## 3 Numerical Results
### Macroscale Analysis
#### 3.2.2 Functionally Graded Sandwich Cantilever Beam  
In this subsection, two cantilever beams with aluminium face sheets with approximately functionally graded PUF cores are analysed. As Figure 17 shows, the PUF core possesses eight distinct layers, each one with its own volume fraction. With its right side clamped, the beam is submitted to a localised load \(P=10^{6}\) N at the top-left corner and it possesses the following dimensions: \(L=4\) m and \(D_{i}=0.1\) m, for \(i=\{1,2,\dots,10\}\).  
For the PUF core, two different approximately functionally graded PUF are considered (FG1 and FG2). The volume fractions and corresponding mechanical properties of each layer are indicated in Table 10 for both FG1 and FG2 beam models.  
Using the FEM-4n and both NNRPIM formulations, the \(Ox\) and \(Oy\) displacements of point A (indicated in Figure 17) were obtained and documented in Tables 11 and 12, respectively. The results show that the difference between the FEM-4n and NNRPIM formulations is around 3.5% for the \(u\) displacement and slightly lower for the \(v\) displacement component. Thus, despite being distinct numerical formulations, these results show that the FEM-4n and NNRPIM provide similar solutions.  
Figure 17: Sandwich cantilever beam, with aluminium face sheets (sheets 1 and 10) and a functionally graded PUF core (sheets from 2 to 9). The geometric characteristics, essential and natural boundary conditions and material distribution are indicated, as well as the interest points A, B and C.  
In Figure 18a,b,e,f, it is possible to observe that the NNRPIMv2 values near the edge surface of the beam (\(y\simeq 0.0\) and \(y\simeq 1.0\)) are smoothed. This effect, in addition to the difference between the location of the integration points of the FEM-4n and NNRPIM formulations, explains why the localised \(\sigma_{xx}\) results of the NNRPIM, shown in Table 13,  
Figure 19: Variation in the shear stress \(\tau_{xy}\) along the beam thickness. (**a**) FG1 beam, for \(y\in[0.9,1.0]\) m. (**b**) FG2 beam, for \(y\in[0.9,1.0]\) m. (**c**) FG1 beam, for \(y\in[0.1,0.9]\) m. (**d**) FG2 beam, for \(y\in[0.1,0.9]\) m. (**e**) FG1 beam, for \(y\in[0.0,0.1]\) m. (**f**) FG2 beam, for \(y\in[0.0,0.1]\) m.  
are so different from the ones obtained with the FEM-4n. Furthermore, with Figure 18, it is possible to visualise that NNRPIMv1 provides very similar results to the FEM-4n. Concerning the normal stress along \(y\in[0.1,0.9]\) m, in Figure 18c,d, all solutions are acceptably close to each other.  
Regarding the shear stress \(\tau_{xy}\), shown in Figure 19, the FEM-4n and NNRPIM solutions are in accordance with previous studies. In Figure 19a,b,e,f, it is visible that at the free edges of the beam (\(y\simeq 0.0\) and \(y\simeq 1.0\)) due to the size of the support domain, the FEM-4n solutions differ from the NNRPIM solutions. Nevertheless, at the PUF core, Figure 19c,d, the FEM-4n and NNRPIM solutions are actually very close to each other.  
Overall, it is possible to observe that the stress distributions obtained with the formulations (depicted in Figures 18 and 19) are generally close to each other, regardless of the volume fraction distribution along the PUF thickness.