A construction method of a multi-layer structure sub-wavelength resonator

Abstract

The present invention discloses a construction method for a multi-layer sub-wavelength resonator, comprising the following steps: constructing a material parameter model of an N-layer sub-wavelength resonator structure based on the material of the sub-wavelength resonator; constructing a capacitance matrix of the N-layer sub-wavelength resonator according to the material parameter model of the N-layer sub-wavelength resonator structure; obtaining the eigenvalue of the capacitance matrix according to the capacitance matrix of the N-layer sub-wavelength resonator; obtaining a parameter selection model corresponding to the resonance frequency of the N-layer sub-wavelength resonator according to the eigenvalue of the capacitance matrix; and selecting material parameters according to the model of the resonance frequency and corresponding parameters of the N-layer sub-wavelength resonator to realize the construction of the N-layer sub-wavelength resonator corresponding to the selected resonance frequency. The present invention improves the accuracy and efficiency of the design of the multi-layer sub-wavelength resonator, reduces the waste of resources and time for repeated experiments, promotes the realization of the customized application of the resonator, and also lays a foundation for its customized application in the fields of optics, acoustics and other fields.

Description

Technical Field

[0001] The present invention belongs to the technical field of resonator structures, and particularly relates to a construction method for a multi-layer sub-wavelength resonator. Background Art

[0002] A phononic crystal is a novel functional material formed by periodically arranging elastic solids in another solid or fluid medium. By analogy with photonic crystals, it was found that when elastic waves propagate in a periodic elastic composite medium, an elastic wave bandgap similar to the photonic bandgap will also be generated, thus proposing the concept of phononic crystals.

[0003] In the research of phononic crystals, most structures usually consist of single-layer resonators. However, due to the narrow bandgap width and poor filtering performance of this configuration, its application in practical engineering is limited. Therefore, researchers have begun to explore the design of metamaterials with wider bandgaps. In particular, multi-layer materials have become an ideal choice for sub-wavelength resonators due to their excellent tunability and high-quality resonance characteristics. Most existing research methods explore materials with specific multi-layer structures through experiments to verify whether they can achieve sub-wavelength resonance. However, this experimental method lacks universality and cannot guide the design of material parameters in advance, and can only be adjusted through continuous experimental attempts. When the number of material layers increases, the limitations of this experimental method become particularly obvious due to the limited prior information it provides. Summary of the Invention

[0004] The purpose of the present invention is to provide a construction method for a multi-layer sub-wavelength resonator, which uses theoretical analysis means to characterize the resonance frequency that causes sub-wavelength resonance when the material structure parameters of any number of layers are selected, promotes the selection of material parameters, thereby inducing the desired resonance, and realizing the construction of a multi-layer sub-wavelength resonator.

[0005] The present invention provides a construction method for a multi-layer sub-wavelength resonator, including the following steps:

[0006] S1. Based on the material of the sub-wavelength resonator, construct a material parameter model of an N-layer sub-wavelength resonator structure;

[0007] S2. According to the material parameter model of the N-layer sub-wavelength resonator structure obtained in step S1, based on the Dirichlet-to-Neumann operator method, construct a capacitance matrix of the N-layer sub-wavelength resonator;

[0008] S3. According to the capacitance matrix of the N-layer sub-wavelength resonator obtained in step S2, based on the characteristics of the tridiagonal capacitance matrix, obtain the capacitance matrix eigenvalues;

[0009] S4. Based on the capacitance matrix eigenvalues obtained in step S3, by solving the generalized capacitance eigenvalue problem, a parameter selection model corresponding to the resonance frequencies of the N-layer subwavelength resonators is obtained;

[0010] S5. According to the model of the resonance frequencies of the N-layer subwavelength resonators and the corresponding parameters obtained in step S4, material parameters are selected to realize the construction of the N-layer subwavelength resonators corresponding to the selected resonance frequencies.

[0011] Step S1 is specifically: Using density and bulk modulus to characterize the material parameters of the N-layer subwavelength resonators, a material parameter model is obtained; where the construction of any N-layer materials changes layer by layer alternately, and the subwavelength resonators are nested in the background matrix; Two dimensionless contrast parameters are introduced, which are used to represent the ratios of the density and bulk modulus of the subwavelength resonators to the density and bulk modulus of the background matrix respectively.

