Ceramic logarithmic-spiral bearing roller phased-array ultrasonic detection method and system

Abstract

This invention discloses a phased array ultrasonic testing method and system for ceramic logarithmic generatrix bearing rollers. The method includes performing surface parameterization and metric calculation on the generatrix curve data to obtain the geometric parameters of the generatrix surface of the ceramic logarithmic roller under test; obtaining the true time delay matrix; obtaining the linear mapping optimization objective function between the physical space and the computational space; solving the linear mapping optimization objective function by augmented Lagrange decoupling and low-dimensional subspace projection to obtain the transformation matrix; performing inverse mapping reconstruction on the distortion-free imaging amplitude and inverting and reconstructing to obtain the physical space inversion reconstruction image of the ceramic logarithmic roller under test that restores the true propagation characteristics; and performing defect detection processing on the physical space inversion reconstruction image to obtain the internal defect detection results of the ceramic logarithmic roller under test. This solves the problem that existing technologies cannot simultaneously achieve high-precision detection of micro-defects on the bearing roller surface and sub-micron-level quantitative evaluation of the logarithmic generatrix profile under non-contact conditions.

Description

Technical Field

[0001] This invention relates to the field of ultrasonic testing technology, and in particular to a phased array ultrasonic testing method and system for ceramic logarithmic busbar bearings. Background Technology

[0002] Rolling bearings are core supporting components of rotating machinery, and their performance directly affects the reliability and lifespan of the entire machine. Ceramic bearings, due to their excellent properties such as high hardness, wear resistance, corrosion resistance, and insulation, exhibit significant advantages under extreme conditions such as high speed, high temperature, corrosion, and strong magnetic fields, and have become a key basic component in high-end equipment. Logarithmic generatrix rollers, as the core rolling elements of ceramic bearings, have generatrix profiles designed according to logarithmic curves to optimize contact stress distribution, reduce edge effects, and improve bearing load capacity and fatigue life. However, the geometric characteristics of these rollers are extremely complex; their profiles change non-linearly along the axial direction, and their surface curvature changes continuously and gradually, posing a significant challenge to traditional ultrasonic testing methods.

[0003] In the field of ultrasonic nondestructive testing, conventional testing methods typically assume that the workpiece under test has a regular geometric shape and isotropic material properties, with the ultrasonic beam propagating in a straight line inside the workpiece. Imaging algorithms are also built based on this simplified model. However, ceramic materials (such as silicon nitride and zirconium oxide) generally exhibit significant anisotropy. Their elastic modulus, sound velocity, and acoustic impedance vary spatially with differences in grain orientation and sintering processes, causing changes in ultrasonic velocity, path bending, and energy scattering during propagation, resulting in severe beam distortion. Simultaneously, the complex curved surface profile of the logarithmic generatrix roller further exacerbates the refraction, reflection, and mode conversion effects of ultrasonic waves, leading to timing discrepancies, amplitude attenuation, and phase distortion in the echo signal. Ultimately, this results in problems such as geometric distortion, reduced resolution, and misjudgment of defects in ultrasonic imaging.

[0004] Existing ultrasonic testing technologies are relatively mature for simple rotating bodies or flat, isotropic workpieces, including phased array ultrasonics, full-focusing methods, and synthetic aperture focusing techniques. However, the application of these methods in the inspection of logarithmic generatrix rollers is greatly limited: on the one hand, complex curved surfaces cause the incident angle of the sound beam to continuously change along the roller axis, rendering the delay law under the conventional plane wave assumption ineffective; on the other hand, the anisotropy of the material causes the sound velocity field to exhibit tensor characteristics, resulting in systematic deviations in sound path calculations and image reconstruction algorithms based on scalar sound velocities. Although some researchers have attempted to correct the sound beam path through ray tracing or finite element simulation, the computational cost is high and it is difficult to meet the real-time inspection requirements in industrial settings; there are also solutions that use water immersion focusing probes in conjunction with mechanical scanning, but these suffer from drawbacks such as unstable coupling, poor surface adaptability, and low inspection efficiency.

[0005] Therefore, existing ultrasonic full-focusing techniques are mostly designed for regular structures in isotropic homogeneous media. When directly applied to the inspection of ceramic logarithmic generatrix rollers, the distortion of sound rays caused by curved surface refraction, anisotropic sound velocity, and in-service stress coupling results in defect location errors exceeding 0.5 mm, which cannot meet the quantitative accuracy requirements for subsurface cracks in some bearing roller standards. How to construct an ultrasonic propagation model suitable for complex geometries and anisotropic material coupling conditions, and develop corresponding beam distortion correction and imaging reconstruction methods to achieve high-precision, high-efficiency non-destructive testing and accurate classification of internal defects in ceramic logarithmic generatrix bearing rollers, has become a pressing technical challenge in this field. Summary of the Invention

[0006] This invention addresses the shortcomings of existing technologies by providing a phased array ultrasonic testing method and system for ceramic logarithmic busbar bearings.

[0007] To solve the above-mentioned technical problems, the present invention provides the following technical solution: A phased array ultrasonic testing method for roller bearings in a ceramic logarithmic busbar bearing includes the following steps: Acquire the generatrix curve data and anisotropic acoustic parameters of the logarithmic roller of the ceramic under test, determine the configuration information of the transducer array, and acquire the time-domain full matrix signal based on the transducer array scan. The configuration information of the transducer array includes at least the transceiver array element arrangement parameters. The surface parameterization and measurement calculation of the bus curve data are performed to obtain the geometric parameters of the bus surface of the logarithmic roller of the ceramic under test. Based on the geometric parameters of the bus surface, the anisotropic acoustic parameters and the arrangement parameters of the transceiver array elements, a real propagation model is constructed. Based on the real propagation model, the sound field characteristics are solved, the sound ray path is traced and the propagation time delay is integrated to obtain the real time delay matrix that characterizes the real propagation characteristics. We construct matrix isomorphism transformation relationships under low-rank sparse constraints for the real and ideal time delay matrices, and obtain the linear mapping optimization objective function between the physical space and the computational space. The objective function of linear mapping optimization is decoupled and spatially projected to obtain a transformation matrix, which represents the mapping relationship between physical space and computational space; The time-domain full matrix signal is converted into the frequency-domain full matrix signal. The propagation distortion correction and full-focus imaging processing are combined with the transformation matrix to obtain the distortion-free imaging amplitude in the computational space. Then, inverse mapping reconstruction is performed to obtain the physical space inversion reconstruction image of the tested ceramic logarithmic roller that restores the true propagation characteristics. Defect detection is performed on the physically inverted and reconstructed image to obtain the internal defect detection results.

[0008] As one possible implementation method, the generatrix curve data is parameterized and metrically calculated to obtain the geometric parameters of the generatrix surface of the ceramic logarithmic roller under test, including the following steps: Based on the logarithmic generatrix curve of the logarithmic roller of the ceramic under test, the logarithmic function equation of the generatrix curve is constructed. Based on the logarithmic function equation, the surface sweep is performed with the roller axis as the rotation axis to determine the surface parameter equation of the logarithmic generatrix surface. By taking partial derivatives of the surface parametric equations along the axial and circumferential parameters respectively, the axial and circumferential basis vectors of the tangent space are obtained. Based on the axial and circumferential basis vectors, the inner product operation is performed to obtain the components of the metric tensor, and then the first basic form of the metric tensor is determined. By calculating surface line elements using metric tensors, the local arc length infinitesimal elements of the logarithmic generatrix surface are quantified. Based on the cross product operation of axial basis vectors and circumferential basis vectors, the normal vector field is determined to characterize the local orientation of the logarithmic generatrix surface, thereby obtaining the geometric parameters of the logarithmic generatrix surface.

[0009] One possible implementation includes the following steps: The wave equation is rewritten based on the constitutive relation of anisotropic elastic media, and the Christofel equation is determined by combining the plane wave solution. The characteristic equation of the Christofel equation is solved to obtain the quasi-longitudinal wave characteristic and two quasi-transverse wave characteristics. The anisotropic phase velocity field and the implicit equation of the slow surface are obtained by combining the slowness vector. The wave equation is transformed and combined with parameterized variables to obtain the characteristic line equation. The Hamiltonian is quantized and simplified in the curvilinear coordinate system to construct the canonical equation. Based on the arrangement parameters of the transceiver array elements and combined with the geometric parameters of the logarithmic generatrix surface, the incident position of the transceiver array elements in the curvilinear coordinate system is calibrated. Directional parameterization and local coordinate transformation are performed to determine the magnitude of the slowness and to construct the calculation relationship between the refractive index and the corresponding partial derivatives, so that the sound propagation path is simultaneously constrained by the surface geometric features and the anisotropic propagation characteristics of the material. The canonical equations are expressed as a set of first-order ordinary differential equations in state vector form. The slope and approximate solution are calculated to obtain the local error. Based on the local error and the allowable error tolerance, the scaling factor is obtained, and then the step size is updated. Based on the incident position of the transceiver array elements and the updated step size, a propagation delay model along the sound ray path is constructed and transformed into a parameterized form. The integral interval is mapped to the standard interval through linear transformation. The real delay matrix integrated along the real ray path and the ideal delay matrix based on the approximation of the plane interface of the uniform medium are constructed to obtain the delay error matrix characterizing the difference between the real propagation characteristics and the ideal propagation characteristics. Based on the time delay error matrix, a correspondence between the real propagation characteristics and the ideal propagation characteristics is constructed, and error analysis and convergence analysis are performed on the time delay error matrix. If the analysis is satisfactory, the real time delay matrix is ​​determined based on the time delay error matrix.

[0010] As one possible implementation method, the following steps are also included: The symmetry of the time delay matrix is ​​verified. If it is not symmetric, outlier interpolation is performed to obtain the final time delay matrix. The outlier interpolation is used to maintain the propagation continuity of the true time delay matrix.

