# Location Detection and Numerical Simulation of Guided Wave Defects in Steel Pipes
## 3 Research Method
### Modeling and Analysis of Steel Pipe
#### 3.3.2 Simulation Analysis  
1. Excitation signal  
The Hanning window has a narrow main lobe width and good smoothness, which effectively reduces frequency domain leakage and ensures the accuracy of frequency domain analysis. The Hanning-windowed signal is a 5-period, 60 kHz sinusoidal signal, as depicted in Figure 6.  
Hanning window expression:  
\[\omega(n)=0.5-0.5 cos(\frac{2\pi n}{N-1})\]  
Among them, \(\omega(n)\) is the time domain value of the window function; \(n\) is the discrete sample point index; and \(N\) is the window length.  
Figure 4: Pipeline defect: (**a**) axial defect; (**b**) inner defect of circumferential weld.  
Figure 5: Flow chart of steel pipe modeling.  
Calculation method  
The ANSYS simulation uses the transient dynamics method. It is used to determine the time-varying displacement, strain, and stress of the structure under a combination of steady, transient, and harmonic loads. The total simulation time t = 0.0033 s satisfies the requirement that guided waves can propagate back and forth throughout the pipeline, ensuring that signals at all positions can be received. This setting is based on the excitation position and propagation speed. The simulation step size is d = 0.0000008 s, which complies with the Moser criterion. We ensured that the curve contains all signal characteristics, calculation speeds, and accuracy. TimeHist Postpro can be used to view the node displacement of the simulation duration of a single node or to extract multi-node data using command scripts for subsequent analysis.  
Four-point torsional excitation is applied to the axial defect model of the pipeline along its circumferential direction. Four-point opposing excitation is applied to the defect model in the circumferential weld of the pipeline along the circumferential direction of the pipeline. The excitations are positioned at the 3, 6, 9, and 12 o'clock positions on the end face of the pipeline.  
T-modal validation  
The guided wave component changes with distance and time (wave velocity and frequency).  
\[\phi=kx_{-}\omega t\]  
where \(k\) is the wave number, \(x\) is the propagation distance, and \(\omega\) is frequency.  
According to two-dimensional fast Fourier transform,  
\[C_{\text{p}}=\frac{\omega}{k},k=\frac{\omega}{C_{p}}=\frac{2\pi}{\lambda} \Rightarrow\frac{1}{x}\]  
At the 12 o'clock position, 501 points were sampled at intervals of 10 mm. FFT transformation was performed on the waveform at each point space position, and then, the frequency/wavenumber data obtained by secondary FFT was converted into frequency/phase velocity data, as shown in Figure 7. The conversion method is detailed in Equation (9).  
\[(\omega_{i},k_{i})\rightarrow\frac{\omega}{k}=\frac{2\pi f}{\frac{2\pi}{ \lambda}}=\lambda f=c\rightarrow(f_{i},C_{i}) \tag{9}\]  
Comparing Figure 1 with Figure 7, it can be seen that the T(0,1) mode is excited near the speed of 3000 m/s.  
L-modal validation  
The axial displacement nephogram of defects in the circumferential weld of pipeline is depicted in Figure 8.  
Figure 6: Hanning window modulation signal.  
Figure 8: Axial displacement nephogram of defects in circumferential weld at different times: (**a**) t = 0.5 ms; (**b**) t = 1 ms; (**c**) t = 1.5 ms; (**d**) t = 2 ms; (**e**) t = 2.5 ms; (**f**) t = 3 ms.  
Figure 7: 2 DFFT result chart.  
When t = 0.5 ms, the guided wave has not yet reached the defect location, and the ultrasonic guided wave propagates in a regular symmetrical mode at the circumferential nodes, as shown in Figure 8a. When t = 1 ms to t = 2 ms, guided waves pass through the circumferential weld. Between t = 2.5 ms and t = 3 ms, partial reflected and transmitted waves propagate in opposite directions. The displacement of transmitted waves is no longer irregular, resulting in asymmetric modes in mode conversion, as shown in Figure 8c-f. In the figure, the red color indicates the distribution of maximum displacement during guided wave propagation, while other colors represent decreasing displacement levels.