Topological optimization method of piezoelectric-viscoelastic phononic crystals based on genetic algorithm

Abstract

The application discloses a piezoelectric-viscoelastic phononic crystal topology optimization method based on a genetic algorithm, and comprises the following steps: in step 1, a phononic crystal rod is constructed based on piezoelectric ceramics and epoxy resin, and parameters of the phononic crystal rod and parameters of the genetic algorithm are obtained; in 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; in 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; in step 4, a maximum absolute band gap width is taken as an objective function, fitness is evaluated, and an optimal target is selected; in step 5, it is judged whether a convergence condition is met, if not, genetic operation is continuously performed to generate new offspring; in step 6, genetic operation is performed; and in step 7, an optimal solution is extracted and result calculation accuracy is verified.

Description

Technical Field

[0001] This invention relates to the field of intelligent phononic crystal optimization design technology, specifically a piezoelectric-viscoelastic phononic crystal topology optimization method based on genetic algorithm. By adjusting the material distribution through topology optimization design, the absolute width of the first bandgap is maximized, which is applicable to engineering fields such as vibration control and acoustic filtering. Background Technology

[0002] Phononic crystals, as artificially designed periodic composite materials, demonstrate significant application value in vibration control and noise suppression due to their bandgap characteristics. Traditional phononic crystal design primarily relies on empirical trial-and-error methods, repeatedly adjusting structural parameters to optimize bandgap performance. This approach is not only inefficient but also struggles to obtain globally optimal solutions. Topology optimization design methods, with their efficiency and flexibility in material distribution optimization, have become a crucial technique for improving the bandgap performance of phononic crystals. Furthermore, the fixed band structure of traditional phononic crystals makes it difficult to meet the requirements of frequency-varying elastic wave modulation. In recent years, the introduction of smart materials has opened new avenues for solving this problem, but several key issues remain to be addressed.

[0003] In terms of optimization algorithms, existing technologies have made some breakthroughs. Patent CN119358345 proposes a phonon crystal topology optimization method based on an improved genetic algorithm, which significantly improves computational efficiency by introducing a hash table to store calculated bandgap information. Patent CN116384138 proposes a phonon crystal topology optimization method with a specific bandgap, which achieves the design of a specific bandgap by establishing frequency band constraints and same-side frequency constraints. However, although these methods have improved the bandgap optimization effect to some extent, they still have obvious limitations. Their optimization objects are limited to a single material system, and although they broaden the bandgap, they cannot achieve dynamic adjustment of the bandgap range. In terms of material design, patent CN101952882 uses the damping properties of viscoelastic materials to broaden the bandgap, which has achieved some effect, but it does not have the ability to actively control the bandgap. The invention patent with publication number CN117912435 discloses an adjustable bandgap piezoelectric phononic crystal and its parameter optimization method. It achieves bandgap control by adjusting the material thickness and electric field strength. Although the method achieves the tunability of the bandgap frequency, it only relies on the adjustment of a single electrical parameter and uses an empirical trial-and-error method for parameter optimization. This is not only inefficient, but also makes it difficult to obtain the global optimal solution, resulting in a limited bandgap adjustment range and making it difficult to fully realize the performance potential of the material. Summary of the Invention

[0004] The purpose of this invention is to optimize the piezoelectric-viscoelastic phononic crystal rod through a genetic algorithm to maximize the absolute width of the first bandgap, thereby improving the phononic crystal rod's ability to control elastic waves and meeting the needs of vibration control and noise isolation in practical engineering.

[0005] This invention discloses a piezoelectric-viscoelastic phonon crystal topology optimization method based on a genetic algorithm. The piezoelectric-viscoelastic phonon crystal topology optimization method based on a genetic algorithm includes the following steps:

[0006] Step 1: Construct a phononic crystal rod based on piezoelectric ceramics and epoxy resin, and obtain the parameters of the phononic crystal rod and the genetic algorithm parameters;

[0007] Step 2: Discretize the phononic crystal rod into multiple phononic crystal rod units, each unit corresponding to a gene locus. Use binary coding to characterize the material distribution and use density interpolation method to construct the mapping relationship between material properties and coding.

