## 3 Results
### Optimization Results  
Material properties for the four considered materials are determined through Equation (15). The Nelder-Mead zero-order algorithm has been considered for the optimization process [22]. As the algorithm can fail in reaching the optimal solution, several repetitions of the optimizations have been performed. In particular, the result of an optimization process has been used as a starting point for the next. After few iterations, the algorithm was able to reach the optimal solution. Table 8 reports the elastic constants calculated through Equation (15) for each analyzed material. The difficulty in reaching an optimal solution was mainly due to the limited dependency of the first resonance frequency on the Poisson's coefficient \(\
u_{12}\) and on the shear modulus \(G_{12}\). Indeed, the first resonant frequency of a clamped composite plate particularly depends on the longitudinal and transverse Young's moduli. In this regard, the rectangular shape of the analyzed region enhances the sensitivity of the first resonant frequency on the elastic properties in the direction of the short edge of the rectangle, i.e., the transverse direction [13]. Indeed, the dependency of the inverse problem on the shear modulus and on the Poisson's coefficient represents a critical aspect, particularly when the elastic properties are calculated considering only the first resonant plate without disposing the mass load. In this case, limits on the design domain are necessary. Instead, by considering further information related to the modal shape, the optimization is able to catch all the four elastic constants without the need of limiting the domain of the elastic properties. This is particularly useful when damage level, i.e., residual elastic properties, are to be assessed. Indeed, in order to guarantee the robustness of the methodology, limits on the design domain of the elastic properties, i.e., \(E_{11},E_{22},G_{12}\), and \(\
u_{12}\), were considered in [13]. This is particularly undesirable if the methodology has to be adopted for detecting local material variations. The results of the optimization problem can be constrained by the limits of the design domain, i.e., the space where the aforementioned material properties can vary, and, in the end, be wrong. Furthermore, as shown in Table 8, even though the objective function of Equation (15) is equal to zero, very limited discrepancies from the nominal values are obtained. Only the shear modulus and the Poisson's ratio are slightly different from the \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(\mathbf{E}_{11}\) [GPa] & \(\mathbf{E}_{22}\) [GPa] & \(\mathbf{G}_{12}\) [GPa] & \(\mathbf{\
u}_{12}\) [-] \\ \hline Unidirectional & & & & \\ \([0^{\circ},0^{\circ},0^{\circ},0^{\circ}]\) & 131.72 & 8.56 & 6.74 & 0.29 \\ Symmetric & & & & \\ \([30^{\circ},-30^{\circ},-30^{\circ},30^{\circ}]\) & 131.7 & 8.55 & 6.75 & 0.29 \\ Anti-symmetric & & & & \\ \([30^{\circ},-30^{\circ},30^{\circ},-30^{\circ}]\) & 131.7 & 8.55 & 6.75 & 0.30 \\ Woven fabric & 59.0 & 59.0 & 3.4 & 0.04 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Optimized material properties at pyle level for each material. nominal values of Tables 1 and 2, with discrepancies lower than 0.3% for the shear modulus and 3% for the Poisson's ratio. By way of example, the optimization has been also performed without retaining the modal shape information for the symmetric and anti-symmetric stacked laminates. The results are reported in Table 9 with relative errors with respect to the elastic properties of Table 1. It is worth noting that even though the resultant first resonant frequencies without mass load for the different material orientations are those reported in the first line of Tables 5 and 6, respectively, the relative errors can be very high. This particularly shows that the inverse problem can be undetermined, i.e., multiple solutions are allowable.