[0011] One possible implementation includes the following steps: A matrix isomorphism transformation relationship is constructed between the real time delay matrix and the ideal time delay matrix to maintain the correspondence between the real propagation characteristics and the ideal propagation characteristics. The transformation matrix is ​​determined so that the real time delay matrix and the ideal time delay matrix satisfy an approximately linear relationship. The transformation matrix is ​​used to construct the mapping relationship between the real propagation characteristics and the ideal propagation characteristics. A sparsity prior is introduced for the approximately linear relationship to construct a sparsity penalty term; a low-rank prior is introduced for the transformation matrix to construct a low-rank regularization term; wherein, the sparsity prior is used to suppress local perturbations of propagation error, and the low-rank prior is used to maintain the overall consistency of propagation characteristics. Based on matrix isomorphism, sparse penalty terms, and low-rank regularization terms, we construct an objective function for optimizing the linear mapping between physical and computational spaces that maintains the correspondence between real and ideal propagation characteristics.

[0012] One possible implementation includes the following steps: An augmented Lagrangian function is constructed based on the real and ideal time delay matrices. The optimization objective function is decoupled into a closed-form update of the singular value threshold of the transformation matrix and an independent Lasso coordinate descent solution of the sparse coefficient matrix through the alternating direction multiplier method. A convergence criterion based on the original residual and the dual residual is constructed to identify the termination condition of the algorithm. By truncating singular value decomposition, the real time delay matrix and the ideal time delay matrix are projected onto a low-dimensional subspace. A subspace transformation matrix is ​​constructed in the low-dimensional subspace, and an upper bound of the low-dimensional subspace approximation error is obtained. The mapping consistency between the real propagation characteristics and the ideal propagation characteristics is maintained by using the subspace transformation matrix. The new data is split into an interpretable part of the old space and entirely new information. After extracting the orthogonal directions of the entirely new information, the original singular values, projected components, and new direction weights are concatenated into a core matrix. Only the core matrix is ​​decomposed, and the decomposition results are mapped back to the high-dimensional subspace to obtain a transformation matrix that satisfies propagation consistency. The high-dimensional subspace consists of new directions and old directions. The transformation matrix is ​​used for subsequent propagation correction processing of the full-matrix signal in the frequency domain.

[0013] As one possible implementation, the method includes the following steps: The time-domain signal captured by the phased array full matrix of the logarithmic roller of the ceramic under test is transformed into the frequency domain to obtain the actual acquired frequency domain full matrix signal; Propagation distortion correction processing is performed on the actual acquired frequency domain full matrix signal and transformation matrix to map the complex boundary of the ceramic logarithmic roller under test and the propagation distortion in the anisotropic medium into an equivalent propagation model in the computational space, and the correction signal in the computational space is obtained. Based on the calculation of the one-way time delay from each array element to the internal imaging point of the logarithmic roller of the ceramic under test in a uniform medium, an ideal steering vector is constructed. The correction signal and the ideal steering vector are subjected to full-focus imaging processing, and the two-way time delay signals of all transmit and receive array element pairs are coherently superimposed to obtain the distortion-free imaging amplitude in the computational space. The distortion-free imaging amplitude is inversely mapped and reconstructed, and the transformation matrix is ​​used to restore the computational spatial imaging results to the physical space to maintain the consistency of the propagation mapping relationship, so as to obtain the physical space inversion reconstruction image of the tested ceramic logarithmic roller that restores the true propagation characteristics.

[0014] As one possible implementation method, the following steps are also included: The physical space inversion and reconstruction image is subjected to surface grayscale correction based on logarithmic generatrix surface geometric parameters to extract defect candidate regions. Based on the defect candidate region, connected components are marked in the generatrix surface coordinate system. Geometric features along the generatrix direction, circumferential direction and radial direction are extracted to determine the axial position, circumferential span and subsurface depth of the defect in the ceramic logarithmic roller under test, and the defect quantification data is obtained. Based on geometric features and the ratio of quasi-longitudinal wave to quasi-transverse wave scattering amplitudes and the deflection angle of the slowness vector corresponding to the anisotropic phase velocity field, the defect candidate region is identified by type, and the defect type identification results distributed along the logarithmic generatrix are obtained. The defect types include at least cracks, near-surface pores or subsurface layer delamination pores. Based on defect quantification data and defect type identification results, defect detection, quantification, and classification results data are generated.

[0015] As one possible implementation, the transformation matrix is ​​determined by the matrix isomorphism transformation relationship between the real time delay matrix and the ideal time delay matrix; The propagation distortion correction process invokes the matrix isomorphism transformation relationship to map the full frequency domain matrix signal to the computation space; The inverse mapping reconstruction process calls the regularized generalized inverse mapping corresponding to the matrix isomorphism transformation relationship to restore the imaging results in the computational space to the physical space, so that the propagation distortion correction process and the inverse mapping reconstruction process can complete the data conversion between the computational space and the physical space based on the same matrix isomorphism relationship.

[0016] A phased array ultrasonic testing system for the rollers of a ceramic logarithmic busbar bearing includes: The data acquisition module is used to acquire the generatrix curve data and anisotropic acoustic parameters of the logarithmic roller of the ceramic under test, determine the configuration information of the transducer array, and acquire the time-domain full matrix signal based on the transducer array scan. The configuration information of the transducer array includes at least the transceiver array element arrangement parameters. The matrix calculation module is used to perform surface parameterization and measurement calculation on the generatrix curve data to obtain the generatrix surface geometric parameters of the logarithmic roller of the ceramic under test; based on the generatrix surface geometric parameters, anisotropic acoustic parameters and transceiver array element arrangement parameters, a real propagation model is constructed, and based on the real propagation model, sound field characteristics are solved, sound ray path is traced and propagation time delay is integrated to obtain the real time delay matrix characterizing the real propagation characteristics; A deterministic module is constructed to establish a matrix isomorphism transformation relationship under low-rank sparse constraints for the real and ideal time delay matrices, thereby obtaining the linear mapping optimization objective function between the physical space and the computational space. The constraint solving module is used to decouple and spatially project the linear mapping optimization objective function to obtain a transformation matrix, which represents the mapping relationship between the physical space and the computational space. The imaging inversion module is used to convert the time-domain full matrix signal into the frequency-domain full matrix signal, and perform propagation distortion correction and full-focus imaging processing in combination with the transformation matrix to obtain the distortion-free imaging amplitude in the computational space. Then, inverse mapping reconstruction is performed to obtain the physical space inversion reconstruction image of the tested ceramic logarithmic roller that restores the true propagation characteristics. The defect detection module is used to detect defects in the physically inverted and reconstructed image to obtain the internal defect detection results.

[0017] This invention, by adopting the above technical solutions, has significant technical effects: This invention achieves high-precision and high-reliability ultrasonic testing of internal defects in the rollers of ceramic logarithmic busbar bearings, with defect location accuracy reaching sub-millimeter level, meeting the engineering requirements for early detection of subsurface fatigue cracks. It solves the problem that existing technologies cannot achieve high-precision detection and accurate classification of internal defects in the rollers of ceramic logarithmic busbar bearings under complex curved surface anisotropic coupling conditions. Attached Figure Description

[0018] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0019] Figure 1 This is a schematic diagram of the overall process of the method of the present invention; Figures 2-4 This is a schematic diagram of the technical process of an embodiment of the method of the present invention; Figure 5 This is a schematic diagram comparing the results of the present invention and the conventional method; Figure 6 This is a classification diagram of a specific embodiment; Figure 7 This is a schematic diagram of the configuration information of the transducer array of the present invention; Figure 8 This is a schematic diagram of the overall structure of the system of the present invention. Detailed Implementation

[0020] The present invention will be further described in detail below with reference to the embodiments. The following embodiments are explanations of the present invention, but the present invention is not limited to the following embodiments.

[0021] Example 1: A phased array ultrasonic testing method for rollers in ceramic logarithmic busbar bearings, such as... Figure 1 As shown, it includes the following steps: S100. Acquire the generatrix curve data and anisotropic acoustic parameters of the logarithmic roller of the ceramic under test, determine the configuration information of the transducer array, and acquire the time-domain full matrix signal based on the transducer array scan. The configuration information of the transducer array includes at least the transceiver element arrangement parameters. A schematic diagram of the array can be found in [reference needed]. Figure 7 , Figure 7 A rough schematic diagram is provided; S200. Perform surface parameterization and measurement calculation on the generatrix curve data to obtain the generatrix surface geometric parameters of the logarithmic roller of the ceramic under test. Based on the generatrix surface geometric parameters, anisotropic acoustic parameters and transceiver array element arrangement parameters, construct a real propagation model, and perform sound field feature solving, sound ray path tracing and propagation delay integration based on the real propagation model to obtain the real delay matrix characterizing the real propagation characteristics. S300. Construct matrix isomorphism transformation relationships under low-rank sparse constraints for the real time delay matrix and the ideal time delay matrix to obtain the linear mapping optimization objective function between the physical space and the computational space. S400. Decouple and spatially project the objective function of the linear mapping optimization to obtain the transformation matrix, which represents the mapping relationship between the physical space and the computational space. S500: Convert the time-domain full matrix signal into the frequency-domain full matrix signal, combine the transformation matrix to perform propagation distortion correction processing and full-focus imaging processing, obtain the distortion-free imaging amplitude in the computational space, and then perform inverse mapping reconstruction to obtain the physical space inversion reconstruction image of the tested ceramic logarithmic roller that restores the true propagation characteristics. S600. Perform defect detection on the physically inverted and reconstructed image to obtain the internal defect detection results.

[0022] The technical solution provided in this application addresses the problems of sound propagation path distortion, phase distortion, and imaging spatial distortion caused by the coupling of the variable curvature geometry of the logarithmic generatrix surface of the rollers in ceramic logarithmic generatrix bearings and the anisotropic acoustic properties of the ceramic material. Through geometric and acoustic coupling modeling, matrix isomorphic transformation optimization, and physical and computational space mapping correction, it ultimately transforms the image into a physically reconstructed image. Geometric features such as the axial position, circumferential span, and subsurface depth of defects are extracted in the generatrix surface coordinate system. Combined with acoustic features such as the quasi-longitudinal wave to quasi-transverse wave scattering amplitude ratio and the slowness vector deflection angle corresponding to the anisotropic phase velocity field, it can effectively distinguish typical defect types such as cracks distributed along the logarithmic generatrix, near-surface porosity, and subsurface delamination. The defect location accuracy can reach sub-millimeter level in the roller generatrix surface coordinate system, meeting the engineering requirements for early detection of fatigue cracks in the subsurface layer of bearing rollers.

