# Propagation, Scattering and Defect Detection in a Circular Edge with Quasi-Edge Waves
## 2 Numerical Study
### Three-Dimensional FE Model for Investigating Propagation Characteristics of \(QES_{0}\)  
The propagation of \(QES_{0}\) along circular edges was modelled using ABAQUS 2017, and the simulations were performed with the Explicit solver. The configuration of 3D FE models is schematically illustrated in Figure 1. The height and wall thickness were fixed at 200 and 5 mm, respectively. To investigate the effect of curvature on the propagation of \(QES_{0}\), FE models were modelled in various radius-to-thickness (R/T) ratios, ranging from 10 to 25 in steps of 5, and from 25 to 100 in steps of 25. It should be noted that the curvature of the edge is inversely proportional to the R/T ratio. The FE models were meshed with eight-node three-dimensional solid elements with reduced integration (C3D8R), and the meshing technique option was'structure', similar to that in previous studies [(20; 23)]. Due to the non-/minor dispersive nature of EGWs, the wave speeds were very similar in the interested \(FTV\) (frequency) range. Hence, the mesh size was selected as 1 mm\({}^{3}\). This mesh size ensures that at least 15 elements exist  
Figure 1: Schematic of the configuration of 3D FE models. per wavelength for the highest frequency of interest (300 kHz or \(FTV\) = 3). The time steps of simulations were automatically determined by the Explicit solver, which employed the central difference method. To maximise the transfer of input energy to the \(QES_{0}\) mode while minimising the influence of other generated wave modes, wedge excitation was used. The angel of the wedge was determined by [PERSON]'s law, as follows: \(\theta_{\text{wedge}}=\sin^{-1}\left(C_{\text{L(wedge}}/C_{\text{EW}}\right)\), where \(\theta_{\text{wedge}}\) is the wedge angle, \(C_{\text{L(wedge}}\) is the longitudinal wave speed of the wedge material and \(C_{\text{EW}}\) is the wave speed of the excited EGW mode. In this study, the longitudinal wave speed of the wedge is around 2300 m/s, and the wave speed of \(ES_{0}\) is about 2870 m/s. Hence, the wedge angle is determined as 53\({}^{\circ}\). The interaction between the wedge and specimen was defined as a surface-based 'Tie' constraint to enable the transmission of input energy to the specimen. The incident wave in the wedge was generated by applying a pressure of 10 MPa perpendicular to the circular excitation area, which is located on the wedge slope. The excitation area has a diameter of 22 mm (the same size as the actual transducer used in the experimental study). The excitation signals are Hanning-window-modulated eight-cycle sinusoidal tone bursts. The material properties of different components of the models are listed in Table 1.  
Figure 2a,b show the wedge excitation and the propagation of \(QES_{0}\) in a curved edge, respectively. Most of the input energy was concentrated and propagated along the curved edge in the clockwise direction. Meanwhile, a backward EGW, which has a much smaller magnitude, was generated and propagated along the anticlockwise direction. Since wave motion can be largely affected by the geometric of the propagation medium, it is necessary to separate the \(QES_{0}\) wave mode from other guided and bulk wave modes. To identify the wave mode of generated and measured signals, a frequency-wavenumber analysis was conducted using a two-dimensional fast Fourier transform (2D FFT). The simulation results obtained from the FE model with an R/T ratio of 10, which is the smallest R/T ratio of interest, were investigated. The normal displacement of the top edge (\(U_{z}\)) was extracted from 35 continuous nodes (starting from measurement point 1) along the wave propagation direction, with intervals of around 1 mm.