Obstacle active stealth method based on generalized non-radiation source in three-dimensional elastic medium

Abstract

A generalized non-radiation source-based obstacle active stealth method in a three-dimensional elastic medium comprises the following steps: describing an elastic wave field meeting a Navier equation based on a wave field and non-radiation source theory in the three-dimensional elastic medium, and establishing a generalized non-radiation source framework of elastic waves; the method comprises the following steps: based on Helmholtz decomposition and a potential theory, performing formulated integral representation on a three-dimensional elastic wave non-radiation field, and in combination with a known incident wave field, obtaining a corresponding relationship between each component of the non-radiation field and the incident wave field on a boundary; according to the integral representation, the stealth problem of obstacles in any shape in the three-dimensional elastic medium is converted into a low-order pole distribution problem on the boundary of a given area, and the target of the low-order pole distribution problem is to generate a non-radiation field with a non-zero internal elastic wave field and zero external elastic wave field in the given area; and performing discretization expression on integral expression of the non-radiation field by adopting a method of combining finite difference and numerical integration, establishing a corresponding relation between an approximate non-radiation field and a monopole, obtaining a structural form of the approximate non-radiation field, and finally realizing active invisibility of the obstacle in any shape in the three-dimensional elastic medium. According to the invention, the structure of an approximate non-radiation field which is high in precision, strong in adaptability and capable of being regulated and controlled instantly is easy to realize.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of obstacle active invisibility in three-dimensional elastic medium, and particularly relates to an obstacle active invisibility method based on generalized non-radiation source in three-dimensional elastic medium. BACKGROUND

[0002] Obstacle invisibility is widely used in many fields such as military, transportation, medicine and environment, and has important research value in safety improvement, medical progress and environmental protection. Realizing target obstacle invisibility has far-reaching significance for equipment life extension, energy consumption reduction, natural environment improvement and related industry development.

[0003] At present, the main methods of obstacle invisibility are divided into two categories: optical transformation method and scattering cancellation method. The optical transformation method is based on the spatial transformation invariance of physical equations, and creates an invisibility "hole" after spatial transformation. A multi-layered super material cloak is usually made to make the wave bypass the specified area and propagate in the original direction, so as to realize obstacle invisibility. The scattering cancellation method reduces or eliminates the obstacle reflection signal by designing special structures and coatings on the surface of the obstacle or placing devices that can actively emit specific waves, so as to realize obstacle invisibility. The spatial transformation method is widely applicable, but the singularity and high cost of the material pose obstacles to production. The invisibility effect of the scattering cancellation method is superior, but the device cannot be adjusted after being made for special situations.

[0004] Although the current obstacle invisibility methods have their own characteristics, they have many shortcomings in practical application. The optical transformation method is suitable for invisibility in different wave fields with the help of super material cloak, but the material is scarce and the cost is high. The scattering cancellation method is limited by the existing shape structure, and the adjustment ability is weak when the wave field changes. The present application aims to propose a new method of constructing a non-radiation field through a generalized non-radiation source, which has superior performance, simple structure, strong applicability, and is suitable for obstacle invisibility in elastic medium. SUMMARY

[0005] The present application overcomes the above-mentioned problems and deficiencies in the prior art, and provides an obstacle active invisibility method based on generalized non-radiation source in three-dimensional elastic medium.

[0006] To achieve the purpose of the application, the technical scheme adopted by the present application is as follows:

[0007] The present application provides an obstacle active invisibility method based on generalized non-radiation source in three-dimensional elastic medium, comprising the following steps:

[0008] S1: Based on the wave field and non-radiation source theory in three-dimensional elastic medium, the elastic wave field satisfying the Navier equation is described, and the generalized non-radiation source framework of elastic wave is established;

[0009] S2: Based on Helmholtz decomposition and potential theory, an integral representation of the non-radiation field of three-dimensional elastic wave is formulated, and the corresponding relationship between each component of the non-radiation field and the incident wave field on the boundary is obtained by combining the known incident wave field;