[0023] The technical solution of this invention can accurately map ultrasonic detection signals from complex curved surfaces and anisotropic coupling media to an ideal computational space and complete physical space reconstruction. This allows the detection results to be directly used for the three-dimensional distribution characterization of internal defects in ceramic logarithmic rollers, the evaluation of equivalent crack size, and the prediction of remaining life. This, in turn, supports online screening, quality grading, and service replacement decisions for rollers, thereby improving the reliability and safety of bearing systems.

[0024] In this application, the matrix isomorphic transformation relationship is a bidirectional structure-preserving mapping between the physical space and the computational space constructed based on regularized generalized inverse, rather than a reversible linear transformation in the usual sense.

[0025] In one embodiment, step S200 involves performing surface parameterization and measurement calculations on the generatrix curve data to obtain the geometric parameters of the generatrix surface of the ceramic logarithmic roller under test, including the following steps: Based on the logarithmic generatrix curve of the logarithmic roller of the ceramic under test, the logarithmic function equation of the generatrix curve is constructed. Based on the logarithmic function equation, the surface sweep is performed with the roller axis as the rotation axis to determine the surface parameter equation of the logarithmic generatrix surface. By taking partial derivatives of the surface parametric equations along the axial and circumferential parameters respectively, the axial and circumferential basis vectors of the tangent space are obtained. Based on the axial and circumferential basis vectors, the inner product operation is performed to obtain the components of the metric tensor, and then the first basic form of the metric tensor is determined. By calculating surface line elements using metric tensors, the local arc length infinitesimal elements of the logarithmic generatrix surface are quantified. Based on the cross product operation of axial basis vectors and circumferential basis vectors, the normal vector field is determined to characterize the local orientation of the logarithmic generatrix surface, thereby obtaining the geometric parameters of the logarithmic generatrix surface.

[0026] See the following specific examples: In this technical field, the surface of a ceramic logarithmic generatrix bearing roller, i.e., a ceramic logarithmic roller, is a surface of revolution, and its generatrix curve is a logarithmic curve. In three-dimensional Euclidean space, the logarithmic function equation containing the generatrix curve is expressed as: The generatrix curve is rotated around the central axis, i.e., the roller axis, i.e., the surface is swept, and the surface parametric equation is determined, which is expressed as: ,in, The logarithmic function equation representing the location of the generatrix curve. Represents radial coordinates, Representing the value boundaries and parameters respectively. It is determined based on the design of the ceramic logarithmic roller. For circumferential angle parameters, Represents a surface of revolution. Represents a position vector. Indicates the circumferential angle parameter; By determining the surface parametric equations, we obtain the axial and circumferential basis vectors of the tangent space, which are expressed as follows: , , This represents the axial basis vector. Represents the circumferential basis vector; The first fundamental form, the metric tensor, is represented as: ; Furthermore, the line elements on the surface can be obtained, represented as: ,because Therefore, the normal vector field is represented as: ,in, These are the components of the metric tensor. This indicates that the parametric curve network is an orthogonal network. Representing the arc length infinitesimal element, since ,but , Represents a unit normal vector field. It is a cross product. This indicates the Euclidean modulus.

[0027] In this embodiment, a surface parameterization method based on differential geometry is used to geometrically model the ceramic logarithmic roller. Compared to the traditional approximation that simplifies the roller to a regular surface such as a cylinder or cone, this method can accurately characterize the curvature characteristics of the logarithmic generatrix surface that continuously changes along the axial direction. Specifically, by constructing a logarithmic function equation and sweeping the surface with the roller axis as the axis of rotation, the resulting surface parameter equation strictly preserves the original geometric shape of the logarithmic generatrix.

[0028] The surface elements calculated in this way provide an accurate geometric kernel for subsequent sound propagation path integration, while the determination of the normal vector field provides a locally orthogonal coordinate framework for describing the incident, reflection, and refraction behavior of sound waves on variable curvature boundaries. This embodiment not only constructs a curvilinear coordinate system that naturally adapts to the logarithmic generatrix surface, enabling subsequent sound field feature solving and sound ray path tracing to be performed under boundary conditions that are strictly consistent with the geometry of the roller entity, but also fundamentally eliminates the systematic deviation of the sound propagation path introduced by geometric approximation, improving the time delay calculation accuracy and imaging spatial positioning accuracy of ultrasonic testing under complex curved surface boundary conditions.

[0029] In one embodiment, based on the geometric parameters of the busbar surface, anisotropic acoustic parameters, and transducer array configuration information, sound field characteristics are solved, sound ray paths are traced, and propagation delay is integrated to obtain the true time delay matrix, including the following steps: The wave equation is rewritten based on the constitutive relation of anisotropic elastic media, and the Christofel equation is determined by combining the plane wave solution. The characteristic equation of the Christofel equation is solved to obtain the quasi-longitudinal wave characteristic and two quasi-transverse wave characteristics. The anisotropic phase velocity field and the implicit equation of the slow surface are obtained by combining the slowness vector. The wave equation is transformed and combined with parameterized variables to obtain the characteristic line equation. The Hamiltonian is quantized and simplified in the curvilinear coordinate system to construct the canonical equation. Based on the arrangement parameters of the transceiver array elements and combined with the geometric parameters of the logarithmic generatrix surface, the incident position of the transceiver array elements in the curvilinear coordinate system is calibrated. Direction parameterization and local coordinate transformation are performed to determine the magnitude of the slowness and to construct the calculation relationship between the refractive index and the corresponding partial derivative. The canonical equations are expressed as a set of first-order ordinary differential equations in state vector form. The slope and approximate solution are calculated to obtain the local error. Based on the local error and the allowable error tolerance, the scaling factor is obtained, and then the step size is updated. Based on the incident position of the transceiver array elements and the updated step size, a propagation delay model along the sound ray path is constructed and transformed into a parameterized form. The integration interval is mapped to the standard interval through linear transformation. The real delay matrix integrated along the real ray path and the ideal delay matrix based on the approximation of the plane interface of the uniform medium are constructed to obtain the delay error matrix. Perform error analysis and convergence analysis on the time delay error matrix. If it passes the test, determine the true time delay matrix based on the time delay error matrix.

[0030] Here, we will first introduce the parameters included in anisotropic acoustic parameters. In existing technologies, the acoustic parameter system of anisotropic materials can be divided into three main categories: basic acoustic parameters, elastic constitutive parameters, and anisotropic characterization parameters. These acoustic parameters generally include longitudinal wave velocity, transverse wave velocity, quasi-longitudinal wave velocity, quasi-transverse wave velocity, acoustic attenuation coefficient, acoustic impedance, and material density. In anisotropic elastic media, the wave equation can be written as: For silicon nitride ceramic materials In general, its symmetry is hexagonal, and the number of independent elastic constants is reduced to 5, then:

[0031] in, Represents the displacement component (i=1,2,3, representing the i-th component in three-dimensional space). Indicates the density of the material. This represents the fourth-order elastic stiffness tensor. Indicates time, They represent the spatial coordinate components, , This represents the k-th displacement component. , The notation for the elastic constant matrix (6×6) has only 5 independent elements for the hexagonal crystal system, namely: , The (6,6) element represents the elastic matrix of the hexagonal crystal system.

[0032] Solving the characteristic equation of the Christofer equation yields the quasi-longitudinal wave characteristic and two quasi-transverse wave characteristics. Combining these with the slowness vector, the anisotropic phase velocity field and the implicit equation of the slowness surface are obtained. The specific process is as follows: Let the plane wave solution be Substituting this into the wave equation, we obtain the Christoffel equation, which can be expressed as: Wherein, the Christofer matrix is The characteristic equation is expressed as: Three real eigenvalues ​​are given. This corresponds to one quasi-longitudinal wave and two quasi-transverse waves; the anisotropic phase velocity field is the characteristic equation. The three eigenvalues ​​obtained by solving The corresponding phase velocity With direction The distribution of changes. Define the slowness vector as... The component is The implicit equation of the slow surface is: .

[0033] In this embodiment, Represents the amplitude vector components (i=1,2,3). Represents the imaginary unit. Represents the wave vector. This indicates that the radial parameters are consistent with the previous radial coordinates. Represents angular frequency. Indicates time, Represents the vector dot product. Let K represent the Christofel matrix (ik=1,2,3). This represents the unit vector indicating the wave vector direction. These represent the wave vector direction components, Indicates the wave vector modulus. Kroneckerdelta ( =1 if i=k, otherwise 0). Represents phase velocity, Indicates density, This represents the amplitude vector (the eigenvector in the Christofel equation). These represent the quasi-longitudinal wave and two quasi-transverse waves, namely the first quasi-transverse wave and the second quasi-transverse wave. These represent the corresponding phase velocities. Represents the slowness vector. Indicates the slowness component, Denotes the direction component of the wave vector, and det represents the determinant. This represents the time-travel function.

[0034] The wave equation is transformed and combined with parameterized variables to obtain the characteristic line equation. The Hamiltonian is then quantized and simplified in curvilinear coordinates to construct the canonical equation. The specific process is as follows: Under the geometric acoustic approximation, the wave equation degenerates into a functional equation: By introducing parameterized variables and substituting them into the equation of the function, the characteristic line equation of the function is: Wherein, Hamilton is taken as: ,in, Represents the time-travel function. Represents the time-travel function Spatial coordinates gradient, For refractive index, The direction of the wavefront normal. For slowness vector, For contravariant metric tensors, Represents the ray parameters (parameterized variables that monotonically increase along the ray). Represents the Hamiltonian. Indicates the gradient magnitude. Representing the slowness vector respectively and spatial coordinates gradient operator, These represent the components of the slowness vector; On the ceramic logarithmic roller revolution surface The above uses curvilinear coordinates Then the Hamiltonian is concretized as: ,in, ,Depend on Simplified to: ,in addition, Therefore, the canonical equation is: ,in, This represents the surface of revolution of a ceramic logarithmic roller. Represents specific curve coordinates ( (These represent the radial parameter and the circumferential angle parameter, respectively). Representing the coordinates of the curve respectively Conjugate generalized momentum (i.e., the slow component). These represent the surface contravariance metric components (the inverse matrix of the first basic form). These are the components of the metric tensor. Indicates refractive index, Represents the square of the refractive index. These represent the polar angle (spherical coordinates) and the azimuth angle (spherical coordinates), respectively. This represents the axial basis vector. This represents the circumferential basis vector.

