## 4 Conclusions  
In this work, a full multiscale study involving sandwich beams with a polyurethane foam (PUF) core is presented. First, the bulk PUF was modified through the inclusion of circular holes, allowing us to create PUFs with distinct volume fractions. Then, the homogenised mechanical properties of the PUF, with respect to its volume fraction, were obtained using a homogenisation technique combined with finite element methods (FEMs) and two versions of the natural neighbour radial point interpolation method (NNRPIM): one using influence domains with the first natural neighbours (NNRPIMv1) and another considering influence domains with the first and second natural neighbours (NNRPIMv2). The obtained results show that the NNRPIM formulations are capable to deliver solutions close to high-order FEM formulations, such as quadratic triangular elements (FEM-6n). However, to achieve variable fields with the same level of smoothness of the FEM-6n, the NNRPIM formulations require an increase in the number of integration points per quadrilateral sub-area integration cell, leading to a very high computational cost and, consequently, decreasing the overall NNRPIM numerical efficiency. Next, after calculating the homogenised mechanical properties of PUFs with respect to their volume fractions, those mechanical properties were applied to large-scale problems: a sandwich cantilever beam with a homogeneous PUF core and a sandwich cantilever beam with an approximated functionally graded PUF core. It was found that near the domain edge, the NNRPIMv2 formulation, having larger influence domains, leads to local lower stress values. However, stress distributions obtained with all the distinct formulations studied tend to agree along the thickness of the beam. The results obtained for the macroscale examples consistently show that NNRPIMv1 is capable to produce results very close to FEM-4n. Such findings are very interesting, since NNRPIMv1 appears to be more efficient than the NNRPIMv2 formulation by showing lower computational costs. The results documented in this work allow us to verify that the NNRPIM formulations, especially the NNRPIMv1 version, are a solid alternative to the FEM. Particularly in the microscale examples, the results also show that if the objective is to obtain a smooth representation of the stress field, the number of integration points per quadrilateral sub-area integration cell should be increased. However, this approach is not very numerically interesting, since it increases the computational cost of the analysis without a significant gain in accuracy. This study presents a comprehensive multiscale investigation of sandwich beams using a polyurethane foam (PUF) core. The research adopts a two-step approach. First, a micromechanical numerical characterisation and then a macroscale analysis were performed. * Micromechanical characterisation: The bulk PUF was modified by incorporating circular holes, enabling the creation of PUFs with varying volume fractions. Subsequently, a homogenisation technique, coupled with finite element methods (FEMs) and two distinct natural neighbour radial point interpolation methods (NNRPIMs), was employed to determine the homogenised mechanical properties of the PUF with respect to its volume fraction. The NNRPIM formulations differed in the consideration of neighbouring nodes within their influence domains: NNRPIMv1 uses only first natural neighbours, while NNRPIMv2 incorporated both first and second natural neighbours. The results demonstrated that both NNRPIM formulations delivered solutions comparable to high-order FEM formulations (e.g., quadratic triangular elements, FEM-6n). However, to achieve variable fields exhibiting a smoothness level equivalent to FEM-6n, NNRPIM formulations require a significant increase in the number of integration points per quadrilateral sub-area integration cell. This resulted in a substantial rise in computational cost, ultimately compromising the overall numerical efficiency of the NNRPIM.