## 5 Numerical Simulation and Verification
### Simulation Investigation and Verification
#### 5.3.2 Investigation for Case 1  
The numerical model of the test Case 1 with parameters shown in Table 3 was established in ABAQUS/Explicit, as illustrated in Figure 10b. This reflector was set on the outer surface of the pipe. The C3D8R type elements with mesh size \(l=1\) mm were used in the model to ensure convergence and solution accuracy. Steps (1) to (3) were performed first. As illustrated in Figure 2, the size of the transducer elements is approximately 48\({}^{\circ}\), and the step is \(\eta=12^{\circ}\), the circumferential displacements (\(u\theta\)) were recorded and 30 sets of data were obtained after scanning along the pipe for 360\({}^{\circ}\). The received signals formed a 2D data matrix \(\mathbf{H}_{i7j}\). The upper envelopes of \(\mathbf{H}_{i7j}\) shown in Figure 11a were obtained by applying the Hilbert transform. Subsequently, the 3D representation of the scan results was illustrated in Figure 11b, and was constructed with the upper envelopes. In order to obtain the axial and circumferential positions of the reflector, the original scanned image was processed in order to improve the detection accuracy by applying a median filter, bilinear interpolation algorithm and binary processing; the enhanced 2D scanned image, smoothed 2D scanned image and enhanced 2D scanned binary image obtained are illustrated in Figure 11c,d respectively. The images were displayed as an unwrapped pipe, with the top and bottom of the images being at the same circumferential location, the vertical axis as the circumferential location, and the horizontal axis as the axial position. The axial and circumferential positions of the reflector are shown clearly in the images. Figure 11: _Cont._The end of the pipe and the incident signals are continuous from the bottom to the top in the scanned images, while the reflector is discontinuous and is located within a small circumferential range. To identify the reflector, steps (4) to (10) were further performed. Based on the wave structural characteristics of the torsional guided wave mode [29], the incident signal \(S_{i}(n)\) of the T(0,1) mode was generated by uniformly applying the excitation displacement of the Gabor pulse with parameters (\(\sigma\), \(\mu\), \(\theta_{in},f_{c}\)) = (\(3\times 10^{-5}\) s, \(1.5\times 10^{-5}\) s, \(\pi\)/2 rad, 64 kHz) depicted in Equation (10) on the excitation nodes. The centre frequency \(f_{c}\) = 64 kHz is below the cut-off frequency of the T(0,2) mode shown in Figure 12. Therefore, the mode conversion was not considered in this study. The reflected signals \(S_{r}(n)\) were obtained through the receiving nodes. Figure 11: Analysis results based on the proposed method: (**a**) the upper envelopes of the guided-wave simulation signals: (**b**) 3D representation of the reflector; (**c**) original 2D scanned image (**Up**), enhanced 2D scanned image (**Middle**) and smoothed 2D scanned image (**Down**); (**d**) enhanced 2D scanned binary image. The observation time is \(T=6\times 10^{-4}\) s. The dimensions of \(S_{i}(n)\) and \(S_{r}(n)\) are \(\Gamma_{I}=43\) and \(\Gamma_{r}=574\) respectively. The analysis step \(\gamma\) is equal to 1, so the total number \(M=531\). Through steps (6) and (7), the standard Euclidian distance curves \(\Psi_{1}\) and \(\Psi_{2}\) were calculated in MATLAB and shown in Figure 13a. The minimum values of the distance curves were \(D_{1}=33.75\) and \(D_{2}=34.11\), respectively. \(D_{1}<D_{2}\), therefore the front-edge of the reflector was a step down, indicating that the reflector was a defect feature. The value of the \(RICR\) was equal to 1.43 (\(RICR>1.05\)), thus the judgement is reliable. Finally, the enhanced 2D scanned binary image was updated and as shown in Figure 13b, the reflector was marked in red. The final Figure 13b presented the location information and character of the concerned reflector in an intuitive way. Figure 12: Dispersion curves of pipes illustrated in Table 1: (**a**) phase velocity curves; (**b**) group velocity curves.