[0008] Step 3: Calculate the band structure of each phononic crystal bar element using the finite element method, and obtain the target band gap value based on the band structure;

[0009] Step 4: Using maximizing the absolute bandgap width as the objective function, evaluate the fitness and select the optimal objective;

[0010] Step 5: Determine if the convergence condition is met. If the convergence condition is not met, continue the genetic operation to generate new offspring, and repeat steps 3 and 4.

[0011] Step 6: Perform genetic manipulation;

[0012] Step 7: Extract the optimal solution and verify the accuracy of the calculation results.

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

[0014] Genetic algorithm parameters include population size, maximum number of iterations, crossover probability, and mutation probability.

[0015] Furthermore, in step 2, the phonon crystal rods are characterized using binary encoding to represent the material distribution:

[0016] ;

[0017] In this context, "0" indicates that the unit is a viscoelastic material, and "1" indicates that it is a piezoelectric material. and The distribution is the feasible region of the distributions of material A and material B. The number of units; each It represents a gene locus.

[0018] Furthermore, in step 3, to ensure population diversity, the initial structure before optimization is a randomly generated binary encoding matrix. The finite element method is used to solve the band structure of the initial structure, the objective function is calculated, and genetic optimization design is performed based on the calculation results.

[0019] During the calculation, the actual material properties of each element are determined using density interpolation and their encoded values. Mapped to the finite element model:

[0020] ;

[0021] in, and These are the interpolated Young's modulus and density matrix. , and , These are the actual material parameters for viscoelastic and piezoelectric materials, respectively. It is a chromosome matrix.

[0022] Furthermore, step 3 also includes:

[0023] Step 31: Construct constitutive relations and fundamental equations;

[0024] The constitutive relation matrix of linear piezoelectric materials is expressed as follows:

[0025] ;

[0026] in, The elastic matrix under a constant electric field is The piezoelectric stress constant matrix is... Here is the dielectric constant matrix. and These are stress and strain tensors, respectively. and Electric displacement and electric field intensity vector , These are spatial coordinates along the length of the rod; Electric potential; electric displacement under conditions of no free charges. Satisfies Gauss's Law: .

[0027] Based on Newton's second law, establish the equilibrium equation:

[0028] ;

[0029] Substituting the constitutive equation into the equilibrium equation, we get:

[0030] ;

[0031] in, It is a displacement field variable. It is displacement , These are spatial coordinates along the length of the rod.

[0032] Furthermore, step 3 also includes:

[0033] Step 32: Determine the boundary conditions and interpolation function;

[0034] Phononic crystals are periodic structures, and the left and right boundaries of their unit cells satisfy the Bloch periodic boundary conditions.

[0035] ;

[0036] The length of a single cell. For wave vector, These are spatial coordinates along the length of the rod; within the element, displacement... and electric potential Represented using linear function interpolation:

[0037] ;

[0038] Let the unit length be Shape function This can be expressed as:

[0039] ;

[0040] Therefore, the response It can be expressed as a shape function:

[0041] ;

[0042] Notation: ,in Similarly, we can obtain the electric field. .

[0043] Furthermore, step 3 also includes:

[0044] Step 33: Calculate the characteristic equation and the solution framework for the band curve;

[0045] Based on the principle of minimum potential energy, the electromechanical coupling problem is simplified into an eigenvalue problem, and the characteristic equation of the element is:

[0046] ;

[0047] The expressions for the element stiffness matrix and the element mass matrix are as follows:

[0048] ;

[0049] ;

[0050] ;

[0051] ;

[0052] By assembling the global stiffness matrix and mass matrix, we obtain the final expression for the global generalized characteristic equation:

[0053] ;

[0054] in, and These are the displacement matrix and electric potential matrix of the element nodes, respectively. For mechanical stiffness matrix, This is the piezoelectric coupling stiffness matrix. Here is the dielectric stiffness matrix. For the quality matrix, For characteristic frequencies, For the wave vector; the wave vector obtained by scanning the first irreducible Brillouin zone. By solving the above global generalized characteristic equation, the band structure of the piezoelectric elastic phonon crystal rod can be obtained.

