# Adhesive Single-Lap Joint Evaluation Using Ultrasound Guided Waves
## 4 Results
### Guided Wave Propagation Simulation  
The k-Wave function used for the simulation is pstdElastic3D, based on the pseudo spectral time domain (PSTD) method, which is able to simulate the time-domain propagation of elastic waves through a three-dimensional homogeneous or heterogeneous medium [68]. The simulation model presented in Section 3.1 was implemented on a computer with an Intel(r) Core(tm) i7-4790 processor and 16 GB RAM, and a Nvidea GeForce GTX1080 Ti GPU, 11.26 GB RAM. For the available hardware and the presented geometry, the maximum resolution (minimum grid point spacing) was limited to 150 \(\upmu\)m. This resolution gave rise to 21.47 \(\times\) 10\({}^{6}\) voxels (840 \(\times\) 213 \(\times\) 120). The source was excited with a three-cycle burst at 500 kHz central frequency, as shown in Figure 8a. Figure 8b illustrates the respective spectrum. For model validation, first the propagation in an aluminum single plate was tested. For that purpose, the theoretical phase velocity extracted from Figure 3 for 500 kHz was used in Equation (9). The resulting incident angle for S0 mode, which is the same as the receiving angle, is 18.2\({}^{\circ}\). The signal collected at the receiver is presented in Figure 9. It is observed that there is an increase in the pulse length due to the dispersion behavior of the S0 mode phase velocity, when compared with the original three-cycle excitation burst (Figure 8a). Besides the direct signal propagated between the transmitter and the receiver, there is also an interference signal corresponding to the round trip in the water path at the emitter. To avoid that interference, the distance between the emitter transducer and the plate should be increased; however, such an increase was not possible in the present simulation due to  
Figure 7: Displacements of the four existing modes in the bonded region: (**a**) S0; (**b**) A0; (**c**) S1; (**d**) A1. limited computational resources. Nevertheless, this fact has not prevented the peak-to-peak amplitude value from being correctly evaluated. To prove that the excited propagation mode is indeed S0, the phase velocity \(c_{p}\) was calculated through the phase spectrum approach [66], as  
\[c_{p}=\frac{2\pi fL}{\Delta\varphi}. \tag{11}\]  
where \(f\) is the frequency, \(L\) is the path difference between two collected signals, and \(\Delta\varphi\) is the respective difference in the phase spectrum. Figure 10 illustrates the phase velocity obtained by this method for \(L=20\) mm. There is a clear agreement between the presented phase velocity in Figure 10 and the theoretical one shown in Figure 3.  
Figure 8: Excitation signal: (a) time domain; (b) frequency domain. Figure 9: Received signal after propagation in the bonded plate. Comparing the peak-to-peak amplitudes of the two above-mentioned signals, the attenuation was also computed, resulting in 27.5 Np/m. Again, this value is in full agreement with the theoretical one shown in Figure 4.  
The same methodology was used to simulate the propagation of S0 mode in the aluminum-bonded plates presented in Figure 1.  
Figure 11 illustrates snapshots of the normalized normal and shear stresses in the XZ plane during the simulation. The emitter transducer is represented by a dashed line on the left. It is clear from the high stress values inside the plates. It is also evident from the energy leakage surrounding the plates and the absence of shear stress in the water. The received signal amplitudes corresponding to the propagation in a single plate and in the bonded plates obtained using the same test conditions were compared. An amplitude decrease of about 10% was observed for the bonded plates compared with the single plate, which can be justified by the higher attenuation values in the bonded region when compared to the attenuation of S0 mode in a single plate.