[0012] The density is expressed by the following formula: where x is the spatial position; is the resonator density; is the background matrix density; is the subwavelength resonator; is the background matrix; The bulk modulus is expressed by the following formula: where, is the resonator bulk modulus; is the background matrix bulk modulus; is the inner radius of the j-th layer resonator; is the outer radius; The wave speed of the subwavelength resonator is expressed by the following formula: ;

[0013] The wave speed of the background matrix is expressed by the following formula: ;

[0014] The wave number of the subwavelength resonator is expressed by the following formula: ;

[0015] where, is the angular frequency; The wave number of the background matrix is expressed by the following formula: ;

[0016] The background matrix is expressed by the following formula: The dimensionless contrast parameters include the first contrast parameter and the second contrast parameter , which is expressed by the following formula: and , takes the value of a bounded positive constant, indicating that the wave velocities in the resonator and the background matrix are comparable.

[0017] Step S2 is specifically as follows: Based on the material parameter model obtained in Step S1, for the material results with any number of layers and thicknesses, by the boundary conditions satisfied by the acoustic wave propagation in the material structure, the resonance phenomenon of the N-layer sub-wavelength resonator is characterized according to the Dirichlet-to-Neumann operator method, and the capacitance matrix related to the number of material layers and thicknesses is obtained.

[0018] The capacitance matrix is a tridiagonal positive definite matrix C, which is expressed by the following formula: where is the outer radius of the j-th layer of material, and j takes values of 1, 2, 3,..., N; is the inner radius of the j-th layer of material, and j takes values of 1, 2, 3,..., N - 1.

[0019] Step S3 is specifically as follows: Based on the capacitance matrix obtained in Step S2, since the capacitance matrix is a tridiagonal positive definite matrix, according to the positive definiteness and irreducibility of the tridiagonal positive definite matrix, N different eigenvalues of the capacitance matrix are obtained, and .

[0020] Step S4 is specifically as follows: According to the eigenvalues of the capacitance matrix obtained in Step S3, the generalized capacitance eigenvalue problem is solved to obtain the parameter selection model corresponding to all sub-wavelength resonance frequencies in the material structure of any N-layer sub-wavelength resonator.

[0021] The generalized capacitance eigenvalue problem is expressed by the following formula: where is the i-th eigenvalue; is the i-th eigenvector; V is a diagonal matrix expressed as , and ; the eigenvectors form an orthogonal basis with respect to the following inner product, which is expressed by the following formula: where is the Kronecker symbol.

[0022] The parameter selection model corresponding to all sub-wavelength resonance frequencies in the material structure of any N-layer sub-wavelength resonator is expressed by the following formula: where is the i-th sub-wavelength resonance frequency; is the first component of the i-th eigenvector; is the contrast parameter in step S1; is a constant, and its value is a bounded positive constant.

[0023] Specifically, step S5 is as follows: Select a model according to the parameters obtained in step S4, calculate the material parameters of the N-layer sub-wavelength resonators for the preset target resonance frequency, construct the N-layer sub-wavelength resonators based on the obtained material parameters and verify their resonance frequencies, and finally adjust the parameters according to the preset application scenario to complete the construction of the multi-layer sub-wavelength resonators.

[0024] The present invention discloses a method for constructing multi-layer sub-wavelength resonators. Through theoretical analysis and calculation, the resonance frequencies that can trigger sub-wavelength resonance phenomena under sub-wavelength resonator structures with any specific number of layers are accurately deduced, and then the construction of multi-layer sub-wavelength resonator structures is realized by means of experimental verification. In addition, the present invention can provide a scientific basis for material selection and structural design, thus greatly improving the accuracy and efficiency of design. Through pre-calculation and theoretical analysis, blind and repeated experiments can be effectively avoided, and the waste of resources can be significantly reduced. Especially in the development process of multi-layer sub-wavelength resonators, the number of tests and the required time can be greatly reduced. The present invention can also greatly reduce the design and production costs of multi-layer sub-wavelength resonator materials. Through the combination of accurate theoretical guidance and experimental verification, the unnecessary trial-and-error costs in the development process are reduced, making the technical application more economical and feasible. This not only provides new impetus for the rapid development of related technologies, but also lays a more solid foundation for future customized applications in the fields of optics, acoustics and other fields. Brief Description of the Drawings

[0025] Figure 1 is a schematic flow chart of the method of the present invention;

