Construction method of multi-layer structure sub-wavelength harmonic oscillator

Abstract

The invention discloses a construction method of a multi-layer structure sub-wavelength harmonic oscillator, and the method comprises the following steps: constructing a material parameter model of an N-layer sub-wavelength harmonic oscillator structure based on the material of the sub-wavelength harmonic oscillator; constructing a capacitance matrix of the N layers of sub-wavelength harmonic oscillators according to the material parameter model of the N layers of sub-wavelength harmonic oscillator structures; obtaining a capacitance matrix characteristic value according to the capacitance matrix of the N layers of sub-wavelength harmonic oscillators; obtaining a parameter selection model corresponding to the resonant frequency of the N layers of sub-wavelength harmonic oscillators according to the capacitance matrix eigenvalues; and selecting material parameters according to the resonant frequency of the N layers of sub-wavelength harmonic oscillators and the model of the corresponding parameters to realize the construction of the N layers of sub-wavelength harmonic oscillators corresponding to the selected resonant frequency. The design accuracy and efficiency of the multi-layer structure sub-wavelength harmonic oscillator are improved, waste of repeated test resources and time is reduced, implementation of customized application of the harmonic oscillator is promoted, and a foundation is laid for customized application of the harmonic oscillator in the fields of optics, acoustics and other fields.

Description

Technical Field

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

[0002] Phononic crystals are a new type of functional material formed by periodic arrangement of elastic solids in another solid or fluid medium. By analogy with photonic crystals, people found that when elastic waves propagate in periodic elastic composite media, elastic wave band gaps similar to photonic band gaps will also be generated, thus proposing the concept of phononic crystals.

[0003] In the study of phononic crystals, most structures are usually composed 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 band gaps. In particular, multilayer materials are ideal for subwavelength resonators due to their excellent tunability and high-quality resonance characteristics. Most existing research methods experimentally explore materials with specific multilayer structures to verify whether they can achieve subwavelength resonance. However, this experimental method lacks universality and cannot guide the design of material parameters in advance. It can only be adjusted through continuous experimental attempts. When the number of material layers increases, the limitations of this experimental method become particularly obvious because it provides limited prior information. Summary of the invention

[0004] The purpose of the present invention is to provide a method for constructing a multi-layer subwavelength resonator. The method adopts theoretical analysis to characterize the resonant frequency that causes the subwavelength resonance phenomenon when the material structure parameters of any number of layers are selected, thereby promoting the selection of material parameters, thereby inducing the desired resonance and realizing the construction of a multi-layer subwavelength resonator.

[0005] The present invention provides a method for constructing a multi-layer sub-wavelength resonator, comprising the following steps: 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 resonance frequency and corresponding parameters of the N-layer sub-wavelength resonator obtained in step S4, material parameters are selected to realize the construction of the N-layer sub-wavelength resonator corresponding to the selected resonance frequency.

[0006] 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 changed alternately layer by layer, and the subwavelength resonator Nested in the background matrix; 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.

[0007] 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: 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.

[0008] Step S2 is specifically as follows: 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 the capacitance matrix related to the number of material layers and thickness is obtained.

[0009] The capacitance matrix is ​​a tridiagonal positive definite matrix C, which is represented by the following formula: in, 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.

[0010] 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 .

[0011] 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.

[0012] 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.

[0013] 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 constant.

[0014] 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.

[0015] The present invention discloses a construction method of a multi-layer sub-wavelength resonator. Through theoretical analysis and calculation, the resonant frequency that can induce the sub-wavelength resonance phenomenon under the sub-wavelength resonator structure of any specific number of layers is accurately derived, and then the construction of the multi-level wavelength resonator structure is realized by experimental verification. In addition, the present invention can provide a scientific basis for material selection and structural design, thereby greatly improving the accuracy and efficiency of the design. Through advance 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-level wavelength resonators, the number of experiments and the required time can be greatly reduced. The present invention can also greatly reduce the design and production costs of multi-level wavelength resonator materials. Through the combination of precise theoretical guidance and experimental verification, 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 customized applications in optics, acoustics and other fields in the future. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] Figure 1 It is a schematic diagram of the process of the present invention; Figure 2 It is a schematic diagram of the structure of a multilayer sub-wavelength resonator material; Figure 3 It is a resonance mode diagram of a multilayer subwavelength resonator material; Figure 4 This is a diagram of the resonance effect of multilayer subwavelength resonator materials. DETAILED DESCRIPTION

[0017] The present invention provides a method for constructing a multilayer sub-wavelength resonator, the flow diagram of which is shown in FIG. Figure 1 As shown, the following steps are included: The present invention provides a method for constructing a multi-layer sub-wavelength resonator, comprising the following steps: S1. Based on the material of the subwavelength resonator, a material parameter model of an N-layer subwavelength resonator structure is constructed, specifically: Use 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 changed alternately layer by layer, and the subwavelength resonator Nested in the background matrix; 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.

