Efficient full-matrix imaging method for internal defects of multi-layer annular structure

Abstract

The invention provides a high-efficiency full-matrix imaging method for internal defects of a multilayer annular structure, which comprises the following steps of: acquiring full-matrix data of the multilayer annular structure by using a flexible ultrasonic phased array, obtaining a wave field of a frequency-wavenumber domain through three-dimensional Fourier transform, and reconstructing a wave field of a target layer by using a designed wave field reconstruction operator to obtain the internal defects of the multilayer annular structure. Imaging the reconstructed wave field to obtain an imaging result of the target layer; and repeating the steps to obtain an imaging result of the whole annular structure. Compared with a traditional TFM method based on rays, the imaging method has the advantages that the resolution ratio is higher, and the effect efficiency is effectively improved. In simulation and experiments of the two-layer annular structure, the imaging time of the imaging method is only 1 / 309 and 1 / 35 of that of a TFM method based on rays, and it is indicated that the imaging method has obvious advantages in the detection efficiency of the annular structure and has huge application potential in annular measurement.

Description

Technical Field

[0001] The present invention belongs to the technical field of detection methods, and in particular relates to a high-efficiency full-matrix imaging method for internal defects of a multi-layer annular structure. Background Art

[0002] As a classic non-destructive testing method, ultrasound is a harmless mechanical wave that is often used to measure defects in parts. In order to more clearly identify defects inside parts, ultrasonic signals measured at multiple points are used to image internal defects. Furthermore, annular measurement is a common application scenario depending on the workpiece or sensor. On the one hand, defects inside annular workpieces can be adaptively measured by flexible ultrasonic phased arrays and laser ultrasonics without the need for wedges, and the detection points are distributed in an annular pattern on the workpiece, such as Figure 2 On the other hand, as shown in (a) and (b) Figure 2 In (c), in order to better adapt to the skin, the medical ultrasound array is specially designed with a circular detection surface, and the array elements are distributed in a circular shape on the detection surface. In addition, in order to perform some annular target detection, some ultrasonic phased arrays with circular array elements are designed and manufactured. Figure 2 As shown in (d), the ring distribution of detection points provides some unique advantages and has attracted some scholars.

[0003] Full-matrix data capture (FMC) is a typical ultrasonic phased array inspection data acquisition method, in which the array elements of the ultrasonic phased array sequentially excite ultrasound and simultaneously receive signals. Because FMC provides stable spatiotemporal resolution to address the challenges encountered, it is widely used in ultrasonic phased array measurement of internal defects in workpieces. Existing FMC-based post-processing primarily focuses on planar ultrasonic phased array inspection. In industrial applications, wedges or liquids are used to couple to annular workpieces. However, the highly non-uniform nature of the medium in the inspection area severely impacts imaging efficiency and signal-to-noise ratio, while also complicating the inspection process. To address these deficiencies, flexible ultrasonic phased arrays have emerged as an option for detecting the health of annular structural parts by acquiring and post-processing FMC data.

[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. The system uses the total focusing imaging method (TFM) to sense changes of 0.1 mm in defects. Shih et al. designed and manufactured a flexible ultrasonic phased array that can measure the thickness of a 150°C steel pipe and internal defects of an aluminum alloy using A-scan and FMC signals, respectively. In addition, because laser ultrasonic testing can adapt to the shape of the workpiece and enable the monitoring points to be adaptively distributed on annular structural workpieces, it has attracted the attention of many scholars. When the detection points are distributed in an annular pattern, the wave field theory of the planar ultrasonic phased array based on the plane time theory is no longer fully applicable.

