A method for active camouflage of obstacles in three-dimensional elastic medium based on generalized non-radiation source

Abstract

The application discloses a method for active camouflage of obstacles in three-dimensional elastic medium based on generalized non-radiation source, which comprises the following steps: based on wave field in three-dimensional elastic medium and non-radiation source theory, describing elastic wave field satisfying Navier equation, and establishing generalized non-radiation source framework of elastic wave; based on Helmholtz decomposition and potential theory, formulating integral representation of three-dimensional elastic wave non-radiation field, and combining with known incident wave field to obtain corresponding relationship between each component of non-radiation field and incident wave field on the boundary; according to the integral representation, converting the problem of active camouflage of obstacles in three-dimensional elastic medium into the problem of low-order monopole distribution on the boundary of a given region, and the target of the problem of low-order monopole distribution is to generate non-radiation field which is non-zero inside the given region and zero outside; using the method combining finite difference and numerical integration, discretizing the integral representation of the non-radiation field, establishing the corresponding relationship between the approximate non-radiation field and monopole, obtaining the construction form of the approximate non-radiation field, and finally realizing the active camouflage of obstacles in three-dimensional elastic medium with any shape. The method is easy to realize the construction of the approximate non-radiation field with high precision, strong adaptability and instant regulation and control.

Description

Technical Field

[0001] This invention belongs to the field of active cloaking of obstacles in three-dimensional elastic media, and specifically relates to an active cloaking method for obstacles in three-dimensional elastic media based on a generalized non-radiative source. Background Technology

[0002] Obstacle stealth technology has wide applications in numerous fields such as military, transportation, medicine, and environment, and has significant research value for improving safety, advancing medical care, and protecting the environment. Achieving target obstacle stealth has profound implications for extending equipment lifespan, reducing energy consumption, improving the natural environment, and developing related industries.

[0003] Currently, the main methods for obstacle cloaking are divided into two categories: optical transformation and destructive scattering. Optical transformation is based on the spatial transformation invariance of physical equations, creating a cloaking "hole" through spatial transformation. It often involves fabricating multi-layered metamaterial cloaks to allow waves to bypass a designated area and propagate in their original direction, thus achieving obstacle cloaking. Destructive scattering, on the other hand, reduces or eliminates obstacle reflection signals by designing special structures and coatings on the obstacle's surface, or by placing devices that actively emit specific waves. While spatial transformation methods have a wider range of applications, the unusual nature of the materials and high cost pose obstacles to their fabrication. Destructive scattering offers superior cloaking effects, but once fabricated, the device cannot be adjusted for specific situations.

[0004] While current methods for obstacle cloaking each have their own characteristics, they also have many shortcomings in practical applications. Optical transformation methods, utilizing metamaterial cloaks, are suitable for cloaking in different wave fields, but materials are scarce and expensive; scattering destructive methods are limited by existing shapes and structures, and have weak adjustment capabilities when the wave field changes. This application addresses the obstacle cloaking problem in elastic media by proposing a novel method with superior performance, simple construction, and strong applicability, which constructs a non-radiative field through a generalized non-radiative source. Summary of the Invention

[0005] The present invention aims to overcome the above-mentioned problems and deficiencies in the prior art and provide an active stealth method for obstacles in a three-dimensional elastic medium based on a generalized non-radiative source.

[0006] To achieve the objectives of this invention, the technical solution adopted is as follows: This invention provides an active cloaking method for obstacles in a three-dimensional elastic medium based on a generalized non-radiative source, comprising the following steps: S1: Based on the theory of wave field and non-radiative source in three-dimensional elastic medium, the elastic wave field that satisfies the Navier equation is described, and a generalized non-radiative source framework for elastic waves is established. S2: Based on Helmholtz decomposition and potential theory, the non-radiative field of three-dimensional elastic waves is expressed in a formulaic integral. Combined with the known incident wave field, the correspondence between each component of the non-radiative field and the incident wave field on the boundary is obtained. S3: According to the integral representation, the problem of stealth of an arbitrary-shaped obstacle in a three-dimensional elastic medium is transformed into a low-order pole distribution problem on the boundary of a given region. The goal of the low-order pole distribution problem is to generate a non-radiative field with a non-zero elastic wave field inside the given region and a zero field outside the region. S4: By combining finite difference and numerical integration, the integral representation of the non-radiative field is discretized, and the correspondence between the approximate non-radiative field and the monopole is established to obtain the construction form of the approximate non-radiative field, ultimately realizing the active stealth of obstacles of arbitrary shape in a three-dimensional elastic medium.

