Method for controlling wave field in elastic thin plate based on potential theory
Through the single-dipole distribution method based on potential theory, the problem of insufficient flexibility in existing wavefield control methods in complex environments is solved, and high-precision and easy-to-structure wavefield regulation is achieved, and it is suitable for acoustics, electromagnetics, optics and other fields.
Patent Information
- Application Number
- CN202510489926.X
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-04-18
- Publication Date
- 2025-08-08
- Estimated Expiration
- 2045-04-18
AI Technical Summary
The existing wavefield control methods are insufficient in complex wavefield environments, the active control method is expensive and difficult to maintain, and the passive control method is poorly adaptable.
Based on the potential theory, by designing the distribution of single-dipoles, it is transformed into the single-dipole distribution problem on the boundary, and a method of combining finite difference and numerical integral is used to establish a boundary element model, solve the position and amplitude of the single-dipole, and realize the precise regulation of the wave field.
It realizes high-precision, easy to construct, and highly flexible wavefield control. It is suitable for complex environments, has good versatility and scalability, and is suitable for acoustics, electromagnetics, optics and other fields.
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Figure CN120449549A_ABST
Abstract
Description
Technical Field
[0001] The present invention belongs to the field of wave field control in elastic thin plates, and in particular relates to a method for controlling wave fields in elastic thin plates based on potential theory. Background Art
[0002] Wavefield control is widely used in a wide range of fields, including aviation, automotive, architecture, acoustics, and nondestructive testing. It plays a crucial role in performance optimization, functional implementation, and safety assurance. Precisely controlling the wavefield is crucial for structural design optimization, performance improvement, fault diagnosis, life extension, and the development of related industries.
[0003] Currently, the main methods of wavefield control are divided into two categories: active control and passive control. Active control uses an external excitation device to apply specific waves to achieve wavefield control, enabling relatively precise adjustment of wavefield characteristics. Passive control achieves wavefield control by adding damping materials and wave-absorbing structures to the structure, changing the wave propagation path and energy dissipation within the material. This method is commonly used for general wavefield control under conventional operating conditions. While active control methods offer significant control effects, they are complex, costly, and difficult to maintain. Passive control methods are relatively simple to modify structurally, but their control effectiveness is limited by the inherent characteristics of the added materials and structures, making them difficult to adapt to complex and changing wavefield environments and lacking flexibility in responding to sudden or special wavefield conditions.
[0004] While existing wavefield control methods each have their own unique characteristics, they suffer from numerous deficiencies in practical application. Active control methods rely on complex systems and precise feedback regulation, resulting in high costs and difficult maintenance. Passive control methods, however, are limited by the inherent limitations of additive materials and structural characteristics, making them less adaptable to complex wavefields. This application addresses the issue of controlling wavefields in thin plates and proposes a novel method that achieves high precision, is easy to construct, and is highly scalable. This method utilizes a distributed structure of single and double dipoles to control wavefields. Summary of the Invention
[0005] In view of the problems and shortcomings in the prior art, the present invention aims to provide a method for controlling the wave field in an elastic thin plate based on potential theory.
[0006] To achieve the purpose of the invention, the technical solution adopted by the present invention is as follows:
[0007] The present invention provides a method for controlling wave fields in 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 the elastic thin plate generated by the source term with compact support is described. Combined with some known data on the boundary of a given region and the radiation condition at infinity, the external problem of wave field control is established.
[0009] S2: Based on the basic solution and potential theory, the wave field that satisfies the external problem is formulated as an integral expression to obtain the corresponding relationship between the boundary data and the wave field;
[0010] S3: Based on the corresponding relationship, the wave field control problem in the elastic thin plate is converted 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 using a method combining finite differences and numerical integration, and a corresponding relationship between the approximate control field of the wave field and the single-dipole is established.
