## 4 Numerical Simulated Data  
To fully utilize the combination of LWs and machine learning methods, it is necessary to create a large dataset that can be used to train and test the algorithms. As there is a need for a large volume of data, creating these experimentally would be labor-intensive, costly, and difficult to obtain consistently. Therefore, the alternative of using simulation data from accurate and experimentally validated numerical models has considerable advantages. Herein, a model of an SLJ composed of two aluminum substrates bonded with the epoxy-modified Nagase T-836/R-810 was adopted. This adhesive was chosen as it has great potential for industrial applications, specifically in the aeronautical and automotive sectors. However, any other epoxy-based adhesive with similar mechanical properties can be used with the same methodology. One actuator was chosen to excite the LWs, and one sensor captured the SLJ response. The specimen dimensions and the positions of the actuator and sensor can be seen in Figure 6.  
Figure 6: Top and side view of the adhesive joint simulation setup with the position of the sensor and actuator and the overall dimensions of each part.  
The model was developed and simulated using the FEM in the software Abaqus. The two SLJ aluminum plates were modeled with a density \(\rho=2500\) kg/m\({}^{3}\), a Poisson's ratio of \(\
u=0.33\), and a Young's modulus of \(E=72.4\) GPa. Weak adhesion defects were simulated by creating a thin layer of adhesive where Young's modulus was reduced in comparison to the rest of the adhesive material. A total of 1000 cases were simulated with linearly distributed values between 600 and 270,000 kPa. For exciting the SLJ, a distributed discretized tangential force was applied to the substrate plane, as illustrated in Figure 7. A 5-cycle Hanning-windowed sinusoidal burst centered at the frequency of 100 kHz was adopted [35; 36; 37; 38]. The amplitude of the force was adjusted in accordance with the experimental observations, by successive trials. The explicit dynamic analysis was adopted, due to its computational efficiency and because the excitation is impulsive and the time-dependent reaction needs to be captured. The total simulation time was limited to 0.5 ms, which corresponds to enough time for the LW to interact with the weak adhesion defects and arrive at the sensor [39; 40]. Following a mesh convergence study, a mesh size of 1.5 mm with elements of the type C3D8R was utilized. This was performed to reduce the computational burden and, thus, the time necessary for each simulation, while still maintaining a small enough mesh to not cause any numerical divergence. This combination allowed a simulation time inferior to 20 min for each case.  
Due to the explicit dynamic analysis carried out in Abaqus, the time series generated presented varying lengths, as each simulation was performed without a fixed time step. Thus, it was necessary to interpolate the data to have a fixed number of points. The value of 5000 points was chosen, this being enough to represent the LW signals well while keeping the amount of data limited.  
It should be noted that the use of only one actuator/sensor pair on the SLJ is the minimum necessary to detect damage. However, it does not allow for damage localization. Indeed, this work focused only on determining the level of weak adhesion in an SLJ of reduced dimensions. For localizing damage in larger parts, extra sensors are needed, along with some triangulation methods for data processing.