[0035] In a specific embodiment, based on the array element arrangement parameters and combined with the logarithmic generatrix surface geometric parameters, the incident position of the array elements in the curvilinear coordinate system is calibrated. Then, direction parameterization and local coordinate transformation are performed to determine the magnitude of the slowness and to establish the calculation relationship between the refractive index and the corresponding partial derivatives. The specific process is as follows: For anisotropic media, the refractive index It is the normal direction of the wavefront. The function, let the wavefront normal direction be... The direction cosine in the local coordinate system is: Then, in the local frame of the curved surface Below, the direction cosine transform is: The magnitude of the slowness is given by the solution to the Christoffel equation: The refractive index is: Its partial derivative is: Define the state vector: A system of first-order ordinary differential equations, i.e., a system of differential equations, is: ,in: ;in, This represents the axial basis vector. Represents the circumferential basis vectors. Let J denote the unit normal vector, and let J denote the Jacobian matrix from the global coordinate system to the local coordinate system. Represents the direction cosine in the local coordinate system. The eigenvalues ​​represent the phase velocity (slowness magnitude) obtained from the solutions of the Christophier equations, and the eigenvalues ​​represent the eigenvalues ​​(real numbers) of the Christophier matrix. Indicates the reference speed of sound. Representing the refractive index versus surface coordinates The partial derivatives, Represents the state vector. This represents the vector field function on the right-hand side of the state equation. Represents the ray parameters (independent variables).

[0036] In another embodiment, the canonical equations are represented as a system of first-order ordinary differential equations in state vector form. The slope and approximate solution are calculated to obtain the local error. Based on the local error and the allowable error tolerance, the scaling factor is obtained, and then the step size is updated. The specific process is as follows: First, let's introduce the Butcher coefficient table, which is represented as follows:

[0037] The leftmost column (c) of this table: the position of the parameters for each calculated "stage". ; Intermediate matrix (A): coefficients , indicating the calculation of the first Level 1 to the 2nd The weights that depend on the slope of the level; Bottom first line ( Fifth-order solution Weighting coefficients ; Second line from the bottom (b): Fourth-order solution Weighting coefficients , used for error estimation.

[0038] Detailed step-by-step calculation formulas are provided, based on the Butcher table, for initial value problems of differential equations. ,from arrive The calculation process is as follows: Calculate the slope:

[0039] Calculate the 4th and 5th order solutions:

[0040] Error estimation:

[0041] In addition, an adaptive step size control strategy is adopted, with an allowable error tolerance of . Define the scaling factor: ,in, For the Euclidean norm. Step size update rule: Initial step size selection: ,in, .

[0042] Here, This represents the standard form of the initial value problem of the differential equation (where t corresponds to the ray parameter). ), They represent discrete nodes, Indicates the integration step size. Indicates the slope of the RKF45 stage. This represents the numerical solution at step n. The Euclidean norm (2-norm) of the error vector is denoted. and These represent the new step size and the old step size, respectively. Represents the radial parameter. Indicates radial difference. These represent the maximum and minimum radial coordinates, respectively.

[0043] In one embodiment, based on the incident positions of the transceiver array elements and the updated step size, a propagation delay model along the sound ray path is constructed and transformed into a parameterized form; the integration interval is mapped to a standard interval through a linear transformation, and the real delay matrix integrated along the real ray path and the ideal delay matrix approximated based on the uniform medium plane interface are constructed to obtain the delay error matrix. The specific process is as follows: Constructing from the position of array elements To the focal point The propagation delay model is expressed as: After conversion in parameterized form, we get: ,in, To measure the covariant components of a tensor, This represents the distance from array element m to the focal point. The propagation delay Indicates the integration path (ray from element m to the focal point). (trajectory), integration path Determined by the ray equation, Represents the arc length infinitesimal element. Represents the path trajectory. Represents phase velocity (position of array element m) (and a function of the wavefront normal direction p̂). These represent the start and end points of the ray parameters, respectively. Let represent the derivatives of the curve coordinates with respect to the ray parameters, respectively. This represents the arc length parameter.

[0044] Integration interval Mapping to standard interval Then we have: ,at this time, ,in, The Gauss-Legend quadrature formula is used. , for Legendre polynomial The root, B represents the number of Gaussian points, and the weights are: ,for Gaussian points, with an accuracy of The error estimate is: .

[0045] In the above formula, Describe the variable on the standard integration interval [-1, 1]. These represent the start and end points of the ray parameters, respectively. Denotes the integrand. G1 indicates the order of the Gauss-Legendé integral formula, with only the subscript indicating the order for easy distinction. This represents the weight corresponding to the B-th Gaussian point. Representing Legendre polynomials in The first derivative at point h, where h represents the step size; Based on this, assuming the phased array has M elements and the detection area is discretized into N focal points, the true time delay matrix is ​​defined as follows: ,in: Define the ideal time delay matrix (approximate for homogeneous medium and planar interface): ,in: Then the time delay error matrix is: M represents the total number of phased array elements, and N represents the total number of focal points in the detection area. Indicates the actual propagation delay. Indicates the ideal propagation delay. Indicates the reference speed of sound. Indicates the position of array element m To the focal point Location The straight path, Represents an M×N dimensional real matrix space. Indicates the focal point. Indicates the direction of the wavefront normal.

[0046] Through the above embodiments, the surface parameters Material parameters and material density The position of array element m Focus Location The surface metric tensor and the anisotropic phase velocity field are calculated as inputs; an initial step size is set. Error tolerance Furthermore, based on each array element-focal point pair Determine the initial conditions: ,in, Representing the coordinates of the curve respectively The conjugate generalized momentum (i.e., the slow component) is determined by Snell's law; calculate the RKF45 step size h and update the state: ; Check for boundary violations or divergence, and adjust the step size h; sample along the ray path. For each point, calculate the Gaussian integral weight; finally, obtain the time delay matrix: .

[0047] In one embodiment, boundary condition processing is also included, where the reflection condition is satisfied when the incident ray reaches the surface boundary: , These represent the incident slowness vector and the outgoing slowness vector, respectively; when the ray encounters a defect, the scattering coefficient is calculated based on the defect type (pore, inclusion, etc.). Considering the impact of material degradation, we have: ,in, For quality factors, The unit normal vector representing the boundary of the surface. Indicates the density of the material. Indicates acoustic impedance. Indicates phase wave velocity, These represent the acoustic impedance of the matrix material and the acoustic impedance of the defect material, respectively. Indicates the distance of propagation The amplitude of the wave afterward Indicates the initial amplitude. Represents angular frequency. Indicates the attenuation coefficient. This indicates that the path length and arc length parameters of wave propagation are consistent; Based on this, the analytical solution is verified and its accuracy is analyzed. For simple curved surfaces such as spheres or cylinders, there exists an analytical solution for ray tracing. The relative error is defined as follows: , These represent the analytical solution and the calculated solution, respectively. This represents the relative error in verifying the analytical solution.

[0048] Convergence tests can also be performed, and the convergence order can be verified through relative error. Specifically, the relative error under different step lengths h is calculated as follows: Take the logarithm of the relative error: Then, linear fitting is performed to obtain the convergence order. Theoretical value We check the conserved quantities to ensure that the Hamiltonian H remains constant along the ray, i.e.: The result satisfies ,in, It is a constant. Indicates the order of convergence. Represents ray parameters, denoted by , h represents the step size, and H represents the Hamiltonian in the conservation check.

[0049] Finally, the accuracy is calculated using the delay time: And ray path accuracy: And the efficiency of serial / parallel time calculation: To verify whether the above data is accurate. These represent numerical delay and reference delay, respectively. Denotes the Frobenius norm. , , These represent the arc lengths along the ray path obtained from numerical calculations. The position vector / coordinates at the location and the corresponding arc length parameter on the reference path The position vector / coordinates at that location represents serial time and parallel time, respectively, and x represents the spatial position vector.

[0050] This embodiment, by embedding the constitutive relation of anisotropic elastic media into the wave equation and solving the Christofel equation, can accurately characterize the propagation characteristics of quasi-longitudinal waves and two quasi-transverse waves in ceramic materials and the anisotropic phase velocity field, overcoming the technical defect that the traditional isotropic sound velocity model cannot describe the orientation-dependent sound velocity of ceramic crystals.

[0051] Building upon this foundation, a differential geometric ray-tracking framework is constructed using characteristic line equations and Hamiltonian canonical equations. By combining the transceiver array element arrangement parameters with the logarithmic generatrix surface geometric parameters, the incident positions of the transceiver array elements in the curvilinear coordinate system are calibrated. This achieves precise spatial coupling between the transducer array and the variable curvature roller surface, enabling ray path tracking to be performed under boundary conditions strictly consistent with the roller's solid geometry. This avoids the errors in calculating the incident angle and refraction path caused by traditional planar interface or cylindrical approximations. By transforming the canonical equations into a system of first-order ordinary differential equations in state vector form, and adaptively updating the step size based on local errors and allowable error tolerances, the numerical accuracy and computational stability of the time delay integral along the true ray path are guaranteed. Finally, a time delay error matrix is ​​constructed between the true time delay matrix integrated along the true ray path and the ideal time delay matrix based on the homogeneous medium planar interface approximation. Error analysis and convergence identification are performed to ensure that the determined true time delay matrix meets the preset accuracy requirements.

[0052] In one embodiment, the method further includes the following steps: verifying the symmetry of the time delay matrix; if it is not symmetric, performing outlier interpolation to obtain the final time delay matrix.

[0053] Specifically, let the numerical integration error be... ray tracing error is The total error then satisfies: Proof: By the triangle inequality and the mean value theorem for integrals: ; in, Indicates numerical integration error. Indicates the total number of array elements. This represents the Gauss-Legendé integral points. This represents the ray tracing error, where L is the maximum ray path length. Represents phase velocity, This represents the numerical approximation of the phase velocity. Indicates the first Gaussian point weights, Indicates the first The arc length of each sub-interval Indicates the first Phase velocity at a Gauss point This represents the distance from array element m to the focal point. The propagation delay Represents the arc length infinitesimal element. Denotes the Frobenius norm. These represent the numerical approximation path and the actual path, respectively. These represent the approximate error and the actual error, respectively. This represents the arc length parameter.