[0055] Furthermore, step 3 also includes:

[0056] Step 34: Solve the characteristic equations and band structure of the piezoelectric-viscoelastic phonon crystal rod;

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

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

[0059] ;

[0060] in, For the initial modulus, For relaxation modulus, For relaxation time, For time; through Fourier transform, the constitutive relation in the time domain is transformed to the frequency domain, further obtaining the complex modulus, and finally separating it into real and imaginary parts:

[0061] ;

[0062] Storage modulus, characterizing the ability of a material to store elastic strain energy. The loss modulus reflects the viscous loss of the material.

[0063] After introducing viscoelastic materials, the Young's modulus of the material changes. The composite modulus of the viscoelastic material replaces the Young's modulus of the original elastic material and is substituted into the final characteristic equation, using the same wave vector scanning method. Solving the eigenvalue problem is equivalent to obtaining the band structure of the piezoelectric-viscoelastic phonon crystal rod.

[0064] Furthermore, in step 34, the method also includes:

[0065] Step 35: Based on the band structure curve, solve for the target band gap value.

[0066] The target bandgap value is determined based on the band structure curve. The bandgap is the width of the non-overlapping frequency range between adjacent bands in the band structure curve, expressed as:

[0067] ;

[0068] This is the band gap value. and The first and the First-order characteristic frequencies, Represents a chromosome matrix. It is a wave vector.

[0069] Furthermore, in step 4, with maximizing the absolute bandgap width as the objective function, the absolute width of the first bandgap is defined as the fitness function. That is, the upper boundary frequency of the first bandgap minus the lower boundary frequency. According to the fitness function, the following objective function can be obtained:

[0070] ;

[0071] ;

[0072] in, Represents a chromosome matrix. It is the feasible region for material distribution; and and are the (n+1)th and nth order characteristic frequencies, respectively, which are functions of the material distribution x and the wave vector k. Represent the objective function;

[0073] Calculate the fitness value of the topology corresponding to each individual in the population. The higher the fitness value, the better the topology represented by that individual is, and the closer it is to the desired goal of maximizing the bandgap width.

[0074] The beneficial effects achieved by this invention are:

[0075] This invention significantly widens the absolute width of the first bandgap of a phononic crystal bar by incorporating viscoelastic and piezoelectric materials into its topology optimization design. During the optimization process, maximizing the absolute width of the first bandgap is used as the objective function, and variables such as material distribution, viscoelastic parameters, and the electric boundary conditions of the piezoelectric phase are comprehensively considered. An improved genetic algorithm-based optimization model is established. The improved genetic algorithm effectively enhances optimization efficiency and stability by introducing an elite retention strategy. Numerical simulation results show that the optimized piezoelectric-viscoelastic phononic crystal bar exhibits excellent bandgap characteristics, with a significantly improved absolute width of the first bandgap compared to traditional designs.

[0076] Furthermore, the electrical boundary conditions (including short-circuit and open-circuit conditions) applied to piezoelectric materials have a significant modulating effect on bandgap characteristics. To address this characteristic, the evolution of bandgap characteristics under different electrical boundary conditions was systematically investigated during the optimization process, and the optimization strategy was adjusted accordingly, further improving the optimization effect. The optimized structure effectively suppressed elastic wave propagation within a specific frequency range, providing a new solution for vibration control and noise isolation in engineering applications.

[0077] Compared with the prior art, the beneficial effects of the present invention are: the present invention significantly broadens the bandgap range of the phononic crystal rod by introducing viscoelastic materials, and at the same time, by reasonably selecting electrical boundary conditions and using an improved genetic algorithm to perform topology optimization on the piezoelectric-viscoelastic phononic crystal rod under different electrical boundary conditions, the material distribution is optimized, and the frequency range of the specified bandgap is significantly improved. Attached Figure Description

[0078] Figure 1 It is an optimized design flowchart that illustrates the iterative process of the genetic algorithm.

[0079] Figure 2 This is a schematic diagram of the structure of a piezoelectric-viscoelastic phonon crystal rod.

