Bent plate defect reconstruction method based on SH and Lamb guided wave inverse scattering method

Abstract

The invention relates to the technical field of nondestructive testing, and particularly discloses a bent plate defect reconstruction method based on an SH and Lamb guided wave inverse scattering method, and the method comprises the following steps: S1, solving a semi-analytical Green function in a bent plate; S2, constructing an equation between the Green function and a scattering coefficient; s3, reconstructing the defects in the bent plate: introducing Born approximation by utilizing an integral equation between the deduced semi-analytic Green function and scattering displacement, replacing a total field with a background field, introducing a defect index function as an unknown number into the integral equation, solving the unknown number by utilizing a moment method, and calculating the defects in the bent plate; and reconstructing the position and the shape of the internal hole or the surface thinning defect of the bent plate. According to the method, only a small number of ultrasonic excitation probes and receiving probes need to be arranged at the port, the position and shape of the defect can be accurately positioned through the change of the excitation frequency range and related mathematical analysis, and the method has very high utility value in engineering.

Description

Technical Field

[0001] The present invention relates to the technical field of nondestructive testing, and in particular to a method for reconstructing defects of a curved plate based on SH and Lamb guided wave inverse scattering methods. Background Art

[0002] With the development of science and technology, in industrial production, plate structures with a large number of bends need to be non-destructively tested to ensure the normal operation of the structural parts. As the requirements for testing are getting higher and higher, it is no longer enough to just detect the approximate location and shape of the defects, but it is necessary to quantitatively give the exact location and specific shape of the defects.

[0003] In recent years, many scholars have participated in research in this area and have produced many results, including eddy current testing, ultrasonic C-scanning, ultrasonic testing, etc. Among them, ultrasonic guided waves are widely used in non-destructive testing due to their long propagation distance, low attenuation, and concentrated energy. However, when ultrasonic guided waves propagate in the curved section of the plate, even if there are no defects, mode conversion and scattering will occur. This will have a certain impact on non-destructive testing. Therefore, it becomes very difficult and important to use ultrasonic guided waves to accurately reconstruct the location and specific shape of the defect. Summary of the invention

[0004] In order to solve the problems existing in the prior art, the present invention provides a bent plate defect reconstruction method based on SH and Lamb guided wave inverse scattering method, which can quantitatively detect internal and surface defects in bent plates, thereby providing a reference for defect detection of three-dimensional bent plates and solving the problems mentioned in the above background technology.

[0005] To achieve the above object, the present invention provides the following technical solution: a method for reconstructing defects of a curved plate based on SH and Lamb guided wave inverse scattering method, comprising the following steps:

[0006] S1. Solve the semi-analytical Green's function in the curved plate: solve the scattered wave field by using the DtN finite element method, and then use the analytical Green's function in the straight plate to derive the semi-analytical Green's function in the assembled curved plate;

[0007] S2. Constructing the equation between Green's function and scattering coefficient: using the Helmholtz equation and radiation conditions satisfied by ultrasonic guided waves to obtain the linearized volume integral equation between Green's function and scattering coefficient;

[0008] S3. Reconstruction of defects in the bent plate: Using the derived integral equation between the semi-analytical Green's function and the scattering displacement, the Born approximation is introduced, the background field is used instead of the total field, the defect index function is introduced as an unknown in the integral equation, and the moment method is used to solve the unknown, and the position and shape of the internal holes or surface thinning defects of the bent plate are reconstructed.

[0009] Preferably, step S1 specifically includes: firstly solving the wave field generated by the unit waveguide mode incident on the curved part by the DtN finite element method; then multiplying it by the modal coefficient of the mode in the far-field Green's function of the straight plate; and then obtaining the semi-analytical Green's function in the curved plate by superposition according to the orthogonality of the modes.

[0010] Preferably, step S2 specifically includes: establishing wave equations in the intact structure and in the defective structure respectively, and using the difference as the scattered wave equation; using the Green function and the reciprocity theorem to obtain the integral expression of the scattered field at any point in the structure; combining the boundary conditions and the Born approximation, simplifying the integral expression into a linearized volume integral equation of the far-end scattered field with respect to the incident field and the defect index function.

[0011] Preferably, the step S3 specifically includes: dividing the area to be inspected into uniform grids and discretizing the linearized volume integral equation; establishing an inverse scattering equation using the moment method, with the defect index function value in each grid as an unknown quantity and the scattering coefficient at each frequency point as a known quantity; and drawing the defect position and image according to the distribution of the obtained defect index function value in space.

[0012] Preferably, in step S1, solving the semi-analytical Green's function in the curved plate specifically comprises the following steps:

[0013] S11. Assume that the SH wave Green's function G in the straight plate 0 (Y,X) is:

[0014]

[0015] Where Y and X are the coordinates of the field point and source point respectively, and their horizontal and vertical coordinate components are (Y1, Y2) and (X1, X2) respectively. The symbol i represents the imaginary unit, J is the mode number, and β J =Jπ / 2b, is the wave number, b is the half plate thickness, μ is the material parameter, and the wave structure function

[0016] Calculate the SH wave Green's function in the curved plate: Assume that the source point X is located on the left side of the boundary Γ1, and the field point Y is located in the curved part surrounded by the boundaries Γ1~Γ4; compared with the Green's function in the straight plate, the part related to X remains unchanged, and the part related to Y becomes the wave field generated by the unit SH wave mode incident on the curved part. Therefore, the semi-analytical Green's function G(Y,X) of the SH wave in the curved plate is:

[0017]

[0018] u J (Y) is the elastic wave displacement field generated by the J-th unit mode SH wave incident on the bending part, which is numerically calculated using the DtN finite element method;

[0019] S12. Assume that the Lamb wave Green function in the straight plate is

[0020]

[0021]

