Piezoelectric-viscoelastic photonic crystal topological optimization method based on genetic algorithm

Abstract

The invention discloses a piezoelectric-viscoelastic photonic crystal topological optimization method based on a genetic algorithm, and the method comprises the steps: 1, constructing a photonic crystal rod based on piezoelectric ceramic and epoxy resin, and obtaining the parameters of the photonic crystal rod and the parameters of the genetic algorithm; step 2, dispersing the phononic crystal rod into a plurality of phononic crystal rod units, each unit corresponding to a gene bit, representing material distribution by adopting binary coding, and constructing a mapping relation between material attributes and coding by adopting a density interpolation method; 3, calculating an energy band curve of each photonic crystal rod unit by using a finite element method, and obtaining a target band gap value based on the energy band curve; 4, taking the maximized absolute band gap width as a target function, evaluating the fitness, and selecting an optimal target; 5, judging whether a convergence condition is met or not, and continuing genetic manipulation if the convergence condition is not met to generate a new filial generation; step 6, carrying out genetic manipulation; and 7, extracting an optimal solution and verifying result calculation accuracy.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of intelligent phononic crystal optimization design, in particular to a piezoelectric-viscoelastic phononic crystal topology optimization method based on a genetic algorithm, which adjusts material distribution through topology optimization design to maximize the absolute width of the first band gap, and is suitable for engineering fields such as vibration control and sound wave filtering. BACKGROUND

[0002] As a kind of artificially designed periodic composite material, the band gap characteristics of phononic crystals show important application value in the field of vibration control and noise suppression. The design of traditional phononic crystals mainly relies on the trial-and-error method, which optimizes the band gap performance by repeatedly adjusting the structural parameters. This method is not only inefficient, but also difficult to obtain a global optimal solution. Topology optimization design method has become an important technical means for improving the band gap performance of phononic crystals due to its high efficiency and flexibility in material distribution optimization. In addition, the traditional phononic crystal is difficult to meet the demand of elastic wave regulation due to its fixed band structure. In recent years, the introduction of smart materials has opened up a new path to solve this problem, but there are still some key problems to be solved.

[0003] In terms of optimization algorithm, the existing technology has made certain breakthroughs. Patent CN119358345 proposes a phononic crystal topology optimization method based on an improved genetic algorithm, which significantly improves the calculation efficiency by introducing a hash table to store the calculated band gap information. Patent CN116384138 proposes a phononic crystal topology optimization method with specific band gaps, which realizes the design of specific band gaps by establishing frequency band constraints and same-side frequency constraints expressions. However, these methods, although to some extent, improve the band gap optimization effect, still have obvious limitations. The optimization object is limited to a single material system, although the band gap is widened, but the dynamic adjustment of the band gap range cannot be realized. In terms of material design, patent CN101952882 uses the damping characteristics of viscoelastic materials to widen the band gap, although it achieves certain effect, but it does not have the ability of active regulation. The invention patent with publication number CN117912435 discloses a tunable band gap piezoelectric phononic crystal and its parameter optimization method, which realizes the regulation of band gap by adjusting the material thickness and electric field intensity. Although this method realizes the tunability of the band gap frequency, it only relies on a single electrical parameter adjustment, and uses the trial-and-error method for parameter optimization, which is not only inefficient, but also difficult to obtain a global optimal solution, resulting in a limited band gap adjustment range and difficulty in fully utilizing the performance potential of the material. SUMMARY

[0004] The purpose of the present application is to optimize the topology of piezoelectric-viscoelastic phononic crystal rods through genetic algorithm, maximize the absolute width of the first band gap, improve the regulation ability of phononic crystal rods for elastic waves, and meet the demand of vibration control and noise isolation in practical engineering.

[0005] The application discloses a piezoelectric-viscoelastic phononic crystal topology optimization method based on a genetic algorithm, and the piezoelectric-viscoelastic phononic crystal topology optimization method based on the genetic algorithm comprises the following steps:

[0006] Step 1: piezoelectric ceramics and epoxy resin are used to construct a phononic crystal rod, and parameters of the phononic crystal rod and parameters of a genetic algorithm are obtained;

[0007] Step 2: the phononic crystal rod is discretized into a plurality of phononic crystal rod units, each unit corresponds to a gene position, a binary coding is used to represent material distribution, and a density interpolation method is used to construct a mapping relationship between material properties and coding;

[0008] Step 3: a finite element method is used to calculate a band curve of each phononic crystal rod unit, and a target band gap value is obtained based on the band curve;

[0009] Step 4: a maximum absolute band gap width is used as an objective function, fitness is evaluated, and an optimal target is selected;

[0010] Step 5: whether a convergence condition is met is judged, genetic operation is continuously performed if the convergence condition is not met, new offspring is generated, steps 3 and 4 are repeated, and the new offspring is generated.

