## 1 Introduction  
In deep excavation, the cavern is located at a high in situ stress level, and subjected to various dynamic disturbances, leading to the occurrence of unconventional disasters such as rockburst, spalling, and zonal disintegration [1; 2; 3]. Exploring the mechanisms of dynamic disasters is quite important for deep engineering. To investigate the failure characteristics of the caverns under the coupled static-dynamic loading condition, many studies have been conducted by laboratory experiment. [PERSON] et al. [4] developed a dynamic true-triaxial experimental system to simulate the impact rockburst, and reproduced the rockburst process on the rock specimens with a circular hole. They concluded that the damage extent under dynamic loading is more severe than that under static loading. [PERSON] et al. [5] analyzed the failure characteristics around the circular opening by using an improved true-triaxial test system, and explained the failure mechanism of the remotely triggered rockburst from an energy perspective. [PERSON] et al. [6] captured the strain rockburst process of the circular opening under dynamic impact loading by using the biaxial Hopkinson pressure bar equipment and high-speed camera. They found that the lateral pressure is an important factor for the range of strain concentration zone around the cavern, and the severity of strain rockburst decreases with increasing lateral pressure. [PERSON] et al. [7] analyzed the dynamic response around a horseshoe-shaped opening under blast loading and pointed out that the loading direction significantly affects the crack propagation around the opening. Due to the limitation of the testing device, it is difficult to replicate the complex geological and stress conditions of practical engineering by laboratory experiments. Withthe rapid development of computer technology, numerical methods such as the finite element method (FEM), the finite difference method (FDM), and the distinct element method (DEM) have extensively been used to investigate the dynamic response and failure characteristics of underground engineering under dynamic disturbance [8; 9; 10]. [PERSON] [11] and [PERSON] et al. [12] studied the blast-induced dynamic response of U-shaped caverns by PFC (Particle Flow Code) and FLAC (Fast Lagrangian Analysis Code) and analyzed the influence of cavern depth and blasting position on the dynamic response of surrounding rock. [PERSON] et al. [13] investigated the damage characteristics around the cavern under multiple explosion sources by LS-DYNA3D and found that the crack density in the middle part of the two explosion sources is much higher than other rock masses. [PERSON] and [PERSON] [14] compared the blast damage around the cavern with different rock strengths and pointed out that the higher rock strength would increase the durability of the cavern but decrease the wave attenuation rate on the surrounding rock. [PERSON] and [PERSON] [15] concluded that the discontinuities accelerate wave attenuation in the surrounding rock, and the stress wave with higher frequency induced large displacements around the boundary of the cavern. [PERSON] and [PERSON] [16] simulated the dynamic failure of deep caverns under coupled static-dynamic loads by using ABAQUS and found that the presence of fault leads the rock failure to become more violent. With the development of underground engineering, double-cavern or even multi-cavern structures became common designs [17; 18]. When the distance between adjacent caverns is much larger than the cavern size, the influence of adjacent caverns can be neglected, and they can be treated as a single-cavern structure [19]. However, in practical engineering, the limited space and complex geological conditions result in the limited spacing between adjacent caverns. Thus, the stability of the caverns would be affected by neighboring caverns. On the one hand, the interaction effect between caverns exacerbates static stress concentration in the surrounding rock [20].