Location Detection and Numerical Simulation of Guided Wave Defects in Steel Pipes

Abstract

At present, researchers in the field of pipeline inspection focus on pipe wall defects while neglecting pipeline defects in special situations such as welds. This poses a threat to the safe operation of projects. In this paper, a multi-node fusion and modal projection algorithm of steel pipes based on guided wave technology is proposed. Through an ANSYS numerical simulation, research is conducted to achieve the identification, localization, and quantification of axial cracks on the surface of straight pipelines and internal cracks in circumferential welds. The propagation characteristics and vibration law of ultrasonic guided waves are theoretically solved by the semi-analytical finite element method in the pipeline. The model section is discretized in one-dimensional polar coordinates to obtain the dispersion curve of the steel pipe. The T(0,1) mode, which is modulated by the Hanning window, is selected to simulate the axial crack of the pipeline and the L(0,2) mode to simulate the crack in the weld, and the correctness of the dispersion curve is verified. The results show that the T(0,1) and L(0,2) modes are successfully excited, and they are sensitive to axial and circumferential cracks. The time–frequency diagram of wavelet transform and the time domain diagram of the crack signal of Hilbert transform are used to identify the echo signal. The first wave packet peak point and group velocity are used to locate the crack. The pure signal of the crack is extracted from the simulation data, and the variation law between the reflection coefficient and the circumferential and radial dimensions of the defect is calculated to evaluate the size of the defect. This provides a new and feasible method for steel pipe defect detection.

1. Introduction

Steel pipe pipelines are an indispensable part of production and life, but internal corrosion, cracks, and welding defects endanger the safety of these pipelines. Ultrasonic guided waves (UGWs) have the advantages of long detection distances and cover the whole cross-sectional area of the pipeline body. They are widely used in pipeline defect detection and cracked pipeline location [1,2,3,4,5]. However, due to the multimode and dispersion characteristics of ultrasonic guided waves and the complexity of welding structure, it is more difficult to detect weld defects by guided waves. At present, many scholars have made many achievements in theoretical research, numerical simulation, and engineering applications for weld defect detection. Duan and Kirby [6] used a semi-analytical finite element method to solve the dispersion curve of a pipeline and analyze a situation in which the number of modes increases due to a high detection frequency. References [7,8,9] established the T(0,1) mode to locate single crack defects, He et al. [10] located defects in a straight pipe structure through the L(0,2) mode, and Carlos Quiterio G M et al. [11] used the Hilbert transform and automatic peak detection algorithm to locate the pipeline notch. However, none of these studies analyzed positioning accuracy. Wang et al. [12] extracted mode L(0,1) and mode L(0,2) from a reflected signal based on the mode shape to identify defects, Yu et al. [13] used superposition technology to enhance the amplitude of modes to detect defects, and Shah Jay et al. [14] evaluated the degree of crack growth based on the energy transmission index of the T(0,1) mode, but none of them determine the location of cracks. Wu et al. [15] expanded the detection range of defects in pipelines through time shift window transformation, which has the problem of frequency domain leakage. Zhang et al. [16] used the sensitivity of torsional bending-guided waves to detect axial cracks and quantitatively analyzed the circumferential cracks of pipelines, but they did not analyze radial cracks. Shah Jay et al. [17] detected defects with different radial depths in the welding zone without considering the circumferential length cracks. Jiang et al. [18] used the COMSOL Multiphysics software to analyze the axial error of pipelines and obtain the axial positioning error of defects, but they did not evaluate the size of the defects. Chua et al. [19] studied the interaction between the T(0,1) mode and circumferential cracks in welds. They concluded that circumferential cracks with larger circumferential ranges and deeper profiles will produce a great change in the amplitude of the reflected torsional T(0,1) signal, but the relationship between the axial and radial directions and torsional guided waves was not further analyzed. Yang et al. [20] proposed an adjustable Q-factor wavelet transform algorithm to solve the difficulty in extracting the echo characteristics of weld defects, but they failed to locate and evaluate the weld defects. Ye et al. [21] fitted and analyzed weld data using a polynomial model and an expectation maximization algorithm, and they proposed a weld classification method based on the polynomial model, which can calculate weld width and height. Li et al. [22] adjusted the AlexNet model according to the characteristics of weld defects and adopted a deep learning algorithm. The detection success rate of AlexNet is as high as 97%, and it improves the accuracy of pipeline weld defect detection, but increasing the depth and width of the model leads to increases in calculation and model training times.

