Optimization design method of reflective broadband achromatic focusing elastic metasurface

Abstract

The invention discloses an optimization design method of a reflective broadband achromatic focusing elastic metasurface, and belongs to the field of vibration and noise control. Aiming at the problems of narrow bandwidth, serious dispersion and the like of the existing focusing elastic metasurface, the invention provides an optimal design method of a reflective broadband achromatic elastic metasurface for realizing elastic longitudinal wave focusing by combining a variable density method based on a wave control principle. According to the method, a calculation model of a metasurface unit is established, a mode of efficiently and accurately calculating a reflection coefficient R of the metasurface unit based on a transfer matrix method is established, and a topological optimization formula for designing and generating the broadband achromatic focusing elastic metasurface unit is constructed based on a real part and an imaginary part of the reflection coefficient R. According to the method, manual participation is not needed, customization according to needs can be achieved, the designed metasurface unit has high efficiency and precision in a broadband, and design of the broadband achromatic elastic metasurface is achieved. The invention is especially suitable for the engineering fields of energy acquisition, wave sensing, communication, nondestructive testing and the like.

Description

Technical Field

[0001] The invention relates to an optimization design method for a reflective broadband achromatic focusing elastic metasurface, and belongs to the field of vibration and noise control. Background Art

[0002] The manipulation of elastic waves is an important topic because of its important applications in vibration reduction, ultrasonic imaging, energy harvesting, nondestructive testing, structural health testing, etc. Metasurfaces offer great possibilities for the manipulation of elastic waves due to their extraordinary wavefront shaping capabilities. Regardless of the specific function, the core of elastic metasurface design is to construct elements with unique refractive index and impedance to customize the amplitude and phase shift curves of transmitted or reflected waves. However, most elastic metasurfaces suffer from large chromatic aberrations due to the inherent velocity dispersion characteristics of elastic waves and the dispersion of the accumulated phase during propagation. The ubiquitous chromatic aberration of metasurfaces leads to problems of narrow bandwidth and dispersed functions, limiting practical engineering applications.

[0003] In order to achieve arbitrarily customized broadband achromatic functionality, each metasurface unit must provide the required local amplitude or phase modulation over the entire working bandwidth. Although optical and acoustic achromatic processing schemes have been proposed, ultra-wideband achromatic processing of elastic waves is more challenging due to its complex physical properties. Some coded or tunable metasurfaces have been proposed that increase the bandwidth by adjusting the configuration of subunits. However, these methods require manual changes in the geometry of the structure or require external elements to control the state of the system, which is difficult to implement in practical applications or is expensive.

[0004] Topology optimization has the highest degree of design freedom among structural optimization methods. It is not only applicable to static problems represented by the problem of maximizing the stiffness of linear elastic bodies, but also to dynamic problems including the problem of elastic wave propagation. Therefore, the present invention develops an optimization design method for a reflective broadband achromatic elastic metasurface suitable for elastic wave manipulation based on the variable density method, which can efficiently reverse-design elastic metasurfaces with different characteristics according to wave control requirements. The present invention provides a new paradigm for the design of broadband achromatic metasurfaces, which overcomes the limitations of traditional intuitive design methods, breaks through narrowband limitations, and can arbitrarily design elastic metasurface units with customized functions. The optimization design method provided by the present invention has important theoretical significance and broad engineering application value for the design of various broadband achromatic components. Summary of the invention

[0005] The purpose of the present invention is to provide an optimization design method for a reflective broadband achromatic focusing elastic metasurface that can achieve precise control of elastic waves in view of the deficiencies of the prior art. The optimization design method provided by the present invention is based on the generalized Snell's law and uses a topology optimization method based on a variable density method to reversely independently design a metasurface unit with a target phase, thereby achieving precise control of elastic waves.

[0006] In order to achieve the above object, the present invention adopts the following technical solution:

[0007] A method for optimizing the design of a reflective broadband achromatic focusing elastic metasurface, the method comprising the steps of:

[0008] (1) Determine the desired elastic longitudinal wave control function.

