Plate structure defect Lamb wave full waveform inversion method based on U-Net neural network regularization

Abstract

The invention discloses a plate structure defect Lamb wave full waveform inversion method based on U-Net neural network regularization, and belongs to the field of nondestructive testing. Firstly, a two-dimensional sound wave equation is solved through a frequency domain finite difference method, and analog signals of all receiving points are obtained; the method comprises the following steps: constructing a target function to represent a relative deviation between an analog signal and a sensor actual measurement signal, and converting an inversion problem of defects in a metal plate into an optimization problem of the target function; performing derivation on the target function by using an adjoint state method to obtain a gradient, updating the velocity field along the negative direction of the gradient, and performing regularization processing on the velocity field through a specific coding-decoding structure of the U-Net neural network to suppress noise and artifacts; a frequency serial inversion strategy is adopted, and inversion is carried out step by step from low frequency to high frequency. According to the method, artifacts in the inversion process are suppressed by using the U-Net neural network, dependence of inversion on initial model precision is reduced by using a frequency domain serial inversion strategy, and high-precision inversion of plate structure defects is realized.

Description

Technical Field

[0001] The present invention relates to a method for inverse analysis of plate structure defects, in particular to a full waveform inverse analysis method of defect Lamb waves based on U-Net neural network regularization. This method is applicable to the inverse analysis and quantitative evaluation of defects in metal plates and belongs to the field of non-destructive testing. Background Art

[0002] Due to advantages such as high strength, corrosion resistance, and plasticity, metal plates are widely used in fields such as aerospace, automotive manufacturing, shipbuilding industry, and construction engineering. However, due to the influence of factors such as material fatigue, corrosion, and external force impact, defects may form inside the metal plate, reducing the integrity of the plate structure and even causing safety accidents. Therefore, it is necessary to develop an effective non-destructive testing method for metal plate defects to ensure the safe operation of industrial structural components.

[0003] As a conventional detection method, ultrasonic detection technology can quickly, accurately and over a large range locate and identify defects in structures by analyzing the characteristics of ultrasonic waves during propagation, such as reflection, scattering, amplitude attenuation and phase shift. The ultrasonic detection method based on inversion can not only determine whether there are defects in the metal plate, but also determine information such as the location and size of the defects. Up to now, many scholars have proposed various methods for the inversion problem of metal plate defects. Among them, full waveform inversion is a method that effectively realizes the inversion of medium parameters by iteratively adjusting model parameters to minimize the difference between the observed waveform and the simulated waveform. This method can make full use of the full wave field information such as the velocity, phase, travel time and amplitude of ultrasonic waves for multi-parameter inversion, improving the inversion accuracy and spatial resolution. At present, the full waveform inversion method has been applied to the field of non-destructive testing of plate structure defects. For example, Rao J. et al. [Rao J, Ratassepp M, Fan Z. Investigation of the reconstruction accuracy of guided wave tomography using full waveform inversion[J]. Journal of Sound and Vibration, 2017, 400: 317-328.] realized the inversion of corrosion defects in plate-like structures through numerical simulation and experiments. The results show that this method still has high accuracy when inverting metal plates with multiple defects, and the inversion resolution can reach 0.7 wavelengths. However, in actual detection, the detection signal inevitably contains interference components such as electromagnetic interference and boundary reflection, resulting in a low signal-to-noise ratio of the detection signal and affecting the accuracy of the inversion result. In addition, full waveform inversion has high requirements for the quality of the initial model. When the difference between the initial velocity model and the actual structure is large, the phase difference between the forward waveform and the measured waveform may exceed half a cycle, resulting in the objective function falling into a local minimum and the optimization process deviating from the true solution.

