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

Abstract

The application discloses a honeycomb-shaped local area resonance topological insulator and an optimization design method thereof, comprising a honeycomb-shaped substrate with C-shaped openings, six pairs of oscillators with the same radius arranged on both sides of nodes, and a band gap inversion realized by changing the height difference of the oscillators, so as to construct a boundary state with different shapes. The topological transmission has backscattering suppression and channel defect immunity characteristics. Based on a particle swarm algorithm, key geometric parameters of a unit cell are optimized, and a topological insulator with a target band gap is customized. The honeycomb-shaped local area resonance topological insulator provides a new idea for flexural wave regulation, and auxiliary design is conducted in combination with a particle swarm optimization algorithm, so that the regulation range of the topological metamaterial on the flexural wave can be increased, and the vibration reduction effect can be improved.

Description

Technical Field

[0001] This invention relates to a honeycomb-shaped local resonant topological insulator for low-frequency flexural wave modulation transmission and an optimization design method based on particle swarm optimization algorithm, belonging to the field of vibration control technology. Background Technology

[0002] Compared to traditional topological insulators, locally resonant systems can achieve structural unit design at the subwavelength scale, making the structural size much smaller than the controlled elastic wave wavelength. This allows for further miniaturization of prototypes when performing low-frequency elastic wave modulation. More importantly, locally resonant systems can localize vibrational energy within the oscillator, reducing vibration on the matrix. This has a natural advantage for achieving highly energy-localized topological states. In recent years, realizing topological boundary states with subwavelength modulation characteristics based on the principle of locally resonant systems has gradually become a new research hotspot.

[0003] Current research primarily focuses on the design of Bragg scatterer structures, concentrating on the mid-to-high frequency range. Research on resonant topological metamaterial plates suitable for low frequencies is relatively limited. Meanwhile, artificial periodic structures, such as stiffened plates and honeycomb materials, which possess lightweight and high-strength properties, are widely used in engineering. These structures are sensitive to dynamic loads and are more prone to inducing elastic wave propagation under external vibration sources. Vibrations in civil engineering typically occur in the low-frequency range, with the natural frequencies of structures or vibration sources often below 100Hz. Traffic-induced vibrations range from 30 to 80Hz. Due to their large wavelengths and high energy, low-frequency vibration reduction has always been a challenge in engineering practice. Therefore, research is needed on methods for controlling low-frequency elastic waves to fully utilize the wave modulation capabilities of topological insulators and regulate low-frequency elastic waves in civil engineering structures. Simultaneously, to meet engineering application needs, there is a lack of efficient and stable optimization design methods for cellular structures, making it difficult to achieve on-demand quantitative design of the operating frequency band of topological insulators. Therefore, in-depth exploration of locally resonant topological insulators remains of significant research value in order to continuously promote the development of subwavelength elastic wave modulation technology to meet the actual needs of engineering applications. Summary of the Invention

[0004] Technical issues:

[0005] This invention aims to at least partially address one of the technical problems existing in related technologies. Therefore, the objective of this invention is to propose a honeycomb-shaped locally resonant topological insulator for low-frequency flexural wave modulation and propagation, and its optimized design method. This honeycomb-shaped locally resonant topological insulator achieves band reversal by changing the oscillator height difference Δh, constructing paths with different shapes to realize boundary state propagation of low-frequency elastic waves. Furthermore, based on particle swarm optimization, it increases the modulation range of the topological metamaterial for flexural waves, improves vibration reduction, and provides a new approach to flexural wave modulation.

[0006] Technical solution:

[0007] To achieve the above objectives, the present invention adopts the following technical solution:

[0008] A local resonant topological insulator is disclosed, wherein the insulator has a honeycomb structure, and the cell constituting the insulator includes a honeycomb substrate with uniformly distributed C-shaped openings and oscillators symmetrically arranged on the upper and lower sides of the inner ring platform of the C-shaped openings; the C-shaped openings are located on the line connecting the corner of the honeycomb and the center of the cell; the radius of the oscillator is the same as the inner diameter of the C-shaped opening, and the height of the oscillators on the left and right sides of each node is the same; band reversal is achieved by synchronously adjusting the height difference between the two oscillators above and below the node and the adjacent oscillators on the left and right.

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

[0010] Preferably, when the heights of adjacent oscillators are the same, the cell has C 6v Symmetry, meaning that structures can coincide after a 60-degree rotation, double degeneracy occurs at point K in the Brillouin zone, resulting in a Dirac cone; when the heights of adjacent oscillators are not the same, the cell has C 3v Symmetry, meaning that the structures can coincide after a 120-degree rotation, and the Dirac cone opens at point K in the Brillouin zone, forming a band gap.

