## 5 Numerical Examples
### Comparison with Pixelated Meshes  
In this section, we compare the two CPFEM simulations performed on PolyCISAMR conforming meshes with their pixelated counterparts. As noted previously, pixelated meshes are widely used in the literature for CPFEM studies due to the easy process of creating such meshes [13; 14]. However, the jagged or stair-stepped representation of grain boundaries in pixelated meshes leads to the introduction of artificial stress concentrations in these regions, which can undermine the accuracy of CPFEM simulations [27]. In the comparison study presented in this section, CPFEM results on two PolyCISAMR meshes generated on a background mesh with \(h=5\)\(\upmu\)m, one with no SAMR (18,705 elements) and another with 1 level of SAMR (36,680 elements), are compared to simulations conducted on pixelated meshes with \(h=5\)\(\upmu\)m (15,625 elements) and \(h=2.5\)\(\upmu\)m (62,500 elements), respectively. Small portions of these meshes are illustrated in Figure 25, which clearly shows the difference in the representation of grain boundaries in conforming versus pixelated meshes. Note that, unlike in the conforming meshes, the geometrical representation of grains in the pixelated meshes depends on the mesh size; although the mesh refinement reduces the stair-stepped appearance of grain boundaries, it does not completely eliminate this artifact, as shown in Figure 26.  
The CPFEM approximations of the variation of normal stress \(\sigma_{xx}\) in the PolyCISAMR conforming and pixelated meshes at an applied strain of 0.02 are illustrated in Figure 25. As shown in the inset of each figure, these plots show element-wise values of \(\sigma_{xx}\) in each mesh, which converge more slowly compared to a smooth stress field obtained using nodal averaging or more sophisticated techniques. Therefore, rather than comparing the stress values, studying the pattern of stress variations would be more useful in understanding the effect of mesh type (conforming vs. pixelated) on the accuracy. The drastic difference between simulated normal stresses in the PolyCISAMR and pixelated meshes can clearly be seen in the highlighted regions marked A, B, and C in the inset of the meshes shown in Figure 25. Note the unrealistic stress concentrations in the pixelated meshes compared to the conforming meshes in these regions, which emanate from the jagged representation of grain boundaries in the former. Although a different window of the microstructure is inspected in Figure 24, even on the coarse conforming PolyCISAMR mesh in Figure 24a  
Figure 24: Effect of mesh refinement levels on the CPFEM approximation axial stress \(\sigma_{xx}\) in Slice 1: (a) Coarse mesh; (b) Medium mesh; (c) Fine mesh. there are no oscillations in color such as are prevalent in Figure 25. The comparison between the highlighted region C in the conforming mesh with one level of SAMR (cf. Figure 25b) and the fine pixelated mesh (cf. Figure 25d) is especially interesting, as it shows a single element in the latter with erroneously high stress due to the inaccurate representation of the grain boundary geometry in this region. It is worth noting that such artifacts of a pixelated mesh only affect the stress/strain approximation in a small portion of elements of the whole mesh along grain boundaries; thus, their impact on averaged values such as homogenized stress-strain curves is muted. However, such localized errors could be devastating when approximating phenomena such as intergranular damage using cohesive elements, which requires accurate recovery of stresses along grain boundaries.