[0004] To solve the above problems, Esser, et al. [Esser E, Guasch L, Van L T, et al. Total variation regularization strategies in full-waveform inversion[J]. SIAM Journal on Imaging Sciences, 2018, 11(1): 376–406.] proposed a full-waveform inversion method based on total variation regularization. By introducing a penalty term into the objective function, this method applies a certain weight to the regions with drastic changes in sound velocity to suppress local gradient outliers. The results show that this method improves the reconstruction ability of the inversion algorithm for complex velocity models and can well suppress artifacts. However, since total variation regularization only uses the first derivative for inversion, the accuracy of the inversion results is limited. On this basis, Wu, et al. [Wu F X, He Q L, Zou J, et al. Rotation total variation regularization for full waveform inversion[J]. Journal of Inverse and Ill-posed Problems, 2025.] proposed a hybrid total variation regularization method that simultaneously considers the first and second derivatives for geological structure inversion. The results show that this method can improve the inversion accuracy of traditional total variation regularization while suppressing inversion artifacts. However, this method introduces additional regularization parameters and requires hyperparameter setting for the model, increasing the complexity of the inversion.

[0005] In recent years, with the development of deep learning, regularization methods based on neural networks have been widely applied in the field of image processing. For example, Ulyanov D., et al. [Ulyanov D, Vedaldi A, Lempitsky V. Deep image prior[J]. International Journal of Computer Vision. 2020: 1867-1888.] proposed an unsupervised regularization method based on the U-Net neural network for image denoising and restoration. This method does not rely on external training data and directly uses the encoding-decoding structure of the network itself to spontaneously capture the variation rules of adjacent pixel points in the image to achieve image regularization processing. However, currently, this method is rarely applied in the field of industrial non-destructive testing. In the full-waveform inversion method of Lamb waves in plate structures, regularizing the inversion process through the U-Net neural network is expected to improve the robustness of the inversion.

[0006] In summary, the full waveform inversion technology can achieve the detection and quantitative evaluation of defects and is applicable to the non-destructive testing of metal plate defects. However, this method is vulnerable to noise interference during engineering applications, and the inversion accuracy depends on the quality of the initial input model. Given that the U-Net neural network can suppress noise and artifacts during the inversion process without pre-training, the present invention proposes a Lamb wave full waveform inversion method for plate structure defects based on U-Net neural network regularization for high-precision inversion of defects in metal plates. Summary of the Invention

[0007] The purpose of the present invention is to propose a Lamb wave full waveform inversion method for plate structure defects based on U-Net neural network regularization. This method regularizes the velocity model of the full waveform inversion through the U-Net neural network, suppresses noise and artifacts, and achieves high-precision inversion of plate structure defects.

[0008] The present invention proposes a Lamb wave full waveform inversion method for plate structure defects based on U-Net neural network regularization, and its implementation process is as Figure 1 shown. This method first establishes an initial thickness model of the metal plate, selects multiple inversion frequency points, and converts the thickness model into a velocity model at the initial frequency points; substitutes the velocity model into the wave equation for forward modeling to obtain the simulated signal and the forward propagation wave field; uses an ultrasonic detection system to obtain the measured signal, calculates the residual between the simulated signal and the measured signal, and backpropagates the residual to obtain the backpropagation wave field; calculates the gradient of the objective function based on the forward propagation wave field and the backpropagation wave field, uses a local optimization algorithm to update the velocity model, and uses the U-Net neural network to suppress the artifacts in the velocity model. Substitute the updated velocity model into the wave equation for the next round of iteration. When the iteration convergence condition is met, output the updated velocity model and convert it into a thickness model. Use this thickness model as the initial model for the next frequency point and perform inversion in sequence until the convergence criterion is reached at the highest frequency to obtain the final defect inversion result.

[0009] The basic principle of a Lamb wave full waveform inversion method for plate structure defects based on U-Net neural network regularization proposed by the present invention lies in:

[0010] The full waveform inversion method inversely calculates the defect parameters by iteratively adjusting the simulation model parameters to minimize the difference between the simulated signal and the measured signal of the sensor. Among them, the simulated signal can be obtained by numerically solving the wave equation. In a two-dimensional homogeneous isotropic medium, the acoustic wave equation can be expressed as:

[0011]

[0012] where, is the Laplace operator; ω is the angular frequency; v is the acoustic wave propagation speed; r(x,y) is the spatial position in the Cartesian coordinate system; ψ(r,ω) is the displacement field; S(ω) is the Fourier transform of the sound source signal at the excitation point.

