A high-efficiency full-matrix imaging method for internal defects of a multi-layer ring structure

Abstract

The application provides a kind of efficient full matrix imaging method for internal defects of multilayer annular structure, first using flexible ultrasonic phased array to collect the full matrix data of multilayer annular structure, then the wave field in frequency-wavenumber domain is obtained by three-dimensional Fourier transform, and then the wave field of target layer is reconstructed by using the designed wave field reconstruction operator, and the imaging result of target layer is obtained by imaging the reconstructed wave field;Repeat the preceding steps to obtain the imaging result of the entire annular structure.Compared with the traditional ray-based TFM method, the imaging method of the application has higher resolution, and the efficiency is effectively improved.In the simulation and experiment of two-layer annular structure, the imaging time of the imaging method of the application is only 1 / 309 and 1 / 35 of the ray-based TFM method, indicating that the imaging method of the application has obvious advantages in annular structure detection efficiency, and has great application potential in annular measurement.

Description

Technical Field

[0001] This invention belongs to the field of detection method technology, specifically relating to an efficient full-matrix imaging method for internal defects in a multi-layered ring structure. Background Technology

[0002] As a classic non-destructive testing method, ultrasound is a harmless mechanical wave frequently used to measure defects in parts. To more clearly identify internal defects, multi-point ultrasonic signal measurements are used to image these internal defects. Furthermore, ring-shaped measurement is a common application scenario depending on the workpiece or sensor. On one hand, defects inside ring-shaped workpieces can be adaptively measured using flexible ultrasonic phased arrays and laser ultrasound without the need for wedges, with the detection points distributed in a ring on the workpiece. Figure 2 As shown in (a) and (b). On the other hand, as... Figure 2 (c) To better adapt to the skin, medical ultrasound arrays are specially designed with an arc-shaped detection surface, and the array elements are distributed in an arc shape on the detection surface. Furthermore, to perform the detection of some ring-shaped targets, some ultrasound phased arrays with circularly distributed elements are designed and manufactured as follows: Figure 2 As shown in (d), the circular distribution of detection points offers some unique advantages and has attracted the attention of some scholars.

[0003] Full-matrix data acquisition (FMC) is a typical data acquisition method for ultrasonic phased array testing, where the elements of the ultrasonic phased array sequentially excite ultrasound and simultaneously receive signals. Because FMC provides stable spatiotemporal resolution to address challenges, it is widely used in ultrasonic phased array measurements of internal defects in workpieces. Existing FMC-based post-processing primarily focuses on planar ultrasonic phased array testing. In industrial applications, wedges or liquids are used to couple ring-shaped workpieces. However, the inhomogeneity of the medium in the testing area severely affects imaging efficiency and signal-to-noise ratio, while also complicating the testing process. To overcome these shortcomings, flexible ultrasonic phased arrays have emerged as an option for acquiring and post-processing FMC data to inspect the health of ring-shaped parts.

[0004] To achieve continuous structural health monitoring, Sun et al. proposed a monitoring system based on a flexible phased array and improved its long-term sensitivity. This system uses total focusing imaging (TFM) to detect defect changes as small as 0.1 mm. Shih et al. designed and fabricated a flexible ultrasonic phased array that can measure the thickness of steel pipes and internal defects in aluminum alloys at 150°C using A-scan and FMC signals, respectively. Furthermore, laser ultrasonic testing has attracted considerable attention because it can adapt to the shape of the workpiece, allowing monitoring points to be adaptively distributed on ring-shaped workpieces. However, when the monitoring points are distributed in a ring, the wave field theory of planar ultrasonic phased arrays based on plane-time theory is no longer entirely applicable.

[0005] However, imaging methods for detecting internal defects in ring-shaped workpieces based on FMC are mainly developed using total focusing imaging (TFM). Nakahata et al. acquired an FMC dataset using a flexible ultrasonic phased array and used TFM with digital apodization and scattering amplitude to image the defects. Due to the differences in ultrasonic sidelobes among different types of flexible ultrasonic phased arrays, the macroscopic capabilities of digital apodization and scattering amplitude are affected. Willey et al. developed an imaging method that improves the FMC-TFM imaging resolution of ring-shaped parts by locating the relative positions between elements of the flexible ultrasonic phased array. In this method, a more complex travel time discriminator is used to filter the travel time of indirect arrivals. However, traditional ray-based FMC-TFM suffers from significant disadvantages in multi-layer structure imaging efficiency due to the limitations of the ray tracing process. Since the detection points are distributed in a ring, the wavefield can be modeled in polar coordinates, and the wavefield of the sound source and the received FMC wavefield can be reconstructed in polar coordinates using the wave equation. Therefore, the reverse time migration (RTM) method can be used for defect detection. However, due to the high time cost and computational complexity of solving the full wave equation, RTM is generally unacceptable in industrial inspection. Imaging methods developed in the frequency-wavenumber domain are typically significantly more efficient than time-domain methods. Therefore, there is a high demand for an efficient FMC post-processing method suitable for ring measurements. Summary of the Invention

[0006] Measurement of ring structures is widely used in medical and industrial fields. In the industrial field, total focusing imaging algorithms are dominant in the field of full-matrix imaging, but the imaging efficiency of traditional total focusing imaging methods in multi-layer structures is limited by their high computational complexity.

