# Shape Optimization of Single-Curvature Arch Dam Based on Sequential Kriging-Genetic Algorithm
## 3 Case Study
### Optimization Results and Discussion  
When a GA-KSM was used to optimize the shape of a single-curvature arch dam, the optimization results depended predominantly on the approximate accuracy of the KSM, which is directly related to the number of training samples. In this study, to meet the accuracy requirements of the KSMs of \(\sigma_{1}\), \(\sigma_{3}\), \(\sigma_{1 eqv}\), \(\sigma_{3 eqv}\), and \(V\), any of the _MAREs_ between \(\sigma_{1}\) and \(\hat{\sigma}_{1}\), \(\sigma_{3}\) and \(\hat{\sigma}_{3}\), \(\sigma_{1 eqv}\) and \(\hat{\sigma}_{1 eqv}\), \(\sigma_{3 eqv}\), and \(V\) and \(\hat{V}\) must not exceed 1%. Figure 4 illustrates the variation of _MAREs_ between \(\sigma_{1}\) and \(\hat{\sigma}_{1}\), \(\sigma_{3}\) and \(\hat{\sigma}_{3}\), \(\sigma_{1 eqv}\) and \(\hat{\sigma}_{1 eqv}\), \(\sigma_{3 eqv}\) and \(\hat{\sigma}_{3 eqv}\), and \(V\) and \(\hat{V}\) with the number of sample points increasing. As illustrated in Figure 4, the _MAREs_ between \(\sigma_{1}\) and \(\hat{\sigma}_{1}\), \(\sigma_{3}\) and \(\hat{\sigma}_{3}\), \(\sigma_{1 eqv}\) and \(\hat{\sigma}_{1 eqv}\), \(\sigma_{3 eqv}\) and \(\hat{\sigma}_{3 eqv}\), and \(V\) and \(\hat{V}\) decreased as the sample size increased. When the accuracy of these KSMs satisfied the requirements, the number of training samples was 500. Such a high sampling cost is unacceptable in engineering optimization problems. Figure 5 depicts the variation of relative errors (\(RE\)) between \(\sigma_{1}\) and \(\hat{\sigma}_{1}\), \(\sigma_{3}\) and \(\hat{\sigma}_{3}\), \(\sigma_{1 eqv}\) and \(\hat{\sigma}_{1 eqv}\), \(\sigma_{3 eqv}\) and \(\hat{\sigma}_{3 eqv}\), and \(V\) and \(\hat{V}\) in GA-SKSM with number of the iterations increasing. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Material**} & **Linear** & **Elastic** & **Mass** & **Poisson’s** & **The Allowable** & **The Allowable** \\  & **Coefficient** & **Modulus** & **Density** & **Ratio** & **Compressive** & **Tensile Stress** \\  & **(10\({}^{-6}\)/\({}^{\circ}\)C)** & **(MPa)** & **(Kg/m\({}^{3}\))** & **Stress (MPa)** & **(MPa)** \\ \hline Dam body & 7 & 10 & 2300 & 0.25 & \(-\)5.0 & 1.5 \\ Foundation rock & 0 & 8 & 2600 & 0.21 & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Properties of dam body and foundation rock. Figure 3: **(a)** Finite element model of the foundation rock system of the Jiancaogou single-curvature arch dam. **(b)** Finite element model of the dam body. As the number of iterations increased, in other words, as the samples in the initial sample set of KSM increased, the relative errors between \(\sigma_{1}\) and \(\hat{\sigma}_{1}\), \(\sigma_{3}\) and \(\hat{\sigma}_{3}\), \(\sigma_{1\text{e}py}\) and \(\hat{\sigma}_{1\text{e}py}\), \(\sigma_{3\text{e}py}\) and \(\hat{\sigma}_{3\text{e}py}\), and \(V\) and \(\hat{V}\) generally decreased, but this reduction was volatile (Figure 5). This volatility can be ascribed to the fact that new optimal solutions were discovered continuously in the process of infilling new samples into the sample set. When the optimal solution tended to be stable, the relative errors between \(\sigma_{1}\) and \(\hat{\sigma}_{1}\), \(\sigma_{3}\) and \(\hat{\sigma}_{3}\), \(\sigma_{1\text{e}py}\) and \(\hat{\sigma}_{1\text{e}py}\), \(\sigma_{3\text{e}py}\) and \(\hat{V}\) decreased gradually until the convergence criteria were satisfied. This indicates that upon application of the MP criterion and the LCB criterion, the algorithm collected sample points from the region of interest only, and finally, the SKSM achieved high approximation accuracy in the region of interest. Table 3 presents a summary of the optimal designs of the single-curvature arch dam obtained using the GA-FEM, GA-KSM, and GA-SKSM. As shown in Table 3, for optimizing the single-curvature arch dam example considered in this study, the GA-FEM and GA-KSM required 1150 and 500 numerical simulations, respectively.