CN116227259B - Correction boundary element solution method for lamb wave mode conversion and scattering in bending plate - Google Patents
Correction boundary element solution method for lamb wave mode conversion and scattering in bending plate Download PDFInfo
- Publication number
- CN116227259B CN116227259B CN202211542887.8A CN202211542887A CN116227259B CN 116227259 B CN116227259 B CN 116227259B CN 202211542887 A CN202211542887 A CN 202211542887A CN 116227259 B CN116227259 B CN 116227259B
- Authority
- CN
- China
- Prior art keywords
- boundary
- integral
- field
- displacement
- matrix
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Active
Links
- 238000000034 method Methods 0.000 title claims abstract description 50
- 235000019687 Lamb Nutrition 0.000 title claims abstract description 28
- 238000012937 correction Methods 0.000 title claims abstract description 25
- 238000005452 bending Methods 0.000 title claims abstract description 21
- 238000006243 chemical reaction Methods 0.000 title claims abstract description 13
- 238000006073 displacement reaction Methods 0.000 claims abstract description 47
- 239000011159 matrix material Substances 0.000 claims abstract description 36
- 239000000243 solution Substances 0.000 claims abstract description 23
- 238000004364 calculation method Methods 0.000 claims abstract description 21
- 230000010354 integration Effects 0.000 claims abstract description 21
- 239000003637 basic solution Substances 0.000 claims abstract description 11
- 239000013598 vector Substances 0.000 claims description 13
- 239000002585 base Substances 0.000 claims description 8
- 230000000694 effects Effects 0.000 claims description 6
- 230000004044 response Effects 0.000 claims description 4
- 230000005540 biological transmission Effects 0.000 claims description 3
- 230000009466 transformation Effects 0.000 claims description 3
- 238000004458 analytical method Methods 0.000 claims description 2
- 230000003068 static effect Effects 0.000 claims 1
- 230000007547 defect Effects 0.000 description 17
- 230000004048 modification Effects 0.000 description 11
- 238000012986 modification Methods 0.000 description 11
- 230000006870 function Effects 0.000 description 8
- 238000010586 diagram Methods 0.000 description 4
- 230000006872 improvement Effects 0.000 description 4
- 239000000463 material Substances 0.000 description 3
- 238000009659 non-destructive testing Methods 0.000 description 3
- 238000012795 verification Methods 0.000 description 3
- 230000006978 adaptation Effects 0.000 description 2
- 238000004590 computer program Methods 0.000 description 2
- 230000008569 process Effects 0.000 description 2
- 238000001228 spectrum Methods 0.000 description 2
- 230000007704 transition Effects 0.000 description 2
- 241000579895 Chlorostilbon Species 0.000 description 1
- 230000004075 alteration Effects 0.000 description 1
- 230000007423 decrease Effects 0.000 description 1
- 239000010976 emerald Substances 0.000 description 1
- 229910052876 emerald Inorganic materials 0.000 description 1
- 238000005265 energy consumption Methods 0.000 description 1
- 230000005284 excitation Effects 0.000 description 1
- 238000009776 industrial production Methods 0.000 description 1
- 238000009434 installation Methods 0.000 description 1
- 239000010977 jade Substances 0.000 description 1
- 230000000644 propagated effect Effects 0.000 description 1
- ZLIBICFPKPWGIZ-UHFFFAOYSA-N pyrimethanil Chemical compound CC1=CC(C)=NC(NC=2C=CC=CC=2)=N1 ZLIBICFPKPWGIZ-UHFFFAOYSA-N 0.000 description 1
- 238000011160 research Methods 0.000 description 1
- 239000010979 ruby Substances 0.000 description 1
- 229910001750 ruby Inorganic materials 0.000 description 1
- 230000035945 sensitivity Effects 0.000 description 1
- 238000004088 simulation Methods 0.000 description 1
- 230000003595 spectral effect Effects 0.000 description 1
Classifications
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
- G06F30/20—Design optimisation, verification or simulation
- G06F30/23—Design optimisation, verification or simulation using finite element methods [FEM] or finite difference methods [FDM]
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
- G06F17/10—Complex mathematical operations
- G06F17/11—Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
- G06F17/10—Complex mathematical operations
- G06F17/16—Matrix or vector computation, e.g. matrix-matrix or matrix-vector multiplication, matrix factorization
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
- G06F2119/14—Force analysis or force optimisation, e.g. static or dynamic forces
-
- Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
- Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
- Y02T—CLIMATE CHANGE MITIGATION TECHNOLOGIES RELATED TO TRANSPORTATION
- Y02T90/00—Enabling technologies or technologies with a potential or indirect contribution to GHG emissions mitigation
Landscapes
- Engineering & Computer Science (AREA)
- Physics & Mathematics (AREA)
- General Physics & Mathematics (AREA)
- Mathematical Physics (AREA)
- Theoretical Computer Science (AREA)
- Data Mining & Analysis (AREA)
- Computational Mathematics (AREA)
- Mathematical Analysis (AREA)
- Mathematical Optimization (AREA)
- Pure & Applied Mathematics (AREA)
- General Engineering & Computer Science (AREA)
- Algebra (AREA)
- Software Systems (AREA)
- Databases & Information Systems (AREA)
- Evolutionary Computation (AREA)
- Computer Hardware Design (AREA)
- Computing Systems (AREA)
- Geometry (AREA)
- Operations Research (AREA)
- Investigating Or Analyzing Materials By The Use Of Ultrasonic Waves (AREA)
Abstract
The invention provides a correction boundary element solution method for lamb wave modal conversion and scattering in a bending plate, which comprises the following steps: s1: constructing a boundary integral formula; the method comprises the following steps: converting the volume integral into area integral by the dynamics reciprocity theorem to obtain a boundary integral equation; s2: solving a far-field basic solution; the method comprises the following steps: adding a virtual boundary, dividing a primary boundary into independent models, and representing the integration of a far field by using near field boundary integration by using a dynamics reciprocity theorem; s3: constructing and solving a global boundary element system matrix; the method comprises the following steps: discretizing the boundary integral equation, and introducing a displacement continuity condition to obtain a final corrected BEM matrix. The invention improves the calculation efficiency, and simultaneously, the far-field information is incorporated into the calculation, thereby improving the calculation accuracy.