[0011] S4: Based on the correspondence between the approximate control field and the single dipole, a boundary element model is established by combining finite difference and moment method, and the boundary element model is solved to obtain the calculated values of the single dipole position and amplitude, which are then brought 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 realizing the regulation 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 is a biharmonic operator, where κ is the wave number related to the angular frequency ω and density β, h is the thickness of the plate, is the bending stiffness of the plate, E is Young's modulus, and γ is Poisson's ratio;
[0016] Since the source term has compact support, it is known that the wave field v is at the boundary The given partial data on the elastic plate are v(x)=q0(x), Δv(x)=q1(x), and the external problem of wave field control in the elastic thin plate is:
[0017]
[0018] According to the above method, further, the step S2 of formulating an integral representation of the wave field that satisfies the external problem based on the basic solution and potential theory specifically includes:
[0019] First, based on the basic solution and potential theory, the wave field in Ω is established. ρThe integral equation on ∪Ω including boundary data is:
[0020]
[0021] and
[0022]
[0023] in here, is the Green function of the two-dimensional time-harmonic biharmonic wave equation, is the solution of the two-dimensional Helmholtz equation, is the solution of the modified two-dimensional Helmholtz equation; is the zero-order Hankel function of the first kind, K0 is the 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, Pointing to the outside of the region Ω, at the boundary Upward pointing area Ω ρ external;
[0027] Then, based on the Cauchy–Schwarz inequality and the Sommerfeld radiation condition, we perform asymptotic analysis at infinity and obtain:
[0028]
[0029] as well as
[0030]
[0031] Next, let ρ→∞, and the limiting forms of the two integral equations can be expressed as:
[0032]
[0033] and
[0034]
[0035] Finally, the limit forms of the two integral equations are subtracted to obtain the integral that satisfies the external problem wave field:
[0036]
[0037] According to the above method, further, the step S2 of obtaining the corresponding relationship between the boundary data and the wave field based on the basic solution and potential theory specifically includes:
[0038] The following potential operator is introduced:
[0039] Where p(x,y) is a multivariate differentiable function with x and y as independent variables, which can be G(x,y), g H (x,y), q(y) represents the boundary The density function on 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 corresponding relationship described in step S3 transforms the wave field control 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 inside the given region is zero and the control field outside is non-zero, specifically including:
[0044] Introducing monopole g H (x,y), G(x,y) and dipole Then the integral representation of the wave field can be transformed into a single-dipole distribution problem, and the control field of the wave field is obtained as:
[0045]
[0046] According to the above method, the control field of the wave field is further discretized to establish a corresponding relationship 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 the approximation of the control field of the wave field:
[0048]
[0049] Where J(t) is the Jacobian function;
[0050] Next, the approximate control field of the wave field is obtained by using the numerical integration method:
[0051]
[0052] where τ p ∈(-1,1) is the node of the Gauss-Legendre quadrature formula, N p Take a non-negative integer, W p corresponds to N p +1 node weight for the Gauss-Legendre quadrature formula.
[0053] According to the above method, further, step S4 is based on the correspondence between the approximate control field and the single-dipole, and a boundary element model is established by combining finite difference and moment method, which specifically includes:
[0054] First, based on the integral operator theory, v(x) is in the region The integral inside is expressed as:
[0055]
[0056] get In the area The integral inside is expressed as:
[0057]
[0058] Then, the finite difference and moment method were combined to establish The boundary element model is:
[0059]
[0060] in Take a non-negative integer and
[0061] The boundary element model of v(x) is established as:
[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 of .
[0064] According to the above method, step S4 further solves the boundary element model to obtain the calculated values of the single-dipole position and amplitude, which are then brought into the discretized expression of the wave field control field to obtain the calculated value of the approximate control field of the wave field, and finally realize the control of the wave field in the elastic thin plate, which specifically includes:
[0065] S401: in the In the boundary element model, x=y s +δn(y s ),s=1,···,N(N P +1), get The linear algebraic equations satisfied are in is N×(N p +1)-order square matrix, is N×(N p +1)-order column vector, the specific form of its elements is as follows:
[0066]
[0067]
[0068] Solving the linear algebraic equations, we get exist The calculated value on ;
[0069] S402: Substitute the calculated value of v into vec Boundary element model, and take x = y s +δn(y s ),s=1,···,N(N P +1), get v vec The linear algebraic equations satisfied are Mv vec =b, where v vec =(v1,v2…v s ) Τ , M is N×(N p +1)-order square matrix, b is N×(N p +1)-order column vector, the specific form of its elements is as follows:
[0070]
[0071] Solve the linear algebraic equation system Mv vec =b, we get v vec exist The calculated value on ;
[0072] S403:Use and vvec The calculated values of , the amplitudes of the single dipole are:
[0073]
[0074] Substituting the amplitude into the discrete expression of the wave field control field, the approximate control field of the wave field is obtained as follows:
[0075]
[0076] Through the calculations in steps S401 to S403 , the relationship between the approximate control field of the wave field and the single-dipole is established, and finally the control of the wave field in the elastic thin plate is achieved.
