Design method and system of piezoelectric metamaterial for realizing higher-performance energy collection by utilizing topological optimization

Abstract

According to the design method and system of the piezoelectric metamaterial capable of achieving energy collection with higher performance through topological optimization, the piezoelectric performance can be remarkably improved by improving the rigidity and energy collection capacity of the material through the design of the still water coupling coefficient, and therefore the energy conversion efficiency is improved. Experiment and numerical simulation results show that the design scheme taking the static pressure coupling coefficient as the target can significantly improve the static pressure coupling coefficient of the piezoelectric material. Compared with a traditional negative Poisson's ratio material, the optimized piezoelectric microstructure shows better rigidity and energy collection capability.

Description

TECHNICAL FIELD

[0001] The present application belongs to the field of metamaterial design, and particularly relates to a design method of piezoelectric metamaterials. BACKGROUND

[0002] Negative Poisson's ratio materials have always been a focus of research on piezoelectric energy harvesting. It is a class of materials that exhibit different deformation characteristics from conventional materials when subjected to stress. For conventional materials, stretching will cause contraction in the transverse direction, while compression will cause expansion in the transverse direction, and the Poisson's ratio is positive. However, negative Poisson's ratio materials will expand in the transverse direction when stretched, and will contract in the transverse direction when compressed, exhibiting a negative Poisson's ratio. Negative Poisson's ratio materials have great potential in the field of piezoelectric energy harvesting, because their unique structural design can usually cause a larger range of deformation. The use of negative Poisson's ratio effect can greatly enhance the transverse piezoelectric coefficient, enhance the output performance of piezoelectric nanogenerators (PENG), and even develop new wearable PEH metamaterials. Especially in the field of hydrophones, the application of negative Poisson's ratio materials in hydrophones significantly improves the performance of the device in ocean exploration, acoustic monitoring and underwater communication by enhancing the sensitivity, expanding the frequency response range, improving the pressure resistance and durability, and also optimizes the stress distribution and improves the energy harvesting efficiency, which is an important innovation in the design of future underwater acoustic devices. A piezoelectric active metamaterial with negative Poisson's ratio and zero Poisson's ratio characteristics is proposed to be applied to hydrophones, and a lightweight actuator and sensor with adjustable properties are realized.

[0003] The above method for designing piezoelectric energy harvesting metamaterials mainly focuses on optimizing the macrostructure and known targets (such as negative Poisson's ratio). For some applications, such as hydrophones, although negative Poisson's ratio materials can improve their performance, they are not necessarily the optimal solution to improve their performance. SUMMARY

[0004] The purpose of the present application is to provide a piezoelectric metamaterial optimization design method with performance improvement.

[0005] To achieve the above purpose, some embodiments of the present application take the z direction as the polarization direction, and propose a high static pressure coupling coefficient structure. Optimization of the static pressure coupling coefficient can obtain a piezoelectric microstructure with higher stiffness and better piezoelectric performance.

[0006] In some embodiments, a method for designing a piezoelectric metamaterial with higher performance energy harvesting using topology optimization is presented, which includes the steps of: defining a design domain of the piezoelectric metamaterial, with polarization direction along z-axis; applying topology optimization to optimize hydrostatic coupling coefficient of the piezoelectric metamaterial; discretizing the design domain into grid cells, each grid cell assigned a density value representing material distribution; interpolating material properties including elastic, piezoelectric and dielectric tensors from cell density using a solid isotropic material with penalization (SIMP) model; introducing a penalization factor to enforce density value to be binary to ensure manufacturability; imposing symmetry constraints and volume constraints to guarantee manufacturability and structural stability; solving the optimization problem using numerical algorithm to maximize hydrostatic coupling coefficient, thereby improving stiffness and energy harvesting efficiency of the material.

[0007] In some embodiments, the topology optimization targets hydrostatic coupling coefficient to achieve higher stiffness and better piezoelectric performance than negative Poisson's ratio materials.

[0008] In some embodiments, the topology optimization considers different stiffness and volume constraints to generate a variety of piezoelectric microstructures suitable for specific application requirements.

[0009] In some embodiments, the method further includes: predicting effective performance of the piezoelectric metamaterial by homogenization method; and simulating mechanical and electrical responses using a two-dimensional finite element model.

[0010] In some embodiments, wherein the SIMP model interpolates material properties using the following equations: elastic tensor c H , piezoelectric tensor e H , and dielectric tensor k H :

[0011]

[0012] wherein p is cell density, P c , P e , P k are penalization factors, C mat1 , e mat1 , k mat1 , and C mat2 , e mat2 , k mat2 are properties of piezoelectric material and void material, respectively.

[0013] In some embodiments, wherein the optimization problem is formulated as: objective function is to maximize hydrostatic coupling coefficient; constraints are volume constraint, stiffness constraint, and equilibrium equation.

[0014] In some embodiments, the method further includes: verifying the optimized piezoelectric microstructure through numerical simulation and experiment; comparing the performance of the optimized piezoelectric metamaterial with that of a negative Poisson's ratio material to demonstrate superior stiffness and energy collection efficiency.

