A quantitative imaging method for internal defects of scattering SH waveguide plates based on the method of moments.

Abstract

This invention provides a quantitative imaging method for internal defects of a scattering SH-guided plate based on the method of moments, belonging to the field of ultrasonic damage detection technology. The method includes the following steps: S1: An ultrasonic guided wave of a specific mode is incident on the area to be detected in the plate structure. The time-domain signal of the reflected wave displacement field received by the ultrasonic transducer in the far field is converted into a frequency-domain signal by Fourier transform; S2: The frequency-domain reflected wave displacement field is modally decomposed using the modal orthogonality of the guided wave to obtain the reflection coefficients of each mode at each frequency. This invention has been experimentally tested for different types of defects in the plate structure based on scattered field information. The reconstruction results show that the higher the detection frequency, the higher the reconstruction accuracy, exhibiting high precision. It can achieve precise quantitative reconstruction of defects in plate structures, thus providing an accurate and reliable quantitative damage detection technology for practical engineering.

Description

Technical Field

[0001] This invention relates to the field of ultrasonic damage technology, and more specifically to a quantitative imaging method for internal defects of a scattering SH waveguide plate based on the method of moments. Background Technology

[0002] Conventional non-destructive testing (NDT) and evaluation methods in industry include magnetic particle testing, X-ray testing, eddy current testing, and ultrasonic testing, among which ultrasonic NDT is a widely used method. Since 1985, ultrasonic guided wave technology has been widely used in the national economy due to its long-distance detection and non-destructive characteristics, especially in the field of physical sensors, such as mechanical sensors, velocity sensors, electrical sensors, voice sensors, tactile sensors, image sensors, temperature sensors, accelerometers, distance sensors, and magnetic sensors. As an active detection method, guided wave testing has advantages such as requiring fewer probes, being suitable for detecting thin plate structures, being able to detect defects in inaccessible areas, and having excellent cost-effectiveness. In guided wave testing, the difficulty in obtaining pure Lamb wave modes and the fact that Lamb wave modes are all dispersive limits the widespread application of Lamb wave-based detection technologies. In contrast, the fundamental modes of SH waves are completely non-dispersive, which can significantly simplify the processing of related signals; SH wave propagation is less affected by the surrounding medium, and particle displacement remains in the plane, thereby extending the detection distance; at the same time, SH waves experience fewer mode transitions when encountering defects, reducing the complexity of signal acquisition. Therefore, SH waves are very suitable for detecting defects in board structures.

[0003] With the continuous development of science and technology, the demand for defect detection and assessment technologies in industries such as aerospace, machinery manufacturing, and bridge construction is increasing. Chinese Patent 2016103024416 discloses an SH guided wave method for quantitative detection of thinning defects in flat plates. However, this method can only determine the defect location and fuzzy shape, failing to accurately reconstruct the precise location and shape of the defect in the longitudinal direction of the plate structure. Therefore, to address the shortcomings of this method, this paper proposes an SH guided wave method for quantitative reconstruction of arbitrary defects in flat plates. Compared with existing methods, this method can not only locate the defect on the surface of the flat plate structure but also obtain the thickness information of the defect in the longitudinal direction, as well as the geometric type and number of defects. Simulation results show that this method can accurately reconstruct various defects with appropriate threshold selection; the number of sampling frequency points in the detection domain determines the reconstruction accuracy. Summary of the Invention

[0004] This invention provides a quantitative imaging method for internal defects of scattering SH waveguide plates based on the method of moments. It can invert and reconstruct the stiffness or geometric thickness information of defects by using an ultrasonic transducer array to collect multi-frequency scattered field information, thereby realizing long-distance damage detection of plate structure materials.

[0005] To achieve the above objectives, the present invention adopts the following technical solution: a quantitative imaging method for internal defects of scattering SH waveguide plates based on the method of moments, characterized by comprising the following steps:

[0006] S1. An ultrasonic guided wave of a specific mode is incident on the area to be tested of the plate structure. The time-domain signal of the displacement field of the reflected wave is received by the ultrasonic transducer in the far field and converted into a frequency-domain signal by Fourier transform.

[0007] S2. Based on the modal orthogonality of ultrasonic guided waves, modal decomposition is performed on the frequency domain signal of the reflected wave displacement field to obtain the reflection coefficients corresponding to each guided wave mode.

[0008] S3. Obtain the displacement Green's function of the defect-free plate structure and construct the inverse scattering volume integral equation; introduce the defect index function to obtain the volume integral equation of the defect index function and the reflection coefficient; introduce the Born approximation to linearize the volume integral equation and obtain the linearized volume integral equation.

[0009] S4. The structure of the area to be inspected is homogenized according to the medium at the midpoint of the grid using the method of moments. The volume integral equation is discretized and linearized to obtain a system of linear equations for the defect index function. Multi-frequency data is obtained by frequency sweeping, and a defect reconstruction equation matrix is ​​constructed.

[0010] S5. Use Tikhonov regularization to solve the defect reconstruction equation matrix to obtain a stable solution for the defect index function; use global threshold segmentation for post-processing of the results to reconstruct the precise shape of the plate structure defect.

[0011] In step S1, a piezoelectric ultrasonic transducer is arranged in a comb array at one end of the plate structure. An external electric field perpendicular to the polarization direction is applied to the transducer in an in-plane mode, which can excite shear mode elastic guided waves (SH waves) into the plate structure under test. The modulated windowed cosine function f(t) = w(t)cos(ωt) is used as the excitation function of the guided wave, where ω(t) is the window function, ω is the angular frequency, and t is the time term.

