## 2 Basic Principles
### Basic Principle of Q8-DIC in Boundary Deformation Measurement  
Using the concept of the FEM, Q8-DIC divides the AOI into several elements connected through nodes, thereby ensuring the continuity of the displacement field within the entire AOI. The boundary deformation can be calculated directly when the element nodes are placed on the edges (Figure 4). Instead of matching the subsets individually to obtain the displacements of each point, an objective function combining the reference and deformed  
Figure 3: Conventional methods for boundary displacement calculation: (**a**) Extrapolation method and (**b**) half-subset method.  
Figure 2: Standard deviation (SD) error of u-displacement and area loss near specimen edge vs. subset size.  
images was constructed to resolve the displacements of all searched points in Q8-DIC. A brief introduction to Q8-DIC is provided below.  
An 8-node isoparametric quadrangular element in the global coordinate system _xoy_ can be mapped onto a square in the local coordinate system \(\zeta o\eta\) (Figure 5). The relationship between the two coordinate systems is expressed as follows:  
\[\begin{array}{l}x=\sum\limits_{i=1}^{8}N_{i}(\xi,\eta)x_{i}\\ y=\sum\limits_{i=1}^{8}N_{i}(\xi,\eta)y_{i}\end{array} \tag{2}\]  
where \(\xi\) and \(\eta\) are the horizontal and vertical coordinates in the local coordinate system, \(x\) and \(y\) are the corresponding coordinates in the global coordinate system and \(N_{i}(\xi,\eta)\) is the shape function of the \(i\)th node of the element, which has the same form as the shape function in the FEM.  
The displacements of a point in the global coordinate system can be obtained by the corresponding shape function and displacements of the elemental nodes in the local coordinate system:  
\[\begin{array}{l}u_{\Omega_{m}}(x,y)=\sum\limits_{i=1}^{8}N_{i}(\xi,\eta)u_{mi }\\ v_{\Omega_{m}}(x,y)=\sum\limits_{i=1}^{8}N_{i}(\xi,\eta)v_{mi}\end{array} \tag{3}\]  
where \(\Omega_{m}\) represents the \(m\)th element, \(u_{\Omega_{m}}\) and \(v_{\Omega_{m}}\) are the displacement components of the point with the coordinate (\(x\), \(y\)) in \(\Omega_{m}\), and \(u_{mi}\) and \(v_{mi}\) are the displacement components of the \(i\)th node in \(\Omega_{m}\). The displacements of each node are essentially the parameters to be optimized. By selecting an appropriate objective function, the solution of the displacement  
Figure 4: Original and deformed meshes in 8-node quadrilateral elements (Q8-DIC).  
Figure 5: Relationship of global (**a**) and local (**b**) coordinates in the Q8-element.  
field can finally be transformed into an optimization problem. The objective function can be expressed as follows:  
\[loss=\sum_{m=1}^{M}\sum_{\Omega_{m}}\left(f(x,y)-g(x+u_{\Omega_{m}}(x,y),y+v_{ \Omega_{m}}(x,y))\right)^{2} \tag{4}\]  
where \(f\) and \(g\) represent the gray intensity of the AOI in the reference and deformed images, respectively. An iterative algorithm (such as the Newton-Raphson algorithm [13], Levenberg-Marquardt (LM) algorithm [14], genetic algorithm [15], and the inverse compositional Gauss-Newton algorithm [16]) can be used to minimize the objective function. The displacement vectors near or even on the edges can be obtained by Equation (4) once the solution is complete.