[0015] This application proposes a method for designing piezoelectric metamaterials using topology optimization to achieve higher-performance energy harvesting. By optimizing the hydrostatic coupling coefficient through topology optimization, the energy conversion efficiency of the piezoelectric material is significantly improved. Designing for the hydrostatic coupling coefficient significantly enhances piezoelectric performance, increasing the material's stiffness and energy harvesting capabilities. Experimental and numerical simulation results confirm the effectiveness of the proposed optimization method, demonstrating that piezoelectric metamaterials designed using these optimizations outperform conventional materials, including those with a negative Poisson's ratio. The optimized piezoelectric microstructures exhibit superior mechanical properties and enhanced energy harvesting performance. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] Figure 1 is a schematic diagram of a piezoelectric film including an upper electrode and a lower electrode;

[0017] Figure 2 Diagram of the process for designing piezoelectric metamaterials using topology optimization;

[0018] Figure 3 Optimize flow chart for piezoelectric microstructures;

[0019] Figure 4(a) is a schematic diagram of an optimized single unit cell.

[0020] Figure 4(b) is a schematic diagram of the optimized 3×3 unit cell array;

[0021] Figure 4(c) is a schematic diagram of the optimized design sample;

[0022] Figure 5(a) is a schematic diagram of a single negative Poisson's ratio unit cell;

[0023] Figure 5(b) is a schematic diagram of a 3×3 negative Poisson's ratio unit cell array;

[0024] Figure 5(c) is a schematic diagram of the negative Poisson's ratio design sample;

[0025] Figure 6(a) is a schematic diagram showing the results of the topology optimization of the water coupling coefficient verified by finite element software;

[0026] Figure 6(b) is a schematic diagram showing the results of Poisson's ratio topology optimization verified by finite element software;

[0027] Figure 7(a) is a schematic diagram of the stress performance verification of a large-scale piezoelectric structure with optimized hydrostatic coupling coefficient;

[0028] Figure 7(b) is a schematic diagram of displacement performance verification;

[0029] Figure 7(c) is a schematic diagram of the verification of the electric potential performance;

[0030] Figure 7(d) is a schematic diagram of the verification of the electric field norm performance.

[0031] Figure 8(a) is a schematic diagram of the verification of the stress performance of the large-scale piezoelectric structure with optimized negative Poisson's ratio;

[0032] Figure 8(b) is a schematic diagram of the verification of the displacement performance;

[0033] Figure 8(c) is a schematic diagram of the verification of the electric potential performance;

[0034] Figure 8(d) is a schematic diagram of the verification of the electric field norm performance.

[0035] Figure 9(a) is a schematic diagram of the verification of the stress performance of the large-scale piezoelectric structure without optimization;

[0036] Figure 9(b) is a schematic diagram of the verification of the displacement performance;

[0037] Figure 9(c) is a schematic diagram of the verification of the electric potential performance;

[0038] Figure 9(d) is a schematic diagram of the verification of the electric field norm performance.

[0039] Figure 10 Optimized configuration under different stiffness constraints (direction 1 is horizontal and direction 3 is vertical)

[0040] Figure 11 Optimized configuration under different volume constraints (direction 1 is horizontal and direction 3 is vertical)

[0041] Figure 12(a) is a schematic diagram of the size of the piezoelectric sheet raw material;

[0042] Figure 12(b) is a schematic diagram of the piezoelectric sheet raw material;

[0043] Figure 13 Schematic diagram of the performance comparison of the optimized piezoelectric sheet and the negative Poisson's ratio piezoelectric sheet under d31, d33, and dh coefficients;

[0044] Figure 14(a) is a comparison curve diagram of the performance of the optimized piezoelectric sheet and the experiment under dh coefficient;

[0045] Figure 14(b) is a comparison curve diagram of the performance of the optimized piezoelectric sheet and the experiment under d31 coefficient;

[0046] Figure 14(c) is a comparison curve diagram of the performance of the optimized piezoelectric sheet and the experiment under d33 coefficient. DETAILED DESCRIPTION

[0047] The embodiments of the present application will be described in detail below with reference to the accompanying drawings.

[0048] The structure of the following description is as follows. First, in Section I, the homogenization method for predicting the effective properties of piezoelectric microstructure composites is introduced. In Section II, the optimization formulation and numerical implementation are described. The optimization results are given and discussed in Section III. In Section IV, the piezoelectric properties of the optimized piezoelectric microstructure composites are verified by numerical simulation and experiment.

[0049] I. Piezoelectricity

[0050] In this application, both direct and inverse piezoelectric effects are used. Direct piezoelectric effect refers to the phenomenon that certain crystalline materials generate electric charges inside the material when subjected to mechanical stress. On the contrary, inverse piezoelectric effect refers to the phenomenon that the material undergoes mechanical deformation when subjected to electric field.

[0051] In this application, the assumption of silva for piezoelectric materials is adopted for physical modeling, i.e., piezoelectric materials only respond linearly to electric field, electric displacement, mechanical stress and mechanical strain, and the changes of temperature and magnetic field are not considered in this application. The following theoretical formulas are outlined according to IEEE piezoelectric standards. The stress-charge form of piezoelectric medium constitutive relation is:

[0052] T = c E ε - eE,

[0053] D = e t ε + κ S E, (1)

[0054] In the above equations, the superscript t is used to denote the transpose of a matrix. The symbol T represents the mechanical stress tensor, ε represents the mechanical strain tensor, D represents the electric charge vector, and E represents the electric field vector. c E represents the elastic stiffness tensor under constant electric field, e and κ S are the piezoelectric tensor and dielectric tensor under constant strain, respectively.

