## 4 Results and Discussion
### Quantitative Evaluation of SBFEM Method  
Once the dispersion curves are generated, a quantitative study of convergence is performed for the SBFEM method with isoparametric and spectral elements. The discretization is set up as described in Figure 1. The phase velocities are presented in Figure 8 for wave mode S1 and a plate thicknesses t between 1 mm and 100 mm. Figure 8 shows the phase velocity estimation for different nodes per wavelength, where the curves converge for six nodes per wavelength (\(\delta\)) for all studied plate thicknesses. The same convergence is evident for high-frequency modes, such as the S3 mode in a 1 mm-thick plate.  
On the other hand, in Figure 9, the plate thickness is independent of the order of interpolation for spectral elements greater than six for the phase velocity estimations. The  
Figure 8: Number of nodes per wavelength required for convergence with plate thicknesses up to 100 mm.  
Figure 7: Dispersion curves given by the SBFEM method with isoparametric and spectral elements for a 50 mm-thick isotropic plate. (**a**) Phase velocity. (**b**) Group velocity. Symmetric \(Sn\) and antisymmetric \(An\) (\(n=0,1,2,\dots\)) modes.  
same behavior is observed for the rest of the dispersion curves. In the same figure, a higher slope for orders less than p = 4 is seen. Based on the previous results, the spectral discretization is independent of the plate thickness in terms of computational time and accuracy. For lower-order modes, convergence is achieved with fewer nodes. The computational efficiency is improved by implementing the spectral elements, which is much more noticeable with increasing plate thickness. Once the method presents convergence, the value obtained does not present modifications due to the increase in the interpolation order.  
The combined results of Figures 9 and 10 provide an appropriate basis for formulating a guideline to generate an efficient discretization. In all cases in this work, a selection of more than six nodes per wavelength, or an interpolation order greater than eight, are adequate to generate the dispersion curves. However, in the present work, a relationship to determine the interpolation order as a function of frequency for higher-order modes is not included because, in practical terms, integrity evaluation by guided waves is always preferred to launch the lowest-mode waves, due to the possibility of multimode activation at high frequencies.  
Figure 10 summarizes the behavior of the semi-analytical and analytical methods studied in this work, where the computational time is related to the thickness of the plate in the case of the semi-analytical approach. The two approaches were tested on a PC (Intel Ci7-6820 HQ CPU at 2.7 GHz, 8 GB RAM) with one stepwise increment of k for the SBFEM approaches. As expected, as the thickness increases in the SBFEM isoparametric scheme, the number of elements increases, increasing the running time for the calculation of the dispersion curve. The minimum number of isoparametric elements required for the calculation of the dispersion curves in this study is six nodes per wavelength, which corresponds to an optimized value of the number of elements. The numerical results show a quadratic relationship between the number of elements and the running time, as shown in Figure 10.  
On the other hand, the spectral elements have a good convergence, which is independent of the plate thickness. The computational time is the same across the range of thicknesses studied.  
Since our objective is to investigate the performance of the SBFEM method in capturing the wave dynamics, the spectral elements within the Gaussian-Lobatto-Legendre  
Figure 9: Comparison of the convergence curves for spectral elements of different orders of interpolation p and thicknesses between 1 mm and 100 mm.  
quadrature yield achievable computational times for the practical range of plate thicknesses, making this a recommendable option.