[0080] Figure 3 The diagrams show a comparison of the band structure curves of piezoelectric-elastic phononic crystal rods and piezoelectric-viscoelastic phononic crystal rods under different electrical boundary conditions. (a) Open circuit, (b) Short circuit.

[0081] Figure 4 These are comparison diagrams of the structure before and after optimization under open-circuit conditions: (a) before optimization, (b) after optimization.

[0082] Figure 5 The image shows a comparison of the band structure curves before and after optimization under open-circuit conditions. (a) Before optimization, (b) After optimization.

[0083] Figure 6These are comparison diagrams of the structure before and after optimization under short-circuit conditions: (a) before optimization, (b) after optimization.

[0084] Figure 7 These are comparison diagrams of the band structure before and after optimization under short-circuit conditions; (a) before optimization, (b) after optimization. Detailed Implementation

[0085] The present invention will be further described below with reference to specific embodiments, and the advantages and features of the present invention will become clearer as a result. However, these embodiments are merely exemplary and do not constitute any limitation on the scope of the present invention. Those skilled in the art should understand that modifications or substitutions can be made to the details and form of the technical solutions of the present invention without departing from the spirit and scope of the present invention, but all such modifications and substitutions fall within the protection scope of the present invention.

[0086] like Figure 1 As shown, this invention provides a piezoelectric-viscoelastic phonon crystal topology optimization method based on a genetic algorithm. This method drives material distribution optimization through a genetic algorithm, and combined with the special properties of piezoelectric-viscoelastic composite materials, ultimately obtains a phonon crystal structure with excellent bandgap characteristics. The specific implementation process is as follows:

[0087] Step 1: Set parameters and build the model

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

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

[0090]

[0091] is the vacuum permittivity.

[0092] Step 2: Population coding.

[0093] Based on the phonon crystal unit cell size in step 1, the phonon crystal rods are discretized into 30 phonon crystal rod units, each unit corresponding to a gene locus. To facilitate subsequent genetic operations to optimize material distribution, binary encoding is used to characterize the material configuration: "0" indicates that the unit is a viscoelastic material, and "1" indicates that the unit is a piezoelectric material. The specific encoding rules are as follows:

[0094]

[0095] in, and The distribution is the feasible region for the distribution of viscoelastic material A and piezoelectric material B. This represents the total number of units. Each Represents a gene locus, consisting of all Composition The matrix is ​​a chromosome, representing a phononic crystal rod topology.

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

[0097] To ensure population diversity, the initial structure before optimization is determined by a randomly generated binary encoding matrix. The finite element method is used to solve for the band structure of the initial structure. After calculating the objective function, genetic optimization design is carried out based on the calculation results. During the calculation process, density interpolation is used, combined with the encoded values ​​of the x matrix from step 2. The actual material properties of each element are mapped to the finite element model using the following interpolation formula:

[0098]

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

[0100] Based on the above material property mapping method, the description logic of the properties and mechanical behavior of the phonon crystal bar unit is further clarified: each phonon crystal bar unit contains two nodes, and its material properties (density, Young's modulus, piezoelectric constant, relative permittivity) can be obtained according to the encoded values ​​in step 2. Confirmed. For a one-dimensional phononic crystal rod composed of alternating piezoelectric and elastic materials, its mechanical behavior needs to be described by coupled field equations due to the electromechanical coupling effect. The derivation process is as follows:

[0101] Step 31, Constitutive relations and fundamental equations

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

[0103]

[0104] in, The elastic matrix under a constant electric field is The piezoelectric stress constant matrix is... Here is the dielectric constant matrix. and These are stress and strain tensors, respectively. and Electric displacement and electric field intensity vector , These are spatial coordinates along the length of the rod; Electric potential; electric displacement under conditions of no free charges. Satisfies Gauss's Law: .

[0105] Based on Newton's second law, establish the equilibrium equation:

[0106]

[0107] Substituting the constitutive equation into the equilibrium equation, we get:

[0108]

[0109] in, It is a displacement field variable. It is displacement , These are spatial coordinates along the length of the rod.

[0110] Step 32, Boundary conditions and interpolation functions

[0111] Phononic crystals are periodic structures, and the left and right boundaries of their unit cells satisfy the Bloch periodic boundary conditions.