Description
Technical Field
The invention relates to the technical field of nondestructive testing, in particular to a correction boundary element solution method for lamb wave modal conversion and scattering in a bending plate.
Background
In industrial production, plate structures containing a large number of bends require non-destructive testing to ensure proper operation of the structure. However, ultrasonic guided waves, even if free of defects, produce modal transformation and scattering as they propagate through the curved section of the plate. This has a certain impact on the non-destructive testing. Thus, it is important to study the propagation problem of ultrasonic wave in the curved plate.
In recent years, many students have participated in this study, yielding many achievements, such as: finite element method, frequency domain spectral domain finite element method, boundary element method, etc. However, these methods have disadvantages. The finite element method requires a fine grid, which limits the simulation calculation for a large area. Although the frequency domain spectrum finite element method has improved calculation efficiency compared with the finite element method, the frequency domain spectrum finite element method is still slower, the adaptability of the unit to complex structures is poor, and great inconvenience is brought to subsequent researches. Although the boundary element method solves the above problems, the use of the cut-off boundary causes reflected waves to be generated at the cut-off boundary, and ignoring far-field information affects the accuracy of the result.
Disclosure of Invention
The invention provides a correction boundary element solution method for lamb wave mode conversion and scattering in a curved plate, which improves the calculation efficiency, simultaneously brings far-field information into calculation, and improves the calculation accuracy.
In order to achieve the above purpose, the invention adopts the following technical scheme:
A method for modifying boundary element solutions for lamb wave mode conversion and scattering in a curved plate, comprising the steps of: s1: constructing a boundary integral formula; the method comprises the following steps: converting the volume integral into area integral by the dynamics reciprocity theorem to obtain a boundary integral equation; s2: solving a far-field basic solution; the method comprises the following steps: adding a virtual boundary, dividing a primary boundary into independent models, and representing the integration of a far field by using near field boundary integration by using a dynamics reciprocity theorem; s3: constructing and solving a global boundary element system matrix; the method comprises the following steps: discretizing the boundary integral equation, and introducing a displacement continuity condition to obtain a final corrected BEM matrix.
In some embodiments, in step S1, the kinetic reciprocity theorem relates to two elastic power states 1 and 2 of the same bounded or unbounded object, expressed in the form:
wherein, Representing the physical strength in the kth direction under the elastic power states 1 and 2 respectively, Representing the displacement in the kth direction under elastic power states 1 and 2 respectively, Representing the traction of the kth component under elastomehc states 1 and 2, respectively, n k is the kth component of the external surface unit vector perpendicular to S; x represents the field point, ω represents the circular frequency of the guided wave, and V and S are the three-and two-dimensional boundaries of the bounded object. The volume fraction is then converted to a surface fraction, resulting in a boundary integral equation.
In some embodiments, from the solved two-dimensional elastohydrodynamic problem in isotropic homogeneous media, we get:
wherein S represents a smooth integration boundary, And (3) withRepresenting the total field displacement and total field drag on the boundary element,And (3) withThe green's function at field point x (x 1,x2) is the fundamental solution in the full-space elastohydrodynamic frequency domain, i.e., when a unit concentrated load acts on source point ζ (ζ 1,ξ2), subscript k, i represents the response obtained in the direction of field point i when a unit concentrated load acts on source point k;
In some embodiments, in step S2, the Lamb wave displacement of the far field is written as the sum of a series of displacements of single mode per unit amplitude multiplied by an unknown coefficient, the far field at the left end Far field at right endWherein (x 1,x2) ε S;
wherein the method comprises the steps of The displacement vector of the m-order mode representing the lamb wave unit amplitude is marked as negative when the propagation direction is from right to left, and is marked as positive conversely,Is the corresponding amplitude coefficient;
After the guided wave is scattered, the amplitude coefficient in the propagation direction is the reflection coefficient and the transmission coefficient which are used for describing far-field scattered wave and are solved respectively;
The displacement vector representing the unit amplitude of the incident lamb wave, the propagation direction is from left to right, the upper mark is positive, and A inc is the corresponding amplitude coefficient.
In some embodiments, the left far field is to be Far field at right end Wherein the far field of (x 1,x2) ε S is brought into the basic boundary integral equation Ζ εS; can obtain
Definition of integral form
The method comprises the following steps:
The first two terms of the integration on the right of the equation represent the effect of the infinitely long part on the scattered wavefield, which corresponds to the mode order m; the last term represents the effect of an infinitely long portion on the incoming wavefield.
In some embodiments of the present invention, in some embodiments, To the right of (a) is the integral over the infinite boundary;
The basic integral equation expression obtained from the kinetic reciprocity theorem:
wherein the area surrounded by x is V, and the boundary is S.
