A method for controlling wave field in elastic thin plate based on potential theory

Abstract

A wave field control method in an elastic thin plate based on potential theory, comprising: based on wave propagation principle and partial differential equation mathematical theory in an elastic medium, describing the wave field in the elastic thin plate generated by a source term with compact support property, combining partial known data on the boundary of a given region and radiation conditions at infinity, establishing an external problem of wave field control; based on the basic solution and potential theory, formulating an integral representation of the wave field satisfying the external problem, obtaining the corresponding relationship between the boundary data and the wave field; according to the corresponding relationship, converting the wave field control problem in the elastic thin plate into a single-dipole distribution problem on the boundary of the given region, the goal of the single-dipole distribution problem is to generate a control field with zero inside the given region and non-zero outside, using the method combining finite difference and numerical integration, discretizing the control field of the wave field, establishing the corresponding relationship between the approximate control field of the wave field and the single-dipole, based on the corresponding relationship between the approximate control field and the single-dipole, using the method combining finite difference and moment method to establish a boundary element model, and solving the boundary element model to obtain the calculated values of the single-dipole position and amplitude, then bringing them into the discretization expression of the control field of the wave field to obtain the calculated value of the approximate control field of the wave field, finally realizing the control of the wave field in the elastic thin plate.

Description

Technical Field

[0001] This invention belongs to the field of wave field control in elastic thin plates, and specifically relates to a wave field control method in elastic thin plates based on potential theory. Background Technology

[0002] Wave field control is widely used in many fields such as aerospace, automotive, construction, acoustics, and non-destructive testing, and is of paramount importance for performance optimization, functional implementation, and safety assurance. Precise wave field control is crucial for structural design optimization, performance improvement, fault diagnosis, lifespan extension, and the development of related industries.

[0003] Currently, wavefield control methods are mainly divided into two categories: active control and passive control. Active control uses an external excitation device to apply a specific wave, achieving precise regulation of the wavefield characteristics. Passive control, on the other hand, adds damping materials or absorbing structures to the structure, altering the wave propagation path and energy dissipation within the material to control the wavefield. This method is commonly used for general wavefield regulation under normal operating conditions. While active control offers significant control effects, it is complex, costly, and difficult to maintain. Passive control involves relatively simple structural modifications; however, its control effect is limited by the inherent characteristics of the added materials and structures, making it difficult to adapt to complex and variable wavefield environments and lacking flexibility in dealing with sudden or special wavefield situations.

[0004] While existing wavefield control methods each have their own characteristics, they also have many shortcomings in practical applications. Active control methods rely on complex systems and precise feedback regulation, resulting in high costs and difficult maintenance; passive control methods are limited by the added materials and structural characteristics, and are not well adapted to complex wavefields. This application addresses the wavefield control problem of thin plates by proposing a novel method that is highly accurate, easy to construct, and highly scalable, and that controls the wavefield by designing the distribution of single-dipoles. Summary of the Invention

[0005] In view of the problems and shortcomings of the existing technology, the purpose of this invention is to provide a method for controlling the wave field in an elastic thin plate based on potential theory.

[0006] To achieve the objectives of this invention, the technical solution adopted is as follows:

[0007] This invention provides a wave field manipulation method for an elastic thin plate based on potential theory, comprising the following steps:

[0008] S1: Based on the wave propagation principle in elastic media and the mathematical theory of partial differential equations, the wave field in an elastic thin plate generated by a source term with compact support properties is described. Combined with some known data on the boundary of a given region and the radiation conditions at infinity, an external problem for wave field control is established.

[0009] S2: Based on the fundamental solution and potential theory, the wave field that satisfies the external problem is given a formulaic integral representation to obtain the correspondence between the boundary data and the wave field.

[0010] S3: Based on the correspondence, the wave field control problem in the elastic thin plate is transformed into a single-dipole distribution problem on the boundary of a given region. The goal of the single-dipole distribution problem is to generate a control field in which the wave field is zero inside the given region and non-zero outside. The control field of the wave field is discretized by using a combination of finite difference and numerical integration, and a correspondence is established between the approximate control field of the wave field and the single-dipole.