[0010] S3: According to the integral representation, the cloaking problem of an obstacle with an arbitrary shape in a three-dimensional elastic medium is converted into a low-order pole distribution problem on the boundary of a given region, and the goal of the low-order pole distribution problem is to generate a non-radiation field that is non-zero inside the given region and zero outside;

[0011] S4: By using a combination of finite difference and numerical integration, the integral representation of the non-radiation field is discretely expressed, the corresponding relationship between the approximate non-radiation field and the monopole is established, the construction form of the approximate non-radiation field is obtained, and finally the active cloaking of an obstacle with an arbitrary shape in a three-dimensional elastic medium is realized.

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

[0013] S101: Let D be a bounded region in Consider the propagation of elastic waves in an isotropic homogeneous medium, and the three-dimensional elastic wave field in region D is controlled by the following Navier equation:

[0014]

[0015] Where μ, λ are Lamé constants, satisfying μ>0, λ+μ>0, ρ is the medium density, ω is the angular frequency, is the Laplace operator, is the gradient operator.

[0016] S102: Based on the non-radiation source theory, the generalized non-radiation source of the elastic wave is a low-order pole distribution on the boundary The low-order poles in the distribution can generate a non-radiation field u nrf (x) = (u nrf,1 (x), u nrf,2 (x), u nrf,3 (x)) Τ That is,

[0017]

[0018] Where u(x) satisfies equation (1) in D.

[0019] According to the above method, further, the integral representation of the non-radiation field of three-dimensional elastic wave based on Helmholtz decomposition and potential theory in step S2 includes:

[0020] S201: In region D, consider the solution of Navier equation (1) make Based on the Helmholtz decomposition, the elastic wave field u is obtained i In D, the following conditions are met:

[0021]

[0022] Where curl is a three-dimensional curl operator, v S , They satisfy the following Helmholtz equations:

[0023]

[0024] S202: Based on potential theory, using the fundamental solution of the Helmholtz equation Establish the integral representation of the above Helmholtz equation respectively:

[0025]

[0026] and

[0027]

[0028] in represents the wave number, is the first kind of zero-order Hankel function, n=(n1,n2,n3) Τ Representing boundaries The unit outward normal vector on points to the outside of region D. The normal derivative on the boundary is defined as follows:

[0029]

[0030] S203: Combine (3) with (5) and (6) to obtain the non-radiative field u nrf (x) The integral on :

[0031]

[0032] According to the above method, further, step S2 combines the known incident wave field to obtain the corresponding relationship between each component of the non-radiated field and the incident wave field on the boundary, specifically including:

[0033] When u i When the incident field is known, the gradient, divergence, and curl in (8) are calculated to obtain the components u of the non-radiative field. nrf,l(x), l=1,2,3, and the corresponding relationship between the incident wave field on the boundary:

[0034]

[0035] According to the above method, further, in step S3, based on the integral representation, the problem of cloaking an obstacle of arbitrary shape in a three-dimensional elastic medium is converted into a low-order pole distribution problem on the boundary of a given region, wherein the goal of the low-order pole distribution problem is to generate a non-zero elastic wave field inside the given region and a zero non-radiative field outside the given region, specifically including:

[0036] Introducing low-order poles and l=1,2,3,σcan be S or Then (9) can be transformed into a low-order polaron and Distribution problem:

[0037]

[0038] According to the above method, further, the method described in step S4 using a combination of finite differences and numerical integration is used to discretize the integral representation of the non-radiative field and establish a corresponding relationship between the approximate non-radiative field and the monopole, specifically including:

[0039] S401: Using the basic solution The nature of

[0040]

[0041] S402: Yes and v=1,2,3, respectively, using the first-order difference and second-order difference approximation, we get:

[0042]

[0043] Substituting (11) and (12) into (10) and combining (13) and (14), and parameterizing and discretizing the integration interval, we can calculate an approximation of each component of the non-radiative field:

[0044]

[0045] where δ0=(0,0,0) Τ , δ1=(δ,0,0) Τ , δ2=(0,δ,0) Τ , δ3=(0,0,δ) Τ , δ4=δ1+δ2, δ5=δ1+δ3, δ6=δ2+δ3, δ7=-δ1, δ8=-δ2, δ9=-δ3, The form is as follows:

[0046]

[0047]

[0048]

[0049]

[0050] J(t1,t2) is the Jacobian function.

