## 1 Introduction  
Discontinuity is a significant mechanical break or fracture of negligibly small tensile strength in a rock, and it has a lower shear strength and higher conductivity of fluids than the rock itself. The properties of discontinuities are closely related to the mechanical properties of the rock mass, and are often used as a controlling factor to determine the stability and safety of rock engineering [1; 2; 3]. Therefore, the development of a reasonable and effective DFN model of rock mass has attracted research interest because it can provide a clear distribution of fractures in the context of stability analysis and calculations in rock engineering. In general, the DFN model consists of three basic parts, an orientation distribution model of the discontinuities, a dimension model of the discontinuities and a spatial distribution model of the discontinuities [4; 5]. For the dimension model, although many scholars have attempted to describe the relationship between the trace of the outcrop of discontinuities and the dimension of discontinuities, our understanding of the shape of the fracture is not entirely clear due to the complexity of fracture formation and the limitations associated with field observations [6; 7]. Two main types of models have been used thus far to formulate the DFN: the Baecher disk model and the polygon model. The former was introduced by [PERSON] et al. [8], and is characterized by the assumption that the fracture is shaped as a disk or an ellipsoidal disk while ignoring its thickness. The polygon model was derived by [PERSON] [9] in the process of generating a fracture model through a system of Poisson lines. [PERSON] and [PERSON] [10] reached a conclusion consistent with that of [PERSON]. They assumed that the fracture is a polygon with three to twenty sides, the side lengths of these polygons can be equal or extensible, and the lengths of these extensions can be defined by the aspect ratio. Because the modeling is simple and can satisfy the needs of the project, the Baecher disk model has been widely used in the analysis of rock mechanics [11; 12; 13; 14; 15; 16; 17]. However, the rock mass is an inhomogeneous, discontinuous, and anisotropic medium. The shape of the fracture is related to the fracture mechanics and energy at the tip of the crack. It usually extends to form polygons rather than disks. Moreover, field investigations have indicated that a polygon is a more appropriate shape to represent fractures. [PERSON] and [PERSON] [18] assumed that the shape of a single fracture tends to be circular in homogeneous rock masses, but when the rock mass contains multiple fractures and its anisotropy is prominent, the fractures should be assumed to be polygonal due to the influence of mutual cutting among them. [PERSON] et al. [19] coupled a network of polygonal fractures with a large-scale geological model to establish a refined model of the rock structure to identify and analyze random blocks. [PERSON] et al. [20] introduced a statistical method for generating 3D polygonal fractures in order to identify a Complex Block System. [PERSON] et al. [21] used an improved control circle algorithm to describe the spatial dimensions and shapes of fractures, and used this to establish a fracture network to provide a base model for the simulation-based analysis of the diffusion of grout. [PERSON] et al. [22] proposed a new method to generate realistic 3D Stochastic DFN by getting fracture plane points from Virtual Outcrop Models and defining clusters using a K-means algorithm. Current field observations suggest that polygonal fractures often do not have more than six sides. The number of sides of a fracture in polygon models proposed by researchers is also generally set at between four and six [5; 19; 23].