[0054] In this embodiment, by verifying the symmetry of the true time delay matrix, the reciprocity principle of the ultrasonic sound field is fully utilized to constrain and verify the physical consistency of the time delay data. This effectively identifies and eliminates non-physical time delay anomalies caused by surface parameterization discretization errors, local numerical oscillations in anisotropic sound velocity fields, or cumulative deviations in adaptive step sizes. For asymmetric elements discovered during the verification process, local repair is performed through outlier interpolation, which preserves the overall calculation accuracy of the true ray path integral while suppressing the amplification effect of individual array elements on time delay errors in subsequent matrix isomorphic transformations. This improves the solution accuracy and convergence reliability of the mapping relationship between physical space and computational space.

[0055] In one embodiment, a matrix isomorphism transformation relationship under low-rank sparse constraints is constructed for the real time delay matrix and the ideal time delay matrix to obtain the linear mapping optimization objective function between the physical space and the computational space, including the following steps: Construct a matrix isomorphism transformation relationship between the real time delay matrix and the ideal time delay matrix, and determine the transformation matrix such that the real time delay matrix and the ideal time delay matrix satisfy an approximately linear relationship; We introduce a sparsity prior to the approximately linear relationship and construct a sparsity penalty term; we introduce a low-rank prior to the transformation matrix and construct a low-rank regularization term. Based on matrix isomorphism, sparse penalty terms, and low-rank regularization terms, a linear mapping optimization objective function between physical space and computational space is constructed.

[0056] In this embodiment, the actual time delay matrix is ​​represented as: The ideal time delay matrix is ​​expressed as: Assuming that the effects of complex curved surfaces and anisotropic media on sound wave propagation can be considered as a linear distortion of the wavefront phase, and that this distortion is linear in the element space, then there exists a transformation matrix. This makes the two matrix spaces satisfy an approximately linear relationship: If for a given and Find the transformation matrix P such that: , Describing the Frobenius norm, Treating it as a dictionary, with its column vectors as base atoms, the transformation matrix P can be approximated by a sparse linear combination of dictionary atoms: ,in, Let be the sparse coefficient matrix. Combining the prior knowledge of sparsity and low rank, we obtain the complete optimization problem, namely the objective function for the linear mapping between the physical space and the computational space, expressed as: ,in, Representing the L1 norm promotes the sparsity of the coefficient matrix. It is the nuclear norm, which promotes the low-rank property of the transformation matrix. and Let these represent the actual time delay matrix and the ideal time delay matrix, respectively. Let A represent the transformation matrix (the sparse coefficient matrix, which is the matrix isomorphic transformation part), M represent the total number of matrix elements, and N represent the total number of focal points. This represents the objective function to be optimized. Let represent the i-th singular value of the transformation matrix P.

[0057] Here, both the real time delay matrix and the ideal time delay matrix are M×N dimensional real matrices, and the transformation matrix is ​​an M×M dimensional square matrix; the sparse coefficient matrix is ​​an N×M dimensional matrix, so that the result of multiplying the ideal time delay matrix and the sparse coefficient matrix is ​​an M×M dimensional square matrix, which then matches the dimension of the transformation matrix.

[0058] In another embodiment, the objective function for optimizing the linear mapping is solved by augmented Lagrangian decoupling and low-dimensional subspace projection. The transformation matrix is ​​obtained through sparse dictionary learning and low-rank constraints, including the following steps: An augmented Lagrangian function is constructed based on the real and ideal time delay matrices. The optimization objective function is decoupled into a closed-form update of the singular value threshold of the transformation matrix and an independent Lasso coordinate descent solution of the sparse coefficient matrix through the alternating direction multiplier method. A convergence criterion based on the original residual and the dual residual is constructed to identify the termination condition of the algorithm. By truncating singular value decomposition, the real time delay matrix and the ideal time delay matrix are projected onto a low-dimensional subspace. A subspace transformation matrix is ​​constructed in the low-dimensional subspace, and the upper bound of the error of the low-dimensional subspace approximation is obtained. The new data is split into the interpretable part of the old space and the new information. After extracting the orthogonal directions of the new information, the original singular values, projected components and new direction weights are concatenated into a core matrix. Only the core matrix is ​​decomposed, and the decomposition result is mapped back to the high-dimensional subspace to obtain the transformation matrix. The high-dimensional subspace is composed of the new direction and the old direction.

[0059] In this embodiment, Lagrange multipliers are introduced. and penalty parameters Construct the augmented Lagrangian function: After sorting, we get: ; in, Represents the matrix inner product. Let A denote the transformation matrix, and let A denote the sparse coefficient matrix. Represents the Lagrange multipliers. and Let these represent the actual time delay matrix and the ideal time delay matrix, respectively. Denotes the Frobenius norm. Represents the matrix inner product. Represents the trace of a matrix.

[0060] The ADMM algorithm solves the problem by alternately updating variables. The specific process is as follows: initialization: , ; The k-th iteration includes updating the transformation matrix P: fixed. Please solve: ; Update A: Fixed Solve ; Update the multipliers: ; Update penalty parameters: ; Among them, the original residual Dual residual , This represents the initial value of the transformation matrix, with the superscript 0 indicating the 0th iteration, i.e., the initial state. Let represent the M-order identity matrix (1s on the diagonal, 0s elsewhere), which indicates that the initial assumption of an identity mapping between the ideal time delay and the actual time delay is valid. This represents the initial value of the sparse coefficient matrix, set to 0 (zero matrix). Indicates the density of the material; The initial value of the Lagrange multipliers is set to 0 (zero matrix); This represents the initial value of the penalty parameter. This indicates the penalty parameter update rate. , , , Let represent the sparse coefficient matrix of the k-th / k+1-th iteration, the Lagrange multiplier matrix of the k-th / k+1-th iteration, the transformation matrix of the k-th / k+1-th iteration, and the penalty parameter of the k-th / k+1-th iteration, respectively. and represents the regularization coefficient and the nuclear norm regularization coefficient, respectively.

[0061] During the process of updating the transformation matrix P, some sub-problems will be encountered, which can be solved in the following way: For example, if the subproblem of updating the transformation matrix P has a closed-form solution, it can be obtained using the singular value thresholding (SVT) operator. Let The objective function is rewritten as: ; Ignoring the nuclear norm terms, calculate the gradient and set it to zero:

[0062] remember ,but: Considering the nuclear norm regularization term, the transformation matrix... The update is a singular value threshold operation: The singular value threshold operator is defined as: for a matrix ,have ,in, Indicates intermediate variables. The symbol represents the temporary function of the objective function of the subproblem P, which represents the transformation matrix. Represents the intermediate matrix. This represents a temporary solution to the transformation matrix P. This represents the singular value threshold (SVT) operator symbol. The threshold parameter represents SVD. Represents the maximum singular value. This represents the gradient operator of the transformation matrix P. Represents the matrix unit. This represents the singular value threshold operator. Indicates soft threshold operation. These represent the SVD decomposition of matrix X; In the process of updating the sparse coefficient matrix A, some subproblems will be encountered. For example, the subproblem of updating the sparse coefficient matrix A can be decomposed into N independent Lasso problems.

[0063] make The objective function is: The Frobenius norm of a matrix can be decomposed into the sum of the 2-norms of its column vectors: Therefore, each column can be solved independently: This is a standard The problem is solved using the coordinate descent method, defined as follows: Then the coordinate descent iteration formula is: Among them, the soft threshold function ,in, Indicates the first intermediate variable. and They are The j-th column, Represents a row vector. Let A be the objective function of the subproblem involving the sparse coefficient matrix A. Represents the i-th column vector. arrive , Represents the soft threshold function. Represents a symbolic function. Represents the square of the norm. Indicates soft threshold operation. Let i represent the update value of the element in the j-th column and i-th row of the sparse coefficient matrix A at the (k+1)-th iteration, where i represents the row index and j represents the column index. In one embodiment, convergence analysis can also be performed, assuming the optimization problem is convex and has a saddle point, then the ADMM iteration satisfies: 1. Residual convergence is represented as: ; 2. The objective function converges, expressed as: ; 3. Variable convergence, expressed as: if the optimization problem has a unique solution, then Then the convergence analysis can be written as: ,in, Then, under appropriate conditions, the algorithm can converge and stop when the following conditions are met simultaneously: ,in, , usually take , , , , These represent the original residual tolerance, the dual residual tolerance, and the absolute tolerance (i.e., the relative tolerance), respectively. To indicate the limit, This represents the objective function to be optimized. This represents the optimal value. Optimal transformation matrix, Represents a sparse coefficient matrix. Indicates constraints. This represents the original residual, and M represents the number of array elements. It represents the Lagrange multiplier.

[0064] By truncating singular value decomposition, the true time delay matrix and the ideal time delay matrix are projected onto a low-dimensional subspace. A subspace transformation matrix is ​​then constructed within this low-dimensional subspace, and an upper bound on the error of the low-dimensional subspace approximation is obtained. The specific steps are as follows: For large-scale problems (where M and N are large), the full ADMM iteration is computationally intensive. To accelerate this process, we utilize the low-rank property of the matrix and employ truncated SVD. , here means and , which represents the real time delay matrix and the ideal time delay matrix, and its rank r-truncated SVD representation is: ,in, It was before A singular value, The i-th left singular vector and the i-th right singular vector are respectively. These are the corresponding left and right singular vector matrices. The rank selection criterion is to choose the smallest r such that the formula holds: ,in, This represents the energy retention rate, which is usually taken as 0.95.

[0065] pass Calculating the truncated SVD separately, we have: .

[0066] Suppose there exists a transformation matrix P such that Then, on the subspace, it is approximated as follows: Left multiplication right multiplication Then we have: ,remember The subspace transformation matrix is ​​obtained as follows:

[0067] then: , Let represent the subspace transformation matrix, and i represent the quantity.

[0068] A complete SVD can be: Then the truncated SVD (using a random algorithm) is: In other words, storage requirements have shifted from .

[0069] Furthermore, we can obtain the upper bound of the error for the low-dimensional subspace approximation, let... and The truncated SVD approximation errors are respectively and ,Right now: Then the transformation matrix of the subspace approximation satisfy: The intermediate terms have an upper bound: This can be proven using the triangle inequality, as follows: ,in, Represents the norm.