In this article, the semi-analytical finite element method is used to study the propagation law of guided waves in a pipeline. By establishing the T(0,1) mode and L(0,2) mode, the UGW detection method for pipeline axial defects and weld defects is studied using numerical simulation. Wavelet transform and Hilbert transform are used to locate the pipeline cracks accurately and analyze the error level accurately. The algorithm of multi-node fusion and modal projection is adopted for defects in pipeline welds, and the influence law of the defect reflection coefficient with the change in the circumferential and radial dimensions of defects is obtained through simulation.

2. Theoretical Basis

2.1. Semi-Analytic Finite Element Algorithm for Steel Pipes

The semi-analytical finite element method is a numerical calculation method which combines the finite element method and the semi-analytical method to solve the dispersion curve of UGWs in steel pipes. The displacement, strain, and stress vectors at any point in the pipeline are expressed as $\mathit{u}$, $\epsilon$, and $\mathit{\sigma }$.

According to the principle of virtual work, the weak form of the wave control equation of a free-boundary pipeline is as follows:

$${\displaystyle \underset{V}{\int }\delta {\mathit{u}}^{\mathrm{T}}}·\ddot{\mathit{u}}\mathrm{d}V+{\displaystyle \underset{V}{\int }\delta {\epsilon }^{\mathrm{T}}}·\mathit{\sigma }\mathrm{d}V=0$$

(1)

In the formula, $\mathit{u}$ is displacement; $\epsilon$ is strain; and $\mathit{\sigma }$ is stress, which can be expressed as

$$\left\{\begin{array}{c}\mathit{u}(r,\theta ,z,t)={(u_r\hspace{1em}{u}_{\theta }\hspace{1em}{u}_z)}^{\mathrm{T}}\\ \epsilon (r,\theta ,z,t)={({\epsilon }_r\hspace{1em}{\epsilon }_{\theta }\hspace{1em}{\epsilon }_z\hspace{1em}{\gamma }_{\theta z}\hspace{1em}{\gamma }_{rz}\hspace{1em}{\gamma }_{r\theta })}^{\mathrm{T}}\\ \mathit{\sigma }(r,\theta ,z,t)={({\sigma }_r\hspace{1em}{\sigma }_{\theta }\hspace{1em}{\sigma }_z\hspace{1em}{\tau }_{\theta z}\hspace{1em}{\tau }_{rz}\hspace{1em}{\tau }_{r\theta })}^{\mathrm{T}}\end{array}\right.$$

(2)

The stress and strain relationship is as follows:

$$\sigma =\mathit{C}\epsilon$$

In the formula, $\mathit{C}$ represents the material stiffness matrix.

The correlation between strain and displacement is as follows:

$$\left\{\begin{array}{l}{\epsilon }_r={\displaystyle \frac{\partial u_r}{\partial r}}\\ {\epsilon }_{\theta }={\displaystyle \frac{u_r}{r}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial u_{\theta }}{\partial \theta }}\\ {\epsilon }_z={\displaystyle \frac{\partial u_z}{\partial z}}\\ {\gamma }_{\theta z}={\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial u_z}{\partial \theta }}+{\displaystyle \frac{\partial u_{\theta }}{\partial z}}\\ {\gamma }_{rz}={\displaystyle \frac{\partial u_r}{\partial z}}+{\displaystyle \frac{\partial u_z}{\partial r}}\\ {\gamma }_{r\theta }={\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial u_r}{\partial \theta }}+{\displaystyle \frac{\partial u_{\theta }}{\partial r}}-{\displaystyle \frac{u_{\theta }}{r}}\end{array}\right.$$

