Honeycomb-shaped local resonance topological insulator and optimization design method thereof

Abstract

The invention discloses a honeycomb-shaped local resonance topological insulator and an optimization design method thereof, the honeycomb-shaped local resonance topological insulator comprises a honeycomb-shaped substrate provided with C-shaped holes and six pairs of oscillators which are arranged on two sides of a node and have the same radius, energy band inversion is realized by changing the height difference h of the oscillators, and boundary states with different shapes are constructed. The topological transmission has the characteristics of back scattering suppression and channel defect immunity, and on the basis of a particle swarm algorithm, cellular key geometric parameters are optimized, and a topological insulator of a target band gap is customized. According to the honeycomb-shaped local resonance topological insulator, a new thought is provided for bending wave regulation and control, aided design is carried out in combination with the particle swarm optimization algorithm, the regulation and control range of the topological metamaterial on bending waves can be enlarged, and the vibration reduction effect can be improved.

Description

Technical Field

[0001] The present invention relates to a honeycomb-shaped local resonance topological insulator for low-frequency bending wave control transmission and an optimization design method based on a particle swarm algorithm, belonging to the technical field of vibration control. Background Art

[0002] Compared with traditional topological insulators, the local resonance type can realize the design of structural units on a subwavelength scale, making the structural size much smaller than the wavelength of the controlled elastic wave, thereby further miniaturizing the size of the sample when performing low-frequency elastic wave regulation; more importantly, the local resonance system can localize the vibration energy in the oscillator and reduce the vibration on the substrate, which has a natural advantage in realizing topological states with highly localized energy. In recent years, realizing topological boundary states with subwavelength wave regulation characteristics based on the principle of local resonance has gradually become a new research hotspot.

[0003] At present, the research is mainly based on the design of Bragg scatterer structure, focusing on the medium and high frequency range. There are relatively few related studies on resonant topological metamaterial plates suitable for low frequencies. Artificial periodic structures represented by stiffened plates and honeycomb materials with lightweight and high strength are widely used in engineering. They are sensitive to dynamic loads and are more likely to excite the propagation of elastic waves under the action of external vibration sources. The vibrations existing in civil engineering are usually in the low-frequency range. The natural frequency of the structure or vibration source is often below 100Hz, and the vibration caused by traffic is 30-80Hz. Due to its large wavelength and large energy, low-frequency vibration reduction has always been a difficult problem in engineering practice. Therefore, it is necessary to study the control method of low-frequency elastic waves, give full play to the wave control ability of topological insulators, and control the low-frequency elastic waves of civil engineering structures. At the same time, facing the needs of engineering applications, there is a lack of efficient and stable optimization design methods in cellular structure design, and it is difficult to achieve on-demand quantitative design of the working frequency band of topological insulators. Therefore, in order to continuously promote the development of subwavelength elastic wave control technology to meet the actual needs of engineering applications, in-depth exploration of local resonant topological insulators is still of great research value. Summary of the invention

[0004] Technical issues:

[0005] The present invention aims to solve at least one of the technical problems existing in the related art to a certain extent. To this end, the purpose of the present invention is to propose a honeycomb-shaped local resonance topological insulator for low-frequency bending wave control transmission and its optimization design method. The honeycomb-shaped local resonance topological insulator achieves band inversion by changing the oscillator height difference △h, constructs paths with different shapes, realizes the boundary state transmission of low-frequency elastic waves, and based on the particle swarm optimization method, increases the control range of topological metamaterials for bending waves, improves the vibration reduction effect, and provides a new idea for bending wave control.

[0006] Technical solution:

[0007] To achieve the above object, the present invention adopts the following technical solutions:

[0008] A local resonance topological insulator, the insulator has a honeycomb structure, and the cells constituting the insulator include a honeycomb substrate with uniformly provided C-shaped openings and vibrators symmetrically arranged on the upper and lower sides of an inner ring platform of the C-shaped opening; the C-shaped opening is located on a connecting line between a honeycomb corner point and a cell center; the vibrator radius is the same as the inner diameter of the C-shaped opening, and the vibrator heights on the left and right sides of a node where each vibrator is located are the same; and energy band inversion is achieved by synchronously adjusting the height difference between the upper and lower vibrators of the node and the left and right adjacent vibrators.