[0070] Finally, the new data is split into the interpretable part of the old space and entirely new information. After extracting the orthogonal directions of the new information, the original singular values, projected components, and new direction weights are concatenated into a core matrix. Only the core matrix is ​​decomposed, and the decomposition result is mapped back to the high-dimensional subspace to obtain the transformation matrix. The steps are as follows: Obtain an existing SVD: New data matrix ; By performing projection, we obtain the projection transformation matrix, which is expressed as: And perform residual processing to obtain the residual transformation matrix: ; The QR decomposition is represented as follows: ,in, For columns to be orthogonal, ; Construct the intermediate matrix: ;calculate The SVD is represented as: , This represents the intermediate concatenated matrix; updated to: Complexity: Far lower than recalculating SVD ; When detection conditions change, the transformation matrix needs to be updated adaptively. A recursive update with a forgetting factor is used: Given a transformation matrix The old real / ideal latency data pair is ( , ), new data pairs ( , Weighted least squares problem: ,in: , It is a forgetting factor. It is a regularization parameter. , These represent the newly added latency data pairs, and W represents the weighting matrix. Represents the identity matrix.

[0071] The new transformation matrix is ​​obtained, represented as: ; This can be further computed efficiently using the Woodbury matrix inversion lemma. In this process, the regularization parameter is chosen as follows: (Sparseness control): Selected through cross-validation or based on theoretical recommendations. , (Low-rank control): Usually set c is a constant determined experimentally, and the penalty parameter is... initial value Initialization strategy: ,in, Represents a small constant. By constructing a matrix isomorphism transformation relationship between the real and ideal time delay matrices, the nonlinear distortion mapping between the physical and computational spaces is transformed into a constrained linear optimization problem, enabling the time delay deviation caused by complex surfaces and anisotropic coupling to be structurally corrected within a unified algebraic framework.

[0072] In one embodiment, a sparsity prior is introduced to construct a sparsity penalty term, which fully utilizes the physical characteristics of the sparse distribution of internal defects of ceramic rollers in the imaging space. This effectively suppresses the interference of background noise and artifacts on the transformation matrix solution. Furthermore, a low-rank regularization term is constructed through a low-rank prior, which characterizes the high correlation and compressibility of the sound propagation structure of the roller body. This ensures that the transformation matrix retains the main sound field structure information during the dimensionality reduction mapping process without overfitting. The matrix isomorphic transformation relationship, the sparsity penalty term, and the low-rank regularization term are integrated to construct a linear mapping optimization objective function, realizing the joint optimization of defect sparse features and background low-rank structure under a unified objective.

[0073] This embodiment significantly improves the generalization ability and physical interpretability of the transformation matrix through dual prior constraints, so that the transformation matrix obtained by subsequent solutions can not only accurately represent the structure-preserving mapping relationship between physical space and computational space, but also adaptively distinguish defect signals from background media, providing a robust and reliable mathematical foundation for subsequent signal correction and distortion-free imaging.

[0074] As one possible implementation, propagation distortion correction processing is performed on the actual acquired frequency domain full matrix signal and transformation matrix to obtain the corrected signal in the computational space. Full-focus imaging processing is then performed on the corrected signal and the ideal steering vector to obtain the distortion-free imaging amplitude in the computational space. Subsequently, inverse mapping reconstruction processing is performed, and the physical space inverted and reconstructed image is obtained. This includes the following steps: The time-domain signal captured by the phased array full matrix of the logarithmic roller of the ceramic under test is transformed into the frequency domain to obtain the actual acquired frequency domain full matrix signal; Propagation distortion correction processing is performed on the actual acquired frequency domain full matrix signal and transformation matrix to transform the complex boundary of the ceramic logarithmic roller under test and the wavefront distortion in anisotropic media into a simplified geometry and an easily solvable model in isotropic media, thus obtaining the correction signal in the computational space. Based on the calculation of the one-way time delay from each array element to the internal imaging point of the logarithmic roller of the ceramic under test in a uniform medium, an ideal steering vector is constructed. The correction signal and the ideal steering vector are subjected to full-focus imaging processing, and the two-way time delay signals of all transmit and receive array element pairs are coherently superimposed to obtain the distortion-free imaging amplitude in the computational space. The distortion-free image amplitude is inversely mapped and reconstructed to obtain the physical space inversion and reconstruction image of the logarithmic roller of the ceramic under test.

[0075] In a specific embodiment, considering the elastic wave scattering problem in the rollers of the ceramic bearing, the displacement field in the frequency domain satisfies the Navier equation: For small defects, using the Born approximation, the total displacement field can be decomposed into the incident field. and scattered field : The scattered field satisfies the integral equation: .

[0076] Assuming the phased array has M elements, the full matrix captures... Signal matrix, time-domain signal The signal transmitted and received by array element i is represented in the frequency domain as: Under the Born approximation, the signal can be represented as: , The kernel function is: .

[0077] Define two spaces: physical space (real space). : It has a complex logarithmic generatrix surface boundary, anisotropic medium (ceramic material), and Green's function. Satisfying complex boundary conditions; computational space (ideal space) Simplified geometry (such as planar boundaries or regular curved surfaces), isotropic homogeneous media, and Green's functions. It has an analytical solution or is easy to calculate.

[0078] Based on the transformation matrix P, a wavefront mapping relationship was constructed between the two spaces for any point. The incident wavefront satisfies: At the position of the array element At that location, the scattered wavefront satisfies: ; Assuming the transformation is linear and reciprocal, more generally, two transformations are required: transmission and reception.

[0079] Full-focus imaging exists in computational space, specifically as follows: In computational space, the time delay from an array element to point y is: ,in, The reference wave velocity (usually the longitudinal wave velocity) is used; the steering vector is defined as: ; The TFM image in computational space is: ,in, For two-way delay, It is weight; Frequency domain representation is: ,in, It is a signal matrix with elements of .

[0080] For homogeneous media, TFM can be calculated rapidly in the wavenumber domain: ,in, It is the spatial Fourier transform input of the signal. The computation of each imaging point is independent and can be fully parallelized. Represents the spatial Fourier transform.

[0081] Based on the transformation matrix Then the relationship between the actual signal and the ideal signal is: ; Therefore, the corrected signal to be used in the computational space is: In practice, It may be irreversible; use generalized inverse or regularized inverse: ,in, It is a Moore-Penrose pseudo-reversal.

[0082] Another method to directly correct latency is as follows: the relationship between actual latency and ideal latency is: , where the function From the time delay matrix Decision. A simple linear model is: The corrected delay is then used for TFM: .

[0083] Therefore, by performing inverse mapping reconstruction on the distortion-free image amplitude and then reconstructing it, the physical space inversion reconstruction image of the logarithmic roller of the ceramic under test is obtained, specifically: Physical Space Point Computational space points That is, the mapping of spatial position vectors is determined by surface parameterization: , where F is the mapping function from the computational space to the physical space, and F is known for the logarithmic generatrix surface.

[0084] The mapping of image values ​​needs to take into account the Jacobian determinant: .

[0085] The mapping from a computational spatial image to a physical spatial image can be represented by an integral equation: ; Kernel functions include wavefront transformation and coordinate transformation: Discretization yields a linear system: ,in, It is a mapping matrix. Because... This is usually pathological and requires regularization to solve: , in the above formula, Let C represent the ideal steering vector, and let C represent the fourth-order elastic tensor / elastic constant matrix. This represents a disturbance in the elastic constant. Denotes the regularization parameters for the inverse problem. Represents the Green's function tensor. and Let i and j represent the Green's functions from the transmitting element i to the receiving element j, respectively. Let these represent the Green's function in real space and the Green's function in ideal space, respectively. Represents an image vector (discretized). Indicates the density of the material. Represents angular frequency. It is a displacement field. It is the source item. It is a disturbance of the elastic constant. It is the scattering potential function; Represents the TFM image (computation space). Let K represent the Jacobian determinant, K represent the imaging kernel function / inverse mapping kernel function, and M represent the total number of array elements. P represents the number of imaging points, and P represents the transformation matrix. This indicates the Moore-Penrose pseudo-inverse. Represents the source term (Navier equation). To represent a regularization term, commonly used Regularization: , Represents the regularization parameter. Represents the frequency domain signal matrix. Represents a time-domain signal. Represents frequency domain signals, Indicates computational spatial delay. Indicates two-way delay. Indicates the reference wave velocity. Represents the total displacement field. Represents the incident displacement field. Represents the scattering displacement field. Represents the imaging weighting coefficient. Represents angular frequency. Represents the coordinates of a point in physical space. This indicates the calculation of the coordinates of a point in space.

[0086] In one embodiment, the transformation matrix is ​​determined by the matrix isomorphic transformation relationship between the real time delay matrix and the ideal time delay matrix, and is repeatedly called in the frequency domain full matrix signal correction processing and physical space inversion reconstruction process to maintain the consistency of the mapping relationship between the real propagation characteristics and the ideal propagation characteristics, so that the propagation constraints constructed based on the real propagation model can simultaneously act on the computational space imaging and physical space restoration processes.

[0087] Specifically, the transformation matrix is ​​determined by the matrix isomorphism transformation relationship between the real time delay matrix and the ideal time delay matrix. That is, the transformation matrix representing the mapping relationship between the physical space and the computational space is obtained by solving an optimization problem under low-rank sparse constraints. In the propagation distortion correction stage, the matrix isomorphism transformation relationship is invoked to map the actual acquired frequency domain full matrix signal from the physical space to the computational space, obtaining the corrected computational space signal. At this time, the transformation matrix performs the forward mapping function. In the inverse mapping reconstruction stage, the regularized generalized inverse mapping corresponding to the matrix isomorphism transformation relationship is invoked to restore the distortion-free imaging amplitude in the computational space to the physical space, obtaining the physical space inversion and reconstruction image. At this time, the transformation matrix performs the inverse mapping function. Through the repeated invocation of the above forward and inverse mappings, it is ensured that the propagation distortion correction and inverse mapping reconstruction processes complete the data conversion between the computational space and the physical space based on the same matrix isomorphism relationship. This ensures that the propagation constraints corresponding to the real propagation model simultaneously apply to both the computational space imaging and physical space restoration processes, maintaining the consistency of the mapping relationship between the physical space and the computational space. Here, the transformation matrix, after being solved by low-rank sparse constraints, exhibits rank deficiency and does not possess an inverse matrix in the usual sense. Therefore, the inverse mapping reconstruction process uses Moore-Penrose generalized inverse or Tikhonov regularized inverse to realize the restoration mapping from computational space to physical space. The generalized inverse is obtained by truncating singular value decomposition to retain the principal singular value components, so as to suppress the numerical oscillations introduced by low-rank approximation.

