf{\Phi}}^{T}=\{\phi_{1}(\mathbf{x})\ \phi_{2}(\mathbf{x})\cdots\phi_{n+m}( \mathbf{x})\}. \tag{18}\]  
Thus, Equation (17) is reformulated as  
\[u(\mathbf{x})=\mathbf{\Phi}^{T}(\mathbf{x})\mathbf{U}_{\mathbf{S}}=\sum_{i=1}^ {n}\phi_{i}u_{i}. \tag{19}\]  
Notice that the \(\mathbf{\Phi}^{T}(\mathbf{x})\) is called the shape function corresponding to the displacement of the node. \[\mathbf{\Phi}^{T}(\mathbf{x})=\Big{\{}\ \phi_{1}(\mathbf{x})\ \ \ \phi_{2}( \mathbf{x})\ \ \cdots\ \phi_{n}(\mathbf{x})\Big{\}}. \tag{20}\]  
The shape function \(\Phi\)T(x) is a mathematical function that determines how the surrogate model interpolates between known data points. It acts as a weighting function that defines how much influence each sample point has on the prediction at any given location in the design space. \(\Phi(x)\) represents the contribution of the i-th sample point to the prediction at location x.