[0007] To improve the efficiency of full-matrix ultrasonic phased array measurements, this invention mathematically models annular ultrasonic measurements in polar coordinates and proposes a highly efficient full-matrix ultrasonic imaging method. In this method, the full-matrix dataset is endowed with five-dimensional information, and the wavefield is reconstructed in the frequency-wavenumber domain using a designed wavefield reconstruction operator. Finally, imaging of the reconstructed wavefield yields the imaging results for the detection area. When imaging annular structures, this invention's imaging method exhibits a significant advantage in detection efficiency compared to existing methods, and the imaging results are clear.

[0008] An efficient full-matrix imaging method for internal defects in a multi-layered ring structure includes the following steps:

[0009] (1) Obtain the full matrix data of the circumferential surface of the multi-layer ring structure;

[0010] (2) The obtained full matrix data is transformed to the frequency-wavenumber domain through three-dimensional Fourier transform to obtain the initial surface wave field;

[0011] (3) Traverse the ring structure layer by layer along the wave propagation direction, take any untraversed layer as the current layer, apply the wave field reconstruction operator to the initial surface wave field to obtain the surface wave field of the current layer.

[0012] If the current layer is a defective layer, proceed to step (4);

[0013] If not, proceed to step (8);

[0014] (4) Discretize the current layer along the polar radius direction to obtain multiple discrete layers;

[0015] (5) Traverse the current layer layer by layer along the wave propagation direction, take any untraversed discrete layer as the current discrete layer, apply the wave field reconstruction operator to the surface wave field of the previous discrete layer, and obtain the surface wave field of the current discrete layer.

[0016] (6) Image the surface wave field of the current discrete layer to obtain the imaging result of the current discrete layer;

[0017] (7) Repeat steps (5) and (6) to obtain the imaging results of all discrete layers, and then superimpose and integrate them to obtain the imaging result of the current layer;

[0018] (8) Perform imaging based on the surface wave field of the current layer to obtain the imaging result of the current layer;

[0019] (9) Repeat steps (3) to (8) to obtain the imaging results of all media layers, and then superimpose and integrate them to obtain the imaging results of the multi-layer ring structure.

[0020] In the above imaging method, the ring structure can be either hollow or solid. When the ring structure is a single layer, the initial surface wave field is used as the surface wave field of the first discrete layer, and steps (4) to (7) are performed directly from step (2). Furthermore, the imaging method of the present invention is applicable to all cases where the ultrasonic phased array elements are distributed in a ring, including ring-shaped materials being detected and ultrasonic sensors with ring structures (such as medical ultrasound probes).

[0021] In actual operation, imaging can be performed only on the known defect layer as needed; that is, the surface wave field of the known defect layer is obtained by directly applying the wave field reconstruction operator corresponding to the initial surface wave field, and the imaging results of the known defect layer are obtained by performing steps (4) to (7).

[0022] In step (1) above, the full matrix data of the multi-layer ring structure is acquired by a flexible ultrasonic phased array.

[0023] Preferably, in step (3), the wave field reconstruction operator is applied to the initial surface wave field to obtain the calculation formula for the surface wave field of the current layer:

[0024]

[0025] in, Represents the initial surface wave field; The surface wave field of the current layer is represented by r; r0 represents the polar radius of the measurement surface; r y θ represents the polar radius of the y-th surface layer; e θ g These represent the polar angles of the transmitting and receiving elements, respectively. Represent θ e θ g The corresponding parameters after three-dimensional Fourier transform; ω represents the angular frequency; y∈[1,Y], Y represents the number of layers of the ring structure; G(r0,r q ,k q (,n) represents the wavefield reconstruction operator corresponding to the q-th layer; k q represents the wave number corresponding to the q-th layer; n represents the number of cycles in the circumferential direction.

[0026] The formula for calculating the wave number k is:

[0027]

[0028] In the formula, sign() is the sign function; c represents the speed of sound.

[0029] Preferably, in step (5), the surface wave field reconstruction operator is applied to the surface wave field of the current layer, and the calculation formula for the surface wave field of the current discrete layer is as follows:

[0030]

[0031] in, This represents the surface wave field of the current layer; Represents the surface wave field of the current discrete layer; r y θ represents the polar radius of the current layer surface. e θ g These represent the polar angles of the transmitting and receiving elements, respectively. Represent θ e θ g The corresponding parameters after three-dimensional Fourier transform; ω represents the angular frequency; j∈[1,M] y ], M y G(r) represents the number of discrete layers in the current layer; Δr represents the discrete step size; G(r) y ,r y +jΔr,k y (j, n) represents the wavefield reconstruction operator corresponding to the j-th discrete layer; k y This indicates the wave number corresponding to the current layer; n represents the number of cycles in the circumferential direction.