[0005] However, FMC-based imaging methods for internal defect detection in annular workpieces are primarily based on the total focusing imaging method (TFM). Nakahata et al. acquired an FMC dataset using a flexible ultrasonic phased array and imaged defects using TFM with digital apodization and scattering amplitude. Due to differences in ultrasonic sidelobes between 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 annular parts by localizing the relative positions of flexible ultrasonic phased array elements. In this method, a more complex travel time discriminator is used to filter out indirect travel times. However, traditional ray-based FMC-TFM is limited by the ray tracing process and has significant disadvantages in imaging efficiency of multilayer structures. Because the inspection points are distributed in an annular pattern, the wave field can be modeled in polar coordinates. The wave field of the sound source and the received FMC wave field can be reconstructed in polar coordinates using the wave equation. Therefore, reverse time migration (RTM) can be used for defect detection. However, due to the high time cost and computational complexity of solving the full wave equation, RTM is often unacceptable in industrial inspections. Imaging methods developed in the frequency-wavenumber domain often offer significantly greater efficiency than time-domain methods. Therefore, efficient FMC post-processing methods for annular measurements are highly desirable. Summary of the Invention

[0006] Ring structure measurement is widely used in the medical and industrial fields. In the industrial field, the all-focus imaging algorithm has an advantage in the field of full-matrix imaging. However, the imaging efficiency of traditional all-focus imaging methods for multi-layer structures is limited by its high computational complexity.

[0007] To improve the efficiency of full-matrix ultrasonic phased array measurements, this paper mathematically models annular ultrasonic measurements in a polar coordinate system and proposes a highly efficient full-matrix ultrasonic imaging method. In this imaging method, the full-matrix dataset is assigned five-dimensional information. A designed wavefield reconstruction operator is then used to reconstruct the wavefield in the frequency-wavenumber domain. Finally, the reconstructed wavefield is imaged to obtain the image of the detection area. When imaging annular structures, this imaging method offers significant advantages in detection efficiency over existing imaging methods, and produces clearer images.

[0008] An efficient full-matrix imaging method for internal defects of a multi-layer annular structure comprises the following steps:

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

[0010] (2) converting the obtained full matrix data into the frequency-wavenumber domain through three-dimensional Fourier transform to obtain the initial surface wavefield;

[0011] (3) Traverse the annular 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, and obtain the surface wave field of the current layer;

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

[0013] If not, go 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) imaging 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 perform superposition and integration to obtain the imaging result of the current layer;

[0018] (8) Imaging is performed 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 imaging results of all dielectric layers, and superimpose and integrate to obtain imaging results of the multi-layer annular structure.

[0020] In the above-mentioned imaging method, the annular structure can be a hollow structure or a solid structure. When the annular 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 directly performed from step (2). In addition, the imaging method of the present invention is applicable to all situations where the ultrasonic phased array elements are distributed in an annular shape, including an annular object to be detected and an annular ultrasonic sensor (such as a medical ultrasonic probe).

[0021] In actual operation, only the known defect layer can be imaged as needed; that is, the wave field reconstruction operator corresponding to the initial surface wave field is directly applied to obtain the surface wave field of the known defect layer, and steps (4) to (7) are performed to obtain the imaging result of the known defect layer.

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

[0023] Preferably, in step (3), the wavefield reconstruction operator is applied to the initial surface wavefield to obtain the surface wavefield of the current layer as follows:

[0024]

[0025] in, represents the initial surface wavefield; Represents the surface wave field of the current layer; r0 represents the polar diameter of the measurement surface; r y represents the polar diameter of the y-th layer surface; θ e ,θ g Respectively represent the polar angles of the transmitting oscillator and the receiving oscillator; Represents θ respectively 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 qth layer; k q represents the wave number corresponding to the qth layer; n represents the number of cycles in the circumferential direction.

[0026] The calculation formula for wave number k is:

[0027]

[0028] Where sign() is the sign function and c represents the speed of sound.

[0029] Preferably, in step (5), a wavefield reconstruction operator is applied to the surface wavefield of the current layer to obtain the calculation formula of the surface wavefield of the current discrete layer as follows:

[0030]

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

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

[0033] a. Exact 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 diameter r; r0 represents the polar diameter of the measurement surface; k represents the wave number corresponding to the polar diameter r; They represent the n-order first-kind Hankel function and the n-order second-kind Hankel function 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 corresponding imaging results:

[0042]

[0043] Among them, I(x,z) represents the imaging result; θ e ,θ g Respectively represent the polar angles of the transmitting oscillator and the receiving oscillator; Represents θ respectively e ,θ g The corresponding parameters after three-dimensional Fourier transform; ω represents the angular frequency; represents the surface wave field corresponding to the polar radius r; i represents the imaginary unit; n represents the number of cycles in the circumferential direction.

