## 3 Analysis Method of End Face Stress for Cycloid Gear
### SACGES-CMMD Method  
By utilizing the dual quaternion cycloid gear tooth profile model delineated by [PERSON] [29], a collection of vertex coordinates was garnered. With the aid of the Plgcirmap toolbox, the bounded connected polygonal domain was adeptly transformed into a bounded connected unit circle domain. Subsequent to this transformation, boundary vertices were systematically numbered. During the meshing phase, the areas of the cycloid gear teeth subject to force warranted a localized refinement. Consequently, a mesh gradient of 1.5 was set, and the GeometricOrder was designated as 'quadratic'. For the boundary conditions, the nodes associated with the central hole were anchored with fixed constraints. Following these configurations, the stress analysis results of the unit circle were then deduced. Ultimately, the coordinates endowed with stress details were inversely mapped from the unit circle back to the polygonal domain. The procedural workflow of the stress analysis method of the cycloid gear end surface based on conformal mapping in multi-connected domains (SACGES-CMMD) is vividly depicted in Figure 4.  
Figure 4: SACGES-CMMD workflow. **(a)** Cycloidal gear profile polygon; **(b)** Unit circle; **(c)** A unit circle with vertex labeling information; **(d)** Unit circle mesh division; **(e)** Unit circle shear stress results; **(f)** Cycloidal gear end face stress results.By amalgamating the Boundary Integral Method and [PERSON]’s iterative technique, this research ventured into computations employing hierarchical grid points, the Nystrom approach, and the GMRES solver. The method undertakes the discretization and resolution of polygons via boundary integral equations, capitalizing on the generalized Neumann kernel, and thereby facilitating the derivation of numerical results for both the mapping function and its inverse. In the framework of this analysis, a bounded polygonal connected domain boasting a connectivity value of 10 was explored. To capture the conformal mapping of this domain, the 'plgcirmap' function in MATLAB 2020b was employed, leveraging the vertices of the polygon and selected points within the domain. During the mapping computation, normalization was invoked. The resultant conformal mapping, \(f\), and its inverse, _f_\({}^{-1}\), were pictorially represented using the 'plotmap' function, as evinced in Figure 5.  
In this investigation, an intricate mapping procedure was carried out, converting the node information from a bounded multiply connected region, specifically the epitrochoid area, to a unit circle. Building on this, the MATLAB PDE toolbox was employed to execute a stress analysis on the unit circle, delineating the process from stages (c) to (d), as shown in Figure 4. The subsequent inverse mapping process from the unit circle back to the polygonal region, embedding the stress information, necessitates the computation of the Jacobian matrix, executed as follows:  
1. Initialization: For a designated polygonal region coupled with its associated stress tensor distribution, the position of each node (polygon vertex) and its corresponding first-order derivative are computed. Special attention is devoted to instances where the first-order derivative is zero, circumventing potential numerical complications in ensuing computations. 2. Computation of Jacobian Matrix for Inverse Mapping: Upon obtaining the node data and first-order derivatives, the Jacobian matrix for inverse mapping at each node is ascertained. The Jacobian is informed by the complex derivative of the inverse mapping function. By inputting complex variable details representing the unit circle node into the inverse mapping function, the correlated polygon node data are derived, and the differential from this reverse mapping (from the unit circle to the polygon) is secured. When constructing the Jacobian matrix, the derivative of the complex variable, normalized by its magnitude, is bifurcated into real and imaginary parts,  
Figure 5: Bounded multiconnected domain mapping results for cycloid contours.