# Shape Optimization of Single-Curvature Arch Dam Based on Sequential Kriging-Genetic Algorithm
## Methodology
### Solution of Mathematical Optimization Model
#### 2.3.2 Kriging Surrogate Model  
A KSM essentially uses the response information of known points to predict the response information of unknown points. The general steps involved in constructing a KSM are as follows. The first step is sampling. Sampling methods (such as central composite design, orthogonal experimental design, uniform design, and Latin hypercube sampling) can be used to generate samples of the design variables. The second step is KSM construction. A set of input-output data can be obtained by analyzing samples using a high-precision analysis model (e.g., ANSYS). Subsequently, a KSM can be constructed by fitting the input-output relationship of these sample data. The third step is accuracy evaluation. **Sampling methods. ** Common sampling methods include central composite design, orthogonal experimental design, uniform design, Latin hypercube sampling (LHS), and inherited LHS (ILHS). LHS [23] is easy to implement and has good consistency; therefore, it is widely used. ILHS [24] can improve the sampling efficiency and ensure the uniformity of LHS. In this study, ILHS was used in the GA-KSM for sample collection. **Constructing KSM. ** The general mathematical expression of a KSM is as follows:  
\[g(X)=\Phi(\varepsilon,X)+m(X)=f^{T}(X)\varepsilon+m(X) \tag{12}\]  
where \(X\) is a vector of design variables; \(\Phi\) (\(\varepsilon\), \(X\)) is the polynomial regression model, generally a zero-, first-, or second-order polynomial; \(\varepsilon\) is the regression coefficient; \(f(X)\) is the basic function; and \(m(X)\) is a stochastic process and has the following statistical properties:  
\[E[m(X)]=0 \tag{13}\]  
\[Var[m(X)]=\sigma^{2} \tag{14}\]  
\[Cov[m(X_{i}),m(X_{j})]=\sigma^{2}Q(\theta,X_{i},X_{j}) \tag{15}\]where \(\sigma^{2}\) is the constant process variance, \(Q(\theta,\,X_{i},\,X_{j})\) is a correlation function of \(X_{i},\,X_{j}\), and \(\theta\). The Gaussian function presented in the following formula is commonly used as a correlation function:  
\[Q(\theta,X_{i},X_{j})=\prod_{k=1}^{n_{0}}e^{-\theta_{k}|X_{i}^{k}-X_{j}^{k}|^{ 2}} \tag{16}\]  
where \(X_{i}^{k}\) and \(X_{j}^{k}\) are the \(k\)th components of the training samples \(X_{i}\) and \(X_{j}\), respectively. \(\theta_{k}\) represents the \(k\)th element of the correlation vector parameter \(\theta\). The response values of each sample can then be subjected to linear weighted superposition interpolation to calculate the response values of unknown samples:  
\[\hat{g}(X)=w(X)^{T}G \tag{17}\]  
where \(w(X)=(w_{1},\,w_{2},\,\ldots\,\,,\,w_{n})^{T}\) is the weighting factor to be determined, and \(G=(g_{1},\,g_{2},\,\ldots\,\,,\,g_{n})^{T}\). Based on the unbiased estimation condition, the following KSM expression can be obtained through collation:  
\[\hat{g}(X)=f^{T}(X)\varepsilon^{*}+q(X)^{T}\gamma^{*} \tag{18}\]  
where \(\varepsilon^{*}=(F^{T}Q^{-1}F)^{-1}F^{T}Q^{-1}G\), \(\gamma^{*}=Q^{-1}(G-F\varepsilon^{*})\), \(\varepsilon^{*}\) and \(\gamma^{*}\) are fixed values and \(q=(Q(\theta,\,X,\,X_{1}),\,\ldots\,\,,\,Q(\theta,\,X,\,X_{n}))^{T}\). \(F\) is the regression matrix. **Accuracy evaluation. ** Accuracy evaluation is essential for determining whether a KSM is credible. If the accuracy meets the required threshold, the KSM can be used instead of FEM in the optimization process; otherwise, the KSM should be reconstructed through the addition of sample points. In this study, the maximum absolute relative error (_MARE_) was used to evaluate the accuracy of the KSM. _MARE_ can be defined as follows:  
\[\textit{MARE}=\max\biggl{\{}100\times\biggl{|}\frac{g_{t}-\hat{g}_{t}}{g_{t}} \biggr{|}\biggr{\}}t=1,\,2,\cdots\,n \tag{19}\]  
where \(g\) and \(\hat{g}\) denote the actual value and predicted value, respectively, and \(n\) denotes the number of evaluation samples. When a GA-KSM is used to solve optimization models, numerous samples are required to establish a KSM while satisfying the accuracy requirements over the entire design space.