[0011] Step 6: genetic operation is performed;

[0012] Step 7: an optimal solution is extracted and result calculation accuracy is verified.

[0013] Further, in step 1, the epoxy resin is used as a viscoelastic material, a standard linear solid model is used, and the phononic crystal rod parameters include a unit cell length, a rod diameter, a viscoelastic material modulus ratio and a relaxation time.

[0014] The genetic algorithm parameters include a genetic algorithm population size, a maximum iteration times, a crossover probability and a mutation probability.

[0015] Further, in step 2, the phononic crystal rod is represented by binary coding to represent material distribution:

[0016] ;

[0017] wherein “0” represents that the unit is a viscoelastic material, and “1” represents that the unit is a piezoelectric material, and The distribution is a feasible region of the material A and the material B distribution, is the number of units; each represents a gene position.

[0018] Further, in step 3, to ensure the population diversity, the initial structure before optimization is a randomly generated binary coded matrix, the initial structure band curve is solved by using the finite element method, the objective function is calculated, and genetic optimization design is performed according to the calculation results;

[0019] During the calculation, the actual material properties of each unit are mapped into the finite element model through the density interpolation method through the coded value of the unit:

[0020] ;

[0021] wherein, and are the interpolated Young's modulus and density matrices, 、 and 、 are the actual material parameters of the viscoelastic material and the piezoelectric material respectively, is a chromosome matrix.

[0022] Further, in step 3, it also includes:

[0023] Step 31, constructing the constitutive relation and the basic equation;

[0024] The constitutive relation matrix form of the linear piezoelectric material is expressed as:

[0025] ;

[0026] wherein, is the elastic matrix under a constant electric field, is the piezoelectric stress constant matrix, is the dielectric constant matrix, and are the stress and strain tensors respectively, and are the electric displacement and electric field intensity vectors respectively, , is the spatial coordinate along the length direction of the rod; is the electric potential; under the condition of no free charge, the electric displacement satisfies the Gauss law: .

[0027] According to Newton's second law, the balance equation is established:

[0028] ;

[0029] The constitutive equation is brought into the balance equation, and the following equation is obtained:

[0030] ;

[0031] wherein, is a displacement field variable, is a displacement , is a spatial coordinate along the length of the rod.

[0032] Further, in step 3, further comprising:

[0033] Step 32, determining the boundary conditions and the interpolation function;

[0034] A phononic crystal is a periodic structure, the cell satisfies the Bloch periodic boundary condition on the left and right boundaries:

[0035] ;

[0036] is the cell length, is the wave vector, is a spatial coordinate along the length of the rod; inside the cell, the displacement and the electric potential are expressed by linear function interpolation:

[0037] ;

[0038] Let the cell length be , the shape function can be expressed as:

[0039] ;

[0040] Thus, the strain can be expressed by the shape function as:

[0041] ;

[0042] denoted as: wherein ; similarly, the electric field .

[0043] Further, in step 3, further comprising:

[0044] Step 33, calculating the eigenvalue equation and the band curve solving framework;

[0045] According to the principle of minimum potential energy, the electromechanical coupling problem is simplified into an eigenvalue solving problem, and the characteristic equation of the cell is:

[0046] ;

[0047] wherein, the expressions of the cell stiffness matrix and the cell mass matrix are as follows:

[0048] ;

[0049] ;

[0050] ;

[0051] ;

[0052] Assemble the global stiffness matrix and mass matrix, and get the final global generalized eigen-equation expression:

[0053] ;

[0054] where, and are the displacement matrix and electric potential matrix of the unit node, is the mechanical stiffness matrix, is the piezoelectric coupling stiffness matrix, is the dielectric stiffness matrix, is the mass matrix, is the characteristic frequency, is the wave vector; by scanning the first irreducible Brillouin zone of the wave vector , and solving the above global generalized eigen-equation, the band curve of the piezoelectric elastic phononic crystal rod can be obtained.

