# Delamination Localization in Multilayered CFRP Panel Based on Reconstruction of Guided Wave Modes
## 2 Materials and Methods
### Numerical Simulation of Guided Waves  
Numerous mock-up trials are necessary for the experimental evaluation of each GW propagation scenario for various inspection setups, different types of flaws, loading, and environmental conditions. Numerical simulation of guided waves is an effective method to understand the propagation of GW and the interaction of various modes with delamination in composite materials.  
In this 2D model, the \(x\)-axis represents the length of the plate, the \(y\)-axis represents the thickness of the plate, and the \(z\)-axis is infinitely long. The velocity of the guided wave varies due to anisotropic properties and ply direction. Numerical simulation using a 3D  
Figure 7: Generated excitation signal for the analytical model.  
model could model guided wave propagation in all directions. However, 20 elements per wavelength criterion in 3D modeling of ultrasonic GW propagation would consume a significant amount of time and computational resources.  
Therefore, in this research, a numerical simulation of GW propagation was performed using the 2D plane strain model. The simulation involved 3750 time steps, with each step lasting 0.08 s. The duration of the propagation of the GW in the simulation was 300 \(\upmu\)s. In the finite element method (FEM), the mesh size and shape are key features. The simulation was performed using CPE4R-type elements, which are quadrilateral elements specifically designed for plain-strain models. These elements offer faster hourglass control and faster integration, resulting in reduced mesh deformation and a finer mesh.  
The accuracy of the GW simulation depends on the mesh size. A finer mesh gives accurate results but increases computation time. Twenty elements per wavelength (\(\lambda\)) criterion was selected to maintain the balance between accuracy and computation time. Therefore, the mesh size was 0.6 mm.  
By applying normal force to the excitation zone, the fundamental asymmetrical (A\({}_{0}\)) mode was excited. The input signal was a three-cycle sine wave with a driving frequency of 150 kHz. The selected frequency provides an optimal balance between dispersion, attenuation, and mode complexity. To extract the A\({}_{0}\) mode, signals were recorded on the plate surface at every 0.6 mm, and the corresponding B-scan data were obtained. The B-scan data were then subjected to a 2D fast Fourier transform to determine the phase velocity at each frequency, as shown in Figure 8.  
The dispersion characteristics of the guided waves in the CFRP plate were obtained using a 2D fast Fourier transform. The propagation of guided waves along the plate is represented by \(u(x,t)\) in terms of distance and time and was transformed into the wavenumber, \(k\), at each frequency using the 2D FFT method [59]:  
\[H(k,f)=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}u(x,t)\mathrm{e}^{- \mathrm{i}(kx+\omega t)}\mathrm{d}x\mathrm{d}t \tag{5}\]  
where \(x\) is the distance, \(t\) is the time, \(\omega\) is the angular frequency, and \(k\) is the wavenumber.  
Figure 8: The fundamental A\({}_{0}\) mode phase velocity dispersion.