[0007] According to the above method, the specific process of step S1 includes: S101: Let... for In a bounded region, considering the propagation of elastic waves in an isotropic homogeneous medium, in the region... The three-dimensional elastic wave field is governed by the following Navier equations: (1) in , It is a Lamé constant, satisfying , , It is the density of the medium. It is angular frequency. It is the Laplace operator. It is the gradient operator.

[0008] S102: Based on the nonradiative source theory, the generalized nonradiative source of the elastic wave is a kind of source at the boundary. The distribution of low-order poles on the surface, in which low-order poles can be generated in the region Non-zero inside, in the region Non-radiative field with zero external radiation ,Right now (2) in exist The interior satisfies equation (1).

[0009] Based on the above method, further, step S2, which involves formulating an integral representation of the three-dimensional elastic wave nonradiative field based on Helmholtz decomposition and potential theory, specifically includes: S201: In the area Inside, consider the solution of Navier equation (1). ,make , Based on Helmholtz decomposition, the elastic wave field is obtained. exist Internal satisfaction: (3) in It is a three-dimensional curl operator. , They respectively satisfy the following Helmholtz equations: (4) S202: Based on potential theory, utilizing the fundamental solution of the Helmholtz equation Establish the integral representations of the Helmholtz equations described above: (5) and (6) in , Indicates wave number, It is a zeroth-order Hankel function of the first kind. Indicates boundary The unit outward normal vector on, Up pointing area Externally. The normal derivative on the boundary is defined as follows: (7) S203: Combining (3) with (5) and (6) yields a nonradiative field. exist Integral representation on: (8) Based on the above method, further, step S2, which combines the known incident wave field to obtain the correspondence between each component of the non-radiative field and the incident wave field on the boundary, specifically includes: when When the incident field is known, the gradient, divergence, and curl in (8) are calculated to obtain the components of the non-radiative field. Correspondence between the incident wave field on the boundary and the incident wave field: (9) According to the above method, further, step S3, based on the integral representation, transforms the cloaking problem of an arbitrary-shaped obstacle in a three-dimensional elastic medium into a low-order pole distribution problem on the boundary of a given region. The goal of the low-order pole distribution problem is to generate a non-radiative field with a non-zero elastic wave field inside the given region and a zero field outside, specifically including: Introducing low-order poles and Desirable or Then (9) can be transformed into a lower-order pole. and Distribution problem: (10) Based on the above method, further, step S4, which uses a combination of finite difference and numerical integration to discretize the integral representation of the non-radiative field and establish the correspondence between the approximate non-radiative field and the monopole, specifically includes: S401: Using the aforementioned basic solution The properties obtained (11) or (12) S402: Yes and Using first-order and second-order difference approximations respectively, we obtain: (13) , (14) Substituting (11) and (12) into (10) and combining them with (13) and (14), and parameterizing and discretizing the integration interval, we can calculate an approximation of each component of the non-radiative field: (15) in , , , , , , , , , , , The format is as follows: (16) (17) (18) (19) (20) (twenty one) (twenty two) (twenty three) (twenty four) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38) (39) (40) (41) (42) (43) (44) (45) It is the Jacobian function.

[0012] S403: Using numerical integration on (15), the components of the approximate nonradiative field and the monopole are obtained. The correspondence between them is as follows: (46) in , , , For the nodes of the Gauss-Legendre quadrature formula, They are intervals and The number of parts to be divided, Take non-negative integers. and The number of nodes in the Gauss-Legendre quadrature formula. and Corresponding to Each node and The weights of the quadrature formulas at each node.

[0013] By calculating the components of the approximate nonradiative field, the construction form of the approximate nonradiative field is obtained. Using the constructed approximate nonradiative field, the total field, which is zero inside the region and equal to the incident field outside the region, can be obtained. ,Right now (47) Ultimately, this will enable active cloaking of obstacles of arbitrary shapes in a three-dimensional elastic medium.

[0014] Compared with the prior art, the beneficial effects of the present invention are as follows: The innovative aspects of this invention: 1. This invention enables the construction of an approximate nonradiative field by employing a small number of low-order generalized nonradiative sources in the form of monopoles. Compared to high-order generalized nonradiative sources in the form of series, this method more easily achieves the construction of a high-precision, highly adaptable, and instantly controllable approximate nonradiative field.

[0015] 2. This invention can construct a non-radiative field according to different stealth purposes to achieve stealth of multiple obstacles of arbitrary shape, and can adapt to applications with larger scale or higher precision requirements; 3. The proposed non-radiative field construction method can be used to analyze stealth performance and predict the impact of different parameter settings on stealth effect. This invention is also applicable to multiple fields such as acoustics, optics, and electromagnetics, and has broad application prospects.