[0077] Compared with the prior art, the present invention has the following beneficial effects:
[0078] (1) The present invention controls the wave field by designing a distributed structure of single-dipoles, which has high control accuracy, strong flexibility, high energy efficiency, easy implementation, and real-time control. This is the first time in the world.
[0079] (2) The method provided by the present invention has excellent versatility and scalability. By increasing or adjusting the number and distribution of mono-dipoles, the system can be easily expanded to accommodate larger-scale or more demanding applications. It can achieve effective wave field control in complex environments and is applicable to a variety of fields such as acoustics, electromagnetics, and optics, with broad application prospects. BRIEF DESCRIPTION OF THE DRAWINGS
[0080] Figure 1 It is a flow chart of the wave field control method in elastic thin plates based on potential theory.
[0081] Figure 2(a)-Figure 2(b) Figure 2(a) is a schematic diagram of the wave field of this embodiment. The real part of the wave field Re(v1) generated, Figure 2(b) is the source The real part of the generated wave field Re(v2), where
[0082] Figure 3(a)-Figure 3(b) Figure 3(a) shows the wave field control situation under different area sizes. When 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. Figures (a)-(d) are the real parts of the wave field when N=20, 30, 40, and 50, respectively.
[0084] Figure 5(a)-Figure 5(b) When the number of single-dipoles is the same, the circumference of different radii outside the Ω region Upper control field v cf The relative error and absolute error diagram of , where the norm is infinite norm, and the calculation formula is absolute error Relative error FIG5(a) and FIG5(b) are for N=300 and N=350 respectively.
[0085] Figure 6(a)-Figure 6(b) is a circle with the same radius outside the Ω region Upper control field v cf The relative error and absolute error plots vary with the number of single-dipoles, where the norm is infinite and the calculation formula is the same as above. Figure 6(a) and Figure 6(b) are for R = 10 and R = 15, respectively. DETAILED DESCRIPTION
[0086] In order to enable those skilled in the art to more clearly understand the technical solution of the present invention, the technical solution of the present invention will be described in detail below in conjunction with specific implementations.
[0087] Example 1
[0088] This embodiment specifically provides a method for controlling the wave field in an elastic thin plate based on potential theory. Figure 1 As shown, the following steps are included:
[0089] (1) Modeling of external wave field control issues
[0090] Assume that 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 is a biharmonic operator, where κ is the wave number related to the angular frequency ω and density β, h is the thickness of the plate, is the bending stiffness of the plate, E is Young's modulus, and γ is Poisson's ratio;
[0093] Since the source term has compact support, it is known that the wave field v is at the boundary The given partial data on is v(x)=q0(x), Δv(x)=q1(x) ， The external problem for establishing wave field control in an elastic thin plate is:
[0094]
[0095] (2) Based on the fundamental solution and potential theory, the relationship between boundary data and wave field is established
[0096] First, based on the basic solution and potential theory, the wave field in Ω is established. ρ The integral equation on ∪Ω including boundary data is:
[0097]
[0098] in here, is the Green function of the two-dimensional time-harmonic biharmonic wave equation, is the solution of the two-dimensional Helmholtz equation, is the solution of the modified two-dimensional Helmholtz equation; is the zero-order Hankel function of the first kind, K0 is the 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. Pointing to the outside of the region Ω, at the boundary Upward pointing area 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 is as follows:
[0103] Applying radiation conditions to Δv yields:
[0104]
[0105] From the radiation conditions (4) and (29), as well as the boundary conditions v(x) = q0(x), Δv(x) = q1(x), we can obtain:
[0106]
[0107] Then, from formula (30) and the radiation condition, we can get:
[0108]
[0109] right Combining the Cauchy–Schwarz inequality and The results are as follows:
[0110]
[0111] Similarly, we can get:
[0112]
[0113] Next, let ρ→∞, and the limiting forms of the two integral equations can be expressed as:
[0114]
[0115] and
[0116]
[0117] Finally, subtracting equation (10) from equation (11) yields:
[0118]
[0119] So far, 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, which can be G(x,y), g H (x,y), q(y) represents the boundary The density function on q2(y),q0(y).
[0123] Therefore, the integral representation of the wave field can be further expressed as:
[0124]
[0125] Thus, the present invention has established the corresponding relationship between the boundary data and the wave field. In order to realize the wave field control, the present invention has further developed the establishment of an approximate control field based on a single-dipole.
