## 4 Analysis of Progressive Failure Process of Surrounding Rock and Tunnel Structure
### Establishment of Numerical Simulation Model  
In tunneling engineering, regional discretization is a fundamental process for both the finite element method (FEM) and the volume of fluid (VOF) method. This procedure involves partitioning a continuous physical domain into a finite number of discrete subregions to enable numerical simulation and analysis. For FEM, discretization is achieved by generating a mesh that divides the computational domain into finite elements. These elements may be triangular, quadrilateral, cubic, or other polyhedral shapes, depending on the problem's dimensionality and complexity, and are defined by nodes. The quality and density of the mesh are critical as they influence both the accuracy and computational efficiency of the simulation. Within each element, interpolation functions, also known as shape functions, approximate the solution. These functions determine how the solution varies within each element based on node values, thereby converting continuous physical fields--such as stress, strain, or temperature--into discrete sets. Consequently, partial differential equations (PDEs) are transformed into algebraic equations, which are then solved using numerical methods applied to the global system of equations formed by all elements. Similarly, in the VOF method, regional discretization entails dividing the computational domain into grid cells. Each cell represents a certain volume of fluid or gas, with the primary goal being to track the fluid volume fraction within each cell. This approach is particularly effective for simulating complex fluid dynamics and free surface behavior. The VOF method leverages these grid cells to monitor fluid interface changes and manage flow dynamics. By discretizing the mass and momentum equations into algebraic equations for each cell, the VOF method accurately simulates fluid movement and interface dynamics. In summary, regional discretization in the FEM and VOF methods is essential for numerical simulations. In the FEM, it involves converting continuous fields into finite elements and using interpolation functions to approximate solutions, resulting in solvable algebraic equations. In the VOF method, discretization facilitates the tracking of fluid volume fractions within grid cells, enhancing the simulation of intricate fluid interfaces. These discretization techniques transform real-world physical problems into computational models, allowing for precise analysis and resolution of engineering challenges. To mitigate the influence of boundary effects, the tunnel section is positioned centrally within the model, with the distance from the boundary set at three times the tunnel diameter. The Mohr-Coulomb constitutive model is employed for the steep rock strata, while the elastic constitutive model is applied to the primary support, secondary lining, and invert filling layer (as shown in Figure 15). The analysis adopts the plane strain criterion. In the vertical stress field, a uniform load is applied to the top of the model, accounting for the self-weight of the rock mass. In the horizontal stress field, the initial stress values vary along a vertical gradient. The bottom of the model is constrained with fixed supports, while force constraints are applied at the upper boundary to simulate the tunnel's burial depth. Additionally, force constraints are applied to the left and right boundaries to represent different lateral pressure coefficients. The physical and mechanical parameters of the surrounding rock, informed by previous research, are presented in Table 2. As mentioned above, the groundwater in the tunnel area is developed, and the physical and mechanical parameters of the fluid are shown in Table 3. Figure 2 illustrates the dimensions of the tunnel structure and its support parameters. This study employs a numerical simulation with a 1:1 scale relative to the actual tunnel dimensions. Additionally, the self-developed software (v1.0) allows for the input of the rock dip angle.