[0013] For the convenience of computer processing, the method needs to be discretized when numerically solving the wave equation. The nine-point difference grid is used to discretize Equation (1). After discretization, the wave equation is:

[0014]

[0015] In the formula, is the second-order finite difference operator in the x and y directions; is the second-order finite difference operator rotated by 45°; a is the weighting coefficient used to balance the contributions in the x, y directions and the diagonal direction. By solving Equation (2) through LU decomposition, the simulated signal at each receiving point and the forward propagation wave field can be obtained.

[0016] The objective function E(v) is constructed to represent the relative deviation between the above-mentioned simulated signal and the measured signal of the sensor, so as to transform the inversion problem of defects in the metal plate into an optimization problem of the objective function. The expression of the objective function is:

[0017]

[0018] In the formula, d obs is the measured signal; d sim is the simulated signal; Δd is the residual; v is the velocity model at each frequency; x s is the position of the excitation sensor, x r is the position of the receiving sensor; E(v) is the objective function.

[0019] To obtain the optimal solution of the objective function, the adjoint state method is used to calculate the gradient of the objective function. This method obtains the backpropagation wave field by backpropagating the residual on the velocity model, and calculates the zero-delay cross-correlation between the forward propagation wave field and the backpropagation wave field to obtain the gradient, which can avoid taking partial derivatives of the objective function at each grid point during the inversion process, reduce the calculation cost, and improve the inversion efficiency. The calculation expression of the objective function gradient is:

[0020]

[0021] In the formula, u is the forward propagation wave field; p is the backpropagation wave field of the residual, and B represents the forward operator of the wave equation.

[0022] The quasi-Newton method is used to update the model parameters along the negative direction of the gradient direction. The update process of the model parameters can be expressed as:

[0023] m (k+1) = m (k)-αH -1(k) g (k) (5)

[0024] where H -1(k) is the inverse of the Hessian matrix; m (k) represents the model parameters at the k-th iteration; α is the optimization step size; g (k) represents the gradient value in the k-th iteration.

[0025] To suppress the artifacts caused by the low signal-to-noise ratio of the detection signal, a U-Net neural network is introduced to regularize the velocity model. By using the encoding-decoding structure of the U-Net neural network for cross-scale skip connections, the correlation between multi-level features of the velocity model is captured, and the artifacts caused by local gradient mutations are suppressed. The input of the U-Net neural network is a random matrix of the same size as the velocity model, and the output is the regularized velocity model. The regularization process is represented by the following formula:

[0026]

[0027] where z is a random matrix of the same size as the velocity model; θ k is the network parameter in the k-th iteration; is the regularized velocity model; θ * is the optimal solution of the parameters obtained through training; is the loss function of the network, and the L2 loss is used. The network parameters are updated by the Adam algorithm.

[0028] During the inversion process, when the difference between the initial velocity model and the actual structure is large, the phase difference between the simulated waveform and the measured waveform exceeds half a cycle, which will cause the objective function to fall into a local minimum. This phenomenon is called "cycle skipping". To solve the above problem, this method adopts a frequency serial inversion strategy, that is, inversion is carried out step by step from low frequency to high frequency.

[0029] During low-frequency inversion, due to the long wavelength, the phase difference caused by the same time error is small, and the phase is not sensitive to the velocity error. Therefore, by adopting the low-frequency priority inversion strategy, the phase matching between the simulated waveform and the measured waveform can be maintained when the deviation between the initial model and the true model is large, and the dependence on a high-precision initial model for inversion can be reduced. Taking the low-frequency inversion result as the initial model for high-frequency inversion realizes the step-by-step optimization of the velocity model. Since the wavelength of the high-frequency component is short and it is more sensitive to the velocity error, using the high-frequency component for inversion can more accurately capture the subtle changes of the velocity model, reduce the velocity model error, and thus improve the inversion accuracy. Through the frequency serial inversion strategy, while ensuring the inversion accuracy, the requirement for the accuracy of the initial model in inversion can be reduced, and the stability of inversion is improved.

