## 4 Numerical Simulation and Engineering Verification  
### Numerical Model and Initial Conditions  
The geological conditions of the mine were simplified, and a 1000 m \(\times\) 1200 m \(\times\) 1200 m (\(x\times y\times z\)) numerical model was constructed using COMSOL Multiphysics 6.2 software, as shown in Figure 5a. The thickness of the roof strata in the upper part of the model is 720 m, the thickness of the ore body in the middle part of the model is 25 m, and the thickness of the floor rocks in the bottom part of the model is 255 m. Figure 5b,c illustrate the location of the ore body in the model and the layout of the slopes within the mine, respectively. The model was meshed using free tetrahedral units. Fine mesh was used in the mining area to ensure the accuracy of the model while coarse mesh was used in areas far away from the slopes to reduce the computational cost, resulting in a total of 640,207 elements. The minimum cell mass is 0.278, the average cell mass is 0.721, and the cell volume ratio is 2.43 \(\times\) 10\({}^{-4}\), giving the mesh a high overall mass. In this case, the Drucker-Prager criterion was used. This criterion is a modified version of the Mohr-Coulomb criterion, retaining the ability to use the Mohr-Coulomb parameters while avoiding the problem of sharp corners that tend to occur in the principal stress space when the Mohr-Coulomb criterion is used. The parameters used in the simulations are listed in Table 1. As the model is large and the excavated area is located in the center of the model, far from the boundary, it can be assumed that there is no displacement at the bottom and only vertical displacement around the surrounding area. The following boundary conditions were used: fixed constraints on the bottom surface of the model, fixed displacement constraints in the normal directions of the four side surfaces of the model, and no constraint for the top surface of the model. Only gravitational loads were considered. The model is the result of a simplification of the actual mine, and in the simulation calculations, it is assumed that all material properties are homogeneous and isotropic, ignoring the anisotropy of materials in the actual geological environment. Moreover, the boundaries are set to be fixed or free, which may deviate from the actual situation. The current model is mainly applicable to laminated rock structures. For geological environments that are unstratified, highly fractured, or where there are a large number of faults, the predictive power of the model may be reduced.  
\begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline
**Name** & **Density (kg.m\({}^{-3}\))** & **Elastic Modulus (GPa)** & **Cohesion (MPa)** & **Internal Friction Angle (\({}^{\circ}\))** & **Poisson Ratio** & **Compressive Strength (MPa)** & **Tensile Strength (MPa)** \\ \hline Ores & 3020 & 28.14 & 25.63 & 46.2 & 0.301 & 102.86 & 2.02 \\ Roof & 2600 & 6.13 & 15.20 & 35.1 & 0.327 & 51.32 & 1.16 \\ Floor & 2620 & 13.48 & 20.37 & 43.8 & 0.210 & 86.33 & 1.47 \\ Cemented & 1740 & 0.85 & 1.02 & 34.6 & 0.311 & 2.19 & 0.23 \\ Backfill & 1640 & 0.41 & - & 22.1 & 0.403 & 0 & - \\ \hline \hline \end{tabular}
\end{table}
Table 1: Physical and mechanical parameters of different materials.