In some embodiments, the left and right boundaries of the curved segment are divided by a virtual boundary;
for the two fields of the elastic dynamics reciprocal theorem, one is the green's function frequency domain basic solution The other is the incident wavefield u m± (x, ω) and u inc (x, ω) of Lamb waves of single mode per unit amplitude; according toAnalyzing each decomposed part to obtain the correction terms of the reflection fields on the left side and the right side Ζ - ε -S and left-side incident field correction term
The far field correction term for the right end is:
The right side of the equation comprises a green's function basic solution and an artificially added incident wave field with single mode unit amplitude, which are known quantities capable of being directly solved; the calculation of the correction term is notified, converting the integral on the infinite boundary into an integral on a finite boundary that can be solved.
In some embodiments, in step S3, the integral equation Discretizing into a matrix form:
Where α represents the cell in which ζ is located, ne represents the total number of nodes of each cell, β represents the node in which x is located, phi β is the shape function of each cell, and η represents the local coordinates of the integration cell.
In some embodiments, the integral of the base solution over the whole cell is set:
Then:
Writing: wherein, T αβ of all units are assembled to form an overall matrix H, the displacement overall matrix of the units is U, the correction term matrix is I ±, and the amplitude matrix is R ±, abbreviated as HU+I ±R±＝IincAinc;
Expanding the matrix can obtain:
U＝[u1(ξ1,ω),u2(ξ1,ω),…,u1(ξN,ω),u2(ξN,ω)]T;
Wherein the total number of modes is M, and the total number of units is N; subscripts 1 and 2 represent the directions at x 1 and x 2, respectively;
Assembling the unknown coefficient matrix R ± into the modified boundary element system, and adding 2 degrees of freedom to the boundary element system of the original equation; introducing finite sequence cut-off points in a far field area, wherein the number of the cut-off points is2 times of the total number of modes;
Based on far field assumptions, the displacement of the chosen intercept points ζ i and ζ N-i+1 (i=1, 2, …, M) can be written as:
Can be written in the form of a matrix:
wherein,
And then, utilizing displacement boundary conditions to assemble a global boundary element system to obtain a scattering coefficient and displacement:
in summary, the invention has at least the following advantages:
The invention provides an improved boundary element method (m-BEM) for simulating the guided wave mode conversion and scattering of a bending part of a plate structure, and the grid and calculation cost are reduced on the premise of ensuring the accuracy. Compared with the traditional boundary element method, the far-field guided wave image is added into the boundary element equation set as a correction term, so that artificial edge reflection can be effectively eliminated. The correction term is directly calculated by the boundary element grid model, and other physical models are not introduced. The near field displacement field and the far field reflection coefficient are obtained simultaneously, and post-treatment is not needed. And the m-BEM is adopted, so that the grid division and calculation cost is greatly reduced. The method also has the following advantages: (1) high calculation efficiency; (2) high calculation accuracy; (3) Lamb excitation is simpler; (4) higher sensitivity and accuracy to defects; (5) low energy consumption and economy.
Drawings
In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings required for the description of the embodiments will be briefly described below, and it is apparent that the drawings in the following description are only some embodiments of the present invention, and other drawings may be obtained according to these drawings without inventive effort for a person skilled in the art.
FIG. 1 is a schematic diagram showing the steps of a method for modifying boundary element solution for lamb wave mode conversion and scattering in curved plates according to the present invention.
FIG. 2 is a schematic diagram of a lamb wave fringe field computing domain with dividing interface boundaries in accordance with the present invention.
FIG. 3 is a schematic diagram of far field correction value calculation according to the present invention, wherein (a) calculation(B) Calculation I inc (ζ, ω), (c) calculation
FIG. 4 is a schematic representation of a lamb wave fringe field computing domain with demarcation interface boundaries for a defect-containing flexural plate structure in accordance with the present invention.
Fig. 5 is a schematic diagram of a point source verification problem involved in the present invention.
Fig. 6 is a graph of the point source verification curved segment outer wall displacement referred to in the present invention, wherein (a) x 1 direction displacement values and (b) x 2 direction displacement values are below.
Fig. 7 is a graph of scattering coefficients at different frequencies of incidence of the A0 mode, in which (a) the S0 mode reflection coefficient and (b) the A0 mode reflection coefficient are involved in the present invention.
Fig. 8 is a graph of scattering coefficients at different depths for defect locations a=pi/4 and widths w=h/5, where (a) S0 mode reflectance and (b) A0 mode reflectance are referred to in the present invention.
Fig. 9 is a graph of scattering coefficients at different widths of defect positions a=pi/4 and depths d=h/5, in which (a) S0 mode reflection coefficient and (b) A0 mode reflection coefficient are related in the present invention.
Detailed Description
For the purpose of the following description, only certain exemplary embodiments are briefly described. As will be recognized by those of skill in the pertinent art, the described embodiments may be modified in numerous different ways without departing from the spirit or scope of the embodiments of the present invention. Accordingly, the drawings and description are to be regarded as illustrative in nature and not as restrictive.
The following disclosure provides many different implementations, or examples, for implementing different configurations of embodiments of the invention. In order to simplify the disclosure of embodiments of the present invention, components and arrangements of specific examples are described below. Of course, they are merely examples and are not intended to limit embodiments of the present invention. Furthermore, embodiments of the present invention may repeat reference numerals and/or letters in the various examples, which are for the purpose of brevity and clarity, and which do not themselves indicate the relationship between the various embodiments and/or arrangements discussed.
Embodiments of the present invention will be described in detail below with reference to the accompanying drawings.
As shown in fig. 1, this embodiment provides a method for correcting boundary element solutions for lamb wave mode conversion and scattering in a curved plate, which first constructs a boundary integral formula, then solves a far-field basic solution, and finally constructs and solves a global boundary element system matrix.