[0032] Preferably, in steps (3) and (5), the wavefield reconstruction operator G(r0,r,k,n) is independently selected from the following forms:

[0033] a. Precise Hankel function form:

[0034]

[0035] b. Asymptotically expanded Hankel function form:

[0036]

[0037] z = kr

[0038] c. Based on the Rayleigh-Sommerfeld diffraction formula:

[0039]

[0040] In the above formula, G(r0,r,k,n) represents the wavefield reconstruction operator corresponding to the polar radius r; r0 represents the polar radius of the measurement surface; and k represents the wavenumber corresponding to the polar radius r. Let i represent the nth order Hankel function of the first kind and the nth order Hankel function of the second kind, respectively; i represents the imaginary unit; and n represents the number of periods in the circumferential direction.

[0041] Preferably, in steps (6) and (8), the surface wave field is imaged using the following imaging conditions to obtain the corresponding imaging results:

[0042]

[0043] Where I(x,z) represents the imaging result; θ e θ g These represent the polar angles of the transmitting and receiving elements, respectively. Represent θ e θ g The corresponding parameters after three-dimensional Fourier transform; ω represents the angular frequency; This represents the surface wave field at a polar radius of r; i represents the imaginary unit; and n represents the number of periods in the circumferential direction.

[0044] In this paper, the number of periods n in the circumferential direction is obtained by the following formula:

[0045]

[0046] Wherein, Δφ represents a vector consisting of the difference in polar angles between any two elements in the measurement plane in a flexible ultrasonic phased array.

[0047] As a preferred option, in steps (3) and (5), the traversal is performed along the wave propagation direction.

[0048] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0049] This invention presents a highly efficient full-matrix imaging method for internal defects in multi-layered ring structures. First, a flexible ultrasonic phased array is used to acquire the full-matrix data of the multi-layered ring structure. Then, a three-dimensional Fourier transform is used to obtain the wavefield in the frequency-wavenumber domain. Next, a designed wavefield reconstruction operator is used to reconstruct the wavefield of the target layer, and the reconstructed wavefield is imaged to obtain the imaging result of the target layer. The aforementioned steps are repeated to obtain the imaging result of the entire ring structure. Compared to the traditional ray-based TFM method, the imaging method of this invention has higher resolution and significantly improved efficiency. In simulations and experiments with two-layered ring structures, the imaging time of the imaging method of this invention is only 1 / 309 and 1 / 35 of that of the ray-based TFM method, indicating that the imaging method of this invention has a significant advantage in the detection efficiency of ring structures and has great application potential in ring measurement. For example, it has wide applicability in the rapid detection of small defects inside industrial pipes, rapid non-destructive detection of cracks inside curved parts, and focused imaging of small tumors and other point-like protrusions using medical ultrasonic probes (usually curved contact surfaces). Attached Figure Description

[0050] Figure 1 This is a flowchart of an embodiment of the present invention;

[0051] Figure 2 The diagram illustrates the application scenarios of ring measurement; where: (a) is flexible ultrasonic phased array detection; (b) is laser ultrasonic detection; (c) is medical ultrasonic detection; and (d) is angular ultrasonic phased array detection.

[0052] Figure 3 This is a schematic diagram of the wave propagation direction; where: (a) is the wave field propagating outward; (b) is the wave field propagating inward;

[0053] Figure 4 This is a schematic diagram of wave propagation in a medium; where: (a) the measurement surface is located on the outer surface of the isotropic medium; (b) the measurement surface is located on the inner surface of the isotropic medium; (c) the measurement surface is located on the outer surface of the multilayer medium; (d) the measurement surface is located on the inner surface of the multilayer medium.

[0054] Figure 5 (d), (e), (j), and (k) are measurement diagrams of single-layer ring structures made of different materials; (a), (b), (g), and (h) are dimensional information of the structures corresponding to (d), (e), (j), and (k); (f) and (l) are measurement diagrams of double-layer structures made of different materials; (c) and (i) are dimensional information of the structures corresponding to (f) and (l).

[0055] Figure 6 The imaging results of the present invention using three wavefield reconstruction operators and the TFM method are shown below. Specifically: (a), (b), (c), and (d) are the imaging results of the outer surface of the aluminum ring component using the four methods; (e), (f), (g), and (h) are the imaging results of the inner surface of the aluminum ring component using the four methods. The imaging results of the ray-based full focusing method (TFM) are (a) and (e); the imaging results of the proposed method based on the exact Hankel function are (b) and (f); the imaging results of the proposed method based on the asymptotic expansion of the Hankel function are (c) and (g); and the imaging results of the proposed method based on Rayleigh-Sommerfeld diffraction are (d) and (h).

[0056] Figure 7 The imaging results of the present invention, using three wavefield reconstruction operators and the TFM method, are shown for a single-layer polyetheretherketone (PEEK) ring structure. Specifically: (a), (b), (c), and (d) are the imaging results of the outer surface of the PEEK ring component using the four methods; (e), (f), (g), and (h) are the imaging results of the inner surface of the concave curved block using the four methods. The imaging results of the ray-based full focusing method (TFM) are (a) and (e); the imaging results of the proposed method based on the exact Hankel function are (b) and (f); the imaging results of the proposed method based on the asymptotic expansion of the Hankel function are (c) and (g); and the imaging results of the proposed method based on Rayleigh-Sommerfeld diffraction are (d) and (h).