[0012] Simultaneously, the transmitting transducer is used as a receiving probe to receive the guided wave reflection signal caused by the defect scatterer. This ensures the transducer is sufficiently far from the measured area so that the received SH reflected waves are all in propagation modes. Furthermore, the wavelength of the incident wave is much larger than the equivalent size of the defect. Therefore, the scattering phenomenon caused by the defect is weak, and the resulting scattering displacement is much smaller than the magnitude of the incident wave, satisfying the Born approximation. The received reflected wave time-domain signal is then converted to a frequency-domain signal using an inverse Fourier transform.

[0013] In step S2, the scattering signal is modally separated according to modal orthogonality to obtain the reflection coefficients of each mode. Specifically, for the SH wave in the isotropic plate structure medium, x1 direction is taken as the axial direction along the plate structure, x2 direction is taken as the thickness direction along the plate structure, and x3 direction is taken as the anti-plane direction away from the plate structure plane.

[0014] In the SH waveguide, the displacement of a particle is zero in the planes containing the x1 and x2 directions, while the displacement u3 in the x3 direction is not zero and satisfies the Helmholtz equation of motion: in The shear wave velocity of the material. This represents the sum of the second partial derivatives of the variable with respect to the x1 and x2 directions. This represents the second partial derivative of the variable with respect to time.

[0015] For the propagation modes of the SH wave, combined with the stress-free boundary conditions at the upper and lower boundaries of the plate structure, the expression for the displacement of the nth-order SH wave mode is obtained as follows:

[0016] Where ω is the angular frequency, k n Let x be the wavenumber of this mode in the x1 direction. Let x be the wavenumber of this mode in the x2 direction, and let the wavenumbers in both directions satisfy the following conditions: h is the thickness of the plate structure, A n The undetermined coefficients for the amplitude of this mode are given by the wave structure function of the SH wave. Since u3 is independent of x3, the SH wave extends infinitely in the x3 direction.

[0017] For two different modes of the SH guided wave, such as the m-th and n-th modes, the corresponding wave numbers are denoted as k. m and k n Based on the modal orthogonality of SH waves, we can obtain

[0018] Among them, u m,n and τ m,n These represent modal displacement and modal shear stress, respectively, with the asterisk and superscript indicating conjugate. Dirac function δ mn It is 1 when m = n, and 0 otherwise. m That is the integral value of the modal displacement when m=n.

[0019] In the defect detection process, a single-mode incident wave is selected, with a displacement of... A inc This represents the amplitude coefficient of the incident wave.

[0020] The reflected wave signal received by the transducer always contains the total field displacement of various modes with different amplitudes:

[0021] Where Σ m This represents the summation of modal orders m, where m = 1, 2, ... m. ω This corresponds to the order of all modes present in the guided wave at the current incident frequency. Based on the orthogonality of SH wave modes, the reflection amplitude corresponding to the desired mode can be extracted from the signal received by the transducer.

[0022] Where u ref and τ ref This represents the total field displacement and total field shear stress of the reflected field.

[0023] A ref and A inc The frequency-dependent far-field reflection coefficient contains information about the scattered field from structural defects in the index. It is a function of the incident frequency.

[0024] For a flat plate structure, the far-field solution of the Green's function in step S3 specifically includes: for the flat plate structure, a time-harmonic lumped unit load δ'=δ(xx)e is applied at any source point position X. iωt The displacement signal obtained at the transducer receiving point x is denoted as U(x,y).

[0025] The Helmholtz wave equation corresponding to the Green's function of the flat plate structure and the corresponding free surface boundary conditions can be obtained as follows:

[0026]

[0027] By using Fourier transform and the residue theorem to solve for the Green's function of the SH wave and taking the far-field approximation, the expression for the Green's function can be obtained as follows: in It is the wave number of the m-th mode of the SH wave in the x1 direction. c T denoted as the shear wave velocity of the material, b as the half-plate thickness, and μ as the shear modulus.

[0028] Solving for the numerical Green's function of a general plate structure specifically involves either obtaining the numerical Green's function from the scattering forward problem numerically or establishing a corresponding displacement database by pre-measuring the scattering information of the undamaged structure. In the numerical calculation of the forward problem, based on the self-adjoint nature of the Green's function corresponding to the Helmholtz equation, the numerical Green's function for the plate structure can be obtained by applying a unit time-harmonic load at the far field to obtain the displacement response at each point inside the structure. For solving the SH wave propagation forward problem of plate structures, computational efficiency is improved by artificially truncating the solution domain and introducing DtN operator boundary conditions on the virtual boundary.

[0029] Step S4, which constructs the defect reconstruction equation matrix for the SH0 mode, specifically includes: the scattered displacement field at the probe point X can be represented by an integral over the surface of the defect. Here, we consider X to be on the incident side, meaning the observed scattered wave is a reflected wave. According to the reciprocity theorem of elastic dynamics, we have:

[0030]

[0031] Among them, u ref (X) is the reflection displacement at the detection point X, u tot (X)=u ref +u inc Let n be the total field displacement field on the outer surface S of the defect. k Let U(xX) represent the normal vector of the outer surface. Let U(xX) represent the Green's function of the SH wave in the plate structure.

[0032] When the probe point is far from the defect region, the far-field Green's function of the flat plate structure can be analytically written as follows:

[0033] The far-field Green's function can be represented as a linear superposition of a series of different guided wave modes of the SH wave. The first term on the right side of the equation is the 0th order symmetric mode. The summation in Table j below represents all propagation modes, including symmetric and antisymmetric modes.

[0034] When the wavelength of the incident wave is much larger than the equivalent size of the defect, the scattering phenomenon caused by the defect is weak, and the magnitude of the scattering displacement is much smaller than the magnitude of the incident wave. In this case, the incident wave field u can be used. inc Replace the total wave field u tot This means introducing the Born approximation hypothesis.