[0051] S403: Apply numerical integration method to (15) to obtain the components of the approximate non-radiative field and the monopole G(x,y+δ m κ σ ),m=0,1,2,3,4,5,6,7,8,9, the corresponding relationships are as follows:

[0052]

[0053] in Nodes of the Gauss-Legendre quadrature formula, l1=1,···,N1, l2=1,···,N2,N1, are the number of partitions of the interval [0,π] and [0,2π], p=0,···,N p ,q=0,···,N q ,N p ,N q Take a non-negative integer, N p +1 and N q +1 is the number of nodes in the Gauss-Legendre quadrature formula, A p and B q Corresponding to N p +1 node and N q +1 weight for the quadrature formula at each node.

[0054] By calculating the components of the approximate non-radiative field, the construction form of the approximate non-radiative field is obtained. Using the constructed approximate non-radiative field, the total field u can be obtained, which is zero inside the region and the incident field outside the region. tol (x)=(u tol,1 (x),u tol,2 (x),u tol,3 (x)) Τ ,Right now

[0055]

[0056] Finally, the active invisibility of the arbitrary shape obstacle in three-dimensional elastic medium is realized.

[0057] Compared with the prior art, the present application has the following advantages:

[0058] The innovation of the present application is:

[0059] 1. The present application can realize the construction of the approximate non-radiation field by using a small number of monopole form low-order generalized non-radiation sources. Compared with the high-order generalized non-radiation source in series form, this method is more easy to realize the construction of the approximate non-radiation field with high precision, strong adaptability and instant control.

[0060] 2. The present application can construct the non-radiation field according to different invisibility purposes to realize the invisibility of multiple arbitrary shape obstacles, and can adapt to larger scale or higher precision requirements of applications.

[0061] 3. For the proposed non-radiation field construction method, the invisibility performance analysis can be carried out to predict the influence of different parameter settings on the invisibility effect. The present application is also applicable to acoustics, optics, electromagnetics and other fields, and has wide application prospect.

[0062] The working principle of the present application is:

[0063] The present application realizes the invisibility of multiple arbitrary shape obstacles in the case of known incident field. The invisibility area is designed according to the shape, size and position of the obstacle, and the approximate non-radiation field is generated by setting the generalized non-radiation source distribution constructed by a small number of monopoles on the boundary of the invisibility area. The approximate non-radiation field is negative incident field inside the invisibility area and zero outside, which can cancel the incident field inside the invisibility area, so as to obtain the total field information that the inside of the invisibility area is zero and the outside is equal to the incident field. In this way, when the incident field irradiates the obstacle, the existence of the scattering field cannot be detected outside the invisibility area, that is, the information of the obstacle cannot be detected, and finally the invisibility target is realized. BRIEF DESCRIPTION OF DRAWINGS

[0064] Figure 1 It is a flowchart of the method of the present application.

[0065] Figures 2(a)-2(d) The numerical result graph of the imaginary part Im(u tol,2 ) of the second component of the total field when (N1, N2) takes (3, 6), (5, 10), (7, 14) and (15, 30) respectively.

[0066] Figure 3 The point-by-point error lg graph of the approximate non-radiation field component In the area Ω1∪Ω2.

[0067] Figure 4When (N1, N2) is different, the approximate non-radiation field component In the ring area The absolute error norm on picture.

[0068] Figures 5(a)-5(c) When the obstacle is located in different stealth areas designed according to its shape and density, the real part of the first component of the total field Re(u tol,1 ) numerical results. DETAILED DESCRIPTION

[0069] 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.