[0016] Working principle of the invention: This invention achieves stealth against multiple obstacles of arbitrary shapes under a known incident field. A stealth region is designed based on the shape, size, and location of the obstacle. An approximate non-radiative field is generated by distributing a generalized non-radiative source composed of a small number of monopoles along the boundary of the stealth region. This approximate non-radiative field is a negative incident field inside the stealth region and zero outside, effectively canceling out the incident field inside the stealth region. This results in a total field where the field is zero inside the stealth region and equal to the incident field outside. Thus, when the incident field illuminates the obstacle, no scattered field can be detected outside the stealth region, meaning the obstacle cannot be detected, ultimately achieving stealth. Attached Figure Description

[0017] Figure 1 This is a flowchart illustrating the method of the present invention.

[0018] Figures 2(a)-2(d) They are respectively Pick , , , At that time, the imaginary part of the second component of the total field The numerical results are shown in the figure.

[0019] Figure 3 for Pick At that time, the approximate nonradiative field component In the region Point-by-point error picture.

[0020] Figure 4 for At the same time, the approximate nonradiative field components In the annular region absolute error norm picture.

[0021] Figures 5(a)-5(c) The first component of the real part of the total field when the obstacle is located in different stealth zones designed according to its shape and density. The numerical results are shown in the figure. Detailed Implementation

[0022] 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.

[0023] Example 1

[0024] This embodiment specifically provides an active cloaking method for obstacles in a three-dimensional elastic medium based on a generalized non-radiative source, such as... Figure 1 As shown, it includes the following steps: (1) Describe the elastic wave field that satisfies the Navier equation and establish the generalized non-radiative source framework of elastic waves. set up for In a bounded region, considering the propagation of elastic waves in an isotropic homogeneous medium, in the region... The three-dimensional elastic wave field is governed by the following Navier equations: (1) in , It is a Lamé constant, satisfying , , It is the density of the medium. It is angular frequency. It is the Laplace operator. It is the gradient operator.

[0025] Based on the theory of nonradiative sources, the generalized nonradiative source of the elastic wave is a kind of source at the boundary. The distribution of low-order poles on the surface, in which low-order poles can be generated in the region Non-zero inside, in the region Non-radiative field with zero external radiation ,Right now (2) in exist The interior satisfies equation (1).

[0026] (2) Based on Helmholtz decomposition and potential theory, and combined with the incident wave field, establish the relationship between each component of the nonradiative field and the incident wave field on the boundary.

[0027] First, consider the Navier equation (1) in the region solution within Based on the relationship between the gradient operator and the curl operator in three-dimensional space: (48) in (49) (50) right Perform Helmholtz decomposition: (51) in , Indicates wave number, , They respectively satisfy the following Helmholtz equations: (4) Then, based on potential theory, the fundamental solution of the Helmholtz equation is used. Establish the integral representations of the Helmholtz equations described above: (5) and (6) in It is a zeroth-order Hankel function of the first kind. Indicates boundary The unit outward normal vector on, Up pointing area external.

[0028] The normal derivative on the boundary is defined as follows: (7) Next, by combining (5) and (6) with (51), the nonradiative field is obtained. exist Integral representation on: (52) Thus, the present invention has obtained the boundary integral representation of the non-radiative field.

[0029] Finally, when When the incident field is known, calculate the gradient, divergence, and curl in (52): (53) (54) (55) (56) This leads to the various components of the non-radiative field. Correspondence between the incident wave field on the boundary and the incident wave field: (9) Thus, this invention establishes the correspondence between the non-radiative field and the incident field data on the boundary. To achieve obstacle stealth, this invention further develops a method for establishing an approximate non-radiative field based on monopoles.

[0030] (3) Establishment of an approximate nonradiative field based on monopoles

[0031] Since this invention aims to achieve obstacle invisibility, it requires generating a non-radiative field with a non-zero wave field inside a given region and a zero wave field outside. Therefore, a low-order pole is first introduced. and Desirable or Then (9) can be transformed into a lower-order pole. and The distribution problem yields the following components of the non-radiative field: (10) The specific process of discretizing each component of the nonradiative field and establishing the correspondence between the approximate nonradiative field and the monopole is as follows: First, using the aforementioned fundamental solution Properties, obtained (11) or (12) Then, to and Using first-order and second-order difference approximations respectively, we obtain: (13) , (14) Substituting (11) and (12) into (10) and combining them with (13) and (14), and parameterizing and discretizing the integration interval, we can calculate an approximation of each component of the non-radiative field: (15) in , , , , , , , , , , , The format is the same as (16)~(45). It is the Jacobian function.