[0026] Figure 2 is a schematic structural diagram of the multi-layer sub-wavelength resonator material;

[0027] Figure 3 is a resonance mode diagram of the multi-layer sub-wavelength resonator material;

[0028] Figure 4 is a resonance effect diagram of the multi-layer sub-wavelength resonator material. Detailed Embodiments

[0029] The present invention provides a method for constructing multi-layer sub-wavelength resonators, and its schematic flow chart is as Figure 1 shown, including the following steps:

[0030] The present invention provides a method for constructing multi-layer sub-wavelength resonators, including the following steps:

[0031] S1. Based on the materials of sub - wavelength resonators, construct a material parameter model of an N - layer sub - wavelength resonator structure, specifically as follows:

[0032] Use density and bulk modulus to characterize the material parameters of the N - layer sub - wavelength resonator, and obtain the material parameter model; where the construction of any N - layer materials changes layer - by - layer alternately, and the sub - wavelength resonator is nested in the background matrix; introduce two dimensionless contrast parameters, which are used to represent the ratios of the density and bulk modulus of the sub - wavelength resonator to those of the background matrix respectively.

[0033] The density is expressed by the following formula: where x is the spatial position; is the resonator density; is the background matrix density; is the sub - wavelength resonator; is the background matrix; The bulk modulus is expressed by the following formula: where, is the resonator bulk modulus; is the background matrix bulk modulus; is the inner radius of the j - th layer resonator; is the outer radius; The wave speed of the sub - wavelength resonator is expressed by the following formula: ;

[0034] The wave speed of the background matrix is expressed by the following formula: ;

[0035] The wave number of the sub - wavelength resonator is expressed by the following formula: ;

[0036] where, is the angular frequency; The wave number of the background matrix is expressed by the following formula: ;

[0037] The background matrix is expressed by the following formula: The dimensionless contrast parameters include the first contrast parameter and the second contrast parameter , and are expressed by the following formula: and , The value of is a bounded positive constant, indicating that the wave speeds in the resonator and the background matrix are comparable.

[0038] S2. Based on the material parameter model of the N-layer subwavelength resonator structure obtained in step S1, construct the capacitance matrix of the N-layer subwavelength resonator based on the Dirichlet-to-Neumann operator method, specifically as follows:

[0039] Based on the material parameter model obtained in step S1, for material structures with any number of layers and thicknesses, according to the boundary conditions satisfied by the propagation of sound waves in the material structure, characterize the resonance phenomenon of the N-layer subwavelength resonator based on the Dirichlet-to-Neumann operator method, and obtain the capacitance matrix related to the number of material layers and thicknesses.

[0040] The capacitance matrix is a tridiagonal positive definite matrix and is represented by the following formula: where, is the outer radius of the j-th layer of material, and the value of j is 1, 2, 3,..., N; is the inner radius of the j-th layer of material, and the value of j is 1, 2, 3,..., N - 1.

[0041] S3. Based on the capacitance matrix of the N-layer subwavelength resonator obtained in step S2, obtain the eigenvalues of the capacitance matrix based on the characteristics of the tridiagonal capacitance matrix, specifically as follows:

[0042] Based on the capacitance matrix obtained in step S2, since the capacitance matrix is a tridiagonal positive definite matrix, according to the positive definiteness and irreducibility of the tridiagonal positive definite matrix, obtain N different eigenvalues of the capacitance matrix , and .

[0043] S4. Based on the eigenvalues of the capacitance matrix obtained in step S3, obtain the parameter selection model corresponding to the resonance frequency of the N-layer subwavelength resonator by solving the generalized capacitance eigenvalue problem, specifically as follows:

[0044] According to the eigenvalues of the capacitance matrix obtained in step S3, solve the generalized capacitance eigenvalue problem to obtain the parameter selection model corresponding to all subwavelength resonance frequencies in the material structure of any N-layer subwavelength resonator.

[0045] The generalized capacitance eigenvalue problem is represented by the following formula: where, is the i-th eigenvalue; is the i-th eigenvector; V is a diagonal matrix represented as , and ; The eigenvectors form an orthogonal basis with respect to the following inner product and are represented by the following formula: where, is the Kronecker symbol.