[0035] At this point, by applying Gauss's integral theorem, the surface integral in the scattering displacement expression can be transformed into a volume integral:

[0036] Taking the SH0 mode as the incident wave, by substituting the incident wave displacement field and the corresponding Green's function expression for the 0th order symmetric mode, and taking the far-field case X1→∞, the reflected wave of the 0th order symmetric mode is... It can be written as:

[0037] Therefore, the scattering coefficient c ref This can be represented as an integral over the defect region:

[0038] Introducing a defect index function O(X), we set it to 1 within the defect region V and 0 elsewhere, thus defining a compact support set over the defect region V. The integration region then extends to the reconstructed region V' of the defect to be plotted, yielding an expression for the scattering coefficients with respect to the defect index function O(X):

[0039] Discretize the reconstructed region and define it within the defect region O. p =1, in other parts of the reconstructed region V' O p =0, we can obtain a discrete expression for the scattering coefficient with respect to the defect index function O(X):

[0040] For the above equation, the same expression can be obtained for every incident frequency. Therefore, calculations can be performed using incident waves of different frequencies (i.e., using different weighting functions w). m By conducting tests, we can obtain a system of test equations, which can be written in matrix form as follows:

[0041]

[0042] Among them, C m and These represent incident angular frequencies ω and ω, respectively. m The reflection coefficient and longitudinal wavenumber of the 0th symmetric mode at time , O (p) and dS (p) Let x1 represent the x1 coordinate value, defect index function value, and area size of the p-th reconstructed element, respectively. In the matrix equation, the number of unknowns is the same as the number of elements in the reconstructed region. Theoretically, by acquiring enough scattered field displacement signals, the unknown defect index vector can be solved, meaning all the geometric information of the defect region in the reconstructed area can be plotted. The accuracy of the reconstruction result depends on the acquisition frequency; the higher the frequency, the higher the accuracy.

[0043] For the case of constructing a reconstructed equation system for higher-order modes of SH waves, specifically including: the incident wave displacement and the corresponding Green's function of the higher-order modes of SH waves can be expressed as follows:

[0044]

[0045]

[0046] At this point, the volumetric discretization expression for the scattering coefficient with respect to the defect index function O(X) after discretizing the reconstructed region is as follows:

[0047]

[0048] The test equations were obtained by testing with incident waves of different frequencies, where [G] mp Difference from the 0th mode:

[0049] The defect index values ​​are obtained by solving the reconstruction matrix using Tikhonov regularization. Global thresholding is then applied to the defect index function for post-processing to reconstruct the precise shape of the defects in the plate structure. Specifically, this includes:

[0050] Due to the introduction of the Born approximation and errors in calculation and observation, the above equations may be ill-posed. To solve this ill-posed system of equations, the Tikhonov regularization method is employed, by adding a regularization term λ||ξ||. 2 using least squares As the loss function, the Tikhonov regularization solution is obtained. Where λ is the regularization parameter.

[0051] The L-curve method is used here to select the value of λ. Since when λ is very small, ||Ax-u|| 2 The value is also small, indicating that the regularized solution matches the perturbed data well. However, x is quite sensitive to changes in the λ parameter, thus indicating an underregularized state, where data error dominates the total error. As λ increases, ||Ax-u|| 2 As the λ parameter increases, the magnitude of x's change with λ decreases, indicating an over-regularized state where regularization error dominates. Therefore, to balance under-regularization and over-regularization, we use ||Ax-u|| 2 and ||x|| 2 Plot the L-curve, and determine the optimal regularization intensity λ from the minimum point of the L-curve. p .

[0052] Finally, the optimal regularization strength λ is determined. p Substituting into the regular expression yields a stable defect index function solution, which indicates the distribution of stiffness in the structure within the reconstructed region.

[0053] The defect index values ​​are then post-processed. Based on the range of the discrete defect index function obtained after reconstructing the reflection coefficients, specific thresholds T are defined for different types of defects:

[0054] The global threshold is obtained through threshold iteration calculation of the defect index function, and the steps are as follows:

[0055] S51. Select an initial estimate for the global threshold T0;

[0056] S52. Use the initial value T0 to perform threshold segmentation, and divide the defect index function into two groups: the function group G1 that is greater than the threshold and the function group G2 that is less than the threshold.

[0057] S53. Calculate the average function values ​​m1 and m2 of G1 and G2 respectively;

[0058] S54. Calculate a new threshold T for m1 and m2. * = (m1+m2) / 2;

[0059] S55. Repeat steps S52-S54 until the difference between two consecutive thresholds T calculated in the iteration is less than the predetermined threshold error e. T Until then, the final global threshold T is determined.

[0060] Using a global threshold T, the reconstructed defect index function is segmented into a complete background object and a defect target object, thus completing the precise quantitative numerical reconstruction of the defect shape.

[0061] The beneficial effects of this invention are:

[0062] This invention constructs a defect reconstruction equation based on the elastic reciprocity theorem, which is universally applicable to general plate structures. Secondly, since the receiver is sufficiently far from the scatterer, it only receives the propagating mode signal; therefore, introducing the Born approximation is reasonable, as it transforms the nonlinear inverse problem into a linear problem for solution. Finally, by iteratively calculating a threshold, the defect index function is globally segmented to obtain a clearer understanding of the defect's specific shape. Testing on different defects ensures the accuracy of the invention's calculations under various conditions, consistently reconstructing the actual defect location and shape with high precision. This invention belongs to the field of quantitative defect reconstruction technology in guided wave detection, overcoming the shortcomings of traditional methods that can only vaguely locate defects. During the detection process, it can simultaneously detect defects over a large area of ​​components, providing an efficient and accurate solution for quantitative ultrasonic guided wave detection, and has significant application value in engineering. Attached Figure Description

[0063] Figure 1 This is a flowchart of the method steps of the present invention;

[0064] Figure 2 This is a model of a defective flat plate structure and a wavelength diagram involved in this invention;

[0065] Figure 3 This is a schematic diagram of the mesh and defect index function of the defect reconstruction region involved in this invention;

[0066] Figure 4 The diagram shows the reflection coefficient data of the SH0 guided wave propagating in a rectangular surface defect, as per the present invention.

