## 3 Kriging-Based Adaptive Surrogate Modeling for Limit Capacity of the Tower
### Kriging Method  
The kriging method assumes that the target function relating the input (vector \(\mathbf{x}\)) to the output (vector \(\mathbf{y}\)) is a sample path of a Gaussian process whose mean and auto-covariance functions are determined by observation points \(\mathbf{X}=[\mathbf{x}^{1}\ \dots\ \mathbf{x}^{p}]^{\text{T}}\) and the corresponding observation results \(\mathbf{Y}=[\mathbf{y}^{1}\ \dots\ \mathbf{y}^{p}]^{\text{T}}\) (\(p\) is the number of observations) [24]. The observation result \(\mathbf{Y}\) is usually solved by high-fidelity simulations or experiments, and can be viewed as the true solutions for the input observation point set \(\mathbf{X}\). The observation set \(\mathbf{Y}(\mathbf{X})\) is named as the design of experiment (DoE) and the set \([\mathbf{X}\ \mathbf{Y}]\) is also called the experiment set. The kriging method constructs a Gaussian process based on the DoE, and then predicts the output \(y\) corresponding to an arbitrary input \(x\). The Gaussian process for the approximation of a target function can be written as:  
\[\mathbf{y}(\mathbf{x})=\mathbf{f}(\mathbf{x})^{\text{T}}\beta+z(\mathbf{x}) \tag{13}\]  
where \(\mathbf{f}(\mathbf{x})\) is a functional set used for regression, \(\beta\) is the regression coefficient vector, and \(z(\mathbf{x})\) is a zero-mean stationary Gaussian process with the variance \(\sigma_{\mathbf{y}}^{2}\) and correlation function \(R\). A kriging model involves three steps [25; 26]: (i) generating the observation points \(\mathbf{X}=[\mathbf{x}^{1}\ \dots\ \mathbf{x}^{p}]^{\text{T}}\) and results \(\mathbf{Y}=[\mathbf{y}^{1}\ \dots\ \mathbf{y}^{p}]^{\text{T}}\); (ii) selecting the kriging functions, i.e., the regression functional set \(\mathbf{f}(\mathbf{x})\) and the correlation function \(R\); (iii) producing the kriging prediction model. Here, the Latin-hypercube sampling are used to generate the initial observation points \(\mathbf{X}\), and the observation results \(\mathbf{Y}\) are solved based on finite element simulations. The regression functional set \(\mathbf{f}(\mathbf{x})\) and correlation function \(R\) are selected as:  
\[\mathbf{f}(\mathbf{x})=\left[ \begin{array}{cccc}1&x_{1}&x_{2}&\dots&x_{n} \end{array} \right] \tag{14}\]  
\[R(\mathbf{x}^{j},\mathbf{x}^{k})=\prod_{i}^{n}\exp\left(-\theta_{i}\Big{|}x_{ i}^{j}-x_{i}^{k}\Big{|}^{2}\right) \tag{15}\]  
where \(x_{i}\) (\(i=1,\,2,\,\dots,\,n\)) is the element of the input vector \(\mathbf{x}\), \(n\) is the number of input variables that contained in the vector \(\mathbf{x}\), \(j\), \(k=1,\,\dots,\,p\), and \(\theta_{i}\) is the correlation function parameter which is optimized using the principle of maximum likelihood estimation (MLE) [24]. Then, the mean and error variance of the kriging prediction in terms of a new input vector \(\mathbf{x}\) are, respectively, expressed as [23]:  
\[\overset{\mathbf{x}}{\mathbf{y}}(\mathbf{x})=\mathbf{f}(\mathbf{x})^{\text{T} }\beta+\mathbf{r}(\mathbf{x})^{\text{T}}R^{-1}(\mathbf{Y}-\overset{\mathbf{ \Lambda}}{\mathbf{F}\beta}) \tag{16}\]  
\[\sigma_{\mathbf{y}}^{2}=\sigma_{\mathbf{y}}^{2}\left[1+\mathbf{u}(\mathbf{x}) ^{\text{T}}\Big{(}\mathbf{F}^{\text{T}}\mathbf{R}^{-1}\mathbf{F}\Big{)}^{-1} \mathbf{u}(\mathbf{x})-\mathbf{r}(\mathbf{x})^{\text{T}}\mathbf{R}^{-1} \mathbf{r}(\mathbf{x})\right] \tag{17}\]where \(\overset{\overset{\alpha}{\beta}}=\left(\mathbf{F}^{\mathsf{T}}\mathbf{R}^{-1} \mathbf{F}\right)^{-1}\mathbf{F}^{\mathsf{T}}\mathbf{R}^{-1}\mathbf{Y}\) is the predicted value for \(\beta\), which is determined by the observation of \(\mathbf{Y}(\mathbf{X})\). \(\mathbf{F}=\left[\mathbf{f}\left(\mathbf{x}^{1}\right)\dots\mathbf{f}\left( \mathbf{x}^{p}\right)\right]^{\mathsf{T}}\), \(\mathbf{R}=\left\{R\left(\mathbf{x}^{j},\ \mathbf{x}^{k}\right)\right\}\). \(\mathbf{r}(\mathbf{x})=\left[R\left(\mathbf{x},\ \mathbf{x}^{j}\right)\dots\ R\left( \mathbf{x},\ \mathbf{x}^{p}\right)\right]^{\mathsf{T}}\) indicates the correlation between the observation input points \(\mathbf{x}\) and the new input \(\mathbf{x}\).