## 3 Numerical Results
### Microscale Analysis
#### 3.1.1 Convergence Study  
Being this one of the first works on meshless methods combined with a multiscale analysis, particularly using void subdomains, it is necessary to understand if the technique under such assumptions shows a convergent behaviour. Thus, considering a radius of \(R=2.821\) mm (corresponding to a volume fraction of \(v_{f}=0.75\)), five distinct meshes with an increasing number of nodes are built; see Figure 4.  
After constructing the models represented in Figure 4, the homogenised mechanical properties of each model (RVE) were obtained applying the procedure described in Section 2.4. Thus, the following elastic components of the homogenised constitutive material matrix, Equation (44), were obtained: \(C_{11}\), \(C_{22}\), \(C_{12}\), \(C_{21}\), \(C_{33}\), \(C_{44}\). Additionally, the homogenised material properties of the RVE were calculated, such as the Young's modulus \(E_{x}\), \(E_{y}\), and \(E_{z}\); the in-plane elastic shear modulus, \(G_{xy}\); and the Poisson's ratio, \(\
u_{xy}\). The results obtained with each mesh, for each formulation, are shown in Figure 5.  
With Figure 5, it is possible to visualise that all formulations converge. The FEM presents an upper bound convergence path, and, in opposition, the NNRPIM possesses a lower bound convergence path. The final converged value is not the same for the FEM and NNRPIM formulations. However, both FEM formulations converge to the same value, as well as both NNRPIM versions. It is possible to observe with Figure 5 that FEM-6n  
Figure 4: Numerical discretisation models used for the NNRPIM and FEM analyses: (**a**) 234 triangular elements and 138 nodes; (**b**) 884 triangular elements and 483 nodes; (**c**) 3408 triangular elements and 1785 nodes; (**d**) 7768 triangular elements and 4005 nodes; (**e**) 13,664 triangular elements and 6993 nodes. Observing the numerical results of Table 1 and comparing the NNRPIM and FEM solutions, it is possible to understand that the difference between the NNRPIM and FEM solutions is about 4.0% for the [PERSON]'s modulus \(E_{x}\) and \(E_{y}\), and 1.5% for \(E_{z}\). Regarding the elastic shear modulus \(G_{xy}\) and the [PERSON]'s ratio, the differences are about 7.7% and 2.0%, respectively. Such differences were expected because the NNRPIM and FEM formulations are very different from each other (background integration scheme, nodal connectivity, base functions of the shape functions). Similarly, for the elastic constitutive constants \(C_{ij}\), differences between the results of the NNRPIM and FEM formulations can be found. With Table 2, it is possible to visualise the following differences: 4.0% for \(C_{11}\) and \(C_{22}\), 3.0% for \(C_{12}\), 2.0% for \(C_{33}\), and 7.7% for \(C_{44}\). Regarding the stress field distribution, in Figure 6 the [PERSON] equivalent stress (\(\sigma_{ef}=\sqrt{3\cdot J_{2}}\)) field is presented for FEM-6n, FEM-3n, NNRPIMv1, and NNRPIMv2. Analysis of Figure 6 reveals a clear distinction in stress distribution patterns. As anticipated, the FEM-6n formulation exhibits a remarkably smooth stress field. This behaviour is in accordance with the presence of quadratic terms within its shape functions, which inherently promote smoothness. Conversely, the stress distributions obtained for both NNRPIMv2 and NNRPIMv1 formulations display a level of granularity similar to FEM-3n. Previous studies [22; 24] have demonstrated the ability of meshless methods to achieve smoother stress fields compared to lower-order finite element methods. However, the results presented in Figure 6 for the effective stress fields obtained using NNRPIMv1 and NNRPIMv2 formulations appear to contradict such expectation. Both NNRPIMv1 and NNRPIMv2 formulations exhibit stress distributions with a similar level of granularity as the three-node triangular finite element method (FEM-3n). This behaviour becomes even more pronounced with NNRPIMv1, which uses a lower number of nodes within its influence domains.