(3)

The finite element method is used to discretize the radial direction of the pipeline. The number of node degrees of freedom at all units is three, and the node displacement u(r,θ,z,t) is as follows:

$$\mathit{u}\left(r,\theta ,z,t\right)=\mathit{N}\left(r\right){\mathit{U}}^j\exp \left(\mathrm{i}\left(kz+n\theta -\omega t\right)\right)$$

(4)

In the formula, the displacement vector at the element j node is expressed as ${\mathit{U}}^{\mathit{j}}$, $\mathit{N}\left(r\right)$: a node shape function.

For the first-order bar element, ${\mathit{U}}^{\mathit{j}}$ is the 6 × 1 column vector and $\mathit{N}\left(r\right)$ is a matrix with three rows and six columns.

Substituting (4) into (3), the strain vector is expressed as follows:

$$\epsilon =\left({\mathit{B}}_1+\mathrm{i}n{\mathit{B}}_2+\mathrm{i}k{\mathit{B}}_3\right){\mathit{U}}^j\exp \left(\mathrm{i}\left(kz+n\theta -\omega t\right)\right)$$

(5)

In the formula, B₁, B₂, and B₃ are all six rows and six columns with their parameters. Substitute Formulas (2), (4) and (5) into (1) to obtain the wave control equation of the unit j:

$$\left({\mathit{K}}_1^j+\mathrm{i}n{\mathit{K}}_2^j+\mathrm{i}k{\mathit{K}}_3^j+n^2{\mathit{K}}_4^j+nk{\mathit{K}}_5^j+k^2{\mathit{K}}_6^j-{\omega }^2{\mathit{M}}^j\right){\mathit{U}}^j=0$$

(6)

$$\left\{\begin{array}{l}{\mathit{K}}_1^j={\displaystyle \int {\mathit{B}}_1^{\mathrm{T}}\mathit{C}{\mathit{B}}_{1}r\mathrm{d}r {\mathit{K}}_2^j={\displaystyle \int \left({\mathit{B}}_1^{\mathrm{T}}\mathit{C}{\mathit{B}}_2-{\mathit{B}}_2^{\mathrm{T}}\mathit{C}{\mathit{B}}_1\right)r\mathrm{d}r}}\\ {\mathit{K}}_3^j={\displaystyle \int \left({\mathit{B}}_1^{\mathrm{T}}\mathit{C}{\mathit{B}}_3-{\mathit{B}}_3^{\mathrm{T}}\mathit{C}{\mathit{B}}_1\right)r\mathrm{d}r {\mathit{K}}_4^j={\displaystyle \int {\mathit{B}}_2^{\mathrm{T}}\mathit{C}{\mathit{B}}_{2}r\mathrm{d}r}}\\ {\mathit{K}}_5^j={\displaystyle \int \left({\mathit{B}}_2^{\mathrm{T}}\mathit{C}{\mathit{B}}_3+{\mathit{B}}_3^{\mathrm{T}}\mathit{C}{\mathit{B}}_2\right)r\mathrm{d}r {\mathit{K}}_6^j={\displaystyle \int {\mathit{B}}_3^{\mathrm{T}}\mathit{C}{\mathit{B}}_{3}r\mathrm{d}r}}\\ {\mathit{M}}^j={\displaystyle \int N^{\mathrm{T}}\rho Nr\mathrm{d}r}\end{array}\right.$$

(7)

Assemble the control equations of all elements in the r direction to obtain the pipeline model:

$$\left({\mathit{K}}_1+\mathrm{i}n\right.{\mathit{K}}_2+\mathrm{i}k{\mathit{K}}_3+n^2{\mathit{K}}_4+nk{\mathit{K}}_5+\left.k^2{\mathit{K}}_6-{\omega }^2\mathit{M}\right)\mathit{U}=0$$

(8)

In the formula, ${\mathit{K}}_1,{\mathit{K}}_2,{\mathit{K}}_3,{\mathit{K}}_4,{\mathit{K}}_5,{\mathit{K}}_6$ are the stiffness matrices; M is the mass matrix; the dimensions are N × N; and N is the degree of freedom of the nodes.