[0057] Figure 8 The imaging results of the present invention using three wavefield reconstruction operators and the TFM method on the bilayer polyetheretherketone / aluminum ring structure are shown below. Specifically: (a), (b), (c), and (d) are the imaging results of the outer surface of the bilayer polyetheretherketone / aluminum ring structure using the four methods, respectively; (e), (f), (g), and (h) are the imaging results of the inner surface of the bilayer polyetheretherketone / aluminum ring structure using the four methods, respectively; the imaging results of the ray-based total focusing method (TFM) are (a) and (e); the imaging results of the proposed method based on the exact Hankel function are (b) and (f); the imaging results of the proposed method based on the asymptotic expansion of the Hankel function are (c) and (g); and the imaging results of the proposed method based on Rayleigh-Sommerfeld diffraction are (d) and (h).

[0058] Figure 9The visualization results of the evaluation factors for all experimental imaging results are shown in the figure; where (a) is the array directivity; (b) is the average relative position error range Δx / Δr; (c) is the average relative position error range Δz / Δθ; and (d) is the time cost. Detailed Implementation

[0059] An efficient full-matrix imaging method for internal defects of a multi-layered ring structure, as follows: Figure 1 As shown in the flowchart, the imaging method is constructed into two parts: an inner loop that traverses discrete layers of known defect layers and an outer loop that traverses multilayer dielectric layers of a ring structure. After obtaining FMC data in the time-space domain, the FMC data is converted to the frequency-wavenumber domain using a three-dimensional Fourier transform. In the outer loop, located at interface r y Boundary (surface) wave field between two materials Based on the wavefield reconstruction operator G(r0,r) q ,k q The initial surface wave field measured at n) and the polar radius r0 Calculated. Once the boundary (surface) wavefield of the target medium is reconstructed through the outer loop, It is input into the inner loop. Within the inner loop, the boundary (surface) wavefield of the target medium and the wavefield reconstruction operator G(r) are... y ,r y +jΔr,k y ,n) is used to reconstruct the wavefield on the discrete polar paths. Depending on the imaging conditions, the wavefield on the discrete polar paths is converted into the imaging result I of the target medium. y (r,θ). Finally, after all target media have been processed, the individual imaging results are combined to produce the imaging result I(r,θ).

[0060] Specifically, such as Figure 1 As shown, an efficient full-matrix imaging method for internal defects of a multi-layered ring structure includes the following steps:

[0061] (1) Obtain the full matrix data of the circumferential surface of the multi-layer ring structure;

[0062] Specifically, a flexible ultrasonic phased array was used to acquire full-matrix ultrasonic data of the multi-layered ring structure under test in the time-space domain, and denoted as D(θ). e ,θ g ,t); where θ e Represents the polar angle and θ of the transmitting element. g The polar angle of the receiving element is represented by , and t represents the time of the time-domain signal.

[0063] (2) The obtained full matrix data is transformed to the frequency-wavenumber domain through three-dimensional Fourier transform to obtain the initial surface wave field;

[0064] In this step, the three-dimensional full matrix data in the time-space domain is transformed into the five-dimensional frequency-wavenumber domain by a three-dimensional Fourier transform to obtain the initial surface wave field, denoted as . in, and Represent θ e θ g The corresponding parameters after Fourier transform, where ω is the angular frequency;

[0065] (3) Traverse the ring structure layer by layer along the wave propagation direction, take any untraversed layer as the current layer, apply the wave field reconstruction operator to the initial surface wave field to obtain the surface wave field of the current layer.

[0066] If the current layer is a defective layer, proceed to step (4);

[0067] If not, proceed to step (8);

[0068] In this step, the wavelength reconstruction operator is applied to the initial surface wave field to obtain the calculation formula for the surface wave field of the current layer:

[0069]

[0070] in, Represents the initial surface wave field; The surface wave field of the current layer is represented by r; r0 represents the polar radius of the measurement surface; r y G(r0, r) represents the polar radius of the y-th layer surface; y∈[1,Y], where Y represents the number of layers in the ring structure; q ,k q (,n) represents the wavefield reconstruction operator corresponding to the q-th layer; k q This represents the wave number corresponding to the q-th layer; n represents the number of cycles in the circumferential direction.

[0071] The formula for calculating the wave number k is:

[0072]

[0073] Where sign() is the sign function; c represents the speed of sound;

[0074] The wave field reconstruction operator G(r0,r,k,n) is selected from the following forms:

[0075] a. Precise Hankel function form:

[0076]

[0077] b. Asymptotically expanded Hankel function form:

[0078]

[0079]

[0080] z = kr

[0081] c. Based on the Rayleigh-Sommerfeld diffraction formula:

[0082]

[0083] In the formulas for the wave field reconstruction operator above, Let i represent the nth order Hankel function of the first kind and the nth order Hankel function of the second kind, respectively; i represents the imaginary unit; n represents the number of periods in the circumferential direction.