[0011] S4: Based on the correspondence between the approximate control field and the single-dipole, a boundary element model is established using a combination of finite difference and moment method. The boundary element model is then solved to obtain the calculated values ​​of the position and amplitude of the single-dipole. These values ​​are then substituted into the discretized expression of the wave field control field to obtain the calculated value of the approximate control field of the wave field, thus achieving the control of the wave field in the elastic thin plate.

[0012] According to the above method, the specific process of step S1 includes:

[0013] S101: Assume that the wave field v in the elastic thin plate is caused by the source term f and satisfies the following equation:

[0014]

[0015] And it satisfies the Sommerfeld radiation condition at infinity; the source term f∈L 2 (Ω), with compact support It is a bitone operator. Where κ is the wavenumber related to the angular frequency ω and the density β, and h is the thickness of the plate. Let E be the bending stiffness of the plate, E be Young's modulus, and γ be Poisson's ratio.

[0016] Since the source term has compact support, the wave field v is known to be at the boundary. Given partial data v(x) = q0(x) and Δv(x) = q1(x), establish the external problem of wave field control in an elastic thin plate as follows:

[0017]

[0018] Based on the above method, further, step S2, which involves formulating an integral representation of the wave field satisfying the external problem based on the fundamental solution and potential theory, specifically includes:

[0019] First, based on the fundamental solution and potential theory, the wave field in Ω is established. ρThe integral equation over ∪Ω that includes boundary data is:

[0020]

[0021] and

[0022]

[0023] in here, The Green's function is the two-dimensional time-harmonic biharmonic wave equation. The solution to the two-dimensional Helmholtz equation is... For the solution of the modified two-dimensional Helmholtz equation; It is a zero-order Hankel function of the first kind, and K0 is a zero-order modified Hankel function;

[0024] The normal derivative on the boundary is defined as follows:

[0025]

[0026] Where n(x) represents the outward unit normal vector on the boundary, in The upper part points to the outside of region Ω, at the boundary. The upper point points to the region Ω ρ external;

[0027] Then, based on the Cauchy–Schwarz inequality and the Sommerfeld radiation condition, an asymptotic analysis at infinity is performed, yielding the following results:

[0028]

[0029] as well as

[0030]

[0031] Next, letting ρ→∞, the limit form of the two integral equations can be expressed as:

[0032]

[0033] and

[0034]

[0035] Finally, subtracting the limit forms of the two integral equations, we obtain the integral expression that satisfies the external problem wave field as follows:

[0036]

[0037] Based on the above method, further, step S2, which involves obtaining the correspondence between boundary data and wave field based on fundamental solutions and potential theory, specifically includes:

[0038] Introduce the following potential operator:

[0039] Where p(x,y) is a multivariate differentiable function with x and y as independent variables, and can take the values ​​G(x,y) and g(y), respectively. H (x,y), q(y) represents the boundary The density function on can be taken as follows: q2(y),q0(y).

[0040] Therefore, the integral representation of the wave field can be further expressed as:

[0041]

[0042] Where q2(x)=q1(x)+κ 2 q0(x),

[0043] According to the above method, further, the correspondence described in step S3 transforms the wave field manipulation problem in the elastic thin plate into a single-dipole distribution problem on the boundary of a given region. The goal of the single-dipole distribution problem is to generate a control field in which the wave field is zero inside the given region and non-zero outside, specifically including:

[0044] Introducing a monopole g H (x,y), G(x,y) and dipoles The integral representation of the wave field can then be transformed into a single-dipole distribution problem, yielding the control field of the wave field as follows:

[0045]

[0046] Based on the above method, the control field of the wave field is further discretized to establish the correspondence between the approximate control field of the wave field and the single-dipole, specifically including:

[0047] First, the control field of the wave field is discretized using the finite difference method to obtain an approximation of the wave field control field:

[0048]

[0049] Where J(t) is the Jacobian function;

[0050] Next, using numerical integration, the approximate control field of the wave field is obtained as follows:

[0051]

[0052] Where τ p ∈(-1,1) are nodes in the Gauss-Legendre quadrature formula. N p Take a non-negative integer, W p It corresponds to N p The weights of the Gauss-Legendre quadrature formula with +1 nodes.