[0022] Where Y, X and their components (Y1, Y2) and (X1, X2) are defined as above. It represents the displacement response in the i direction at Y when a unit force is applied in the k direction at X. The displacement and stress wave structures of the J-th unit Lamb guided wave mode propagating to the right can be obtained by the numerical calculation method of the guided wave dispersion equation (where the positive and negative signs before J represent propagation to the right and left, respectively, the same below); is the coefficient related to the source point position X and the mode number +J; the physical quantity marked with * represents the complex conjugate, and the integration range Γ of the denominator is the integration on the cross section;

[0023] Find the Lamb wave Green's function in the curved plate: Assume that the source point X is located on the left side of the boundary Γ1, and the field point Y is located in the curved part surrounded by the boundaries Γ1~Γ4; then compared with the Green's function in the straight plate, the part related to X remains unchanged, and the part related to Y becomes the wave field generated by the unit Lamb wave mode incident on the curved part. Therefore, the semi-analytical Green's function of the Lamb wave in the curved plate is G ik (Y,X) is:

[0024]

[0025] is the displacement field in the i direction of the elastic wave generated by the J-th unit mode Lamb wave incident on the curved part, calculated using the DtN finite element method;

[0026] S13. Use the DtN finite element method to solve the wave field generated by the unit mode incident wave passing through the curved area of ​​the flat plate. Suppose a certain order unit mode incident wave u inc It is transmitted from the left side to the curved part, and the scattered wave field u generated by the curved part is calculated. sca , the modeling area of ​​the DtN finite element is the upper and lower free surfaces Γ3 and Γ4 of the curved part, and the part surrounded by the cross sections Γ1 and Γ2 on the left and right sides;

[0027] The elastic dynamics equation of the finite element model is: ([K]-ω 2 [M])·[U tot ]=[F tot ]

[0028] [K] and [M] are the overall stiffness matrix and mass matrix assembled according to the general finite element method, respectively.tot ] and [F tot ] are the total wavefield displacement vector and the external force vector caused by the incident wave, respectively. 2 [M] is written as

[0029] Applying the DtN finite element method, the above equations are operated as follows:

[0030] (1) Label the node on the left boundary Γ1 as L, the node in the middle as M, and the node on the right boundary Γ2 as R; then The rows and columns of are divided into sub-matrices according to "nodes on the left boundary Γ1", "nodes in the middle", and "nodes on the right boundary Γ2";

[0031] (2) Will [U tot ] is divided into the known incident wave field displacement vector [U inc ] and the desired scattering wave field displacement vector [U sca ] and is divided into sub-matrices according to L, M, and R;

[0032] (3) Based on the orthogonality of the elastic wave field and the mathematical relationship between the displacement and the surface force on Γ1 and Γ2, the DtN boundary condition is introduced, namely:

[0033]

[0034] in, is the scattered field to be determined, D L With D R is the DtN condition matrix of ultrasonic guided waves, and the right side of the equal sign is are the nodal displacements and nodal loads satisfied by the incident wave.

[0035] Preferably, in step S2, constructing an equation between the Green's function and the scattering coefficient specifically comprises the following steps:

[0036] S21. If the guided wave excited in the curved plate is an SH wave, its scattered field displacement u sca (X) satisfies the Helmholtz equation of the following form:

[0037] ▽ 2 u sca (X)+K0 2 u sca (X)+K0 2 o(X)u tot (X) = 0

[0038] Among them, u tot (X) is the total wavefield displacement, K0 2 =ρω 2 / μ, ω is the circular frequency, ρ is the material density, μ is the Lame constant of the material, o(X) refers to the defect index function, which is a function with a range of -1 to 0, -1 is a defect, and 0 is intact; using the Green function method, K0 2 o(X)u tot As an incentive, we can conclude that:

[0039] ▽ 2 G(X,Y)+K0 2 G(X,Y)+δ(XY)=0

[0040] ▽ 2 u(X)+K0 2 u(X)+f(X)=0

[0041] Multiply the first equation by u(X) and the second equation by G(X,Y), perform volume integration in the curved plate, and then use partial integration to eliminate the boundary integral term according to the radiation boundary condition to obtain:

[0042] u sca (Y)≈∫ V K0 2 o(X)u inc G(X,Y)dX

[0043] By exchanging the signs of X and Y and substituting into the semi-analytical Green's function expression G(X,Y) of SH waveguide, we get:

[0044]

[0045] Fix the observation point on the left, and finally use the relationship between the scattered field and each standard mode guided wave α J is the reflection coefficient of the J-th mode.

[0046] To u sca (X) Perform modal decomposition to obtain the mathematical relationship between the reflection coefficient of the J-th mode of SH guided wave and the defect index function:

[0047]

[0048] Among them, u inc,I (Y) refers to the displacement of the first-order mode incident at Y, u J (Y) refers to the displacement of the J-th mode incident on point Y.

[0049] S22. If the guided wave excited in the curved plate is a Lamb wave, its scattered field and stress Total Field The elastic wave equation satisfies the following form:

[0050]

[0051] in, is the total wavefield displacement, o(X) refers to the defect index function, which is a function with a value range of -1 to 0, -1 is a defect, and 0 is intact; using the Green function method, As an incentive, we can conclude that:

[0052] S ijk,j (X,Y)+ω 2 G ik (X,Y)+δ(XY)δ ik =0

[0053] σ ij,j (X)+ω 2 u i (X)+f i (X) = 0

[0054] Among them G ik (X,Y) is the displacement Green's function, S ijk,j (X,Y) is the stress Green's function, u i (X),σ ij (X) is the i The displacement field and stress field generated by the excitation of (X) are obtained by multiplying the first equation of the above equation by u i (X), multiply the second formula by G ik (X, Y), perform volume integration in the curved plate, and then use partial integration to eliminate the boundary integral term according to the radiation boundary condition to obtain:

[0055]

[0056] Exchange the signs of X and Y and substitute them into the semi-analytical Green's function expression G of Lamb waveguide. ik (X,Y), we get:

[0057]

[0058] Fix the observation point on the left, and finally use the relationship between the scattered field and each standard mode guided wave α J is the reflection coefficient of the J-th mode.