[0088] Finally, the numerical values ​​were verified by simulation. The propagation of ultrasound in the ceramic bearing was simulated using the finite element method, and simulation data with known defects were generated. The results can be evaluated by positioning error, dimensional error and contrast-to-noise ratio (CNR). The positioning error, dimensional error and contrast-to-noise ratio (CNR) can be calculated using general methods, which will not be elaborated here.

[0089] In addition, real ceramic bearing roller specimens with artificial defects (flat-bottomed holes, side-drilled holes, etc.) were used for experimental verification. The detection probability (POD) curve was plotted to evaluate the system's ability to detect defects of different sizes. The false alarm rate (FAR) was calculated to quantify the proportion of false defect indications. The consistency of multiple measurement results was analyzed to verify the system's repeatability, thereby comprehensively evaluating the reliability of the detection method.

[0090] The following tables show data from a specific embodiment. The experimental procedure can be found in the appendix. Figure 2 - Figure 4 The comparison results can be used as a reference. Figure 5 The final classification and recognition results can be referenced. Figure 6 As shown in Table 1. Table 1 shows the acoustic parameters of the selected ceramic materials.

[0091] Table 1 shows the acoustic parameters of the ceramic materials.

[0092] In this embodiment, the ultrasonic array and acquisition parameters are shown in Table 2 below.

[0093] Table 2 shows the ultrasonic array and acquisition parameters.

[0094] This embodiment provides specific samples and corresponding experimental datasets for materials and defects, as shown in Table 3.

[0095] Table 3 shows the experimental data and test results dataset.

[0096] Finally, a comparison between this method and traditional methods was also drawn. Figure 5 The diagram and Table 4 below, Table 4 and Figure 5 This fully demonstrates that our method is superior to traditional methods. Figure 5 The first figure is a schematic diagram of traditional TFM imaging (uncorrected), the second figure is a schematic diagram of the corrected (computational space) using this method, and the third figure is a schematic diagram of the inverse mapping reconstruction (physical space) using this method.

[0097] Table 4 shows the comparison results between our method and the traditional method.

[0098] Specific experiments show that the entire technical solution realizes a complete signal processing link from actual distortion signal acquisition to real image reconstruction in physical space. It retains the high resolution advantage of full-focus imaging and fundamentally eliminates image stretching, distortion and artifacts caused by anisotropic coupling of curved surfaces through mapping correction of physical and computational spaces, which significantly improves the imaging resolution and positioning accuracy of internal defects in ceramic logarithmic busbar bearings.

[0099] In one specific embodiment, the following steps are also included: The physical space inversion and reconstruction image is subjected to surface grayscale correction based on logarithmic generatrix surface geometric parameters to extract defect candidate regions. Based on the defect candidate region, connected components are marked in the generatrix surface coordinate system. Geometric features along the generatrix direction, circumferential direction and radial direction are extracted to determine the axial position, circumferential span and subsurface depth of the defect in the ceramic logarithmic roller under test, and the defect quantification data is obtained. Based on geometric features and the ratio of quasi-longitudinal wave to quasi-transverse wave scattering amplitudes and the deflection angle of the slowness vector corresponding to the anisotropic phase velocity field, the defect candidate region is identified by type, and the defect type identification results distributed along the logarithmic generatrix are obtained. The defect types include at least cracks, near-surface pores or subsurface layer delamination pores. Based on defect quantification data and defect type identification results, defect detection, quantification, and classification results data are generated.

[0100] Alternatively, the following approach can be taken: First, further multimodal fusion and intelligent enhancement processing is applied to the physically reconstructed image. Specifically, this involves combining information from longitudinal and transverse waves. Since longitudinal waves are more sensitive to volumetric defects, while transverse waves are better at detecting crack-like defects, fusing these two modalities together yields more reliable detection results. Then, deep learning enhancement using a convolutional neural network is applied, including image denoising and enhancement, automatic defect identification and classification, and image quality evaluation. This results in clearer and more accurate defect characterization. Finally, based on the optimized defect characterization results and combined with actual bearing operating data, the development trend of defects is predicted, the remaining service life is assessed, and finally, decision-making recommendations on whether and when to perform maintenance are provided, forming complete service status decision support data.

[0101] This technical solution not only directly embeds defect detection into the intrinsic geometric coordinate frame of the logarithmic generatrix surface and the physical characteristics of the anisotropic acoustic field of ceramics, fundamentally eliminating the defects in location and quantification errors caused by the traditional planar imaging assumption, but also significantly improves the detection rate, classification accuracy, and engineering applicability of internal defects in ceramic logarithmic generatrix bearing rollers by fusing geometric and acoustic features.

[0102] Example 2: A phased array ultrasonic testing system for rollers of ceramic logarithmic busbar bearings, such as Figure 8 As shown, it includes: The data acquisition module 100 is used to acquire the generatrix curve data and anisotropic acoustic parameters of the logarithmic roller of the ceramic under test, determine the configuration information of the transducer array, and acquire the time-domain full matrix signal based on the transducer array scan. The configuration information of the transducer array includes at least the transceiver array element arrangement parameters. The matrix calculation module 200 is used to perform surface parameterization and measurement calculation on the generatrix curve data to obtain the generatrix surface geometric parameters of the logarithmic roller of the ceramic under test; based on the generatrix surface geometric parameters, anisotropic acoustic parameters and transceiver array element arrangement parameters, a real propagation model is constructed, and based on the real propagation model, sound field characteristics are solved, sound ray path is traced and propagation time delay is integrated to obtain the real time delay matrix characterizing the real propagation characteristics; A determination module 300 is constructed to construct a matrix isomorphism transformation relationship under low-rank sparse constraints for the real time delay matrix and the ideal time delay matrix, so as to obtain the linear mapping optimization objective function between the physical space and the computational space. The constraint solving module 400 is used to decouple and spatially project the linear mapping optimization objective function to obtain a transformation matrix, which represents the mapping relationship between the physical space and the computational space. The imaging inversion module 500 is used to convert the time-domain full matrix signal into the frequency-domain full matrix signal, and perform propagation distortion correction and full-focus imaging processing in combination with the transformation matrix to obtain the distortion-free imaging amplitude in the computational space. Then, it performs inverse mapping reconstruction to obtain the physical space inversion reconstruction image of the tested ceramic logarithmic roller that restores the true propagation characteristics. The defect detection module 600 is used to perform defect detection on the physical space inversion and reconstruction image to obtain the internal defect detection results.

[0103] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on the differences from other embodiments. The same or similar parts between the various embodiments can be referred to each other.

[0104] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, apparatus, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0105] This invention is described with reference to flowchart illustrations and / or block diagrams of the method, terminal device (system), and computer program product according to the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing terminal device to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing terminal device, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0106] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing terminal device to operate in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0107] These computer program instructions can also be loaded onto a computer or other programmable data processing terminal equipment, causing a series of operational steps to be performed on the computer or other programmable terminal equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable terminal equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0108] It should be noted that: The phrase "an embodiment" or "an embodiment" used in this specification means that a particular feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment of the invention. Therefore, the phrase "an embodiment" or "an embodiment" appearing in various places throughout the specification does not necessarily refer to the same embodiment.

[0109] Furthermore, it should be noted that the shapes and names of the parts and components described in the specific embodiments described in this specification may differ. All equivalent or simple variations made to the structure, features, and principles described in this patent concept are included within the protection scope of this patent. Those skilled in the art to which this invention pertains may make various modifications or additions to the described specific embodiments or use similar methods to replace them, as long as they do not depart from the structure of this invention or exceed the scope defined in these claims, they should all fall within the protection scope of this invention.

Claims

1. A phased array ultrasonic testing method for rollers of ceramic logarithmic busbar bearings, characterized in that, Includes the following steps: Acquire the generatrix curve data and anisotropic acoustic parameters of the logarithmic roller of the ceramic under test, determine the configuration information of the transducer array, and acquire the time-domain full matrix signal based on the transducer array scan. The configuration information of the transducer array includes at least the transceiver array element arrangement parameters. The surface parameterization and measurement calculation of the bus curve data are performed to obtain the geometric parameters of the bus surface of the logarithmic roller of the ceramic under test. Based on the geometric parameters of the bus surface, the anisotropic acoustic parameters and the arrangement parameters of the transceiver array elements, a real propagation model is constructed. Based on the real propagation model, the sound field characteristics are solved, the sound ray path is traced and the propagation time delay is integrated to obtain the real time delay matrix that characterizes the real propagation characteristics. We construct matrix isomorphism transformation relationships under low-rank sparse constraints for the real and ideal time delay matrices, and obtain the linear mapping optimization objective function between the physical space and the computational space. The objective function of linear mapping optimization is decoupled and spatially projected to obtain a transformation matrix, which represents the mapping relationship between physical space and computational space; The time-domain full matrix signal is converted into the frequency-domain full matrix signal. The propagation distortion correction and full-focus imaging processing are combined with the transformation matrix to obtain the distortion-free imaging amplitude in the computational space. Then, inverse mapping reconstruction is performed to obtain the physical space inversion reconstruction image of the tested ceramic logarithmic roller that restores the true propagation characteristics. Defect detection is performed on the physically inverted and reconstructed image to obtain the internal defect detection results.

2. The method for ultrasonic testing of ceramic logarithmic busbar bearing rollers using a phased array according to claim 1, characterized in that, The generatrix curve data is parametrically analyzed and metrically calculated to obtain the geometric parameters of the generatrix surface of the ceramic logarithmic roller under test. This includes the following steps: Based on the logarithmic generatrix curve of the logarithmic roller of the ceramic under test, the logarithmic function equation of the generatrix curve is constructed. Based on the logarithmic function equation, the surface sweep is performed with the roller axis as the rotation axis to determine the surface parameter equation of the logarithmic generatrix surface. By taking partial derivatives of the surface parametric equations along the axial and circumferential parameters respectively, the axial and circumferential basis vectors of the tangent space are obtained. Based on the axial and circumferential basis vectors, the inner product operation is performed to obtain the components of the metric tensor, and then the first basic form of the metric tensor is determined. By calculating surface line elements using metric tensors, the local arc length infinitesimal elements of the logarithmic generatrix surface are quantified. Based on the cross product operation of axial basis vectors and circumferential basis vectors, the normal vector field is determined to characterize the local orientation of the logarithmic generatrix surface, thereby obtaining the geometric parameters of the logarithmic generatrix surface.