[0053] Based on the above method, further, step S4, which establishes the boundary element model using a combination of finite difference and method of moments based on the correspondence between the approximate control field and the single-dipole, specifically includes:

[0054] First, based on the theory of integral operators, v(x) in the region Integral representation within:

[0055]

[0056] get In the region The integral within is expressed as:

[0057]

[0058] Next, a method combining finite difference and method of moments is used to establish... The boundary element model is:

[0059]

[0060] in Take non-negative integers and

[0061] The boundary element model of v(x) is established as follows:

[0062]

[0063] in t s =t j,p , s=(j-1)N+(p+1),j=1,…,N,p=0,…,N p , yes In the boundary element model Approximate value.

[0064] Based on the above method, further, step S4 solves the boundary element model to obtain the calculated values ​​of the single-dipole position and amplitude, and then substitutes them into the discretized expression of the wave field control field to obtain the calculated value of the approximate control field of the wave field, thus realizing the control of the wave field in the elastic thin plate, specifically including:

[0065] S401: In the above In the boundary element model, x = y s +δn(y s ),s=1,···,N(N P +1), get The system of linear algebraic equations satisfied is in It is N×(N p +1) order square matrix, It is N×(N p +1) order column vector, the specific form of each element is as follows:

[0066]

[0067]

[0068] Solving the system of linear algebraic equations yields... exist The calculated value above;

[0069] S402: Will Substitute the calculated value into v vec The boundary element model, and taking x = y s +δn(y s ),s=1,···,N(N P +1), to get v vec The system of linear algebraic equations satisfied is Mv vec =b, where v vec =(v1,v2…v s ) Τ M is N×(N p +1) order square matrix, b is an N×(N) matrix p +1) order column vector, the specific form of each element is as follows:

[0070]

[0071] Solve the system of linear algebraic equations Mv vec =b, thus obtaining v vec exist The calculated value above;

[0072] S403:Use and vvec The calculated values ​​yielded the amplitudes of the single-dipole as follows:

[0073]

[0074] Substituting the amplitude into the discretized expression of the wave field control field, we obtain the approximate control field of the wave field as follows:

[0075]

[0076] Through the calculations in steps S401-S403, the relationship between the approximate control field of the wave field and the single-dipole is established, ultimately realizing the control of the wave field in the elastic thin plate.

[0077] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0078] (1) This invention controls the wave field by designing the distribution of single-dipoles. It has high control precision, high flexibility, high energy efficiency, is easy to implement, and can be adjusted in real time. This is the first time in the world.

[0079] (2) The method provided by this invention has good multifunctionality and scalability. By increasing or adjusting the number and distribution of mono-dipoles, the system can be easily expanded to adapt to larger-scale or more demanding applications. It can achieve effective wave field control in complex environments and is applicable to multiple fields such as acoustics, electromagnetics, and optics, with broad application prospects. Attached Figure Description

[0080] Figure 1 This is a flowchart illustrating a wave field manipulation method for elastic thin plates based on potential theory.

[0081] Figure 2(a)-Figure 2(b) This is a schematic diagram of the wave field in this embodiment. Figure 2(a) shows the source. The real part of the generated wave field Re(v1), as shown in Figure 2(b) is the source. The real part of the generated wave field is Re(v2), where

[0082] Figures 3(a)-3(b) Figure 3(a) shows the wavefield control situation under different region sizes. At that time, the real part of the wave field Figure 3(b) shows When, the real part of the wave field

[0083] Figure 4 shows the wave field control under different numbers of single-dipole distributions. In the figure, (a)-(d) represent the real parts of the wave field when N = 20, 30, 40, and 50, respectively.

[0084] Figures 5(a)-5(b) For the same number of mono-dipoles, the circles with different radii outside the Ω region Upper control field v cf The relative and absolute error plots are shown, where the norm is the infinity norm, and the calculation formula is the absolute error. relative error Figures 5(a) and 5(b) show N=300 and N=350, respectively.

[0085] Figures 6(a)-6(b) A circle of the same radius outside the Ω region Upper control field v cf The relative and absolute errors vary with the number of mono-dipoles, where the norm is the infinite norm and the calculation formula is the same as above. Figures 6(a) and 6(b) show R=10 and R=15, respectively. Detailed Implementation

[0086] To enable those skilled in the art to better understand the technical solution of the present invention, the technical solution of the present invention will be described in detail below with reference to specific implementations.

