## 3 Results and Analysis
### Mutual Influence between Adjacent Roadways
#### 3.3.1 Solution of Stress Function for a Plane Containing Two Openings
To enhance the accuracy of the solution, a refined method was introduced, employing the schwarz alternating method to obtain analytical solutions for the external stress around complex dual-opening systems without considering the inverse mapping functions. The detailed solution process is introduced below. Let the mapping functions of the plane containing Opening 1 in the \(x_{1}O_{1}y_{1}\) coordinate system and the plane containing Opening 2 in the \(x_{2}O_{2}y_{2}\) coordinate system be denoted as \(z_{1}=\omega_{1}(\zeta_{1})\) and \(z_{2}=\omega_{2}(\zeta_{2})\), respectively. \(\zeta_{1}\) and \(\zeta_{2}\) are the coordinates of the planes containing Opening 1 and Opening 2 on the plane of the unit circle mapping, respectively. Starting from Opening 1, in the absence of any opening, the stress state at any point within the plane remains the same. The excess surface force \(f_{1}^{0}(t_{1})\) on the virtual boundary of the yet-to-be excavated Opening 1 is expressed by the following:  
\[f_{1}^{0}(t_{1})=\varphi^{0}(t_{1})+t_{1}\overline{\varphi^{0}}(t_{1})+ \overline{\psi^{0}}(t_{1})=-2 Bt_{1}-\big{(}B^{\prime}-iC^{\prime}\big{)} \overline{t_{1}} \tag{14}\]  
where \(t_{1}\) represents the point on the boundary of Opening 1. Upon substituting the mapping function of Opening 1 into the above equation, the corresponding excess surface force on the unit circle can be obtained as follows:  
\[f_{1}^{0}(\sigma_{1})=2B\omega_{1}(\sigma_{1})+\big{(}B^{\prime}-iC^{\prime} \big{)}\overline{\omega_{1}}(\sigma_{1}) \tag{15}\]  
where \(\sigma_{1}\) represents the point on the mapped unit circle corresponding to Opening 1. Therefore, after the excavation of Opening 1, the stress boundary condition on the unit circle line on the \(\zeta_{1}\) plane is expressed as follows:  
\[\varphi_{1}^{0}(\zeta_{1})+\frac{\omega_{1}(\zeta_{1})}{\omega_{1}^{\prime}( \zeta_{1})}\overline{\varphi_{1}^{0}}(\zeta_{1})+\overline{\psi_{1}^{0}}( \zeta_{1})=-f_{1}^{0}(\zeta_{1}),\zeta_{1}=\sigma_{1} \tag{16}\]  
where \(\varphi_{1}^{0}(\zeta_{1})\) and \(\psi_{1}^{0}(\zeta_{1})\) are the relevant analytic functions corresponding to Opening 1 on the plane. This equation represents the stress boundary condition on the unit circle line for a single opening consistent with Equation (13). The superposition of \(\varphi^{0}(\zeta_{1})\) and \(\varphi_{1}^{0}(\zeta_{1})\) as well as \(\psi^{0}(\zeta_{1})\) and \(\psi_{1}^{0}(\zeta_{1})\) results in the analytic solutions \(\varphi_{10}(\zeta_{1})\) and \(\psi_{10}(\zeta_{1})\) for the stress functions and for the plane with only Opening 1 present. After obtaining the analytic functions for the \(\zeta_{1}\) plane when only a single opening (Opening 1) is present, it is necessary to eliminate the excess surface force on the unit circle boundary corresponding to Opening 2. The core issue is to utilize \(\varphi_{10}(\zeta_{1})\) and \(\psi_{10}(\zeta_{1})\) to obtain the desired excess surface force distribution function. During this process, three coordinate transformations need to be performed on the point \(\sigma_{2}\) on the unit circle corresponding to Opening 2. Let the excess surface force on the boundary of the unit circle on the \(\xi_{2}\) plane corresponding to Opening 2 be denoted as \(f^{1}_{12}(\sigma_{2})\), then it can be obtained as follows:  
\[f^{1}_{12}(\sigma_{2})=\varphi_{10}(\gamma_{1})+\frac{\omega_{1}(\sigma)}{ \omega_{1}^{\prime}(\sigma)}\overline{\varphi_{10}^{\prime}}(\gamma_{1})+ \overline{\psi_{10}}(\gamma_{1}) \tag{17}\]  
where \(\gamma_{1}\) represents the point on the unit circle corresponding to Opening 2 in the \(x_{1}O_{1}y_{1}\) coordinate system after undergoing a series of transformations to obtain its position in the \(x_{2}O_{2}y_{2}\) coordinate system. Taking Equation (17) as an example, let the inverse mapping function of the plane where Opening 1 is located be denoted as \(\zeta_{1}=w_{1}^{-1}(z_{1})\).