[0059] Will By performing modal decomposition, the mathematical relationship between the reflection coefficient of the J-th mode of Lamb guided wave and the defect index function is obtained:

[0060]

[0061] in, Refers to the displacement in the i direction when the I-order mode is incident on Y. Refers to the displacement in the i direction when the J-th mode is incident on point Y.

[0062] Preferably, in step S3, the following steps are specifically included:

[0063] S31. Moment method grid division: Before using the moment method for discretization, the area to be measured is gridded, and the grid of the area to be measured is divided into uniform convex quadrilateral grids, and the size of each grid is less than 1 / 20 of the maximum wavelength; the modulus of o(Y) in each grid is taken, so that Displacement All use the displacement of the center point of the quadrilateral instead;

[0064] S32. For the case of exciting SH guided waves, the integral equation of the reflection coefficient and the defect index function is discretized according to the moment method grid as follows:

[0065]

[0066] Among them, N P Refers to the number of isoparametric units in the reconstruction area, o (p) , With ΔS (p) They are the defect index vector on the p-th unit (defect is defined as -1, and no defect is defined as 0), the displacement at the center of the p-th unit (the I-th order pure guided wave mode incident on the bent plate structure), the displacement at the center of the p-th unit (the J-th order pure guided wave mode incident on the bent plate structure), and the area of ​​the p-th unit;

[0067] The test equations are calculated by using radio waves of different frequencies and written in matrix form as: Ax = b

[0068] For SH guided waves,

[0069]

[0070]

[0071] S33. For the case of exciting Lamb guided waves, the integral equation of the reflection coefficient and the defect index function is discretized according to the moment method grid as follows:

[0072]

[0073] Among them, N P Refers to the number of isoparametric units in the reconstruction area, o (p) , With ΔS (p)They are the defect index vector on the p-th unit (defect is defined as -1, and no defect is defined as 0), the displacement at the center of the p-th unit (the I-th order pure guided wave mode incident on the bent plate structure), the displacement at the center of the p-th unit (the J-th order pure guided wave mode incident on the bent plate structure), and the area of ​​the p-th unit;

[0074] The test equations are calculated by using radio waves of different frequencies and written in matrix form as: Ax = b

[0075] For Lamb guided waves,

[0076]

[0077]

[0078] in ω J Refers to the circular frequency selected for the Jth mode.

[0079] S34, solve the ill-conditioned system of equations Ax = b, select the Tikhonov regularization method, add a regularization term, and use the least squares 1 / 2 (||Aξ-b|| 2 +λ||ξ 2 ||) as the loss function, we get the regularization solution ξ, where ξ=(A H A+λI) -1 A H b, where λ is the regularization parameter.

[0080] The beneficial effect of the present invention is that after obtaining the scattering field displacement, the present invention can directly solve the specific position and shape of the defect in the bent plate. Before defect detection, it is not necessary to know the approximate position of the defect, and the relationship between the scattering coefficient and the defect index function is directly derived from the theory to obtain the specific position and shape of the defect. As long as enough scattering field displacement data is collected, a high defect reconstruction accuracy can be obtained, providing a valuable reference for the detection of defects in the bent plate using ultrasonic guided waves. The feasibility and accuracy of the present invention are illustrated by numerical simulation, the defects in the bent plate are reconstructed, and relatively accurate position and shape information is obtained. BRIEF DESCRIPTION OF THE DRAWINGS

[0081] Figure 1 Schematic diagram of the steps of the defect reconstruction method of the curved plate based on SH and Lamb guided wave inverse scattering method;

[0082] Figure 2 This is a schematic diagram of the structure of a curved plate in an embodiment of the present invention;

[0083] Figure 3 A mesh division diagram of a reconstruction area of ​​a curved plate in an embodiment of the present invention;

[0084] Figure 4 is the Tikhonov regularized L curve in the present invention;

[0085] Figure 5 is the dispersion curve of Lamb wave in the present invention, ωd is the dimensionless frequency, kd is the dimensionless wave number, and b is the half plate thickness;

[0086] Figure 6 Reconstructing a scattering coefficient diagram for the scattering coefficient of the 0th order SH wave mode incident on the curved plate and the known defect shape involved in the present invention;

[0087] Figure 7 Reconstructing a scattering coefficient diagram for the scattering coefficient of the 0th-order mode of Lamb wave incident on the curved plate and the known defect shape involved in the present invention;

[0088] Figure 8 To use Figure 5 Schematic diagram of elliptical defect reconstructed from the scattering coefficient in;

[0089] Fig. 9 To use Figure 6 Schematic diagram of elliptical defect reconstructed from the scattering coefficient in;

[0090] Fig.10 Schematic diagram of a double elliptical defect reconstructed using the scattering coefficient;

[0091] Fig.11 Schematic diagram of the semi-elliptical defect on the surface reconstructed using the scattering coefficient;

[0092] Fig.12 Schematic diagram of surface rectangular defects reconstructed using the scattering coefficient. DETAILED DESCRIPTION

[0093] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0094] The present invention provides a technical solution: a method for reconstructing defects of a curved plate based on SH and Lamb guided wave inverse scattering method, such as Figure 1 As shown, the following steps are included:

[0095] S1. Solve the semi-analytical Green's function in the curved plate: Utilize the modal orthogonality and use the DtN finite element method to solve the scattered wave field. Then use the analytical Green's function in the straight plate to derive the semi-analytical Green's function in the assembled curved plate.

[0096] Firstly, the wave field generated by the unit guided wave mode incident on the curved part is solved by the DtN finite element method; then it is multiplied by the modal coefficient of the mode in the far-field Green's function of the straight plate; and then the semi-analytical Green's function in the curved plate is obtained according to the orthogonality of the modes.

