## 3 Verification of Q8-DIC in Boundary Deformation Measurement  
To verify the accuracy and efficiency of the three methods described in Section 2 for deformation measurement near curved edges, a simple simulated test was conducted in the form of a square plate under open-hole compression (Figure 6a). The reference image, with a size of 1001 \(\times\) 1001 pixels and hole radius of \(a=240\) pixels, was cropped from a real speckle image. Distributed loading was applied on the top and bottom with a load density of \(q\). Based on the solution of the displacement field of the problem, the deformed image was obtained from the reference image by cubic interpolation. The pixel intensities of the deformed image were encoded into 8-bit integers with values ranging from 0 to 255. The theoretical elastic solution of the displacement field of the open-hole compression problem can be expressed by Equation (5):  
\[\left[ \begin{array}{c}u\\ v\end{array} \right]=\left[ \begin{array}{cc}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{array} \right]\left[ \begin{array}{c}u_{r}\\ u_{\theta}\end{array} \right] \tag{5}\]  
where  
\[u_{r}=\frac{(1+v)qr}{2E}\left\{\frac{1-v}{1+v}+\frac{a^{2}}{r^{2}}+\left[1+ \frac{4}{1+v}\frac{a^{2}}{r^{2}}-\frac{a^{4}}{r^{4}}\right]\right\}\cos 2\theta \tag{6}\]  
\[u_{\theta}=-\frac{(1+v)qr}{2E}\left[1+\frac{2-2v}{1+v}\frac{a^{2}}{r^{2}}+ \frac{a^{4}}{r^{4}}\right]\sin 2\theta \tag{7}\]  
in which \(r\) and \(\theta\) are the polar coordinates with the origin at the hole center, and \(a\) represents the hole radius. In this simulated test, the load density \(q=1.4\times 10^{7}\) N/pixel, [PERSON]'s modulus \(E=10\) GPa, and [PERSON]'s ratio \(v=0.33\). Figure 6: The geometry of open-hole specimen (**a**) and meshing scheme in Q8-DIC (**b**). Based on the reference and deformed images, the \(v\) displacement field was first solved by the subset-based DIC. The commercial image correlation software VIC-2D was used with a subset size of 47 pixels to assess the displacement distribution. The displacements near the specimen edges that could not be directly calculated by the subset-based DIC were obtained using cubic extrapolation and the half-subset method. The displacement field was also calculated by Q8-DIC with the meshing scheme illustrated in Figure 6b. In total, there were 8 elements and 40 nodes. The LM optimization algorithm was employed, and the optimization iteration was stopped when the change of _loss_ in Equation (4) was less than \(1\times 10^{-6}\) or the variations in the parameters to be optimized were less than \(1\times 10^{-3}\). It can be observed from Figure 7 that the displacement field calculated by Q8-DIC was smoother and more consistent with the theoretical elastic solution than those obtained by the extrapolation and half-subset methods. This is because the continuity of the entire displacement field was considered in the Q8-DIC method. For a clearer comparison of the accuracy of the three methods in the boundary deformation measurement, the displacement distribution around different circles near the hole can be plotted. Figure 8a,b presents the vertical displacements around two circles with radii of 241 and 255 pixels, respectively. It can be observed that even on the hole edge (\(r=241\) pixels), Q8-DIC could yield a sufficiently accurate result, but the extrapolation method exhibited significant deviation from the theoretical solution, while the half-subset method provided completely unreasonable displacements. The accuracy of the calculated displacements improved with an increasing distance from the hole edge. When \(r=255\) pixels, the displacement distributions already agreed strongly with the theoretical solution, except for the relatively large error near the top (\(\theta=90^{\circ}\)) and bottom (\(\theta=270^{\circ}\)) of the circle in the extrapolation method. Figure 8c illustrates the absolute errors of the three methods compared with the theoretical solution; the non-smoothness of the displacement field obtained by the half-subset method can be clearly observed.