CN115035965A - Single-phase phonon crystal plate band gap optimization method based on geometric shape optimization - Google Patents

Single-phase phonon crystal plate band gap optimization method based on geometric shape optimization Download PDF

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CN115035965A
CN115035965A CN202210729358.2A CN202210729358A CN115035965A CN 115035965 A CN115035965 A CN 115035965A CN 202210729358 A CN202210729358 A CN 202210729358A CN 115035965 A CN115035965 A CN 115035965A
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尹硕辉
白忠义
陈睿
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Xiangtan University
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Abstract

The invention discloses a band gap optimization method of a single-phase photonic crystal plate based on geometric shape optimization, and belongs to the field of photonic crystal plate band gap optimization design. The band gap of the phononic crystal plate is calculated by utilizing a NURBS (non-Uniform rational B-spline) equal geometric analysis method and a Mindlin plate theory, the NURBS is adopted to describe a plate thickness curved surface, a NURBS control node is taken as a design variable, and the shape optimization design is carried out through a particle swarm algorithm to obtain a phononic crystal unit cell structure with the maximum relative band gap and the low-frequency central frequency, so that the defects of blindness, low efficiency and the like of the shape design of the conventional phononic crystal plate structure are overcome, and meanwhile, the problem of unstable values of sawteeth and the like when the traditional finite element is used for carrying out the plate thickness shape optimization can be avoided.

