## 5 Numerical Examples
### Mesh Refinement Study  
First, we investigate the accuracy of stress calculations on three meshes generated using PolyCISAMR with varying levels of refinement on background meshes with \(h=7.5\)\(\upmu\)m (coarse mesh), 5.0 \(\upmu\)m (medium mesh), and 2.5 \(\upmu\)m (fine mesh) on one of the image slices (Slice 1). The medium mesh is illustrated in Figure 21. All meshes are generated using two levels of SAMR, resulting in a total number of 46,500, 75,721, and 189,448 elements, respectively. In none of these meshes does the maximum aspect ratio of the elements exceed 3. More details regarding the characteristics and computational cost associated with the construction of these meshes are provided in Table 1. The time required for transforming the initial pixelated image to a NURBS model (input file for mesh generation) was 15 s. The total PolyCISAMR mesh generation time (\(t_{\text{tot}}\)) scales in a superlinear fashion with the number of elements (\(N_{\text{e}}\)) for meshes with different refinement levels (see \(r_{1}=t_{\text{tot}}/N_{\text{e}}\) values for different meshes). Similarly, according to Table 1, the time required for constructing individual grain meshes (\(t_{\text{g}}\)) scales superlinearly with \(N_{\text{e}}\). This is attributed to the relatively small time required for merging segregated grain meshes (\(t_{\text{m}}\)) compared to the overall computational time \(t_{\text{tot}}\), as reported in Table 1. Additionally, this table shows that \(t_{\text{m}}\) scales well with the number of nodes (\(N_{\text{n}}\)) in the mesh. These performance metrics demonstrate the efficiency of the PolyCISAMR algorithm for meshing large polycrystalline domains composed of many grains. Plane stress CPFE simulations under uniaxial tension were conducted on these meshes. The resulting stress, total strain, plastic strain, and hardening fields for the fine mesh are \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Mesh Resolution** & \(N_{\text{n}}\) & \(N_{\text{e}}\) & \(t_{\text{tot}}\) & \(t_{\text{g}}\) & \(t_{\text{m}}\) & \(r_{1}=t_{\text{tot}}/N_{\text{e}}\) & \(r_{2}=t_{\text{s}}/N_{\text{e}}\) \\ \hline Coarse & 34,044 & 51,534 & 22.55 s & 21.02 s & 1.53 s & \(4.38\times 10^{-4}\) & \(4.08\times 10^{-4}\) \\ Medium & 56,910 & 84,382 & 32.53 s & 29.72 s & 2.81 s & \(3.86\times 10^{-4}\) & \(3.52\times 10^{-4}\) \\ Fine & 148,413 & 206,024 & 74.86 s & 65.31 s & 9.55 s & \(3.63\times 10^{-4}\) & \(3.17\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Total mesh generation time (\(t_{\text{tot}}\)), individual grain meshing time (\(t_{\text{g}}\)), and merging time (\(t_{\text{m}}\)) using PolyCISAMR for meshing Slice 1 model with different refinement levels, where \(N_{\text{n}}\) and \(N_{\text{e}}\) denote the number of nodes and the number of elements, respectively. Figure 21: Conforming meshes generated using PolyCISAMR for the medium mesh (background mesh: \(h=5.0\)\(\upmu\)m) in the mesh refinement study on the Slice 1 microstructure consisting of 60 grains. illustrated in Figures 22 and 23. Two applied strain levels were considered: a lower value of 0.0026, at which some grains begin to develop plastic flow, and a larger value of 0.02, at which most grains have hardened and reached saturation stress. At the first load level, there is significant variation in the axial stress \(\sigma_{xx}\) contour shown in Figure 22a, as certain grains are favorably oriented for plastic slip while others are not. On the other hand, at the higher applied strain level in Figure 23, the axial stress is more uniform across the domain and mostly equal to the saturation stress of 310 MPa. It is important to note that these contour plots are not projected nodal stress averages; they are the elemental stress averaged over the quadrature points. The lack of color discontinuities in each figure indicates the high quality of the mesh.