## 2 Methodology
### RSM Model
#### 2.3.1 Sample Point Design  
In the seismic response optimization design of small-radius curved tunnel, the deformation of the tunnel structure due to seismic excitation should be reduced; therefore, the displacement parameter of small-radius curved tunnel was selected as the first objective to minimize. At the same time, the vibration energy generated by the seismic excitation should not be excessive, so that the acceleration parameter of small-radius curved tunnel was selected as the second objective to minimize. In the parametric design, three main designed parameters are used, buried depth for small-radius curved tunnel, angle degree for curved tunnels and the curve radius \(R\), so that they vary within a defined range. The seismic response of the tunnel structure is, therefore, formulated as follows:  
\[\left\{\begin{array}{rl}Max:(depth,\text{deg},R)\\ Min:(depth,\text{deg},R)\\ S.t\ x^{l}\leq(depth,\text{deg},R)\leq x^{u}\\ x=(depth,\text{deg},R)\end{array}\right. \tag{1}\]  
where, \(x=(x_{1},x_{2},\cdots,x_{k})\) is a vector of \(k\) design variables of structural geometry \(x^{l}=(x^{l}_{1},x^{l}_{2},\cdots,x^{l}_{k})\) and \(x^{u}=(x^{u}_{1},x^{u}_{2},\cdots,x^{u}_{k})\) are the respective lower and upper bounds of the design variables. Due to high computational cost of quasi-static compression finite element simulation, it cannot be directly used to address the MOOD issues that requires several hundred performance evaluations. RSM model is widely used as an alternative model instead of non-linear finite element simulations to solve MOOD problems that  
Figure 2: Modeling process of RSM model.  
require fast iterations. In this paper, Design-Expert V12 statistical software is used for sample point design. A range of designed parameter values is selected through several FEM tests. The buried depth for the small-radius curved tunnel is 10-20 m, the angle deg of the curved tunnel is 45-60\({}^{\circ}\)and the curve radius R is 300-350 m, as shown in Table 1.  
The experimental design used a triple Box-Behnken response surface design type without embedding factors or partial factor designs. Through the experimental design, 17 sets of finite element simulations corresponding to different combinations of geometric parameters were conducted to derive the response values of each geometry combination to the displacement and acceleration of key points. The key point is to select the entrance endpoint A (Figure 3, Point A) of the small-radius curve tunnel and the connection point B (Figure 3, Point B) of the small-radius curve tunnel and the straight tunnel. The different combinations of design geometries and the corresponding response values are shown in Tables 2 and 3. Geometrical combinations for 17 different sets of parameters and the corresponding displacement (s) and acceleration (a) response values drawn from the FEM were input into Box-Behnken model of the statistical software Design-Expert V12 using the RSM. The polynomial equations (RSM model) were fitted to the FEM results using stepwise regression and identifying the relevant model terms. In this way, the constructed model and the individual terms in the regression equation can be checked for best fit results.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**No.** & **Depth (m)** & **Angle (deg)** & **R (m)** & **s (m)** & **_a (m/s\({}^{2}\))** \\ \hline  
5 & 325 & 0.12005 & 4.152 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The design matrix of Point A.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variable** & \(-\)**1** & **0** & **1** \\ \hline Depth of the curved tunnel, U (m) & 10 & 15 & 20 \\ Angle of the curved tunnel, V (deg) & 45 & 52.5 & 60 \\ Radius of curve, W (m) & 300 & 325 & 350 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameter setting and test design level.