[0097] The method for solving the semi-analytical Green's function in the curved plate specifically comprises the following steps:

[0098] S11. Assume that the SH wave Green's function in the straight plate is:

[0099]

[0100] Where Y and X are the coordinates of the field point and source point respectively, and their horizontal and vertical coordinate components are (Y1, Y2) and (X1, X2) respectively. The symbol i represents the imaginary unit, J is the mode number, and β J =Jπ / 2b, is the wave number, b is the half plate thickness, μ is the material parameter, and the wave structure function

[0101] Now ask for Figure 2 The SH wave Green's function in the curved plate is shown in Figure 1. Assume that the source point X is located on the left side of the boundary Γ1, and the field point Y is located in the curved part surrounded by the boundaries Γ1 to Γ4; compared with the Green's function in the straight plate, the part related to X remains unchanged, and the part related to Y becomes the wave field generated by the unit SH wave mode incident on the curved part. Therefore, the semi-analytical Green's function G(Y,X) of the SH wave in the curved plate is:

[0102]

[0103] u J (Y) is the elastic wave displacement field generated by the J-th unit mode SH wave incident on the bending part, which is numerically calculated using the DtN finite element method;

[0104] S12. Assume that the Lamb wave Green function in the straight plate is

[0105]

[0106]

[0107] Where Y, X and their components (Y1, Y2) and (X1, X2) are defined as above. It represents the displacement response in the i direction at Y when a unit force is applied in the k direction at X. The displacement and stress wave structures of the J-th unit Lamb guided wave mode propagating to the right can be obtained by the numerical calculation method of the guided wave dispersion equation (where the positive and negative signs before J represent propagation to the right and left, respectively, the same below);

[0108] is the coefficient related to the source point position X and the mode number +J; the physical quantity marked with * represents the complex conjugate, and the integration range Γ of the denominator is the integration on the cross section;

[0109] Now ask for Figure 2 The Lamb wave Green's function in the curved plate is shown. Assume that the source point X is located on the left side of the boundary Γ1, and the field point Y is located in the curved part surrounded by the boundaries Γ1 to Γ4; then compared with the Green's function in the straight plate, the part related to X remains unchanged, and the part related to Y becomes the wave field generated by the unit Lamb wave mode incident on the curved part. Therefore, the semi-analytical Green's function of the Lamb wave in the curved plate is G ik (Y,X) is:

[0110]

[0111] is the displacement field in the i direction of the elastic wave generated by the J-th unit mode Lamb wave incident on the curved part, calculated using the DtN finite element method;

[0112] S13. Use the DtN finite element method to solve the wave field generated by the unit mode incident wave passing through the curved area of ​​the flat plate, such as Figure 2 As shown in the figure, the left and right parts of the curved plate structure are straight, with only a curved part in the middle at an angle of θ. DtN boundary conditions are applied to the left and right sides, and free boundary conditions are applied to the upper and lower sides. This area is the actual simulation area. Compared with the general finite element method, the scattering coefficient obtained by this method has the advantages of not requiring a perfect matching layer and only requiring simple post-processing. The meshing of the moment method can also be expanded by it.

[0113] Assume that the incident wave u is a unit mode of a certain order inc It is transmitted from the left side to the curved part, and the scattered wave field u generated by the curved part is calculated. sca , the modeling area of ​​the DtN finite element is the upper and lower free surfaces Γ3 and Γ4 of the curved part, and the part surrounded by the cross sections Γ1 and Γ2 on the left and right sides;

[0114] The elastic dynamic equation of the finite element model is:

[0115] ([K]-ω 2 [M])·[U tot ]=[F tot ]

[0116]

[0117] [K] and [M] are the overall stiffness matrix and mass matrix assembled according to the general finite element method, respectively, which are composed of the unit stiffness matrix [k e ] I With the unit mass matrix [m e ]I Synthesize by node number, where the subscript I represents the unit number, I = 1, 2, ..., N e , N e is the number of units. In the Ith unit, [B (I) ] is the geometry matrix, [D] is the elasticity matrix, [N (I) ] is the shape function matrix, ρ is the material density, S I Represents the unit area. tot ] and [F tot ] are the total wavefield displacement vector and the external force vector caused by the incident wave, respectively. 2 [M] is written as

[0118] For the SH wave scattering problem, [k e ] I With [m e ] I All are 8×8 matrices, the unit uses 8-node quadrilateral isoparametric units, and the shape function A quadratic shape function is used.

[0119]

[0120] [k e ] and [m e ] are all 8×8 matrices.

[0121] For Lamb wave scattering, [k e ] I With [m e ] I All are 16×16 matrices, and the units also use 8-node quadrilateral isoparametric units. The shape function N i Quadratic shape functions are also used.

[0122]

[0123]

[0124] [k e ] and [m e ] are all 16×16 matrices.

[0125] Where E is the elastic modulus and μ is Poisson's ratio.

[0126] Applying the DtN finite element method, the above equations are operated as follows:

[0127] (1) Label the node on the left boundary Γ1 as L, the node in the middle as M, and the node on the right boundary Γ2 as R; Figure 2 As shown, then The rows and columns of are divided into sub-matrices according to "nodes on the left boundary Γ1", "nodes in the middle", and "nodes on the right boundary Γ2";

[0128] (2) Will [U tot ] is divided into the known incident wave field displacement vector [U inc ] and the desired scattering wave field displacement vector [U sca ] and is divided into sub-matrices according to L, M, and R;

[0129] (3) Based on the orthogonality of the elastic wave field and the mathematical relationship between the displacement and the surface force on Γ1 and Γ2, the DtN boundary condition is introduced, namely:

[0130]

[0131] in, is the scattered field to be determined, D L With D R is the DtN condition matrix of ultrasonic guided waves, and the right side of the equal sign is are the nodal displacements and nodal loads satisfied by the incident wave.