[0084] The number of periods n in the circumferential direction is obtained by the following formula:

[0085]

[0086] Wherein, Δφ represents a vector consisting of the difference in polar angles between any two elements in the measurement plane in the flexible ultrasonic phased array;

[0087] (4) Discretize the current layer along the polar radius direction to obtain multiple discrete layers;

[0088] Specifically, setting the step size to Δr, we obtain M discrete layers. y ;

[0089] (5) Traverse the current layer layer by layer along the wave propagation direction, take any untraversed discrete layer as the current discrete layer, apply the wave field reconstruction operator to the surface wave field of the current layer, and obtain the surface wave field of the current discrete layer.

[0090] In this step, the wavefield reconstruction operator is applied to the surface wavefield of the current layer, and the calculation formula for the surface wavefield of the current discrete layer is as follows:

[0091]

[0092] in, This represents the surface wave field of the current layer; Represents the surface wave field of the current discrete layer; j∈[1,M] y ];G(r y ,r y +jΔr,k y (j, n) represents the wavefield reconstruction operator corresponding to the j-th discrete layer; k y This indicates the wave number corresponding to the current layer; n represents the number of periods in the circumferential direction.

[0093] The wave field reconstruction operator is selected from any of the forms in step (3);

[0094] (6) Image the surface wave field of the current discrete layer to obtain the imaging result of the current discrete layer;

[0095] The imaging conditions for imaging surface wavelengths are:

[0096]

[0097] Where I(x,z) represents the imaging result; This represents the surface wave field at a polar radius of r; i represents the imaginary unit; n represents the number of periods in the circumferential direction.

[0098] (7) Repeat steps (5) and (6) to obtain the imaging results of all discrete layers, and then superimpose and integrate them to obtain the imaging result of the current layer;

[0099] (8) Apply imaging conditions to the surface wave field of the current layer to obtain the imaging results of the current layer;

[0100] In this step, the imaging conditions in step (6) are used to perform imaging and obtain the imaging results;

[0101] (9) Repeat steps (3) to (8) to obtain the imaging results of all layers, and then superimpose and integrate them to obtain the imaging results of the multi-layer ring structure.

[0102] The computational complexity of the above imaging method is shown in Table 1, where Y represents the number of layers in the multi-ring structure; It refers to the number of transmitting units; N is the number of receiving units; ω It is the number of discrete frequency points within the selected frequency band; I n This is an image of the nth layer of the ring structure, with a grid of M. r ×N θ Pixel; N θ M represents the number of oscillating elements; n It is the number of different dielectric layers; M r It is the number of discretizations along the thickness direction of the region of interest (ROI); I is composed of multiple ROIs. y The final image is composed of I and M. r ×N θ Grid; in FMC data, The wave field is reconstructed using MATLAB's built-in Hankel function. The computational complexity of a single Hankel function is α times that of a dot product of the same dimension. The symbol o(.) is used to denote the order of computational complexity.

[0103] Table 1 shows the computational complexity of each step of the proposed method.

[0104]

[0105]

[0106] Theoretical Derivation

[0107] 1. Modeling of the wave field in polar coordinates

[0108] The wave equation in polar coordinates can be expressed as equation (1):

[0109]

[0110] Where P(r,φ,t) is the sound pressure field; c is the longitudinal wave velocity in the medium; φ,r and t represent the propagation time of the polar angle, polar radius and wavelength, respectively.

[0111] If there is no change in the speed of sound in the circumferential direction, the wave equation can be transformed into the frequency-space domain and can be written as:

[0112]

[0113] Where P(r,φ,ω) represents the wave field in the frequency-space domain; ω is the angular frequency; and k is the wave number, expressed as:

[0114]

[0115] Equation (2) can be simplified to an ordinary differential equation by separating variables P(r,φ,ω)=R(r)Φ(φ); where R(r) and Φ(φ) represent functions of r and φ, respectively.

[0116] Considering the periodic variation of Φ along the circumference, it can be expressed using trigonometric functions as follows:

[0117] Φ=Acos(nφ)+Bsin(nφ)(3)

[0118] Where A and B are uncertain coefficients, and n is the number of periods in the circumferential direction.

[0119] Therefore, the partial differential equation in equation (2) can be rewritten as:

[0120]

[0121] Let z = kr, then equation (4) can be transformed into:

[0122]

[0123] Therefore, the solution to equation (5) can be expressed as an nth-order Hankel equation with z as the independent variable. This analytical solution can be expressed as the outward wave field P. out and the inward wave field Pin Schematic diagrams of traveling waves to the outside and inside are shown below. Figure 3 As shown in the diagram. Here, wave propagation away from the origin is defined as an outward wave field, while wave propagation inward indicates wave field propagation towards the origin.

[0124] P in and P out The expressions are written as follows:

[0125]

[0126] Among them, A n (φ,ω) represents the complex amplitude; r0 represents the polar radius of the measuring surface; and These represent the nth-order Hankel function of the first kind and the nth-order Hankel function of the second kind at a radius of r0, respectively; since A n (φ,ω) is independent of the polar radius, once A n Once (φ,ω) is determined, the wave field at any polar radius position can be reconstructed.