[0055] Further, in step 3, further comprising:

[0056] Step 34, solving the eigen-equation and band curve of the piezoelectric-viscoelastic phononic crystal rod;

[0057] When studying the piezoelectric-viscoelastic phononic crystal rod based on the standard linear solid model, it is necessary to solve the modulus expression of the viscoelastic material to replace the Young's modulus of the elastic material in the general framework;

[0058] The modulus of the viscoelastic material in the time domain is:

[0059] ;

[0060] where, is the initial modulus, is the relaxation modulus, is the relaxation time, is the time; by Fourier transform, the constitutive relation in the time domain is converted to the frequency domain, and further the complex modulus is obtained, and finally it is separated into real and imaginary parts:

[0061] ;

[0062] is the storage modulus, which represents the ability of the material to store elastic strain energy, G'' is loss modulus, which embodies the viscous loss of the material;

[0063] After introducing the viscoelastic material, the Young's modulus of the material changes, the complex modulus of the viscoelastic material replaces the Young's modulus of the original elastic material, and is substituted into the final characteristic equation, and the same wave vector scanning method is used to solve the eigenvalue problem, that is, the band curve of the piezoelectric-viscoelastic phononic crystal rod is obtained.

[0064] Further, in step 34, further comprising:

[0065] Step 35, based on the band curve, solving the target band gap value.

[0066] According to the band curve, the target band gap value is solved, and the band gap is the width of the non-overlapping frequency interval between adjacent energy bands in the band curve, which is expressed as:

[0067] ;

[0068] G is the band gap value, and are the n+1th and nth characteristic frequencies, respectively, and are the n+1th and nth characteristic frequencies, respectively, denotes the chromosome matrix, is the wave vector.

[0069] Further, in step 4, the absolute band gap width is defined as the fitness function, and the maximum absolute band gap width is taken as the objective function, that is, the absolute width of the first band gap is the upper boundary frequency minus the lower boundary frequency, according to the fitness function, the following objective function can be obtained:

[0070] ;

[0071] ;

[0072] wherein, denotes the chromosome matrix, is the feasible region of material distribution; and are the n+1th and nth characteristic frequencies, respectively, which are functions of material distribution x and wave vector k, denotes the objective function;

[0073] The fitness value of the topological structure corresponding to each individual in the population is calculated, and the higher the fitness value, the better the topological structure represented by the individual, and the closer to the expected maximum band gap width.

[0074] The beneficial effects achieved by the present application are:​​

[0075] The present application significantly widens the absolute width of the first band gap of the structure by introducing viscoelastic material and piezoelectric material into the topological optimization design of phononic crystal rods. In the optimization process, the absolute width of the first band gap is maximized as the objective function, and variables such as material distribution, viscoelastic parameters, and electric boundary conditions of the piezoelectric phase are comprehensively considered, and an optimization model based on an improved genetic algorithm is established. The improved genetic algorithm effectively improves the optimization efficiency and stability by introducing the elite reservation strategy. The numerical simulation results show that the optimized piezoelectric-viscoelastic phononic crystal rod exhibits excellent band gap characteristics, and the absolute width of the first band gap is significantly improved compared with the traditional design.

[0076] In addition, the electric boundary conditions (including short circuit and open circuit conditions) applied to the piezoelectric material have a significant regulating effect on the band gap characteristics. In view of this feature, the optimization process systematically investigates the variation of the band gap characteristics under different electric boundary conditions, and accordingly adjusts the optimization strategy, further improving the optimization effect. The optimized structure realizes effective suppression of elastic wave propagation in a specific frequency range, which provides a new solution for vibration control and noise isolation in engineering applications.

[0077] Compared with the prior art, the present application has the beneficial effects that: the present application significantly widens the band gap range of the phononic crystal rod by introducing viscoelastic material, and at the same time, by reasonably selecting the electric boundary conditions, the piezoelectric-viscoelastic phononic crystal rod under different electric boundary conditions is topologically optimized by using an improved genetic algorithm, the material distribution is optimized, and the frequency range of the specified band gap is significantly improved. BRIEF DESCRIPTION OF DRAWINGS

[0078] Figure 1 is an optimization design flowchart, showing the iteration process of the genetic algorithm.

[0079] Figure 2 is a structural schematic diagram of a piezoelectric-viscoelastic phononic crystal rod.