[0046] The parameter selection model corresponding to all sub - wavelength resonance frequencies in the material structure of any N - layer sub - wavelength resonator is represented by the following formula: Wherein, is the i - th sub - wavelength resonance frequency; is the first component of the i - th eigenvector; is the contrast parameter in step S1; is a constant, and its value is a bounded positive constant.

[0047] S5. According to the model of the resonance frequency and corresponding parameters of the N - layer sub - wavelength resonator obtained in step S4, material parameter selection is carried out to realize the construction of the N - layer sub - wavelength resonator corresponding to the selected resonance frequency. Specifically:

[0048] According to the parameter selection model obtained in step S4, calculate the material parameters of the N - layer sub - wavelength resonator with the preset target resonance frequency. Construct the N - layer sub - wavelength resonator according to the obtained material parameters and verify its resonance frequency. Finally, adjust the parameters according to the preset application scenario to complete the construction of the multi - layer sub - wavelength resonator.

[0049] The technical solution of the present invention will be described below in conjunction with an embodiment:

[0050] In this embodiment, the spherical wave expansion method and the solution of the present invention are used to construct the N - layer sub - wavelength resonator simultaneously. Specifically:

[0051] When using the spherical wave expansion method to calculate the resonance frequency of the N - layer sub - wavelength resonator, it is necessary to calculate the eigenvalues of a 4N - order matrix composed of spherical functions and apply a root - finding method (such as Muller's method) to solve the complex roots, which usually requires multiple iterations to converge.

[0052] For each resonance frequency, usually 10 - 20 iterations are required to ensure the accuracy of root - value solution. Since the convergence of the root - value method is affected by the initial value selection, the number of construction times for each calculation may be different.

[0053] Using the spherical wave expansion method, it takes 2.740 seconds to calculate the resonance frequency of a 15 - layer structure; it takes 32.817 seconds to calculate the resonance frequency of a 45 - layer structure. The calculation time increases significantly with the increase of the number of structure layers. The calculation cost is relatively high, especially when multiple iterations are required. The time and resource consumption for each calculation are relatively large.

[0054] The method of the present invention directly calculates the resonance frequency through the eigenvalues of the N - order tridiagonal generalized capacitance matrix. Only one calculation of eigenvalues is required, that is, only 1 operation is needed for the calculation of each resonance frequency. Using the method of the present invention to calculate the resonance frequency of a 15 - layer structure only takes 0.005 seconds; calculating the resonance frequency of a 45 - layer structure only takes 0.009 seconds. The calculation cost is significantly lower than the existing methods. And as the number of structure layers increases, the increase in calculation time is very small, which is suitable for rapid calculation and large - scale structures.

[0055] The comparison results of the two methods are shown in Table 1.

[0056] As can be seen from Table 1, the existing spherical wave expansion method needs to perform 10N - 20N iterations to calculate the resonance frequency of the N - layer sub - wavelength resonator. As the number of structure layers increases, the calculation time and cost will increase significantly. It has relatively high stability, but the choice of initial value has a certain impact on the result, which may lead to non - convergence or slow convergence of the calculation. While the method of the present invention only needs to perform 1 calculation for each resonance frequency. The calculation time is hardly affected by the number of layers and is very suitable for large - scale structures. The stability is very high because it does not rely on the root - finding process and the calculation results are accurate. The method of the present invention has significant advantages in terms of calculation speed, stability and cost. Especially when dealing with large - scale structures, it can greatly improve the efficiency and reduce the consumption of calculation resources. This makes the method of this application very suitable for engineering applications that require a large number of rapid calculations.

Claims

1. A method for constructing a multi-layer sub-wavelength resonator, characterized in that: The following steps are involved: S1. Based on the material of subwavelength resonator, a material parameter model of N-layer subwavelength resonator structure is constructed; S2. constructing a capacitance matrix of the N-layer subwavelength resonator based on the material parameter model of the N-layer subwavelength resonator structure obtained in step S1 and based on the Dirichlet-to-Neumann operator method; S3. Obtaining the capacitance matrix eigenvalues ​​of the capacitance matrix based on the characteristics of the tridiagonal capacitance matrix of the N-layer sub-wavelength resonator obtained in step S2; S4. According to the capacitance matrix eigenvalues ​​obtained in step S3, by solving the generalized capacitance eigenvalue problem, the parameter selection model corresponding to the resonant frequency of the N-layer sub-wavelength resonator is obtained; S5. According to the model of the resonant frequency and corresponding parameters of the N-layer sub-wavelength resonator obtained in step S4, material parameters are selected to achieve the structure of the N-layer sub-wavelength resonator corresponding to the selected resonant frequency; Step S2 specifically includes: based on the material parameter model obtained in step S1, for material results with any number of layers and thickness, the boundary conditions satisfied by the propagation of sound waves in the material structure are used to characterize the resonance phenomenon of N-layer subwavelength resonators according to the Dirichlet-to-Neumann operator method, and a capacitance matrix related to the number of material layers and thickness is obtained; The capacitance matrix is ​​a tridiagonal positive definite matrix, which is represented by the following formula: in, is the outer radius of the j-th layer of material; is the inner radius of the jth layer of material.