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

[0012] Preferably, the specific steps for achieving band reversal by adjusting the height difference between adjacent oscillators include: when h1 = 8 mm and h2 gradually decreases from 8 mm, Δh > 0, at which point the insulator is a topologically trivial crystal; when h2 = 8 mm and h1 gradually decreases from 8 mm, Δh < 0, at which point the insulator is a topologically nontrivial crystal; by changing the height difference of the oscillators, band reversal is achieved, transforming the insulator from a topologically trivial crystal to a topologically nontrivial crystal.

[0013] Preferably, the substrate material is selected as photosensitive resin, and the oscillator material is selected as copper.

[0014] The optimization design method for local resonant topological insulators based on particle swarm optimization includes the following steps:

[0015] (1) Construct a honeycomb-shaped local resonant topological insulator cell structure, set the initial key geometric parameters of the structure: cell height h1, outer diameter of C-shaped opening R1, inner diameter of C-shaped opening R2, fix the oscillator height difference Δh, and determine the target bandgap frequency value.

[0016] (2) Determine the particle swarm size N, maximum number of iterations T, inertia weight ω, learning factors c1 and c2, randomly assign binary codes to the key geometric parameters R1, R2 and h1 of the cell 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 cell structure that meet the requirements into decimal code to obtain the new key parameters R1, R2, h1 of the structure. Substitute them into the cell structure in step (1) and 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 between the i-th generation and the target band gap frequency value to obtain the fitness value of each particle. Then obtain the optimal pbesti of the individual and the optimal gbesti of the population in the i-th generation.

[0018]

[0019] Among them, F upper F represents the upper boundary frequency value of the actual bandgap of the cell. lower F represents the upper boundary frequency value of the actual bandgap of the cell; l This represents the lower boundary frequency of the target bandgap, with a value of 50Hz; F u This represents the upper boundary frequency, with a value of 65Hz.

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

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

[0022] (6) Repeat steps (3) to (5) until the maximum number of iterations is met, and obtain the group optimal value and the 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 this invention can realize the design of structural units at the subwavelength scale, so that the structural size is much smaller than the controlled elastic wave wavelength, thereby further miniaturizing the sample size when performing low-frequency elastic wave modulation.

[0026] (2) The honeycomb-shaped local resonant topological insulator proposed in this invention achieves band reversal and completes the topological phase transition by changing the height difference Δh of the oscillator, and the degenerate band changes from open to closed and then back to open. This results in structures with different properties. The topologically trivial and topologically nontrivial cells are used to form paths of different shapes, which can be used to control low-frequency elastic waves (54-65Hz) caused by rail traffic. This topological boundary state has backscatter suppression and robust transmission characteristics, providing a new idea for bending wave control.

[0027] (3) The honeycomb-shaped local resonant topological insulator proposed in this 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 parameters R1, R2, and Δh.

[0028] (4) This invention selects the particle swarm optimization algorithm, which has advantages such as simplicity, fewer parameters, and fast convergence speed. After optimization, the bandgap width of the structure is increased to 1.52 times the original value, and the average attenuation intensity is increased from 35.1 dB to 39.06 dB. Therefore, by optimizing the key size parameters of the unit cell, the range of topological metamaterials for controlling bending waves can be increased, and the vibration reduction effect can be improved. Attached Figure Description

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

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

[0031] Figure 3 This is a schematic diagram of the band structure 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 This is a schematic diagram illustrating the effect of the height difference Δh on the intrinsic frequency of point K in an embodiment of the present invention.

[0033] Figure 5 This is a schematic diagram of the supercell energy band structure of a honeycomb-shaped localized resonant topological insulator according to an embodiment of the present invention.

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

[0035] Figure 7 This is a schematic diagram illustrating the robust transmission of topological boundary states with different shapes according to an embodiment of the present invention.

[0036] Figure 8 This is a schematic diagram of the particle swarm optimization algorithm according to an embodiment of the present invention.

[0037] Figure 9 This is a schematic diagram illustrating the particle swarm optimization algorithm and the comparison of structural performance before and after optimization in an embodiment of the present invention. Detailed Implementation

[0038] The present invention will now be described in more detail with reference to the accompanying drawings, which illustrate preferred embodiments of the invention. It should be understood that those skilled in the art can modify the invention described herein while still achieving its advantageous effects. Therefore, the following description should be understood as being of general knowledge to those skilled in the art and is not intended to limit the invention.

[0039] The following description, in conjunction with the accompanying drawings, further illustrates an embodiment of the present invention of a honeycomb-shaped local resonant topological insulator for low-frequency flexural wave modulation transmission and an optimization design method based on particle swarm optimization.

