## 5 Damage Model and Numerical Test
### Numerical Test  
Based on the proposed damage model, this work developed a numerical calculation program using the finite volume method (FVM), the numerical calculation program flow is shown in Figure 18.  
1. Define constants, variables, and arrays. Input basic parameters and determine the solution area. 2. Grid division can be carried out according to certain rules within the solution area, radiating from the inner boundary to the outer boundary of the solution area. The grid gradually changes from dense to sparse from the tunnel wall to the surrounding rock in the distance. After dividing the grid cells, associate the cells with nodes. 3. Select an appropriate control volume, with node displacement as an unknown variable, and integrate and form a stiffness matrix using the finite volume method. Figure 17: Comparison between model curves and experimental results. According to the geometric equations of elasticity mechanics, the element strain matrix can be expressed as:  
\[\epsilon(x,y)=B(x,y)\mathbf{d} \tag{17}\]  
where \(\epsilon(x,y)\) represents the element strain matrix, \(\mathrm{B(x,y)}\) represents the element geometry matrix, and \(\mathrm{d}\) represents the node displacement matrix. The element stress matrix can be expressed as:  
\[\sigma(x,y)=(1-D)\mathbf{M_{0}}\epsilon(x,y) \tag{18}\]  
where \(\sigma(x,y)\) represents the element stress matrix. When solving, first set boundary conditions and constraints. Start the iterative calculation with a preset time step; the damage can be solved using Equation (15) so that the elastic modulus can be updated. Then, calculate the displacement, strain, and stress of each unit at the new time step. When the running time is greater than the set time or the damage is greater than or equal to 1, the iterative calculation is ended and the result is output. In order to explore the applicability of the theoretical models and numerical calculation programs in engineering practice, the Yimaqianqiu Coal Mine in Henan Province, China is taken as the engineering background. The area between the goat of the 21,201 working face and the 21,181 working face is selected as the research object, and the numerical calculation program developed in this work is used for simulation. The average thickness of the 2# coal seam in the mining area is set to 10.5 m, with an elastic modulus of 3.5 GPa and a Poisson's ratio of 0.19. The direct top is composed of conglomerate and sandy mudstone, with an average thickness of about 310 m, an elastic modulus of 20.4 GPa, and a Poisson's ratio of 0.23. The bottom plate is fine sandstone, with an average thickness of about 183 m, an elastic modulus of 18.5 GPa, and a Poisson's  
Figure 18: Numerical calculation program flow. ratio of 0.22. The on-site environment has been simplified, ignoring the influence of the north-south F16 fault. The spatial relationship of the solution area is shown in Figure 19.  
The solution area is set to be 250 m long in the vertical direction and 250 m long in the horizontal direction. In coal seams, the grid is divided into equal lengths. In the rock layer, the grid is divided radially around the coal seam as the center. There are a total of 2376 units and 1295 nodes in the grid division. After conducting a time independence test and taking into account the impact of time step size on calculation accuracy and computational complexity, the time step size of the model can be determined to be 10 s. The lower boundary of the solution area is fixed with vertical displacement, and a geostress load is applied above the solution area, with a lateral coefficient of 1.2. A support force is applied in the goaf to simulate the support effect of gauge in the goaf. The grid division result is shown in Figure 20.  
The initial state is taken as the moment when the 21,201 working face completes mining and reaches an equilibrium state. The numerical calculation results are shown in Figures 21-23. Figure 19: The spatial relationship of solution area. Figure 20: The grid division result.