## 1 Introduction  
Underground rock engineering frequently involves the construction and maintenance of structures such as tunnels, mines, and storage caverns, where the mechanical behavior of rock masses around openings is a critical factor for stability and safety. The presence of openings in rock masses significantly alters the stress distribution, leading to potential failure and instability of the surrounding rock mass [1; 2; 3]. Understanding the stress distribution of surrounding rock under different stress conditions and its impact on tunnel stability is crucial for ensuring the stability of engineering projects. Traditional experimental studies have extensively investigated the mechanical behavior, failure characteristics, and energy evolution processes of rock masses with openings. Comprehensive research has been carried out to analyze the effects of different parameters,such as loading conditions, the properties of rock-like materials, and the geometry of openings [4; 5; 6; 7]. These studies provide valuable insights into the macroscopic mechanical properties of the rock and the associated failure patterns. However, experimental conditions impose limitations; the sample size may significantly affect the results, and laboratory tests cannot accurately replicate the actual boundary conditions of underground rock engineering. Additionally, experiments primarily investigate the impact of openings on the mechanical properties and failure characteristics of rock masses, but they struggle to comprehensively explain the mechanical behavior characteristics of the surrounding rock based on stress mechanisms. The inability to precisely simulate in situ stress conditions and the scale effects further complicate the extrapolation of laboratory findings to real-world scenarios. Therefore, while traditional experiments are invaluable for understanding basic mechanical responses, they need to be complemented with analytical or numerical methods to provide a more holistic understanding of the mechanical behavior of rock masses. Analytical solutions offer a theoretical approach for systematically analyzing stress distributions. They can precisely describe the changes in the stress field of rock masses under different stress conditions and, when combined with numerical simulations, provide better predictions of failure modes and stability of the surrounding rock. The stress state of the rock mass surrounding circular openings has been extensively investigated due to its simple geometry. For rock masses with openings of intricate shapes, the complex variable function method has been extensively employed for stress analysis because of its distinct advantages in managing complex geometrical problems [8]. This theoretical framework enables more precise analytical solutions for rock stress, particularly around tunnels in homogeneous, isotropic elastic rock masses [9; 10; 11]. [PERSON] et al. [12] investigated the analytical solution for the rock stress surrounding a square tunnel under various confining stress conditions. For stress analysis of rock masses with such openings, the mapping function can be determined using formulas, significantly reducing the computational complexity. For stress analysis of rock masses with complex-shaped openings in underground engineering, optimization algorithms are also a powerful method for solving the corresponding mapping functions. [PERSON] et al. [13] used the Box complex algorithm to solve the mapping function for rock masses with commonly encountered underground openings such as horseshoe-shaped tunnels. They analyzed the stress concentration factor of the surrounding rock through analytical stress solutions. In this study, the fracture behavior of the samples, demonstrated by mechanical tests, was accurately predicted using analytical solutions, highlighting the reliability of the complex variable function method. [PERSON] et al. [14] utilized complex variable theory to derive the stress distribution around rock masses with complex-shaped openings.