[0112]

[0113] The length of a single cell. For wave vector, These are spatial coordinates along the length of the rod. Within the element, the displacement... and electric potential Represented using linear function interpolation:

[0114]

[0115] Let the unit length be Shape function This can be expressed as:

[0116]

[0117] Therefore, the response It can be expressed as a shape function:

[0118]

[0119] Notation: ,in

[0120]

[0121] Similarly, we can obtain the electric field .

[0122] Step 33: Framework for solving the characteristic equation and band curve

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

[0124]

[0125] The expressions for the element stiffness matrix and the element mass matrix are as follows:

[0126]

[0127]

[0128]

[0129]

[0130] By assembling the global stiffness matrix and mass matrix, we can obtain the final expression for the global generalized characteristic equation:

[0131]

[0132] in, and These are the displacement matrix and electric potential matrix of the element nodes, respectively. For mechanical stiffness matrix, This is the piezoelectric coupling stiffness matrix. Here is the dielectric stiffness matrix. For the quality matrix, For characteristic frequencies, The wave vector is obtained by scanning the wave vector of the first irreducible Brillouin zone. By solving the aforementioned global generalized characteristic equation, the band structure of the piezoelectric elastic phononic crystal rod can be obtained. This method serves as a general framework for solving the band structure of phononic crystal rods; subsequent solutions for piezoelectric-viscoelastic systems require adjustments to material parameters based on this framework.

[0133] Step 34: Solving the characteristic equations and band structure of the piezoelectric-viscoelastic phonon crystal rod.

[0134] When studying piezoelectric-viscoelastic phonon crystal rods based on the standard linear solid model, it is necessary to solve for the modulus expression of viscoelastic materials to replace the Young's modulus of elastic materials in the general framework. The specific process is as follows:

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

[0136]

[0137] in, For the initial modulus, The relaxation modulus can be determined based on the modulus ratio provided in step 1. Please solve. Let t represent the relaxation time. The constitutive relation in the time domain is transformed to the frequency domain using a Fourier transform:

[0138]

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

[0140]

[0141] To better understand the mechanical behavior of materials, it is separated into real and imaginary parts, namely:

[0142]

[0143] Storage modulus, characterizing the ability of a material to store elastic strain energy. The loss modulus reflects the viscous loss of the material. The Young's modulus changes after the introduction of viscoelastic materials. Therefore, in studying piezoelectric-viscoelastic phonon crystals, the composite modulus of the viscoelastic material replaces the Young's modulus of the original elastic material and is substituted into the final characteristic equation, using the same wave vector scanning method. The band structure of the piezoelectric-viscoelastic phonon crystal rod can be obtained by solving the eigenvalue problem.

[0144] Step 35, Bandgap value calculation

[0145] The target bandgap value can be calculated from the band structure obtained using the above method. The bandgap is defined as the width of the non-overlapping frequency range between adjacent bands in the band structure, and its expression is:

[0146]

[0147] in, This is the band gap value. and The first and the First-order characteristic frequencies, The chromosome matrix represents the material distribution variables. It is a wave vector.

[0148] To investigate the effects of piezoelectric and viscoelastic properties on band structure, Figure 2 This paper compares the band structure characteristics of piezoelectric-elastic and piezoelectric-viscoelastic rods under both open-circuit and short-circuit electrical boundary conditions. Under the open-circuit boundary condition, the first and second band gaps of the piezoelectric-elastic phononic crystal rod are 14.15 kHz and 16.73 kHz, respectively. After introducing viscoelastic properties, the first band gap becomes 13.37 kHz and the second band gap becomes 17.91 kHz. Under the short-circuit boundary condition, the first and second band gaps are 13.90 kHz and 14.85 kHz, respectively, under the elastic condition; under the viscoelastic condition, the first band gap is 13.20 kHz and the second band gap is 16.64 kHz. Analysis shows that the addition of viscoelastic material changes the complex modulus characteristics of the material, causing the frequency range of all three bands to shift towards lower frequencies overall, with the frequency shift of the first and second bands being particularly significant. This change directly leads to a significant widening of the second band gap. Meanwhile, under different electrical boundary conditions, the bandgap curves of both piezoelectric-viscoelastic phononic crystal rods and piezoelectric-elastic phononic crystal rods changed, with the bandgap range under short-circuit boundary conditions being smaller than that under open-circuit conditions. The variation law of the bandgap curves under different electrical boundary conditions confirms the important regulatory role of electrical boundary conditions on the bandgap characteristics of phononic crystals.