[0132]

[0133] For SH waves:

[0134]

[0135]

[0136]

[0137] n is the SH waveguide mode number (n=1…N), k N is the corresponding wave number. (or ) The matrix elements are the integrals of a unit modal displacement field and a node shape function on the left (or right) boundary. The row number i is the modal number, and the column number j is the overall number of the node on the cross section. When the node with the overall number j is shared by several units, the integrated shape function needs to traverse these units and superimpose. (I) J = j means that the node with the internal number of the Jth node (J = 1...8) of the Ith unit is exactly the node with the overall number j.

[0138]

[0139] For Lamb waves:

[0140] is the unit nth-order Lamb wave modal stress. P is the number of cross-sectional nodes.

[0141]

[0142] N [j] is the node shape function matrix of the jth unit. Similarly, it is also necessary to traverse and superimpose in the units that share the node:

[0143]

[0144] is the elastic coefficient matrix, is the nth order Lamb wave modal displacement. λ and μ are Lamé constants. B [j] is the node geometry matrix of the jth unit. Similarly, it is also necessary to traverse and superimpose the units that share the node:

[0145] S2. Construct an equation between the Green's function and the scattering coefficient: Use the Helmholtz equation satisfied by ultrasonic guided waves and the radiation conditions to obtain the linearized volume integral equation between the Green's function and the scattering coefficient.

[0146] The wave equations in the intact structure and the defective structure are established respectively, and the difference is used as the scattered wave equation. The integral expression of the scattered field at any point in the structure is obtained using Green's function and the reciprocity theorem. Combined with the boundary conditions and Born approximation, the integral expression is simplified into a linearized volume integral equation of the far-end scattered field with respect to the incident field and the defect index function.

[0147] The constructing of the equation between the Green's function and the scattering coefficient specifically comprises the following steps:

[0148] S21. If the guided wave excited in the curved plate is an SH wave, its scattered field displacement u sca (X) satisfies the Helmholtz equation of the following form:

[0149]

[0150] Among them, u tot (X) is the total wavefield displacement, K0 2 =ρω 2 / μ, ω is the circular frequency, ρ is the material density, μ is the Lame constant of the material, o(X) refers to the defect index function, which is a function with a range of -1 to 0, -1 is a defect, and 0 is intact; using the Green function method, K0 2 o(X)u tot As an incentive, we can conclude that:

[0151] ▽ 2 G(X,Y)+K0 2G(X,Y)+δ(XY)=0

[0152] ▽ 2 u(X)+K0 2 u(X)+f(X)=0

[0153] Multiply the first equation by u(X) and the second equation by G(X,Y), perform volume integration in the curved plate, and then use partial integration to eliminate the boundary integral term according to the radiation boundary condition to obtain:

[0154]

[0155] The first term on the left side of the above equation can be converted into a surface integral using Gauss's theorem, and the boundary integral is 0 at the far field boundary (DtN condition) and the free surface, so the above equation can be simplified to:

[0156] ∫ V (δ(XY)u(X)-f(Y)G(X,Y))dV=0

[0157] Using the symmetry of the Green function G(X,Y)=G(Y,X), and the Born approximation, we use the incident field u inc (X) replaces the total field u tot (X), and substitute it into f(Y) to get:

[0158] u sca (Y)≈∫ V K0 2 o(X)u inc (X)G(X,Y)dX

[0159] Exchange the signs of X and Y and substitute the semi-analytical Green's function expression G(X,Y) of SH waveguide into it: curved ,get:

[0160]

[0161] Fix the observation point on the left, and finally use the relationship between the scattered field and each standard mode guided wave α J is the reflection coefficient of the J-th mode.

[0162] To u sca (X) Perform modal decomposition to obtain the mathematical relationship between the reflection coefficient of the J-th mode of SH guided wave and the defect index function:

[0163]

[0164] Among them, u inc,I (Y) refers to the displacement of the first-order mode incident at Y, u J (Y) refers to the displacement of the J-th mode incident on point Y.

[0165] S22. If the guided wave excited in the curved plate is a Lamb wave, its scattered field and stress Total Field The elastic wave equation satisfies the following form:

[0166]

[0167] in, is the total wavefield displacement, o(X) refers to the defect index function, which is a function with a value range of -1 to 0, -1 is a defect, and 0 is intact; using the Green function method, As an incentive, we can conclude that:

[0168] S ijk,j (X,Y)+ω 2 G ik (X,Y)+δ(XY)δ ik =0

[0169] σ ij,j (X)+ω 2 u i (X)+f i (X) = 0

[0170] Among them G ik (X,Y) is the displacement Green's function, S ijk,j (X,Y) is the stress Green's function, u i (X),σ ij (X) is the i The displacement field and stress field generated by the excitation of (X) are obtained by multiplying the first equation of the above equation by u i (X), multiply the second formula by G ik (X,Y), and subtract them to get:

[0171]

[0172] The first term on the left side of the above equation can be converted into a surface integral using Gauss's theorem, and the boundary integral is 0 at the far field boundary (DtN condition) and the free surface, so the above equation can be simplified to:

[0173]

[0174] Using the symmetry of Green's function G ik (X,Y)=G ki (Y,X), similar to the Born approximation, using the incident field Replace the total field And substitute f i (X) can be obtained:

[0175]

[0176] Exchange the signs of X and Y and substitute them into the semi-analytical Green's function expression G of Lamb waveguide. ki (Y,X), we get:

[0177]

[0178] Fix the observation point on the left, and finally use the relationship between the scattered field and each standard mode guided wave α J is the reflection coefficient of the J-th mode.

[0179] Will By performing modal decomposition, the mathematical relationship between the reflection coefficient of the J-th mode of Lamb guided wave and the defect index function is obtained:

[0180]

[0181] in, Refers to the displacement in the i direction when the I-order mode is incident on Y. Refers to the displacement in the i direction when the J-th mode is incident on point Y.

[0182] S3. Reconstruction of defects in the bent plate: Using the derived integral equation between the semi-analytical Green's function and the scattering displacement, the Born approximation is introduced, the background field is used instead of the total field, the defect index function is introduced as an unknown in the integral equation, and the moment method is used to solve the unknown, and the position and shape of the internal holes or surface thinning defects of the bent plate are reconstructed.