Specifically including steps S1, S2 and S3 as described below.
S1: constructing a boundary integral formula: and converting the volume integral into the area integral through the principle of dynamics reciprocity to obtain a boundary integral equation.
The principle of kinetic reciprocity involves two states of elastic dynamics 1 and 2 of the same bounded or unbounded object, which can be expressed in the form of
Here the number of the elements is the number, Representing the physical strength in the kth direction under the elastic power states 1 and 2 respectively, Representing the displacement in the kth direction under elastic power states 1 and 2 respectively, Traction representing the ki-th component under elastomehc states 1 and 2, respectively; n k is the kth component of the outer surface unit vector perpendicular to S. x represents the field point, ω represents the circular frequency of the guided wave, and V and S are the three-and two-dimensional boundaries of the bounded object. This converts the volume fraction into a surface fraction.
The basic boundary integral equation can be obtained, and according to the solved two-dimensional elastic dynamics problem in the isotropic uniform medium, the following form can be written
Where S represents a smooth integration boundary.And (3) withRepresenting the total field displacement and total field drag on the boundary element.And (3) withThe subscript k, i represents the response that would be obtained in the direction of field point i when a unit load acts in the direction of source point k, which is the basic solution in the full space elastohydrodynamic frequency domain, i.e., the green's function at field point x (x 1,x2) when a unit load acts in the direction of source point ζ (ζ 1,ξ2).
As in fig. 2, we divide the surface of the whole plate structure into different parts. The solid line represents the finite length boundary, indicated by a numerical subscript. The dashed line represents an infinite boundary farther from the bending position, and the upper and lower surfaces at both ends thereof are represented by S -∞ and S +∞, respectively. In addition, two virtual boundaries S 4 and S 5 which are nearer to the bending section are included and divide the whole structure into a left part and a right part. The superscripts "-" and "+" indicate that the boundaries are located on the left and right sides of the curved section, respectively.
In the conventional boundary element method, integration on an infinite boundary is usually omitted, so that artificial truncation boundary can cause false echo to influence calculation accuracy. And we next consider the far field part as well.
According to And surface free boundary conditions, the basic integral equation can be written as: wherein,
ξ∈S0∪S1∪S2∪S3。
S2: solving a far-field basic solution: by adding virtual boundaries, the original boundaries are divided into independent models, and once again, the integration of the far field is represented by near field boundary integration using the reciprocity theorem.
Since the bulk wave propagates in the waveguide with attenuation, it is assumed that only the propagation mode of the waveguide is sufficiently far from the bending position in the far field portion of the scattered wave, i.e. the cutoff boundary. Whereas the Lamb wave displacement of the far field can be written as the sum of a series of displacements of single mode of unit amplitude multiplied by an unknown coefficient, the far field at the left end Far field at right endWherein (x 1,x2)∈S0∪S1∪S2∪S3).
Wherein the method comprises the steps ofThe displacement vector of the m-order mode representing the lamb wave unit amplitude is marked as negative when the propagation direction is from right to left, and is marked as positive conversely,Is the corresponding magnitude coefficient. After the guided wave scattering, the amplitude coefficients in the propagation direction are respectively the reflection coefficient and the transmission coefficient to be solved for describing the far-field scattered wave.The displacement vector representing the unit amplitude of the incident lamb wave, the propagation direction is from left to right, the upper mark is positive, and A inc is the corresponding amplitude coefficient.
Far field of left end Far field at right end Wherein (x 1,x2)∈S0∪S1∪S2∪S3's far field hypothesis is brought into the basic boundary integral equation:
The method can obtain the following steps:
Definition of integral form
Then, there are:
the first two terms of the integration on the right part of the equation represent the effect of the infinitely long part on the scattered wavefield, which can be seen to correspond to the mode order m, i.e. the correction term that needs to be processed next. The last term represents the effect of an infinitely long portion on the incoming wavefield.
It can be seen that To the right of (a) is the integration over the infinite boundary and cannot be solved directly. Therefore, the basic integral equation expression obtained by the reciprocity theorem is introduced again
Where x encloses a region of V, the boundaries S, u i (x, ω) and t i (x, ω) are boundary displacements and tractive forces, and C (ζ) is a constant related to the source point.
The left and right boundaries of the curved segment are divided by virtual boundaries as shown in fig. 2, as shown in fig. 3.
At this time, for the two fields of the elastic dynamics reciprocity theorem, one is the green's function frequency domain basic solutionThe other is the incident wavefield u m± (x, ω) and u inc (x, ω) of Lamb waves of single mode per unit amplitude, and when attention is paid, the incident waves of single mode are artificially added and the virtual boundary is artificially introduced.
Thus, according to: Analyzing each decomposed part to obtain the correction terms of each reflection field on the left side and the right side:
ζ - ε S 0∪S1∪S2∪S3 and left-side incident field correction term
The far field correction term for the right end is: ξ∈S0∪S1∪S2∪S3。
The incident wavefield, including the green's function base solution and the artificially added single mode unit magnitude, to the right of the equation, are both known quantities that can be directly solved. The calculation of the correction term is notified, converting the integral on the infinite boundary into an integral on a finite boundary that can be solved.
S3: constructing and solving a global boundary element system matrix: discretizing the boundary integral equation, and introducing a displacement continuity condition to obtain a final corrected BEM matrix.
Integral equation May be discretized into the form of a matrix,
Where α represents the cell in which ζ is located, ne represents the total number of nodes of each cell, β represents the node in which x is located, φ β is a shape function of each cell, and η represents the local coordinates of the integration cell.
