## 1 Introduction  
Rockburst is a dynamic geological hazard in which elastic strain energy accumulated in rock mass is suddenly and violently released due to an external disturbance, resulting in the bursting and ejection of the surrounding rock [1,2]. With the rapid development of the domestic economy in China, various rock mass engineering projects, such as hydroelectric power stations, transportation tunnels, metal and non-metal mines, and underground repositories for nuclear waste disposal, are developing to great depths at an unprecedented speed [3]. More than 40 percent of underground mines built since the 1950s have transitioned or are gradually transitioning to deep mining, with about 30 mines across the country involved in underground construction projects being deeper than 1000 m, and approximately 10 of these mines were constructed at depths exceeding 1300 m, making ultra-deep mining increasingly common [4]. However, as the depth of the mine increases, the in situ stress rises dramatically, and the challenges of controlling the stability of the surrounding rock in affected roadways, chambers, and mining areas become increasingly apparent. High stress levels intensify ground pressure activity and, thus, increase the likelihood of rockburst disasters occurring, posing substantial challenges to the safety of deep mining. More than 20 underground metal mines in China have experienced or are currently experiencing rockburst disasters, and the number of mines affected by rockburst in the past 30 years has shown an upward trend [5]. Therefore, given the practical requirements of ultra-deep mining, the risk assessment of the rockburst hazard has become both necessary and urgent. With the rapid development of computational technology and a greater understanding of the rockburst mechanism, many scholars and engineers have employed numerical simulation methods to assess rockburst risk [6, 7, 8, 9]. The central concept of this method is to establish a quantitative relationship between rockburst risk parameters (e.g., rockburst location, rockburst severity, etc.) and numerical indexes; by analyzing the spatial distribution range of these numerical indexes, the location of a potential rockburst and the extent of anticipated damage can be effectively determined [10]. [PERSON] et al. [11] combined model experiments and numerical simulation techniques to study the mechanical response characteristics of rock surrounding tunnels during the fracture process, and they found that the ratio of compressive strength to induced stress (the maximum tangential stress around the surrounding rock) varies during the progressive destabilization of the rock surrounding the tunnel. The critical value for this ratio (a function related to the uniaxial compressive strength) was proposed to identify the cause of the instability and failure, based on which the surrounding rock failure process could be predicted. [PERSON] et al. [12] first proposed the Failure Approach Index (FAI), a model which they used to evaluate the stability of the underground powerhouse, water inlet, and diversion tunnel of Jinping II Hydropower Station. The FAI was later used to simulate the evolution of the \"11-28\" structure-type rockburst at Jinping II Hydropower Station [13]. To study the rockburst risk and dynamic response characteristics of the surrounding rock under dynamic loading, [PERSON] et al. proposed a damage risk index based on the Mohr-Coulomb criterion, which enables the prediction of potential rockburst locations and their intensity [14]. [PERSON] et al. [15] used ANSYS software (16.0) combined with rockburst index criteria to predict the possible locations and intensity of potential rockbursts in the rock surrounding a deep-buried tunnel in a folded tectonic zone rock mass. [PERSON] et al. [16] evaluated the rockburst risk of horizontally lined and vertically lined models of strongly and weakly interbedded rock mass in a comparison based on the finite element method combined with the Turchaninov criterion. [PERSON] et al.