## 3 Numerical Results
### Macroscale Analysis
#### 3.2.1 Sandwich Cantilever Beam  
A sandwich cantilever, with the the geometry and boundary conditions shown in Figure 10, is analysed. The beam presents the dimensions: \(L=4\) m, \(D_{1}=D_{3}=0.1\) m and \(D_{2}=0.8\) m, and a unitary thickness (\(H=1\) m) along \(Oz\) direction. At the beam's right edge, all degrees of freedom are constrained (clamped condition) and at its left-top corner, a localised load is applied: \(P=10^{6}\)N. Six distinct volume fractions (\(v_{f}=\{0.5,0.6,0.7,0.8,0.9,1.0\}\)) were considered for the PUF core, which will allow us to analyse the influence on the beam's displacement and stress fields of using PUF cores with distinct volume fractions. For each considered volume fraction, the corresponding homogenised mechanical properties (obtained with FEM-3n) are extracted from Table 5.  
Respectively, in Tables 6 and 7 are presented the displacement components along \(Ox\) (\(u\)) and \(Oy\) (\(v\)) obtained in point A (represented in Figure 10). The last two columns correspond to the difference between the NNRPIM and FEM solutions following the expression  
Figure 10: Sandwich cantilever beam, with aluminium face sheets (layers 1 and 3) and PUF core (layer 2). The geometric characteristics, essential and natural boundary conditions and material distribution are indicated, as well as the interest points A, B and C.  
\[d_{v1}=\left\|\frac{f_{FEM}-f_{NNRPIMv1}}{f_{FEM}}\right\| \tag{57}\]  
being \(f_{FEM}\) and \(f_{NNRPIMv1}\) the variable values obtained with FEM and NNRPIMv1, respectively. For NNRPIMv2, the same Equation (57) is applied considering NNRPIMv2 instead NNRPIMv1. It is possible to observe that NNRPIM solutions present lower displacement values than FEM solutions. Thus, the NNRPIM leads consistently to more rigid solutions than the FEM. Regarding the vertical displacement \(v\), depending on the density of the PUF core, the solutions of the FEM and NNRPIMv1 present differences between 5.6% and 6.1% and between 5.5% and 6.6% for NNRPIMv2. Comparing the NNRPIMv1 and NNRPIMv2 results, it is possible to find lower differences, around 1%. On the other hand, observing the horizontal displacement \(u\), the overall difference between the FEM and NNRPIM techniques is significantly reduced to 3% to 7.6%. Local stress values were also documented and analysed. The normal stress \(\sigma_{xx}\) at point B (in the top aluminium layer) and the shear stress \(\tau_{xy}\) at point C (in the PUF core, at \(y=0.5\) m), both represented in Figure 10, were obtained and included in Tables 8 and 9. As expected, being more rigid when compared with the FEM, the NNRPIM presents lower stress values. In this case, the differences are significant. Regarding the normal stress \(\sigma_{xx}\), the difference is about 8.4% to 11.2% for NNRPIMv1 and betwe \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 & 5.193 & 4.849 & 4.799 & 0.066 & 0.076 \\
0.6 & 3.641 & 3.416 & 3.394 & 0.062 & 0.068 \\
0.7 & 2.830 & 2.678 & 2.666 & 0.054 & 0.058 \\
0.8 & 2.368 & 2.261 & 2.254 & 0.045 & 0.048 \\
0.9 & 2.077 & 2.000 & 1.996 & 0.037 & 0.039 \\
1.0 & 1.882 & 1.826 & 1.822 & 0.030 & 0.032 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Displacement obtained at point A along direction \(Ox\) for sandwich cantilever beams with PUF cores with distinct volume fractions. \(u\) 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 & \(-\)284.042 & \(-\)268.188 & \(-\)265.192 & 0.056 & 0.066 \\
0.6 & \(-\)185.147 & \(-\)174.062 & \(-\)173.102 & 0.060 & 0.065 \\
0.7 & \(-\)130.169 & \(-\)122.259 & \(-\)121.992 & 0.061 & 0.063 \\
0.8 & \(-\)96.8805 & \(-\)91.0714 & \(-\)91.028 & 0.060 & 0.060 \\
0.9 & \(-\)74.7106 & \(-\)70.3729 & \(-\)70.3885 & 0.058 & 0.058 \\
1.0 & \(-\)58.9077 & \(-\)55.6522 & \(-\)55.6656 & 0.055 & 0.055 \\ \hline \hline \end{tabular}
\end{table}