[0055] In addition, the constitutive equation can also be expressed in another form:

[0056] ε = s E T - dE,

[0057] D = d t T + κ T E, (2)

[0058] where s E is the compliance tensor under short circuit condition, k T is the dielectric tensor of the clamping body, and d is the piezoelectric stress tensor. The relationship between equations (1) and (2) is as follows:

[0059] s E = (c E ) -1 , κ T= κ s + d t < s E ) -1 d, d = (s E ) e. (3)

[0060] 1.1 Piezoelectric homogenization

[0061] Considering the standard homogenization procedure, the two-dimensional cell is defined as Y = [0, Y1] x [0, Y2] and the material functions and are defined as Y-periodic functions:

[0062] c Eα (x) = c E (x, y), e α (x) = e(x, y), κ Sα (x) = κ S (x, y),

[0063] c Eα (x, y) = c E (x, y + Y), e α (x, y) = e(x, y + Y),

[0064] k Sα (x, y) = κ S (x, y + Y) (4)

[0065] where y = x / α, α > 0 is a small parameter representing the microscale (composite microstructure scale) at which the properties change, and x and y are the coordinates associated with the macro- and micro-dimensions of the composite, respectively.

[0066] The asymptotic expansion of the piezoelectric composite displacement u and electric potential φ gives:

[0067] u α (x, y) = u0(x) + αu1(x, y),

[0068] φ α (x, y) = φ0(x) + αφ1(x, y), (5)

[0069] Since the working wavelength is much larger than the assumed cell size, only the first order variation terms are considered. Considering that u1 and φ1 are periodic, the strain gradient and electric displacement gradient can be expressed as:

[0070]

[0071] where ε α is the mechanical strain, then we have:

[0072]

[0073] According to some theoretical derivations and algebraic manipulations, see

[46] for details, the uniform property equation can be obtained:

[0074]

[0075] Where χ(x,y) is the characteristic displacement of the unit cell, R(x,y) is the characteristic potential of the unit cell, and ψ(x,y) and Φ(x,y) are the characteristic coupling functions of the unit cell. All of these functions are Y-periodic characteristic functions and the relationship between u1 and φ1 is as follows:

[0076]

[0077] 1.2 Piezoelectric finite element model

[0078] In this work, a two-dimensional model is considered. As a convention, the polarization axis of the piezoelectric material is assumed to be in the z (or 3) direction. This application assumes that the electromechanical system is linear, the electrodes are perfectly conductive, and the potential variation through the thickness direction is linear. The potential on each electrode is constant (equipotential condition), and the lower electrode is grounded.

[0079] In this section, the physical model includes the design domain and the upper and lower electrodes, such as Figure 1 As shown in Figure 2. The design domain is discretized using four-node rectangular elements. However, in the finite element modeling, the thickness of the piezoelectric patch is assumed to be h = 1e-4m, the thickness of the electrode is ignored, and only the piezoelectric patch is discretized. Consider that each node has two mechanical degrees of freedom and one electrical degree of freedom in the 1-direction and the 3-direction. The strain field and electric field of each element can be expressed using the mechanical and electrical degrees of freedom:

[0080]

[0081] Where, and are the strain field and electric field of each unit, B u is the strain displacement matrix, B φ is the electric displacement matrix, u and φ are the mechanical displacement vector and electric potential scalar. u and B φ The specific expression is as follows:

[0082]

[0083] B φ =1 / h, (14)

[0084] Among them B iFor details on (i=1, 2, 3, 4) and the value of the Jacobi determinant |J|, see "Kattan PI. MATLAB guide to finite elements: an interactive approach. Springer Science & Business Media; 2010".

[0085] Ignoring the damping effect, the linear differential equation of a single element can now be obtained by Hamilton's variation principle:

[0086]

[0087] Among them, k uu is the mechanical stiffness matrix, k uφ is the piezoelectric coupling matrix, k φu It is k uφ The transpose of k φφ is the dielectric stiffness matrix, f is the external force, and q is the charge.

[0088] These components are derived in the following form:

[0089]

[0090] Where A is the unit area, ξ and η are natural coordinates. This application uses rectangular units, and the values ​​of shape functions are not given here.

[0091] After assembling Equation (15), K uu It is k uu The assembled matrix, K uφ It is k uφ The assembled matrix, K φu It is k φu The assembled matrix, K φφ It is k φφ After assembling the matrix, the finite element equation of the entire design domain is obtained as:

[0092] 2. Piezoelectric Topology Optimization

[0093] A key concept of topology optimization is to first discretize the design domain, then calculate the material distribution of each unit in the design domain through the optimization algorithm. After post-processing the optimized configuration, the unit is periodically arrayed and finally manufactured. Note that during the manufacturing process, the positions of the upper and lower electrodes need to be reserved, such as Figure 2 shown.

[0094] In a discrete problem, the design space is usually discretized into grid cells. The density value of each cell usually represents whether the cell contains material or the distribution of material in the cell. The density value of each cell can only take 0 or 1 (i.e. void material and piezoelectric material). In the actual solving process, the direct use of binary density design of 0 or 1 may lead to computational difficulties, especially in the solving process of optimization problem, the optimization space will become discontinuous. The restriction can be relaxed, and intermediate density is adopted, that is, the density value of a certain cell is allowed to be between 0 and 1, and then the density value of 0 and 1 is restored by applying a penalty coefficient to the material model.