[0009] (2) Determine the phase profile that the broadband achromatic metasurface needs to provide to achieve the desired wave control function. Based on the generalized Snell's law, the phase distribution provided by the broadband achromatic metasurface used to focus the incident elastic longitudinal wave at a specified position conforms to the following profile:

[0010]

[0011] in, represents the phase, y represents the ordinate, ω represents the circular frequency of the longitudinal wave, c represents the longitudinal wave velocity, and F0 represents the focal length.

[0012] (3) Discretize the broadband achromatic metasurface into metasurface units of the same width, and determine the specific value corresponding to the phase profile in step (2), i.e., the target phase value, according to the ordinate y of the center position of each metasurface unit.

[0013] (4) A finite element calculation model of the super surface unit is established. The control equation of the finite element calculation model is expressed as:

[0014] (K P -ω 2 M P ) P =F P

[0015] Where:

[0016] K P =Λ T KΛ,M P =Λ T MΛ,F P =Λ T F

[0017] Where: K represents the overall stiffness matrix, M represents the overall mass matrix, F represents the exciting force vector, UP represents a periodic displacement field.

[0018] The specific form of matrix Λ is:

[0019]

[0020] Among them, I U , I I , I L Represents the identity matrix of the corresponding dimension.

[0021] (5) Periodic displacement field U calculated based on the finite element model P , using the transfer matrix method, the reflection coefficient R of the metasurface unit is solved as:

[0022]

[0023] Where Z1 represents the mechanical impedance, T 21 、T 22 represents the element in the transfer matrix, ω represents the circular frequency of the longitudinal wave; i explains the imaginary unit.

[0024] (6) The topology optimization formula is constructed using the real and imaginary parts of the reflection coefficient R:

[0025] For: i ∈[f min ,f max ](i=1,2,…,N f )

[0026]

[0027] Where:

[0028]

[0029] Among them, f i represents the longitudinal wave frequency, μ represents the design variable, F(μ) represents the objective function, G(μ) represents the constraint function, Re(R) represents the real part of R, Im(R) represents the imaginary part of R, |R| represents the modulus of R, Represents the target phase corresponding to the i-th metasurface unit.

[0030] (7) Solve the sensitivity of the objective function and constraint function in the topology optimization formula obtained in step (6).

[0031] (8) Based on the variable density method, the sensitivity of step (7) is utilized and the moving asymptote method (MMA) is used to update the design variables to obtain the metasurface unit that meets the target phase value.

[0032] The elastic metasurface specifically refers to a phase gradient metasurface, which can realize wave control functions including abnormal deflection, focusing, beam splitting, etc.

[0033] Beneficial Effects

[0034] The present invention provides an optimization design method for a reflective broadband achromatic elastic metasurface for realizing elastic longitudinal wave focusing. The method adopts a topological optimization method based on a variable density method to perform independent reverse design of a metasurface unit. The designed elastic metasurface can convert an incident elastic longitudinal wave field into a desired reflected longitudinal wave field within a broadband range. The optimization design provided by the present invention does not require manual participation and can be customized on demand. Its beneficial effects mainly include:

[0035] 1. In step (3), an air domain interval with a width of 1 mm is set between discrete metasurface units to effectively reduce the vibration coupling between metasurface units.

[0036] 2. In step (4), according to the characteristics of the finite element calculation model of the super surface unit, the periodic boundary conditions are introduced into the control equation by constructing the matrix Λ to simplify the model and improve the calculation efficiency.

[0037] 3. In step (6), the real and imaginary parts of the reflection coefficient R calculated using the transfer matrix method are used to construct the topology optimization formula to avoid the increased complexity of the sensitivity solution caused by the non-uniqueness of the phase value.

[0038] 4. In step (7), the sensitivity is solved by combining the direct difference method and the adjoint method, avoiding the complexity of solving the sensitivity by the direct difference method.