In the steel pipe pipeline, k is the wave number (a real number), and the dispersion characteristics and vibration characteristics are obtained from the eigenvalue of (8).

2.2. Multi-Node Fusion and Modal Projection Algorithm

In order to extract ultrasonic guided wave signals of different modes, m calculation nodes are evenly distributed around the pipeline, and the signal of each node is expressed as $S_i\left(t\right)$. The steps for extracting modal signals are as follows:

(1)

Signal preprocessing

The time domain signal $S_i\left(t\right)$ is transformed into the time–frequency domain using the wavelet transform algorithm. The signal components of different frequency bands are separated, which lays the foundation for subsequent modal extraction. The converted signal is expressed as $S_{i,\omega }\left(t\right)$.

(2)

Calculation of the modal basis function

Based on the properties of a steel pipe and the dispersion curve, the basis function $B_m\left(\theta \right)$ of each mode is determined, where m represents the mode number.

(3)

Modal projection

Using the basis function $B_m\left(\theta \right)$, the signal of each node is modal-projected, and the corresponding modal signal is extracted. The mth mode of signal projection is calculated on the modal basis function through angle integration, and the projection operation is defined as follows:

$$S_{m,i}(t)={\int }_0^{2\pi } S_{i,\omega }(t,\theta )B_m(\theta )d\theta$$

(4)

Multi-node signal fusion

The weighted average or weighted sum operation is performed on the modal projections of all node signals, and the global m-th modal signal $S_m\left(t\right)$ is obtained. The specific formula is as follows:

$$S_m(t)={\displaystyle \frac{1}{M}}\sum _{i=1}^M w_i{S}_{m,i}(t)$$

where $w_i$ is the weight factor of the sensor, which depends on the sensor position or signal-to-noise ratio.

(5)

Modal signal reconstruction

Based on the extracted modal signal $S_m\left(t\right)$, the original modal signal is reconstructed using back projection or deconvolution techniques, and the pure signals of each mode are reconstructed. The reconstructed signal can be expressed as follows:

$$S_m\left(t\right)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }} S_m\left(\omega \right)e^{j\omega t}d\omega$$

3. Research Method

3.1. Solving the Dispersion Curve of the Steel Pipe

A dispersion curve of the steel pipe was drawn based on the semi-analytical finite element method. Material parameters of the pipeline: Q235b. Specifications: diameter, 219 mm; wall thickness, 5.5 mm; elastic modulus, 2.1 × 10¹¹ Pa; Poisson’s ratio, 0.32; and density, 7850 kg/m³. The drawn dispersion curves of the group velocity and phase velocity are presented in Figure 1.

When the frequency was 0–60 KHz, a variety of modes were generated during the propagation of UGWs: axial symmetry longitudinal mode L(0,m), m = 1, 2, where m reflects the vibration shape in the thickness direction of the pipeline; axisymmetric torsional mode T(0,m), m = 1; non-axis symmetric bending mode F(n,m), n = 1, 2, 3; m = 1, 2, 3…, where n represents the circumferential order, reflecting the spiral propagation pattern of UGWs around the pipe wall.

The group velocity of the T(0,1) mode is relatively stable, and its dispersion curve shows no dispersion. Both the phase velocity curve and the group velocity curve are straight lines parallel to the horizontal axis. Compared with other modes, it can accurately carry important defect information, and its propagation is mainly in the circumferential direction. Therefore, the T(0,1) mode was selected for the numerical simulation of axial defects in the pipeline.