[0032] Finally, numerical integration is used for (15) to obtain the components of the approximate nonradiative field and the monopole. The correspondence between them is as follows: (57) in , , , For the nodes of the Gauss-Legendre quadrature formula, They are intervals and The number of parts to be divided, Take non-negative integers. and The number of nodes in the Gauss-Legendre quadrature formula. and Corresponding to Each node and The weights of the Gauss-Legendre quadrature formula at each node.

[0033] Thus, this invention establishes the correspondence between the approximate nonradiative field and the monopole. By calculating the components of the approximate nonradiative field, its structural form is obtained. Using the constructed approximate nonradiative field, the total field, which is zero inside the region and equal to the incident field outside the region, can be obtained. ,Right now (47) Ultimately, this will enable active cloaking of obstacles of arbitrary shapes in a three-dimensional elastic medium.

[0034] Example 2

[0035] This embodiment uses the method of Embodiment 1 to test the stealth effect of obstacles in a three-dimensional elastic medium. The obstacles in the three-dimensional elastic medium are used to construct a non-radiative field using a generalized non-radiative source to achieve stealth.

[0036] The incident field in the experiment has the following form: Compressed plane waves: (58) Shear plane wave: (59) in . above All in the region It satisfies the three-dimensional Navier equation (1).

[0037] In Example 1, the incident field in equation (52) can take the specific forms of (58) and (59). After calculating the gradient, divergence, and curl, the components of the non-radiative field are obtained. The correspondence between the incident wave field on the boundary and (9).

[0038] The finite difference method is used to obtain the relationship between the approximate values ​​of each component of the monopole and nonradiative field (15). Then, the numerical integration method is used to obtain the approximate nonradiative field components constructed with the monopole. The calculated value is (57).

[0039] Numerical experiments in three-dimensional space sections Bounded regions on In the implementation, the specific parameter settings in Example 2 are as follows: , .

[0040] Figures 2(a)-2(d) The stealth of the obstacle is shown for different numbers of monopoles. In the experiment, an incident field of type (58) was selected, and the obstacle... Included in the stealth area Among them

[0041] Figures 2(a)-2(d) They are respectively Pick At that time, the imaginary part of the second component of the total field The numerical results show that as the number of monopoles increases, the total field outside the stealth region gets closer to the incident field, and the total field inside gets closer to zero. This indicates that by applying a non-radiative field, when the incident field irradiates the obstacle, less and less scattering occurs on the obstacle surface, thus improving the stealth effect.

[0042] Figure 3 for Pick At that time, the approximate nonradiative field component In the region Point-by-point error Figure, in which when At that time, the point-by-point error is ,when At that time, the point-by-point error is The stealth zone setup and the selection of the incident field are the same as above. Figure 3 It can be seen that in the region The order of magnitude of the pointwise error is smaller than This demonstrates that the approximate non-radiative field constructed by the present invention has good stealth performance.

[0043] Figure 4 for At the same time, the approximate nonradiative field components In the annular region absolute error norm ,in The stealth zone setup and the selection of the incident field are the same as above. Figure 4 This indicates that as the number of monopoles increases, the error of the approximate non-radiative field decreases, and the error also decreases as the distance from the stealth region increases. This demonstrates that the present invention can achieve a good stealth effect by flexibly utilizing different numbers of monopoles.

[0044] Figures 5(a), 5(b), and 5(c) illustrate the design of the camouflage zone when there are multiple obstacles of different shapes. The experiment used an incident field of type (59), in which the obstacles included heart-shaped pillars. cube triangular prism Figures 5(a)(b)(c) show individual designs based on the location and density of the obstacles. ,two Or three The stealth area, and the total number of monopoles set in the three diagrams gradually decreases.

[0045] The parameter settings for the obstacle, cloaked area, and mesh in Figure 5(a) are as follows: The vertex is , , , , The vertex is , , , , The subdivision fraction Pick .

[0046] The parameter settings for the obstacle, cloaking area, and subdivision in Figure 5(b) are as follows: , The vertex is , , , The vertex is , , , , , The subdivisions are respectively .

[0047] The parameter settings for the obstacle, cloaking area, and subdivision in Figure 5(c) are as follows: The vertex is , , , , The vertex is , , , , The subdivisions are respectively

[0048] Figures 5(a), 5(b), and 5(c) respectively show the three obstacles located in a single ,two and three When in the stealth zone, the first component of the total field is real. The numerical results. Figures 5(a)-5(c) This invention demonstrates that different stealth zones can be designed based on the location and density of obstacles, in order to achieve the goal of generating good stealth effect by constructing a non-radiative field using a small number of monopoles.