[0067] Figure 5 This is a schematic diagram of the L-curve method for reconstructing rectangular surface defects based on SHO waveguides, which is involved in this invention.

[0068] Figure 6 The image shows the reconstruction result of a rectangular surface defect based on SH0 guided wave incident, as per the present invention.

[0069] Figure 7 The image shows the reconstruction result of a circular surface defect based on SHO guided wave incident, as per the present invention.

[0070] Figure 8 The image shows the reconstruction result of a rectangular surface defect based on SH1 guided wave incident, as per the present invention.

[0071] Figure 9 The image shows the reconstruction result of a circular surface defect based on SH1 guided wave incident, as per the present invention.

[0072] Figure 10 The diagram shows the reconstruction result of an internal shape defect based on SHO guided wave incident, as per the present invention.

[0073] Figure 11 The figure shows the reconstruction result of two internal shape defects based on SHO guided wave incident, which is the subject of this invention.

[0074] Figure 12 The figure shows the reconstruction result of three internal shape defects based on SH0 guided wave incidence, which is the subject of this invention.

[0075] Figure 13 The image shows the reconstruction result of three internal shape defects based on SH0 guided wave incident, as per the present invention.

[0076] Figure 14 The diagram shows the scattering coefficient data of the SH0 guided wave propagating in a double circular defect, which is the subject of this invention.

[0077] Figure 15 The image shows the reconstruction result of a double circular defect based on SH0 guided wave incident, as per the present invention. Detailed Implementation

[0078] In the following description, only certain exemplary embodiments are briefly described. As those skilled in the art will recognize, the described embodiments can be modified in various ways without departing from the spirit or scope of the embodiments of the invention. Therefore, the drawings and description are considered to be exemplary in nature and not restrictive.

[0079] The embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0080] like Figure 1 As shown, this example provides a quantitative imaging method for internal defects of a scattering SH waveguide plate based on the method of moments. In some examples, the method includes the following steps:

[0081] S1: An ultrasonic guided wave of a specific mode is incident on the area to be tested of the flat plate structure. The time-domain signal of the displacement field of the reflected wave received by the ultrasonic transducer in the far field is converted into a frequency-domain signal by Fourier transform.

[0082] A piezoelectric ultrasonic transducer is arranged in a comb array at one end of the plate structure. An external electric field perpendicular to the polarization direction is applied to the transducer in an in-plane mode, which can excite shear mode elastic guided waves (SH waves) into the plate structure under test. The modulated windowed cosine function f(t) = ω(t)cos(ωt) is used as the excitation function to excite the SH0 mode guided wave, where ω(t) is the window function, ω is the angular frequency, and t is the time term. Simultaneously, the transmitting transducer is used as the receiving probe to receive the guided wave reflection signal caused by the defect scatterer. The transducer array needs to be arranged far enough from the defect to ensure that the Born approximation assumption is satisfied, so that the received SH reflected waves are all propagation modes. Then, the inverse Fourier transform is performed. The received reflected wave time-domain signal is converted into a frequency-domain signal for processing and analysis.

[0083] S2: Based on the modal orthogonality of the guided wave, the total field of reflection displacement is decomposed into modes to obtain the reflection coefficients corresponding to each mode.

[0084] For SH waves in an isotropic plate structure medium, let x1 be the axial direction along the plate structure, x2 be the thickness direction along the plate structure, and x3 be the antiplane direction away from the plane of the plate structure, such as... Figure 2 As shown. The displacement of a particle in the SH waveguide is zero in the planes containing the x1 and x2 directions, but the displacement u3 in the x3 direction is not zero, and displacement u3 satisfies the Helmholtz equation of motion: in The shear wave velocity of the material. This represents the sum of the second partial derivatives of the variable with respect to the x1 and x2 directions. This represents the second partial derivative of the variable with respect to time.

[0085] For the propagation modes of the SH wave, combined with the stress-free boundary conditions at the upper and lower boundaries of the plate structure, the expression for the displacement of the nth-order SH wave mode is obtained as follows: Where ω is the angular frequency, k n Let x be the wavenumber of this mode in the x1 direction. Let x be the wavenumber of this mode in the x2 direction, and let the wavenumbers in both directions satisfy the following conditions: h is the thickness of the plate structure, A n The undetermined coefficients for the amplitude of this mode are given by the wave structure function of the SH wave. Since u3 is independent of x3, the SH wave extends infinitely in the x3 direction.

[0086] For two different modes of the SH guided wave, such as the m-th and n-th modes, the corresponding wave numbers are denoted as k. m and k n Based on the modal orthogonality of SH waves, we can obtain Among them, u m,n and τ m,n These represent modal displacement and modal shear stress, respectively, with the asterisk and superscript indicating conjugate. Dirac function δ mn It is 1 when m = n, and 0 otherwise. m That is the integral value of the modal displacement when m=n.

[0087] In the defect detection process, a single-mode incident wave is selected, with a displacement of... A inc This represents the amplitude coefficient of the incident wave.

[0088] The reflected wave signal received by the transducer always contains the total field displacement of various modes with different amplitudes: Where Σ m This represents the summation of modal orders m, where m = 1, 2, ... m. ω This corresponds to the order of all modes present in the guided wave at the current incident frequency. Based on the orthogonality of SH wave modes, the reflection amplitude corresponding to the desired mode can be extracted from the signal received by the transducer. Where u ref and τ ref This represents the total field displacement and total field shear stress of the reflected field.