As the excitation frequency is 60 kHz, the displacement wave structures of the L(0,2) and L(0,1) modes are shown in Figure 2. The circular direction displacement of these two modes is 0, the displacement of L(0,2) guided waves is large in the axial direction of the in–outside wall surfaces of the pipeline, and the radial displacement is evenly distributed in the pipeline, sensitive to defects across the entire wall thickness direction, so the L(0,2) guided waves are suitable for detecting annular defects. The displacement of the L(0,1) mode is primarily radial, with larger radial displacements at both ends along the in–outside walls, while it is zero in the middle. The in–outside walls of the pipe have large axial displacement values, so the L(0,1) mode is suitable for detecting the defects of the in–outside surfaces of the pipe. The energy leakage of the UGW mode in the propagation process is related to its radial displacement distribution on the inner and outer surfaces, and the energy leakage of mode L(0,2) is less than that of mode L(0,1). It is easy to identify in many modes and mainly detects circumferential defects. Therefore, the L(0,2) mode ultrasonic guided wave is selected for detecting crack defects in the welds of circular pipes.

3.2. Ultrasonic Guided Wave Defect Detection Principle

The propagation of UGWs is shown in Figure 3. Mode T(0,1) and mode L(0,2) are used to detect the defects on the outer surface of the pipeline and cracks in the circumferential weld of the pipeline, respectively. During the propagation process, reflection, transmission, and mode conversion occur at the defect locations. By analyzing the reflected waves, the defect position can be determined. The damage at circumferential welds is typically axisymmetric, and the reflected waves of the L(0,2) mode at the circumferential weld include only L(0,2) and L(0,1) axisymmetric modes. The bending mode F(n, m) is also generated, which can be used to identify circumferential weld damage.

3.3. Modeling and Analysis of Steel Pipe

3.3.1. Three-Dimensional Modeling

The empty pipe model established by SOLIDWORKS 2022 software is shown in Figure 4: the outer-diameter is 219 mm, and the wall thickness is 5.5 mm. Axial defect model of pipeline: the length is 5 m, the defect length is 10 mm, the axial width is 1 mm, and the radial depth is 2 mm. The internal defect model of the pipeline circumferential weld: the length is 2 m, the axial width of the defect is 5 mm, and the radial depth is 1 mm. The finite element analysis of pipeline was gridded by Hypermesh 2020 software, Tetrahedral element type SOLID45 with 5 mm mesh. The transient dynamics module of the Ansys Mechanical APDL 2020 R2 software was used for simulation analysis and calculation. The three-dimensional modeling process of pipeline is depicted in Figure 5.

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:

$$\omega (n)=0.5-0.5cos({\displaystyle \frac{2\pi n}{N-1}})$$

Among them, $\omega \left(n\right)$ is the time domain value of the window function; $n$ is the discrete sample point index; and N is the window length.

2.

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.

3.

T-modal validation

The guided wave component changes with distance and time (wave velocity and frequency).

$$\varphi =kx\_\omega t$$

where k is the wave number, x is the propagation distance, and $\omega$ is frequency.

According to two-dimensional fast Fourier transform,

$$C_{\mathrm{p}}={\displaystyle \frac{\omega }{k}},k={\displaystyle \frac{\omega }{C_p}}={\displaystyle \frac{2\pi }{\lambda }}\Rightarrow {\displaystyle \frac{1}{x}}$$

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).

$$\left({\omega }_i,{\mathrm{k}}_i\right)\to {\displaystyle \frac{\omega }{k}}={\displaystyle \frac{2\pi f}{\frac{2\pi }{\lambda }}}=\lambda f=c\to \left(f_i,C_i\right)$$

(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.

4.

L-modal validation

The axial displacement nephogram of defects in the circumferential weld of pipeline is depicted in Figure 8.

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.

4. Pipeline Crack Location Algorithm and Quantitative Analysis

4.1. Crack Location

1.

Wavelet transform

Wavelet transform:

$$\gamma \left(s,\tau \right)=\int f\left(t\right){\psi }_{s\tau }\left(t\right)dt$$

In the formula, s is scale and $\mathsf{\tau }$ is translation.

