## 3 A Post-Processing Algorithm to Enforce Traction Continuity along the Interface
### Inverting for the Polynomial Coefficients  
To obtain the polynomial coefficients, we aim to minimize the misfit between the observed displacements \(\mathrm{u}_{2}^{ob+}\), \(\mathrm{u}_{2}^{ob-}\), \(\mathrm{u}_{1}^{ob+}\), and \(\mathrm{u}_{1}^{ob-}\), produced by the DIC analysis, and the displacements predicted by the polynomials \(\mathrm{u}_{2}^{p+}\), \(\mathrm{u}_{2}^{p-}\), \(\mathrm{u}_{1}^{p+}\), and \(\mathrm{u}_{1}^{p-}\) at the pixels closer to the center of the subset. Recall that the DIC solution is computed up to half a subset away from the interface and extrapolated to the interface by the \"fill Boundary\" algorithm of Vic-2D. Since we expect such extrapolation to be reliable close to the center of the subset, but not close to the interface, we can use some of the extrapolated values by taking \(n_{x2,\mathrm{inter}}<20\) (Figure 3c). We can also exclude the displacement extrapolated by Vic-2D and consider only the computed values (up to the subset center) by taking \(n_{x2,\mathrm{inter}}\geq 20\) (Figure 3d). The minimization condition leads to the following system of equations:\[\left[\left[ \begin{array}{cccc}X_{[m\times n]}^{+}&\cdots&\cdots&0\\ \vdots&X_{[m\times n]}^{-}&\ddots&\vdots\\ \vdots&\ddots&X_{[m\times n]}^{+}&\vdots\\ 0&\cdots&\cdots&X_{[m\times n]}^{-}&\vdots\\ \end{array} \right]=\left[ \begin{array}{c}u_{2[m]}^{0+}\\ u_{2[m]}^{0-}\\ u_{1[m]}^{0+}\\ u_{1[m]}^{0-}\end{array} \right] \tag{12}\]  
where  
\[X_{[m\times n]}^{+}=\left[ \begin{array}{cccc}{x_{11}}^{3}&{x_{11}}^{2}x_{12 }&\cdots&1\\ \vdots&\vdots&\ddots&\vdots\\ {x_{m1}}^{3}&{x_{m1}}^{2}x_{m2}&\cdots&1\end{array} \right]x_{2}>0\]  
and  
\[X_{[m\times n]}^{-}=\left[ \begin{array}{cccc}{x_{11}}^{3}&{x_{11}}^{2}x_{12 }&\cdots&1\\ \vdots&\vdots&\ddots&\vdots\\ {x_{m1}}^{3}&{x_{m1}}^{2}x_{m2}&\cdots&1\end{array} \right]x_{2}<0\]  
are the sensitivity sub-matrices above and below the interface, respectively,  
\[A_{2[n]}^{+}=\left[ \begin{array}{cccc}a_{2}^{10+}&\cdots&a_{2}^{1+}\\ \end{array} \right]^{T}\] \[A_{2[n]}^{-}=\left[ \begin{array}{cccc}a_{2}^{10-}&\cdots&a_{2}^{ 1-}\\ \end{array} \right]^{T}\] \[A_{1[n]}^{+}=\left[ \begin{array}{cccc}a_{1}^{10+}&\cdots&a_{1}^ {1+}\\ \end{array} \right]^{T}\] \[A_{1[n]}^{-}=\left[ \begin{array}{cccc}a_{1}^{10-}&\cdots&a_{1}^ {1-}\\ \end{array} \right]^{T}\]  
are the assemblies of polynomial coefficients, and \(m=n_{x1}\times n_{x2,cnt}\) is the number of pixels within the group closer to the center of the subset. The polynomials must also fulfill the constraints for continuity of tractions across the interface. The condition for continuity of normal tractions is enforced by  
\[\frac{E^{+}}{1-\
u^{+2}}\Bigg{(}\frac{\partial u_{2}^{PQ+}}{\partial x_{2}}+ \
u^{+}\frac{\partial u_{1}^{PQ+}}{\partial x_{1}}\Bigg{)}=\frac{E^{-}}{1-\
u ^{-2}}\Bigg{(}\frac{\partial u_{2}^{PQ-}}{\partial x_{2}}+\
u^{-}\frac{ \partial u_{1}^{PQ-}}{\partial x_{1}}\Bigg{)} \tag{13}\]  
and can be assembled into the following matrix form:  
\[\left[\frac{E^{+}}{1-\
u^{+2}}\frac{\partial X_{[n_{x1}\times n]}^{0+}}{ \partial x_{2}}-\frac{E^{-}}{1-\
u^{-2}}\frac{\partial X_{[n_{x1}\times n]}^{0 -}}{\partial x_{2}}\frac{E^{+}}{1-\
u^{+2}}\frac{\partial X_{[n_{x1}\times n]} ^{0+}}{\partial x_{1}}-\frac{E^{-}}{1-\
u^{-2}}\
u\frac{\partial X_{[n_{x1} \times n]}^{0-}}{\partial x_{1}}\Bigg{]}\Bigg{[} \begin{array}{c}A_{2,[n]}^{ -}\\ A_{2,[n]}^{+}\\ A_{1,[n]}^{-}\\ A_{1,[n]}^{-}\\ \end{array} \Bigg{]}=0, \tag{14}\]  
where, for example,  
\[\frac{\partial u_{2}^{PQ+}}{\partial x_{2}}=\frac{\partial X_{[n_{x1}\times n ]}^{0+}}{\partial x_{2}}A_{2,[n]}^{+}=\left[ \begin{array}{cccc}0&{x_{11}}^{2 }&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&{x_{n_{x1}}}^{2}&\cdots&0\end{array}