## 4 Optimization of FEM  
The optimization of the FEM model has been carried out by the RSM to know the present condition of the Bohyeonsan CGD. This will give the correct estimation of the degraded modulus of elasticity for showing the aging effect on safety evaluation. RSM is a statistical and mathematical technique for system identification initially developed and explained by [PERSON] and [PERSON] [51]. In generating an empirical model building, RSM plays an important role [52]. The objective of this method is to optimize the output (\(y\)) variables generally affected by the input variables (\(x\)), which can be expressed by Equation (17) [53]. \[y=\ f(x_{1},\ x_{2},\ \ldots\ x_{k})\ +\varepsilon \tag{17}\]  
where the responses of the structure are indicated by \(f(x_{1},\ x_{2},\ \ldots\ x_{k})\), and error of that response is described by \(\varepsilon\). In order to skip the unnecessary true function, for many reasons, a flexible mark off polynomial function can be adopted. The two most common forms of a polynomial function are as follows:  
\[y=\ \beta_{0}+\beta_{1}x_{1}+\beta_{2}x_{2}+\ \cdots\cdots\cdots+\beta_{k}x_{k} \tag{18}\]  
\[y=\ \beta_{0}+\sum_{j=1}^{k}\beta_{j}x_{j}+\sum_{j=1}^{k}\beta_{jj}x_{j}^{2}+ \sum_{i}^{k-1}\sum_{j}^{k}\beta_{ij}x_{i}x_{j} \tag{19}\]  
The coefficients \(\beta_{0},\ \beta_{1}\cdots\cdots\) are parameters that need to be estimated from the data. Several mathematical tools are available for the DoE, and CCD developed by [PERSON] and [PERSON] [51] is a very useful tool to reduce the experimental phase. This equation is also used recently for evaluating the seismic capacity and risk prediction of CGD in [PERSON] et al. [23]; [PERSON] et al. [39] and [PERSON] et al. [54]. The CCD is classified into three types according to the axial points (an indication of the curvature) as central composite circumscribed, central composite inscribed and central composite face-centered. In this study, using CCD, the total number of samples for running the experiment is calculated by the following Equation (20). \[N=2^{p}+2p+n_{c} \tag{20}\]  
where \(p\) is the factor number, and \(n_{c}\) denotes the center-point number of the sample, which is generally considered as 5 or 6. Using Equation (20) the number of numerical simulations is calculated as nine with two input variables such as \(CoE\) (coefficient of modulus of elasticity) and \(\rho\) (mass density). To get the optimized modulus of elasticity, the \(CoE\) will help by multiplying with the present variables. The recorded acceleration data was collected for the Pohang Earthquake, which was the strongest earthquake in the Korean Peninsula [55]. The simulation had been done with the nine combinations of two material properties (input variables), and after optimization by RSM, the present value of those two parameters has been come out as 0.787 and 2.32 (tone/m\({}^{3}\)), respectively. In addition, the final inside and outside modulus of elasticity is 10,835 MPa and 13,269 MPa, respectively. The response spectra comparison between the before and after optimization has been carried out and showed in Figure 10. Figure 10 indicates that the similarities of the top RS after optimization is acceptable for the structural system identification. The FEM with the present parameters then verified by comparing the fundamental frequencies by the frequency domain decomposition (FDD) of the experimental data. According to [PERSON] et al. [56] the FDD can decompose the system response from recorded data by following a simple technique that is related to the decomposition of each of the estimated spectral density matrices explained in [PERSON] et al. [57]. The FDD from the sensor database is visualized in Figure 11a, where the first peak depicts the fundamental frequency of the CGD. Analysis of the mode shape of 3D FEM considering the optimized material properties is also illustrated in Figure 11b. From the modal analysis, the most dominating mode shape shows the fundamental frequency of the FEM.