[0070] Example 1

[0071] This embodiment specifically provides an active cloaking method for obstacles based on a generalized non-radiative source in a three-dimensional elastic medium. Figure 1 As shown, the following steps are included:

[0072] (1) Describe the elastic wave field that satisfies the Navier equations and establish a generalized non-radiative source framework for elastic waves

[0073] Let D be In the bounded region, consider the propagation of elastic waves in an isotropic homogeneous medium. The three-dimensional elastic wave field in region D is governed by the following Navier equations:

[0074]

[0075] Where μ, λ are Lamé constants, satisfying μ>0, λ+μ>0, ρ is the medium density, ω is the angular frequency, is the Laplace operator, is the gradient operator.

[0076] Based on the non-radiative source theory, the generalized non-radiative source of elastic waves is a The low-order poles in the distribution can generate a non-radiative field u that is non-zero inside the region D and zero outside the region D. nrf (x)=(u nrf,1 (x),u nrf,2 (x),u nrf,3 (x)) Τ ,Right now

[0077]

[0078] where u(x) satisfies equation (1) in D.

[0079] (2) Based on Helmholtz decomposition and potential theory, combined with the incident wave field, the relationship between the components of the non-radiative field and the incident wave field on the boundary is established

[0080] First, consider the solution of Navier equation (1) in region D According to the relationship between the gradient operator and the curl operator in three-dimensional space:

[0081]

[0082] in

[0083]

[0084] (50)

[0085] To u i Perform Helmholtz decomposition:

[0086]

[0087] in represents the wave number, They satisfy the following Helmholtz equations:

[0088]

[0089] Then, based on potential theory, the fundamental solution of the Helmholtz equation is used Establish the integral representation of the above Helmholtz equation respectively:

[0090]

[0091] and

[0092]

[0093] in is the first kind of zero-order Hankel function, n=(n1,n2,n3) Τ Representing boundaries The unit outward normal vector on Up points to outside of area D.

[0094] The normal derivative on the boundary is defined as follows:

[0095]

[0096] Next, combine (51) with (5) and (6) to obtain the non-radiative field u nrf (x) The integral on :

[0097]

[0098] So far, the present invention has obtained the boundary integral representation of the non-radiative field.

[0099] Finally, when u i When the incident field is known, the gradient, divergence, and curl in (52) are calculated as follows:

[0100]

[0101]

[0102] Then we can get the components u of the non-radiation field nrf,l (x), l=1,2,3, and the corresponding relationship between the incident wave field on the boundary:

[0103]

[0104] Thus, the present invention has established the corresponding relationship between the non-radiative field and the incident field data on the boundary. In order to achieve the invisibility of obstacles, the present invention has also developed the establishment of an approximate non-radiative field based on monopoles.

[0105] (3) Establishment of an approximate non-radiative field based on monopoles

[0106] Because the present invention is to achieve the invisibility of obstacles, it is necessary to generate a non-zero wave field inside a given area and a zero non-radiative field outside, so the low-order pole is first introduced. and l=1,2,3,σcan be S or Then (9) can be transformed into a low-order polaron and The distribution problem of the non-radiation field is obtained as follows:

[0107]

[0108] The specific process of discretizing the components of the non-radiative field and establishing the corresponding relationship between the approximate non-radiative field and the monopole is as follows:

[0109] First, using the basic solution Nature, get

[0110]

[0111] Then, yes and v=1,2,3, respectively, using the first-order difference and second-order difference approximation, we get:

[0112]

[0113] Substitute (11) and (12) into (10) and combine them with (13) and (14), parameterize and discretize the integration interval, and calculate an approximation of each component of the non-radiative field:

[0114]

[0115] where δ0=(0,0,0) Τ , δ1=(δ,0,0) Τ , δ2=(0,δ,0) Τ , δ3=(0,0,δ) Τ , δ4=δ1+δ2, δ5=δ1+δ3, δ6=δ2+δ3, δ7=-δ1, δ8=-δ2, δ9=-δ3, The form is the same as (16) to (45), and J(t1, t2) is the Jacobi function.