[0039] 5. In step (8), the variable density method is adopted, and the SIMP interpolation method and Heaviside projection method are used to facilitate the stability of the optimization process.

[0040] 6. In step (8), the sensitivity of analytical calculation is used to update the design variables through the moving asymptote method (MMA), which significantly improves the optimization efficiency, saves calculation cost, and is suitable for the calculation of large models.

[0041] The optimization design method provided by the present invention will promote the practical application of elastic wave metasurfaces in engineering and has important engineering application value in the design of various broadband achromatic components. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] The drawings described herein are used to provide a further understanding of the present invention and constitute a part of the present invention. The schematic implementation examples of the present invention and their descriptions are used to explain the present invention and do not constitute an improper limitation of the present invention. In the drawings:

[0043] Figure 1A schematic flow chart of an optimization design method for a reflective broadband achromatic elastic metasurface for realizing elastic longitudinal wave focusing is provided for the present invention;

[0044] Figure 2 Schematic diagram of the finite element model of the incident wave field and the reflected wave field in the elastic medium;

[0045] Figure 3 A theoretical model for describing the transfer matrix method to calculate the phase delay of metasurface units;

[0046] Figure 4 A finite element discretization model is developed to describe the transfer matrix method for calculating the phase delay of the metasurface unit;

[0047] Figure 5 15 elastic metasurface unit structures designed for realizing elastic longitudinal wave ultra-wideband achromatic focusing optimization in the embodiment of the present invention;

[0048] Figure 6 The actual phase provided by the target phase profile and the optimized metasurface unit for realizing ultra-wideband achromatic focusing of elastic longitudinal waves in the examples of the present invention;

[0049] Figure 7 This is a picture of the wave control effect of broadband achromatic focusing achieved by the reflective broadband achromatic focusing elastic metasurface optimized and designed in the implementation example of the present invention. DETAILED DESCRIPTION

[0050] The present invention is further described in detail below with reference to embodiments, but is not intended to limit the present invention.

[0051] like Figure 1 As shown, a schematic diagram of a flow chart of an optimization design method for a reflective broadband achromatic focusing elastic metasurface disclosed in this embodiment, the specific implementation steps are as follows:

[0052] 1. In this embodiment, the optimization design method proposed in the present invention is applied to realize broadband achromatic focusing of elastic longitudinal waves. In this embodiment, the frequency range is selected as 80 to 120 kHz. Based on the generalized Snell's law in step (2), the phase distribution provided by the broadband achromatic metasurface for focusing the incident elastic longitudinal wave at a specified position should conform to a specific profile, namely:

[0053]

[0054] in, represents the phase, y represents the ordinate, ω represents the circular frequency of the longitudinal wave, c represents the longitudinal wave velocity, and F0 represents the focal length. In this embodiment, the focal point is (-0.30m, 0.1645m).

[0055] 2. In this implementation example, the metasurface is discretized into 30 metasurface units with a width of 10 mm (due to the symmetry of the structure, only 15 metasurface units need to be optimized in practice), and an air domain interval with a width of 1 mm is set between adjacent units. According to the ordinate y of the center position of each metasurface unit, its specific value corresponding to the phase profile, i.e., the target phase value, is determined.

[0056] 3. Establish the finite element calculation model of the super surface unit. Figure 2 The figure shows a schematic diagram of the finite element calculation model of the metasurface unit. The model size is 10mm*310mm, the excitation position is set at x=-240mm, the upper and lower boundaries are set with periodic boundary conditions, and the left and right boundaries are set with free boundaries. For the design of each unit cell, the finite element discretization is performed through a 4-node plane stress linear unit, which is divided into 50*50 rectangular units, and the size of each rectangular unit is 0.2mm*0.2mm. Aluminum is used as the matrix material of the design domain and the non-design domain, and its material elastic modulus is 70GPa and the density is 2700kg / m 3 , Poisson’s ratio is 0.33; to avoid numerical singularities, smaller non-zero material property values ​​are used to represent the voids, with a material elastic modulus of 10 MPa and a density of 0.0001 kg / m 3 , the Poisson's ratio is 0.33.