Description

Single-phase phonon crystal plate band gap optimization method based on geometric shape optimization
Technical Field
The invention belongs to the field of band gap optimization design of a phononic crystal plate, and particularly relates to a geometric shape optimization method for a single-phase phononic crystal plate structure and the like.
Background
The phononic crystal plate is an artificial periodic plate structure with elastic wave band gap characteristics. By designing the thickness curved surface of the single cell structure of the phonon crystal plate, elastic waves in a certain frequency range cannot be transmitted in the phonon crystal plate. The elastic wave band gap characteristic enables the elastic wave band gap to have wide application prospects in the aspects of projects such as sound insulation, vibration reduction, noise reduction and the like. The band gap size, the band gap center frequency and the like of the phononic crystal are important indexes in engineering application, and the wider the band gap is, the wider the application range is. Therefore, designing the maximum band gap phononic crystal plate structure is a hot point of research in the field of phononic crystals in recent decades.
The crystal is designed mainly by designing the configuration and material parameters of the single-cell multiphase material under the condition of a specific lattice type, so that the single cell with better band gap characteristic is obtained. The design method mainly comprises a bionic design and a topological optimization design, wherein the bionic design is to combine the existing structure in the nature with experience so as to design a better crystal configuration; the topological optimization design is to find the optimal material topological layout of the phononic crystal by adopting an optimization algorithm according to a band gap target and constraint conditions. Furthermore, single phase periodic materials have also been demonstrated to have a viable vibration attenuation profile in both one and two dimensional propagation, and for phononic crystal plate structures, variations in plate thickness can also reduce the vibration of bending waves to produce a band gap. Therefore, the thickness curved surface of the phononic crystal plate can be optimized to obtain the better configuration of the phononic crystal plate, but no thickness shape optimization method for the phononic crystal plate is used for widening the band gap of the phononic crystal plate at present. In addition, for the optimization problem of the thickness curved surface shape, a smooth function is required to be used for description so as to avoid the unstable problems such as saw-tooth shape and the like.
However, it is a problem to be studied to construct a reasonable single-phase phononic crystal plate thickness curve to obtain the desired optimal band gap. Therefore, it is necessary to develop a geometrically accurate isogeometric optimization method to construct a thickness curve of the phononic crystal plate to obtain an optimal band gap.
Disclosure of Invention
The invention aims to provide a band gap optimization method of a single-phase phononic crystal plate based on geometric shape optimization, which aims to overcome the defects of blindness, low efficiency and the like of structural shape design of the conventional phononic crystal plate; meanwhile, the problem of unstable numerical values such as sawteeth and the like when the traditional finite element is used for optimizing the thickness and the shape of the plate can be avoided.
In order to solve the problems, the invention calculates the band gap of the phononic crystal plate by utilizing a NURBS (non-uniform rational B-spline) equal geometric analysis method and a Mindlin plate theory, describes a plate thickness curved surface by adopting the NURBS, takes a NURBS control node as a design variable, and performs shape optimization design by a particle swarm algorithm to obtain a phononic crystal unit cell structure with the maximum relative band gap and the low-frequency central frequency. The method comprises the following specific steps:
the method comprises the following steps: a geometric model was constructed using NURBS and the geometry of the phononic crystal plate was described.
Step two: initializing a population of thickness particles.
Step three: in the particle swarm optimization process, the minimum value hmin and the maximum value hmax allowed by the thickness of the control node are used as constraint conditions, the optimization target is that the band gap of the phononic crystal plate is widest, meanwhile, the central frequency of the band gap is minimized, and the thickness of the NURBS control node is used as a design variable in the optimization process.
Step four: and performing energy band calculation and transmission analysis calculation by using geometric analysis methods such as NURBS (non-uniform rational B-spline) and the like, verifying the final phononic crystal unit cell thickness optimization structure, and displaying the optimized phononic crystal unit cell thickness curved surface.
Further, in the process of initializing the thickness particle swarm, setting a minimum value hmin and a maximum value hmax allowed by the thickness of the control node, and selecting an initially designed phononic crystal plate unit cell thickness curved surface function.
Further, the particle swarm optimization process comprises the steps of constructing a phononic crystal unit cell shape optimization problem model, calculating the individual fitness of particles, continuously comparing and adjusting the particles, continuously evolving the particles, continuously updating to generate new particles, and checking whether the particle swarm meets the convergence condition.