[0087] Example 1

[0088] This embodiment specifically provides a wave field manipulation method for elastic thin plates based on potential theory, such as... Figure 1 As shown, it includes the following steps:

[0089] (1) Modeling of external problems of wave field control

[0090] Assume the wave field v in the elastic thin plate is caused by the source term f and satisfies the following equation:

[0091]

[0092] The wave field v satisfies the Sommerfeld radiation condition at infinity; the source term f∈L 2 (Ω), with compact support It is a bitone operator. Where κ is the wavenumber related to the angular frequency ω and the density β, and h is the thickness of the plate. Let E be the bending stiffness of the plate, E be Young's modulus, and γ be Poisson's ratio.

[0093] Since the source term has compact support, the wave field v is known to be at the boundary. Given partial data in the above, v(x) = q0(x) and Δv(x) = q1(x). ， The external problem for wave field control in an elastic thin plate is:

[0094]

[0095] (2) Based on the fundamental solution and potential theory, establish the relationship between boundary data and wave field.

[0096] First, based on the fundamental solution and potential theory, the wave field in Ω is established. ρ The integral equation over ∪Ω that includes boundary data is:

[0097]

[0098] in here, The Green's function is the two-dimensional time-harmonic biharmonic wave equation. The solution to the two-dimensional Helmholtz equation is... For the solution of the modified two-dimensional Helmholtz equation; It is a zero-order Hankel function of the first kind, and K0 is a zero-order modified Hankel function;

[0099] The normal derivative on the boundary is defined as follows:

[0100]

[0101] Where n(x) represents the outward unit normal vector on the boundary, in The upper part points to the outside of region Ω, at the boundary. The upper point points to the region Ω ρ external;

[0102] Then, based on the Cauchy–Schwarz inequality and the Sommerfeld radiation condition, an asymptotic analysis at infinity is performed, the specific process of which is as follows:

[0103] Applying radiation conditions to Δv yields:

[0104]

[0105] From the radiation conditions (4) and (29), and the boundary conditions v(x) = q0(x) and Δv(x) = q1(x), we can obtain:

[0106]

[0107] From equation (30) and the radiation condition, we can obtain:

[0108]

[0109] right Combining Cauchy–Schwarz inequalities and The following results were obtained:

[0110]

[0111] Similarly, we can conclude that:

[0112]

[0113] Next, letting ρ→∞, the limit form of the two integral equations can be expressed as:

[0114]

[0115] and

[0116]

[0117] Finally, subtracting equation (10) from equation (11) yields:

[0118]

[0119] Thus, the present invention has obtained an integral representation that satisfies the external problem wave field.

[0120] Next, the following potential operator is introduced:

[0121]

[0122] Where p(x,y) is a multivariate differentiable function with x and y as independent variables, and can take the values ​​G(x,y) and g(y), respectively. H (x,y), q(y) represents the boundary The density function on can be taken as follows: q2(y),q0(y).

[0123] Therefore, the integral representation of the wave field can be further expressed as:

[0124]

[0125] Thus, this invention establishes the correspondence between boundary data and wavefield. To achieve wavefield manipulation, this invention further develops a method for establishing an approximate control field based on a single-dipole.

[0126] (3) Establishment of approximate control wave field based on single-dipole

[0127] Since this invention aims to achieve wavefield manipulation, it requires generating a control field where the wavefield is zero within a given region and non-zero outside. Therefore, a monopole g is first introduced. H (x,y), G(x,y) and dipoles The integral expression of equation (18) can then be transformed into a single-dipole distribution problem, yielding the control field of the wave field as:

[0128]

[0129] The specific process of discretizing the control field of the wave field and establishing the correspondence between the approximate control field of the wave field and the single-dipole is as follows:

[0130] First, the control field is discretized using the finite difference method to obtain an approximation of the wave field control field:

[0131]

[0132] Where J(t) is the Jacobian function;

[0133] Next, by applying numerical integration to equation (15), the approximate control field of the wave field is obtained as follows:

[0134]

[0135] Where τ p ∈(-1,1) is a node of the Gauss-Legendre quadrature formula. N p Take a non-negative integer, W p It corresponds to N p The weights of the Gauss-Legendre quadrature formula with +1 nodes.

[0136] Thus, this invention establishes the correspondence between the approximate control field of the wave field and the single-dipole.