[0095] 2.1 Material model

[0096] The material interpolation model used herein is the popular SIMP method. Therefore, the relaxed element density: ρ e ∈ [0, 1] n is introduced to describe the relationship between the piezoelectric tensor and the element density ρ e .

[0097]

[0098] where c mati , e mati and κ mati correspond to the elastic, piezoelectric and dielectric tensors of material 1 and material 2 in equation (1), respectively. The 0-1 design can be restored by applying a penalty factor p c , p e and p κ to the material model. These penalty factors are proposed according to two conditions that meet the convergence condition:

[0099] P F = 2p e - (p c + p κ )>0, (19)

[0100] P S = p e - p c >0. (20)

[0101] where P F is the first condition that should be met, i.e. the condition that is independent of the objective function. P S is the second condition that should be met, i.e. the condition when applied to energy harvesting. Unlike the penalty mechanism of linear elastic materials, the choice of penalty factor is particularly important for convergence in piezoelectric materials. The penalty factor can be chosen as:

[0102] p C = 3, p e = 6, p κ= 4, P F = 5, P S = 3) (21)

[0103] In the subsequent examples, the selection of different penalty factors is discussed based on the results of topology optimization.

[0104] 2.2 Formulating the design problem

[0105] In the design of piezoelectric energy harvesting, the static hydrostatic coupling coefficient is chosen as the objective function. This coefficient measures the strength of the coupling between mechanical stress and electric displacement, directly affecting the ability of piezoelectric materials to convert mechanical energy into electrical energy. By optimizing this coefficient, the electromechanical coupling performance of the material can be maximized, improving energy harvesting efficiency and helping to select the appropriate piezoelectric material and structural design for specific working conditions.

[0106] For a 1-3 in-plane two-dimensional plane stress microstructure, the static hydrostatic coupling coefficient can be expressed as:

[0107] d h = vc -1 e, (22)

[0108] where

[0109]

[0110] In this way, the optimization problem can be formulated as:

[0111] Maximize: F(p(x)) = d h

[0112]

[0113] 0 < p(x) < 1

[0114] symmetry conditions (24)

[0115] where F(p(x)) is the objective function, ξ vol is the given volume constraint value, V is the volume of the unit, is the volume average of all units, is the given stiffness constraint value. To facilitate manufacturing, a mandatory symmetry constraint is adopted. Heaviside projection is used to reduce the gray transition zone, and the optimal solution of optimization is sought through moving asymptotes (MMA). The optimization algorithm flowchart describing the steps involved is shown in Figure 3 . This method is implemented in MATLAB.

[0116] 2.3 Sensitivity analysis

[0117] The sensitivity of the objective function with respect to the design variable p i is given in this section. Sensitivity analysis plays a very important role in topology optimization, which determines the direction and efficiency of the optimization design, and directly affects the quality of the design results.

[0118] By taking the derivative of equation (17) and applying the chain rule, we have

[0119]

[0120] Then we have

[0121]

[0122] where N M and N E represent the index of the mechanical displacement and the electrical displacement, respectively. After that, the derivative of equation (8), equation (9), and equation (10) can be obtained as

[0123]

[0124]

[0125] The solution of and is the same as above, and is not repeated here.

[0126] The value of and can be obtained by equation (16) and equation (18) directly, and is not repeated here.

[0127] After obtaining and , the sensitivity of the objective function d h can be obtained by the chain rule as

[0128] III. Numerical results

[0129] In this section, numerical examples of the optimization of piezoelectric microstructures are presented. First, the optimization of the static pressure coupling coefficient is performed, and compared with the negative Poisson's ratio structure. Then, in order to customize the range of metamaterials, numerical examples of piezoelectric microstructures with different stiffness constraints and different volume fractions are given.

[0130] The design domain is discretized into N = 6400 (80 x 80) cells, and the material used is PZT-5A, whose properties are shown in Table 1. For topology optimization, the selection of initial solution has a significant impact on the final optimization result. In this paper, the initial solution in Xia L, Breitkopf P. Design of materials using topology optimization and energy-based homogenization approach in Matlab. Struct Multidiscip Optim 2015; 52: 1229-41. https: / / doi.org / 10.1007 / s00158-015-1294-0 is chosen after comprehensive consideration. The d h values are as follows:

[0131]

[0132] Table 1: PZT-5A material properties

[0133]

[0134]

[0135] 3.1 Comparison with negative Poisson's ratio materials

[0136] As mentioned above, negative Poisson's ratio materials have always been a focus of piezoelectric energy harvesting. In this section, the method of optimizing negative Poisson's ratio materials described above is used to optimize piezoelectric materials for negative Poisson's ratio, and the performance of the resulting results is compared with the configuration of the present application. Since the rigidity of PZT-5A material is large, it cannot form a negative Poisson's ratio structure at a large volume fraction, so the volume fraction is selected as ξ vol = 50%, and to avoid the chessboard pattern, the filter radius r min is set. For the homogenization of piezoelectric microstructures, the selection of the penalty factor is particularly important. In this example, the penalty factor p c = 3, p e = 8, and p κ = 6. A discussion on the penalty factor will be given in subsequent examples.

