# An Adaptive Kriging-Based Fourth-Moment Reliability Analysis Method for Engineering Structures
## 2 Materials and Methods
### Adaptive Kriging Fourth-Moment Method  
Suppose the sample input parameter set of a mechanical structure is \(x=(x_{1},x_{2},\cdots,x_{n})^{T}\), and the corresponding response set is \(g=g(x)\). The Kriging model [39] between the two can be expressed as follows:  
\[g(x)=f^{T}(x)\beta+z(x) \tag{16}\]  
where \(f^{T}(x)\) represents the polynomial function of \(x\), which is the regression model used to establish the overall approximate relationship between the input and output. \(\beta\) denotes the regression coefficient of \(x\), while \(z(x)\) is a random factor utilized to approximate the deviation of local simulation. The statistical characteristics of \(z(x)\) can be expressed as  
\[\left\{\begin{aligned} E[z(x)]&=0\\ Var[z(x)]&=\sigma_{z}{}^{2}\\ Cov[z(x_{i}),z(x_{j})]&=\sigma_{z}{}^{2}\big{[}R_{ij} \big{(}x_{i},x_{j}\big{)}\big{]}\end{aligned}\right. \tag{17}\]  
In (17), \(\sigma_{z}{}^{2}\) represents the variance. \(R_{ij}\big{(}x_{i},x_{j}\big{)}\) denotes the spatial correlation function between the sample input parameter points \(x_{i}\) and \(x_{j}\). The widely used expression for the Gaussian spatial correlation function is adopted as  
\[R_{ij}\big{(}x_{i},x_{j}\big{)}=\prod_{r=1}^{s}\exp\Big{(}-\lambda_{r}\big{|} x_{i}{}^{r}-x_{j}{}^{r}\big{|}^{2}\Big{)} \tag{18}\]  
where \(\lambda_{r}(r=1,2,\cdots,s)\) is the correlation coefficient, \(r\) is the variable dimension ordinal number, and \(s\) denotes the total variable dimension.  
The variance \(\sigma_{z}{}^{2}\) can be expressed as  
\[\sigma_{z}{}^{2}=\frac{1}{N}(y-J\beta)^{T}R^{-1}(y-J\beta) \tag{19}\]  
where \(J\) is the sample point regression matrix, \(N\) is the total number of sample points, and \(R\) is the matrix consisting of \(R_{ij}\big{(}x_{i},x_{j}\big{)}\).  
The regression coefficient \(\beta\) in (16) can be estimated as  
\[\hat{\beta}=\left(J^{T}R^{-1}J\right)J^{T}R^{-1}y \tag{20}\]  
Therefore, the output Kriging estimate at point \(x_{u}\) can be expressed as  
\[g_{K}(x_{u})=f^{T}(x_{u})\hat{\beta}+r^{T}(x_{u})R^{-1}\big{(}g-J\hat{\beta} \big{)} \tag{21}\]  
where \(r\) is a vector of the correlation function between the unknown input parameter point, \(x_{u}\) and the set of the training sample set.  
The correlation coefficient \(\lambda_{r}=[\lambda_{1},\lambda_{2},\cdots,\lambda_{s}]^{T}\) can be estimated by maximum likelihood estimation (MLE); the maximum likelihood function \(\text{max}F(\lambda)\) is denoted as  
\[\text{max}F(\lambda)=\frac{\ln(\sigma_{z}{}^{2})+\ln|R|}{2},\quad\lambda_{r} \geq 0(r=1,2,\cdots,s) \tag{22}\]  
Substituting the obtained \(\lambda_{r}\) into Equation (21) yields a surrogate model with an optimized fitting accuracy. To establish an accurate enough Kriging model, it is necessary to constantly update the model. In this paper, an adaptive Kriging model is established using a U function [34], which is expressed as  
\[\text{U}(x)=\left|\frac{\mu_{g_{K}}(x)}{\sigma_{g_{K}}(x)}\right| \tag{23}\]  
The convergence condition of the updating process is typically chosen as min U(x) \(\geq\) 2, when the maximum probability of the model identification error is approximately 2.28%. The specific flow of the adaptive Kriging fourth-moment method (AK-FM) proposed in this paper is illustrated in Figure 1.  
Figure 1: Process of AK-FM method.