[0137] Based on the correspondence between the approximate control field and the single-dipole, the specific process of establishing the boundary element model using a combination of finite difference and method of moments is as follows:

[0138] First, based on the theory of integral operators, v(x) in the region Integral representation within:

[0139]

[0140] get In the region The integral within is expressed as:

[0141]

[0142] Next, the finite difference method is applied to equations (17) and (18), namely:

[0143]

[0144] and

[0145]

[0146] Then, the integral in equation (33) is parameterized, that is:

[0147]

[0148] The integral in equation (34) is parameterized and combined with get:

[0149]

[0150] Finally, numerical integration and the method of moments are used to establish equation (35). Boundary element model:

[0151]

[0152] in Take non-negative integers and

[0153] Using numerical integration and the method of moments, a boundary element model of v(x) is established for equation (36):

[0154]

[0155] in t s =t j,p , s=(j-1)N+(p+1),j=1,…,N,p=0,…,N p , yes In the boundary element model Approximate value.

[0156] Thus, the present invention has been established. The boundary element model of v(x) is used. To ultimately achieve wavefield manipulation in the elastic thin plate, the boundary element model needs to be solved to obtain the calculated values ​​of the single-dipole position and amplitude. These values ​​are then substituted into the discretized expression of the wavefield control field to obtain the calculated values ​​of the approximate control field. The specific process is as follows:

[0157] ①In the In the boundary element model, x = y s +δn(y s ),s=1,···,N(N P +1), get The system of linear algebraic equations satisfied is in It is N×(N p +1) order square matrix, It is N×(Np +1) order column vector, the specific form of each element is as follows:

[0158]

[0159] Solving the system of linear algebraic equations yields... exist The calculated value above;

[0160] ② Substitute the calculated value into v vec The boundary element model, and taking x = y s +δn(y s ),s=1,···,N(N P +1), to get v vec The system of linear algebraic equations satisfied is Mv vec =b, where v vec =(v1,v2…v s ) Τ M is N×(N p +1) order square matrix, b is an N×(N) matrix p +1) order column vector, the specific form of each element is as follows:

[0161]

[0162] Solve the system of linear algebraic equations Mv vec =b, thus obtaining v vec exist The calculated value above;

[0163] ③Use and v vec The calculated values ​​yielded the amplitudes of the single-dipole as follows:

[0164]

[0165] Substituting the amplitude into the discretized expression of the wave field control field, we obtain the approximate control field of the wave field as follows:

[0166]

[0167] Through calculations ①-③, the relationship between the approximate control field of the wave field and the single-dipole is established, ultimately realizing the control of the wave field in the elastic thin plate.

[0168] Example 2

[0169] The wave field modulation test in the elastic thin plate was carried out using the method of Example 1, and the wave field generated by different source terms was controlled.

[0170] In this numerical experiment, v(x) is taken as v1(x) - v2(x). (37)

[0171] In equation (37) v j (x) represents the value of equation (1) in Example 1. The wave field generated at time, where f j (j=1,2) is a source with compact support property, satisfying f j ∈L 2 (D j ), From source f j The generated wave field v j Satisfy the following equation:

[0172]

[0173] In equation (38), the wave number κ = 0.3 and the region Ω is a circle with a center at (0,0) and a radius of 3.

[0174] Tron v j At the border The following conditions must be met:

[0175]

[0176] In equation (39)

[0177] The wave field v is calculated using equation (12) in Example 1. j It has the following integral representation:

[0178]

[0179] Using equation (14) from Example 1, and combining it with equation (37), the control field of the following form can be obtained:

[0180]

[0181] in q j,1 (x)+κ 2 q j,0 (x)=q j,2 (x),q l =q 1,l -q 2,l ,l=0,1,2.

[0182] By combining finite difference and numerical integration methods, an approximate control field can be obtained from equation (41). The correspondence between the single-dipole and the single-dipole is given by equation (26). Then, by solving the algebraic linear equations from equations (21) to (24), we can obtain v. s , The value of is used to further obtain the amplitude of the mono-dipole.