[0137] Figure 4(a) to Figure 4(c) and Figure 5(a) to Figure 5(c) represent the optimization results targeting the static pressure coupling coefficient and the negative Poisson's ratio, respectively. Figure 4(a) is the unit cell structure obtained by projecting after topology optimization, where d h= 181 (C / N). Fig. 4(b) is a structure obtained by arranging the 3x3 array of the structure; Fig. 4(c) is a physical diagram obtained by manufacturing and cutting the structure. As can be seen from the result of the topology optimization, Fig. 4(a) shows that the optimization aiming at the static pressure coupling coefficient reduces the stiffness in the 3 direction (z direction) to improve the value of the objective function. The thin rods are formed at the boundary of the configuration, and the column at the center of the configuration provides most of the stiffness after the load is applied, while the thin rods at the two sides have a large deformation to ensure the release of the 3 direction electric energy. In the 1 direction, a high stiffness is formed compared with the 3 direction, and the deformation is smaller than the 3 direction after the load is applied, which also reflects the increase of d h from the side. The increase is realized by reducing the release of the 1 direction electric energy and increasing the release of the 3 direction electric energy.

[0138] Fig. 5(a) is a negative Poisson's ratio unit cell structure obtained after projection, wherein d h = 125 (C / N). Fig. 5(b) is a configuration obtained by arranging the 3x3 array of the structure; Fig. 5(c) is a physical diagram obtained by manufacturing and cutting the structure. As can be seen from Fig. 5(a), compared with Fig. 4(a), the configuration obtained by optimizing the negative Poisson's ratio is obviously smaller than the structure of Fig. 4(a) in the stiffness of the 3 direction and the 1 direction, but since there is no obvious column-shaped rod in the structure in the 3 direction, the value of e 33 is basically close to the value of e 13 , that is, the release of the 1 direction electric energy and the 3 direction electric energy is increased at the same time. In the physical sense, if a load is applied in the 3 direction, assuming that the 3 direction contraction produces a positive charge, due to the characteristics of the negative Poisson's ratio material and its low stiffness, a large contraction will also occur in the 1 direction. Since d 31 <0, d 33 > 0, the 3 direction and the 1 direction will produce opposite charges and offset a part. For Fig. 4(a), the 1 direction has a small deformation due to the large stiffness, the 3 direction has a large deformation due to the small stiffness, and the 1 direction is elongated when the 3 direction is contracted, so the 3 direction and the 1 direction will produce the same charge and accumulate, thereby increasing the objective function.

[0139] In order to verify the mechanical properties and piezoelectric properties of the optimized unit cell, the performance of the microstructure is verified and compared, and the results are shown in Figs. 6(a) and 6(b). Fig. 6(a) is the result of the topology optimization of the static water coupling coefficient verified by the finite element software; Fig. 6(b) is the result of the topology optimization of the negative Poisson's ratio verified by the finite element software.

[0140] Overall, the error of the two is not big. But the finite element software will calculate the larger mechanical properties and piezoelectric properties. The reasons for the error are as follows: 1. The grid of the finite element software is inconsistent with the square grid in matlab. 2. The boundary of the configuration needs to be processed when importing into the finite element software. 3. The finite element software uses a three-dimensional solid model, and matlab is a two-dimensional model.

[0141] After that, the application will construct a 3x3 array, due to the particularity of piezoelectric materials, the electrode is an indispensable part of simulation verification, the uppermost side and the lowermost side of the structure are added with electrode devices, the right boundary and the lower boundary are applied with fixed constraints, and the upper boundary and the left boundary are applied with boundary loads with a size of 1N. The optimized configuration and the negative Poisson's ratio macrostructure are analyzed:

[0142] Fig. 7(a) and Fig. 8(a) show the stress distribution diagram of the macrostructure, and it can be found that the maximum stress of Fig. 7(a) is 55.4N / m 2 , and the maximum stress of Fig. 8(a) is 17.3N / m 2 , although the configuration with static pressure coupling coefficient as the target has larger stiffness, it shows larger stress value under the same constraints and loads. The position is concentrated on the thin rod component in the 3 direction, that is, the rod component that generates electric energy analyzed before. Fig. 7(b) and Fig. 8(b) show the displacement distribution diagram of the macrostructure, and similarly, the maximum displacement of Fig. 7(b) is greater than that of Fig. 8(b). The structure with static pressure coupling coefficient as the optimization target produces displacement uniformly, while the displacement of the negative Poisson's ratio structure is mostly concentrated in the upper left corner of the unit cell, which is one of the reasons why the negative Poisson's ratio produces less electric energy. Fig. 7(c) and Fig. 8(c) represent the electric potential of the two structures. It can be seen that the highest electric potential of the former is greater than that of the latter, and the lowest electric potential is less than that of the latter, that is, the potential difference is larger. Fig. 7(d) and Fig. 8(d) represent the electric field mode of the two structures. The basic idea of electric field mode is to regard the electric field as being generated by a series of charges, and to calculate their average value in space. The maximum electric field mode of the former is larger, and the area with large electric field mode is mainly concentrated in the position with smaller stiffness, which is consistent with the previous analysis. In order to facilitate comparison, Figure 9(a) to Figure 9(d) the stress, displacement, electric potential and electric field mode distribution diagrams of the unoptimized piezoelectric macro model are given. Overall, the optimization results show that the negative Poisson's ratio structure can improve the static pressure coupling coefficient and to a certain extent improve the electric energy generated by the structure; but the optimization with static pressure coupling coefficient as the target can achieve greater improvement than negative Poisson's ratio, and has higher stiffness than negative Poisson's ratio structure, and can generate more electric energy in the macrostructure under the same loads and constraints.