3. The method for ultrasonic testing of ceramic logarithmic busbar bearing rollers using a phased array according to claim 1, characterized in that, The method includes the following steps: The wave equation is rewritten based on the constitutive relation of anisotropic elastic media, and the Christofel equation is determined by combining the plane wave solution. The characteristic equation of the Christofel equation is solved to obtain the quasi-longitudinal wave characteristic and two quasi-transverse wave characteristics. The anisotropic phase velocity field and the implicit equation of the slow surface are obtained by combining the slowness vector. The wave equation is transformed and combined with parameterized variables to obtain the characteristic line equation. The Hamiltonian is quantized and simplified in the curvilinear coordinate system to construct the canonical equation. Based on the arrangement parameters of the transceiver array elements and combined with the geometric parameters of the logarithmic generatrix surface, the incident position of the transceiver array elements in the curvilinear coordinate system is calibrated. Directional parameterization and local coordinate transformation are performed to determine the magnitude of the slowness and to construct the calculation relationship between the refractive index and the corresponding partial derivatives, so that the sound propagation path is simultaneously constrained by the surface geometric features and the anisotropic propagation characteristics of the material. The canonical equations are expressed as a set of first-order ordinary differential equations in state vector form. The slope and approximate solution are calculated to obtain the local error. Based on the local error and the allowable error tolerance, the scaling factor is obtained, and then the step size is updated. Based on the incident position of the transceiver array elements and the updated step size, a propagation delay model along the sound ray path is constructed and transformed into a parameterized form. The integral interval is mapped to the standard interval through linear transformation. The real delay matrix integrated along the real ray path and the ideal delay matrix based on the approximation of the plane interface of the uniform medium are constructed to obtain the delay error matrix characterizing the difference between the real propagation characteristics and the ideal propagation characteristics. Based on the time delay error matrix, a correspondence between the real propagation characteristics and the ideal propagation characteristics is constructed, and error analysis and convergence analysis are performed on the time delay error matrix. If the analysis is satisfactory, the real time delay matrix is ​​determined based on the time delay error matrix.

4. The ultrasonic testing method for the roller phased array of ceramic logarithmic busbar bearings according to claim 3, characterized in that, It also includes the following steps: The symmetry of the time delay matrix is ​​verified. If it is not symmetric, outlier interpolation is performed to obtain the final time delay matrix. The outlier interpolation is used to maintain the propagation continuity of the true time delay matrix.

5. The method for ultrasonic testing of ceramic logarithmic busbar bearing rollers using a phased array according to claim 1, characterized in that, To construct matrix isomorphism transformation relations under low-rank sparse constraints for the real and ideal time delay matrices, the objective function for optimizing the linear mapping between the physical and computational spaces is obtained, including the following steps: A matrix isomorphism transformation relationship is constructed between the real time delay matrix and the ideal time delay matrix to maintain the correspondence between the real propagation characteristics and the ideal propagation characteristics. The transformation matrix is ​​determined so that the real time delay matrix and the ideal time delay matrix satisfy an approximately linear relationship. The transformation matrix is ​​used to construct the mapping relationship between the real propagation characteristics and the ideal propagation characteristics. A sparsity prior is introduced for the approximately linear relationship to construct a sparsity penalty term; a low-rank prior is introduced for the transformation matrix to construct a low-rank regularization term; wherein, the sparsity prior is used to suppress local perturbations of propagation error, and the low-rank prior is used to maintain the overall consistency of propagation characteristics. Based on matrix isomorphism, sparse penalty terms, and low-rank regularization terms, we construct an objective function for optimizing the linear mapping between physical and computational spaces that maintains the correspondence between real and ideal propagation characteristics.

6. The method for ultrasonic testing of ceramic logarithmic busbar bearing rollers using a phased array according to claim 1, characterized in that, The process of decoupling and spatially projecting the objective function of the linear mapping optimization to obtain the transformation matrix includes the following steps: An augmented Lagrangian function is constructed based on the real and ideal time delay matrices. The optimization objective function is decoupled into a closed-form update of the singular value threshold of the transformation matrix and an independent Lasso coordinate descent solution of the sparse coefficient matrix through the alternating direction multiplier method. A convergence criterion based on the original residual and the dual residual is constructed to identify the termination condition of the algorithm. By truncating singular value decomposition, the real time delay matrix and the ideal time delay matrix are projected onto a low-dimensional subspace. A subspace transformation matrix is ​​constructed in the low-dimensional subspace, and an upper bound of the low-dimensional subspace approximation error is obtained. The mapping consistency between the real propagation characteristics and the ideal propagation characteristics is maintained by using the subspace transformation matrix. The new data is split into an interpretable part of the old space and entirely new information. After extracting the orthogonal directions of the entirely new information, the original singular values, projected components, and new direction weights are concatenated into a core matrix. Only the core matrix is ​​decomposed, and the decomposition results are mapped back to the high-dimensional subspace to obtain a transformation matrix that satisfies propagation consistency. The high-dimensional subspace consists of new directions and old directions. The transformation matrix is ​​used for subsequent propagation correction processing of the full-matrix signal in the frequency domain.

7. The method for ultrasonic testing of ceramic logarithmic busbar bearing rollers using a phased array according to claim 1, characterized in that, The method includes the following steps: The time-domain signal captured by the phased array full matrix of the logarithmic roller of the ceramic under test is transformed into the frequency domain to obtain the actual acquired frequency domain full matrix signal; Propagation distortion correction processing is performed on the actual acquired frequency domain full matrix signal and transformation matrix to map the complex boundary of the ceramic logarithmic roller under test and the propagation distortion in the anisotropic medium into an equivalent propagation model in the computational space, and the correction signal in the computational space is obtained. Based on the calculation of the one-way time delay from each array element to the internal imaging point of the logarithmic roller of the ceramic under test in a uniform medium, an ideal steering vector is constructed. The correction signal and the ideal steering vector are subjected to full-focus imaging processing, and the two-way time delay signals of all transmit and receive array element pairs are coherently superimposed to obtain the distortion-free imaging amplitude in the computational space. The distortion-free imaging amplitude is inversely mapped and reconstructed, and the transformation matrix is ​​used to restore the computational spatial imaging results to the physical space to maintain the consistency of the propagation mapping relationship, so as to obtain the physical space inversion reconstruction image of the tested ceramic logarithmic roller that restores the true propagation characteristics.

8. The method for ultrasonic testing of ceramic logarithmic busbar bearing rollers using a phased array according to claim 1, characterized in that, It also includes the following steps: The physical space inversion and reconstruction image is subjected to surface grayscale correction based on logarithmic generatrix surface geometric parameters to extract defect candidate regions. Based on the defect candidate region, connected components are marked in the generatrix surface coordinate system. Geometric features along the generatrix direction, circumferential direction and radial direction are extracted to determine the axial position, circumferential span and subsurface depth of the defect in the ceramic logarithmic roller under test, and the defect quantification data is obtained. Based on geometric features and the ratio of quasi-longitudinal wave to quasi-transverse wave scattering amplitudes and the deflection angle of the slowness vector corresponding to the anisotropic phase velocity field, the defect candidate region is identified by type, and the defect type identification results distributed along the logarithmic generatrix are obtained. The defect types include at least cracks, near-surface pores or subsurface layer delamination pores. Based on defect quantification data and defect type identification results, defect detection, quantification, and classification results data are generated.

9. The method for ultrasonic testing of ceramic logarithmic busbar bearing rollers using a phased array according to claim 7, characterized in that, The transformation matrix is ​​determined by the matrix isomorphism transformation relationship between the real time delay matrix and the ideal time delay matrix; The propagation distortion correction process invokes the matrix isomorphism transformation relationship to map the full frequency domain matrix signal to the computation space; The inverse mapping reconstruction process calls the regularized generalized inverse mapping corresponding to the matrix isomorphism transformation relationship to restore the imaging results in the computational space to the physical space, so that the propagation distortion correction process and the inverse mapping reconstruction process can complete the data conversion between the computational space and the physical space based on the same matrix isomorphism relationship.

10. A phased array ultrasonic testing system for the rollers of a ceramic logarithmic busbar bearing, characterized in that, include: The data acquisition module is used to acquire the generatrix curve data and anisotropic acoustic parameters of the logarithmic roller of the ceramic under test, determine the configuration information of the transducer array, and acquire the time-domain full matrix signal based on the transducer array scan. The configuration information of the transducer array includes at least the transceiver array element arrangement parameters. The matrix calculation module is used to perform surface parameterization and measurement calculation on the generatrix curve data to obtain the generatrix surface geometric parameters of the logarithmic roller of the ceramic under test; based on the generatrix surface geometric parameters, anisotropic acoustic parameters and transceiver array element arrangement parameters, a real propagation model is constructed, and based on the real propagation model, sound field characteristics are solved, sound ray path is traced and propagation time delay is integrated to obtain the real time delay matrix characterizing the real propagation characteristics; A deterministic module is constructed to establish a matrix isomorphism transformation relationship under low-rank sparse constraints for the real and ideal time delay matrices, thereby obtaining the linear mapping optimization objective function between the physical space and the computational space. The constraint solving module is used to decouple and spatially project the linear mapping optimization objective function to obtain a transformation matrix, which represents the mapping relationship between the physical space and the computational space. The imaging inversion module is used to convert the time-domain full matrix signal into the frequency-domain full matrix signal, and perform propagation distortion correction and full-focus imaging processing in combination with the transformation matrix to obtain the distortion-free imaging amplitude in the computational space. Then, inverse mapping reconstruction is performed to obtain the physical space inversion reconstruction image of the tested ceramic logarithmic roller that restores the true propagation characteristics. The defect detection module is used to detect defects in the physically inverted and reconstructed image to obtain the internal defect detection results.

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