Furthermore, in the optimization process, a characteristic equation of a discrete form phononic crystal kinetic equation is calculated by using a NURBS (non-uniform rational B-spline) and other geometric analysis methods, and the Floquet periodic boundary condition is substituted into the characteristic value equation to calculate the energy band dispersion relation of the phononic crystal unit cell.
Further, after optimization, energy band calculation and check are carried out by using geometric analysis methods such as NURBS and the like, and the optimized thickness curved surface of the phononic crystal unit cell is obtained.
Compared with the prior art, the invention has the following beneficial effects: the band gap obtained by using the isogeometric analysis shape optimization method has the characteristic of wide low-frequency band gap, the band gap before optimization appears between the third-order frequency and the fourth-order frequency, the band gap range is 2889.2-3208.7Hz, and the center frequency is 3049 Hz. After optimization, the band gap from the third-order frequency to the fourth-order frequency is obviously widened, and the central frequency of the band gap is also reduced; at this time, the band gap range is 658.8-1989Hz, which is widened by 316.3% compared with that before optimization, and the center frequency is 1323.9, which is reduced by 56.6%. In addition, several new bandgaps appear above this bandgap. The geometric analysis shape optimization method provides an efficient and systematic method for obtaining the design of the phononic crystal plate with the widened band gap frequency band, and the optimized phononic crystal plate has good manufacturability and can be applied to vibration reduction and noise reduction of common bearing members.
Drawings
FIG. 1 is an analysis flowchart.
FIG. 2a is the initial shape diagram of the phononic crystal plate unit cell.
FIG. 2b is the diagram of initial band gap of phonon crystal plate unit cell.
FIG. 3 is a graph of unit cell node classification and periodic boundary conditions.
Fig. 4 is a diagram of an iterative process of a particle swarm algorithm.
FIG. 5a is the curved surface shape diagram of the unit cell thickness of the optimized phononic crystal plate.
FIG. 5b is the contour line top view of the optimized thickness curve of the single cell of the phononic crystal plate.
FIG. 6 is a diagram of the band gap of the phononic crystal unit cell after optimization.
Detailed Description
A band gap optimization method of a single-phase phonon crystal plate based on isogeometric shape optimization is characterized in that a band gap of the phonon crystal plate is calculated by utilizing an NURBS isogeometric analysis method and a Mindlin plate theory, a NURBS is adopted to describe a plate thickness curved surface, a NURBS control node is used as a design variable, shape optimization design is carried out through a particle swarm optimization, and a phonon crystal single cell structure with the maximum relative band gap and low-frequency center frequency is obtained. The present invention is described in detail below with reference to the accompanying drawings and specific examples, wherein the following steps are provided, and the analysis flow is shown in fig. 1:
the method comprises the following steps: constructing a thickness curved surface of the phononic crystal plate by adopting NURBS; describing phonon crystal plate geometry using NURBS to control nodal thickness h k For design variables, the thickness of the phononic crystal plate at any point uses the NURBS basis function R k (xi, eta) is approximated by
Figure BDA0003712384080000031
Step two: initializing a thickness particle swarm; setting the minimum value hmin and the maximum value hmax allowed by the thickness of the control node, and initially designing a single-cell thickness curved surface function of the phonon crystal plate as
Figure BDA0003712384080000032
Wherein L is x And L y The geometrical curves of the thickness of the plate are described by NURBS, which is shown in figure 2a, and the band gap of the phonon crystal plate can be calculated by adopting isogeometric analysis Mindlin plate theory, which is shown in figure 2 b. And when the particle swarm optimization is adopted to carry out band gap optimization design, the thickness of the control node is taken as a design variable. In order to reduce design variables, iteration times and calculation amount and improve optimization efficiency, two layers of NURBS grids are adopted, a thicker NURBS control grid describes a thickness curved surface, and the middle surface geometry of a thinner NURBS control grid description plate is used for calculating the band gap of the phononic crystal plate and the like. The thickness of the control node is used as a design variable, the thickness function is initialized to control the thickness of the control node, and the initial value of the design variable is given to improve the calculation efficiency. In addition, the size of the particle search population is taken as N to be 100.
Step three: particle swarm optimization process;
(1) and constructing a phononic crystal unit cell shape optimization problem model.
Using NURBS control node thickness for constructing phonon crystal plate unit cell geometry as design variable in optimization process to control nodeMinimum value h allowed for point thickness min And a maximum value h max For constraint conditions, the optimization target is that the band gap of the phononic crystal plate is as wide as possible, and meanwhile, the central frequency of the band gap is as small as possible, and the particle swarm optimization method is adopted to define the thickness shape optimization model of the phononic crystal unit cell as follows:
Figure BDA0003712384080000041
wherein, ω is 1l And omega 1u Respectively, the lower limit frequency and the upper limit frequency of the first order bandgap in the initial design.
(2) Calculating the individual fitness of the particles;