[0080] Figure 3 is a comparison diagram of the energy band curves of the piezoelectric-elastic phononic crystal rod and the piezoelectric-viscoelastic phononic crystal rod under different electric boundary conditions, (a) open circuit, (b) short circuit;

[0081] Figure 4 is a comparison diagram of the structure before and after optimization under open circuit condition, (a) before optimization, (b) after optimization;

[0082] Figure 5 is a comparison diagram of the energy band curves before and after optimization under open circuit condition, (a) before optimization, (b) after optimization;

[0083] Figure 6are the structure comparison diagrams before and after optimization under short circuit condition, (a) before optimization, (b) after optimization;

[0084] Figure 7 are the energy band curve comparison diagrams before and after optimization under short circuit condition, (a) before optimization, (b) after optimization. DETAILED DESCRIPTION

[0085] The advantages and features of the present application will become more apparent with the description. However, these examples are only exemplary and do not constitute any limitation on the scope of the present application. Those skilled in the art should understand that the details and forms of the technical solutions of the present application can be modified or replaced without departing from the spirit and scope of the present application, and such modifications and replacements all fall within the protection scope of the present application.

[0086] As shown in Figure 1 , the present application provides a piezoelectric-viscoelastic phononic crystal topology optimization method based on genetic algorithm. The method drives material distribution optimization through genetic algorithm, combines the special performance of piezoelectric-viscoelastic composite material, and finally obtains a phononic crystal structure with excellent band gap characteristics. The specific implementation process is as follows:

[0087] Step 1: Set parameters and establish a model

[0088] Taking a one-dimensional piezoelectric-viscoelastic phononic crystal rod as shown in Figure 2 , this embodiment selects PZT-7A piezoelectric ceramic and epoxy resin as viscoelastic material to construct a one-dimensional phononic crystal rod, and the specific parameters are shown in Table 1. The viscoelastic material epoxy resin adopts a standard linear solid model, the modulus ratio is set to 2, and the relaxation time is 0.00001s. The cell length is set to 0.1m, and the rod diameter is 0.01m. These size parameters will directly determine the grid division accuracy of subsequent finite element analysis. In order to ensure the optimization effect in the subsequent, the population size of genetic algorithm is set to 50, the maximum iteration number is 500, the crossover probability is 0.8, and the mutation probability is 0.05. The above parameters are used for genetic operation in step 6.

[0089] Table 1 Material parameters of PZT-7 and epoxy resin

[0090] Density young's modulus piezoelectric constant Relative dielectric constant PZT-7A 7890 131 10.99 235 Epoxy resin 1180 1.35 0 2.8

[0091] is the vacuum dielectric constant.

[0092] Step 2: Population coding.

[0093] According to the phononic crystal unit size in step 1, the phononic crystal rod is discretized into 30 phononic crystal rod units, each of which corresponds to a gene site. In order to facilitate subsequent genetic operation optimization of material distribution, binary coding is used to represent material configuration, "0" represents that the unit is viscoelastic material, and "1" represents that the unit is piezoelectric material. The specific coding rules are as follows:

[0094]

[0095] wherein, and is the feasible region of the distribution of viscoelastic material A and piezoelectric material B, is the total number of units. Each represents a gene site, which is composed of all The matrix composed of all is a chromosome, representing a phononic crystal rod topology structure.

[0096] Step 3: Calculate the band curve of the initial structure using the finite element method.

[0097] In order to ensure population diversity, the initial structure before optimization is determined by a randomly generated binary coding matrix. After calculating the objective function using the finite element method to solve the band curve of the initial structure, genetic optimization design is carried out based on the calculation results. During the calculation process, the actual material properties of each unit are mapped to the finite element model by the density interpolation method combined with the coding value of the x matrix in step 2 , and the interpolation formula is as follows:

[0098]

[0099] wherein, and are the interpolated Young's modulus matrix and density matrix, respectively, , are the actual material parameters of the viscoelastic material, , are the actual material parameters of the piezoelectric material, is a chromosome matrix.

[0100] Based on the above material property mapping method, the property and mechanical behavior description logic of the phononic crystal rod unit are further clarified: each phononic crystal rod unit contains two nodes, and its material properties (density, Young's modulus, piezoelectric constant, relative dielectric constant) can be determined according to the coding value of step 2. For a one-dimensional phononic crystal rod composed of piezoelectric material and elastic material alternately, due to the existence of electromechanical coupling effect, its mechanical behavior needs to be described by coupling field equations. The derivation process is as follows:

[0101] Step 31, constitutive relation and basic equation

[0102] The constitutive relation matrix of linear piezoelectric material is expressed as:

[0103]

[0104] where, is the elastic matrix under constant electric field, is the piezoelectric stress constant matrix, is the dielectric constant matrix, and are stress and strain tensors, and electric displacement and electric field intensity vector, , is the spatial coordinate along the length direction of the rod; is the electric potential; under the condition of no free charge, the electric displacement satisfies the Gauss law: .