[0183] The area to be inspected is divided into uniform grids, and the linearized volume integral equation is discretized; the inverse scattering equation of the moment method is established, with the defect index function value in each grid as the unknown quantity and the scattering coefficient at each frequency point as the known quantity; the defect position and image are drawn according to the distribution of the obtained defect index function value in space.

[0184] The specific steps include:

[0185] S31. Moment method grid division: Before using the moment method for discretization, the area to be measured is gridded and the grid of the area to be measured is divided into uniform convex quadrilateral grids, such as Figure 3 As shown, and the size of each grid is less than 1 / 20 of the maximum wavelength; modulo o(Y) in each grid, so that Displacement All use the displacement of the center point of the quadrilateral instead;

[0186] S32. For the case of exciting SH guided waves, the integral equation of the reflection coefficient and the defect index function is discretized according to the moment method grid as follows:

[0187]

[0188] Among them, N P Refers to the number of isoparametric units in the reconstruction area, o (p) , With ΔS (p) They are the defect index vector on the p-th unit (defect is defined as -1, and no defect is defined as 0), the displacement at the center of the p-th unit (the I-th order pure guided wave mode incident on the bent plate structure), the displacement at the center of the p-th unit (the J-th order pure guided wave mode incident on the bent plate structure), and the area of ​​the p-th unit;

[0189] For this formula, the same expression can be obtained for each incident frequency. The test equation group is calculated by using the incident waves of different frequencies and written in matrix form as: Ax = b

[0190] For SH guided waves,

[0191]

[0192]

[0193] S33. For the case of exciting Lamb guided waves, the integral equation of the reflection coefficient and the defect index function is discretized according to the moment method grid as follows:

[0194]

[0195]

[0196] Among them, N P Refers to the number of isoparametric units in the reconstruction area, o (p) , With ΔS (p) They are the defect index vector on the p-th unit (defect is defined as -1, and no defect is defined as 0), the displacement at the center of the p-th unit (the I-th order pure guided wave mode incident on the bent plate structure), the displacement at the center of the p-th unit (the J-th order pure guided wave mode incident on the bent plate structure), and the area of ​​the p-th unit;

[0197] For this formula, the same expression can be obtained for each incident frequency. The test equation group is calculated by using the incident waves of different frequencies and written in matrix form as: Ax = b

[0198] For Lamb guided waves,

[0199]

[0200]

[0201] in ω JRefers to the circular frequency selected for the Jth mode.

[0202] Theoretically, as long as enough scattered field reflection coefficients are collected, the unknown defect index vector can be solved, and all the geometric information of the defect area in the reconstruction area can be drawn. The accuracy of reconstruction depends on the number of frequencies collected. The more frequencies, the higher the accuracy.

[0203] like Figure 4 As shown, the defect index vector is obtained by solving the reconstruction matrix using Tikhonov regularization, and the defect index vector is post-processed using threshold segmentation to reconstruct the precise shape of the plate structure defect. Specifically, due to the introduction of the Born approximation and calculation errors, observation errors, etc., the above equation may be ill-posed.

[0204] S34, solve the ill-conditioned system of equations Ax = b, select the Tikhonov regularization method, add a regularization term, and use the least squares 1 / 2 (||Aξ-b|| 2 +λ||ξ 2 ||) as the loss function, we get the regularization solution ξ, where ξ=(A H A+λI) -1 A H b, where λ is the regularization parameter.

[0205] Here, the L-Curve method is used to select λ. The minimum point of the L-curve is the best λ value. The threshold division of the solution is

[0206] like Figure 6 and Figure 7 As shown in the figure, they are the reflection coefficients obtained after the 0th-order pure SH guided wave and Lamb wave are incident on the curved plate. It can be seen that the same scattering coefficient can be obtained for both the direct problem and the inverse problem, which is sufficient to verify the accuracy of this method to some extent.

[0207] like Figure 8 , 9 , 10, 11, and 12 show the defect shape and position quantitatively reconstructed by using the equation of the scattering coefficient and the defect index in the present invention. Fig.10 , Fig.11 , Fig.12 The following are schematic diagrams of double ellipse, surface semi-ellipse and surface rectangular defects reconstructed using the scattering coefficient. It can be seen that this method can also reconstruct single defects and double defects well.

[0208] The present invention has the following advantages and beneficial effects: 1. The ultrasonic nondestructive testing method is time-saving and fast in operation, and does not require a large coverage of the test object. Only a small number of ultrasonic excitation probes and receiving probes need to be set at the port. By changing the excitation frequency range and related mathematical analysis, the position and shape of the defect can be accurately located, which has a high utility value in engineering. 2. The method of the present invention is not only applicable to simple single-layer isotropic bending plates, but also to complex multi-layer anisotropic bending plates in engineering. It has strong universality.

[0209] The terms used in the embodiments of the present invention are only for the purpose of describing specific embodiments, and are not intended to limit the present invention. The singular forms "a", "said" and "the" used in the embodiments of the present invention and the appended claims are also intended to include plural forms, unless the context clearly indicates other meanings.

[0210] It should be understood that the term "and / or" used in this article is only a description of the association relationship of associated objects, indicating that there can be three relationships. For example, A and / or B can represent: A exists alone, A and B exist at the same time, and B exists alone. In addition, the character " / " in this article generally indicates that the associated objects before and after are in an "or" relationship.

[0211] The word "if" as used herein may be interpreted as "at the time of" or "when" or "in response to determining" or "in response to detecting", depending on the context. Similarly, the phrases "if it is determined" or "if (stated condition or event) is detected" may be interpreted as "when it is determined" or "in response to determining" or "when detecting (stated condition or event)" or "in response to detecting (stated condition or event)", depending on the context.