based on the Mindlin board theory and the fluctuation theory, the particle individual fitness is calculated by solving the fluctuation equation by using a NURBS (non-uniform rational B-spline) and other geometric analysis numerical methods. The displacement component of any point on the middle plane of the Mindlin board is
u 1 (x,y,z)＝zβ x (x,y)
u 2 (x,y,z)＝zβ y (x,y)
u 3 (x,y,z)＝w(x,y)
The isogeometric analysis method is a high-order finite element method, and adopts high-order continuous NURBS basis function for describing geometry to replace shape function in finite element analysis, and adopts NURBS to describe displacement mode of plate as
Figure BDA0003712384080000042
Like finite elements, the discrete form of the phononic crystal kinetic equation can be derived as a characteristic equation of the form:
[K-ω 2 M]U＝0
wherein K is a phonon crystal rigidity matrix, M is a phonon crystal quality matrix, and U is displacement vectors of each node of a phonon crystal unit cell.
Considering a square unit cell, the nodes on the unit cell can be divided into 4 angular point nodes u according to positions BL ,u BR ,u TL ,u TR Class 4 boundary node u L ,u R ,u B ,u T And an internal central node u I A total of 9 classes are shown in FIG. 3. According to the Bloch theorem, the unit cell boundary satisfies the periodic boundary condition as follows:
Figure BDA0003712384080000051
Figure BDA0003712384080000052
Figure BDA0003712384080000053
Figure BDA0003712384080000054
Figure BDA0003712384080000055
wherein, a x And a y Are the basis vectors of phononic crystal unit cells, respectively; k is a radical of formula x And k y The components of the wave vector k in the x and y directions, respectively.
The Floquet periodic boundary conditions are:
Figure BDA0003712384080000056
substituting the Floquet periodic boundary condition into a characteristic value equation to obtain
Figure BDA0003712384080000057
The equation comprises a Floquet boundary condition related to wave vectors, which is called a frequency dispersion equation of the phononic crystal unit cell, and the energy band dispersion relation of the phononic crystal unit cell can be obtained by solving the frequency dispersion equation under different wave vectors. The goal of shape optimization is to initially design the first band gap as wide as possible, and simultaneously to make the center frequency of the band gap as small as possible, so that the objective function is used to measure the applicability of individual particles.
(3) Continuously comparing and adjusting the particles to make the particles continuously evolve and continuously update to generate new particles.
(4) Checking whether the particle swarm meets a convergence condition, if so, outputting an optimal result, and exiting; otherwise, returning to the second step in the optimization process.
Step four: and performing energy band calculation and transmission analysis calculation by using geometric analysis methods such as NURBS (non-uniform rational B-spline) and the like, verifying the final phononic crystal unit cell thickness optimization structure, and displaying the optimized phononic crystal unit cell thickness curved surface.
Specific examples are listed below:
taking L as the unit cell geometry of the phonon crystal plate x ＝L y Minimum thickness h allowed for 100mm min 1mm, maximum h max 40 mm. The material is epoxy resin with the material properties of the elastic modulus E of 2300Mpa, the Poisson ratio of 0.35 and the density of 1200kg/m 3 . The thickness NURBS curved surface is a square control grid of 9 multiplied by 9, the initial value of the thickness is initialized by adopting a single-cell thickness curved surface function of the phononic crystal plate, and only 1/4 control nodes of the control grid are required to be design variables in consideration of symmetry. The phononic crystal plate unit cell was discretely divided with a 29 × 29 square control grid using third-order NURBS splines to calculate the band gap.
The iterative process of optimization by particle swarm optimization is shown in fig. 4. The shape of the single cell thickness curved surface of the phonon crystal plate before optimization is shown in figure 2a, the shape of the single cell thickness curved surface of the phonon crystal plate after optimization is shown in figure 5a, and the contour diagram of the single cell thickness of the phonon crystal plate viewed from top to bottom is shown in figure 5 b. From the result of optimizing the shape, it is equivalent to obtaining a convex phononic crystal. The band gap before optimization is between the third order frequency and the fourth order frequency as shown in figure 2b, the band gap range is 2889.2-3208.7Hz, and the center frequency is 3049 Hz. The optimized bandgap diagram is shown in fig. 6, and it is apparent from the diagram that the bandgaps from the third order frequency to the fourth order frequency are widened significantly, and the bandgap center frequency is also lowered; at this time, the band gap range is 658.8-1989Hz, which is widened by 316.3% compared with that before optimization, and the center frequency is 1323.9, which is reduced by 56.6%. In addition, several new bandgaps appear above this bandgap.
The band gap optimization method of the single-phase phononic crystal plate based on the isogeometric optimization is described in detail above, a specific example is applied in the embodiment to explain the principle and the implementation mode of the invention, and the description of the embodiment is only used for helping to understand the method and the core idea of the invention; meanwhile, for those skilled in the art, according to the idea of the present invention, there may be variations in the specific implementation and application scope, and in summary, the content of the embodiment should not be construed as a limitation to the present invention.