[0105] According to Newton's second law, the balance equation is established:

[0106]

[0107] The constitutive equation is brought into the balance equation, and the following equation is obtained:

[0108]

[0109] where, is the displacement field variable, is the displacement , is the spatial coordinate along the length direction of the rod.

[0110] Step 32, boundary conditions and interpolation functions

[0111] The phononic crystal is a periodic structure, and the left and right boundaries of its unit cell satisfy the Bloch periodic boundary condition:

[0112]

[0113] is the unit cell length, is the wave vector, is the spatial coordinate along the length direction of the rod. Inside the unit, the displacement and the electric potential are expressed by linear function interpolation:

[0114]

[0115] Let the unit length be , and the shape function Can be expressed as:

[0116]

[0117] Thus, the strain Can be expressed by the shape function as:

[0118]

[0119] Noted as: Where

[0120]

[0121] Similarly, the electric field .

[0122] Step 33, characteristic equation and band curve solving framework

[0123] Based on the above equation, according to the principle of minimum potential energy, the electromechanical coupling problem can be simplified to an eigenvalue problem, and the characteristic equation of the unit is:

[0124]

[0125] Wherein, the expression of the unit stiffness matrix and the unit mass matrix is as follows:

[0126]

[0127]

[0128]

[0129]

[0130] Assemble the global stiffness matrix and the mass matrix, and the final global generalized characteristic equation expression can be obtained:

[0131]

[0132] Wherein, And The displacement matrix and the electric potential matrix of the unit node are respectively, The mechanical stiffness matrix is, The piezoelectric coupling stiffness matrix is, The dielectric stiffness matrix is, The mass matrix is, The characteristic frequency is, The wave vector is. By scanning the wave vector of the first irreducible Brillouin zone and solve the above global generalized eigen-equation, the band structure curve of the piezoelectric phononic crystal rod can be obtained. This method is a general framework for solving the band structure curve of the phononic crystal rod, and the subsequent solution of the piezoelectric-viscoelastic system needs to adjust the material parameters based on this framework.

[0133] Step 34, eigen-equation and band structure curve solving of piezoelectric-viscoelastic phononic crystal rod

[0134] When studying piezoelectric-viscoelastic phononic crystal rods based on the standard linear solid model, the modulus expression of viscoelastic material needs to be solved to replace the Young's modulus of elastic material in the general framework. The specific process is as follows:

[0135] The modulus of viscoelastic material in the time domain is:

[0136]

[0137] where, is the initial modulus, is the relaxation modulus, which can be obtained according to the modulus ratio provided in step 1 solving, is the relaxation time, t represents time. By Fourier transform, the constitutive relation in the time domain is converted to the frequency domain:

[0138]

[0139] Further, the complex modulus expression is obtained:

[0140]

[0141] In order to more clearly understand the mechanical behavior of the material, it is separated into real and imaginary parts, i.e.:

[0142]

[0143] is the storage modulus, which characterizes the ability of the material to store elastic strain energy, is the loss modulus, which reflects the viscosity loss of the material. After introducing the viscoelastic material, the Young's modulus of the material changes, so when studying the piezoelectric-viscoelastic phononic crystal, the complex modulus of the viscoelastic material is replaced by the original Young's modulus of the elastic material, and is substituted into the final eigen-equation. The band structure curve of the piezoelectric-viscoelastic phononic crystal rod can be obtained by solving the eigenvalue problem using the same wave vector scanning method ( ).

[0144] Step 35, band gap value calculation

[0145] The band gap value can be solved according to the band curve obtained by the above method. The band gap is defined as the width of the non-overlapping frequency interval between adjacent energy bands in the band curve, and the expression is:

[0146]

[0147] wherein, is the band gap value, and are the first and second characteristic frequencies, respectively, and the second characteristic frequency, is the chromatic matrix representing the material distribution variable, is the wave vector.