[0040] like Figures 1 to 2 As shown, a honeycomb-shaped localized resonant topological insulator according to an embodiment of the present invention includes the following: six C-shaped openings are uniformly formed on a honeycomb substrate, the openings being located on the connecting line between the honeycomb corner and the cell center, each C-shaped opening coinciding with a 60° rotation around the cell center; there are six pairs of oscillators, the oscillator radius being the same as the inner diameter of the opening, and the oscillator heights on both sides of each node being the same. The cell lattice constant is a = 100 mm, the outer diameter of the opening R1 = 14 mm, the inner diameter of the opening R2 = 7 mm, the width of the connecting beam w = 2 mm, the initial oscillator heights h1 = 8 mm, h2 = 2 mm, the oscillator height difference Δh = h1 - h2 = 6 mm, the substrate thickness H = 3 mm, and the distance d between the inner and outer diameters of the oscillators and openings and the cell center is a / 3. The substrate material is photosensitive resin, and the oscillator material is copper.

[0041] like Figure 3 As shown, (a) and (b) are the energy band diagrams of the structure when the oscillator heights are h1 = 8 mm, h2 = 2 mm, Δh = 6 mm, h1 = 2 mm, h2 = 8 mm, and Δh = -6 mm, respectively. The C1 of the structure at these conditions is... 6v The symmetry is broken, and the Dirac cone opens and the double degeneracy disappears at point K, creating a new band gap; (c) is the band diagram of the structure when the oscillator heights h1 = 8 mm, h2 = 8 mm, and Δh = 0 mm. The cellular structure has C 6v Symmetry occurs, resulting in doubly degenerate at point K in the Brillouin zone, leading to the appearance of a Dirac cone.

[0042] like Figure 4 As shown, as the height difference Δh changes, the eigenfrequency at point K changes when Δh = 0, the direction of mechanical energy flow changes, and band reversal occurs. Furthermore, the energy of the eigenmode mode is concentrated on the oscillator, which is a typical local resonance mode.

[0043] like Figure 5As shown, (b) is a supercell model consisting of 6 topologically trivial crystals (TTCs), 6 topologically nontrivial crystals (TNCs), and 6 topologically trivial crystals (TTCs), with the upper and lower boundaries 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, where gray dots represent bulk states and red and blue dots represent boundary states. The displacement field distribution in the z-direction outside the plate surface is extracted from boundary state points A and B, as shown in (c). The maximum values ​​of the displacement amplitude are concentrated at the junction of TNCs and TTCs, and decay rapidly to both sides. At the same time, the vortex directions are opposite, indicating the existence of elastic wave boundary state transmission related to valley vortex.

[0044] like Figures 6 to 7 As shown, straight lines and Zigzag paths 1-2 (shown in red) were constructed by changing the arrangement of TNCs and TTCs. By introducing missing and disordered oscillators into the paths, the curved wave propagated along the paths, mainly concentrated on the boundaries, verifying that the topological boundary states have robust transmission characteristics of backscatter suppression and channel defect immunity.

[0045] like Figures 8 to 9 As shown, based on the particle swarm optimization algorithm, the vibration range caused by rail transit is used as the optimization objective. The target bandgap width is set to 50–65 Hz, and the upper and lower frequency values ​​of the target are determined. The particle swarm size N = 20, and the weight function ω is determined using a linear weighting method. The maximum inertial weight ω is... max =0.8, minimum inertia weight ω min =0.4. For tasks 1-3, the learning factors c1=c2=1.5, and the maximum number of iterations are 100, 150, and 200 respectively; for tasks 4-6, the learning factors 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. By optimizing structural parameters related to the oscillator's intrinsic frequency to widen the bandgap, while fixing other geometric parameters, the outer diameter R1 and inner diameter R2 of the C-shaped aperture and the initial height h1 of the oscillator were selected as the optimization objects (h2 = h1 - Δh, Δh = 6mm). The smaller the difference between the actual frequency and the target frequency, the closer it is to the target frequency. Tasks 1-6 were calculated by changing the maximum number of iterations in the particle swarm optimization algorithm and the learning factors c1 and c2. The results show that as the optimization generation increases, the actual structural bandgap frequency gradually decreases from the target frequency, and the fitness value gradually stabilizes at 3.5706, verifying the stability of the optimization program and the correctness of the optimized solution. Furthermore, tasks 1-6 show that the parameter settings of the particle swarm optimization algorithm affect the convergence speed. The initial structure had a bandgap width of 60.5–67.3 Hz, a bandgap length of 6.8 Hz, and an average attenuation intensity of 35.16 dB. The optimized structure had a bandgap width of 54–64.3 Hz, a bandgap length of 10.3 Hz, and an average attenuation intensity of 39.06 dB. The bandgap width was increased to 1.52 times the original value. By optimizing the key dimension parameters of the unit cell, the range of topological metamaterial's control over bending waves can be increased, thereby improving the vibration reduction effect.