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

[0045]

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

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

[0048] Compared with the prior art, the present invention has the following beneficial effects:

[0049] The present invention's efficient full-matrix imaging method for internal defects in multi-layer annular structures first uses a flexible ultrasonic phased array to acquire full-matrix data of the multi-layer annular structure. A three-dimensional Fourier transform is then used to obtain a wavefield in the frequency-wavenumber domain. A designed wavefield reconstruction operator is then used to reconstruct the wavefield of the target layer. The reconstructed wavefield is then imaged to obtain an imaging result of the target layer. These steps are repeated to obtain an imaging result of the entire annular structure. Compared to traditional ray-based TFM methods, the present invention's imaging method offers higher resolution and significantly improved efficiency. In simulations and experiments on a two-layer annular structure, the imaging time of the present invention was only 1 / 309 and 1 / 35 of that of ray-based TFM methods, respectively. This demonstrates a significant advantage in annular structure detection efficiency and holds great potential for application in annular measurement. For example, the present invention has a wide range of applications, including rapid detection of small defects within industrial pipelines, rapid nondestructive detection of cracks within curved surfaces, and focused imaging of point-like protrusions such as small tumors using medical ultrasound probes (typically at the contact surface of a curved surface). BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Figure 1 is a flow chart of an embodiment of the present invention;

[0051] Figure 2 Schematic diagram of the application scenario of ring measurement; among them: (a) is flexible ultrasonic phased array detection; (b) is laser ultrasonic detection; (c) is medical ultrasonic detection; (d) is angular ultrasonic phased array detection;

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

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

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

[0055] Figure 6 The imaging method of the present invention uses three wave field reconstruction operators and the TFM method to image a single-layer aluminum ring structure. (a), (b), (c), and (d) are the imaging results of the outer surface of the aluminum ring component by the four methods, respectively; (e), (f), (g), and (h) are the imaging results of the inner surface of the aluminum ring component by 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); the imaging results of the proposed method based on Rayleigh-Sommerfeld diffraction are (d) and (h);

[0056] Figure 7 The imaging method of the present invention uses three wavefield reconstruction operators and the TFM method to image a single-layer polyetheretherketone (PEEK) annular structure. (a), (b), (c), and (d) are the imaging results of the outer surface of the PEEK annular component by the four methods; (e), (f), (g), and (h) are the imaging results of the inner surface of the PEEK concave curved block by the four methods; 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).

[0057] Figure 8 The imaging method of the present invention uses three wave field reconstruction operators and the TFM method to image a double-layer polyetheretherketone / aluminum ring structure; wherein: (a), (b), (c), and (d) are the imaging results of the outer surface of the double-layer polyetheretherketone / aluminum ring structure by the four methods; (e), (f), (g), and (h) are the imaging results of the inner surface of the double-layer polyetheretherketone / aluminum ring structure by the four methods; 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 evaluation factor visualization results of all experimental imaging results are shown in the figure. (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 DESCRIPTION

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

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

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

[0062] Specifically, a flexible ultrasonic phased array is used to collect ultrasonic full matrix data of the multi-layer annular structure in the time-space domain, and it is recorded as D(θ e ,θ g ,t); where θ e represents the polar angle and θ of the emitting element g represents the polar angle of the receiving oscillator, and t represents the moment of the time domain signal;

[0063] (2) converting the obtained full matrix data into the frequency-wavenumber domain through three-dimensional Fourier transform to obtain the initial surface wavefield;