[0143] 3.2 Different constraints

[0144] In this section, in order to show the generality of the optimization, the paper gives the topology optimization results of the static pressure coupling target under different stiffness constraints and volume constraints, and discusses the results.

[0145] In the topology optimization problem of optimizing the static pressure coupling coefficient, the stiffness constraint is necessary. Because the solver will tend to reduce the stiffness in the 3 direction to improve the value of the objective function, if the stiffness constraint is not applied, the value of will be close to 0, and the structure will be disconnected in the 3 direction, which is not pressure-resistant and has manufacturing difficulties. For this reason, the paper first discusses different stiffness constraints. In this example, the volume fraction ξ vol = 50%, the filter radius r min , the penalty factor p c = 3, p e = 8, p κ = 6, the results are as follows:

[0146] In all the configurations obtained by subsequent optimization, blue represents PZT-5A material and white represents empty material. It can be seen that the left and right sides of the structure in the example have thin rod components extending in the 1 direction, which is caused by the application of mandatory symmetry constraints. The connection mode of the 1 direction structure is roughly the same, but the connection mode of the 3 direction is greatly different due to the existence of the stiffness constraint. It can be seen that when , the structure produces two thin rods in the 3 direction to connect the main body part; When the 3 direction stiffness increases, more material is allowed to be arranged in the 3 direction, and the 3 direction thin rod will become thick, but at the same time the value of the objective function will also decrease. When the 3 direction stiffness increases, more material is allowed to be arranged in the 3 direction, and the 3 direction thin rod will become thick, but at the same time the value of the objective function will also decrease. When the 3 direction stiffness increases, more material is allowed to be arranged in the 3 direction, and the 3 direction thin rod will become thick, but at the same time the value of the objective function will also decrease. The 3 direction is connected with the rod in the 1 direction, forming a relatively stable structure.

[0147] In addition to the stiffness constraint, the volume constraint is also a key factor affecting the optimization configuration and optimization result. In this paper, the volume fraction ξ vol = 60%, and the other conditions are the same as in this example, the topology structure is given as follows:

[0148] In this example, by increasing the volume fraction (i.e. increasing the amount of piezoelectric material), significantly different results can be obtained, and the value of the objective function also increases. When , the stiffness can only be provided by the thin rods on the left and right sides. When , the structure near the center also forms a thin rod. The configuration after the transformation forms a similar structure as the previous configuration. However, it should be noted that due to the larger amount of piezoelectric material used at this time, the center part of the horizontal bar becomes thicker, which is equivalent to an increase in stiffness in direction 1. Compared to the previous example, d h The improvement method is to improve the stiffness in direction 1, that is, under the action of hydrostatic pressure load, direction 1 will produce smaller deformation, that is, d 31 The increase of d 31 is negative). Since d h = d 31 + d 33 , it will lead to the increase of the objective function. In summary, the optimization with the hydrostatic coupling coefficient as the target is of wide applicability, and different constraint conditions can be applied in different application scenarios to obtain different configurations.

[0149] Four, experimental investigation

[0150] 4.1 Manufacturing

[0151] The manufacturing process of the design starts with the manufacture of a piezoelectric sheet with a specific polarization direction. PZT raw materials are selected. First, the PZT is granulated and then the PZT is qualified. After qualification, PZT-5A raw materials are obtained. After firing, a paste-like material is obtained, which is then shaped and fixed in shape using a template or jig, after which the glue in the granulated powder is removed. After hardening pretreatment, polarization is performed, and after aging (to make the parameters of the piezoelectric sheet consistent in all directions), a piezoelectric sheet is obtained for cutting raw materials.

[0152] It should be noted that in order to facilitate cutting and experiments later, the size of the piezoelectric sheet raw material is 22mm*20mm*1mm. The common polarization direction is the z-axis piezoelectric sheet, with a thickness of about 100μm to 2mm. In this paper, the polarization direction of the piezoelectric sheet is the z-axis, and the length to be polarized is 22mm, the polarization time is 0.5h, and the polarization voltage required per millimeter is 1600V, that is, the piezoelectric sheet needs to be polarized with a voltage of 35.2KV for half an hour. Figure 12(a) is a size diagram, and Figure 12(b) is a finished product diagram after processing. The black dots in the red box position in Figure 12(b) are the indications of the electrode edges, that is, the 1mm*20mm rectangular area where the black dots are located is the upper electrode. The electrode is formed by a magnetron sputtering technique, in which high-energy ions bombard the target surface to sputter atoms or molecules. These particles migrate and deposit on the surface of the piezoelectric sheet to form a silver thin film.