[0116] Finally, the numerical integration method is used for (15) to obtain the components of the approximate non-radiative field and the monopole G(x,y+δ m κ σ ),m=0,1,2,3,4,5,6,7,8,9, the corresponding relationships are as follows:

[0117]

[0118] in Nodes of the Gauss-Legendre quadrature formula, l1=1,···,N1, l2=1,···,N2,N1, are the number of partitions of the interval [0,π] and [0,2π], p=0,···,N p ,q=0,···,N q ,N p ,N q Take a non-negative integer, N p +1 and N q +1 is the number of nodes in the Gauss-Legendre quadrature formula, A p and B q Corresponding to N p +1 node and N q +1 weight for the Gauss-Legendre quadrature formula at the node.

[0119] So far, the application establishes the corresponding relationship between the approximate non-radiation field and the monopole. By calculating the components of the approximate non-radiation field, the construction form of the approximate non-radiation field is obtained, and the total field u tol (x) = (u tol,1 (x), u tol,2 (x), u tol,3 (x)) Τ , that is

[0120]

[0121] Finally, the active stealth of the obstacle with any shape in the three-dimensional elastic medium is realized.

[0122] Embodiment 2

[0123] In this embodiment, the method of embodiment 1 is used to test the stealth effect of the obstacle in the three-dimensional elastic medium, and the non-radiation field is constructed by using the generalized non-radiation source to realize the stealth of the obstacle in the three-dimensional elastic medium.

[0124] The incident field in the experiment has the following form:

[0125] Compressed plane wave:

[0126]

[0127] Shear plane wave:

[0128]

[0129] Where d = (sinφcosθ, sinφsinθ, cosφ) Τ ,d ⊥ = (cosφcosθ, cosφsinθ, -sinφ) Τ , θ ∈ [0, 2π], φ ∈ [0, π]. The above u i satisfy the three-dimensional Navier equation (1) in the region D.

[0130] The incident field in formula (52) of embodiment 1 can adopt the specific form of (58) and (59), and after calculating the gradient, divergence and curl, the corresponding relationship (9) between each component u nrf,l (x) of the non-radiation field and the incident wave field on the boundary is obtained.

[0131] The finite difference method is used to obtain the relationship (15) between the approximate values of the monopole and each component of the non-radiation field, and then the numerical integration method is used to obtain the calculated values (57) of each component of the approximate non-radiation field constructed by the monopole. ​

[0132] The numerical experiment is implemented in a bounded region Ω = [-50, 50] × [-50, 50] on the three-dimensional space cross section x3 = 0. The specific parameters in Example 2 are set as follows: δ=0.001。

[0133] Figure 2 shows the invisibility of obstacles with different numbers of monopoles. In the experiment, the incident field of the form (58) is selected, and the obstacle O1 is included in the invisibility area D, where

[0134] Figures 2(a)-2(d) When (N1, N2) are (3, 6), (5, 10), (7, 14), (15, 30), the second component imaginary part of the total field is As can be seen from Figure 2, as the number of monopoles increases, the total field outside the stealth region becomes closer and closer to the incident field, and the total field inside the stealth region becomes closer and closer to zero. This indicates that by applying a non-radiative field, when the incident field hits the obstacle, the scattering on the obstacle surface becomes less and less, thus improving the stealth effect.

[0135] Figure 3 When (N1, N2) takes (15, 30), the approximate non-radiation field component A point-by-point error lg map over the region Ω1∪Ω2, where When x∈Ω1, the point-by-point error is When x∈Ω2, the point-by-point error is The setting of the stealth area and the selection of the incident field are the same as above. Figure 3 It can be seen that the magnitude of the point-by-point error in the region Ω1∪Ω2 is less than 10 -3 , which shows that the approximate non-radiation field constructed by the present invention has better stealth performance.

[0136] Figure 4 When (N1, N2) is different, the approximate non-radiation field component In the ring area The absolute error norm on in r=8,10,20,40, The setting of the stealth area and the selection of the incident field are the same as above. Figure 4 It shows that as the number of monopoles increases, the error of the approximate non-radiating field decreases. At the same time, the error becomes smaller and smaller as the distance from the stealth area increases. This shows that the present invention can achieve good stealth effect by flexibly utilizing different numbers of monopoles.