[0089] A ref and A inc The frequency-dependent far-field reflection coefficient contains information about the scattered field from structural defects in the index. It is a function of the incident frequency.

[0090] S3: The displacement Green's function of the defect-free plate structure is written as the sum of series of guided wave modes of each order. Using this Green's function, the far-field scattering coefficient is expressed as the volume integral equation of the defect index function. The Born approximation is introduced to linearize it.

[0091] Because the fundamental solution of the Green's function for elastic waves is complex and inconvenient to directly use to construct the reconstruction equation, and because the incident wave and the defect in actual detection generate rapidly attenuating body waves and unattenuated guided waves, and the distance between the defect and the sensor is relatively far, the sensor only receives the unattenuated portion, the far-field approximation solution can be used for the far-field Green's function. The Green's function for a flat plate is defined as follows: At any source point X, a time-harmonic lumped unit load δ' = δ(xX)e iωt The displacement signal obtained at the transducer receiving point x is denoted as U(x,y).

[0092] The Helmholtz wave equation corresponding to the Green's function of the flat plate structure and the corresponding free surface boundary conditions can be obtained as follows:

[0093]

[0094] The Green's function of the SH wave is solved using Fourier transform and residue theorem, and its far-field approximation is taken. The final expression is: in It is the wave number of the m-th mode of the SH wave in the x1 direction.

[0095] For general plate structures, the Green's function can be obtained through numerical calculation or by establishing a displacement database of the corresponding lossless structure in advance. In numerical calculation, based on the self-adjoint nature of the Green's function corresponding to the Helmholtz equation, the numerical Green's function for the plate structure can be obtained by applying a unit time-harmonic load at the far field and obtaining the displacement response at each point inside the structure. For solving the forward problem of SH wave propagation in plate structures, computational efficiency is improved by artificially truncating the solution domain and introducing DtN operator boundary conditions on the virtual boundary.

[0096] S4: Discretize the volume integral equations using the method of moments to form a system of linear equations about the defect index function. Use frequency sweeping to obtain multi-frequency data to supplement the equations and construct the defect reconstruction equation matrix.

[0097] The scattered displacement field at the detection point X can be represented by an integral over the surface at the defect. Here, we consider X to be on the incident side, meaning the observed scattered wave is a reflected wave. According to the reciprocity theorem of elastic dynamics, we have:

[0098] Among them, uref (X) is the reflection displacement at the detection point X, u tot (X)=u ref +u inc Let n be the total field displacement field on the outer surface S of the defect. k Let U(xX) represent the normal vector of the outer surface. Let U(xX) represent the Green's function of the SH wave in the plate structure.

[0099] When the probe point is far from the defect region, the far-field Green's function of the flat plate structure can be analytically written as follows:

[0100] The far-field Green's function can be represented as a linear superposition of a series of different guided wave modes of the SH wave. The first term on the right side of the equation is the 0th order symmetric mode. The summation in Table j below represents all propagation modes, including symmetric and antisymmetric modes.

[0101] When the wavelength of the incident wave is much larger than the equivalent size of the defect, the scattering phenomenon caused by the defect is weak, and the magnitude of the scattering displacement is much smaller than the magnitude of the incident wave. In this case, the incident wave field u can be used. inc Replace the total wave field u tot This means introducing the Born approximation hypothesis.

[0102] At this point, by applying Gauss's integral theorem, the surface integral in the scattering displacement expression can be transformed into a volume integral:

[0103] Taking the SH0 mode as the incident wave, by substituting the incident wave displacement field and the corresponding Green's function expression for the 0th order symmetric mode, and taking the far-field case X1→∞, the reflected wave of the 0th order symmetric mode is... It can be written as:

[0104] Therefore, the scattering coefficient c ref This can be represented as an integral over the defect region:

[0105] Introducing a defect index function O(X), we set it to 1 within the defect region V and 0 elsewhere, thus defining a compact support set over the defect region V. The integration region then extends to the reconstructed region V' of the defect to be plotted, yielding an expression for the scattering coefficients with respect to the defect index function O(X):

[0106] Discretize the reconstructed region, such as Figure 3 As shown, O is defined within the defect region. p =1, in other parts of the reconstructed region V' O p=0, we can obtain a discrete expression for the scattering coefficient with respect to the defect index function O(X):

[0107] For the above equation, the same expression can be obtained for each incident frequency. Therefore, frequency sweep detection is used to perform calculations using incident waves of different frequencies (i.e., using different weighting functions w). m By conducting tests, we can obtain a system of test equations, which can be written in matrix form as follows:

[0108]

[0109] Among them, C m and These represent incident angular frequencies ω and ω, respectively. m The reflection coefficient and longitudinal wavenumber of the 0th symmetric mode at time , O (p) and dS (p) Let x1 represent the x1 coordinate value, defect index function value, and area size of the p-th reconstructed element, respectively. In the matrix equation, the number of unknowns is the same as the number of elements in the reconstructed region. Theoretically, by acquiring enough scattered field displacement signals, the unknown defect index vector can be solved, meaning all the geometric information of the defect region in the reconstructed area can be plotted. The accuracy of the reconstruction result depends on the acquisition frequency; the higher the frequency, the higher the accuracy.

[0110] The above system of equations may be an ill-conditioned system of equations. The solution to this ill-conditioned system of equations uses the Tikhonov regularization method, and the final solution formula can be expressed as:

[0111] Where λ is the regularization parameter, which can be obtained through the L-curve method, such as... Figure 5 As shown;

[0112] Finally, a stable equation solution defect index function can be obtained, which is used to indicate the distribution of stiffness of the structure in the reconstruction region, and complete the precise quantitative reconstruction of the defect.

