## 3 Surrogate Modeling  
### The Kriging Surrogate Modeling Method  
Shape optimization is necessary and important in engineering, such as vehicle shapes, airfoil shapes, and building shapes. Surrogate-based optimization has been shown to be a practical approach in aeronautics and astronautics engineering [47; 48], civil engineering [49; 50], and ocean engineering [51; 52]. The surrogate model can replace many expensive objective evaluations with an approximation model. [PERSON] and [PERSON] [53] provided an overview of surrogate models, focusing on solving optimization problems such as global optimization, multi-objective optimization, and probabilistic design optimization. [PERSON] and [PERSON] [54] discussed details of surrogate modeling methodology, focusing on sampling, surrogate model building, and validation. [PERSON] and [PERSON] [55] explained the use of surrogate modeling for building design regarding applications in the conceptual design stage, sensitivity and uncertainty analysis, and building design optimization. Many surrogate models have been adopted in the field of optimization, such as polynomial regression, radial basis functions, neural networks, and support vector regression [56]. In the present work, the widely used Kriging model was adopted [49]. For a stationary random process, the statistical prediction of the unknown function \(y\) at an untried \(x\) is given by a linear combination of the function values at the original samples and their gradients at observed points, as follows:  
\[\hat{y}(x)=\sum_{i=1}^{n}w^{(i)}y^{(i)} \tag{5}\]  
where \(\hat{y}(x)\) is the predicted value at an untried site \(x\), \(w^{(i)}\) is the weight coefficient for the function value \(y^{(i)}\). To determine the weight coefficient, the output of a deterministic experiment is treated as a realization of a stochastic process, i.e.,  
\[Y(\mathbf{x})=\beta_{0}+Z(\mathbf{x}) \tag{6}\]  
where \(\beta_{0}\) is an unknown constant mean value and the stationary random processes \(Z(x)\) having zero mean and a covariance Cov,  
\[\text{Cov}\big{[}Z(\mathbf{x}),Z\big{(}\mathbf{x}^{\prime}\big{)}\big{]}=\sigma^{2}R \big{(}\mathbf{x},\mathbf{x}^{\prime}\big{)} \tag{7}\]  
where \(\sigma^{2}\) is the process variance of \(Z(x)\), \(R\) is the spatial correlation function, which depends only on the spatial distance between two sites. \(\hat{y}(x)\) is treated as random, and we try to minimize its mean squared error (MSE)  
\[\text{MSE}[\hat{y}(\mathbf{x})]=E\bigg{[}\Big{(}\mathbf{w}^{\text{T}}\mathbf{Y_{S}}-Y (\mathbf{x})\Big{)}^{2}\bigg{]} \tag{8}\]  
subject to the unbiasedness constraint  
\[E\bigg{[}\sum_{i=1}^{n}w^{(i)}Y\Big{(}\mathbf{x}^{(i)}\Big{)}\bigg{]}=E[Y(\mathbf{x})] \tag{9}\]where \(\mathbf{Y}\)s is the response of the sample points. Solving this constrained minimization problem, \(w^{(i)}\) can be found by solving the following linear equations in matrix form:  
\[\left[ \begin{array}{cc}\mathbf{R}&\mathbf{F}\\ \mathbf{F}^{\mathbf{\Gamma}}&0\end{array} \right]\left[ \begin{array}{c}\mathbf{w}\\ \widetilde{\mu}\end{array} \right]=\left[ \begin{array}{c}\mathbf{r}\\ 1\end{array} \right] \tag{10}\]  
where  
\[\mathbf{F}=\left[ \begin{array}{cccc}1&1&\cdots&1\end{array} \right]^{\text{T}} \in\mathbb{R}^{n} \tag{11}\]  
\[\widetilde{\mu}=\mu/\left(2\sigma^{2}\right) \tag{12}\]  
\[\mathbf{R}=\left[ \begin{array}{ccc}R\Big{(}\mathbf{x}^{(1)},\mathbf{x}^{(1)}\Big{)}& \cdots&R\Big{(}\mathbf{x}^{(1)},\mathbf{x}^{(n)}\Big{)}\\ \vdots&&\vdots\\ R\Big{(}\mathbf{x}^{(n)},\mathbf{x}^{(1)}\Big{)}&\cdots&R\Big{(}\mathbf{x}^{(n)},\mathbf{x}^{( n)}\Big{)}\end{array} \right]\in\mathbb{R}^{n\times n} \tag{13}\]  
\[\mathbf{r}=\left[ \begin{array}{c}R\Big{(}\mathbf{x}^{(1)},\mathbf{x}\Big{)}\\ \vdots\\ R\Big{(}\mathbf{x}^{(n)},\mathbf{x}\Big{)}\end{array} \right]\in\mathbb{R}^{n} \tag{14}\]  
where \(\mathbf{R}\) is the correlation matrix representing the correlation between the observed points, and \(\mathbf{r}\) is the correlation vector representing the correlation between the untried point and the observed points.