[0137] Figure 5(a), Figure 5(b), and Figure 5(c) show the design of the stealth area when there are multiple obstacles of different shapes. The incident field of the form (59) is selected in the experiment. The obstacles in this experiment include a heart-shaped column O1, a cube O2, and a triangular prism O3. Figure 5(a)(b)(c) designs a single D, two Ds, and a spherical shape of the ... j (j=1,2) or three D j (j=1, 2, 3) stealth areas, and the total number of monopoles set in the three figures gradually decreases.

[0138] The parameter settings of the obstacle, stealth area and segmentation in Figure 5(a) are as follows:

[0139] The vertices of O2 are (6,3,±1), (6,5,±1), (8,3,±1), (8,5,±1), and the vertices of O3 are (1,5,±1), (2,7,±1), (4,5,±1), The subdivision number (N1, N2) is (30, 60).

[0140] The parameter settings of the obstacle, stealth area and segmentation in Figure 5(b) are as follows:

[0141] The vertices of O2 are (-7,-1,±1), (-7,1,±1), (-5,-1,±1), (-5,1,±1), and the vertices of O3 are (-7,6,±1), (-6,8,±1), (-4,6,±1), The splitting numbers are

[0142] The parameter settings for the obstacle, stealth area, and segmentation in Figure 5(c) are as follows:

[0143] The vertices of O2 are (1,-9,±1), (1,-11,±1), (-1,-11,±1), (-1,-9,±1), and the vertices of O3 are (7,9,±1), (8,11,±1), (10,9,±1), The splitting numbers are

[0144] Figure 5(a), Figure 5(b), and Figure 5(c) show three obstacles located in a single D, two D j (j=1,2) and three Dj (j=1,2,3) In the stealth region, the real part of the first component of the total field Figure 5 shows that the present invention can design different stealth areas according to the location and density of obstacles, so as to achieve the goal of using a small number of monopoles to construct a non-radiation field to produce a good stealth effect.

[0145] 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 active cloaking of obstacles in a three-dimensional elastic medium based on a generalized non-radiative source, characterized in that: The following steps are involved: S1: Based on the theory of wave fields and non-radiative sources in three-dimensional elastic media, the elastic wave field that satisfies the Navier equations is described, and a generalized non-radiative source framework for elastic waves is established; S2: Based on the Helmholtz decomposition and potential theory, the three-dimensional elastic wave non-radiating field is formulated as an integral representation. Combined with the known incident wave field, the corresponding relationship between each component of the non-radiating field and the incident wave field on the boundary is obtained; S3: Based on the integral representation, the problem of cloaking an obstacle of arbitrary shape in a three-dimensional elastic medium is transformed into a low-order pole distribution problem on the boundary of a given region, wherein the goal of the low-order pole distribution problem is to generate a non-zero elastic wave field inside the given region and a zero non-radiative field outside the given region; S4: Using a method combining finite differences and numerical integration, the integral representation of the non-radiating field is discretized, a correspondence between the approximate non-radiating field and the monopole is established, and the structural form of the approximate non-radiating field is obtained, ultimately realizing the active cloaking of obstacles of arbitrary shapes in three-dimensional elastic media.

2. The method according to claim 1, characterized in that The specific process of step S1 is: S101: Let D be In the bounded region, consider the propagation of elastic waves in an isotropic homogeneous medium. The three-dimensional elastic wave field in region D is governed by the following Navier equations: μΔu(x)+(λ+μ)▽▽·u(x)+ρω 2 u(x)=0,x∈D, (1) Where μ and λ are Lamé constants, satisfying μ>0, λ+μ>0, ρ is the medium density, ω is the angular frequency, is the Laplace operator, is the gradient operator. S102: Based on the non-radiative source theory, the generalized non-radiative source of elastic waves is a The low-order poles in the distribution can generate a non-radiative field u that is non-zero inside the region D and zero outside the region D. nrf (x)=(u nrf,1 (x),u nrf,2 (x),u nrf,3 (x)) Τ ,Right now where u(x) satisfies equation (1) in D.

