## 2 Methods
### Determination of Mapping Function  
The mapping function that transforms the region outside the opening in the z-plane to the region outside the unit circle in the \(\zeta\)-plane is expressed as:  
\[Z=\omega(\zeta)=\sum_{k=1}^{\infty}C_{k}\zeta^{2-k} \tag{6}\]  
where \(C_{k}\) is a complex number, but when the opening is symmetric about the x-axis, \(C_{k}\) is real. The more complex the shape of the opening, the more terms of \(C_{k}\) are needed to improve the accuracy of the mapping shape. Generally, only the first few terms of \(C_{k}\) are sufficient to achieve adequate accuracy. The mapping function, taking only the first m terms, can be expressed as:  
Figure 1: Far-field stress state in a plane: (**a**) non-principal stress state; (**b**) principal stress state. \[Z=\omega(\zeta)=\sum\limits_{k=1}^{m}C_{k}\zeta^{2-k} \tag{7}\]  
In this study, the three-center arch-shaped opening is an axially symmetric figure; therefore, \(C_{k}\) is real. For any point \(\sigma\) at the boundary of the unit circle in the \(\zeta\)-plane with polar coordinates (1, 0), it can be expressed as:  
\[\sigma=\cos\theta+i\sin\theta \tag{8}\]  
In a similar manner, for the corresponding point t of \(\sigma\) on the boundary of the opening in the z-plane, represented in polar coordinates as (r, \(\alpha\)), it can be expressed as:  
\[t=r\cos\alpha+ir\sin\alpha \tag{9}\]  
Substituting Equations (8) and (9) into Equation (7) yields:  
\[r\cos\alpha+ir\sin\alpha=\sum\limits_{k=1}^{m}\{C_{k}\cos[(k-2)\theta]-iC_{k} \sin[(k-2)\theta]\} \tag{10}\]  
By extracting the real and imaginary parts, the following equation is obtained:  
\[\left\{ \begin{array}{l}r\cos\alpha=\sum\limits_{k=1}^{m}\{C_{k}\cos[(k-2) \theta]\}\\ r\sin\alpha=\sum\limits_{k=1}^{m}\{-C_{k}\sin[(k-2)\theta]\}\end{array} \right. \tag{11}\]  
Based on Equation (11), the trigonometric interpolation method can be used to solve for \(C_{k}\), thereby determining the mapping function. [PERSON] et al. [18] have conducted a detailed study on this method, which is not elaborated here. Assuming the mapping point of \(\sigma_{i}\) is (r, \(\alpha_{\sigma}\)), its corresponding point on the opening boundary is (r, \(\alpha_{\sigma}\)). By uniformly sampling \(n\) points \(\sigma_{i}\) (1, \(\theta_{i}\)) from the unit circle, the average absolute relative error of the \(n\) points can be defined as [15]:  
\[f_{error}=\frac{1}{n}\sum\limits_{j=1}^{n}\frac{\left|r_{\sigma}-r_{\sigma}^{ (0)}\right|}{r_{\sigma}^{(0)}} \tag{12}\]  
Generally, the greater the number of sampling points \(n\), the higher the accuracy of the mapping function. However, too many sampling points can lead to a decrease in computational efficiency. In this study, it was found that the accuracy of the mapping function stabilizes when \(n\) exceeds 50. Ultimately, \(n\) was set to 200 in all cases to balance computational efficiency and accuracy. Figure 2 shows the variation in the mapping function error \(f_{error}\) as the number of mapping function coefficients \(m\) increases. It can be seen that the error is as high as 0.065 at \(m=4\). As the number of terms increases from 4 to 8, the error significantly decreases to below 0.015. This phase shows the most notable reduction in error, indicating that increasing the number of terms has a significant effect on reducing the error. When \(m\) increases from 8 to 15, the error continues to decrease slowly. During this phase, the error gradually approaches zero, but the rate of decrease diminishes considerably. Although increasing the number of terms still reduces the error, the effect is markedly less significant compared to the initial phase. When \(m\) exceeds 15, the error stabilizes and nearly stops changing, approaching zero. In this phase, increasing the number of terms has almost no effect on improving the error, and the error curve becomes flat. As shown in Figure 3a, when \(m=6\), the mapped shape already closely matches the original shape. When m is greater than or equal to 10, the mapped shape is almost identical to the original shape.