## 3 A Post-Processing Algorithm to Enforce Traction Continuity along the Interface
### The Effects of the Geometrical Parameters of the Polynomial  
It is important to choose appropriate values for the geometrical parameters \(n_{x1}\), \(n_{x2,inter}\), and \(n_{x2,cent}\) in order not to introduce unwanted field distortions. For instance, a choice of a small value of \(n_{x2,inter}\) together with a large value of \(n_{x2,cent}\) or vice versa can lead to stress fields that are continuous on the interface, but have sharp discontinuities (Figure 2e,f) or spurious oscillations (Figure 2g,h) in the bulk. In this section, we study the effects of different parameter combinations in more details, aiming to find the parameter sets that avoid these distortions in the stress field. We ran the algorithm 324 times, varying the geometrical parameters in the following ranges: \(3\leq n_{x1}\leq 9\), \(4\leq n_{x2,inter}\leq 20\), and \(5\leq n_{x2,cent}\leq 21\). Note that for \(n_{x1}\geq 13\) the system was over constrained. Overall, we found that parameters in ranges of \(8\leq n_{x2,inter}\leq 14\) (1/5 to 1/3 of the subset size) and \(7\leq n_{x2,cent}\leq 15\) gave similar results, independent of the value of \(n_{x1}\). Figure 4: Interface parallel (**a**) and normal (**b**) displacements at the central columns of the subsets shown in Figure 3a. The black lines represent the displacements obtained with Vic-2D, while the blue lines are the displacements modified with the traction continuity algorithm, using \(n_{x1}=5\), \(n_{x2,inter}=12\), and \(n_{x2,cent}=11\). The red circles represent the observed displacements that are used to invert for the coefficients of the displacement polynomials \(u_{2}^{p+},u_{2}^{p-},u_{1}^{p+}\), and \(u_{1}^{p-}\). The black diamonds correspond to the centers of the subsets. We found that the stress fields obtained with the algorithm were insensitive to the value of \(n_{x1}\). For example, there was no observable difference between full-field maps of stresses obtained with \(n_{x1}=3\) and \(n_{x1}=9\) (Figure 5a-d). To further examine the effect of \(n_{x1}\), we plot the normal and shear stresses vs position along two lines perpendicular to the interface at \(x_{1}=170\) and 220 pixels (Figure 5e-h). For both lines there was a \"jump\" larger than 5.5 MPa in the normal stress at the interface when the calculation was done with the \"fill boundary\" algorithm of Vic-2D without TC enforcement. The TC algorithm enforced the normal stress to be continuous across the interface, with a normal stress at the interface that was about the average of the normal stresses at the pixels immediately above and below the interface when TC was not enforced. The results were not sensitive to \(n_{x1}\). A small effect of \(n_{x1}\) on the shear stress was observed for the line located at x1 = 220 pixels (Figure 5h), with a difference of 0.12 MPa between the results for \(n_{x1}=3\) and 9. The shear stress at the interface when TC was enforced were smaller than the shear stresses at the pixels immediately above and below the interface when TC was not enforced. As shown in Figure 2e-h, a large difference between the parameters \(n_{x2,inter}\) and \(n_{x2,cent}\) resulted in artifacts in the stress field. On the one hand, taking \(n_{x2,inter}=4\) and \(n_{x2,cent}=21\) (with \(n_{x1}=7\)) resulted in discontinuities at the boundary between the Vic-2D solution and polynomial extrapolation (Figure 2e,f), \(n_{x2,inter}+n_{x2,cent}\) pixels away from the interface due to a poor fit between the stresses obtained by the polynomial and the original stresses at this region. On the other hand, taking \(n_{x2,inter}=20\) and \(n_{x2,cent}=5\) (with \(n_{x1}=7\)) resulted in spurious oscillations (Figure 2g,h). Some effects of \(n_{x2,inter}\)  
Figure 5: The effect of parameter \(n_{x1}\) on the stress field. (**a–d**) Full-field maps of normal (left) and shear (right) stresses for \(n_{x1}=3\) and 9. (**e–h**) Normal (left) and shear (right) stresses vs position along paths perpendicular to the interface at \(x_{1}=170\) and 220 pixels (grey lines in (**c**)) for different values of \(n_{x1}\).