[0057] The control equation of the established finite element calculation model is expressed as:

[0058] (K P -ω 2 M P ) P =F P

[0059] Where:

[0060] K P =Λ T KΛ,M P =Λ T MΛ,F P =Λ T F

[0061] Where: K represents the overall stiffness matrix, M represents the overall mass matrix, F represents the exciting force vector, U P represents the periodic displacement field;

[0062] The specific form of matrix Λ is:

[0063]

[0064] Among them, I U , I I , I LRepresents the identity matrix of the corresponding dimension.

[0065] Based on the established finite element calculation model of the metasurface unit, the periodic displacement field U can be calculated using the control equation P .

[0066] 4. Based on the periodic displacement field U calculated in the previous step P , use the transfer matrix method to solve the reflection coefficient R.

[0067] Figure 3 The theoretical model describing the use of the transfer matrix method to calculate the phase delay of the metasurface unit is shown in FIG. The transfer matrix T is used to establish the metasurface unit x = L x The force F on the boundary with x=0 x , displacement U x The relationship between:

[0068]

[0069] Figure 4 To describe the finite element discretization model for calculating the phase delay of the metasurface unit using the transfer matrix method, based on this model, the metasurface unit x = L x The force F on the boundary with x=0 x , displacement U x It can be solved as:

[0070]

[0071] Where Ly represents the side length of the hypersurface unit, u x , f x Represents the periodic displacement field U P Computed forces and displacements at the nodes of a discrete finite element.

[0072] Using the hypersurface unit x = L x The force F on the boundary with x=0 x , displacement U x , each element in the transfer matrix T can be expressed as:

[0073]

[0074] Using the elements in the transfer matrix, the reflection coefficient of a cell of the metasurface can be expressed as:

[0075]

[0076] Wherein, Z1 represents the mechanical impedance, ω represents the circular frequency of the longitudinal wave, and i represents the imaginary unit.

[0077] 5. Based on the real and imaginary parts of the reflection coefficient R calculated in the previous step, the topology optimization formula is constructed as follows:

[0078] For: i ∈[f min ,f max ](i=1,2,…,N f )

[0079]

[0080] Where:

[0081]

[0082] Among them, f i represents the longitudinal wave frequency, μ represents the design variable, F(μ) represents the objective function, G(μ) represents the constraint function, Re(R) represents the real part of R, Im(R) represents the imaginary part of R, |R| represents the modulus of R, Represents the target phase corresponding to the i-th metasurface unit.

[0083] 6. Solve the sensitivity of the objective function and constraint function in the topology optimization formula obtained in the previous step. First, use the direct difference method to solve. The sensitivity of the objective function and constraint function can be expressed as:

[0084]

[0085] Using the chain rule, we can further solve Re(R i ) and Im(R i ) sensitivity to the design variable μ:

[0086]

[0087] In the above formula, we also need to solve T 21 、T 22 Sensitivity to design variable μ:

[0088]

[0089] in

[0090]

[0091] Next, we need to solve for dF x | x＝x’ / dμ,dU x | x＝x’ / dμ, using the chain rule:

[0092]

[0093] Considering the complexity of directly solving dU / dμ, we consider using the adjoint method to solve it. First, the control equation of the finite element calculation model is differentiated with respect to the design variable μ:

[0094]

[0095] The above formula is transformed into:

[0096]

[0097] because

[0098]

[0099] At this time, dF x | x＝x’ / dμ,dU x | x＝x’ / dμ can be solved as:

[0100]

[0101] in

[0102]

[0103] dF x | x＝x’ / dμ,dU x | x＝x’ The expression of / dμ is simplified to:

[0104]

[0105] Among them, the S and R matrices respectively establish F x | x＝x’ , U x | x＝x’ The relationship between and displacement field U is:

[0106] F x | x＝x' =S T U

[0107] U x | x＝x' =R T U

[0108] Introducing the accompanying variable λ F and λ U :

[0109]

[0110] dF x | x＝x’ / dμ,dU x |x＝x’ The expression of / dμ is finally simplified to

[0111]

[0112] In summary, by combining the direct difference method with the adjoint variable method, the sensitivity of the analytical calculation of the objective function and the constraint function is obtained.

