## 2 Methodology
### Multi-Dimensional Joint Distribution Model
#### 2.1.3 The Construction Method of a Multi-Dimensional Joint Distribution Model for Parameters
The joint probability density function and distribution function of a multi-dimensional Gaussian Copula are as follows:  
\[D(F_{1}(x_{1}),F_{2}(x_{2}),\ldots,F_{n}(x_{n});\theta)=|\theta|^{-1/2}\exp[- \frac{1}{2}\zeta^{\prime}(\theta^{-1}-\mathbf{I})\zeta] \tag{4}\]  
\[C(F_{1}(x_{1}),F_{2}(x_{2}),\ldots,F_{n}(x_{n});\theta)=\Phi_{n}(\Phi^{-1}(F_ {1}(x_{1})),\Phi^{-1}(F_{2}(x_{2})),\ldots,\[PERSON]^{-1}(F_{n}(x_{n}));\theta) \tag{5}\]  
where \(\theta\) is the Copula correlation parameter matrix of \(n\)-dimensional random vectors; \(|\theta|\) is the determinant of the correlation parameter; \(\zeta^{\prime}=(\[PERSON]^{-1}(u_{1}),\[PERSON]^{-1}(u_{2}),\ldots,\Phi^{-1}(u_{n}))\) is the \(n\)-dimensional standard normal distribution variable; \(\Phi_{n}(\bullet,\ldots,\bullet;\theta)\) is the \(n\)-dimensional standard normal distribution function of the correlation parameter matrix \(\theta\); \(\Phi^{-1}(\bullet)\) is the inverse function of the one-dimensional standard normal distribution; and I is the unit matrix.  
The joint probability density function and distribution function of the multi-dimensional \(t\) Copula are as follows:  
\[D(F_{1}(x_{1}),F_{2}(x_{2}),\ldots,F_{n}(x_{n});\theta)=|\theta|^{-1/2}\frac{ \Gamma(\frac{\
u+n}{2})(\Gamma(\frac{\
u}{2}))^{n-1}}{\big{(}\Gamma(\frac{\
u +1}{2})\big{)}^{n}}\frac{(1+\frac{1}{\
u}\zeta^{\prime}\theta^{-1}\zeta)\frac{ \
u+n}{2}}{\prod\limits_{i=1}^{n}{(1+\frac{\zeta_{i}^{2}}{\
u})^{-\frac{\
u+1 }{2}}}} \tag{6}\]  
\[C(F_{1}(x_{1}),F_{2}(x_{2}),\ldots,F_{n}(x_{n});\theta)=T_{n}(T_{v}^{-1}(F_{1} (x_{1})),T_{v}^{-1}(F_{2}(x_{2})),\ldots,T_{v}^{-1}(F_{n}(x_{n}));\theta,\
u) \tag{7}\]  
where \(\zeta^{\prime}=(T^{-1}(F_{1}(x_{1})),T^{-1}(F_{2}(x_{2})),\ldots,T^{-1}(F_{n} (x_{n}))\) is the \(t\) distribution variable with \(v\) degrees of freedom; \(T_{n}(\bullet,\ldots,\bullet;\theta,\
u)\) is the \(n\)-dimensional standard distribution function with correlation parameter matrix \(\theta\) and \(v\) degrees of freedom; and \(T_{v}^{-1}(\bullet)\) is the inverse function of the one-dimensional \(t\) distribution of \(u_{i}\) with \(v\) degrees of freedom. The other symbols have the same meaning as before. The elements \(\theta\) of the correlation parameter matrix \(\theta\) in the Copula function can be obtained by transforming Pearson correlation coefficients \(\rho\) as follows:  
\[\rho=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{ \infty}\frac{x_{1}-\mu_{1}}{\sigma_{1}}\frac{x_{2}-\mu_{2}}{\sigma_{2}}\ldots \frac{x_{n}-\mu_{n}}{\sigma_{n}}D(F_{1}(x_{1}),F_{2}(x_{2})\ldots F_{n}(x_{n}) ;\theta)f_{1}(x_{1})f_{2}(x_{2})\ldots f_{n}(xthe deformation inclination at various positions exceeds the empirical allowable value [33], that is,  
\[f(\delta)=\delta_{0}-\delta \tag{10}\]  
where \(\delta_{0}\) is the allowable value of deformation inclination, and \(\delta\) is the calculation value of deformation inclination. Usually, the surrogate model can be used to reduce the difficulty of solving the above performance function (Equation (10)). Here, a quadratic response surface method without considering the cross term is introduced as a surrogate model, which can be computed as  
\[\sigma_{t}=a_{0}+\sum_{i=1}^{n}b_{i}X_{i}+\sum_{i=1}^{n}c_{i}X_{i}^{2} \tag{11}\]  
where \(\sigma_{t}\) are stress calculation values of characteristic points, \(X_{i}\) is the random variable and \(a_{0}\), \(b_{i}\), and \(c_{i}\) are parameters of the response surface equation to be solved. Based on Equation (11), Combined with the multi-dimensional joint distribution model, the Monte Carlo method [34] is adopted and sample values of \(X=\{x_{1},x_{2},\ldots,x_{n}\}^{T}\) are selected as the random samples. Additionally, an extra function with values of 0 and 1 is constructed:  
\[I[G(x)]= \begin{cases}1,&G(x)<0,\ Cracking\\ 0,&G(x)\geq 0,\ \ \ \ \ Safety\end{cases}  \tag{12}\]  
where \(I[\cdot]\) is the function symbol; and \(G(x)\) is the value of the performance function \(G(X)\).