[0148] In order to study the influence of piezoelectric properties and viscoelastic properties on the band curve, Figure 2 The band characteristics of the piezoelectric-elastic rod and the piezoelectric-viscoelastic rod under open circuit and short circuit two kinds of electrical boundary conditions are compared. Under the open circuit boundary condition, the first band gap and the second band gap width of the piezoelectric-elastic phononic crystal rod are 14.15 kHz and 16.73 kHz, respectively. After introducing the viscoelastic properties, the first band gap width becomes 13.37 kHz, and the second band gap width becomes 17.91 kHz. Under the short circuit boundary condition, the first band gap and the second band gap width of the elastic case are 13.90 kHz and 14.85 kHz, respectively. In the viscoelastic case, the first band gap width is 13.20 kHz, and the second band gap width is 16.64 kHz. The analysis shows that the addition of viscoelastic material changes the complex modulus characteristics of the material, which makes the frequency range of the three energy bands move to the low frequency direction as a whole, and the frequency shift of the first and second energy bands is particularly significant. This change directly leads to the significant widening of the second band gap. At the same time, under different electrical boundary conditions, the band curves of the piezoelectric-viscoelastic phononic crystal rod and the piezoelectric-elastic phononic crystal rod are changed, and the band gap range under the short circuit boundary condition is smaller than that under the open circuit condition. The change rule of the band curve under different electrical boundary conditions confirms the important regulation effect of the electrical boundary condition on the band gap characteristics of the phononic crystal.

[0149] Step 4: Evaluate fitness.

[0150] The larger the band gap range, the wider the sound wave frequency band that can be isolated by the phononic crystal. Therefore, the maximum absolute band gap width is taken as the objective function. According to the band gap value solving method in step 3, the absolute width of the first band gap is defined as the fitness function, that is, the upper boundary frequency of the first band gap minus the lower boundary frequency. According to the fitness function, the expression of the following objective function can be obtained:

[0151]

[0152] ​​​

[0153] wherein, represents a material distribution variable, is a feasible region of the material distribution; and are the first and second order eigenfrequencies, respectively, which are functions of the material distribution variable and wave vector . In this example, the first bandgap of the structure is chosen as the fitness function. The fitness value of the topological structure corresponding to each individual in the population is calculated, and the higher the fitness value, the better the topological structure represented by the individual, and the closer it is to the goal of maximizing the bandgap width we expect.

[0154] Step 5: Determine whether to converge.

[0155] After the target function calculation is completed, the next step of operation needs to be determined according to the convergence condition. When the number of iterations reaches the optimization iteration number set in step 1, the convergence condition is met, the algorithm ends the loop, and the topological structure represented by the current optimal individual is output; if the number of iterations does not reach the set value, the genetic operation of step 6 is continued to produce new offspring, and steps 3 to calculate the energy band curve, step 4 to evaluate the fitness, etc. are repeated until the convergence condition is met.

[0156] Step 6: Genetic operation.

[0157] When step 5 does not meet the convergence condition, the genetic operation is performed. Genetic algorithm is an intelligent optimization algorithm that simulates the natural selection and genetic mechanism in biological evolution, and its core idea is to iteratively optimize the individual genes in the population through selection, crossover and mutation operations. In the individual selection stage, the roulette wheel selection mechanism based on the fitness value is adopted, and the probability of an individual being selected is proportional to its fitness value calculated in step 4, ensuring that high-quality individuals have a greater chance of entering the next generation. To maintain population diversity and avoid loss of good genes, an elite preservation strategy is implemented, and the individual with the highest fitness in the current population is directly preserved to the next generation, and this individual does not participate in the crossover and mutation operations. The single-point crossover method is used for crossover operation, and two parent individuals are randomly selected and a crossover site is randomly determined on their chromosomes, and the gene segments after the crossover site are exchanged to generate new individuals. The mutation operation randomly flips the gene bit state on the chromosome with a preset probability of 0.05, i.e. changing "0" to "1" or "1" to "0", and the mutation operation prevents the algorithm from falling into a local optimal solution by introducing new gene mutations.

[0158] Step 7: Extract the optimal solution and verify the accuracy of the result calculation.

[0159] After the completion of the above-mentioned all optimization processes, the individual with the highest fitness is selected from the set of historical optimal individuals, and the corresponding material distribution is the final optimization result. The finite element method is used to independently verify the optimal solution, draw the energy band structure curve, compare the target function value with the theoretical calculation result, and evaluate the optimization effect.