[0149] Step 4: Assess fitness.

[0150] The larger the bandgap, the wider the acoustic frequency range that the phononic crystal can isolate. Therefore, maximizing the absolute bandgap width is the objective function. Based on the bandgap value calculation method in step 3, the absolute width of the first bandgap is defined as the fitness function. That is, the upper boundary frequency of the first bandgap minus the lower boundary frequency. Based on the fitness function, the following objective function can be obtained. The expression:

[0151]

[0152]

[0153] in, Represents the material distribution variable. It is the feasible region for material distribution; and The first and the The first-order characteristic frequency is related to the material distribution variable. Wawaya The fitness function is set as the first bandgap of the structure. The fitness value of the topology corresponding to each individual in the population is calculated. A higher fitness value indicates a better topology, closer to our desired maximum bandgap width.

[0154] Step 5: Determine if convergence has occurred.

[0155] After the objective function calculation is completed, the next step needs to be determined based on the convergence status. 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 outputs the topology represented by the current best individual; if the number of iterations has not reached the set value, the genetic operation in step 6 is continued to generate new offspring, and the steps of calculating the band structure in step 3 and evaluating fitness in step 4 are repeated until the convergence condition is met.

[0156] Step 6: Genetic manipulation.

[0157] If the convergence condition is not met in step 5, perform the genetic operation. A genetic algorithm is an intelligent optimization algorithm that simulates the natural selection and genetic mechanisms in biological evolution. Its core idea is to manipulate the genes of individuals in a population through operations such as selection, crossover, and mutation. Iterative optimization is performed. During the individual selection phase, a roulette wheel selection mechanism based on fitness values ​​is adopted. 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 the loss of superior genes, an elite retention strategy is implemented, directly retaining the individual with the highest fitness ranking in the current population to the next generation; this individual does not participate in crossover or mutation operations. The crossover operation uses a single-point crossover method, randomly selecting two parent individuals and randomly determining crossover sites on their chromosomes, exchanging the gene segments after the crossover sites to generate new individuals. The mutation operation randomly flips the gene position state on the chromosome with a preset probability of 0.05, that is, changing "0" to "1" or "1" to "0". The mutation operation prevents the algorithm from getting trapped in local optima by introducing new gene mutations.

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

[0159] After all the above optimization processes are completed, the individual with the highest fitness is selected from the set of historical best individuals, and its corresponding material distribution is the final optimization result. The optimal solution is independently verified using the finite element method, the band structure curve is plotted, the objective function value is compared with the theoretical calculation results, and the optimization effect is evaluated.

[0160] Optimization results are as follows Figures 4 to 7 As shown, Figure 4 and Figure 5 The images show the structure and band structure of the piezoelectric-viscoelastic phonon crystal rod under open-circuit conditions, before and after optimization. It can be seen that in the randomly generated structure before optimization, the two materials are discretely distributed, resulting in a narrow band gap; the first band gap width is only 5.53 kHz. After optimization, the structure tends to be continuously distributed, exhibiting a distribution pattern of piezoelectric material encapsulating viscoelastic material, leading to a significant change in the band structure. The optimized band gap width is greatly widened to 25.40 kHz, an increase of 359.3%. Figure 6 and Figure 7 The images show the structure and band structure of the piezoelectric-viscoelastic phononic crystal rod before and after optimization under short-circuit conditions. A similar pattern was observed under short-circuit conditions: the material distribution of the unoptimized structure was discrete, and the band gap was only 4.28 kHz. The optimized structure still exhibited a "sandwich" distribution with piezoelectric material at both ends and viscoelastic material in the middle, but the band gap widened to 21.47 kHz, an increase of 401.6%. In summary, the optimization effect under open-circuit conditions is generally better than under short-circuit conditions. This is because the stronger electromechanical coupling effect under open-circuit conditions is more conducive to band gap widening. The optimized structures all exhibit a "sandwich" characteristic—the piezoelectric phase is distributed at both ends of the structural unit, while the middle layer is composed of a viscoelastic phase. This structure effectively improves the performance of the phononic crystal rod in suppressing elastic wave propagation in a specific frequency band.