Set the integral of the base solution over the whole cell:
Then:
can be written as:
T αβ of all units are assembled to form an overall matrix H, the displacement overall matrix of the units is U, the correction term matrix is I ± and the amplitude matrix is R ±, and the displacement overall matrix can be abbreviated as HU+I ±R±＝IincAinc;
Expanding the matrix can obtain:
U＝[u1(ξ1,ω),u2(ξ1,ω),…,u1(ξN,ω),u2(ξN,ω)]T;
Wherein the total number of modes is M, and the total number of units is N; subscripts 1 and 2 denote the directions x 1 and x 2, respectively.
The modified boundary element matrix system can also be written in a more intuitive form:
wherein T αβ is a block matrix, U represents the displacement of different boundary regions, and subscript values represent the boundary regions.
Fitting the unknown coefficient matrix R ± into the modified boundary element system will add 2 degrees of freedom to the boundary element system of the original equation. Finite sequence cut-off points are introduced into the far field area, and the number of the cut-off points is 2 times of the total number of modes. Based on far field assumptions, see fig. 2, the displacements of the chosen intercept points ζ i and ζ N-i+1 (i=1, 2, …, M) can be written as
Can also be written in the form of a matrix
Wherein the method comprises the steps of
Then, the global boundary element system is assembled by utilizing the displacement boundary condition, and the scattering coefficient and the displacement can be directly obtained.
For a curved plate structure with groove-like defects, fig. 4 is shown.
The symmetry line of the defect is at the angle alpha of the inner wall of the bending section, the depth of the defect is d, the width of the defect is w, the boundary of the defect section is s 6, and the boundaries of the periphery are all free boundaries, so that the formula is similar to a defect-free bending plate structure, and only one boundary is added.
The global boundary element system is as follows:
The effectiveness of the modified boundary element method on the two-dimensional lamb wave bending model is illustrated by numerical calculations. Firstly, comparing the numerical result obtained by correcting the boundary element method with the result obtained by the finite element method by a point source verification method, as shown in fig. 5, after the material parameters of the curved plate are normalized, the material density ρ=1 kg/m 3, the young modulus e= 2.5980Pa, the poisson ratio v= 0.2990, the thickness h=2m of the plate, the inner wall curvature radius r=10m, the bending angle θ=pi/2, and each wavelength is at least 48 units based on the mode A0 to ensure the accuracy of the result. A linear load f= (1, 1) N of one unit is applied to the inner wall of the curved section, and fig. 6 shows the real part of displacement of the outer wall of the curved section in 2 directions of f=0.1 Hz. It can be seen that the results obtained are very consistent with the finite element results, proving the accuracy of the method.
Then, an A0 mode with the frequency of 100kHz-300kHz is selected to be incident from the left end of the bending section and is propagated in the defect-free bending plate structure, and the guided wave mode conversion and scattering conditions of the bending plate structure are studied. The plate density ρ=2700 kg/m 3,cL＝6094m/s,cT =3263 m/S, the thickness h=4.76 mm, the inner wall radius of curvature r=50.8 mm, the bending angle θ=pi/2, the resulting reflection coefficient, as shown in fig. 7, at which two modes can be seen to propagate A0 (zero-order anti-symmetric motion), S0 (zero-order anti-symmetric motion). The guided wave propagates near the 90 degree bend with only a small mode transition, with a slight reflected wave (on the order of 1% but not negligible in subsequent analysis), the reflected wave mode being essentially the fundamental mode, and the mode transition being negligible. The reflection coefficient decreases with increasing frequency.
Finally, an A0 mode with the frequency of 300kHz is selected to be incident from the left end of the bending section and is transmitted in the bending plate structure containing the groove defects, and the mode conversion and scattering conditions of the bending plate structure at different defect depths and defect widths are studied. The density ρ=2700 kg/m 3,cL＝6094m/s,cT =3263 m/s, the thickness h=4.76 mm, the radius of curvature r of the inner wall=50.8 mm, the bending angle θ=pi/2. The resulting reflection coefficients are shown in fig. 8 and 9.
The visible guided wave generates obvious reflection in the plate structure containing the defects, and compared with the defect-free guided wave, the guided wave has obviously increased reflection coefficient through the defects, which indicates that the defects can be effectively detected. The reflection influence of the defect position on the guided wave is negligible, the reflection phenomenon is more and more obvious along with the increase of the depth, and the reflection phenomenon is smaller along with the increase of the width.
The above embodiments are provided to illustrate the present invention and not to limit the present invention, so that the modification of the exemplary values or the replacement of equivalent elements should still fall within the scope of the present invention.
From the foregoing detailed description, it will be apparent to those skilled in the art that the present invention can be practiced without these specific details, and that the present invention meets the requirements of the patent statutes.
While preferred embodiments of the present invention have been described, additional variations and modifications in those embodiments may occur to those skilled in the art once they learn of the basic inventive concepts. It is therefore intended that the following claims be interpreted as including the preferred embodiments and all such alterations and modifications as fall within the scope of the invention. The foregoing description of the preferred embodiment of the invention is not intended to be limiting, but rather to cover all modifications, equivalents, and alternatives falling within the spirit and principles of the invention.
It should be noted that the above description of the flow is only for the purpose of illustration and description, and does not limit the application scope of the present specification. Various modifications and changes to the flow may be made by those skilled in the art under the guidance of this specification. However, such modifications and variations are still within the scope of the present description.