3. The method according to claim 2, characterized in that Step S2, based on the Helmholtz decomposition and potential theory, formulates an integral representation of the three-dimensional elastic wave non-radiative field, specifically including: S201: In region D, consider the solution of Navier equation (1) make Based on the Helmholtz decomposition, the elastic wave field u is obtained i In D, the following conditions are met: Where curl is a three-dimensional curl operator, v S , They satisfy the following Helmholtz equations: S202: Based on potential theory, using the fundamental solution of the Helmholtz equation Establish the integral representation of the above Helmholtz equation respectively: and in represents the wave number, is the first kind of zero-order Hankel function, n=(n1,n2,n3) Τ Representing boundaries The unit outward normal vector on points to the outside of region D. The normal derivative on the boundary is defined as follows: S203: Combine (3) with (5) and (6) to obtain the non-radiative field u nrf (x) The integral on :

4. The method according to claim 3, characterized in that Step S2 combines the known incident wave field to obtain the corresponding relationship between each component of the non-radiation field and the incident wave field on the boundary, specifically including: When u i When the incident field is known, the gradient, divergence, and curl in (8) are calculated to obtain the components u of the non-radiative field. nrf,l (x), l=1,2,3, and the corresponding relationship between the incident wave field on the boundary:

5. The method according to claim 4, characterized in that The step S3 converts the problem of cloaking an obstacle of arbitrary shape in a three-dimensional elastic medium into a low-order pole distribution problem on the boundary of a given region based on the integral representation, wherein the goal of the low-order pole distribution problem is to generate a non-zero elastic wave field inside the given region and a zero non-radiative field outside the given region, specifically including: Introducing low-order poles and σ can be S or Then (9) can be transformed into a low-order polar F l (1,σ) and F l (2,σ) Distribution problem:

6. The method according to claim 5, characterized in that The method described in step S4, which combines finite differences and numerical integration, discretizes the integral representation of the non-radiative field and establishes a corresponding relationship between the approximate non-radiative field and the monopole, specifically includes: S401: Using the basic solution The nature of S402: Yes and Using first-order difference and second-order difference approximation respectively, we get: Substituting (11) and (12) into (10), combining (13) and (14), and parameterizing and discretizing the integration interval, we can calculate an approximation of the components of the non-radiative field: where δ0 = (0, 0, 0) Τ , δ1 = (δ, 0, 0) Τ , δ2 = (0, δ, 0) Τ , δ3 = (0, 0, δ) Τ , δ4 = δ1 + δ2, δ5 = δ1 + δ3, δ6 = δ2 + δ3, δ7 = -δ1, δ8 = -δ2, δ9 = -δ3, The form is as follows: J(t1,t2) is the Jacobian function. S403: Apply numerical integration method to (15) to obtain the components of the approximate non-radiative field and the monopole G(x,y+δ m κ σ ),m=0,1,2,3,4,5,6,7,8,9, the corresponding relationships are as follows: in Nodes of the Gauss-Legendre quadrature formula, l1=1,···,N1, l2=1,···,N2, are the number of partitions of the interval [0,π] and [0,2π], p=0,···,N p ,q=0,···,N q ,N p ,N q Take a non-negative integer, N p +1 and N q +1 is the number of nodes in the Gauss-Legendre quadrature formula, A p and B q Corresponding to N p +1 node and N q +1 weight for the quadrature formula at each node. By calculating the components of the approximate non-radiative field, the construction form of the approximate non-radiative field is obtained. Using the constructed approximate non-radiative field, the total field u can be obtained, which is zero inside the region and the incident field outside the region. tol (x)=(u tol,1 (x),u tol,2 (x),u tol,3 (x)) Τ ,Right now Ultimately, active cloaking of obstacles of arbitrary shapes in three-dimensional elastic media is achieved.

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