[0126] (3) Establishment of approximate control wave field based on single-dipole
[0127] Because the present invention is to realize wave field control, it is necessary to generate a control field in which the wave field inside a given area is zero and the wave field outside is non-zero, so the monopole g is first introduced. H (x,y), G(x,y) and the dipole Then the integral expression of formula (18) can be transformed into the distribution problem of single-dipole, and the control field of the wave field is obtained as:
[0128]
[0129] The specific process of discretizing the control field of the wave field and establishing the corresponding relationship 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 the approximation of the wave field control field:
[0131]
[0132] Where J(t) is the Jacobian function;
[0133] Then, the numerical integration method is used for Equation (15) to obtain the approximate control field of the wave field:
[0134]
[0135] where τ p ∈(-1,1) is the node of the Gauss-Legendre quadrature formula N p Take a non-negative integer, W p corresponds to N p +1 node weight for the Gauss-Legendre quadrature formula.
[0136] Thus, the present invention has established the corresponding relationship 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 by combining the finite difference and moment method is as follows:
[0138] First, based on the integral operator theory, v(x) is in the region The integral inside represents:
[0139]
[0140] get In the area The integral inside is expressed as:
[0141]
[0142] Then, the finite difference method is used for equations (17) and (18), namely:
[0143]
[0144] and
[0145]
[0146] Then, the integral in equation (33) is parameterized as follows:
[0147]
[0148] Parameterize the integral in Eq. (34) and combine get:
[0149]
[0150] Finally, the numerical integration method and moment method are used to establish Equation (35): Boundary element model of:
[0151]
[0152] in Take a non-negative integer and
[0153] The numerical integration method and moment method are used for equation (36) to establish the boundary element model of v(x):
[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 of .
[0156] Thus, the present invention has established In order to ultimately achieve the control of the wave field in the elastic thin plate, it is necessary to solve the boundary element model to obtain the calculated values of the single-dipole position and amplitude, and then bring 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. The specific process is as follows:
[0157] ① In the In the boundary element model, x=y s +δn(y s ),s=1,···,N(N P +1), get The linear algebraic equations satisfied are in is N×(N p +1)-order square matrix, is N×(Np +1)-order column vector, the specific form of its elements is as follows:
[0158]
[0159] Solving the linear algebraic equations, we get exist The calculated value on ;
[0160] ② General Substitute the calculated value of v into vec Boundary element model, and take x = y s +δn(y s ),s=1,···,N(N P +1), get v vec The linear algebraic equations satisfied are Mv vec =b, where v vec =(v1,v2…v s ) Τ , M is N×(N p +1)-order square matrix, b is N×(N p +1)-order column vector, the specific form of its elements is as follows:
[0161]
[0162] Solve the linear algebraic equation system Mv vec = b, we get v vec exist The calculated value on ;
[0163] ③Use and v vec The calculated values of , the amplitudes of the single dipole are:
[0164]
[0165] Substituting the amplitude into the discrete expression of the wave field control field, the approximate control field of the wave field is obtained as follows:
[0166]
[0167] Through the calculations of ①-③, the relationship between the approximate control field of the wave field and the single-dipole is established, and finally the control of the wave field in the elastic thin plate is realized.
[0168] Example 2
[0169] The method of Example 1 was used to conduct a wave field control test in an elastic thin plate to control the wave fields generated by different source terms.
[0170] In this numerical experiment, v(x) = v1(x) - v2(x). (37)
[0171] In formula (37), v j (x) is the value of equation (1) in Example 1. The wave field generated when f j (j=1,2), is a source with compact support property, satisfying f j ∈L 2 (D j ), By source j Generated wave field v j Satisfies the following equation:
[0172]
[0173] In formula (38), the wave number κ = 0.3, and the region Ω is a circle with the center at (0,0) and a radius of 3.
[0174] Tron V j At the border The following conditions are met:
[0175]
[0176] In formula (39)
[0177] The wave field v is calculated using formula (12) in Example 1 j It has the following integral representation:
[0178]
[0179] Using formula (14) in Example 1 and combining it with formula (37), the following control field 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] The finite difference and numerical integration method is used to combine Equation (41) to obtain the approximate control field The corresponding relationship between the single-dipole and the single-dipole is Equation (26), and then the algebraic linear equations (21)-(24) are solved to obtain v s , The value of is further used to obtain the amplitude of the single dipole.