[0113] The inventive concept of this invention is as follows:

[0114] First, the reflection coefficients of each mode of the SH wave in the actual defective plate structure are obtained.

[0115] Taking the reconstruction of surface defects in a monolayer isotropic structure using SH waves as an example, such as Figure 2 As shown. Process overview: As... Figure 2 The diagram shows a flat plate with surface defects, where b is half the plate thickness. The solid-lined area represents the reconstructed region, bounded by boundaries Γ1 and Γ2. The maximum width of the defect is w, and the maximum depth is d. The horizontal axis x1 is located at half the plate thickness and points to the right, while the vertical axis x2 is located on the surface of the plate and points downwards. The incident wave uinc Along the positive x1 direction, the interaction with the defect produces a reflected wave u. ref and transmitted wave u tra .

[0116] The transducer is used to excite SH waves and receive the total field signal of the actual reflected field of the defective plate structure. Based on the orthogonality between different mode guided waves, the modes are separated from the total field information of the SH waves, the amplitude of the guided wave modes is extracted, and the reflection coefficient of the desired mode is obtained.

[0117] Incident wave selects a single mode The reflected wave in the far field is: The reflection coefficient is then expressed as The reflection coefficient is a function of frequency, such as... Figure 4 As shown.

[0118] Using the S1 and S2 steps method, the following is given: Figure 4 The reflection coefficient diagram is shown. For rectangular surface defects, the width w = 0.8b and depth d = 0.4b are taken; for semi-circular surface defects, the radius r = 0.5b is equivalent to width w = b and depth d = 0.5b. During reconstruction, to reduce computation time and memory while ensuring sufficient accuracy in describing the defect shape, the mesh of the reconstruction region is divided into at least 40 elements in the plate thickness direction. The number of sampling frequency points is equal to the number of mesh elements in the reconstruction region, ranging from the dimensionless wavenumber Ω = [1.6, 3], where the dimensionless wavenumber is normalized with respect to the frequency-thickness product Ω = ωh / c. T This corresponds to the reflection coefficient data of the first four modes of the SH wave, such as... Figure 4 As shown.

[0119] The far-field Green's function in the defect-free plate structure is obtained by solving.

[0120] For a flat plate structure, the far-field Green's function refers to the time-harmonic lumped unit load δ' = δ(xX)e at any source point location X. iωt The displacement signal obtained at the transducer receiving point x is denoted as U(x,y).

[0121] For the SH-wave plate, the corresponding far-field Green's function can be obtained using the method described in S3 as follows: Where k m Let m be the wave number of the m-th mode of the SH wave in the x1 direction. denoted as the shear wave velocity of the material, b as the half-plate thickness, and μ as the shear modulus.

[0122] By deriving the volume integral equation of the reflection coefficient using the elastic reciprocity theorem and the Born approximation, a discretized reconstruction equation is constructed by introducing a defect reconstruction index. Frequency sweeping is used to supplement multi-frequency information to obtain the reconstruction linear equation matrix.

[0123] Using the method described in step S4, the numerical far-field Green's function and the reflection coefficient of the total scattered field are obtained by incident with SH0 and SH1 guided waves, respectively, and the defect reconstruction equation set is constructed:

[0124]

[0125] Among them, C m and These represent incident angular frequencies ω and ω, respectively. m The reflection coefficient and longitudinal wavenumber of the 0th symmetric mode at time , O (p) and dS (p) These represent the x1 coordinate value, defect index function value, and area size of the p-th reconstructed unit, respectively.

[0126] The defect index value is obtained by solving the reconstruction matrix using Tikhonov regularization. The result post-processing is performed by using global threshold segmentation on the defect index function to reconstruct the precise shape of the plate structure defect.

[0127] The reconstructed equation system matrix is ​​ill-conditioned, and the method described in step S5 is used, employing the Tikhonov regularization method. The solution is performed, where λ is the regularization parameter, obtained through the L-curve method, as shown below. Figure 5 As shown. Solving the system of equations yields the numerical values ​​of the defect index function O for each element within the reconstructed region mesh. p .

[0128] Based on the range of the discrete defect index function obtained after reflection coefficient reconstruction, a global threshold segmentation method is proposed. The global threshold is obtained through threshold iteration calculation of the defect index function. The global mean of the defect index function within the reconstruction region is taken as the initial estimate T0 to obtain the global threshold for each defect. Using the global threshold T, the reconstructed defect index function is segmented into a complete background object and a defect target object, thus completing the final reconstruction result of the defect, as shown below. Figure 6 and Figure 7 As shown, the dashed line represents the actual shape of the defect.

[0129] As can be seen, the obtained reconstruction results are relatively accurate, with a global threshold of 0.9 for rectangular defects and 0.3 for circular defects.

[0130] Relationship between reconstruction results and SH wave modes: Except for the above calculation examples which use the 0th order symmetric mode of SH wave incident and the corresponding reflection coefficient, Figure 8 and Figure 9The paper also demonstrates the reconstruction results obtained using the first-order anti-symmetric mode of the SH wave and the corresponding emission coefficients under the same defect conditions; the dashed line represents the actual defect shape. Compared to the 0th-order symmetric mode, the reconstruction results obtained using the first-order anti-symmetric mode are not accurate enough in obtaining the depth information of the defect. The comparison of different mode incidence results shows that the SSH0 mode is more suitable for use in defect reconstruction algorithms than ASH1.

[0131] The influence of internal defects and different shapes on defect reconstruction: Examining the reconstruction of internal defects using this method, excluding surface defects. For example... Figures 10-13 The images show the reconstruction results of internal circular defects and prismatic crack defects, with the dashed lines representing the actual defect shapes. The circular defect has a radius of 0.25b and a distance of 0.5b from the central axis of the plate; the prismatic crack defect has a width of 0.1b and a depth of 0.1b.