[0064] In this step, the three-dimensional full matrix data in the time-space domain is converted to the five-dimensional frequency-wavenumber domain by three-dimensional Fourier transform to obtain the initial surface wave field, which is recorded as in, and Represents θ respectively e ,θ g The corresponding parameters after Fourier transform, ω is the angular frequency;

[0065] (3) Traverse the annular 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, and obtain the surface wave field of the current layer;

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

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

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

[0069]

[0070] in, represents the initial surface wavefield; Represents the surface wave field of the current layer; r0 represents the polar diameter of the measurement surface; r y Represents the polar radius of the y-th layer surface; 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 qth layer; k q represents the wave number corresponding to the qth layer; n represents the number of cycles in the circumferential direction;

[0071] The calculation formula for wave number k is:

[0072]

[0073] Among them, sign() is the sign function; c represents the speed of sound;

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

[0075] a. Exact 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 above formula of wavefield reconstruction operator, They represent the n-order first-kind Hankel function and the n-order second-kind Hankel function 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 set of vectors consisting of the difference in polar angles between any two elements in the flexible ultrasonic phased array on the measurement surface;

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

[0088] Specifically, set the discrete step length to Δr, and get the number of discrete layers to be M 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 obtained as follows:

[0091]

[0092] in, 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 ,n) represents the wave field reconstruction operator corresponding to the jth discrete layer; k y Indicates the wave number corresponding to the current layer; n indicates the number of cycles in the circumferential direction;

[0093] The wavefield reconstruction operator is selected from any form in step (3);

[0094] (6) imaging 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 the surface wavelength are:

[0096]

[0097] Where I(x,z) represents the imaging result; represents the surface wave field corresponding to the polar radius r; i represents the imaginary unit; n represents the number of cycles in the circumferential direction;

[0098] (7) Repeat steps (5) and (6) to obtain the imaging results of all discrete layers, and perform superposition and integration to obtain the imaging result of the current layer;

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

[0100] In this step, imaging is performed using the imaging conditions in step (6) to obtain an imaging result;

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

[0102] The computational complexity of the above imaging method is shown in Table 1 , where Y represents the number of layers of the multi-layer annular structure; is the number of transmitting units; is the number of receiving units; N ω is the number of discrete frequency points within the selected frequency band; I n is the image of the n-th layer ring structure, whose grid is M r ×N θ Pixel; N θ is the number of vibration elements; M n is the number of different dielectric layers; M r is the discretization number in the thickness direction of the region of interest (ROI); I is the I of multiple regions of interest y The final image composed of, and I is set to M r ×N θ Grid; In FMC data, The wavefield is reconstructed using MATLAB's built-in Hankel function. The computational complexity of a single Hankel function is α times the computational complexity of the dot product of the same dimension. The symbol o(.) is used to indicate the order of computational complexity.

[0103] Table 1 Computational complexity of each step of the proposed method

[0104]

[0105]

[0106] Theoretical derivation

[0107] 1. Modeling in the wave field polar coordinate system

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

[0109]

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

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

[0112]

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

[0114]

[0115] Formula (2) can be simplified to an ordinary differential equation by variable separation: 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 circumferential direction, it can be expressed by 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 cycles in the circumferential direction.

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

[0120]

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

[0122]

[0123] Therefore, the solution of Equation (5) can be expressed as an n-order Hankel equation with the independent variable z. This analytical solution can be expressed as the outward wave field P out and the inward wave field Pin The schematic diagram of the traveling wave toward the outside and toward the inside is as follows Figure 3 Here, waves propagating away from the origin are defined as wavefields propagating outward, while wavefields propagating inward represent wavefields propagating toward 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 diameter of the measuring surface; and Represent the nth order first kind Hankel function and nth order second kind Hankel function at the pole radius 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 position can be reconstructed.

[0127] 2. Full matrix capture

[0128] Full Matrix Capture (FMC) is a typical phased array signal acquisition method, in which each element is excited in turn and all elements receive the reflected signal simultaneously within a fixed time period. For a phased array with n elements, n elements can be acquired through FMC. 2 Group A scan signal. The full matrix data is represented as D(θ e ,θ g ,t), it means that the ultrasonic signal excited by the e-th array element is received by the g-th array element. e ,θ g They represent the polar angles of the transmitting oscillator and the receiving oscillator respectively; t represents time.