[0183] This numerical experiment is calculated on a 40×40 square area, N, The number of parts of the complex Gauss-Legendre and rectangular quadrature formulas on their respective intervals (here ), in the experiment, x=y s +δn(y s ),y s =y(t s ), where t s For each node in the small interval, the Gauss-Legendre quadrature formula is used. Here, the two-point Gauss-Legendre quadrature formula is used, that is, N p =2, the node is The corresponding weights are 1,1,t l For each node in the small interval, the middle rectangle product formula is used, and the small change δ=0.2 is taken when the difference quotient is replaced by the derivative quotient.
[0184] Substituting the above calculated values into equation (26), we can obtain the approximate control wave field The calculated value of .
[0185] FIG2 is a schematic diagram of the wave field of this embodiment, in which (a) is the source The real part of the wave field Re(v1) is generated, and (b) is the source The real part of the generated wave field Re(v2), where As can be seen from Figure 2, due to the different placement of the source, the real parts of the wave field Re(v1) and Re(v2) generated are also different.
[0186] Figure 3 shows the wave field control situation under different area sizes. Figure (a) is When the real part of the wave field In the figure (b) When the real part of the wave field To an observer outside Ω, wavefield v2 generates the same wavefield as v1 outside Ω, which shows that the method provided by the present invention has a good wavefield control effect, making it possible to achieve the purpose of imitating different wavefields through regulation.
[0187] Figure 4 shows the wave field control under different numbers of single-dipole distributions. Figures (a)-(d) are the real part of the wave field when N=20, 30, 40, and 50, respectively. Obviously, the wave field is controlled better with the increase of the number of mono-dipoles, which shows that the method provided by the present invention has high computational efficiency and can achieve the goal of controlling the wave field with a small number of mono-dipoles.
[0188] Figure 5 shows circles of different radii outside the Ω region when the number of single dipoles is the same. Upper control field v cf The relative error and absolute error diagram of , where the norm is infinite norm, and the calculation formula is absolute error Relative error In the figure (a)-(b) are the control field v when N=300 and 350 respectively. cf The relative error and absolute error diagram shows that the error quickly approaches 10 as the radius outside the Ω region increases. -2 ,This shows that the method provided by the present invention has high ,computational efficiency.
[0189] Figure 6 shows the same radius of the circumference outside the Ω area Upper control field v cf The relative error and absolute error diagrams vary with the number of single-dipoles, where the norm is infinite and the calculation formula is the same as above. Figures (a) and (b) are the control field v when R = 10 and 15 respectively. cf The relative error and absolute error diagram shows that the error quickly approaches 10 as the number of single-dipoles increases. -2 ,This also shows that the method provided by the present invention has high ,computational efficiency.
[0190] The above embodiments are specific implementation methods of the present invention, but the implementation methods of the present invention are not limited to the above embodiments. Any other combination, change, modification, substitution, and simplification that does not exceed the design concept of the present invention shall fall within the scope of protection of the present invention.
Claims
1. A method for controlling wave fields in an elastic thin plate based on potential theory, characterized in that: The following steps are involved: S1: Based on the wave propagation principle in elastic media and the mathematical theory of partial differential equations, the wave field in the elastic thin plate generated by the source term with compact support is described. Combined with some known data on the boundary of a given region and the radiation condition at infinity, the external problem of wave field control is established. S2: Based on the basic solution and potential theory, the wave field that satisfies the external problem is formulated as an integral expression to obtain the corresponding relationship between the boundary data and the wave field; S3: Based on the corresponding relationship, the wave field control problem in the elastic thin plate is converted 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 using a method combining finite differences and numerical integration, and a corresponding relationship between the approximate control field of the wave field and the single-dipole is established. S4: Based on the correspondence between the approximate control field and the single dipole, a boundary element model is established by combining finite difference and moment method, and the boundary element model is solved to obtain the calculated values of the single dipole position and amplitude, which are then brought 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 realizing the regulation 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: S101: Assume that the wave field v in the elastic thin plate is caused by the source term f and satisfies the following equation: And it satisfies the Sommerfeld radiation condition at infinity; the source term f∈L 2 (Ω), with compact support is a biharmonic operator, where κ is the wave number related to the angular frequency ω and density β, h is the thickness of the plate, is the bending stiffness of the plate, E is Young's modulus, and γ is Poisson's ratio; S102: Since the source term has compact support, it is known that the wave field v is at the boundary The given partial data on the elastic plate are v(x)=q0(x), Δv(x)=q1(x), and the external problem of wave field control in the elastic thin plate is:
3. The method according to claim 2, characterized in that The step S2, based on the basic solution and potential theory, is to formulate an integral representation of the wave field that satisfies the external problem, specifically including: First, based on the basic solution and potential theory, the wave field in Ω is established. ρ The integral equation on Ω including boundary data is: and in here, is the Green function of the two-dimensional time-harmonic biharmonic wave equation, is the solution of the two-dimensional Helmholtz equation, is the solution of the modified two-dimensional Helmholtz equation; is the zero-order Hankel function of the first kind, K0 is the zero-order modified Hankel function; the normal derivative on the boundary is defined as follows: Where n(x) represents the outward unit normal vector on the boundary, Pointing to the outside of the region Ω, at the boundary Upward pointing area Ω ρ external; Then, based on the Cauchy–Schwarz inequality and the Sommerfeld radiation condition, we perform asymptotic analysis at infinity and obtain as well as Next, let ρ→∞, and the limiting forms of the two integral equations can be expressed as: and Finally, the limit forms of the two integral equations are subtracted to obtain the integral expression that satisfies the external problem wave field:
4. The method according to claim 3, characterized in that The step S2, based on the basic solution and potential theory, obtains the corresponding relationship between the boundary data and the wave field, specifically including: The following potential operator is introduced: Where p(x,y) is a multivariate differentiable function with x and y as independent variables, which can be G(x,y), g H (x,y), q(y) represents the boundary The density function on q2(y),q0(y). Therefore, the integral representation of the wave field can be further expressed as: where q2(x)=q1(x)+κ 2 q0(x), 5. The method according to claim 4, characterized in that The corresponding relationship described in step S3 transforms the wave field control 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, it includes: Introducing monopole g H (x,y), G(x,y) and dipole Then the integral representation of the wave field can be transformed into a single-dipole distribution problem, and the control field of the wave field is obtained as:
6. The method according to claim 5, characterized in that Discretizing the control field of the wave field and establishing a corresponding relationship between the approximate control field of the wave field and the single-dipole, specifically including: First, the control field of the wave field is discretized using the finite difference method to obtain the approximation of the control field of the wave field: Where J(t) is the Jacobian function; Next, the approximate control field of the wave field is obtained by using the numerical integration method: where τ p ∈(-1,1) is the node of the Gauss-Legendre quadrature formula, N p Take a non-negative integer, W p corresponds to N p +1 node weight for the Gauss-Legendre quadrature formula.
7. The method according to claim 6, characterized in that Step S4, based on the correspondence between the approximate control field and the single-dipole, establishes a boundary element model by combining finite difference and moment method, which specifically includes: First, based on the integral operator theory, v(x) is in the region The integral inside is expressed as: get In the area The integral inside is expressed as: Then, the finite difference and moment method were combined to establish The boundary element model is: in Take a non-negative integer and The boundary element model of v(x) is established as: 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 of .
8. The method according to claim 7, characterized in that Step S4 solves the boundary element model to obtain the calculated values of the single-dipole position and amplitude, which are then introduced into the discretized expression of the wave field control field to obtain the calculated value of the approximate control field of the wave field, ultimately achieving the control of the wave field in the elastic thin plate, which specifically includes: S401: in the In the boundary element model, x=y s +δn(y s ),s=1,···,N(N P +1), get The linear algebraic equations satisfied are in is N×(N p +1)-order square matrix, is N×(N p +1)-order column vector, the specific form of its elements is as follows: Solving the linear algebraic equations, we get exist The calculated value on ; S402: Substitute the calculated value of v into vec Boundary element model, and take x = y s +δn(y s ),s=1,···,N(N P +1), get v vec The linear algebraic equations satisfied are Mv vec =b, where v vec =(v1,v2v s ) Τ , M is N×(N p +1)-order square matrix, b is N×(N p +1)-order column vector, the specific form of its elements is as follows: Solve the linear algebraic equation system Mv vec = b, we get v vec exist The calculated value on ; S403:Use and v vec The calculated values of , the amplitudes of the single dipole are: Substituting the amplitude into the discrete expression of the wave field control field, the approximate control field of the wave field is obtained as follows: Through the calculations in steps S401 to S403 , the relationship between the approximate control field of the wave field and the single-dipole is established, and finally the control of the wave field in the elastic thin plate is achieved.
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