[0018] 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: 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.

[0019] S2. According to the material parameter model of the N-layer subwavelength resonator structure obtained in step S1, based on the Dirichlet-to-Neumann operator method, a capacitance matrix of the N-layer subwavelength resonator is constructed, specifically: 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 the capacitance matrix related to the number of material layers and thickness is obtained.

[0020] 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, 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.

[0021] 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, specifically: 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 .

[0022] S4. According to the capacitance matrix eigenvalues ​​obtained in step S3, by solving the generalized capacitance eigenvalue problem, a parameter selection model corresponding to the resonant frequency of the N-layer sub-wavelength resonator is obtained, specifically: According to the eigenvalues ​​of the capacitance matrix obtained in step S3, the generalized capacitance eigenvalue problem is solved to obtain a parameter selection model corresponding to all sub-wavelength resonance frequencies in the material structure of any N-layer sub-wavelength resonator.

[0023] 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.

[0024] 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 constant.

[0025] 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 realize the construction of the N-layer sub-wavelength resonator corresponding to the selected resonant frequency, specifically: A model is selected based on the parameters obtained in step S4, and material parameters of an N-layer subwavelength resonator with a preset target resonance frequency are calculated. An N-layer subwavelength resonator is constructed based on the obtained material parameters and its resonance frequency is verified. Finally, the parameters are adjusted according to the preset application scenario to complete the construction of a multilayer subwavelength resonator.

[0026] The technical solution of the present invention is described below in conjunction with an embodiment: In this embodiment, the spherical wave expansion method and the solution of the present invention are used to simultaneously construct an N-layer sub-wavelength resonator, specifically: When using the spherical wave expansion method to calculate the resonant frequency of an N-layer subwavelength resonator, it is necessary to calculate the eigenvalues ​​of a 4N-order matrix composed of spherical functions and apply a root value finding method (such as the Muller method) to solve the complex roots, which usually requires multiple iterations to converge.

[0027] For each resonant frequency, 10-20 iterations are usually required to ensure the accuracy of the root value solution. Since the convergence of the root value method is affected by the choice of initial value, the number of constructions for each calculation may vary.

[0028] Using the spherical wave expansion method, it takes 2.740 seconds to calculate the resonant frequency of a 15-layer structure; it takes 32.817 seconds to calculate the resonant frequency of a 45-layer structure. The calculation time increases significantly with the number of layers in the structure. The calculation cost is high, especially when multiple iterations are required. Each calculation consumes a lot of time and resources.

[0029] The method of the present invention is used to directly calculate the resonant frequency through the eigenvalues ​​of the N-order tridiagonal generalized capacitance matrix. It only takes one operation to calculate the eigenvalue, that is, the calculation of each resonant frequency only requires one operation. It only takes 0.005 seconds to calculate the resonant frequency of a 15-layer structure using the method of the present invention; it only takes 0.009 seconds to calculate the resonant frequency of a 45-layer structure. The calculation cost is significantly lower than the existing method. And as the number of structural layers increases, the increase in calculation time is very small, which is suitable for fast calculations and large-scale structures.

[0030] The comparison results of the two methods are shown in Table 1.

[0031] As can be seen from Table 1, the prior art spherical wave expansion method requires 10N-20N iterations to calculate the resonant frequency of an N-layer subwavelength resonator. As the number of structural layers increases, the calculation time and cost will increase significantly. The stability is relatively high, but the initial value selection has a certain impact on the results, which may cause the calculation to not converge or converge slowly. The method of the present invention only requires 1 calculation for each resonant frequency. The calculation time is almost unaffected by the number of layers, which is very suitable for large-scale structures. The stability is very high because it does not rely on the root value finding process, and the calculation results are accurate. The method of the present invention has significant advantages in calculation speed, stability and cost, especially when dealing with large-scale structures, it can greatly improve efficiency and reduce the consumption of computing resources. This makes the application method very suitable for engineering applications that require a large number of fast 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 resonance frequency and corresponding parameters of the N-layer sub-wavelength resonator obtained in step S4, material parameters are selected to realize the construction of the N-layer sub-wavelength resonator corresponding to the selected resonance frequency.

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 the background matrix; 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 S2 is specifically as follows: 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 the capacitance matrix related to the number of material layers and thickness is obtained.

6. The method for constructing a multi-layer sub-wavelength resonator according to claim 5, characterized in that: 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, 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.

7. 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 .

8. 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.

9. The method for constructing a multi-layer sub-wavelength resonator according to claim 8, 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.

10. 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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