[0009] Preferably, six C-shaped openings are evenly formed on the honeycomb substrate, and each C-shaped opening is rotated 60 degrees around the center of the cell and overlaps with each other.

[0010] Preferably, when the heights of the adjacent left and right oscillators are the same, the cell has C 6v Symmetry, that is, the structure can overlap after rotating 60 degrees, double degeneracy occurs at the K point in the Brillouin zone, and a Dirac cone appears; when the heights of the left and right adjacent oscillators are different, the cell has C 3v Symmetry, that is, the structure can overlap after rotating 120 degrees, and the Dirac cone at the K point in the Brillouin zone opens to form a band gap.

[0011] Preferably, the lattice constant of the cell is a=100mm, the outer diameter of the opening is R1=14mm, the inner diameter of the opening is R2=7mm, the width of the connecting beam is w=2mm, the height of adjacent oscillators is h1=h2=8mm, the height difference of the oscillators is △h=h1-h2=0, the thickness of the substrate is H=3mm, and the distance between the oscillator and the inner and outer diameters of the opening and the center of the cell is d=a / 3.

[0012] Preferably, the specific steps of achieving band inversion by adjusting the height difference between adjacent left and right oscillators include: when h1=8mm, h2 gradually decreases from 8mm, △h>0, and the insulator is a topologically trivial crystal; when h2=8mm, h1 gradually decreases from 8mm, △h<0, and the insulator is a topologically non-trivial crystal; band inversion is achieved by changing the height difference of the oscillators, and the insulator is converted from a topologically trivial crystal to a topologically non-trivial crystal.

[0013] Preferably, the substrate material is photosensitive resin, and the vibrator material is copper.

[0014] The optimization design method of local resonance topological insulator based on particle swarm includes the following steps:

[0015] (1) Construct a honeycomb-shaped local resonant topological insulator cellular structure, set the initial key structural geometric parameters: cellular height h1, C-shaped opening outer diameter R1, C-shaped opening inner diameter R2, fix the oscillator height difference △h, and determine the target band gap frequency value;

[0016] (2) Determine the particle swarm size N, the maximum number of iterations T, the inertia weight ω, the learning factors c1 and c2, randomly assign binary codes to the key geometric parameters R1, R2, and h1 of the cellular structure in step (1), and initialize the particle swarm position x1 and velocity v1;

[0017] (3) Convert the binary code of the key geometric parameters of the cellular structure that meets the requirements into decimal code to obtain the new key parameters R1, R2, h1 of the structure, substitute them into the cellular structure in step (1), use the finite element method to perform numerical simulation on it, solve the band gap frequency value of each particle in the i-th generation, i = 1, 2...T, calculate the difference F with the target band gap frequency value, obtain the fitness value of each particle, and then obtain the optimal individual pbesti and group optimal gbesti of the i-th generation;

[0018]

[0019] Among them, F upper Indicates the upper boundary frequency value of the actual band gap of the cell, F lower Indicates the upper boundary frequency value of the actual band gap of the cell; F l Indicates the target bandgap lower boundary frequency, which is 50Hz; F u Indicates the upper boundary frequency, the value is 65Hz;

[0020] (4) Update the speed and position of each particle;

[0021] (5) Calculate the particle fitness value and update the optimal value for individual particles and the optimal value for the group;

[0022] (6) Repeat steps (3) to (5) until the maximum number of iterations is met, and obtain the group optimal value and the structural key geometric parameters of the optimal insulator structure.

[0023] Beneficial effects:

[0024] Compared with the prior art, the present invention has the following advantages:

[0025] (1) The honeycomb-shaped local resonant topological insulator proposed in the present invention can realize the design of structural units on a subwavelength scale, so that the structural size is much smaller than the wavelength of the controlled elastic wave, thereby further miniaturizing the sample size when performing low-frequency elastic wave regulation.

[0026] (2) The honeycomb-shaped locally resonant topological insulator proposed in the present invention realizes the band inversion and completes the topological phase transition by changing the oscillator height difference △h, and the degenerate band changes from open to closed and then to open, thereby obtaining a structure with different properties. The topologically mediocre and topologically non-trivial cells form paths of different shapes, which can be used to regulate the low-frequency elastic waves (54-65Hz) caused by rail transit. This topological boundary state has the characteristics of backscattering suppression and robust transmission, which provides a new idea for the regulation of bending waves.

