# Calculation of Guided Wave Dispersion Characteristics Using a Three-Transducer Measurement System
## 5 Finite Element Analysis  
A three-dimensional FE analysis was performed in Abaqus to simulate signals. While the previous analysis required the input of the dispersion curves to create the propagated signals, this FE analysis does not require prior knowledge of the dispersion characteristics. FE models were created simulating guided wave propagation at five different frequencies in an aluminium plate (1000 \(\times\) 1000 \(\times\) 3 mm). The transmitter and receivers were modelled as ideal point transducers. The transmitting point was placed at the centre of the plate, where the coordinate system was established. A force was applied in the x-direction to simulate a shear transducer. This generated S\({}_{0}\) and A\({}_{0}\) modes along the \(x\)-axis and the SH\({}_{0}\) mode along the \(y\)-axis. The input signal was a 5-cycle sine with a Hanning window at central frequencies of 60, 80, 100, 120 and 140 kHz. Multiple receiver points were located on the positive \(x\)-axis and positive \(y\)-axis from the centre of the plate to the edge every 1 cm, resulting in 50 receivers at each axis. The time and spatial discretization for the finite element model was established at 10\({}^{-2}\) us and 1 mm, respectively, in order to ensure convergence and to obey the Nyquist sample-rate criterion. The mesh was formed by C3D8 elements. The FE analysis software provides the results decoupled for each axis direction, so it is straightforward to evaluate each wave mode separately. In Figure 9, the results have been computed in cylindrical coordinates to show clearly the three wave modes of propagation at each axis. In Figure 9a, showing radial displacement, S\({}_{0}\) and A\({}_{0}\) modes are depicted, S\({}_{0}\) being the faster mode. In Figure 9b showing tangential displacement, the SH\({}_{0}\) mode propagates along the \(y\)-axis, and in Figure 9c, showing out-of-plane displacement, the A\({}_{0}\) mode is clearly present. Since the plate is relatively small, overlaps between wave modes and echoes from edges are produced, as can be seen in Figure 10. This is a particular problem at lower frequencies (60 and 80 kHz) due to the length of the pulses. Therefore, an additional optimisation step has been added to minimize the error introduced by the overlapping. The FE model has 50 receiver points, every 1 cm along the axis so the five least-overlapped signals are selected for analysis. The selection is performed by analyzing the amplitude at the beginning and end of the envelope of the wavepacket of interest; the  
Figure 9: Images of the finite elemenstimulation of the wave propagation in polar-coordinate instantaneous displacement is shown 76 us after excitation. (**a**) Radial displacement, (**b**) tangential displacement and (**c**) out-of-plane displacement. Scale bars in meters. signals with the lowest amplitude at those regions are selected. Then, the signal processing described in Section 3 is applied for all the pair combinations between the five selected receivers. Once the phase and group velocities are calculated for each combination, the velocities are averaged to give the final velocity estimates. Any outliers that are introduced during selection of the five signals are eliminated. As shown in Figure 11, the phase velocity matches very well with the theoretical values from Disperse(r). In the group velocity graph, the results correlate well with the theoretical velocities, having a noticeable variation in the S\({}_{0}\) value at 60 kHz. This variation occurs at 60 kHz due to the overlap between the S\({}_{0}\) mode and the A\({}_{0}\) mode, which is more pronounced for the longer pulses at this frequency. This error does not appear for the A\({}_{0}\) mode because the signals used for the analysis were out-of-plane displacements, where the S\({}_{0}\) mode is practically imperceptible. Note that the results improve as the frequency increases, since the wavelength decreases and mode separation increases.