# Numerical Modeling of 3D Slopes with Weak Zones by Random Field and Finite Elements
## 2 Methodology
### Simulation and Model Verification of Weak Zones in Slopes  
Taking into account the high degree of randomness the parameters such as shape, inclination angle, and the position of the weak zones, in order to perform verification, we used the Monte-Carlo simulation method to statistically analyze the results, generating a large number of random field samples through the modified linear estimation method. The airport was mapped to a finite element grid in order to construct a 3D model of a slope with weak zones. The stability of each slope in 100 examples of Monte-Carlo simulation was calculated. The safety factor was evaluated and compared with the content of the previous research [33] ([PERSON] et al., 2018) for purposes of verification. As shown in Figure 7, the specific steps of combining USDFLD subroutine random field with the software Abaqus were as follows:  
1. Use Abaqus to build a 3D finite element slope model, including defining the material parameters of the rocks and weak zones, and setting the grid and boundary conditions of the model in the analysis step. Then output an inp file (a file executed by the Abaqus software). 2. Associate the yield strength in the material parameters of the slope with the user-defined field variable User Defined Field, and set the state variable Depvar at the same time. Figure 5: Correlation distance of different related structures. (**a**) Schematic diagram of the correlation length of \(z=0\); (**b**) correlation length of rotational anisotropy after rotation of the \(\alpha\) angle about the \(Z\)-axis. Figure 6: Schematic diagram of the distribution of weak zones with different inclination angles. (**a**) Rotation angle of 30\({}^{\circ}\) around the \(X\)-axis; (**b**) rotation angle of 0. 3. Call the random field generated by the USDFLD subroutine to replace the yield strength in the original slope model. 4. Carry out the stability calculation of the slope model with weak zones constructed in (3), and obtain the safety factor of the slope model. 5. According to the above steps, perform 100 Monte-Carlo simulations on the established model, and perform statistical analysis on the calculation results. To verify the rationality of the two-phase random media characterization method, a typical 3D slope was established as shown in Figure 8. The model contains 213,240 C3D8 elements, and two-phase random media was used to characterize the weak zone and rock mass (using the ideal elastoplastic model); the two-dimensional slope model corresponding to the 3D model was in agreement with the content found in the literature, and the boundary conditions of the 3D model were consistent with the two-dimensional slope model found in the literature [33]. The material parameters used in this section are shown in Table 2.  
Figure 7: Flow chart of the stability analysis of slopes with weak zones. On the basis of the parameters shown in Table 2, a Monte-Carlo simulation (100 times) of the model in Figure 8 was performed, and the results obtained for the average safety factor are shown in Figure 9. When the volume fraction of the weak zones was 50%, the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Unit** & **Value** \\ \hline (a) Deterministic parameter & & \\ Yield strength of weak zones & kPa & 10 \\ Yield strength of rocks & kPa & 50 \\ Young’s moduli of weak zones & kPa & \(10^{5}\) \\ Young’s moduli of rocks & kPa & \(5\times 10^{5}\) \\ Poisson’s ratio & – & 0.3 \\ Correlation length along \(x\) direction \(\theta_{x}\) & m & 5 \\ Correlation length along \(y\) direction \(\theta_{y}\) & m & 1 \\ Correlation length along \(z\) direction \(\theta_{z}\) & m & 10,000 \\ (b) Variable parameter & & \\ Volume fraction of weak zones & – & [0\(\sim\)1] \\ \hline \hline \end{tabular} Note. – means that this parameter has no unit.  
\end{table} 
Table 2: Material mechanical parameters of the validation slope model.