## 3 Numerical Simulation Results and Analysis
### Failure Characteristics of the Double U-Shaped Caverns with Different Height Ratios  
In practical engineering, the different engineering requirements and geological conditions contribute to the differences in the geometric dimensions of adjacent caverns. Three cavern height ratios (\(h_{1}\)/\(h_{2}\)) of 0.67, 1.0, and 1.33 were specified in this section to explore the effect of height ratios on the failure characteristics of the caverns. The width of two caverns and the height of cavern 2 remain constant (\(w_{1}\) = \(w_{2}\) = 2.5 m, \(h_{2}\) = 3.75 m), while the corresponding heights of cavern 1 are \(h_{1}\) =2.5, 3.75, and 5.0 m.  
The unequal caverns cause the distribution of the static stress field of the model to be non-symmetrical. Figure 12 presents the strain energy density distribution of double U-shaped caverns with different cavern height ratios under in situ stress field. When  
Figure 11: Failure modes of rock pillar in practical engineering: (**a**) hourglass-shaped pillar [36]; (**b**) the rock pillar after pillar burst [37]. Figure 10: Kinetic energy evolution process of C1 under vertically incident stress wave. \(h_{1}/h_{2}<1.0\), the strain energy density on the left side of the middle rock pillar is higher than that on the right side, and when \(h_{1}/h_{2}>1.0\), the accumulated strain energy on the right side of the rock pillar is higher. Although the geometric dimensions of cavern 2 remain constant, the low-strain-energy-density area around the floor will increase with the increase in the cavern height ratio, and the low-strain-energy-density area around the cavern floor with smaller height-width ratio is more significant. When the stress wave is incoming horizontally (\(P_{d}\) = 60 MPa), the failure modes with different cavern height ratios are illustrated in Figure 13. As the cavern height ratio (\(h_{1}/h_{2}\)) increases from 0.67 to 1.33, the number of microcracks on the incident side of cavern 1 increases. When \(h_{1}/h_{2}\) = 1.0 and 1.33, macroscopic failure occurs in the facing-wave side of cavern 1, and the damage extent and depth of \(h_{1}/h_{2}=1.33\) are greater than \(h_{1}/h_{2}=1.0\). The damage pattern on the facing-wave side of cavern 2 exhibits an opposite trend with the increase in the cavern height ratio. The larger the cavern height ratio \(h_{1}/h_{2}\), the smaller the damage extent on the facing-wave side of cavern 2. For the cavern floor, when \(h_{1}/h_{2}<1.0\), macroscopic failure occurs only at the floor of cavern 2. When \(h_{1}/h_{2}>1.0\), the floor of cavern 2 remains stable, with macroscopic failure occurring only at the floor of cavern 1. These results indicate that under the horizontally incident stress waves, the higher the incident side cavern, the relatively more stable the non-incident side cavern. When the stress wave is vertically incident, the damage modes of double U-shaped caverns with different cavern height ratios are depicted in Figure 14. There are differences in the damage pattern between the two caverns when the heights of the two caverns are inconsistent. The spalling occurs at the floor areas of both caverns due to stress wave reflection. When \(h_{1}/h_{2}\) is equal to 1.0, the damage degree of the floor areas of both caverns is roughly equal; otherwise, more microcracks with larger depths are generated near the floor of the cavern with a smaller height-width ratio. The dynamic stress concentration occurs at the sidewall areas of the caverns. The stress concentration in the sidewalls adjacent to each cavern is more significant compared to those non-adjacent, and macroscopic failure occurs in the sidewalls of both caverns under different height ratios. With increasing height ratio of caverns, the initial strain energy density at the left sidewall of cavern 2 increases, and the depth of damage in the left sidewall of cavern 2 also increases.