[0027] (3) The honeycomb-shaped local resonant topological insulator proposed in the present invention has a Dirac cone that is closely related to the eigenfrequency of the oscillator and has a wealth of adjustable parameters. The position and width of the band gap can be achieved by adjusting the R1, R2, and △h parameters.

[0028] (4) The present invention selects a particle swarm algorithm with advantages such as simplicity, few setting parameters, and fast convergence speed. After optimization, the band gap width of the structure is expanded to 1.52 times of the original one, and the average attenuation intensity is increased from the original 35.1dB to 39.06dB. Therefore, by optimizing the key size parameters of the unit cell, the control range of the topological metamaterial on the bending wave can be increased, and the vibration reduction effect can be improved. BRIEF DESCRIPTION OF THE DRAWINGS

[0029] Figure 1 Schematic diagram (partial schematic diagram) of a honeycomb-shaped local resonant topological insulator according to an embodiment of the present invention.

[0030] Figure 2 Three-view diagram of a honeycomb-shaped local resonant topological insulator according to an embodiment of the present invention.

[0031] Figure 3 Schematic diagram of the energy bands of a honeycomb-shaped local resonant topological insulator with different oscillator height differences Δh according to an embodiment of the present invention.

[0032] Figure 4 Schematic diagram of the effect of the height difference △h on the eigenfrequency of point K according to an embodiment of the present invention.

[0033] Figure 5 Schematic diagram of the energy band of a honeycomb-shaped local resonant topological insulator supercell according to an embodiment of the present invention.

[0034] Figure 6 Schematic diagram of topological boundary state transmission of different shapes according to an embodiment of the present invention.

[0035] Figure 7 Schematic diagram of robust transmission of topological boundary states of different shapes according to an embodiment of the present invention.

[0036] Figure 8 The figure is a flow chart of a particle swarm optimization algorithm according to an embodiment of the present invention.

[0037] Fig. 9 It is a schematic diagram of the particle swarm optimization algorithm and the structural performance comparison before and after optimization according to an embodiment of the present invention. DETAILED DESCRIPTION

[0038] The present invention will be described in more detail below in conjunction with the schematic diagram, wherein the preferred embodiment of the present invention is shown, and it should be understood that those skilled in the art can modify the present invention described herein, and still achieve the beneficial effects of the present invention. Therefore, the following description should be understood as being widely known to those skilled in the art, and not as a limitation of the present invention.

[0039] The following further describes a honeycomb-shaped local resonant topological insulator for low-frequency bending wave controlled transmission and an optimization design method based on a particle swarm algorithm according to an embodiment of the present invention in conjunction with the accompanying drawings.

[0040] like Figure 1 to Figure 2 As shown, a honeycomb-shaped local resonance topological insulator proposed according to an embodiment of the present invention includes the following contents: 6 C-shaped openings are evenly opened on the honeycomb substrate, and the opening positions are on the connecting line between the honeycomb corner point and the cell center, and each C-shaped opening is rotated 60° around the cell center to overlap; there are 6 pairs of vibrators, the vibrator radius is the same as the inner diameter of the opening, and the vibrator heights on both sides of each node are the same. The cell lattice constant is a=100mm, the opening outer diameter R1=14mm, the opening inner diameter R2=7mm, the connecting beam width w=2mm, the initial vibrator height h1=8mm, h2=2mm, the vibrator height difference △h=h1-h2=6mm, the substrate thickness H=3mm, and the distance between the vibrator and the inner and outer diameters of the opening and the cell center is d=a / 3. The substrate material is photosensitive resin, and the vibrator material is copper.

[0041] like Figure 3 As shown in (a) and (b), the energy band diagrams of the structure when the oscillator height is h1 = 8mm, h2 = 2mm, △h = 6mm, h1 = 2mm, h2 = 8mm, △h = -6mm, respectively. 6v The symmetry is broken, and the Dirac cone at point K opens and the double degeneracy disappears, resulting in a new band gap; (c) is the band diagram of the structure when the oscillator height h1 = 8mm, h2 = 8mm, △h = 0mm, and the cell structure has C 6v Symmetry, at this time double degeneracy occurs at the K point in the Brillouin zone, and a Dirac cone appears.