While the basic concepts have been described above, it will be apparent to those of ordinary skill in the art after reading this application that the above disclosure is by way of example only and is not intended to be limiting. Although not explicitly described herein, various modifications, improvements, and adaptations of the application may occur to one of ordinary skill in the art. Such modifications, improvements, and modifications are intended to be suggested within the present disclosure, and therefore, such modifications, improvements, and adaptations are intended to be within the spirit and scope of the exemplary embodiments of the present disclosure.
Meanwhile, the present application uses specific words to describe embodiments of the present application. For example, "one embodiment," "an embodiment," and/or "some embodiments" means a particular feature, structure, or characteristic in connection with at least one embodiment of the application. Thus, it should be emphasized and should be appreciated that two or more references to "an embodiment" or "one embodiment" or "an alternative embodiment" in various positions in this specification are not necessarily referring to the same embodiment. Furthermore, certain features, structures, or characteristics of one or more embodiments of the application may be combined as suitable.
Furthermore, those of ordinary skill in the art will appreciate that aspects of the application are illustrated and described in the context of a number of patentable categories or conditions, including any novel and useful processes, machines, products, or materials, or any novel and useful improvements thereof. Accordingly, aspects of the present application may be implemented entirely in hardware, entirely in software (including firmware, resident software, micro-code, etc.) or a combination of hardware and software. The above hardware or software may be referred to as a "unit," module, "or" system. Furthermore, aspects of the present application may take the form of a computer program product embodied in one or more computer-readable media, wherein the computer-readable program code is embodied therein.
Computer program code required for operation of portions of the present application may be written in any one or more programming languages, including an object oriented programming language such as Java, scala, smalltalk, eiffel, JADE, emerald, C ++, c#, vb net, python, etc., a conventional programming language such as C programming language, visualBasic, fortran2103, perl, COBOL2102, PHP, ABAP, a dynamic programming language such as Python, ruby, and Groovy, or other programming languages, etc. The program code may execute entirely on the user's computer, or as a stand-alone software package, partly on the user's computer and partly on a remote computer or entirely on the remote computer or server. In the latter scenario, the remote computer may be connected to the user's computer through any form of network, such as a Local Area Network (LAN) or a Wide Area Network (WAN), or the connection may be made to an external computer (for example, through the Internet), or the use of services such as software as a service (SaaS) in a cloud computing environment.
Furthermore, the order in which the elements and sequences are presented, the use of numerical letters, or other designations are used in the application is not intended to limit the sequence of the processes and methods unless specifically recited in the claims. While certain presently useful inventive embodiments have been discussed in the foregoing disclosure, by way of example, it is to be understood that such details are merely illustrative and that the appended claims are not limited to the disclosed embodiments, but, on the contrary, are intended to cover all modifications and equivalent arrangements included within the spirit and scope of the embodiments of the application. For example, while the implementation of the various components described above may be embodied in a hardware device, it may also be implemented as a purely software solution, e.g., an installation on an existing server or mobile device.
Likewise, it should be noted that in order to simplify the presentation of the disclosure and thereby aid in understanding one or more inventive embodiments, various features are sometimes grouped together in a single embodiment, figure, or description thereof. This method of disclosure, however, is not to be interpreted as reflecting an intention that the claimed subject matter requires more features than are expressly recited in each claim. Rather, the inventive subject matter should be provided with fewer features than the single embodiments described above.
Claims (4)
1. A method for modifying boundary element solutions for lamb wave mode conversion and scattering in a curved plate, comprising the steps of:
S1: constructing a boundary integral formula;
the method comprises the following steps: converting the volume integral into area integral by the dynamics reciprocity theorem to obtain a boundary integral equation;
S2: solving a far-field basic solution;
the method comprises the following steps: adding a virtual boundary, dividing a primary boundary into independent models, and representing the integration of a far field by using near field boundary integration by using a dynamics reciprocity theorem;
S3: constructing and solving a global boundary element system matrix;
the method comprises the following steps: discretizing a boundary integral equation, and introducing a displacement continuity condition to obtain a final correction BEM matrix;
In step S2, the Lamb wave displacement of the far field is written as the sum of a series of displacements of single mode of unit amplitude multiplied by an unknown coefficient, the far field at the left end
Far field at right end
Wherein (x 1,x2) ε S;
wherein the method comprises the steps of The displacement vector of the m-order mode representing the lamb wave unit amplitude is marked as negative when the propagation direction is from right to left, and is marked as positive conversely,Is the corresponding amplitude coefficient;
After the guided wave is scattered, the amplitude coefficient in the propagation direction is the reflection coefficient and the transmission coefficient which are used for describing far-field scattered wave and are solved respectively;
The displacement vector representing the unit amplitude of the incident lamb wave, the propagation direction is from left to right, the upper mark is positive, and A inc is the corresponding amplitude coefficient;
far field of left end Far field at right end Wherein the far field of (x 1,x2) ε S is brought into the basic boundary integral equation Can obtain
Definition of integral form
The method comprises the following steps:
