## 3 Results and Analysis
### Mutual Influence between Adjacent Roadways  
#### 3.3.1 Solution of Stress Function for a Plane Containing Two Openings  
In practical underground mining engineering, the layout of underground spatial structures tends to be complex, and adjacent spatial structures may affect the stability of existing structures. Therefore, in the design of underground mining stops, the impact of the adjacent roadway on an existing roadway is considered through the double-opening calculation model shown in Figure 3b. The Schwarz alternating method is a common technique for solving the plane stress problems in systems with double openings. The essence of the Schwarz Alternating Method is to transform a multi-opening problem into a series of single-opening problems, each of which can be solved using the method introduced in the previous chapter. As shown in Figure 8, there are two openings (Opening 1 and Opening 2) on the plane. Coordinate system Z1 is centered at the centroid of Opening 1, while coordinate system Z2 is centered at the centroid of Opening 2. Coordinate systems \(\zeta_{1}\) and \(\zeta_{2}\) are the planes containing unit circle openings corresponding to the relevant coordinate systems. Upon the completion of the excavation of Opening 1, assuming that Opening 2 has not yet been excavated, the stress at the hypothetical boundary of Opening 2 after the excavation of Opening 1 can be calculated in the coordinate system Z1. When proceeding to excavate Opening 2, it is essential to apply external forces to the boundary of Opening 2 to balance the surplus surface forces generated by Opening 1 at the boundary of Opening 2. Assuming that Opening 1 has not been excavated and switching the coordinate system to Z2, the stress function corresponding to the stress balancing conditions at the boundary of Opening 2 can be calculated using the method discussed earlier. At this point, Opening 2 would generate surplus surface forces at the boundary of Opening 1, necessitating a switch back to coordinate system Z1 to calculate the stress function for Opening 1 under new boundary balancing conditions. As the number of iterations increases, the surplus surface forces of a particular opening generated at the boundary of the adjacent opening would diminish, and the solution would increasingly approximate the true solution. Since the number of iterations is finite, once the solution accuracy meets the engineering requirements, the iterative calculations can be halted, thereby concluding the stress solution process. Figure 8: Schematic diagram of a plane system with two irregularly shaped openings arranged arbitrarily. Figure 7: Distribution curve of hoop stress around the roadway in Slope 101 before the excavation of the adjacent roadway. (The red solid line indicates the stress curve, and the black dashed line indicates the opening boundary). The early application of the Schwarz alternating method was confined to systems comprising circular or elliptical openings. Even for such simplistic shapes forming a dual-opening system, the complex stress functions derived from iterative calculations using the Schwarz alternating method were highly complex, allowing for only one or two iterations, which hardly met the accuracy requirements. For planes with openings of complex shapes, the highly complex stress functions precluded direct transformations between different coordinate systems, making the calculation of excess surface forces on the opening boundaries during iterations significantly challenging. [PERSON] et al. [18] addressed this issue by introducing inverse mapping functions to transform the coordinates of the opening boundaries, representing the excess surface forces on the boundaries using complex series functions, and effectively solving the mentioned problems to a certain extent. However, when the deviation of the opening shape from a circle is significant, the inverse mapping functions can introduce certain errors.