## 7 Numerical Results
### Convergence Analysis  
A flat panel is considered in order to perform the convergence analysis. The length of each edge of the plate is equal to \(L_{x}=L_{y}=2\)m, whereas the thickness is given by \(h=0.1\)m. Two different lamination schemes are taken into account in order to prove the capability of the current approach to deal with different mechanical configurations. In particular, an isotropic plate made of Aluminum is \begin{table}
\begin{tabular}{c c} \hline \hline
**Material** & **Mechanical properties** \\ \hline Ceramic Foam & \(E=0.232\) GPa, \(\
u=0.2\), \(\rho=320\) kg/m\({}^{3}\) \\ Aluminum & \(E=70\) GPa, \(\
u=0.3\), \(\rho=2707\) kg/m\({}^{3}\) \\ Graphite-Epoxy & \(E_{1}=137.9\) GPa, \(E_{2}=E_{3}=8.96\) GPa, \(G_{12}=G_{13}=7.1\) GPa, \(G_{23}=6.21\) GPa, \(v_{12}=v_{13}=0.3\), \(v_{23}=0.49\), \(\rho=1450\) kg/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mechanical properties of the materials employed in the numerical analyses. considered firstly. Then, the same structure is taken with (90/0/90/0/90) as lamination scheme. In this circumstance, each ply has the same thickness and is made of the same material (Graphite-Epoxy). For conciseness purposes, the mechanical properties of the materials employed in this section are summarized in Table 2. As far as the boundary conditions are concerned, the structure under consideration is fully clamped along each edge (CCCC). Figure 3 shows the convergence characteristics of the first three natural frequencies as a function of the total number of grid points \(I_{N}=I_{M}\) varying the grid distribution. It can be easily noticed that the current approach reaches the maximum level of convergence with a number of point equal to \(I_{N}=I_{M}=11\), for both the two lamination schemes. In a similar manner, the various grids lead to similar tendency. It should be noticed that the graphs depicted in Figure 3 are related to the \(\text{ELD1}_{RS}^{X=1.2}\) model. This means that both the reduced elastic coefficients and the shear correction factor (\(\chi=1.2\)) are employed. Figure 3: Convergence characteristics of the first three frequencies for a completely clamped (CCCC) square plate (\(L_{x}=L_{y}=2\)m) of thickness \(h=0.1\)m for two different lamination schemes as a function of the number of grid points \(I_{N}=I_{M}\), varying the grid distribution. The ELW theory is the \(\text{ELD1}_{RS}^{X=1.2}\) in both the circumstances. (**a**) and (**b**) are related to the first frequency, (**c**) and (**d**) to the second one and (**e**) and (**f**) to the third one. The same graphs could be achieved also for higher-order theories, whose results are just summarized in Table 3 (for the isotropic case) and in Table 4 (for the laminated one) for conciseness purposes. It should be noticed that the natural frequencies presented in Tables 3 and 4 are obtained considering the Cheb-Gau-Lob grid distribution with \(I_{N}=I_{M}=21\). In addition, the 3D-FEM solution (9000 brick elements \"Hexa20\") is also included in these tables. The GDQ solution is evidently in good agreement with the one used as a reference. For the sake of completeness, the analysis is carried out considering HSDTs with and without the Murakami's function for the laminated composite plate (Table 4). In this circumstance, since the mechanical properties of each layer are the same, the benefits introduced by embedding this function are negligible. It should be noticed also that the reduced elastic coefficients are used only for a first order kinematic expansion (\(N=1\)). On the other hand, a constant value of the shear correction factor equal to \(\chi=1.2\) is used for both the first-order and second order theories (\(N=1,2\)) due to the reasons illustrated in the previous sections. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(f\)[Hz] & \(\mathbf{ELD_{RS}^{\mathbf{\chi=1.2}}}\) & \(\mathbf{ELD2^{\mathbf{\chi=1.2}}}\) & \(\mathbf{ELD3}\) & \(\mathbf{ELD4}\) & \(\mathbf{3D-FEM}\) \\ \hline  
\hline \hline \end{tabular}
\end{table}