[0161] This invention, through the specific embodiments described above, utilizes a genetic algorithm to achieve topology optimization of piezoelectric-viscoelastic phononic crystal rods, effectively expanding their first bandgap absolute width. In practical applications, material and algorithm parameters can be flexibly adjusted according to different engineering requirements to obtain optimal phononic crystal rod performance. Furthermore, the methods and techniques involved in this invention are not only applicable to piezoelectric-viscoelastic phononic crystal rods, but also have certain reference value and promotional value in the optimization design of other types of phononic crystal structures.

Claims

1. A piezoelectric-viscoelastic phonon crystal topology optimization method based on genetic algorithm, characterized in that, The piezoelectric-viscoelastic phonon crystal topology optimization method based on genetic algorithm includes the following steps: Step 1: Construct a phononic crystal rod based on piezoelectric ceramics and epoxy resin, and obtain the parameters of the phononic crystal rod and the genetic algorithm parameters; Step 2: Discretize the phononic crystal rod into multiple phononic crystal rod units, each unit corresponding to a gene locus. Use binary coding to characterize the material distribution and use density interpolation method to construct the mapping relationship between material properties and coding. Step 3: Calculate the band structure of each phononic crystal bar element using the finite element method, and obtain the target band gap value based on the band structure; Step 4: Using maximizing the absolute bandgap width as the objective function, evaluate the fitness and select the optimal objective; Step 5: Determine if the convergence condition is met. If the convergence condition is not met, continue the genetic operation to generate new offspring, and repeat steps 3 and 4. Step 6: Perform genetic manipulation; Step 7: Extract the optimal solution and verify the accuracy of the calculation results; Step 3 also includes: Step 31: Construct constitutive relations and fundamental equations; The constitutive relation matrix of linear piezoelectric materials is expressed as follows: ； in, The elastic matrix under a constant electric field is The piezoelectric stress constant matrix is... Here is the dielectric constant matrix. and These are stress and strain tensors, respectively. and Electric displacement and electric field intensity vector , These are spatial coordinates along the length of the rod; Electric potential; electric displacement under conditions of no free charges. Satisfies Gauss's Law: ; Based on Newton's second law, establish the equilibrium equation: ； Substituting the constitutive equation into the equilibrium equation, we get: ； in, It is a displacement field variable. It is displacement ; Step 32: Determine the boundary conditions and interpolation function; Step 33: Calculate the characteristic equation and the solution framework for the band curve; Step 34: Solve the characteristic equations and band structure of the piezoelectric-viscoelastic phonon crystal rod; Step 35: Based on the band structure curve, solve for the target band gap value.

2. The piezoelectric-viscoelastic phonon crystal topology optimization method based on genetic algorithm according to claim 1, characterized in that, In step 1, epoxy resin is used as a viscoelastic material, and a standard linear solid model is adopted. The parameters of the phononic crystal rod include unit cell length, rod diameter, viscoelastic material modulus ratio, and relaxation time. Genetic algorithm parameters include population size, maximum number of iterations, crossover probability, and mutation probability.

3. The piezoelectric-viscoelastic phonon crystal topology optimization method based on genetic algorithm according to claim 1, characterized in that, In step 2, the phonon crystal rods are characterized by binary encoding to represent the material distribution: ； Where "0" indicates that the unit is a viscoelastic material, and "1" indicates that it is a piezoelectric material. and The distribution is the feasible region of the distributions of material A and material B. The number of units; each It represents a gene locus.