[0127] 2. Full matrix capture

[0128] Full matrix acquisition (FMC) is a typical method for acquiring signals from a phased array. Each element is sequentially excited, and the reflected signals are simultaneously received by all elements within a fixed time period. For a phased array with n elements, FMC can acquire n... 2 Group A scan signal. The full matrix data is represented as D(θ) e ,θ g When ,t), it means that the ultrasonic signal excited by the e-th array element is received by the g-th array element. θ e θ g t represents the polar angles of the transmitting and receiving elements, respectively; t represents time.

[0129] like Figure 4 As shown, located at (r0, θ) e The e-th element of the array excites an ultrasonic signal when the ultrasonic wave is in (r x ,θ x When the ultrasonic wave encounters a scattering object at position (r0, θ), the scattering object reflects the ultrasonic wave, and the ultrasonic wave is located at (r0, θ). g The reflected ultrasonic signal is acquired by the g-th element of the array. Therefore, the full matrix wavefield acquired at the polar radius position r0 can be represented as P(r0, θ). e ,r0,θ g ,t), in the full matrix dataset, can be represented as:

[0130] P(r0,θ e ,r0,θ g,t)=D(θ e ,θ g ,t)(7)

[0131] 3. Wave Field Reconstruction

[0132] After collecting the full matrix data P(r0,θ) e ,r0,θ g After ,t), the wave field of the measurement area can be reconstructed.

[0133] First, a three-dimensional Fourier transform is performed to convert the full matrix data into the frequency-wavenumber domain:

[0134]

[0135] In the formula, i represents the imaginary unit; n represents the number of periods in the circumferential direction;

[0136] According to equation 6(a), when the A scan signal P is obtained at the polar radius r0... in The wavefield propagating inside different polar radials can be reconstructed using the first-type Hankel function.

[0137] Therefore, the full-matrix data wavefield reconstruction starting from r0 is... Connecting them:

[0138]

[0139] Where r0 represents the starting position (measurement position) for reconstructing the inward wavefield, and dr represents the polar radius step size in the inward wavefield reconstruction process.

[0140] In FMC data, the nth-order Hankel function of the first kind is related to the angle difference between different array elements and is represented as:

[0141] n = 2π / Δφ

[0142] Wherein, Δφ represents a vector consisting of the difference in polar angles between any two elements in the measurement plane in a flexible ultrasonic phased array.

[0143] Therefore, the wave field propagating inward can be expressed by a first-kind Hankel function as:

[0144]

[0145] Based on the reconstruction process of the inwardly propagating wavefield according to equations (6(b)), (9), and (10), the reconstruction of the outwardly propagating wavefield using the second-type Hankel function can be written as

[0146]

[0147] Although some research has focused on the expansion of the Hankel function, its application in wavefield reconstruction is limited by programming convenience and computational complexity. To reduce the computational complexity of the Hankel function, it can be simplified to the asymptotic expansion of the Hankel function for pressure waves:

[0148] Let kr = z, the asymptotic expansion of a Hankel function H(z,n) is expressed as:

[0149]

[0150] Here, n represents the order of the original Hankel function, and the absolute value of z is exponentially greater than n.

[0151] Furthermore, the Rayleigh-Sommerfeld diffraction formula was introduced to further simplify the Hankel function, and the simplified expression H(r0,r,k,n) was written as

[0152]

[0153] Combining equations (10) and (11), in the outer loop, the wavefield reconstructed from the measured polar radius position r0 to the wavefield of any y layer can be summarized as follows:

[0154]

[0155] For the wavefield reconstruction (inner loop) of the discretized defect layer (assumed to be layer y), the surface wavefield of the defect layer is first obtained by formula (14a), and then the wavefield of each discrete layer is reconstructed by the surface wavefield. The reconstruction formula is as follows:

[0156]

[0157] Where G(r0,r,k,n,ω) represents the wavefield reconstruction operator, and is defined as follows:

[0158] 1) Precise Hankel function form:

[0159]

[0160] (2) Asymptotically expanded Hankel function form:

[0161]

[0162] (3) Based on the Rayleigh-Sommerfeld diffraction formula:

[0163]

[0164] 4. Imaging conditions

[0165] After reconstructing the wavefield at the discrete polar radius positions, converting the wavefield information into imaging results requires suitable imaging conditions. The explosion reflection model assumes that the scatterer emits ultrasonic waves at t=0. Due to the redundant dimensions in the full matrix wavefield, the virtual positions of the migrated sound source and receiver should be equal to match the outward and inward propagating waves. Therefore, the imaging conditions can be defined as:

[0166] I(x,z)=I(r,θ)=∫P(r,θ,r,θ,ω)dω(16)

[0167] Where I(x,z) and I(r,θ) are the matrices of the imaging results in Cartesian and polar coordinate systems, respectively.

[0168] P(r,θ,r,θ,ω) can be obtained by the following formula:

[0169]

[0170] Combining equations (16) and (17), the imaging conditions based on the FMC data measured at position r0 (measurement surface) can be expressed as:

[0171]

[0172] The innermost integrals are superimposed on the ω-axis, and the three-dimensional FMC dataset is reduced in the frequency-wavenumber domain to two-dimensional wavenumber domain data as shown in Equation (18). After performing the Fourier transform, the imaging result matrix is ​​obtained in the spatial domain.