[0030] The present invention is a Lamb wave full waveform inversion method for plate structure defects based on U-Net neural network regularization, aiming to achieve high-precision inversion of defects in metal plates. This method is realized through the following steps:

[0031] Step 1: Build a guided wave detection system as shown in Figure 2 to collect the received signal d obs ;

[0032] Step 2: Establish a two-dimensional metal plate simulation model in MATLAB. Adopt a frequency serial inversion strategy, select multiple inversion frequency points, and convert the thickness of the metal plate into the phase velocity value at the current frequency in combination with the Lamb wave dispersion curve;

[0033] Step 3: Substitute the phase velocity into the wave equation for solution according to Equation 1 and Equation 2 to obtain the numerical simulation signal d sim and the forward propagation wave field u;

[0034] Step 4: Subtract the simulated signal from the actual received signal to obtain the residual d, and calculate the L2 norm of the residual according to Equation 3 to obtain the objective function E(v);

[0035] Step 5: Back-propagate the residual on the model to obtain the back-propagation wave field p. Calculate the zero-delay cross-correlation of the forward propagation wave field and the back-propagation wave field and sum it according to Equation 4 to obtain the gradient g(v);

[0036] Step 6: Substitute the gradient into the local optimization algorithm and update the velocity model m at the current iteration according to Equation 5 (k) ;

[0037] Step 7: Build a U-Net network model as shown in Figure 3 . This model consists of a symmetric encoder and decoder. The encoder contains two downsampling layers, each downsampling layer consists of two 3×3 convolutional layers, an activation function, and a batch normalization layer, followed by a 2×2 max pooling layer. The decoder contains two upsampling layers, each upsampling layer is upsampled through a 2×2 transposed convolution operation, and feature fusion is performed with the corresponding layer of the encoder through skip connections. After the upsampling layer, two 3×3 convolutional layers, an activation function, and a batch normalization layer are connected, and finally, the result is output through a convolutional layer and an activation function layer. Use a random matrix z with the same size as the velocity model as the network input, regularize the velocity model according to Equation 6, and update the network parameters;

[0038] Step 8: If the inversion reaches the iteration convergence condition, stop the inversion at the current frequency, convert the inverted output velocity model into thickness, use it as the initial input for the next frequency point, and return to Step 2 to continue the iteration. If the iteration does not reach the iteration convergence condition, return to Step 3 to continue the iteration. After all the selected frequency points are completely inverted, convert the final velocity model into a thickness distribution to generate a defect imaging result.

[0039] The present invention has the following advantages: 1) The velocity model in the inversion process is regularized through the encoding-decoding structure of the U-Net neural network, suppressing inversion artifacts. 2) The frequency serial inversion strategy from low frequency to high frequency is adopted, reducing the dependence of inversion on the accuracy of the initial model, and improving the inversion accuracy while alleviating the cycle skipping problem. Brief Description of the Drawings

[0040] Figure 1 is a flowchart of the method of the present invention.

[0041] Figure 2 is a schematic diagram of the defect detection system.

[0042] Figure 3 is a schematic diagram of the U-Net neural network.

[0043] Figure 4 Among them, (a) is the received signal when sensor 1 is excited and the other sensors receive; (b) is the experimental signal after windowing and filtering processing.

[0044] Figure 5 Among them, (a) is the defect imaging result of the traditional inversion method. (b) is the profile depth curve of the traditional inversion method at y = 100 mm at different frequencies.

[0045] Figure 6 Among them, (a) is the defect imaging result of the inversion after regularization by the U-Net neural network. (b) is the profile depth curve of the inversion after regularization by the U-Net neural network at y = 100 mm at different frequencies.

[0046] Figure 7 is a comparison diagram of the profile depth curves of the two methods. Detailed Embodiments

[0047] The present invention will be further described below in combination with specific experiments:

[0048] An ultrasonic detection system is built in the laboratory. The specimen to be tested is an aluminum plate with a size of 500 mm × 500 mm × 3 mm, and there is a flat-bottomed hole defect with a diameter of 6 mm and a depth of 1.5 mm at the center. The sensors are arranged on a circle with the center of the plate as the center and a radius of 50 mm, with a step of 10°, and a total of 36 sensors are arranged. The schematic diagram of the ultrasonic detection system is as Figure 2 shown.

