## 1 Introduction
Using a complex function, a stress distribution function was established, and the distribution characteristics of the surrounding rock stress field were determined, which could be instrumental in improving tunnel design and construction practices. [PERSON], [PERSON] et al. [10] explored the design of a flat-top U-shaped section for mining roadways in three-soft coal seams based on the complex function method, presenting the geometric parameters of such design. This work helps in addressing the challenges faced in roadway support in such coal seams, making a contribution toward safer and more efficient mining operations. In addition to the aforementioned research, there are studies that have bridged the gap between rock mechanics and other fields such as seepage and thermodynamics, thereby broadening the application scope of the complex variable theory in rock engineering. [PERSON] [11] elaborated on the stress and deformation analysis for a deep tunnel situated in transversely anisotropic elastic rock when subjected to groundwater seepage, providing closed-form solutions for the stresses and deformations induced both in the ground and the tunnel liner by using the complex variable theory and conformal mapping. The study contributed to the understanding of the complex interplay between geological conditions, groundwater flow the structural integrity of deep tunnels, which is crucial for ensuring the safety and stability of such underground structures. [PERSON] et al. [12] delved into the challenges posed by temperature variations in the stress analysis of deep buried circular tunnels and presented a method to address the stress field of a deep buried circular-lining tunnel under the combined effects of an unsteady temperature field and in situ stress. Their study contributed to the understanding of and methodological advancement in addressingtemperature-induced stress variations in tunnel engineering, which is crucial for ensuring the safety and longevity of tunnel structures under varying thermal conditions. The research introduced above is all targeted towards addressing the issues of a single opening in underground engineering. Utilizing the complex variable functions to compute the stress around complex openings can yield high-precision analytical solutions. With its application in the underground geotechnical field continually increasing, the related theory of solving single complex opening issues using the complex variable functions has become relatively mature. However, when there are two or more openings on the plane, the situation becomes significantly more complex compared to a single opening, due to the mutual interactions between the neighboring openings. The stress functions also become exceedingly complicated. Currently, the Schwarz alternating method is an important method for solving multi-connected domain problems with multiple openings [13; 14]. But early research, limited by the computational conditions at the time, generally only completed one iteration, and the solution accuracy needs further enhancement [14; 15; 16; 17]. Later, [PERSON] et al. [18] employed the Schwarz alternating method to analytically solve the area stress field of a double circular opening system. Through two iterations, the solution accuracy was significantly improved. [PERSON] et al. [19] proposed a modified Schwarz alternating method that disregards the inverse mapping function, further enhancing the accuracy of stress function solutions for planes with multiple openings, especially the planes with irregular non-arch-shaped openings. The studies mentioned above are instrumental in studying stress distribution around a tunnel and understanding the mechanical behavior of both the tunnel and the surrounding rock. The findings can be useful in the design and analysis of tunnel structures in the rock engineering field.