The first two terms of the integration on the right of the equation represent the effect of the infinitely long part on the scattered wavefield, which corresponds to the mode order m; the last term represents the effect of the infinitely long portion on the incoming wavefield;
To the right of (a) is the integral over the infinite boundary;
The basic integral equation expression obtained from the kinetic reciprocity theorem:
Wherein the area enclosed by x is V, the boundary is S, u i (x, omega) and t i (x, omega) are boundary displacement and traction, and C (xi) is a constant related to a source point;
dividing the left and right boundaries of the bending section through virtual boundaries;
for the two fields of the elastic dynamics reciprocal theorem, one is the green's function frequency domain basic solution The other is the incident wavefield u m± (x, ω) and u inc (x, ω) of Lamb waves of single mode per unit amplitude; according toAnalyzing each decomposed part to obtain the correction terms of the reflection fields on the left side and the right side Ζ - ε -S and left-side incident field correction term
The far field correction term for the right end is:
The right side of the equation comprises a green's function basic solution and an artificially added incident wave field with single mode unit amplitude, which are known quantities capable of being directly solved; informing the calculation of correction terms, and converting the integral on an infinite boundary into the integral on a finite boundary which can be solved;
In step S3, the integral equation Discretizing into a matrix form:
wherein, alpha represents the unit where xi is located, ne represents the total number of nodes of each unit, beta represents the node where x is located, phi β is the shape function of each unit, and eta represents the local coordinates of the integral unit;
Set the integral of the base solution over the whole cell:
Then:
Writing into
Wherein, T αβ of all units are assembled to form an overall matrix H, the displacement overall matrix of the units is U, the correction term matrix is I ±, and the amplitude matrix is R ±, abbreviated as HU+I ±R±＝IincAinc;
Expanding the matrix to obtain U＝[u1(ξ1,ω),u2(ξ1,ω),…,u1(ξN,ω),u2(ξN,ω)]T;
Wherein the total number of modes is M, and the total number of units is N; subscripts 1 and 2 represent the directions at x 1 and x 2, respectively;
Assembling the unknown coefficient matrix R ± into the modified boundary element system, and adding 2 degrees of freedom to the boundary element system of the original equation; introducing finite sequence cut-off points in a far field area, wherein the number of the cut-off points is2 times of the total number of modes;
Based on far field assumptions, the displacement of the chosen intercept points ζ i and ζ N-i+1 (i=1, 2, …, M) can be written as:
Can be written in the form of a matrix:
wherein,
And then, utilizing displacement boundary conditions to assemble a global boundary element system to obtain a scattering coefficient and displacement:
2. a method of modifying boundary element solutions for lamb wave modal transformation and scattering in curved plates according to claim 1, characterized in that in step S1 the kinetic reciprocity theorem relates to two elastomehc states 1 and 2 of the same bounded or unbounded object expressed in the form:
wherein, Representing the physical strength in the kth direction under the elastic power states 1 and 2 respectively,Representing the displacement in the kth direction under elastic power states 1 and 2 respectively,Represents the traction force of the kth component under the elastic power states 1 and 2 respectively, n k is the kth component of the external surface unit vector vertical to S, x represents the field point, ω represents the circular frequency of the guided wave, and V and S are the three-dimensional and two-dimensional boundaries of the bounded object; the volume fraction is converted into a surface fraction by this equation, and a boundary integral equation is obtained.
3. The method for modified boundary element solution of lamb wave modal transformation and scattering in curved plates according to claim 2, wherein the two-dimensional elastohydrodynamic problem in the isotropic homogeneous medium is solved to obtain:
wherein S represents a smooth integration boundary, And (3) withRepresenting the total field displacement and total field drag on the boundary element,And (3) withThe green's function at field point x (x 1,x2) is the fundamental solution in the full-space elastohydrodynamic frequency domain, i.e., when a unit concentrated load acts on source point ζ (ζ 1,ξ2), subscript k, i represents the response obtained in the direction of field point i when a unit concentrated load acts on source point k;
Displacement of
Traction force
Wherein,
Η j denotes the j-th component in the direction of the normal outside the boundary, ζ and x denote the position vectors of the source point where the concentrated load acts and the field point as a response, respectively, r denotes the distance vector between the field point and the source point; the expression m-th class i order hanker function.
4. A modified boundary element solution for lamb wave mode conversion and scattering in curved plates according to claim 3, characterized in that the processing of the singular integral, when r→0, Singularities may exist;
calculating a scattering problem by using boundary elements, wherein the unit type selected on the boundary is a constant unit;
when the source point xi and the field point x are not in the same unit, the integral equation has no singularity, and the solution is carried out through Gaussian integration;
when the source point xi and the field point x are in the same unit, the integral equation has singularity, the normal vector n outside the boundary is orthogonal with the distance vector r, and analysis is carried out
Knowing that its integral over the cell is zero;
for the displacement base solution component, the following is written:
i.e. the displacement base solution can be written as elastohydrostatic part With elastic dynamic part
The statics part of the base solution has singularities in the integration due to the presence of ln r;
using the expression of the displacement base solution component:
subtracting out Subtracting outThis term leading to singularities is eliminated and the gaussian integration method is applied.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN202211542887.8A CN116227259B (en) | 2022-12-02 | 2022-12-02 | Correction boundary element solution method for lamb wave mode conversion and scattering in bending plate |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN202211542887.8A CN116227259B (en) | 2022-12-02 | 2022-12-02 | Correction boundary element solution method for lamb wave mode conversion and scattering in bending plate |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CN116227259A CN116227259A (en) | 2023-06-06 |
