## 3 Meta-Modeling
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Design Variable** & **Level I** & **Level II** & **Level III** \\ \hline \(x_{1}\): Length of body(mm) & 150 & 200 & 250 \\ \(x_{2}\): Diameter of tube(mm) & \(\phi\)6 [1] & \(\phi\)8 & \(\phi\)12 \\ \(x_{3}\): Material type & S45C & SUS316L & \\ \(x_{4}\): Insulation density(kg/m\({}^{3}\)) & 96 & 128 & 160 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Design variables and their levels. where \(A_{\text{K}}\) is the unknown coefficient vector, and \(Z(x)\) is the global model vector, which is defined by design variable, \(x\in E^{u_{d}}\). \(E(x)\) is supposed as an independent normal distribution. \(E(x)\) is also the spatial correlation of design data, and defined as follows:  
\[E\big{(}\theta,\;x_{i},\;x_{j}\big{)}=\exp\Bigg{[}-\sum_{k=1}^{n_{d}}\theta_{k} \big{(}x_{i}-x_{j}\big{)}^{2}\Bigg{]}. \tag{6}\]  
The correlation coefficient \(\theta\) can be calculated as the most probabilistic correlation coefficient through the maximum likelihood estimation process that maximizes a likelihood function. The RSM is generally generated as a regression model of second-order polynomials via the least-squares method [12]. The RSM was defined as follows:  
\[g=A_{0}+\sum_{i=1}^{n}A_{i}\;Z_{i}+\sum_{i=1}^{n}A_{ii}\;Z_{i}^{2}+\sum_{i=1}^ {n}\sum_{j=i+1}^{n}A_{i}\;Z_{i}Z_{j}+e. \tag{7}\]  
Given the actual response vector \(g\) and the matrix \(Z\) of \(n\) basis variables from the design of experiment data, the relationship between \(g\) and \(Z\) is as follows:  
\[g=ZA_{R}+e. \tag{8}\]  
In order to minimize the random error vector \(e\) and estimate the unknown coefficient vector \(A_{\text{R}}\), a least-squares function is used as follows:  
\[A_{R}=\;\left(Z^{T}Z\right)^{-1}Z^{T}g. \tag{9}\]  
Applying the coefficient calculated from Equation (9), the quadratic regression metamodel is obtained as follows:  
\[\widetilde{g}(x)_{R}=\;a_{0}+\;\sum_{i=1}^{n}a_{i}x_{i}+\;\sum_{i=1}^{n}a_{ii} x_{i}^{2}+\sum_{i=1}^{n}\sum_{j=i+1}^{n}a_{ij}x_{i}x_{j}, \tag{10}\]  
where the RSM coefficient \(a\) is the element of the estimated \(A_{\text{R}}\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & & & \\ \hline
**Run** & \(x_{1}\)**(mm)** & \(x_{2}\)**(mm)** & \(x_{3}\) & \(x_{4}\)**(kg/m\({}^{3}\))** \\ \hline
1 & 150 & 6 & S45C & 96 \\
2 & 150 & 6 & S45C & 128 \\
3 & 150 & 6 & S45C & 160 \\ � & � & � & � & � \\
52 & 250 & 12 & SUS316L & 96 \\
53 & 250 & 12 & SUS316L & 128 \\
54 & 250 & 12 & SUS316L & 160 \\ \hline \hline \multicolumn{5}{c}{**Output Responses**} \\ \hline
**Run** & \(f\)**(kg)** & \(g_{1}\)**(\({}^{\circ}\)**C)** & \(g_{2}\) & \(g_{3}\)**(1000 \(\times\) W)** \\ \hline
1 & 0.65 & 177.06 & 12.6 & 3.8 \\
2 & 0.65 & 169.56 & 12.6 & 3.8 \\
3 & 0.65 & 169.09 & 12.6 & 3.9 \\
52 & 1.30 & 58.70 & 12 & 37 \\
53 & 1.30 & 56.65 & 12 & 37 \\
54 & 1.30 & 56.45 & 12 & 37 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Design variables and their levels. The RBFN was developed for scattered multivariate data interpolation [13; 14]. The RBFN meta-model generally uses linear combinations of a radially symmetric function based on Euclidean distance to approximate the output response functions. Let \(x_{1},\ \cdots,\ x_{n}\in\Omega\subset U^{n}\) be a given set of nodes, and a set of any RBFN basis function is defined as follows:  
\[g_{j}(x)=\ \varnothing\big{(}||x-x_{j}||\big{)}\in U,\ \ j=1,\ 2,\ 3,\ \cdots,\ n, \tag{11}\]  
where \(\varnothing\) is power spline basis function, and \(||x-x_{j}||\) is Euclidean distance. From the Euclidean distance, the input data layers of RBFN are assorted as a homogeneous cluster, and the \(\varnothing\) is defined as follows:  
\[\varnothing(r,c)=r^{c_{j}}, \tag{12}\]  
where \(c\) is a shape function coefficient with a positive constant variable, and \(r\) is the Euclidean distance.