[0173] Detection experiment

[0174] To evaluate the proposed method in practice, laboratory experiments were designed and conducted. A 64-element flexible phased array (Olympus Corporation, USA) was used as the ultrasonic transducer, with an element spacing of 1 mm and a center frequency of 5 MHz. A data acquisition card (EM2M panther card from IDPIE, France) was used to collect frequency-modulated continuous wave (FMC) data ranging from 2.5 MHz to 7.5 MHz, with a sampling frequency of 62.5 MHz. Figure 5 As shown, imaging was performed on three structures: single-layer aluminum, single-layer polyether ether ketone (PEEK), and polyether ether ketone / aluminum bilayer ring structure. Six sets of experiments were arranged to evaluate the above imaging methods. More experimental parameter settings are listed in Table 2 below.

[0175] Table 2 Key Information Regarding Experimental Setup

[0176]

[0177] The total focusing method based on rays (TFM) and the aforementioned imaging methods employ three forms of wavefield reconstruction operators G(r0,r,k) respectively. r (n) is used for post-processing of the experimental dataset (full matrix data). Imaging results for monolayer aluminum, monolayer polyetheretherketone (PEEK), and bilayer PEEK / aluminum ring structures are shown below. Figure 6 , Figure 7 and Figure 8 As shown. The time taken for each method was determined by running the algorithm five times, and the relative position error range was based on... Figure 5 The calculations were based on the dimensions in the image. The time consumption, array directivity, and relative error range are listed in Table 3.

[0178] like Figure 6 As shown, both the ray-based total focusing method (TFM) and the aforementioned imaging method can identify all side-drilled holes (SDH). However, in the imaging results of the ray-based total focusing method, background artifacts appear on both the outer and inner surfaces. Figure 6 (a) and (e)). Especially in the measurement of the outer surface, although in Figure 6 In (a), all side-drilled holes are detected, but the ray-based full-focusing method shows that the two side-drilled holes on the left are less noticeable than those on the right. In contrast, the imaging method described above, using three forms of wavefield reconstruction operator G(r0,r,k,n), can accurately detect all side-drilled holes with less background and near-field artifacts. Figure 6 (b)~(d) and (f)~(h)). Furthermore, from Figure 6 As can be seen, the imaging results of the above imaging method show good stability in the measurement of both the outer and inner surfaces.

[0179] Compared to aluminum ring components, polyetheretherketone (PEEK) ring components have lower sound velocity and different acoustic impedances. Therefore, although Figure 7 Background ratio of the imaging results Figure 6 It is clearer in the middle, but Figure 7 Near-field artifacts in Figure 6 This is even more pronounced in the external surface measurements. Due to the influence of near-field artifacts on the imaging results of the aforementioned imaging methods, the two side-drilled holes (SDH) located on the left side are difficult to observe. Figure 7 In (b), (c), and (d), these features are not visible in the imaging results of the total focusing method (TFM). Figure 7 (a)). However, in measurements of the inner surface, both the ray-based full-focusing method and the proposed method can accurately identify all side-drilled holes (a). Figure 7 (e), (f), (g), (h)). Figure 6From the imaging results, the X-ray-based full-focusing method is more sensitive to the detection position (the detection results are greatly affected by the detection position), therefore its internal surface measurement is more stable when detecting side-drilled holes. In contrast, the imaging results of the aforementioned methods are largely unaffected by the detection position.

[0180] like Figure 8 As shown, since the double-layered circular structure is composed of different materials, although the above imaging method is effective on the outer surface ( Figure 8 (b)~(d)) and inner surface ( Figure 8 In measurements (f) to (h), side-drilled holes (SDH) can be clearly and accurately detected, but due to the interface between different materials, echo artifacts still appear periodically. However, as mentioned above, the ray-based total focusing method (TFM) is less effective for internal surface measurements (f) to detect side-drilled holes (SDH). Figure 8 The performance in (e) is better than that measured on the outer surface. Figure 8 The method is more stable in (a)). In the measurement of the outer surface, the ray-based full-focusing method cannot identify side-drilled holes. Overall, the proposed method performs well in terms of imaging results, and can clearly and accurately measure side-drilled holes in single-layer and double-layer circular structures.

[0181] Table 3 Evaluation factors for all experimental imaging results

[0182]

[0183]

[0184] To facilitate comparison of evaluation indicators (evaluation factors), Table 3 is visualized, and the results are as follows: Figure 9 As shown.

