## 4 Results and Discussion
### Spectral Elements  
In this case, the dispersion curves have been generated with a six-order spectral element with the nodes located at the GLL points in the material. The results have been compared with the isoparametric elements and with the analytical solution. The phase velocity and group velocity dispersion curves generated with a spectral element are presented in Figures 5 and 6 for modes S and A. The curves generated with spectral elements are in good agreement with those generated with isoparametric elements and analytically. The performance, computing time and accuracy of the spectral approach for a six-order element is preserved even when a 50 mm-thick plate is considered, as shown in Figure 7a,b. As a result, it can be concluded that the spectral elements showed a fast convergence, leading to excellent results with good accuracy. To illustrate numerical results for the spectral element approach, one six-order spectral element is required for obtaining convergence in the S1 mode estimation. Selecting the appropriate element order is fundamental, since the benefit of increasing the element order is imperceptible when convergence is achieved. These results reveal a comparable performance in the dispersion curves' estimation when fewer nodes per wavelength in the spectral scheme are implemented, compared with the isoparametric elements approach.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Number of Elements (\(n\))** & **S0 at** & **S0 at** & **S1 at** & **S1 at** & **S2 at** & **S2 at** & **A0 at** & **A0 at** \(f=\) & **A1 at** & **A1 at** \\  & \(f=\) **400** & \(f=\) **800** & \(f=\) **1000** & \(f=\) **1200** & \(f=\) **1200** & \(f=\) **1400** & \(f=\) **400** & **800** & \(f=\) **1000** & \(f=\) **1400** \\ \hline
6 & 5258.14 & 4193.82 & 6527.95 & 5933.52 & 1673.7626 & 1080.500 & 2447.16 & 2771.13 & 6118.18 & 4848.82 \\ Percentage of variation & 0.46 & 0.26 & 32.5 & 16.5 & 1.69 & 1.88 & 0.16 & 0.05 & 0.08 & 0.41 \\
28 & 5258.13 & 4192.55 & 6524.64 & 5931.94 & 16,672.32 & 10,053.50 & 2446.88 & 2770.32 & 61,160.62 & 4836.97 \\ Percentage of variation & 0.46 & 0.23 & 3.30 & 1.67 & 2.06 & 2.14 & 0.17 & 0.05 & 0.11 & 0.26 \\ \hline Order (\(p\)) & & & & & & & & & & \\ \hline
4 & 5358.32 & 4316.36 & 7020.75 & 6178.83 & 16,076.90 & 10,204.81 & 2476.76 & 2807.94 & 6271.53 & 4979.68 \\ Percentage of variation & 1.44 & 3.19 & 4.06 & 2.42 & 5.56 & 0.67 & 1.05 & 1.28 & 2.43 & 3.22 \\
6 & 5358.31 & 4312.78 & 6981.31 & 6143.07 & 18,798.51 & 10,724.92 & 2476.47 & 2805.44 & 6261.02 & 4952.14 \\ Percentage of variation & 1.43 & 3.11 & 3.47 & 1.83 & 4.30 & 4.39 & 1.04 & 1.19 & 2.26 & 2.65 \\
8 & 5358.36 & 4312.78 & 6981.25 & 6142.48 & 18842 & 10,740.53 & 2476.47 & 2805.44 & 6260.99 & 4952.05 \\ Percentage of variation & 1.44 & 3.11 & 3.47 & 1.82 & 4.54 & 4.54 & 1.04 & 1.19 & 2.26 & 2.65 \\
12 & 5358.36 & 4312.78 & 6981.25 & 6142.48 & 18,842 & 10,740.50 & 2476.47 & 2805.44 & 6260.99 & 4952.05 \\ Percentage of variation & 1.44 & 3.11 & 3.47 & 1.82 & 4.54 & 4.54 & 1.04 & 1.19 & 2.26 & 2.65 \\ Analytical & 5282.51 & 4182.90 & 6747.02 & 6032.88 & 18,023.09 & 10,273.77 & 2450.99 & 2772.41 & 6122.81 & 4824.21 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Percentage of variation of Cp (m/s) for modes S0, S1, A0 and A1 for specific frequencies as a function of the number of elements n. Frequency \(f\) in kHz.  
Figure 6: Group velocity dispersion curves calculated by SBFEM with isoparametric and spectral elements for a 3 mm-thick isotropic plate. Symmetric \(Sn\) and antisymmetric \(An\) (\(n=0,\,1,\,2,\dots\)) modes.