[0113] 7. Use variable density method and SIMP interpolation method:

[0114]

[0115] ρ e =μ e ρ solid +(1-μ e )ρ void

[0116] Use the Heaviside filter:

[0117]

[0118] Among them, η represents the projection level, and β represents the projection intensity. In this example, η is set to 0.5, β takes an initial value of 1, and becomes twice the original value every 50 iterations, and the maximum value is set to 4096.

[0119] In order to update the design variables, the moving asymptote method (MMA) is used. The MMA algorithm is a prior art well known to professionals in the field, and its detailed iterative solution process is not described in detail here.

[0120] result

[0121] like Figure 5 The figure shows 15 elastic metasurface unit structures optimized for realizing elastic longitudinal wave ultra-wideband achromatic focusing in the embodiment of the present invention. By arranging them in the order of #1 to #15 and performing symmetric operations, they can be assembled into a reflective broadband achromatic focusing elastic metasurface containing 30 elastic metasurface units.

[0122] Figure 6 The target phase profile required for realizing ultra-wideband achromatic focusing of elastic longitudinal waves and the actual phase provided by the optimized metasurface unit in the implementation example of the present invention can be found that the actual phase provided by the elastic metasurface unit obtained by executing the optimization design method in the present invention can well match the target phase profile required by the theory.

[0123] Figure 7The wave control effect diagram of the ultra-wideband achromatic focusing of elastic longitudinal waves realized by the reflective broadband achromatic focusing elastic metasurface optimized in the implementation example of the present invention. It can be found that within the set frequency range of 80 to 120 kHz, the normally incident elastic longitudinal wave is reflected and focused at a constant position after being incident on the elastic metasurface obtained by the optimized design, showing a stable broadband achromatic focusing characteristic, which verifies the effectiveness of the reflective broadband achromatic elastic metasurface optimization design method for elastic wave control provided by the present invention.

[0124] In summary, the optimization design method of a reflective broadband achromatic focusing elastic metasurface disclosed in this embodiment is based on the generalized Snell law. The designed elastic metasurface can perfectly convert the incident elastic longitudinal wave field into the desired reflected wave field in a broadband range; the topological optimization method based on the variable density method is used to optimize the design of the metasurface unit, without manual participation, and on-demand customization is achieved; an air domain interval is set between discrete metasurface units, which effectively reduces the vibration coupling between the metasurface units; the real and imaginary parts of the reflection coefficient R calculated by the transfer matrix method are used to construct the topological optimization column formula, which effectively avoids the complexity of the sensitivity solution increased by the non-uniqueness of the phase value; the SIMP material interpolation method and the Heaviside projection method are used to facilitate the stability of the optimization process; the sensitivity of the analytical calculation is used to update the design variables by the moving asymptote method (MMA), which significantly improves the optimization efficiency, saves the calculation cost, and is suitable for the calculation of large models. The optimization design method provided by the present invention will promote the practical application of elastic wave metasurfaces in engineering, and has important theoretical significance and extensive engineering application value in the design of various broadband achromatic components.

[0125] The contents not described in detail in this specification belong to the prior art known to professional and technical personnel in this field.

[0126] The above description shows and describes the preferred embodiments of the present invention, but it should be understood that the present invention is not limited to the form disclosed herein. Modifications and changes made by those skilled in the art should be within the scope of protection of the appended claims if they do not depart from the spirit and scope of the present invention.