4. The piezoelectric-viscoelastic phonon crystal topology optimization method based on genetic algorithm according to claim 1, characterized in that, In step 3, to ensure population diversity, the initial structure before optimization is a randomly generated binary encoding matrix. The finite element method is used to solve the band structure of the initial structure, the objective function is calculated, and genetic optimization design is performed based on the calculation results. During the calculation, the actual material properties of each unit are determined through density interpolation, using each gene locus. The values ​​of are mapped to the finite element model: ； in, and These are the interpolated Young's modulus and density matrix. , and , These are the actual material parameters for viscoelastic and piezoelectric materials, respectively. It is a chromosome matrix.

5. The piezoelectric-viscoelastic phonon crystal topology optimization method based on genetic algorithm according to claim 1, characterized in that, Step 3 also includes: Phononic crystals are periodic structures, and the left and right boundaries of their unit cells satisfy the Bloch periodic boundary conditions. ； The length of a single cell. For wave vector, These are spatial coordinates along the length of the rod; within the element, displacement... and electric potential Represented using linear function interpolation: ； Let the unit length be Shape function This can be expressed as: ； Therefore, strain It can be expressed as a shape function: ； Notation: ,in Similarly, we can obtain the electric field. .

6. The piezoelectric-viscoelastic phonon crystal topology optimization method based on genetic algorithm according to claim 5, characterized in that, Step 3 also includes: Based on the principle of minimum potential energy, the electromechanical coupling problem is simplified into an eigenvalue problem, and the characteristic equation of the element is: ； The expressions for the element stiffness matrix and the element mass matrix are as follows: ； ； ； ； By assembling the global stiffness matrix and mass matrix, we obtain the final expression for the global generalized characteristic equation: ； in, and These are the displacement matrix and electric potential matrix of the element nodes, respectively. For mechanical stiffness matrix, This is the piezoelectric coupling stiffness matrix. Here is the dielectric stiffness matrix. For the quality matrix, For characteristic frequencies, For the wave vector; the wave vector obtained by scanning the first irreducible Brillouin zone. By solving the above global generalized characteristic equation, the band structure of the piezoelectric elastic phonon crystal rod can be obtained.

7. The piezoelectric-viscoelastic phonon crystal topology optimization method based on genetic algorithm according to claim 6, characterized in that, Step 3 also includes: When studying piezoelectric-viscoelastic phonon crystal rods based on the standard linear solid model, it is necessary to solve the modulus expression of viscoelastic materials to replace the Young's modulus of elastic materials in the general framework; The expression for the modulus of a viscoelastic material in the time domain is: ； in, For the initial modulus, For relaxation modulus, For relaxation time, For time; through Fourier transform, the constitutive relation in the time domain is transformed to the frequency domain, further obtaining the complex modulus, and finally separating it into real and imaginary parts: ； Storage modulus, characterizing the ability of a material to store elastic strain energy. The loss modulus reflects the viscous loss of the material. After introducing viscoelastic materials, the Young's modulus of the material changes. The composite modulus of the viscoelastic material replaces the Young's modulus of the original elastic material and is substituted into the final characteristic equation, using the same wave vector scanning method. Solving the eigenvalue problem is equivalent to obtaining the band structure of the piezoelectric-viscoelastic phonon crystal rod.

8. The piezoelectric-viscoelastic phonon crystal topology optimization method based on genetic algorithm according to claim 7, characterized in that, Step 3 also includes: The target bandgap value is determined based on the band structure curve. The bandgap is the width of the non-overlapping frequency range between adjacent bands in the band structure curve, expressed as: ； This is the band gap value. and The first and the First-order characteristic frequencies, Represents a chromosome matrix. It is a wave vector.

9. The piezoelectric-viscoelastic phonon crystal topology optimization method based on genetic algorithm according to claim 1, characterized in that, In step 4, the objective function is to maximize the absolute bandgap width, and the absolute width of the first bandgap is defined as the fitness function. That is, the upper boundary frequency of the first bandgap minus the lower boundary frequency. According to the fitness function, the following objective function can be obtained: ； ； in, Represents a chromosome matrix. It is the feasible region for material distribution; and and are the (n+1)th and nth order characteristic frequencies, respectively, which are functions of the material distribution x and the wave vector k. Represent the objective function; Calculate the fitness value of the topology corresponding to each individual in the population. The higher the fitness value, the better the topology represented by that individual is, and the closer it is to the desired goal of maximizing the bandgap width.

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