[0129] like Figure 4 As shown, located at (r0,θ e )’s e-th array element excites ultrasonic signals. When the ultrasonic wave is at (r x ,θ x ) position encounters a scatterer, the scatterer reflects the ultrasonic wave and the ultrasonic wave is located at (r0,θ g ) collects the reflected ultrasonic signal. Therefore, the full matrix wave field collected at the polar position r0 can be expressed as P(r0,θ e ,r0,θ g ,t), can be expressed in the full matrix data set as:

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

[0131] 3. TRON Reconstruction

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

[0133] First, perform a 3D Fourier transform to convert the full matrix data into the frequency-wavenumber domain:

[0134]

[0135] Where i represents the imaginary unit; n represents the number of cycles in the circumferential direction;

[0136] According to formula 6(a), when the A scan signal P is obtained at the pole radius r0, in , the wave fields propagating inwards at different polar radial directions can be reconstructed using the first-kind Hankel function.

[0137] Therefore, the full matrix data wavefield reconstruction starting from r0 is Get in touch:

[0138]

[0139] Wherein, r0 represents the starting position (measurement position) of reconstructing the inward wavefield, and dr represents the polar step length of the inward wavefield reconstruction process.

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

[0141] n=2π / Δφ

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

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

[0144]

[0145] According to the reconstruction process of the inward propagating wave field in equations (6(b)), (9) and (10), the reconstruction of the outward propagating wave field through the second-kind Hankel function can be written as

[0146]

[0147] Although some research has been conducted on the expansion form of the Hankel function, the application of the Hankel function expansion series in the wave field reconstruction process is limited by programming convenience and computational complexity. In order to reduce the computational complexity of the Hankel function, the Hankel function can be simplified to the asymptotic expansion form of the Hankel function of the pressure wave:

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

[0152]

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

[0154]

[0155] For the wave field reconstruction (inner loop) of the discrete defect layer (assuming it is layer y), the surface wave field of the defect layer is first obtained by formula (14a), and then the wave field of each discrete layer is reconstructed by the surface wave field. The reconstruction formula is:

[0156]

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

[0158] 1) Exact 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 completing the wavefield reconstruction at the discrete polar positions, converting the wavefield information into imaging results requires a suitable imaging condition. The explosive reflection model assumes that the scatterer emits ultrasound at time t = 0. Due to the extra dimensions in the full matrix wavefield, the virtual positions of the migrated sound source and receiver should be equal so that the outward and inward propagating waves can match. Therefore, the imaging condition can be defined as:

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

[0167] Among them, I(x,z) and I(r,θ) are the matrices of the imaging results in the Cartesian coordinate system and the polar coordinate system.

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

[0169]

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

[0171]

[0172] The innermost integral superimposes the ω-axis, and the three-dimensional FMC data set in the frequency-wavenumber domain is degraded to two-dimensional data in the wavenumber domain as shown in Equation (18). After performing Fourier transform, the imaging result matrix is ​​obtained in the spatial domain.

[0173] Detection experiment

[0174] In order to evaluate the proposed method in practice, a laboratory experiment was designed and conducted. A 64-element flexible phased array (Olympus, USA) was used as the ultrasonic transducer, with an element spacing of 1 mm and a center frequency of 5 MHz. At the same time, an acquisition card (M2M panther acquisition card from Aidifi, France) was used to collect frequency modulated continuous wave (FMC) data sets with a frequency range of 2.5 MHz to 7.5 MHz, with a sampling frequency of 62.5 MHz. Figure 5 As shown, three structures, single-layer aluminum, single-layer polyetheretherketone (PEEK), and polyetheretherketone / aluminum double-layer ring structure, were imaged respectively; six groups of experiments were arranged here to evaluate the above imaging method, and more experimental parameter setting details are listed in Table 2 below.