2. The method for constructing a multi-layer sub-wavelength resonator according to claim 1, characterized in that: Step S1 is specifically: using density and bulk modulus To characterize the material parameters of N-layer subwavelength resonators, a material parameter model is obtained; wherein any N-layer material structure is alternating layer by layer, and the subwavelength resonator Nested in a background matrix, where is the outer radius of the j-th layer of material; is the inner radius of the jth layer of material; two dimensionless contrast parameters are introduced to represent the ratio of the density to bulk modulus of the subwavelength resonator and the density to bulk modulus of the background matrix.

3. The method for constructing a multi-layer sub-wavelength resonator according to claim 2, characterized in that: The density Use the following formula to express it: Where x is the spatial position; is the resonator density; is the background matrix density; It is a sub-wavelength resonator; is the background matrix; the bulk modulus Use the following formula to express it: in, is the resonator bulk modulus; is the background matrix bulk modulus; is the inner radius of the j-th layer resonator; is the outer radius; the wave velocity of the subwavelength resonator Use the following formula to express it: ; The wave velocity of the background matrix Use the following formula to express it: ; The wave number of the subwavelength resonator Use the following formula to express it: ; in, is the angular frequency; the wave number of the background matrix Use the following formula to express it: ; The background matrix is ​​represented by the following formula: in, is the outer radius of the j-th layer of material; is the inner radius of the jth layer of material.

4. The method for constructing a multi-layer sub-wavelength resonator according to claim 2, characterized in that: The dimensionless contrast parameter includes a first contrast parameter and the second contrast parameter , expressed using the following formula: , ,and , The value of is a bounded positive constant, indicating that the wave speeds in the resonator and the background matrix are comparable.

5. The method for constructing a multi-layer sub-wavelength resonator according to claim 1, characterized in that: Step S3 is specifically as follows: based on the capacitance matrix obtained in step S2, since the capacitance matrix is ​​a tridiagonal positive definite matrix, according to the positive definiteness and non-simplicity of the tridiagonal positive definite matrix, N different eigenvalues ​​of the capacitance matrix are obtained. ,and .

6. The method for constructing a multi-layer sub-wavelength resonator according to claim 1, characterized in that: Step S4 specifically includes: solving the generalized capacitance eigenvalue problem according to the eigenvalue of the capacitance matrix obtained in step S3, and obtaining a parameter selection model corresponding to all sub-wavelength resonance frequencies in the material structure of any N-layer sub-wavelength resonator.

7. The method for constructing a multi-layer sub-wavelength resonator according to claim 6, characterized in that: The generalized capacitance eigenvalue problem is expressed using the following formula: ,in, is the i-th eigenvalue; is the i-th eigenvector; V is a diagonal matrix expressed as ,and ; The eigenvectors form an orthogonal basis with respect to the following inner product, expressed using the following formula: in, is the Kronecker symbol; The parameter selection model corresponding to all sub-wavelength resonant frequencies in the material structure of any N-layer sub-wavelength resonator is expressed by the following formula: in, is the i-th sub-wavelength resonance frequency; is the first component of the i-th eigenvector; is the contrast parameter in step S1; is a constant whose value is a bounded positive number.

8. The method for constructing a multi-layer sub-wavelength resonator according to claim 1, characterized in that: Step S5 is specifically as follows: a model is selected according to the parameters obtained in step S4, material parameters of an N-layer subwavelength resonator with a preset target resonance frequency are calculated, an N-layer subwavelength resonator is constructed according to the obtained material parameters and its resonance frequency is verified, and finally the parameters are adjusted according to a preset application scenario to complete the construction of a multilayer subwavelength resonator.

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