| CN116227259B true CN116227259B (en) | 2024-08-16 |
Family
ID=86588047
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN202211542887.8A Active CN116227259B (en) | 2022-12-02 | 2022-12-02 | Correction boundary element solution method for lamb wave mode conversion and scattering in bending plate |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN116227259B (en) |
Families Citing this family (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN117648584B (en) * | 2024-01-26 | 2024-04-30 | 中国石油大学(华东) | Guided wave mode classification method and system based on displacement vector similarity |
Citations (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN109187769A (en) * | 2018-07-27 | 2019-01-11 | 南京航空航天大学 | A method of defect of pipeline is detected based on supersonic guide-wave quantification |
| CN112989680A (en) * | 2021-05-14 | 2021-06-18 | 中国空气动力研究与发展中心计算空气动力研究所 | FVFD far-field integration boundary condition calculation method for reducing grid usage |
Family Cites Families (8)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US20090048789A1 (en) * | 2007-04-13 | 2009-02-19 | University Of South Carolina | Optimized Embedded Ultrasonics Structural Radar System With Piezoelectric Wafer Active Sensor Phased Arrays For In-Situ Wide-Area Damage Detection |
| EP2751541A4 (en) * | 2011-08-30 | 2015-04-22 | Georgia Tech Res Inst | Weld analysis using laser generated narrowband lamb waves |
| CN106198721A (en) * | 2016-06-29 | 2016-12-07 | 南京中车浦镇城轨车辆有限责任公司 | A kind of lossless detection method of thin plate electric resistance of stainless tack-weld |
| CN106153724A (en) * | 2016-08-16 | 2016-11-23 | 南京航空航天大学 | Based on Lamb wave without the method with reference to the quantification detection thinning defect of flat board |
| FR3079668B1 (en) * | 2018-03-29 | 2020-03-27 | Frec'n'sys | SURFACE ACOUSTIC WAVE DEVICE ON COMPOSITE SUBSTRATE |
| US10983095B2 (en) * | 2018-05-16 | 2021-04-20 | University Of South Carolina | Combined global-local structural health monitoring |
| CN110941906A (en) * | 2019-11-28 | 2020-03-31 | 湖南科技大学 | Triple reciprocity method conversion method for unknown variable domain integration in boundary integral equation |
| CN114813943B (en) * | 2022-04-19 | 2024-04-26 | 北京航空航天大学 | Adhesive joint debonding profile evaluation method based on lamb wave phase delay |
-
2022
- 2022-12-02 CN CN202211542887.8A patent/CN116227259B/en active Active
Patent Citations (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN109187769A (en) * | 2018-07-27 | 2019-01-11 | 南京航空航天大学 | A method of defect of pipeline is detected based on supersonic guide-wave quantification |
| CN112989680A (en) * | 2021-05-14 | 2021-06-18 | 中国空气动力研究与发展中心计算空气动力研究所 | FVFD far-field integration boundary condition calculation method for reducing grid usage |
Also Published As
| Publication number | Publication date |
|---|---|
| CN116227259A (en) | 2023-06-06 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| Ng et al. | Analytical and finite element prediction of Lamb wave scattering at delaminations in quasi-isotropic composite laminates | |
| Poddar et al. | Scattering of Lamb waves from a discontinuity: An improved analytical approach | |
| Benmeddour et al. | Numerical modeling of guided wave interaction with non-axisymmetric cracks in elastic cylinders | |
| Lee et al. | Parabolic equation development in the twentieth century | |
| Kudela et al. | Parallel spectral element method for guided wave based structural health monitoring | |
| Ge et al. | Numerical modeling of PZT-induced Lamb wave-based crack detection in plate-like structures | |
| CN116227259B (en) | Correction boundary element solution method for lamb wave mode conversion and scattering in bending plate | |
| Li et al. | Modelling of the high-frequency fundamental symmetric Lamb wave using a new boundary element formulation | |
| Prusty et al. | Free vibration analysis of sandwich plates with cut-outs: an experimental and numerical study with artificial neural network modelling | |
| He et al. | Predicting range-dependent underwater sound propagation from structural sources in shallow water using coupled finite element/equivalent source computations | |
| Wang et al. | Inverse Shape Reconstruction of Inner Cavities Using Guided SH‐Waves in a Plate | |
| Asakura et al. | Prediction of low-frequency structure-borne sound in concrete structures using the finite-difference time-domain method | |
| Leclercq et al. | The interaction of a bluff body with a vortex wake | |
| Chronopoulos | Prediction of the vibroacoustic response of aerospace composite structures in a broadband frequency range | |
| Lü et al. | Stochastic simulation of the high-frequency wave propagation in a random medium | |
| Li et al. | Characteristics of very low frequency sound propagation in full waveguides of shallow water | |
| Huynh | A static condensation reduced basis element approximation: Application to three-dimensional acoustic muffler analysis | |
| Huang | Effect of beam diffraction on nonlinear Rayleigh surface wave measurement | |
| Joglekar | Scattering of the fundamental Lamb modes in bent metallic plates | |
| Legault et al. | Statistical energy analysis of inhomogeneous systems with slowly varying properties | |
| Fossat et al. | Evaluation of the acoustic radiation of contrasted orthogonally stiffened plates from homogenized model | |
| Maruyama et al. | Three-dimensional hybrid safe-BEM for elastic guided-wave scattering in a plate with finite width | |
| Maruyama et al. | Steady-state anti-plane shear wave scattering by a crack with friction | |
| Faßbender | Time integration for the spectral cell method with application to the full waveform inversion | |
| Gao et al. | An integrated calculation model for vibroacoustic radiation and propagation from underwater structures in oceanic acoustic environments |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| PB01 | Publication | ||
| PB01 | Publication | ||
| SE01 | Entry into force of request for substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| TA01 | Transfer of patent application right | ||
| TA01 | Transfer of patent application right |
Effective date of registration: 20240708 Address after: No. 29, Qinhuai District, Qinhuai District, Nanjing, Jiangsu Applicant after: Nanjing University of Aeronautics and Astronautics Country or region after: China Applicant after: JIANGSU JK ELECTRICAL EQUIPMENT Co.,Ltd. Address before: No. 29, Qinhuai District, Qinhuai District, Nanjing, Jiangsu Applicant before: Nanjing University of Aeronautics and Astronautics Country or region before: China |
|
| GR01 | Patent grant | ||
| GR01 | Patent grant |