[0049] The specific steps of a Lamb wave full waveform inversion method for plate structure defects based on U-Net neural network regularization proposed by the present invention are as follows:

[0050] Step 1: In as Figure 2Experiments were carried out under the experimental system shown. First, 1 sensor was selected for excitation, and the remaining 35 sensors were used for reception. The excitation signal was selected as a sine signal modulated by a Hanning window with a center frequency of 100 kHz. The sensors were excited cyclically in turn, and finally 35×36 groups of received signals were obtained. The duration of the received signal was 25 ms, and the sampling rate was 1.25 MHz. Among them, when sensor 1 was excited and the other sensors received, the received signal was as shown in Figure 4 (a). The direct wave in the received signal was intercepted by windowing, as shown in Figure 4 (b). The starting point and ending point of the time window were determined by the group velocity of the A0 mode and the transmitted waveform of the signal generator. The detailed formula can be seen in the following formula:

[0051]

[0052] In the formula, t start is the starting point of the time window; L is the distance between the transmitting and receiving transducers; c g is the group velocity of the guided wave; t end represents the ending point of the time window, and T0 is the period of the transmitted waveform of the signal generator. After applying the time window to the waveform, a second-order Butterworth band-pass filter was used for filtering, as shown in Figure 3 . Fast Fourier transform was performed on all processed signals to obtain experimental data.

[0053] Step 2: Establish a simulation model corresponding to the test parameters in MATLAB, and use a uniform and defect-free two-dimensional metal plate thickness model as the initial model for inversion. The frequency serial inversion strategy was adopted, and the inversion frequencies were selected as 70 kHz, 90 kHz, and 120 kHz. Combining with the Lamb wave dispersion curve, the metal plate thickness was converted into phase velocity values.

[0054] Step 3: Substitute the phase velocity into the wave equation, and use the finite difference time domain method to solve the wave equation to solve the simulated signal and the forward propagation wave field, as shown in the following formula:

[0055]

[0056] Step 4: Subtract the simulated signal from the actual received signal to obtain the residual d, and obtain the objective function by calculating the L2 norm of the residual, as shown in the following formula:

[0057]

[0058] Step 5: Back-propagate the residual between the simulated signal and the measured signal on the model to obtain the back-propagation wave field p. According to the adjoint state method, calculate the zero-delay cross-correlation of the forward propagation wave field and the back-propagation wave field and sum it to obtain the gradient g(v), as shown in the following formula:

[0059]

[0060] p = B * [(d sim (x r , x s , v) - d obs (x r , x s ))]

[0061] Step Six: Update the velocity model along the negative direction of the gradient using the quasi - Newton method to obtain the velocity model at the current iteration number, as shown in the following formula:

[0062] m (t+1) = m (t) - αH -1(t) g (t)

[0063] Step Seven: Construct a U - Net network structure and input the currently updated velocity model into the network for regularization. The network input is a random matrix with the same size as the model, and the output is the regularized velocity distribution. The network consists of a symmetric encoder and decoder. The encoder contains two downsampling layers, each of which is composed of two 3×3 convolutional layers, an activation function, and a batch normalization layer, followed by a 2×2 max - pooling layer. The decoder contains two upsampling layers, each of which is upsampled by a 2×2 transposed convolution operation and performs feature fusion with the corresponding layer of the encoder through skip connections. After the upsampling layer, there are two 3×3 convolutional layers, an activation function, and a batch normalization layer, and finally, the result is output through a convolutional layer and an activation function layer. The network parameters are shown in Table 1. Minimize the difference between the network output and the current velocity model through the Adam algorithm to update the network hyperparameters.