[0153] 4.2 Piezoelectric sheet cutting

[0154] For laser cutting of 1 mm thick PZT5A piezoelectric sheets, a fiber laser cutting machine was chosen to ensure high precision cutting. Before cutting, the piezoelectric sheet should be prepared first, keeping its surface clean to avoid dust or oil affecting the cutting effect. Next, set the power and cutting speed of the laser cutting machine, ensuring that the focal point position is parallel to the material surface. After the laser cutting starts, follow the design drawing to perform accurate cutting operations, and the cutting edge should be kept neat to avoid burrs or irregular cutting marks. After cutting, clean up the residues and check the cutting quality, especially whether the cutting surface is smooth, whether there are overheating, melting or cracking phenomena. If there is residual stress, heat treatment or other methods should be taken to remove it. Finally, through electrical performance detection and size precision detection, ensure that the cutting process does not affect the piezoelectric properties of PZT5A piezoelectric sheet and its size requirements. In order to avoid overheating or heat affected zone, the adjustment of laser parameters is crucial, which must ensure that the laser power, cutting speed and focal point position parameters are within the appropriate range, so as to reduce thermal damage and changes in material properties.

[0155] The biggest difficulty in the cutting process is the complexity of the cutting pattern, especially the structure containing many thin rods (about 1.3 mm long, 0.4 mm wide). These thin rods are very prone to breakage during cutting, especially when the laser beam passes through, due to the sharp rise in temperature, and because the piezoelectric sheet is thin (1 mm thick), the piezoelectric material may become brittle due to uneven heating. To address this issue, fine control of laser power and cutting speed is required, as well as appropriate cutting path planning to avoid excessive thermal load or stress concentration in these thin rod sections. In addition, strict monitoring during the cutting process is required to adjust the parameters in time to prevent breakage due to the brittle nature of the material.

[0156] 4.3 Experimental results

[0157] The d33 and d31 coefficients were measured using a Sinocera YE2730A model meter. Before conducting the experiment, the instrument was calibrated using a standard piezoelectric material to ensure the accuracy of the test results. The PZT5A piezoelectric sheet was installed vertically in the electrode clamp of the Sinocera YE2730A meter, ensuring good contact between the electrodes on both sides of the piezoelectric sheet. The silver electrodes on both sides of the sample were aligned and clamped to ensure the accuracy of the electrical signal. The force sensor or loading device was connected to the sample, ensuring that the sample was fixed firmly during the experiment and that the direction of the force was consistent with the axis of the piezoelectric sheet, avoiding the influence of shaking or loosening on the test results. By applying force and measuring the change in electrical signal caused by it, the instrument can record the relationship between the applied force and the electrical signal. According to the known force and charge signal, the d33 coefficient can be calculated. It should be noted that when measuring the d31 coefficient, the electrode layer of the piezoelectric sheet is rotated 90 degrees, i.e. the clamp clamps the 1 direction side of the piezoelectric sheet without electrodes. The electrodes of the piezoelectric sheet are connected to the electrode clamp of the meter with wires, and the d31 coefficient can be measured.

[0158] The measurement results are shown in Figure 13 , Figure 14(a) to Figure 14(c) .

[0159] Figure 13 The piezoelectric performance of the uncut piezoelectric sheet, the piezoelectric sheet after optimization design, and the negative Poisson ratio design piezoelectric sheet is shown. As can be seen from the figure, the optimization design significantly improves the piezoelectric performance of the piezoelectric sheet, especially in terms of static pressure coupling coefficient. Specifically, the key to optimization design is to reduce the value of d31 coefficient by adjusting the internal stress distribution of the material, while trying to maintain the improvement of d33 coefficient. Reducing d31 coefficient helps to reduce the lateral response of the piezoelectric sheet in some directions, thereby enhancing its piezoelectric performance in the axial direction. At the same time, through reasonable adjustment of the structure and geometry of the material during the optimization process, the d33 coefficient is improved to some extent, which effectively improves the overall performance of the piezoelectric sheet, especially in terms of high static pressure coupling coefficient. Finally, the piezoelectric sheet after optimization design exhibits high electromechanical coupling efficiency and low lateral non-ideal response, so that the entire piezoelectric system can more efficiently convert mechanical energy and electrical energy during operation, meeting the actual application requirements.

[0160] Figure 14(a) to Figure 14(c)The comparison of the piezoelectric performance of the optimized piezoelectric sheet and the piezoelectric performance of the piezoelectric sheet obtained through experiments is shown. As can be seen from the figure, the trend of the experimental results and the simulation calculation results is basically consistent, although there is a certain error, but the error is within the acceptable range. This shows that the simulation calculation method of the static pressure coupling coefficient is effective, and can accurately predict the performance of the piezoelectric material in practical application. The possible error sources in the experiment include the slight changes in the experimental environment, the microstructure differences of the material itself and the uncertainties in the experimental operation. However, simulation calculation still provides strong support for the optimization design and provides a theoretical basis for further optimization design. In general, the good consistency between the experiment and the simulation verifies the feasibility of the optimization design in improving the piezoelectric performance, and lays a foundation for subsequent design improvement and practical application.

[0161] The present application proposes a method and system for designing piezoelectric metamaterials using topology optimization to achieve higher performance energy harvesting. By applying topology optimization, the hydrostatic coupling coefficient is optimized, significantly improving the energy conversion efficiency of piezoelectric materials. The design for the hydrostatic coupling coefficient can significantly improve the piezoelectric performance by enhancing the stiffness and energy harvesting capacity of the material. Experimental and numerical simulation results confirm the effectiveness of the proposed optimization method, showing that the piezoelectric metamaterials designed through these optimizations outperform traditional materials including materials with negative Poisson's ratio. The optimized piezoelectric microstructure exhibits superior mechanical properties and higher energy harvesting performance.