[0175] Table 2 Key information of experimental setup

[0176]

[0177] The ray-based total focusing method (TFM) and the above imaging methods use three forms of wave field reconstruction operators G(r0, r, k r ,n) is used to post-process the experimental data set (full matrix data). The imaging results of single-layer aluminum, single-layer polyetheretherketone (PEEK) and double-layer polyetheretherketone / aluminum ring structures are shown in Figure 2. Figure 6 、 Figure 7 and Figure 8 The time taken for each method was determined by running the algorithm five times, and the relative position error range was calculated based on Figure 5 The calculation is based on the size information in . The time consumption, array directivity and relative error range are listed in Table 3.

[0178] like Figure 6 As shown in Figure 2, both the ray-based total focusing method (TFM) and the above imaging methods are able to identify all side drilled holes (SDHs). 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 Figure 6 In (a), all side-drilled holes are detection targets, but the ray-based full-focus method shows that the two side-drilled holes on the left are less obvious than the side-drilled hole on the right. In contrast, the above imaging method can accurately detect all side-drilled holes with less background and near-field artifacts through the three forms of wavefield reconstruction operator G(r0,r,k,n) ( Figure 6 (b) to (d) and (f) to (h)). In addition, from Figure 6 It can be seen from the figure that the imaging results of the above imaging method show good stability in the measurement of both the outer surface and the inner surface.

[0179] Compared to the aluminum ring, the sound velocity of the PEEK ring is lower and the acoustic impedance is different. Figure 7 Background ratio of imaging results Figure 6 It is clearer in Figure 7 Near-field artifact ratio Figure 6 In the external surface measurement, the two side drilled holes (SDH) on the left are difficult to observe because the imaging results of the above imaging method are affected by near-field artifacts ( Figure 7 (b), (c), (d)), whereas they cannot be seen in the imaging results based on the ray-based total focusing method (TFM) ( Figure 7 (a)). However, in the measurement of the inner surface, both the ray-based full-focus method and the proposed method can accurately identify all the side-drilled holes ( Figure 7 (e), (f), (g), (h)). Combined Figure 6Judging from the imaging results in Figure 2, the ray-based full-focus method is more sensitive to the detection position (the detection results are greatly affected by the detection position), so its internal surface measurement is more stable when inspecting side-drilled holes. The imaging results of the above-mentioned imaging methods are basically unaffected by the detection position.

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

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

[0182]

[0183]

[0184] In order to more conveniently compare the evaluation indicators (evaluation factors), Table 3 is visualized, and the results are as follows: Figure 9 shown.

[0185] according to Figure 9 The performance difference between the ray-based total focusing method (TFM) and the above imaging methods becomes more obvious when comparing the evaluation indicators in

[15] . Overall, the above imaging method is able to measure side-drilled holes (SDH) with higher quality in annular structures because its array directivity (APIs) is lower in most cases, such as Figure 9 In addition, the asymptotic expansion of the Hankel function using the wavefield reconstruction operator G(r0,r,k,n) is similar to the exact Hankel function wavefield reconstruction operator G(r0,r,k,n) in terms of measurement capability, as shown in (a). Figure 9 In terms of the average relative position error range in the experiment, the difference between the ray-based full focusing method and the proposed method is small, as shown in (a). Figure 9 As shown in (b) and (c), all methods can locate the side drilling hole with high accuracy. However, the difference in measurement efficiency between the ray-based full focusing method and the above imaging methods is significant, as shown in Figure 9 As shown in (d). In the measurement of a single-layer circular structure, the above imaging method uses three forms of wavefield reconstruction operators G(r0,r,k,n) with a time consumption of only 1 / 2, 1 / 5, and 1 / 42 of the traditional ray-based full-focusing method, respectively. In addition, in the measurement of a two-layer structure, the above imaging method uses three forms of wavefield reconstruction operators G(r0,r,k,n) with a time consumption of 1 / 2, 1 / 6, and 1 / 35 of the traditional ray-based full-focusing method, respectively. Considering measurement efficiency, the above imaging method is superior to the ray-based full-focusing method, and the operator using Rayleigh-Sommerfeld diffraction shows the highest efficiency in both simulations and experiments.

