## 2 Literature Review
### Classical Methods  
Theoretical analysis methods, based on classical mechanics theories, provide profound insights into the behavior of materials and rock under various stress conditions, making them pivotal in the prediction of stress in the roof of mined-out areas. The primary analytical methods include elastic mechanics analysis, limit equilibrium analysis, plastic mechanics analysis, fracture mechanics analysis, and rock mechanics analysis. For example, [PERSON] et al. established an analytical model based on the stress-strain characteristics of collapsed rock mass and voussoir beam theory to study the stress reconstruction problem in mined-out areas, revealing the relationship between fracture deformation and permeability in the rock mass [2]. [PERSON] et al. further expanded the application of rock mechanics through numerical simulations and elastoplastic models to study hydraulic fracturing propagation in deep reservoirs [3]. However, theoretical analysis methods typically rely on simplified assumptions, including simplified geological spatial shapes, the assumption of homogeneous linear materials, and so on. Therefore, their applicability is limited when dealing with complex geometries, nonlinear material properties, and multi-physical field coupling issues in actual engineering [2]. The limitations of theoretical analysis methods, the demand for high-precision predictions, and the development of computational technology have contributed to the emergence of numerical simulation techniques. Since the mid-20 th century, numerical simulation has gradually become the primary means of addressing complex engineering problems. Numerical simulation methods primarily include the finite element method (FEM), finite difference method (FDM), discrete element method (DEM), and boundary element method (BEM). The FEM works by breaking down a complex structure into simpler elements to establish a mathematical model of stress-strain relationships, allowing for the analysis of the structure's response under external forces. [PERSON] et al. employed the finite element method to simulate the stress distribution and roadway displacement under different roof cutting parameters when studying the reasonable width of coal pillars under thick hard roofs [4]. The FDM solves complex engineering problems by discretizing continuous partial differential equations. [PERSON] et al. used the FLAC3D (Version 7.0) finite difference software to study the stress evolution mechanism under dynamic pressure during reverse mining in deep mines [5]. The discrete element method (DEM) is a numerical simulation technique used to analyze the mechanical behavior of granular or block materials. [PERSON]'s study used FLAC3D numerical simulation to analyze the mining damage characteristics and abutment pressure distribution in the overlying rock of an uneven-length working face [6]. The BEM is a boundary-based numerical simulation approach that solves domain problems by applying conditions on the boundaries. Although BEM is used less frequently compared to other methods, it can be highly efficient and accurate, making it a valuable tool under specific conditions. For example, [PERSON] et al. utilized BEM to simulate the mechanical evolution characteristics of overlying rock layers in a coal mine working face, revealing the impact of boundary conditions on stress distribution [7]. However, while numerical simulation methods address the shortcomings of theoretical analysis, they also have drawbacks, such as high computational costs and a strong parameter dependency. The final category of methods is field monitoring and experimental techniques, which provide an accurate representation of geological conditions and the mechanical behavior of rock formations, thereby offering an empirical foundation for stress field analysis and prediction. However, due to the complexity of geological environments, a single monitoring or experimental method often fails to capture all influencing factors comprehensively.