The window function needs to satisfy the following:

$${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }} {\displaystyle \frac{\mid \psi \left(\omega \right){\mid }^2}{\mid \omega \mid }}\mathrm{d}\omega <+\mathrm{\infty }$$

(10)

If ψ($\mathsf{\omega }$) of ψ(t) Fourier transform satisfies Equation (10), ψ(t) can be used as the basis function of Daubechies wavelet.

Based on the simulation results in Section 3.3.2, the aperiodic echo signals received from crack-free and cracked pipelines are plotted in Figure 9a and Figure 9b, respectively. Figure 9c displays the residual signal for both, and the defect is identified based on its characteristics.

Time–frequency analysis of the residual signal at a 60 kHz excitation frequency is performed using wavelet transform to obtain the time–frequency diagrams of the crack-free and cracked pipeline waveforms. The characteristics of the defect signal near 0.0015 s after the difference between them are shown in Figure 10.

2.

Hilbert transform

Hilbert transform:

$$\widehat{x}(t)=H[x(t)]={\displaystyle \frac{1}{\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }} {\displaystyle \frac{x\left(\tau \right)}{t-\tau }}\mathrm{d}\tau$$

The complex signal $\widehat{x}\left(t\right)$ is an analytic signal of (t).

The time domain diagram of the analytic signal generated by Hilbert transform in Figure 10 is shown in Figure 11.

At 1.5 × 10⁻³ s, the first reflected signal when the guided wave encounters the crack is the head wave signal, which carries the position information of the crack. The crack location is determined by the receiving time of the head wave signal and the group velocity of UGW propagation.

3.

Positioning calculation

At different sampling points, according to the group velocity and the first wave packet positioning time obtained by Hilbert transform, the actual position of the crack is determined by calculating the distance between the crack and the excitation-receiving point. The calculation results are shown in

Table 1. Calculation of crack location.

Sampling Point Position (m)	Speed (m/s)	Actual Position (m)	Positioning Time (s)	Positioning Position (m)	Positioning Error (m)	Relative Error of Wavelet–Hilbert Transform Algorithm	Relative Error of Wavelet Transform Algorithm
1.40	3225.8	3.10	0.0015	3.12	0.02	0.6%	2.1%
2.00	3225.8	3.10	0.0013	3.09	0.01	0.3%	1.2%
2.60	3225.8	3.10	0.0011	3.07	0.03	0.9%	2.5%

.

Method for calculating crack position X in Table 1:

$$\mathit{X}=\mathit{t}\mathit{v}$$

where t is the positioning time of the first wave packet and v is the group velocity.

The defect position is determined by comparing the positions of different sampling points, and the relative errors are all less than 1%, indicating that the positioning accuracy is high. Compared to the combination of wavelet and Hilbert transform, the algorithm using wavelet transform alone results in larger errors, which hinder accurate defect localization.

4.2. Quantitative Analysis of Welds

4.2.1. Circumferential Defects

By analyzing the reflection coefficient of UGWs in Section 2.2, the size of defects can be estimated. The radial depth and axial width of the pipeline defect are set to 1 mm and 5 mm, respectively. The circumferential length is changed; 30° is taken as the time step; the defects at 30°, 60°, 90°, 120°, 150°, 180°, 210°, 240°, 270°, and 300° are set; and a simulation is performed in Ansys. The maximum displacement of the initial excitation wave and defect echo is shown in

Table 2. Relationship table of reflection coefficient between defects with different circumferential dimensions and pipeline defects.

Circumferential Length	Maximum Displacement of Initial Wave (mm)	Maximum Echo Displacement (mm)	Reflectance
30°	0.3998	0.0225	0.05629
60°	0.3999	0.0423	0.10578
90°	0.3989	0.0607	0.15214
120°	0.3999	0.0807	0.20179
150°	0.3988	0.1022	0.25631
180°	0.3989	0.1205	0.30211
210°	0.3995	0.1435	0.35931
240°	0.3999	0.1635	0.40881
270°	0.3979	0.1802	0.45278
300°	0.3975	0.1999	0.50301

. The reflection coefficient of defects with different circumferential lengths is calculated, the correlation between the reflection coefficient and the circumferential length of pipeline defects is observed, and defects with different circumferential lengths are quantitatively analyzed to obtain partial data on the pipeline reception signals with different circumferential lengths, as shown in Figure 12a–c.

