# Non-Probabilistic Reliability Analysis of Slopes Based on Fuzzy Set Theory
## 3 Non-Probabilistic Reliability Analysis Method Based on the Ellipsoidal Model
### Construction of the Ellipsoidal Model  
To construct the ellipsoidal model, it is necessary to know the center point and the radius of the ellipsoidal model, and the covariance matrix of the ellipsoidal variables. For the n-dimensional variables \(X_{1}\), \(X_{2}\),..., \(X_{n}\), when the value interval of each variable is known, the coordinates of the centroid \(X_{ic}\) of the value interval and the deviation \(X_{ir}\) in the \(X_{i}(i=1,\;2,\;\cdots,\;n)\)-direction can be obtained from Equation (5):  
\[\left. \begin{array}{l}X_{ic}=\dfrac{\left(\overline{X}_{i}+\underline{X}_{i} \right)}{2}\\ X_{ir}=\dfrac{\left(\overline{X}_{i}-\underline{X}_{i}\right)}{2}\end{array} \right\} \tag{5}\]  
where \(\overline{X}_{i}\) and \(\underline{X}_{i}\) represent the upper and lower bounds of the interval number, respectively. The centroid of the ellipsoidal model \(X_{io}\) coincides with the centroid of the value interval, and the ellipsoidal radius \(e_{i}\) is greater than the absolute value of the difference between the variable and the centroid, i.e.,  
\[\left. \begin{array}{l}X_{io}=X_{ic}=\left(\overline{X}_{i}+\underline{X}_{i }\right)\\ e_{i}>\left|X_{i}-X_{ic}\right|^{2}\end{array} \right\} \tag{6}\]  
The value intervals of all the variables in space represent a hyper-cube, and the ellipsoidal model is constructed by solving for the minimum external hyper-ellipsoid of this hyper-cube. The ellipsoidal covariance matrix is obtained by solving it using the Lagrange multiplier method [21]:  
\[\mathbf{C}_{X}=diag\Big{(}nX^{2}{}_{1r},\;nX^{2}{}_{2r},\;\cdots,\;nX^{2}{}_{ nr}\Big{)} \tag{7}\]  
Let \(\boldsymbol{X}=\left(X_{1},\;X_{2},\;\cdots,\;X_{n}\right)^{T}\). The ellipsoidal model is constructed as  
\[E_{\mathbf{X}_{o},\boldsymbol{\Omega}}=\Big{\{}\boldsymbol{X}|\left( \boldsymbol{X}-\boldsymbol{X}_{o}\right)^{T}\boldsymbol{\Omega}(\boldsymbol{X} -\boldsymbol{X}_{o})\leq 1\Big{\}} \tag{8}\]  
where \(\boldsymbol{\Omega}\) is the inverse matrix of \(\mathbf{C}_{X}\). Introduce an \(n\times 1\) dimensional vector \(\boldsymbol{u}=\left(u_{1},\;u_{2},\;\cdots,\;u_{n}\right)^{T}\) and define \(\boldsymbol{u}\) as  
\[\boldsymbol{u}=\boldsymbol{\Omega}^{0.5}(\boldsymbol{X}-\boldsymbol{X}_{o}) \tag{9}\]  
Substituting Equation (9) into Equation (8), the original ellipsoidal model is transformed to  
\[E=\Big{\{}\boldsymbol{u}|\boldsymbol{u}^{T}\boldsymbol{u  
system (as shown in Figure 2). The non-probabilistic reliability index \(\eta\) is defined as follows [22]:  
\[ \begin{array}{l}\eta=sgn(G(0))\cdot\min\lVert\boldsymbol{u}\rVert\\ \text{s.t.}\;G(u_{1},u_{2},\cdots,u_{n})=0\end{array} \right\} \tag{13}\]  
The value of the non-probabilistic reliability index \(\eta\) can be calculated using the mathematical programming method, the direct iterative method and other methods [23]. The HL-RF iterative algorithm [24,25] has the advantages of a clear concept, simple iteration format, and rapid convergence. The main idea of this method is to select an initial iteration point, and then determine the next iteration point according to the iteration format until the result satisfies the convergence condition (more information about the HL-RF algorithm can be found in the related studies by [PERSON] et al. [26] and [PERSON] [27]). In this paper, this method will be used to calculate the non-probability reliability index of the slope.