## 3 Calibration and Application
### Case 1: A Homogeneous Slope
#### 3.1.2 Result of Homogeneous Slope  
The slope is calculated by running the UBM parallel calculation program on a small work station (processor: AMD Ryzen 3970X, 32 cores, physical memory: 128 GB). Table 2 lists the results obtained by the FEM, LEM and UBM. Figures 3 and 4 give the relationship between top load, bulk density and the stability slope coefficient when \(t_{m}=1\). Figure 5 gives the histogram of the external overload coefficient \(k_{F}(t_{m})\), bulk overload coefficient \(k_{\gamma}(t_{m})\) and slope stability coefficient \(k_{m}(t_{m})\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Shear Strength** & **Mean** & **Variation** & **Correlation** & **Distribution Type** & **Fluctuation Range** \\
**Parameters** & & **Coefficient** & **Coefficient** & **Distribution Type** & **Fluctuation Range** \\ \hline \(c/\mathrm{kPa}\) & 10 & 0.3 & \(\rho_{c,\varphi}=-0.5\) & log-normal & \(L_{h}=40\) m \\ \(\varphi/(^{\circ})\) & 30 & 0.2 & log-normal & \(L_{\circ}=4\) m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Calculation parameters of the slope. Figure 2: Homogeneous slope calculation model. The calculation results show that:  
1. The slope reliability results of different methods are compared in Table 2. It can be seen that the mean values of the external overload coefficient, bulk density over \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Method** & **Mean** & \begin{tabular}{c} **Standard** \\ **Deviation** \\ \end{tabular} & \
\begin{tabular}{c} **Failure** \\ **Probability (\%)** \\ \end{tabular} &
\begin{tabular}{c} **Computing** \\ **Time (h)** \\ \end{tabular} \\ \hline \multirow{2}{*}{\(k_{F}(t_{m})\)} & UBM & 82.361 & 39.120 & 0.6 & 0.52 \\  & FEM & 75.843 & 36.765 & 1.1 & 28.4 \\ \hline \multirow{2}{*}{\(k_{\gamma}(t_{m})\)} & UBM & 2.447 & 2.260 & 0.6 & 0.53 \\  & FEM & 2.158 & 1.871 & 1.1 & 29.1 \\ \hline \multirow{2}{*}{\(k_{m}(t_{m})\)} & UBM & 1.242 & 0.103 & 0.6 & 4.16 \\  & LEM & 1.210 & 0.106 & 1.7 & / \\ \cline{1-1}  & FEM & 1.229 & 0.102 & 0.9 & 127.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Homogeneous slope reliability results of different methods. Figure 4: Relationship between bulk density and slope stability coefficient of homogeneous slope. Figure 5: Histogram of \(k_{F}(t_{m})\), \(k_{\gamma}(t_{m})\) and \(k_{m}(t_{m})\) of homogeneous slope. Figure 3: Relationship between external load and slope stability coefficient of homogeneous slope. load coefficient and slope stability coefficient obtained by the UBM are all greater than the calculated results obtained by the LEM and FEM, but the errors are small. From the upper limit theorem, it is clear that the external load corresponding to the kinematically-admissible velocity field obtained by the UBM is the strict upper bound of the ultimate load; that is, the \(k_{F}(t_{m})\), \(k_{\gamma}(t_{m})\) and \(k_{m}(t_{m})\) obtained by the UBM must be greater than the actual solution. Therefore, when the UBM is used to calculate the slope failure probability, it will be slightly underestimated. However, compared with the traditional LEM, the UBM can fully determine the random field of two-dimensional soil slope shear strength parameters. It is unnecessary to assume the critical sliding surface in advance and it is also unnecessary to convert the super-static problem into the static problem by assuming the magnitude and direction of the inter-slice force. Compared with FEM, the UBM can ignore the complex constitutive relationship of rock and soil mass. The slope complex problem is solved according to the plastic flow constraint of triangular elements, the plastic flow constraint of discontinuities and the velocity boundary condition of triangular elements which must be met simultaneously when the slope is in the limit state and therefore the calculation efficiency is more efficient. 2. Based on the \(k_{F}(t_{m})\), \(k_{\gamma}(t_{m})\) and \(k_{m}(t_{m})\), the slope failure probability can be calculated, which is close or equal.