[0064] Table 1 U - Net neural network structure parameters

[0065]

[0066] Step Eight: Determine whether the current iteration meets the termination condition. When the inversion reaches 30 times or the objective function is less than 10 -6 , stop the inversion at the current frequency and convert the optimized velocity model at this frequency into a thickness model, which is used as the initial input for the next frequency point. After the inversion is completed at all selected frequency points, convert the final velocity model into a thickness distribution to generate the defect imaging result. Figure 5 (a) of... shows the defect reconstruction result of the traditional inversion method, Figure 5 (b) of... shows the profile depth curve of the traditional inversion method at y = 100 mm under different frequencies. Figure 6 (a) of... is the defect reconstruction result of the inversion after regularization by the U - Net neural network, Figure 6b shows the profile depth curves at y = 100 mm at different frequencies after the inversion of the U-Net neural network regularization. From Figure 5 (a) of Figure 6 and (a) of Figure 5 , it can be seen that the traditional inversion method produces deeper artifacts at the sensor, while the inversion method based on the U-Net neural network regularization can effectively suppress such artifacts and improve the structural continuity and authenticity of the imaging. In addition, from Figure 6 (b) of Figure 7 and (b) of Figure 7 , it can be seen that as the inversion frequency increases, the reconstruction error of the defect gradually decreases. At the highest frequency of 120 kHz, the reconstruction error of the defect at (100 mm, 100 mm) is only 0.06 mm. This indicates that the serial inversion from low frequency to high frequency can effectively improve the analytical ability and reconstruction accuracy of the defect boundary.

[0067] The above is a typical application of the present invention, and the application of the present invention is not limited thereto.

Claims

1. A Lamb wave full waveform inversion method for plate structure defects based on U-Net neural network regularization, characterized in that, First, establish the initial thickness model of the metal plate, select multiple inversion frequency points, and convert the thickness model into a velocity model at the initial frequency points; substitute the velocity model into the wave equation for forward modeling to obtain the simulated signal and the forward wave field; use the ultrasonic detection system to obtain the measured signal, calculate the residual between the simulated signal and the measured signal, and backpropagate the residual to obtain the backpropagated wave field; calculate the gradient of the objective function based on the forward wave field and the backpropagated wave field, adopt a local optimization algorithm to update the velocity model, and use the U-Net neural network to suppress the artifacts in the velocity model; substitute the updated velocity model into the wave equation for the next round of iteration. When the iteration convergence condition is met, output the updated velocity model and convert it into a thickness model; use this thickness model as the initial model for the next frequency point and perform inversion in sequence until the convergence criterion is reached at the highest frequency to obtain the final defect inversion result.