[0049] 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 method for active obstacle stealth based on a generalized non-radiative source in a three-dimensional elastic medium, characterized in that, Includes the following steps: S1: Based on the theory of wave field and non-radiative source in three-dimensional elastic medium, the elastic wave field that satisfies the Navier equation is described, and a generalized non-radiative source framework for elastic waves is established. S2: Based on Helmholtz decomposition and potential theory, the non-radiative field of three-dimensional elastic waves is expressed in a formulaic integral. Combined with the known incident wave field, the correspondence between each component of the non-radiative field and the incident wave field on the boundary is obtained. S3: Based on the integral representation, the problem of stealth of an arbitrary-shaped obstacle in a three-dimensional elastic medium is transformed into a low-order pole distribution problem on the boundary of a given region. The objective of the low-order pole distribution problem is to generate a non-radiative field with a non-zero elastic wave field inside the given region and a zero wave field outside; specifically including: Introducing low-order poles and Desirable or Then the correspondence between each component of the nonradiative field and the incident wave field on the boundary is transformed into a low-order pole. and Distribution problem: (10) in, for Boundary region , Indicates wave number; , It is a Lamé constant, satisfying , , It is the density of the medium. It is angular frequency; It is a gradient operator; S4: Using a combination of finite difference and numerical integration, the integral representation of the non-radiative field is discretized, establishing a correspondence between the approximate non-radiative field and the monopole, thus obtaining the construction form of the approximate non-radiative field, ultimately achieving active cloaking of obstacles of arbitrary shapes in a three-dimensional elastic medium; specifically including: S401: Using the fundamental solution The properties obtained ， (11) or (12) S402: Yes and Using first-order and second-order difference approximations respectively, we obtain: ， (13) ， ； (14) 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: (15) in , , , , , , , , , , , These are the piecewise weighted coefficients; S403: Using numerical integration on (15), the components of the approximate nonradiative field and the monopole are obtained. The correspondence between them is as follows: (46) in , , , For the nodes of the Gauss-Legendre quadrature formula, , They are intervals and The number of parts to be divided, Take non-negative integers. and The number of nodes in the Gauss-Legendre quadrature formula. and Corresponding to Each node and The weights of the quadrature formulas at each node; By calculating the components of the approximate nonradiative field, the construction form of the approximate nonradiative field is obtained. Using the constructed approximate nonradiative field, the total field, which is zero inside the region and equal to the incident field outside the region, can be obtained. ,Right now (47) Ultimately, this will enable active cloaking of obstacles of arbitrary shapes in a three-dimensional elastic medium.

2. The method according to claim 1, characterized in that, The specific process of step S1 is as follows: S101: Let for In a bounded region, considering the propagation of elastic waves in an isotropic homogeneous medium, in the region... The three-dimensional elastic wave field is governed by the following Navier equations: (1) in, It is the Laplace operator; S102: Based on the nonradiative source theory, the generalized nonradiative source of the elastic wave is a kind of source at the boundary. The distribution of low-order poles on the surface, in which low-order poles can be generated in the region Non-zero inside, in the region Non-radiative field with zero external radiation ,Right now (2) in exist The interior satisfies equation (1).

3. The method according to claim 2, characterized in that, Step S2, which involves formulating an integral representation of the nonradiative field of a three-dimensional elastic wave based on Helmholtz decomposition and potential theory, specifically includes: S201: In the area Inside, consider the solution of Navier equation (1). ,make , Based on Helmholtz decomposition, the elastic wave field is obtained. exist Internal satisfaction: ， (3) in It is a three-dimensional curl operator. , They respectively satisfy the following Helmholtz equations: (4) S202: Based on potential theory, utilizing the fundamental solution of the Helmholtz equation Establish the integral representations of the Helmholtz equations described above: (5) and (6) in It is a zeroth-order Hankel function of the first kind. Indicates boundary The unit outward normal vector on, Up pointing area The normal derivative on the exterior boundary is defined as follows: (7) S203: Combining (3) with (5) and (6) yields a nonradiative field. exist Integral representation on: (8)。 4. The method according to claim 3, characterized in that, Step S2, which involves combining the known incident wave field to obtain the correspondence between each component of the non-radiative field and the incident wave field on the boundary, specifically includes: when When the incident field is known, the gradient, divergence, and curl in (8) are calculated to obtain the components of the non-radiative field. Correspondence between the incident wave field and the boundary: (9)。

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