[0185] according to Figure 9 A comparison of evaluation metrics reveals a more pronounced performance difference between the ray-based total focusing method (TFM) and the aforementioned imaging methods. Overall, the aforementioned imaging method is able to measure side-drilled holes (SDH) with higher quality in annular structures because, in most cases, its array directivity (APIs) is lower, such as... Figure 9 As shown in (a). Furthermore, the asymptotic expansion of the Hankel function using the wavefield reconstruction operator G(r0,r,k,n) is similar in measurement capability to the exact Hankel function wavefield reconstruction operator G(r0,r,k,n), as shown in [example]. Figure 9 As shown in (a). Regarding the range of average relative position error in the experiment, the difference between the ray-based full-focusing method and the proposed method is small, as... Figure 9 As shown in (b) and (c), all methods can locate side boreholes with high precision. However, the ray-based full-focusing method differs significantly from the aforementioned imaging methods in measurement efficiency, such as... Figure 9 As shown in (d), in the measurement of a single-layer circular structure, the imaging method described above, using three forms of wavefield reconstruction operator G(r0,r,k,n), takes only 1 / 2, 1 / 5, and 1 / 42 of the time required by the traditional ray-based full-focusing method, respectively. Furthermore, in the measurement of a double-layer structure, the imaging method described above, using three forms of wavefield reconstruction operator G(r0,r,k,n), takes only 1 / 2, 1 / 6, and 1 / 35 of the time required by the traditional ray-based full-focusing method, respectively. Considering measurement efficiency, the imaging method described above is superior to the ray-based full-focusing method, and the operator using Rayleigh-Sommerfeld diffraction exhibits the highest efficiency in both simulation and experiments.

Claims

1. A highly efficient full-matrix imaging method for internal defects of a multi-layered ring structure, characterized in that, Includes the following steps: (1) Obtain the full matrix data of the circumferential surface of the multi-layer ring structure; (2) The obtained full matrix data is transformed to the frequency-wavenumber domain through three-dimensional Fourier transform to obtain the initial surface wave field; (3) Take any untraversed layer as the current layer, apply the wave field reconstruction operator to the initial surface wave field, and obtain the surface wave field of the current layer; If the current layer is a defective layer, proceed to step (4); If not, proceed to step (8); (4) Discretize the current layer along the polar radius direction to obtain multiple discrete layers; (5) Take any untraversed discrete layer as the current discrete layer, apply the wave field reconstruction operator to the surface wave field of the current layer, and obtain the surface wave field of the current discrete layer; (6) Image the surface wave field of the current discrete layer to obtain the imaging result of the current discrete layer; (7) Repeat steps (5) and (6) to obtain the imaging results of all discrete layers, and then combine them to obtain the imaging result of the current layer; (8) Image the current layer based on the surface wave field to obtain the imaging result of the current layer; (9) Repeat steps (3) to (8) to obtain the imaging results of all layers, and then superimpose and integrate them to obtain the imaging results of the multi-layer ring structure; In step (3), the wavefield reconstruction operator is applied to the initial surface wavefield to obtain the calculation formula for the surface wavefield of the current layer: ； in, Represents the initial surface wave field; This represents the surface wave field of the current layer; Indicates the polar diameter of the measuring surface; Indicates the first The polar diameter of the layer surface; , These represent the polar angles of the transmitting and receiving elements, respectively. , They represent , The corresponding parameters after three-dimensional Fourier transform; Indicates angular frequency; Indicates the number of layers in a ring structure; This represents the wavefield reconstruction operator corresponding to the q-th layer; This represents the wave number corresponding to the q-th layer; n represents the number of cycles in the circumferential direction. In steps (3) and (5), the calculation formulas for the wavefield reconstruction operator are each independently selected from the following forms: a. Precise Hankel function form: ； b. Asymptotically expanded Hankel function form: ； ； ； c. Based on the Rayleigh-Sommerfeld diffraction formula: ； ； In the above formula, This represents the wavefield reconstruction operator corresponding to the polar radius r; The polar radius of the measurement surface is represented by ; k represents the wavenumber. , Let represent the first-order Hankel function and the second-order Hankel function of the nth order, respectively; i represents the imaginary unit; n represents the number of periods in the circumferential direction.

2. The efficient full-matrix imaging method for internal defects of multi-layer ring structures according to claim 1, characterized in that, In step (5), the wavefield reconstruction operator is applied to the surface wavefield of the current layer, and the calculation formula for the surface wavefield of the current discrete layer is as follows: ； in, This represents the surface wave field of the current layer; This represents the surface wave field of the current discrete layer; Indicates the polar radius of the current layer surface; , These represent the polar angles of the transmitting and receiving elements, respectively. , They represent , The corresponding parameters after three-dimensional Fourier transform; Indicates angular frequency; Indicates the number of discrete layers in the current layer; Indicates the distance from the walk; This represents the wavefield reconstruction operator corresponding to the j-th discrete layer; This indicates the wave number corresponding to the current layer; n represents the number of cycles in the circumferential direction.

3. The efficient full-matrix imaging method for internal defects of a multi-layered ring structure according to claim 1, characterized in that, In steps (6) and (8), the imaging conditions used for imaging based on the surface wave field are as follows: ； in, Indicates the imaging results; , These represent the polar angles of the transmitting and receiving elements, respectively. , They represent , The corresponding parameters after three-dimensional Fourier transform; Indicates angular frequency; This represents the surface wave field at a polar radius of r; i represents the imaginary unit; and n represents the number of periods in the circumferential direction.

4. The efficient full-matrix imaging method for internal defects of a multi-layered ring structure according to claim 1, characterized in that, In steps (3) and (5), the traversal is performed along the wave propagation direction.

Images