With the increase in the circumferential length of pipeline defects, the echo displacement of cracks also shows an increasing trend. According to the relationship between defects with different circumferential sizes and the reflection coefficient of pipeline defects, the relationship curve between the circumferential defect size and the reflection coefficient of defects is drawn as shown in Figure 12d. When the circumferential length of the defects increases, the reflection coefficient increases linearly. The reflection coefficient of defects presents a linear change trend with the circumferential length of the pipeline circumferential weld defects.

4.2.2. Radial Defects

The axial width of the pipeline defect is set to 5 mm, and the radial depth is 1 mm. The radial depth of the pipeline circumferential weld defect is changed, the time step is 0.5 mm, and the defect is quantitatively detected and analyzed. The reflection coefficients of circumferential weld defects with different radial depths are calculated as shown in

Table 3. Relationship between reflection coefficient of defects with different radial dimensions and pipeline defects.

Radial Depth (mm)	Maximum Displacement of Initial Wave (mm)	Maximum Echo Displacement (mm)	Reflectance
1	0.3889	0.0276	0.07093
1.5	0.3888	0.0567	0.14596
2	0.3890	0.1023	0.26321
2.5	0.3890	0.1112	0.39586
3	0.3899	0.1539	0.51239
3.5	0.3897	0.2415	0.61979
4	0.3897	0.2743	0.70392
4.5	0.3901	0.3151	0.80782
5	0.3900	0.3551	0.91051
5.5	0.3952	0.3759	0.95133

, and the relationship between the reflection coefficient and the radial depth of pipeline defects is observed. Some data on the signals received by the pipeline with different radial depth defects are shown in Figure 13a–c.

With the increase in crack radial length, the amplitude of the defect echo increases obviously. According to the data on the echo and initial wave at the defect, the fitting curve of the relationship between the reflection coefficient and the radial size of the defect is drawn. As shown in Figure 13d, the radial depth of the pipeline defect is not linear with the reflection coefficient of the echo at the defect. With the increase in the radial depth of the defect, the reflection coefficient of the defect not only increases but also increases faster and faster.

Compared with the quantitative analysis results of circumferential length change in Section 4.2.1 and the change in reflection coefficient, the change law of the radial depth of defects is quite different.

5. Conclusions

Through a numerical simulation study of ultrasonic guided wave detection for non-defective and defective pipelines, the conclusions are as follows:

(1)

The dispersion curve obtained by the semi-analytical finite element method is in good agreement with the simulation data. The successful excitation of the T(0,1) mode by the torsional excitation mode can be effectively used to detect axial cracks. The L(0,2) mode can be successfully excited by observing the displacement nephogram and can be effectively used to detect the defects in the circumferential weld.

(2)

Through the algorithm of combining wavelet transform and Hilbert transform, the relative errors are all less than 1%, so the positioning accuracy is high.

(3)

The quantitative analysis of defects in circular welds based on multi-node fusion and modal projection algorithm clearly extracted ultrasonic guided wave signals of different modes. L(0,2) guided waves are sensitive to pipeline defects with circumferential and radial dimensions.

(4)

Multi-node fusion improves the sensitivity and noise immunity of defect signals in field detection, which is especially suitable for pipeline detection in complex environments. Modal projection enhances the recognition of defects in complex structures such as welds by separating different guided wave modes. The combination of the two makes the field detection more accurate, adapts to complex working conditions, and supports real-time monitoring.

6. Patents

A patent of the UGW detection method, system, equipment, and medium for pipeline health status has, at present, been accepted and is under review.