[0132] It can be observed that the global threshold for internal defects is smaller than that for surface defects. Therefore, manual fine-tuning of the iteratively calculated global threshold is necessary to obtain more accurate reconstruction results. For crack defects, a defect discrimination threshold of at least 0.15 is required to clearly depict the actual defect shape. Since the defect size differs significantly in both directions at this point, further refinement of the reconstruction mesh is needed to obtain higher-precision reconstruction results, ensuring that the mesh sufficiently reflects the defect size.

[0133] Multi-defect reconstruction case: Two circular holes with a radius of 0.3b are set in a flat plate structure, 0.5b away from the centerline of the plate. Using SH0 mode incidence, the reflection coefficients of the first four modes are as follows: Figure 14 As shown, the final reconstruction result is as follows: Figure 15 As shown, the dashed line represents the actual shape of the defect.

[0134] As can be seen, the reconstruction result roughly matches the actual defect, but the divided defect area is slightly smaller than the actual defect size. At the same time, there are a few additional non-existent false small defects around it. This is due to the more complex scattering phenomenon caused by the two holes, resulting in a small error.

[0135] The above examples verify the effectiveness and correctness of the method proposed in this invention. This method, as a signal data processing technique in the field of ultrasonic damage detection, utilizes scattered field information to quantitatively reconstruct defects in plate structures, providing data support for subsequent fatigue damage analysis and life prediction, and has broad application prospects.

[0136] Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A quantitative imaging method for internal defects of scattering SH waveguide plates based on the method of moments, characterized in that, It includes the following steps: S1. An ultrasonic guided wave of a specific mode is incident on the area to be tested of the plate structure. The time-domain signal of the displacement field of the reflected wave is received by the ultrasonic transducer in the far field and converted into a frequency-domain signal by Fourier transform. S2. Based on the modal orthogonality of ultrasonic guided waves, modal decomposition is performed on the frequency domain signal of the reflected wave displacement field to obtain the reflection coefficients corresponding to each guided wave mode. S3. Obtain the displacement Green's function of the defect-free plate structure and construct the inverse scattering volume integral equation; introduce the defect index function to obtain the volume integral equation of the defect index function and the reflection coefficient; introduce the Born approximation to linearize the volume integral equation and obtain the linearized volume integral equation. S4. The structure of the area to be inspected is homogenized according to the medium at the midpoint of the grid using the method of moments. The volume integral equation is discretized and linearized to obtain a system of linear equations for the defect index function. Multi-frequency data is obtained by frequency sweeping, and a defect reconstruction equation matrix is ​​constructed. S5. Use Tikhonov regularization to solve the defect reconstruction equation matrix to obtain a stable solution for the defect index function; use global threshold segmentation for post-processing of the results to reconstruct the precise shape of the plate structure defect.

2. The quantitative imaging method for internal defects of scattering SH waveguide plates based on the method of moments according to claim 1, characterized in that, In step S1, a piezoelectric ultrasonic transducer is arranged at one end of the plate structure using a comb array. An external electric field perpendicular to the polarization direction is applied to the transducer, and a shear mode elastic guided wave, i.e. SH wave, is excited into the plate structure under test using an in-plane mode. The modulated windowed cosine function f(t) = w(t)cos(ωt) is used as the excitation function of the guided wave, where ω(t) is the window function, ω is the angular frequency, and t is the time term.

3. The quantitative imaging method for internal defects of scattering SH waveguide plates based on the method of moments according to claim 2, characterized in that, In step S2, for the SH wave in the isotropic plate structure medium, the x1 direction is taken as the axial direction along the plate structure, the x2 direction is taken as the thickness direction along the plate structure, and the x3 direction is taken as the anti-plane direction away from the plane of the plate structure. In the SH waveguide, the displacement of a particle is zero in the planes containing the x1 and x2 directions, while the displacement u3 in the x3 direction is not zero and satisfies the Helmholtz equation of motion: in, The shear wave velocity of the material. This represents the sum of the second partial derivatives of the variable with respect to the x1 and x2 directions. This represents the second partial derivative of the variable with respect to time; Based on the propagation modes of the SH wave and the stress-free boundary conditions at the upper and lower boundaries of the plate structure, the expression for the displacement of the nth-order SH wave mode is obtained as follows: Where ω is the angular frequency, k n Let x be the wavenumber of this mode in the x1 direction. Let x be the wavenumber of this mode in the x2 direction, and let the wavenumbers in both directions satisfy the following conditions: h is the thickness of the plate structure, A n The undetermined coefficients for the amplitude of this mode are given by the wave structure function of the SH wave. Since u3 is independent of x3, the SH wave extends infinitely in the x3 direction; The wavenumbers corresponding to the m-th and n-th modes of the SH guided wave are denoted as k, respectively. m and k n Based on the modal orthogonality of SH waves, we can obtain Among them, u m,n and τ m,n These represent modal displacement and modal shear stress, respectively; the asterisk and superscript indicate conjugate; Dirac function δ mn It is 1 when m = n, otherwise it is 0; V m It is the integral value of the modal displacement when m = n; In the defect detection process, a single-mode incident wave is selected, with a displacement of... A inc This represents the amplitude coefficient of the incident wave; The reflected wave signal received by the transducer contains the total field displacement of each mode with different amplitudes: Where, Σ m This represents the summation of modal orders m, where m = 1, 2, ... m. ω The order of all modes present in the guided wave at the current incident frequency corresponds to the wave order. Based on the orthogonality of SH wave modes, the reflection amplitude corresponding to the desired mode can be extracted from the signal received by the transducer. Where u ref and τ ref This represents the total field displacement and total field shear stress of the reflected field; A ref and A inc The frequency-dependent far-field reflection coefficient contains information about the scattered field from structural defects in the index. It is a function of the incident frequency.