[0049] Finally, it should be noted that the accompanying drawings are for illustrative purposes only, representing schematic diagrams rather than actual physical objects, and should not be construed as limiting the invention. To better illustrate the embodiments of the invention, some components in the drawings may be omitted, enlarged, or reduced, and do not represent the actual product dimensions. It is understandable to those skilled in the art that some well-known structures and their descriptions may be omitted in the drawings. The examples of this invention are merely descriptions of preferred embodiments and are not intended to limit the concept and scope of the invention. Various modifications and improvements made to the technical solutions of this invention by those skilled in the art without departing from the inventive concept should fall within the protection scope of this invention.

Claims

1. A honeycomb shaped local resonant topological insulator, characterized in that, The insulator is in a honeycomb structure, and a cell constituting the insulator includes a honeycomb substrate with uniformly opened C-shaped holes and a vibrator symmetrically arranged on the upper and lower sides of a ring platform in the C-shaped hole; the C-shaped hole is located on a connecting line between a honeycomb corner point and a cell center; a radius of the vibrator is the same as an inner diameter of the C-shaped hole; and a height of each vibrator on the left and right sides of a node is the same; and band inversion is realized by synchronously adjusting a height difference between the two vibrators on the node and their left and right adjacent vibrators. The optimization design method of the honeycomb local resonance topological insulator includes the following steps: (1) constructing a honeycomb local resonance topological insulator cell structure, setting initial structure key geometric parameters: vibrator height h1, C-shaped hole outer diameter R1, C-shaped hole inner diameter R2, vibrator height difference Δh, and determining a target band gap frequency value; (2) determining a particle swarm size N, a maximum iteration number T, an inertia weight ω, learning factors c1 and c2, randomly allocating binary codes to the cell structure key geometric parameters R1, R2 and h1 in step (1), and initializing particle swarm positions x1 and velocities v1; (3) converting the binary codes of the cell structure key geometric parameters meeting the requirements into decimal codes to obtain new structure key parameters R1, R2 and h1, substituting the new structure key parameters into the cell structure in step (1), and performing numerical simulation on the cell structure by using a finite element method to solve an i-th generation band gap frequency value of each particle, i = 1, 2…T, calculating a difference F between the target band gap frequency value and the i-th generation band gap frequency value, obtaining an adaptation value of each particle, and further obtaining an i-th generation individual optimal value pbesti and a group optimal value gbesti; ； where F upper represents the upper boundary frequency value of the actual bandgap of the cell, F lower represents the upper boundary frequency value of the actual bandgap of the cell, F l represents the lower boundary frequency of the target bandgap, F u represents the upper boundary frequency of the target bandgap; (4) updating each particle velocity and position; (5) calculating a particle adaptation value, and updating a particle individual optimal value and a group optimal value; (6) repeating steps (3) to (5) until the maximum iteration number is met, and obtaining a group optimal value and a structure key geometric parameter of an optimal insulator structure.

2. A honeycomb-shaped local resonant topological insulator according to claim 1, wherein, Six C-shaped holes are uniformly opened on the honeycomb substrate, and each C-shaped hole is coincided by rotating 60 degrees around the cell center.

3. A honeycomb-shaped local resonant topological insulator according to claim 2, wherein, When the heights of the left and right adjacent oscillators are the same, the unit cell has C 6v Symmetry, i.e. the structure can coincide after rotating 60 degrees, the Dirac cone appears at the double degenerate point in the Brillouin zone K point; when the heights of the left and right adjacent oscillators are different, the unit cell has C 3v Symmetry, i.e. the structure can coincide after rotating 120 degrees, the Dirac cone opens at the Brillouin zone K point, forming a band gap.

4. A honeycomb-shaped local resonant topological insulator according to claim 3, wherein, The cell lattice constant is a = 100 mm, the hole outer diameter R1 = 14 mm, the hole inner diameter R2 = 7 mm, the connecting beam width w = 2 mm, the adjacent vibrator height h1 = h2 = 8 mm, the vibrator height difference Δh = h1-h2 = 0, the substrate thickness H = 3 mm, and the distance between the vibrator and the hole inner and outer diameters and the cell center distance d = a / 3.

5. A honeycomb-shaped local resonant topological insulator according to claim 4, wherein, The specific steps of realizing band inversion by adjusting the height difference between the left and right adjacent vibrators include: when h1 = 8 mm and h2 gradually decreases from 8 mm, Δh > 0, the insulator is a topological trivial crystal; when h2 = 8 mm and h1 gradually decreases from 8 mm, Δh < 0, the insulator is a topological non-trivial crystal; and the band inversion is realized by changing the vibrator height difference, and the insulator is converted from a topological trivial crystal to a topological non-trivial crystal.

6. A honeycomb-shaped local resonant topological insulator according to claim 1, wherein, The substrate material is selected from photosensitive resin, and the vibrator material is selected from copper.

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