[0212] The "first\second" mentioned in the embodiments is only to distinguish similar objects, and does not represent a specific order for the objects. It is understandable that the "first\second" can be interchanged with the specific order or sequence where permitted. It should be understood that the objects distinguished by "first\second" can be interchanged where appropriate, so that the embodiments described herein can be implemented in an order other than those illustrated or described herein.

[0213] Although the present invention has been described in detail with reference to the aforementioned embodiments, it is still possible for those skilled in the art to modify the technical solutions described in the aforementioned embodiments, or to make equivalent substitutions for some of the technical features therein. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A method for reconstructing defects of a curved plate based on SH and Lamb guided wave inverse scattering method, characterized in that: The steps include: S1. Solve the semi-analytical Green's function in the curved plate: solve the scattered wave field by using the DtN finite element method, and then use the analytical Green's function in the straight plate to derive the semi-analytical Green's function in the assembled curved plate; S2. Constructing the equation between Green's function and scattering coefficient: using the Helmholtz equation and radiation conditions satisfied by ultrasonic guided waves to obtain the linearized volume integral equation between Green's function and scattering coefficient; S3. Reconstruction of defects in the bent plate: Using the derived integral equation between the semi-analytical Green's function and the scattering displacement, the Born approximation is introduced, the background field is used instead of the total field, the defect index function is introduced as an unknown in the integral equation, and the moment method is used to solve the unknown, and the position and shape of the internal holes or surface thinning defects of the bent plate are reconstructed.

2. The method for defect reconstruction of a curved plate based on SH and Lamb guided wave inverse scattering method according to claim 1, characterized in that: The step S1 specifically includes: firstly, solving the wave field generated by the unit waveguide mode incident on the curved part by the DtN finite element method; then multiplying the modal coefficient of the mode in the far-field Green's function of the straight plate; and then obtaining the semi-analytical Green's function in the curved plate by superposition according to the orthogonality of the modes.

3. The method for defect reconstruction of a curved plate based on SH and Lamb guided wave inverse scattering method according to claim 1, characterized in that: The step S2 specifically includes: establishing wave equations in the intact structure and in the defective structure respectively, and using the difference as the scattered wave equation; using the Green function and the reciprocity theorem to obtain the integral expression of the scattered field at any point in the structure; combining the boundary conditions and the Born approximation, simplifying the integral expression into a linearized volume integral equation of the far-end scattered field with respect to the incident field and the defect index function.

4. The method for reconstructing defects of a curved plate based on SH and Lamb guided wave inverse scattering method according to claim 1, characterized in that: The step S3 specifically includes: dividing the area to be inspected into uniform grids and discretizing the linearized volume integral equation; establishing an inverse scattering equation using the moment method, with the defect index function value in each grid as an unknown quantity and the scattering coefficient at each frequency point as a known quantity; and drawing the defect position and image according to the distribution of the obtained defect index function value in space.

5. The method for reconstructing defects of a curved plate based on SH and Lamb guided wave inverse scattering method according to claim 1, characterized in that: In step S1, solving the semi-analytical Green's function in the curved plate specifically includes the following steps: S11. Assume that the SH wave Green's function G in the flat plate 0 (Y,X) is: Among them, Y, X are the coordinates of the field point and the source point respectively, and their horizontal and vertical coordinate components are (Y1, Y2) and (X1, X2) respectively. The symbol i represents the imaginary unit, J is the mode number, and β J =Jπ / 2b, is the wave number, b is the half plate thickness, μ is the material parameter, and the wave structure function Calculate the SH wave Green's function in the curved plate: Assume that the source point X is located on the left side of the boundary Γ1, and the field point Y is located in the curved part surrounded by the boundaries Γ1~Γ4; compared with the Green's function in the straight plate, the part related to X remains unchanged, and the part related to Y becomes the wave field generated by the unit SH wave mode incident on the curved part. Therefore, the semi-analytical Green's function G(Y,X) of the SH wave in the curved plate is: u J (Y) is the elastic wave displacement field generated by the J-th unit mode SH wave incident on the bending part, which is numerically calculated using the DtN finite element method; S12. Assume that the Lamb wave Green function in the straight plate is in, It represents the displacement response in the i direction at Y when a unit force is applied in the k direction at X. They refer to the displacement and stress wave structures of the J-th unit Lamb guided wave mode propagating to the right, respectively, which are obtained by the numerical calculation method of the guided wave dispersion equation; is the coefficient related to the source point position X and the mode number +J; the physical quantity marked with * represents the complex conjugate, and the integration range Γ of the denominator is the integration on the cross section; Find the Lamb wave Green's function in the curved plate: Assume that the source point X is located on the left side of the boundary Γ1, and the field point Y is located in the curved part surrounded by the boundaries Γ1~Γ4; then compared with the Green's function in the straight plate, the part related to X remains unchanged, and the part related to Y becomes the wave field generated by the unit Lamb wave mode incident on the curved part. Therefore, the semi-analytical Green's function of the Lamb wave in the curved plate is G ik (Y,X) is: is the displacement field in the i direction of the elastic wave generated by the J-th unit mode Lamb wave incident on the curved part, calculated using the DtN finite element method; S13. Use the DtN finite element method to solve the wave field generated by the unit mode incident wave passing through the curved area of ​​the flat plate. Suppose a certain order unit mode incident wave u inc It is transmitted from the left side to the curved part, and the scattered wave field u generated by the curved part is calculated. sca , the modeling area of ​​the DtN finite element is the upper and lower free surfaces Γ3 and Γ4 of the curved part, and the part surrounded by the cross sections Γ1 and Γ2 on the left and right sides; The elastic dynamics equation of the finite element model is: ([K]-ω 2 [M])·[U tot ]=[F tot ] [K] and [M] are the overall stiffness matrix and mass matrix assembled according to the general finite element method, respectively. tot ] and [F tot ] are the total wavefield displacement vector and the external force vector caused by the incident wave, respectively. 2 [M] is written as Applying the DtN finite element method, the above equations are operated as follows: (1) Label the node on the left boundary Γ1 as L, the node in the middle as M, and the node on the right boundary Γ2 as R; then The rows and columns of are divided into sub-matrices according to "nodes on the left boundary Γ1", "nodes in the middle", and "nodes on the right boundary Γ2"; (2) Will [U tot ] is divided into the known incident wave field displacement vector [U inc ] and the desired scattering wave field displacement vector [U sca ] and is divided into sub-matrices according to L, M, and R; (3) Based on the orthogonality of the elastic wave field and the mathematical relationship between the displacement and the surface force on Γ1 and Γ2, the DtN boundary condition is introduced, namely: in, is the scattered field to be determined, D L With D R is the DtN condition matrix of ultrasonic guided waves, and the right side of the equal sign is are the nodal displacements and nodal loads satisfied by the incident wave.

