## 3 Meta-Modeling  
The meta-model used for the approximate optimization is generated in order to predict the response characteristics of both objective function and constraints in a given design \begin{table}
\begin{tabular}{c c c c} \hline
**No.** & **Analysis (\({}^{\circ}\)C)** & **Test (\({}^{\circ}\)C)** & **Error (\%)** \\ \hline
1 & 172.4 & 167.1 & 3.2 \\
2 & 76.6 & 67.2 & 14.0 \\
3 & 170.0 & 179.4 & 5.2 \\
4 & 58.4 & 53.7 & 8.8 \\ \hline \end{tabular}
\end{table} 
Table 2: Comparison of temperature measurement results. Figure 7: Fire test facility and configuration. range with the minimum error rate and then applied to the optimization analysis process to improve convergence and to reduce numerical cost. In this study, various meta-models to be applied to the approximate optimization were generated using the design of the experiment method, and the verified results of both the finite element model and transient heat transfer analysis were integrated with the design of the experiment. The design variables and their levels applied to the design of the experiment were set to three levels for both the body length and the tube diameter of A60 BPP, two levels for the material type of A60 BPP, and three levels for the insulation density, respectively. The details of the design variables and their levels were summarized in Table 3.  
As represented in Table 3, the levels for the length of the body were defined as 50 mm intervals. The levels for the diameter of the tube were set to \(\phi\)6, \(\phi\)8, and \(\phi\)12 considering the diameter specifications of steel tubes that can be actually manufactured. Since both the S45C and the SUS316L were conventionally used for the fire-resistance apparatus made of steel category were, the material type of A60 BPP was set to the two-level design variable. The levels for the insulation density were set to 96, 128, and 160, considering the insulation specifications that have been actually used for the A60 class insulation. The output responses evaluated from the design of the experiment according to the variation of the levels of design variables were considered the single objective function and a number of constraints. The objective function (\(f\)) was considered the weight of the A60 BPP itself and the insulation attached to the A60 BPP, excluding the fire test specimen. Among the constraints, the fire test temperature (\(g_{1}\)) was considered the results measured from the transient heat transfer analysis. Productivity constraint (\(g_{2}\)) was generated by applying weight factors of 2.0, 1.0, and 0.5 for each design variable of length, diameter, and material type of A60 BPP. For cost constraint (\(g_{3}\)), the price of each material type of the A60 BPP per unit weight was considered in units of 1000 Korea won. A total of 45 full-factorial experiments were composed by applying the level variation of the design variables represented in Table 3. In Table 4, the results of full-factorial experiments for the objective function and constraints are summarized according to the variation of the design variables. Various meta-models such as Kriging, RSM, and RBFN were generated from the results of full-factorial experiments in Table 4, and the accuracy of approximating the design space of A60 BPP was compared for each meta-model. The Kriging is defined as the sum of the global model representing the global characteristics of an actual design space function to be approximated and the local model corresponding to the deviation between the actual function and the global model [11]. The Kriging meta-model was defined as follows:  
\[\widetilde{g}(x)_{K}=Z(x)^{T}A_{K}+E(x) \tag{5}\]  
\[A_{K}=\ \big{(}a_{1},\ a_{2},\ \cdots,\ a_{p}\big{)}^{T}\]  
\[Z(x)=\ \big{[}z_{1}(x),\ z_{2}(x),\ \cdots,\ z_{p}(x)\big{]}^{T},\]