[0160] The optimization result is shown in Figures 4 to 7 , and Figure 4 and Figure 5 are respectively the structure diagram and the energy band curve diagram of the piezoelectric-viscoelastic phononic crystal rod before and after optimization under open circuit condition. It can be seen that in the randomly generated structure before optimization, the two materials are discretely distributed, and the corresponding band gap is narrow, and the first band gap width is only 5.53 kHz. After optimization, the structure tends to be continuous distribution, and presents the distribution form of piezoelectric material wrapping viscoelastic material, which leads to the obvious change of the energy band curve, and the band gap width after optimization is greatly widened to 25.40 kHz, with an increase of 359.3%. Figure 6 and Figure 7 are respectively the structure diagram and the energy band curve diagram of the piezoelectric-viscoelastic phononic crystal rod before and after optimization under short circuit condition. Similar rules are also observed under short circuit condition, and the material distribution of the structure before optimization is discrete, and the band gap width is only 4.28 kHz. After optimization, the structure still presents the "sandwich" distribution of piezoelectric material at both ends and viscoelastic material in the middle, and the band gap is widened to 21.47 kHz, with an increase of 401.6%. As can be seen from the above, the optimization effect under open circuit condition is generally better than that under short circuit condition, because the stronger electromechanical coupling effect under open circuit is more conducive to the widening of the band gap. The optimized structure all presents the "sandwich" feature-piezoelectric phase is distributed at both ends of the structure unit, and the middle layer is composed of viscoelastic phase, and this structure effectively improves the performance of the phononic crystal rod in inhibiting the propagation of elastic waves in a specific frequency band.

[0161] Through the above specific embodiments, the genetic algorithm is used to realize the topological optimization of the piezoelectric-viscoelastic phononic crystal rod, and the first band gap absolute width is effectively expanded. In practical application, the material and algorithm parameters can be flexibly adjusted according to different engineering requirements to obtain the best performance of the phononic crystal rod. At the same time, the method and technology involved in the present application are not only suitable for the piezoelectric-viscoelastic phononic crystal rod, but also have certain reference significance and popularization value in the optimization design of other types of phononic crystal structures.