6. The method for defect reconstruction of a curved plate based on SH and Lamb guided wave inverse scattering method according to claim 1, characterized in that: In step S2, constructing an equation between the Green's function and the scattering coefficient specifically includes the following steps: S21. If the guided wave excited in the curved plate is an SH wave, its scattered field displacement u sca (X) satisfies the Helmholtz equation of the following form: ▽ 2 u sca (X)+K0 2 u sca (X)+K0 2 o(X)u tot (X)=0 Among them, u tot (X) is the total wavefield displacement, K0 2 =ρω 2 / μ, ω is the circular frequency, ρ is the material density, μ is the Lame constant of the material, o(X) refers to the defect index function, which is a function with a range of -1 to 0, -1 is a defect, and 0 is intact; using the Green function method, K0 2 o(X)u tot As an incentive, we can conclude that: Multiply the first equation by u(X) and the second equation by G(X,Y), perform volume integration in the curved plate, and then use partial integration to eliminate the boundary integral term according to the radiation boundary condition to obtain: u sca (Y)≈∫ V K0 2 o(X)u inc G(X,Y)dX By exchanging the signs of X and Y and substituting into the semi-analytical Green's function expression G(X,Y) of SH waveguide, we get: Fix the observation point on the left, and finally use the relationship between the scattered field and each standard mode guided wave is the reflection coefficient of the J-th order SH wave mode; To u sca (X) Perform modal decomposition to obtain the mathematical relationship between the reflection coefficient of the J-th mode of SH guided wave and the defect index function: Among them, u inc,I (Y) refers to the displacement of the first-order mode incident at Y, u J (Y) refers to the displacement of the Jth mode incident at point Y; S22. If the guided wave excited in the curved plate is a Lamb wave, its scattered field and stress field With the total displacement field The elastic wave equation satisfies the following form: in, is the total wavefield displacement, o(X) refers to the defect index function, which is a function with a value range of -1 to 0, -1 is a defect, and 0 is intact; using the Green function method, As an incentive, we can conclude that: S ijk,j (X,Y)+ω 2 G ik (X,Y)+δ(X-Y)δ ik =0 s ij,j (X)+ω 2 you i (X)+f i (X)=0 Among them G ik (X,Y) is the displacement Green's function, S ijk,j (X,Y) is the stress Green's function, u i (X),σ ij (X) is the i The displacement field and stress field generated by the excitation of (X) are obtained by multiplying the first equation of the above equation by u i (X), multiply the second formula by G ik (X, Y), perform volume integration in the curved plate, and then use partial integration to eliminate the boundary integral term according to the radiation boundary condition to obtain: Exchange the signs of X and Y and substitute them into the semi-analytical Green's function expression G of Lamb waveguide. ik (X,Y), we get: Fix the observation point on the left, and finally use the relationship between the scattered field and each standard mode guided wave is the reflection coefficient of the J-th order Lamb wave mode; Will By performing modal decomposition, the mathematical relationship between the reflection coefficient of the J-th mode of Lamb guided wave and the defect index function is obtained: in, Refers to the displacement in the i direction when the I-order mode is incident on Y. Refers to the displacement in the i direction when the J-th mode is incident on point Y.

7. The method for defect reconstruction of a curved plate based on SH and Lamb guided wave inverse scattering method according to claim 1, characterized in that: In step S3, the following steps are specifically included: S31. Moment method grid division: Before using the moment method for discretization, the area to be measured is gridded, and the grid of the area to be measured is divided into uniform convex quadrilateral grids, and the size of each grid is less than 1 / 20 of the maximum wavelength; the modulus of o(Y) in each grid is taken, so that Displacement All use the displacement of the center point of the quadrilateral instead; S32. For the case of exciting SH guided waves, the integral equation of the reflection coefficient and the defect index function is discretized according to the moment method grid as follows: Among them, N P Refers to the number of isoparametric units in the reconstruction area, o (p) , With ΔS (p) are the defect index vector on the p-th unit, the displacement at the center of the p-th unit, the displacement at the center of the p-th unit, and the area of ​​the p-th unit; The test equations are calculated by using waves of different frequencies and written in matrix form as: Ax = b For SH waveguide, S33. For the case of exciting Lamb guided waves, the integral equation of the reflection coefficient and the defect index function is discretized according to the moment method grid as follows: Among them, N P Refers to the number of isoparametric units in the reconstruction area, o (p) , With ΔS (p) are the defect index vector on the p-th unit, the displacement at the center of the p-th unit, the displacement at the center of the p-th unit, and the area of ​​the p-th unit; The test equations are calculated by using radio waves of different frequencies and written in matrix form as: Ax = b For Lamb guided waves, in ω J Refers to the circular frequency selected by the Jth mode; S34, solve the ill-conditioned system of equations Ax = b, select the Tikhonov regularization method, add a regularization term, and use the least squares 1 / 2 (||Aξ-b|| 2 +λ||ξ 2 ||) as the loss function, we get the regularization solution ξ, where ξ=(A H A+λI) -1 A H b, where λ is the regularization parameter.

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