[0183] This numerical experiment was performed on a 40×40 square region, N, These represent the number of parts that the complex Gauss-Legendre and the quadrature formulas for the middle rectangle divide into on their respective intervals (here). In the experiment, x = y s +δn(y s ),y s =y(t) s ), where t s For each interval, the nodes that use the Gauss-Legendre quadrature formula are defined. Here, the two-point Gauss-Legendre quadrature formula is used, i.e., N. p =2, node is The corresponding weights are 1, 1, t l The node in each small interval is calculated using the rectangle quadrature formula, and the small change δ = 0.2 is obtained when the difference quotient is used instead of the derivative.

[0184] Substituting the calculated values ​​into equation (26), we obtain the approximate control wave field. The calculated value.

[0185] Figure 2 is a schematic diagram of the wave field in this embodiment. In the figure, (a) is the source. The real part of the generated wave field Re(v1), and (b) in the figure is the source. The real part of the generated wave field is Re(v2), where As can be seen from Figure 2, the real parts Re(v1) and Re(v2) of the generated wave field are different due to the different positions of the source.

[0186] Figure 3 shows the wavefield control situation under different region sizes. In the figure, (a) is... At that time, the real part of the wave field Figure (b) is When, the real part of the wave field From the perspective of an observer outside Ω, wave field v2 produces the same wave field as v1 outside Ω, which indicates that the method provided by the present invention has a good wave field control effect, enabling the simulation of different wave fields through adjustment.

[0187] Figure 4 shows the wave field control under different numbers of single-dipole distributions. In the figure, (a)-(d) represent the real part of the wave field when N = 20, 30, 40, and 50, respectively. Clearly, the wave field control effect improves with the increase of the number of mono-dipoles, indicating that the method provided by this invention has high computational efficiency and can achieve the goal of controlling the wave field using a small number of mono-dipoles.

[0188] Figure 5 shows the different radii of circles outside the Ω region when the number of single dipoles is the same. Upper control field v cf The relative and absolute error plots are shown, where the norm is the infinity norm, and the calculation formula is the absolute error. relative error Figures (a) and (b) show the control field v when N = 300 and 350, respectively. cf The relative and absolute error plots clearly show that the error quickly approaches 10 as the outer radius of the Ω region increases. -2 This indicates that the method provided by the present invention has high computational efficiency.

[0189] Figure 6 shows a circle of the same radius outside the Ω region. Upper control field v cf The graphs show the relative and absolute errors as a function of the number of monopole-dipoles, where the norm is taken as the infinite norm, and the calculation formula is the same as above. In the graphs, (a)-(b) represent the control field v for R=10 and 15, respectively. cf The relative and absolute error plots clearly show that the error quickly approaches 10^10 as the number of mono-dipoles increases. -2 This also demonstrates that the method provided by the present invention has high computational efficiency.

[0190] The above embodiments are specific implementations of the present invention, but the implementation of the present invention is not limited to the above embodiments. Any other combination, change, modification, substitution, or simplification that does not exceed the design concept of the present invention shall fall within the protection scope of the present invention.

Claims

1. A wave field manipulation method for an elastic thin plate based on potential theory, characterized in that, Includes the following steps: S1: Based on the wave propagation principle in elastic media and the mathematical theory of partial differential equations, the wave field in an elastic thin plate generated by a source term with compact support properties is described. Combining some known data at the boundary of a given region and the radiation conditions at infinity, an external problem controlled by the wave field is established; whereby the wave field... At the border The given partial data is , ; S2: Based on the fundamental solution and potential theory, the wave field satisfying the external problem is expressed in a formulaic integral, yielding the following correspondence between the boundary data and the wave field: (13) in The Green's function is the two-dimensional time-harmonic biharmonic wave equation. The solution to the two-dimensional Helmholtz equation is... For the solution of the modified two-dimensional Helmholtz equation; It is a zeroth-order Hankel function of the first kind. It is the zero-order modified Hankel function; , ; , It is related to angular frequency and density Related wavenumbers, For the thickness of the plate, For the bending stiffness of the plate, For Young's modulus, Poisson's ratio; S3: Based on the aforementioned correspondence, the wave field manipulation problem in the elastic thin plate is transformed into a single-dipole distribution problem on the boundary of a given region. The objective of the single-dipole distribution problem is to generate a control field where the wave field is zero inside the given region and non-zero outside. A combination of finite difference and numerical integration is used to discretize the control field of the wave field, establishing a correspondence between the approximate control field of the wave field and the single-dipole, specifically including: First, the monopole is introduced. and dipole The integral representation of the wave field can then be transformed into a single-dipole distribution problem, yielding the control field of the wave field as follows: (14) Then, the control field of the wave field is discretized using the finite difference method to obtain an approximation of the wave field control field: (15) in It is a Jacobian function; Next, using numerical integration, the approximate control field of the wave field is obtained as follows: (16) in For the nodes of the Gauss-Legendre quadrature formula, , Take non-negative integers. It corresponds The weights of the Gauss-Legendre quadrature formula for each node; S4: Based on the correspondence between the approximate control field and the single-dipole, a boundary element model is established using a combination of finite difference and moment method. The boundary element model is then solved to obtain the calculated values ​​of the position and amplitude of the single-dipole. These values ​​are then substituted into the discretized expression of the wave field control field to obtain the calculated value of the approximate control field of the wave field, thus achieving the control of the wave field in the elastic thin plate.