Claims

1. A genetic algorithm-based piezoelectric-viscoelastic phononic crystal topology optimization method, characterized by, The piezoelectric-viscoelastic phononic crystal topology optimization method based on the genetic algorithm comprises the following steps: Step 1, based on piezoelectric ceramic and epoxy resin as the construction of phononic crystal rod, and obtain the parameters of the phononic crystal rod and the genetic algorithm parameters; Step 2: the phononic crystal rod is discretized into a plurality of phononic crystal rod units, each unit corresponds to a gene site, a binary coding is used to represent the material distribution, and a density interpolation method is used to construct the mapping relationship between the material properties and the coding; Step 3: the band curve of each phononic crystal rod unit is calculated by using the finite element method, and the target band gap value is obtained based on the band curve; Step 4: taking the maximization of the absolute band gap width as the objective function, and evaluating the fitness, the optimal target is selected; Step 5: judge whether the convergence condition is met, if not, continue the genetic operation, generate new offspring, repeat steps 3 and 4; Step 6: genetic operation; Step 7: extract the optimal solution and verify the result calculation accuracy. 2.The method of claim 1, wherein, In step 1, the epoxy resin is used as a viscoelastic material, and a standard linear solid model is used, and the phononic crystal rod parameters include cell length, rod diameter, viscoelastic material modulus ratio and relaxation time; The genetic algorithm parameters include the population size of the genetic algorithm, the maximum iteration times, the crossover probability and the mutation probability. 3.The piezoelectric-viscoelastic phononic crystal topology optimization method based on genetic algorithm according to claim 1, wherein, In step 2, the binary coding is used to represent the material distribution of the phononic crystal rod: ； wherein "0" means that the unit is a viscoelastic material and "1" means that it is a piezoelectric material, and The distribution is the feasible region of the distribution of material A and material B, is the number of units; each represents a gene locus. 4.The method of claim 1, wherein, In step 3, in order to ensure the diversity of the population, the initial structure before optimization is a randomly generated binary coding matrix, the finite element method is used to solve the band curve of the initial structure, the objective function is calculated, and the genetic optimization design is carried out according to the calculation result; During the computation, the actual material properties of each element are mapped into the finite element model by a density interpolation method, using their encoded values to the finite element model: ； wherein and are interpolated Young's modulus and density matrix, , and , are the actual material parameters of the viscoelastic and piezoelectric material, respectively, is a chromosome matrix. 5.The method of claim 1, wherein, In step 3, it also includes: Step 31, construct the constitutive relation and basic equation; The constitutive relation matrix of linear piezoelectric material is expressed as: ； where is the elastic matrix for constant electric field, is the piezoelectric stress constant matrix, is the dielectric constant matrix, and are the stress and strain tensors, respectively, and are the electric displacement and electric field intensity vectors, respectively, , is the spatial coordinate along the length of the rod; is the electric potential; the electric displacement satisfies Gauss' law: ; According to Newton's second law, the balance equation is established: ； The constitutive equation is brought into the balance equation, and the following equation is obtained: ； wherein is the displacement field variable, is the displacement , is the spatial coordinate along the length of the rod. 6.The piezoelectric-viscoelastic phononic crystal topology optimization method based on genetic algorithm according to claim 5, wherein, In step 3, it also includes: Step 32, determine the boundary condition and interpolation function; The phononic crystal is a periodic structure, and the left and right boundaries of its unit cell satisfy the Bloch periodic boundary condition: ； is the cell length, is the wave vector, is the spatial coordinate along the rod length; inside the cell, the displacement and the electric potential are expressed by linear function interpolation: ； Let the cell length be , the shape function can be expressed as: ； Thus, the strain may be expressed by the shape function as: ； is written as: wherein ; by the same token, the electric field . 7.The piezoelectric-viscoelastic phononic crystal topology optimization method based on genetic algorithm according to claim 6, wherein, In step 3, it also includes: Step 33, calculate the characteristic equation and band curve solving framework; According to the principle of minimum potential energy, the electromechanical coupling problem is simplified to an eigenvalue problem, and the characteristic equation of the unit is: ； Wherein, the expression of the unit stiffness matrix and the unit mass matrix is as follows: ； ； ； ； Assemble the global stiffness matrix and mass matrix to obtain the final global generalized characteristic equation expression: ； where, and are the displacement matrix and the potential matrix of the unit cell, respectively, is the mechanical stiffness matrix, is the piezoelectric coupling stiffness matrix, is the dielectric stiffness matrix, is the mass matrix, is the characteristic frequency, is the wave vector; by scanning the wave vector through the first irreducible Brillouin zone and solving the above global generalized eigenvalue equation, the band curves of the piezoelectric elastic phononic crystal rod can be obtained. 8.The method of claim 7, wherein, In step 3, it also includes: Step 34, solve the characteristic equation and band curve of the piezoelectric-viscoelastic phononic crystal rod; When studying the piezoelectric-viscoelastic phononic crystal rod based on the standard linear solid model, the modulus expression of the viscoelastic material needs to be solved to replace the Young's modulus of the elastic material in the general framework; The modulus of the viscoelastic material in the time domain is: ； where, E0is the initial modulus, Erelis the relaxation modulus, trelis the relaxation time, t is time; by Fourier transformation, the constitutive relation in time domain is converted to frequency domain, further obtaining complex modulus, and finally separating it into real and imaginary parts: ； G' is the storage modulus, which characterizes the material's ability to store elastic strain energy, G" is the loss modulus, which embodies the material's viscous losses; After the introduction of the viscoelastic material, the Young's modulus of the material changes, and the complex modulus of the viscoelastic material replaces the Young's modulus of the original elastic material, and is substituted into the final characteristic equation. The eigenvalue problem is solved by using the same wave vector scanning method ( ) to obtain the band curve of the piezoelectric-viscoelastic phononic crystal rod. 9.The method of claim 8, wherein, In step 34, it also includes: Step 35, based on the band curve, the target band gap value is solved; According to the band curve, the target band gap value is solved, and the band gap is the width of the non-overlapping frequency interval between adjacent energy bands in the band curve, which is expressed as: ； is the band gap value, and are the first and second order eigenfrequencies, denotes the chromosome matrix, is the wave vector. 10.The method of claim 1, wherein the method is characterized by, In step 4, the absolute width of the first bandgap is defined as the fitness function with the objective function of maximizing the absolute bandgap width as i.e. the upper boundary frequency minus the lower boundary frequency of the first bandgap, according to the fitness function, the following objective function can be obtained: ； ； wherein, denotes a chromosome matrix, is a feasible region of material distribution; and are the (n+1)th and nth eigenfrequencies, respectively, are functions of the material distribution x and the wave vector k, denotes an objective function; The fitness value of the topological structure corresponding to each individual in the population is calculated, and the higher the fitness value, the better the topological structure represented by the individual, and the closer to the desired maximum band gap width target.

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