[0042] like Figure 4 As shown, with the change of height difference △h, the eigenfrequency of point K is exchanged when △h=0, the direction of mechanical energy flow changes, the energy band inversion phenomenon occurs, and the energy of the eigenstate vibration mode is concentrated on the oscillator, which is a typical local resonance mode.

[0043] like Figure 5As shown in (b), a supercell model consisting of 6 topologically trivial crystals (TTCs), 6 topologically nontrivial crystals (TNCs), and 6 topologically trivial crystals (TTCs) is shown in (c). The upper and lower boundaries are set as low-reflection boundary conditions, and the lattice basis vector k x The direction is set as a periodic boundary condition. (a) is the supercell band diagram, the gray dots represent the body state, the red and blue dots represent the boundary state, and the displacement field distribution in the z direction outside the plate surface is extracted from the boundary state points A and B, as shown in (c). The maximum displacement amplitude is concentrated at the boundary between TNCs and TTCs, and decays rapidly to both sides. At the same time, the vortex directions are opposite, indicating the existence of elastic wave boundary state transmission related to the valley vortex.

[0044] like Figure 6 to Figure 7 As shown, by changing the arrangement of TNCs and TTCs, straight and zigzag paths 1 to 2 shown in red are constructed, and by introducing the absence and disorder of oscillators in the paths, the bending waves are transmitted along the paths and are mainly concentrated on the boundaries, verifying that the topological boundary state has the robust transmission characteristics of backscattering suppression and channel defect immunity.

[0045] like Figures 8 to 9 As shown in the figure, based on the particle swarm optimization algorithm, the vibration range caused by rail transit is taken as the optimization target, the target band gap width is set to 50-65Hz, and the target upper and lower frequency values ​​are determined. The particle swarm scale N = 20, the linear weight reduction method is used to determine the weight function ω, and the maximum inertia weight ω max =0.8, minimum inertia weight ω min =0.4. For tasks 1 to 3, the learning factor c1=c2=1.5, and the maximum number of iterations are 100, 150, and 200 respectively; for tasks 4 to 6, the learning factor c1=c2=1.8, and the maximum number of iterations are 100, 150, and 200 respectively.

[0046] The objective function expression is

[0047]

[0048] Among them, F u =65Hz, F l=50Hz, the band gap is widened by optimizing the structural parameters related to the eigenfrequency of the oscillator, and other geometric parameters are fixed. The outer diameter R1 of the C-shaped hole, the inner diameter R2 of the C-shaped hole and the initial height h1 of the oscillator are selected as the optimization objects (h2=h1-△h, △h=6mm). The smaller the difference between the actual frequency and the target frequency, the closer it is to the target frequency. The maximum number of iterations of the particle swarm, the learning factors c1 and c2 are changed, and tasks 1 to 6 are calculated. The results show that with the increase of the optimization generation, the actual structure band gap frequency and the target frequency gradually decrease, and the fitness value gradually stabilizes at 3.5706, which verifies the stability of the optimization program and the correctness of the optimization solution. At the same time, it can be seen from tasks 1 to 6 that the parameter setting of the particle swarm optimization algorithm will affect the convergence speed. The initial structure has a bandgap width of 60.5-67.3Hz, a bandgap width of 6.8Hz, and an average attenuation strength of 35.16dB; the optimized structure has a bandgap width of 54-64.3Hz, a bandgap width of 10.3Hz, and an average attenuation strength of 39.06dB. The bandgap width is expanded to 1.52 times the original. By optimizing the key size parameters of the unit cell, the control range of the topological metamaterial on the bending wave can be increased, and the vibration reduction effect can be improved.

[0049] Finally, it should be noted that the drawings are only for illustrative purposes and represent only schematic diagrams rather than physical drawings, and should not be construed as limiting the present invention; in order to better illustrate the embodiments of the present invention, some parts of the drawings may be omitted, enlarged or reduced, and do not represent the size of the actual product; it is understandable to those skilled in the art that some well-known structures and their descriptions in the drawings may be omitted. The examples of the present invention are only descriptions of the preferred embodiments of the present invention, and do not limit the concept and scope of the present invention. Without departing from the design concept of the present invention, various modifications and improvements made by engineers and technicians in this field to the technical solution of the present invention should fall within the scope of protection of the present invention.