[0162] Embodiments of the present disclosure can be implemented by computer software executable by a data processor of the mobile device, such as in the processor entity, or by hardware, or by a combination of software and hardware. Computer software or program, also called program product, including software routines, applets and / or macros, can be stored in any apparatus-readable data storage medium and they include program instructions for executing certain tasks. The computer program product can include one or more computer-executable components which, when the program is run, are configured to carry out embodiments. The one or more computer-executable components can be at least one software code or a portion thereof.

[0163] In this regard, it should also be noted that any block of the logical flow of the figure can represent a program step, or an interconnected logic circuit, block and function, or a combination of program steps and logic circuits, blocks and functions. Software can be stored on a physical medium, such as a memory chip or a memory block implemented within a processor, a magnetic medium such as a hard disk or a floppy disk, and an optical medium such as, for example, a DVD and its data variants CD. The physical medium is a non-transitory medium.

[0164] The term "non-transitory" as used herein is a limitation of the medium itself (i.e., tangible, as opposed to signals), not the durability of the data stored thereon (e.g., RAM vs. ROM).

[0165] The memory can be of any type appropriate for the local technical environment and can be implemented using any appropriate data storage technology, such as semiconductor based memory devices, magnetic memory devices and systems, optical memory devices and systems, fixed memory and removable memory. The data processor can be of any type appropriate for the local technical environment, and can include one or more of general purpose computers, special purpose computers, microprocessors, digital signal processors (DSPs), application specific integrated circuits (ASICs), FPGAs, gate level circuits and processors based on multi-core processor architectures, as non-limiting examples.

[0166] Embodiments of the disclosure can be practiced in a variety of components such as integrated circuit modules. The design of integrated circuits is by nature a highly automated process. Complex and powerful software tools are available for converting a logic level design into a semiconductor circuit design ready to be fabricated on semiconductor chips.

[0167] The scope of the protection sought to be afforded by the teachings of the present disclosure, which extend to various embodiments as described in the specification and as set out in the independent claims, is to be interpreted in conformity with the language and practices followed in the United States in implementing 35 U.S.C. § 101. The embodiments described in the specification and as set out in the dependent claims that do not fall within the scope of the independent claims are to be interpreted as examples useful in understanding the various embodiments of the present disclosure.

[0168] The foregoing description has provided by way of non-limiting examples a full and informative description of exemplary embodiments of the present disclosure. However, where appropriate, various modifications and adaptations to those embodiments could become apparent to those skilled in the relevant art in view of the foregoing description, when read in conjunction with the accompanying drawings and the appended claims. However, all such and similar modifications will still fall within the scope of the teachings of the present disclosure as defined in the appended claims. Indeed, other embodiments of the present disclosure will be apparent to those skilled in the art from consideration of the specification and practice of the present disclosure.

Claims

1. A design method for piezoelectric metamaterials for achieving higher performance energy harvesting using topology optimization, characterized by: Including steps: defining a design domain of the piezoelectric metamaterial, wherein the polarization direction is along the z-axis; applying topology optimization to optimize the hydrostatic coupling coefficient of the piezoelectric metamaterial; discretizing the design domain into grid cells, each grid cell being assigned a density value representing material distribution; interpolating material properties from element density using the Solid Isotropic Material Penalty (SIMP) model, including elastic, piezoelectric, and dielectric tensors; A penalty factor is introduced to force the density value to be binary to ensure manufacturability; Imposing symmetry and volume constraints to ensure manufacturability and structural stability; Numerical algorithms are used to solve the optimization problem and maximize the hydrostatic coupling coefficient, thereby improving the stiffness of the material and the energy harvesting efficiency.

2. The method according to claim 1, wherein: The topology optimization targets the hydrostatic coupling coefficient to obtain higher stiffness and better piezoelectric performance than negative Poisson's ratio materials.

3. The method according to claim 1, wherein: The topology optimization considers different stiffness and volume constraints to generate a variety of piezoelectric microstructures suitable for specific application requirements.

4. The method according to claim 1, wherein: The method further includes predicting the effective performance of the piezoelectric metamaterial by a homogenization method; and simulating the mechanical and electrical responses using a two-dimensional finite element model.

5. The method according to claim 1, wherein: The SIMP model interpolates material properties using the following equation: H , piezoelectric tensor e H , dielectric tensor k H : Where ρ is the cell density, P c 、P e 、P k is the penalty factor, C mat1 ,e mat1 ,k mat1 and C mat2 ,e mat2 ,k mat2 are the properties of piezoelectric material and empty material, respectively.

6. The method according to claim 1, wherein: The optimization problem is expressed as follows: the objective function is to maximize the hydrostatic coupling coefficient; the constraints are volume constraints, stiffness constraints and equilibrium equations.

7. The method according to claim 1, wherein: The method further includes: verifying the optimized piezoelectric microstructure through numerical simulation and experiment; comparing the performance of the optimized piezoelectric metamaterial with that of a negative Poisson's ratio material to demonstrate superior stiffness and energy collection efficiency.

8. A design system for piezoelectric metamaterials using topology optimization to achieve higher performance energy harvesting, characterized in that: The computer device includes a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the computer program implements the method according to any one of claims 1 to 8 when executed by the processor.

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