4. The quantitative imaging method for internal defects of scattering SH waveguide plates based on the method of moments according to claim 3, characterized in that, In step S3, a time-harmonic concentrated unit load δ' = δ(xX)e is applied to the flat plate structure at any source point location X. iωt The displacement signal obtained at the transducer receiving point x is denoted as U(x,y); The Helmholtz wave equation corresponding to the Green's function of the flat plate structure and the corresponding free surface boundary conditions are as follows: The Green's function of the SH wave is solved using Fourier transform and residue theorem, and its far-field approximation is taken. The final expression is: in, It is the wave number of the m-th mode of the SH wave in the x1 direction.

5. The quantitative imaging method for internal defects of scattering SH waveguide plates based on the method of moments according to claim 4, characterized in that, In step S4, the scattered displacement field at the detection point X is represented by the integral over the surface of the defect. The detection point X is located on the incident side, meaning the observed scattered wave is a reflected wave. According to the reciprocity theorem of elastic dynamics, we have: Among them, u ref (X) is the reflection displacement at the detection point X, u tot (X)=u ref +u inc Let n be the total field displacement field on the outer surface S of the defect. k U(xX) represents the normal vector of the outer surface; U(xX) represents the Green's function of the SH wave in the plate structure. When the probe point is far from the defect region, the far-field Green's function of the flat plate structure can be written as: The far-field Green's function is represented as a linear superposition of a series of different guided wave modes of the SH wave. The first term on the right side of the equation is the 0th order symmetric mode. The summation in Table j below represents all propagation modes, including symmetric and antisymmetric modes. When the wavelength of the incident wave is much larger than the equivalent size of the defect, the scattering phenomenon caused by the defect is weak, and the magnitude of the scattering displacement is much smaller than the magnitude of the incident wave; in this case, the incident wave field u inc Replace the total wave field u tot This means introducing the Born approximation hypothesis; By applying Gauss's integral theorem, the surface integral in the scattering displacement expression is transformed into a volume integral: Taking the SH0 mode as the incident wave, by substituting the incident wave displacement field and the corresponding Green's function expression for the 0th order symmetric mode, and taking the far-field case X1→∞, the reflected wave of the 0th order symmetric mode is... Written as: Therefore, the scattering coefficient c ref This can be represented as an integral over the defective region: Introducing a defect index function O(X), we set it to 1 within the defect region V and 0 elsewhere, thus defining a compact support set over the defect region V. The integration region is then extended to the reconstructed region V' of the defect to be plotted, yielding an expression for the scattering coefficients with respect to the defect index function O(X): Discretize the reconstructed region and define it within the defect region O. p =1, in other parts of the reconstructed region V' O p =0, thus obtaining the discrete expression for the scattering coefficient with respect to the defect index function O(X): For the above equation, the same expression can be obtained for each incident frequency. By calculating the test equations using incident waves of different frequencies, the system of equations can be written in matrix form as follows: Among them, C m and These represent incident angular frequencies ω and ω, respectively. m The reflection coefficient and longitudinal wavenumber of the 0th symmetric mode at time , O (p) and dS (p) These represent the x1 coordinate value, defect index function value, and area size of the p-th reconstructed unit, respectively.

6. The quantitative imaging method for internal defects of scattering SH waveguide plates based on the method of moments according to claim 5, characterized in that, For higher-order modes of the SH wave, the displacement of the incident wave and the corresponding Green's function are expressed as follows: At this point, the volume integral expression for the scattering coefficient becomes: Similarly, discretizing the reconstructed region, we can write the discrete expression for the scattering coefficient with respect to the defect index function O(X): By using incident waves of different frequencies for testing, a set of test equations was obtained, where [G mp Difference from the 0th order symmetric mode: The solution to the test equations was obtained using the Tikhonov regularization method. Wherein, λ is the regularization parameter, obtained through the L-curve method.

7. The quantitative imaging method for internal defects of scattering SH waveguide plates based on the method of moments according to claim 6, characterized in that, Step S5 specifically includes: Add regularization term λ||ξ|| 2 using least squares As the loss function, the Tikhonov regularized solution is obtained through the minimum gradient method. Where λ≥0 is the regularization parameter; The value of λ is selected using the L-curve method, through ||Ax-u|| 2 and ||x|| 2 Plot the L-curve, and determine the optimal regularization intensity λ from the minimum point of the L-curve. p Finally, the optimal regularization strength λ is determined. p Substituting into the regular expression yields a stable solution for the defect index function. The regularized result of the defect index function obtained after applying Tikhonov regularization is post-processed. A global threshold segmentation method is used to separate the actual structure from the defects, and specific thresholds are defined for different types of defects. Where T is the global threshold; The global threshold is obtained through threshold iteration calculation of the defect index function, and the steps are as follows: S51. Select an initial estimate T0 for the global threshold T; S52. Use the initial value T0 to perform threshold segmentation, and divide the defect index function into two groups: the function group G1 that is greater than the threshold and the function group G2 that is less than the threshold. S53. Calculate the average function values ​​m1 and m2 of G1 and G2 respectively; S54. Calculate a new threshold T for m1 and m2. * = (m1+m2) / 2; S54. Repeat steps S52-S54 until the difference between two consecutive thresholds T calculated in the iteration is less than the predetermined threshold error e. T Until then, determine the final global threshold T; Using a global threshold T, the reconstructed defect index function is segmented into a complete background object and a defect target object, thus completing the precise quantitative numerical reconstruction of the defect shape.

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