2. A Lamb wave full waveform inversion method for plate structure defects based on U-Net neural network regularization according to claim 1, characterized in that The full waveform inversion method realizes the inversion of defect parameters by iteratively adjusting the simulation model parameters to minimize the difference between the simulated signal and the measured signal by the sensor; among them, the simulated signal is obtained by numerically solving the wave equation; in a two-dimensional homogeneous isotropic medium, the acoustic wave equation is expressed as: In the formula, is the Laplace operator; ω is the angular frequency; v is the sound wave propagation velocity; r(x, y) is the spatial position in the Cartesian coordinate system; ψ(r, ω) is the displacement field; S(ω) is the Fourier transform of the sound source signal at the excitation point; For the convenience of computer processing, discretization is required when numerically solving the wave equation; the nine-point difference grid is used to discretize Equation (1). After discretization, the wave equation is: In the formula, is the second-order finite difference operator in the x and y directions; is the second-order finite difference operator rotated by 45°; a is the weighting coefficient used to balance the contributions in the x and y directions and the diagonal direction; by solving Equation (2) through LU decomposition, the simulated signal and the forward propagation wave field at each receiving point can be obtained; Construct the objective function E(v) to represent the relative deviation between the above-mentioned simulated signal and the measured signal by the sensor, so as to transform the inversion problem of defects in the metal plate into an optimization problem of the objective function. The expression of the objective function is: where d obs is the measured signal; d sim is the simulated signal; Δd is the residual; v is the velocity model at each frequency; x s is the position of the excitation sensor, and x r is the position of the receiving sensor; E(v) is the objective function; To obtain the optimal solution of the objective function, the adjoint state method is used to calculate the gradient of the objective function; this method obtains the backpropagated wave field by backpropagating the residual on the velocity model, and calculates the zero-delay cross-correlation between the forward wave field and the backpropagated wave field to obtain the gradient, avoiding taking partial derivatives of the objective function at each grid point during the inversion process, reducing the computational cost, and improving the inversion efficiency; the calculation expression of the objective function gradient is: In the formula, u is the forward wave field; p is the backpropagated wave field of the residual, and B represents the forward operator of the wave equation; The quasi-Newton method is used to update the model parameters along the negative direction of the gradient direction; the update process of the model parameters can be expressed as: m (k+1) =m (k) -αH -1(k) g (k) (5) Where H -1(k) is the inverse of the Hessian matrix; m (k) represents the model parameters at the k-th iteration; α is the optimization step size; g (k) represents the gradient value in the k-th iteration; To suppress the artifacts generated due to the low signal-to-noise ratio of the detection signal, the U-Net neural network is introduced to regularize the velocity model; utilize the cross-scale skip connection of the encoding-decoding structure of the U-Net neural network itself to capture the correlation between multi-level features of the velocity model and suppress the artifacts caused by local gradient mutations; the input of the U-Net neural network is a random matrix with the same size as the velocity model, and the output is the regularized velocity model. The regularization process is expressed by the following formula: where \(z\) is a random matrix of the same size as the velocity model; \(\theta\) k is the network parameter in the \(k\)-th iteration; is the regularized velocity model; \(\theta\) * is the optimal solution of the parameters obtained through training; is the loss function of the network, using the L2 loss; the network parameters are updated by the Adam algorithm; During the inversion process, when the difference between the initial velocity model and the actual structure is large, the phase difference between the simulated waveform and the measured waveform exceeds half a cycle, which will cause the objective function to fall into a local minimum, that is, cycle skipping; adopt the frequency serial inversion strategy, that is, perform inversion step by step from low frequency to high frequency; During low-frequency inversion, a low-frequency first inversion strategy is adopted to maintain the phase matching between the simulated waveform and the measured waveform when the deviation between the initial model and the true model is large, reducing the dependence of inversion on a high-precision initial model; the low-frequency inversion result is used as the initial model for high-frequency inversion to achieve step-by-step optimization of the velocity model; using high-frequency components for inversion can more accurately capture the subtle changes in the velocity model, reducing the velocity model error, thereby improving the inversion accuracy; through the frequency serial inversion strategy, while ensuring the inversion accuracy, the requirement for the accuracy of the initial model in inversion is reduced, and the stability of inversion is improved.

3. A Lamb wave full waveform inversion method for plate structure defects based on U-Net neural network regularization according to claim 2, characterized in that This method is implemented through the following steps: Step 1: Set up a guided wave detection system and collect the received signal d obs ; Step 2: Establish a two-dimensional metal plate simulation model in MATLAB, adopt the frequency serial inversion strategy, select multiple inversion frequency points, and convert the metal plate thickness into the phase velocity value at the current frequency in combination with the Lamb wave dispersion curve; Step 3: Substitute the phase velocity into the wave equation according to Equation 1 and Equation 2 to solve and obtain the numerical simulation signal d sim and the forward wave field u; Step 4: Subtract the simulated signal from the actual received signal to obtain the residual d, and calculate the L2 norm of the residual according to Equation 3 to obtain the objective function E(v); Step 5: Back-propagate the residual on the model to obtain the back-propagated wave field p; calculate the zero-delay cross-correlation of the forward-propagated wave field and the back-propagated wave field and sum them according to Equation 4 to obtain the gradient g(v); Step 6: Substitute the gradient into the local optimization algorithm and update the velocity model m at the current iteration according to Equation 5 (k) ; Step 7: Construct a U-Net network model, which consists of a symmetric encoder and decoder; the encoder contains two downsampling layers, each downsampling layer consists of two 3×3 convolutional layers, an activation function, and a batch normalization layer, followed by a 2×2 max pooling layer; the decoder contains two upsampling layers, each upsampling layer is upsampled through a 2×2 deconvolution operation, and feature fusion is performed with the corresponding layer of the encoder through skip connections; After the upsampling layer, there are two 3×3 convolutional layers, an activation function, and a batch normalization layer, and finally the result is output through a convolutional layer and an activation function layer; use a random matrix z with the same size as the velocity model as the network input, regularize the velocity model according to Equation 6 and update the network parameters; Step 8: If the inversion reaches the iterative convergence condition, stop the inversion at the current frequency, convert the velocity model output by the inversion into thickness, and use it as the initial input for the next frequency point, then return to Step 2 to continue the iteration; if the iteration does not reach the iterative convergence condition, return to Step 3 to continue the iteration; after all the selected frequency points have completed the inversion, convert the final velocity model into a thickness distribution to generate a defect imaging result.