Claims

1. An efficient full-matrix imaging method for internal defects of multi-layer annular structures, characterized by: The following steps are involved: (1) Obtaining full matrix data of the circumferential surface of the multi-layer annular structure; (2) converting the obtained full matrix data into the frequency-wavenumber domain through three-dimensional Fourier transform to obtain the initial surface wavefield; (3) Take any untraversed layer as the current layer, apply the wavefield reconstruction operator to the initial surface wavefield, and obtain the surface wavefield of the current layer; If the current layer is a defective layer, go to step (4); If not, go 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 wavefield reconstruction operator to the surface wavefield of the current layer, and obtain the surface wavefield of the current discrete layer; (6) imaging the surface wave field of the current discrete layer to obtain an imaging result of the current discrete layer; (7) Repeat steps (5) and (6) to obtain the imaging results of all discrete layers, and perform superposition and integration to obtain the imaging result of the current layer; (8) Imaging is performed based on the surface wave field of the current layer to obtain the imaging result of the current layer; (9) Repeat steps (3) to (8) to obtain imaging results of all layers, and superimpose and integrate them to obtain imaging results of the multi-layer annular structure.

2. The high-efficiency full-matrix imaging method for internal defects of a multi-layer annular structure according to claim 1, characterized in that: In step (3), the wavefield reconstruction operator is applied to the initial surface wavefield to obtain the surface wavefield of the current layer as follows: in, represents the initial surface wavefield; Represents the surface wave field of the current layer; r0 represents the polar diameter of the measurement surface; r y represents the polar diameter of the y-th layer surface; θ e ,θ g Respectively represent the polar angles of the transmitting oscillator and the receiving oscillator; Represents θ respectively 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 qth layer; k q represents the wave number corresponding to the qth layer; n represents the number of cycles in the circumferential direction.

3. The high-efficiency full-matrix imaging method for internal defects of a multi-layer annular structure 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 obtained as follows: in, Represents the surface wave field of the current layer; represents the surface wave field of the current discrete layer; r y Represents the polar diameter of the current layer surface; θ e ,θ g Respectively represent the polar angles of the transmitting oscillator and the receiving oscillator; Represents θ respectively e ,θ g The corresponding parameters after three-dimensional Fourier transform; ω represents the angular frequency; j∈[1,M y ],M y represents the number of discrete layers in the current layer; Δr represents the discrete step length; G(r y ,r y +jΔr,k y ,n) represents the wave field reconstruction operator corresponding to the jth discrete layer; k y Indicates the wave number corresponding to the current layer; n indicates the number of cycles in the circumferential direction.

4. The high-efficiency full-matrix imaging method for internal defects of a multi-layer annular structure according to claim 1, characterized in that: In steps (3) and (5), the calculation formulas of the wavefield reconstruction operators are independently selected from the following forms: a. Exact Hankel function form: b. Asymptotically expanded Hankel function form: z=kr c. Based on the Rayleigh-Sommerfeld diffraction formula: 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; k represents the wave number; They represent the n-order first-kind Hankel function and the n-order second-kind Hankel function respectively; i represents the imaginary unit; and n represents the number of periods in the circumferential direction.

5. The high-efficiency full-matrix imaging method for internal defects of a multi-layer annular 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: Among them, I(x,z) represents the imaging result; θ e ,θ g Respectively represent the polar angles of the transmitting oscillator and the receiving oscillator; Represents θ respectively e ,θ g The corresponding parameters after three-dimensional Fourier transform; ω represents the angular frequency; represents the surface wave field corresponding to the polar radius r; i represents the imaginary unit; n represents the number of cycles in the circumferential direction.

6. The high-efficiency full-matrix imaging method for internal defects of a multi-layer annular structure according to claim 1, characterized in that: In steps (3) and (5), traversal is performed along the wave propagation direction.

Images