Claims

1. An optimization design method for a reflective broadband achromatic focusing elastic metasurface, characterized by: The following steps are included: (1) Determine the desired elastic longitudinal wave control function; (2) Determine the phase profile required for the broadband achromatic metasurface to achieve the desired wave control function based on the generalized Snell's law; (3) discretizing the broadband achromatic metasurface into metasurface units with the same width, and determining the target phase value in the phase profile of step (2) corresponding to each metasurface unit; (4) Establish a finite element calculation model of the metasurface unit; (5) Based on the periodic displacement field calculated by the finite element model, the reflection coefficient R of the metasurface unit is calculated using the transfer matrix method; (6) Use the real and imaginary parts of the reflection coefficient R to construct a topology optimization formula; (7) solving the sensitivity of the objective function and constraint function in the topology optimization formula obtained in step (6); (8) Based on the variable density method and utilizing the sensitivity of step (7), the moving asymptote method (MMA) is used to update the design variables and obtain the metasurface unit that meets the target phase value, thereby realizing the optimization of the reflective broadband achromatic focusing elastic metasurface.

2. The optimization design method of the reflective broadband achromatic focusing elastic metasurface according to claim 1, characterized in that: The specific implementation method of step (2) is: based on the generalized Snell's law, the phase distribution provided by the broadband achromatic metasurface for focusing the incident elastic longitudinal wave at a specified position meets the following profile requirements, namely: in, represents the phase, y represents the ordinate, ω represents the circular frequency of the longitudinal wave, c represents the longitudinal wave velocity, and F0 represents the focal length.

3. The optimization design method of the reflective broadband achromatic focusing elastic metasurface according to claim 1, characterized in that: In step (3), there is an air domain gap with a width of 1 mm between adjacent discrete metasurface units.

4. The optimization design method of the reflective broadband achromatic focusing elastic metasurface according to claim 1, characterized in that: In step (3), the specific value corresponding to the phase profile in step (2), i.e., the target phase value, is determined according to the ordinate y of the center position of each metasurface unit.

5. The optimization design method of the reflective broadband achromatic focusing elastic metasurface according to claim 1, characterized in that: The implementation method of step (4) is as follows: the control equation of the finite element calculation model of the super surface unit is established as follows: (K P -ω 2 M P )U P =F P Where: K P =L T KL, M P =Λ T MΛ, F P =L T F Where: K represents the overall stiffness matrix, M represents the overall mass matrix, F represents the exciting force vector, U P represents the periodic displacement field; The specific form of matrix Λ is: Among them, I U , I I , I L Represents the identity matrix of the corresponding dimension.

6. The optimization design method of the reflective broadband achromatic focusing elastic metasurface according to claim 1, characterized in that: The implementation method of step (5) is as follows: Based on the periodic displacement field calculated by the finite element model, the reflection coefficient R is solved using the transfer matrix method as follows: Where Z1 represents the mechanical impedance, T 21 , T 22 represents the element in the transfer matrix, ω represents the circular frequency of the longitudinal wave; i explains the imaginary unit.

7. The optimization design method of the reflective broadband achromatic focusing elastic metasurface according to claim 1, characterized in that: The implementation method in step (6) is: using the real part and imaginary part of the reflection coefficient R obtained in step (5) to construct the topology optimization formula: For:f i ∈[f min ,f max ](i=1,2,…,N f ) Where: Among them, f i represents the longitudinal wave frequency, μ represents the design variable, F(μ) represents the objective function, G(μ) represents the constraint function, Re(R) represents the real part of R, Im(R) represents the imaginary part of R, |R| represents the modulus of R, Represents the target phase corresponding to the i-th metasurface unit.

8. The optimization design method of the reflective broadband achromatic focusing elastic metasurface according to claim 1, characterized in that: The elastic metasurface specifically refers to a phase gradient metasurface, which can realize wave control functions including abnormal deflection, focusing, and beam splitting.

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