4. A Lamb wave full waveform inversion method for plate structure defects based on U-Net neural network regularization according to claim 3, characterized in that Intercept the direct wave in the received signal by windowing, and the start point and end point of the time window are determined by the group velocity of the A0 mode and the transmitted waveform of the signal generator: Where, t start is the starting point of the time window; L is the distance between the transmitting and receiving transducers; c g is the guided wave group velocity; t end represents the end point of the time window, and T0 is the period of the waveform emitted by the signal generator. After adding the time window to the waveform, a second-order Butterworth bandpass filter is used for filtering. Fast Fourier transform is performed on all processed signals to obtain experimental data.

5. A Lamb wave full waveform inversion method for plate structure defects based on U-Net neural network regularization according to claim 3, characterized in that, Substitute the phase velocity into the wave equation, and use the finite difference method in the frequency domain to solve the wave equation to solve the simulated signal and the forward-propagated wave field, as shown in the following formula:

6. A Lamb wave full waveform inversion method for plate structure defects based on U-Net neural network regularization according to claim 3, characterized in that, Subtract the simulated signal from the actual received signal to obtain the residual d, and obtain the objective function by calculating the L2 norm of the residual, as shown in the following formula:

7. A Lamb wave full waveform inversion method for plate structure defects based on U-Net neural network regularization according to claim 3, characterized in that, Back-propagate the residual between the simulated signal and the measured signal on the model to obtain the back-propagated wave field p; according to the adjoint state method, calculate the zero-delay cross-correlation of the forward-propagated wave field and the back-propagated wave field and sum them to obtain the gradient g(v), as shown in the following formula: p = B * [(d sim (x r ,x s ,v)-d obs (x r ,x s ))] 8. A Lamb wave full waveform inversion method for plate structure defects based on U-Net neural network regularization according to claim 3, characterized in that, The velocity model is updated along the negative direction of the gradient using the quasi-Newton method to obtain the velocity model at the current iteration number, as shown in the following equation: m (t+1) = m (t) -αH -1(t) g (t) 。 9. A Lamb wave full waveform inversion method for plate structure defects based on U-Net neural network regularization according to claim 3, characterized in that, A U-Net neural network is constructed, and the velocity model updated currently is input into the network for regularization; the input of the U-Net neural network is a random matrix with the same size as the model, and the output is the regularized velocity distribution; the U-Net neural network consists of a symmetric encoder and decoder; the encoder contains two downsampling layers, each downsampling layer consists of two 3×3 convolutional layers, an activation function, and a batch normalization layer, followed by a 2×2 max pooling layer; the decoder contains two upsampling layers, each upsampling layer is upsampled by 2×2 transposed convolution operation, and feature fusion is performed with the corresponding layer of the encoder through skip connections; After the upsampling layer, there are two 3×3 convolutional layers, an activation function, and a batch normalization layer, and finally the result is output through a convolutional layer and an activation function layer; the difference between the network output and the current velocity model is minimized by the Adam algorithm to update the network hyperparameters.

10. A Lamb wave full waveform inversion method for plate structure defects based on U-Net neural network regularization according to claim 3, characterized in that The frequency serial inversion strategy is adopted, and several frequency points are selected for inversion in sequence. The inversion result at the current frequency point is used as the initial model for the next frequency point until the convergence criterion is reached at the highest frequency, and the final defect inversion result is output.

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