Claims

1. A honeycomb-shaped local resonant topological insulator, characterized in that: The insulator has a honeycomb structure, and the cells constituting the insulator include a honeycomb substrate with uniformly provided C-shaped openings and vibrators symmetrically arranged on the upper and lower sides of an inner ring platform of the C-shaped opening; the C-shaped opening is located on a connecting line between a honeycomb corner point and a cell center; the vibrator radius is the same as the inner diameter of the C-shaped opening, and the vibrator heights on the left and right sides of a node where each vibrator is located are the same; energy band inversion is achieved by synchronously adjusting the height difference between the upper and lower vibrators of the node and its left and right adjacent vibrators.

2. A honeycomb-shaped local resonance topological insulator according to claim 1, characterized in that: The honeycomb substrate is evenly provided with 6 C-shaped openings, and each C-shaped opening is rotated 60 degrees around the cell center to overlap.

3. A honeycomb-shaped local resonance topological insulator according to claim 2, characterized in that: When the heights of the adjacent oscillators on the left and right are the same, the cell has C 6v Symmetry, that is, the structure can overlap after rotating 60 degrees, double degeneracy occurs at the K point in the Brillouin zone, and a Dirac cone appears; when the heights of the left and right adjacent oscillators are different, the cell has C 3v Symmetry, that is, the structure can overlap after rotating 120 degrees, and the Dirac cone at the K point in the Brillouin zone opens to form a band gap.

4. A honeycomb-shaped local resonance topological insulator according to claim 3, characterized in that: The lattice constant of the cell is a=100mm, the outer diameter of the opening is R1=14mm, the inner diameter of the opening is R2=7mm, the width of the connecting beam is w=2mm, the height of adjacent oscillators is h1=h2=8mm, the height difference of the oscillators is △h=h1-h2=0, the thickness of the substrate is H=3mm, and the distance between the oscillator and the inner and outer diameters of the opening and the center of the cell is d=a / 3.

5. A honeycomb-shaped local resonance topological insulator according to claim 4, characterized in that: The specific steps of achieving band inversion by adjusting the height difference between adjacent left and right oscillators include: when h1=8mm, h2 gradually decreases from 8mm, △h>0, and the insulator is a topologically trivial crystal; when h2=8mm, h1 gradually decreases from 8mm, △h<0, and the insulator is a topologically non-trivial crystal; band inversion is achieved by changing the height difference of the oscillators, and the insulator is converted from a topologically trivial crystal to a topologically non-trivial crystal.

6. A honeycomb-shaped local resonance topological insulator according to claim 1, characterized in that: The substrate material is photosensitive resin, and the vibrator material is copper.

7. The optimization design method of the honeycomb-shaped local resonance topological insulator according to any one of claims 1 to 6 based on particle swarms, characterized in that: The following steps are involved: (1) Construct a honeycomb-shaped local resonant topological insulator cellular structure, set the initial key structural geometric parameters: cellular height h1, C-shaped opening outer diameter R1, C-shaped opening inner diameter R2, fix the oscillator height difference △h, and determine the target band gap frequency value; (2) Determine the particle swarm size N, the maximum number of iterations T, the inertia weight ω, the learning factors c1 and c2, randomly assign binary codes to the key geometric parameters R1, R2, and h1 of the cellular structure in step (1), and initialize the particle swarm position x1 and velocity v1; (3) Convert the binary code of the key geometric parameters of the cellular structure that meets the requirements into decimal code to obtain the new key parameters R1, R2, h1 of the structure, substitute them into the cellular structure in step (1), use the finite element method to perform numerical simulation on it, solve the band gap frequency value of each particle in the i-th generation, i = 1, 2...T, calculate the difference F with the target band gap frequency value, obtain the fitness value of each particle, and then obtain the optimal individual pbesti and group optimal gbesti of the i-th generation; Among them, F upper Indicates the upper frequency boundary of the actual band gap of the cell, F lower Indicates the upper frequency boundary of the actual band gap of the cell, F l Indicates the target bandgap lower boundary frequency, F u Indicates the upper boundary frequency of the target bandgap; (4) Update the speed and position of each particle; (5) Calculate the particle fitness value and update the optimal value for individual particles and the optimal value for the group; (6) Repeat steps (3) to (5) until the maximum number of iterations is met, and obtain the group optimal value and the structural key geometric parameters of the optimal insulator structure.

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