## 5 Results and Discussion
### SACGES-CMMD vs. FEM Comparison Results  
The SACGES-CMMD, in contrast to the FEM, brings forward a set of unique advantages. Central to these is the conformal mapping technique, which preserves angular relationships within multi-connected domains. This ensures that the stress patterns on a mapped unit circle accurately reflect those of the original polygonal domain. This geometric preservation, coupled with a scaling factor, streamlines stress computations. Additionally, this method boasts greater numerical stability, minimizing the potential instabilities and singularities that are often associated with direct FEM calculations on polygons. Moreover, applying the FEM on the unit circle guarantees a uniformly distributed grid, benefiting from the circular domain's inherent regularity and thereby heightening the computational precision. Furthermore, the straightforward circular geometry of the stress distribution facilitates an enhanced visualization and comparability of the results. Collectively, the use of conformal mapping for stress distribution in multi-connected domains enhances geometric integrity, numerical robustness, grid uniformity, and the clarity of the visualization and analysis. Figure 9: Accuracy results for conformal mapping. (**a**) Test point set. (**b**) Variation of the error norm during the inverse mapping process. From a comparative analysis of the results obtained using the FEM and SACGES-CMMD, as depicted in Figure 10 and summarized in Table 1, four key observations can be drawn:  
1. Grid Scale and Gradient Consistency: Under equivalent computational conditions, both methodologies exhibited grid sizes and gradients of comparable scales. 2. Optimized Grid Utilization with SACGES-CMMD: Notably, the SACGES-CMMD approach utilized marginally fewer grid nodes than its FEM counterpart. The smallest reduction ratio in the number of nodes was 7.34%. As the mesh size increased, the reduction ratio in the nodes gradually increased as well, reaching up to 19.34% in the end. This suggests that the conformal mapping process underpinning SACGES-CMMD permits the elimination of superfluous grid elements, thereby optimizing computational efficiency within a compact grid layout. 3. Comparative Stress Metrics: The comparison highlighted a close correspondence between the maximal shear stress and the stress distribution's STD for both methods, with no significant disparities discernible. 4. Computational Efficiency: While the computation durations of both techniques were roughly commensurate, a downward trend in the grid element count conferred the SACGES-CMMD with a slight edge in the computational expediency. In the comparison of the computational efficiency, when the mesh was dense, both exhibited similar efficiencies with variations of 0.11% and 0.29%, respectively. As the mesh size increased, the computational efficiency of SACGES-CMMD showed a significant improvement, reaching a peak of 4.75%. When synthesizing these insights, it becomes evident that while SACGES-CMMD parallels the FEM in terms of computational precision and reliability, it prefers certain advantages. Namely, these comprise a leaner grid element composition and a modest reduction in the computation time. Consequently, SACGES-CMMD emerges as a viable strategy for stress distribution evaluations in multi-connected domains, holding potential for elucidating the characteristic stress distribution patterns within such domains. This