## 3 Numerical Results
### Macroscale Analysis
#### 3.2.1 Sandwich Cantilever Beam

Table 7: Displacement obtained at point A along direction \(Oy\) for sandwich cantilever beams with PUF cores with distinct volume fractions. \(v\) in [mm]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Vol.Frac** & **FEM-4n** & **NNRPIMv1** & **NNRPIMv2** & \(d_{v1}\) & \(d_{v2}\) \\ \hline
0.5 & \(-\)316.024 & \(-\)280.512 & \(-\)250.526 & 0.112 & 0.207 \\
0.6 & \(-\)252.946 & \(-\)226.490 & \(-\)204.089 & 0.105 & 0.193 \\
0.7 & \(-\)213.791 & \(-\)192.837 & \(-\)174.953 & 0.098 & 0.182 \\
0.8 & \(-\)187.105 & \(-\)169.779 & \(-\)154.905 & 0.093 & 0.172 \\
0.9 & \(-\)167.088 & \(-\)152.400 & \(-\)139.764 & 0.088 & 0.164 \\
1.0 & \(-\)151.028 & \(-\)138.398 & \(-\)127.560 & 0.084 & 0.155 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Normal stress \(\sigma_{xx}\) obtained at point B for sandwich cantilever beams with PUF cores with distinct volume fractions. \(\sigma_{xx}\) in MPa. for NNRPIMv2. For the shear stress \(\tau_{xy}\), NNRPIMv1 shows differences of around 2.0% to 7.6% to FEM solutions, and NNRPIMv2 presents differences of about 3.1% to 6.7% to FEM results. Although the formulation of NNRPIMv1 leads to much smaller influence domains than NNRPIMv2, both formulations deliver very close results. In the FEM-4n formulation, shape functions are constructed using four nodes (the nodes comprising the finite element). In the NNRPIMv1 and NNRPIMv2 formulations, shape functions are constructed using approximately 5 to 10 nodes and 16 to 32 nodes, respectively. Thus, the NNRPIM possesses a larger nodal connectivity than the FEM. This additional connectivity can have an unwanted effect. When the strain/stress fields are being calculated, an integration point within material 1 domain use nodes belonging to material 2 to construct its shape function because those nodes are inside its influence domain. Thus, the strain/stress fields are averaged by both materials, smoothing (lowing) the strain/stress at the transition zone. Furthermore, since corner nodes posses a lower number of natural neighbours, their integration points will possess influence domains with a lower number of nodes, leading to the construction of shape functions with lower complexity than the shape functions of integration points within the middle of the domain (which possess more nodes inside their influence domains). All these factors contribute to the higher rigidity of the NNRPIM formulations and their lower stress values. An additional factor explaining the lower stress value of normal stress \(\sigma_{xx}\) is the geometric position of the integration point closest to point B. Actually, the normal stress \(\sigma_{xx}\) is obtained on the integration point closer to point B, not at point B exactly. Due to the difference in the construction procedure of the background integration mesh of the FEM-4n and NNRPIM, the closest integration point will be placed in a very different position. In the FEM, the integration point closer to point B will be much closer to point B than the integration point closer to point B of the NNRPIM. Thus, the documented normal stress \(\sigma_{xx}\) near point B of the FEM and NNRPIM will be significantly different. This effect can be observed in the following figures. In Figures 11-13 are presented the variation in the normal stress \(\sigma_{xx}\) along the edge of the beam thickness on the clamped edge (the closest integration points to the clamped edge are selected, and their stress values are documented). It was observed that the magnitude of the normal stress \(\sigma_{xx}\) varies significantly from the aluminium face sheets to the PUF core. Thus, the thickness was divided in three sections: in Figures 11 and 13, the normal stress \(\sigma_{xx}\) on the top and bottom aluminium layers is shown, respectively, and Figure 12 shows the normal stress \(\sigma_{xx}\) on the PUF core. Figure 11: Variation in the normal stress \(\sigma_{xx}\) along the aluminium top face sheet for distinct PUF cores. (**a**) \(v_{f}=0.5\). (**b**) \(v_{f}=0.6\). (**c**) \(v_{f}=0.7\). (**d**) \(v_{f}=0.8\).