## 3 Numerical Simulation Results and Analysis
### Failure Characteristics of the Double U-Shaped Caverns under Horizontally Incident Wave
The static tangential stresses at key points around the caverns under the in situ stress field are presented in Table 3. With decreasing cavern clearance, the tangential stresses at the crown and floor decrease, while the tangential stresses at the midpoints of both sidewalls increase. Notably, the tangential stress at the adjacent sidewall is higher than that at the non-adjacent sidewall, which indicates that the compressive stress concentration occurs in the middle of two caverns. The result is consistent with the analytical solution of [PERSON] et al. [35]; they also found that the hoop stress at the middle pillar decreases with increasing cavern spacing. When the dynamic disturbance is superimposed and the stress wave amplitude is \(P_{d}=\) 60 MPa, the distributions of microcracks and the macroscopic failure modes of surrounding rock with different cavern clearances are shown in Figure 6. The red segments represent the tensile microcracks, and the blue segments represent shear microcracks. The crack number represents the total microcracks in the numerical model. As the incident stress wave propagates to the boundary of the caverns, the reflected tensile stress wave causes the generation of an amount of microcracks around the facing-wave sidewall of cavern 1, leading to macroscopic failure resembling spalling. When \(d=\) 3.75 m, there is a small amount of microcracks on the back-wave side of cavern 1 and the facing-wave side of cavern 2, with the intermediate rock pillar remaining stable and no macroscopic failure occurring. With decreasing cavern clearance, the number of microcracks in the intermediate rock pillar continuously increases. When the clearances between caverns are 1.25 m and 2.50 m, macroscopic failures occur on both sides of the rock pillar, with the severity of failure being more significant at \(d=\) 1.25 m. The presence of the incident side cavern weakens the dynamic stress concentration at the crown and floor of the non-incident side cavern. Moreover, according to the distribution of tangential stress around the cavern under in situ stress field (Table 3), the static tangential stress at the cavern floor decreases with decreasing cavern clearance. Thus, macroscopic failures mainly concentrate on the floor of cavern 1, and the damage extent around the floor of cavern 1 decreases with decreasing cavern clearance, especially as \(d=\) 1.25 m, where there is no macroscopic damage on the cavern floor. The smaller the cavern clearance is, the more stable the cavern floor is. As shown in Figure 6, the failure modes of the intermediate rock pillar between the two caverns varies with the various cavern clearances. Figure 7 illustrates the tangential stress and kinetic energy evolution processes of B2 (the midpoint of the facing-wave sidewall of cavern 2) under various cavern clearances. Because of the diffraction of the stress wave, the tangential stress at the B2 position first increases to the positive peak value. The smaller cavern clearance results in a higher peak tangential stress, and the microcrack density around the facing-wave sidewall of cavern 2 also increases. Subsequently, the reflected tensile wave accelerates the propagation and coalescence of the microcracks in the intermediate rock pillar. The smaller the cavern clearance is, the greater the macroscopic failure that occurs at the intermediate rock pillar. The residual kinetic energy implies \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{
\begin{tabular}{c} **Cavern** \\ **Clearance** \\ \end{tabular} } & **Midpoint of** & **Midpoint of** & **Midpoint of** & **Midpoint of** \\  & **Crown** & **Floor** & **Adjacent Sidewall** & **Non-Adjacent Sidewall** \\  & **(A1)** & **(D1)** & **(C1)** & **(B1)** \\ \hline Single cavern & 38.2 MPa & 19.3 MPa & 29.7 MPa & 30.0 MPa \\
3.75 m & 33.7 MPa & 17.2 MPa & 35.3 MPa & 30.2 MPa \\
2.50 m & 32.9 MPa & 16.1 MPa & 43.1 MPa & 34.4 MPa \\
1.25 m & 29.9 MPa & 15.2 MPa & 59.6 MPa & 36.5 MPa \\ \hline \hline \end{tabular}
\end{table}