## 5 Simulations of UGW Testing  
Comsol Multiphysics software was used to simulate the UGW testing, and the pipeline model was built, as illustrated in Figure 11. The pipe length is 5.5 m, the outer diameter is 48 mm, and the thickness is 5 mm. Defect 1 is 2.7 m away from the excitation/reception end, and the spacing among defects is 0.6 m. The pipe material is low-carbon steel, and the density, Young's modulus and Poisson's ratio are 7.89 g/cm\({}^{3}\), 200 GPa and 0.27, respectively. The circumferential angle of the defect is \(\alpha\), and the depth and width of the defect are both 1.5 mm. At the excitation/reception end, 65 points were taken at r = 19 mm, 21.5 mm and 24 mm (r is the radius of the pipe) at equal distances along the circumference, respectively, of which the sinusoidal signals modulated by a 5-period Hanning window and the wave structure of L(0,2) UGW at 110 kHz were input to each excitation point from axial, circumferential and radial directions, and 65 points along the circumferential direction of r = 24 mm were taken as the receiving points. The simulated UGW signals with 1 defect (\(\alpha\) = 300\({}^{\circ}\)) are shown in Figure 12. It can be seen from the signals in the time domain that the direct and end-reflected echoes with great amplitudes are obvious, and the resolution of defect is weak because of the influence of coherent noise, especially the circumferential signals. It can be observed from the signals in the frequency domain that F(1,3) and F(2,3) UGWs arise in the axial and radial directions when L(0,2) UGW encounters the asymmetric defect, whereas the F(1,3), F(2,3) and F(3,3) UGWs appear in the circumferential direction. This may be because the biggest difference among L(0,2) and F(\(n\),3) UGWs is the circumferential displacement. Considering that L(0,2) UGW is dominated by the axial displacement, the simulated UGW signals below are represented by the axial displacement signals. Figure 11: Schematic diagram of the pipeline and defects. Figure 10: Optimized synthesized UGW signals processed by FBR-RC-SSP. The simulated UGW signals with a multi-defect (\(\alpha=300^{\circ}\)) are shown in Figure 13. It is predictable that the L(0,2) UGW undergoes multiple mode conversions at defects when there are multiple defects in the pipeline, resulting in more coherent noise in the signal. Moreover, the amplitude of the defect echo may decrease because of the increase in the defect number with the consistent excitation energy. It is possible to confuse the defect signals with coherent noise and even give the wrong detection. In addition, it can also be observed that the chirplet transform shows the limitation of distinguishing defects when there are three defects; this is true for frequency domain signals obtained by the chirplet transform with high resolution, not to mention the time domain signals. Comsol Multiphysics software was used to simulate the UGW signals with different number and circumferential angle \(\alpha\), the data of 65 receiving points were averaged and the results are illustrated in Figure 14, where 3-300\({}^{\circ}\) indicates that there are three defects and \(\alpha=300^{\circ}\). It can be seen that the amplitudes of the defect echo and coherent noise become big as \(\alpha\) increases from 60\({}^{\circ}\) to 300\({}^{\circ}\), which is caused by the greater energy reflected from the defect with a larger cross-sectional area loss. In addition, it is similar to Figure 13 in that the defect echoes are polluted by the coherent noise when there are three defects; so, the signal interpretation accuracy decreases. Figure 12: Time-frequency diagrams of simulated UGW signals (axial, circumferential and radial) with defect 1 (\(\alpha=300^{\circ}\)). Figure 13: Time-frequency diagrams of simulated UGW signals (axial, circumferential and radial) with multi-defect (\(\alpha=300^{\circ}\)).