2. The method according to claim 1, characterized in that, The specific process of step S1 is as follows: S101: Assuming a wave field in an elastic thin plate From source item This causes the following equation to be satisfied: (1) And it satisfies the Sommerfeld radiation condition at infinity; the source term It has compact support , It is a bitone operator; S102: Since the source term has compact support, the wave field is known. At the border The given partial data is , The external problem for wave field control in an elastic thin plate is: (2) (3) (4)。 3. The method according to claim 2, characterized in that, Step S2, which involves formulating an integral representation of the wave field satisfying the external problem based on the fundamental solution and potential theory, specifically includes: First, based on the fundamental solution and potential theory, the wave field is established in... The integral equation including the boundary data is: (5) and (6) in The normal derivative on the boundary is defined as follows: , ,in This represents the outward unit normal vector on the boundary. Up pointing area Outside, at the boundary Up pointing area external; Then, based on the Cauchy–Schwarz inequality and the Sommerfeld radiation condition, an asymptotic analysis at infinity is performed, yielding... (7) (8) as well as (9) Next, order The limit form of the two integral equations can be expressed as: (10) and (11) Finally, subtracting the limit forms of the two integral equations, we obtain the integral expression that satisfies the external problem wave field as follows: (12)。 4. The method according to claim 3, characterized in that, Step S2, which describes obtaining the correspondence between boundary data and wave field based on fundamental solutions and potential theory, specifically includes: Introduce the following potential operator: ,in For Let be a multivariable differentiable function of the independent variable, and let each of the following values ​​be possible: , , Indicates boundary The density function on can be taken as follows: .

5. The method according to claim 4, characterized in that, Step S4, based on the correspondence between the approximate control field and the single-dipole, establishes the boundary element model using a combination of finite difference and method of moments, specifically including: First, based on the theory of integral operators, by In the region Integral representation within: (17) get In the region The integral within is expressed as: (18) Next, a method combining finite difference and method of moments is used to establish... The boundary element model is: (19) in , Take non-negative integers and ; Establish The boundary element model is: (20) in , yes In the boundary element model Approximate value.

6. The method according to claim 5, characterized in that, Step S4 solves the boundary element model to obtain the calculated values ​​of the single-dipole position and amplitude. These values ​​are then substituted into the discretized expression of the wave field control field to obtain the calculated values ​​of the approximate control field of the wave field, ultimately achieving wave field manipulation in the elastic thin plate. Specifically, this includes: S401: In the above In the boundary element model, take get The system of linear algebraic equations satisfied is ,in , yes Square array yes A column vector of order, whose elements have the following specific forms: (21) (22) Solving the system of linear algebraic equations yields... exist The calculated value on; S402: Will Substitute the calculated value The boundary element model, and take get The system of linear algebraic equations satisfied is ,in , yes Square array yes A column vector of order, whose elements have the following specific forms: (23) (24) Solve the system of linear algebraic equations ,get exist The calculated value on; S403:Use and The calculated values ​​yielded the amplitudes of the single-dipole as follows: (25) Substituting the amplitude into the discretized expression of the wave field control field, we obtain the approximate control field of the wave field as follows: (26) Through the calculations in steps S401-S403, the relationship between the approximate control field of the wave field and the single-dipole is established, ultimately realizing the control of the wave field in the elastic thin plate.

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