Claims (9)

1. A band gap optimization method of a single-phase phononic crystal plate based on isogeometric optimization is characterized by comprising the following steps:
step one, constructing a thickness curved surface of a phononic crystal plate by adopting NURBS;
initializing a thickness particle swarm;
step three, in the particle swarm optimization process, the minimum value hmin and the maximum value hmax allowed by the thickness of the control node are taken as constraint conditions, the band gap of the phononic crystal plate is widened to the maximum, meanwhile, the center frequency of the band gap is minimized to serve as an optimization target, and the thickness of the NURBS control node is taken as a design variable in the optimization process;
step four: and performing energy band calculation and transmission analysis calculation by using a NURBS (non-uniform rational B-spline) equal geometric analysis method, verifying the final phononic crystal unit cell thickness optimization structure, and displaying the optimized phononic crystal unit cell thickness curved surface.
2. The method for optimizing the band gap of the single-phase phononic crystal plate based on the isogeometric optimization of claim 1, wherein the method comprises the following steps: the first step takes the thickness of the control node as a design variable, and the thickness of any point of the phononic crystal plate is expressed by a NURBS basis function.
3. The method for optimizing the band gap of the single-phase phononic crystal plate based on the isogeometric optimization of claim 1, wherein the method comprises the following steps: in the step of initializing the thickness particle swarm, setting a minimum value hmin and a maximum value hmax allowed by the thickness of the control node and establishing an initial design phononic crystal plate unit cell thickness curved surface function.
4. The method for optimizing the band gap of the single-phase phononic crystal plate based on the isogeometric optimization as claimed in claim 2 or 3, wherein the method comprises the following steps: setting two layers of NURBS grids, describing a thickness curved surface by adopting a thicker NURBS control grid, and describing the middle surface geometry of the plate by adopting a thinner NURBS control grid for calculating the band gap of the phononic crystal plate.
5. The method for optimizing the band gap of the single-phase phononic crystal plate based on the isogeometric optimization of claim 1, wherein the method comprises the following steps: the particle swarm optimization process specifically comprises the steps of constructing a phononic crystal unit cell shape optimization problem model, calculating the individual fitness of particles, comparing and adjusting the particles and checking whether the particle swarm meets the convergence condition.
6. The method for optimizing the band gap of the single-phase phononic crystal plate based on the isogeometric optimization of claim 5, wherein the method comprises the following steps: the step of calculating the individual fitness of the particles is based on a Mindlin board theory and a fluctuation theory, and the particle individual fitness is calculated by solving a fluctuation equation by using a NURBS (non-uniform rational B-spline) and other geometric analysis numerical methods.
7. The method for optimizing the band gap of the single-phase phononic crystal plate based on the isogeometric optimization of claim 5, wherein the band gap optimization method comprises the following steps: in the step of calculating the individual fitness of the particles, a square unit cell is considered, nodes on the unit cell are divided into corner nodes, boundary nodes and internal central nodes, and periodic boundary conditions are established according to the Bloch theorem.
8. The method for optimizing the band gap of the single-phase phononic crystal plate based on the isogeometric optimization of claim 7, wherein the method comprises the following steps: after the periodic boundary condition is established, solving a frequency dispersion equation of the phononic crystal unit cell under different wave vectors to obtain an energy band dispersion relation of the phononic crystal unit cell.
9. The method for optimizing the band gap of the single-phase phononic crystal plate based on the isogeometric optimization of claim 5, wherein the method comprises the following steps: the step of checking whether the particle swarm meets the convergence condition is specifically that if the particle swarm meets the convergence condition, an optimal result is output and quitting; otherwise, returning to the step of calculating the individual fitness of the particles in the optimization process.
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Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN116776752A (en) * 2023-08-24 2023-09-19 中南大学 Phonon crystal band gap design method and system based on intelligent coding

Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN116776752A (en) * 2023-08-24 2023-09-19 中南大学 Phonon crystal band gap design method and system based on intelligent coding
CN116776752B (en) * 2023-08-24 2023-11-24 中南大学 Phonon crystal band gap design method and system based on intelligent coding

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