Design method and system of piezoelectric energy harvesting structure under static load effect

Abstract

The invention relates to a design method and system for a piezoelectric energy harvesting structure under the action of static load, and the method comprises the steps: dispersing a design domain of the piezoelectric energy harvesting structure to be designed into N finite elements, constructing a multiphase material interpolation model which represents the structural distribution of a substrate material, a piezoelectric material and an empty material in the piezoelectric energy harvesting structure for each finite element, wherein the design domain is a two-dimensional cross section of the piezoelectric energy harvesting structure; a target function of the energy conversion efficiency of the piezoelectric energy harvesting structure under the action of the static load is constructed, and a topological model of the piezoelectric energy harvesting structure under the static load is constructed based on the multiphase material interpolation model and the target function of the energy conversion efficiency; performing parameter optimization on the topological model of the piezoelectric energy harvesting structure under the static load; and solving the topological model of the piezoelectric energy harvesting structure under the static load after parameter optimization so as to obtain a designed piezoelectric energy harvesting structure. The piezoelectric energy harvesting structure is good in energy harvesting efficiency.

Description

Technical Field

[0001] The present invention relates to the technical field of piezoelectric energy harvesting structures, and in particular to a design method and system for piezoelectric energy harvesting structures under static load. Background Art

[0002] Piezoelectric materials can generate electrical energy under the action of external mechanical forces and are widely used in fields such as energy harvesting, sensors, and actuators. However, when collecting energy through piezoelectric structures under static load conditions, there is still a large room for improvement in their energy harvesting efficiency. As an effective means to improve the output performance of piezoelectric energy harvesting structures, the topology optimization method aims to maximize the electrical energy output of the structure under given input conditions.

[0003] According to different optimization objectives, existing research mainly focuses on the optimization of energy conversion efficiency or electromechanical coupling coefficient under static conditions. For the design of static load energy harvesting structures, scholars at home and abroad have carried out extensive research in aspects such as optimization methods, design domains, and optimization objectives. Common optimization methods include the variable density method, level set method, homogenization method, and progressive method, etc.; the optimization regions cover single piezoelectric domains, piezoelectric domains and polarization directions, etc.; the optimization objectives include energy conversion efficiency and electromechanical coupling coefficient, etc.

[0004] However, most existing research only considers the layout optimization of single piezoelectric materials or the layout of piezoelectric materials and polarization directions. Although the energy conversion efficiency has been improved to a certain extent, there are still problems with limited efficiency. At present, only a very small number of studies have attempted to explore the collaborative optimization design of the substrate and piezoelectric materials, trying to improve the overall energy harvesting performance by comprehensively considering the material layout, shape, and their interaction with the substrate. However, these studies still face challenges such as high design complexity and large computational volume, and lack a systematic optimization method.

[0005] In summary, the energy harvesting efficiency of the piezoelectric energy harvesting structures obtained by existing research under static load has not reached a relatively high value, and these studies still face challenges such as high design complexity and large computational volume, and lack a systematic optimization method. Summary of the Invention

[0006] Therefore, the technical problem to be solved by the present invention is to overcome the problem that the energy harvesting efficiency of the piezoelectric energy harvesting structures obtained by existing research under static load has not reached a relatively high value.

[0007] To solve the above technical problem, the present invention provides a design method for a piezoelectric energy harvesting structure under static load, including:

[0008] Step S1: Discretize the design domain of the piezoelectric energy harvesting structure to be designed into N finite elements. For each finite element, construct a multi-phase material interpolation model representing the structural distribution of the base material, piezoelectric material, and air material in the piezoelectric energy harvesting structure, where the design domain is the two-dimensional cross-section of the piezoelectric energy harvesting structure.

[0009] Step S2: Construct an objective function for the energy conversion efficiency of the piezoelectric energy harvesting structure under static load. Based on the multi-phase material interpolation model and the objective function of the energy conversion efficiency, construct a topology model of the piezoelectric energy harvesting structure under static load.

[0010] Step S3: Optimize the parameters of the topology model of the piezoelectric energy harvesting structure under static load.

[0011] Step S4: Solve the topology model of the piezoelectric energy harvesting structure under static load after parameter optimization to obtain the designed piezoelectric energy harvesting structure.

[0012] In an embodiment of the present invention, the method for constructing the multi-phase material interpolation model representing the structural distribution of the base material, piezoelectric material, and air material in the piezoelectric energy harvesting structure in step S1 includes:

[0013] Represent each finite element of the piezoelectric energy harvesting structure by design variables ρ E and ρ P The design variable ρ E is used to determine whether there is material in this finite element, and the design variable ρ P is used to distinguish what kind of material this finite element is filled with, and 0 ≤ ρ E,e ≤ 1, 0 ≤ ρ P,e ≤ 1, e = 1, 2,..., N;

[0014] Combine the design variables ρ E and ρ P to construct a multi-phase material interpolation model for each finite element, expressed as:

[0015]

[0016] where C, e, and κ are the effective property matrices of the interpolated materials, C pzt and C nonpzt are the elastic matrices of the piezoelectric material and the base material respectively, e pzt and κ pzt represent the piezoelectric coupling matrix and the dielectric matrix of the piezoelectric material respectively, and p1, p2, p3, and p4 are penalty factors.

[0017] In an embodiment of the present invention, the method of step S2 includes:

[0018] Construct the energy conversion efficiency of the piezoelectric energy harvesting structure under static load, expressed as:

[0019]

[0020] Among them, U T K uu U and φ T K φφ U and Φ respectively represent the strain energy and the generated electrical energy of the piezoelectric energy harvesting structure. The sum of the strain energy and the electrical energy represents the work done by the external force; U represents the mechanical displacement, and Φ represents the equivalent electric potential;

[0021] Rewrite J η as: Π S and Π E respectively represent the strain energy and the generated electrical energy of the piezoelectric energy harvesting structure, corresponding to U T K uu U and Φ T K φφ Φ;

[0022] Transform the problem of maximizing the energy harvesting efficiency into a minimization problem, then rewrite J η again as: ζ = 1 / J η = 1 + Π S / Π E , maximizing the energy conversion efficiency J η is equivalent to minimizing ζ, and taking the minimization of ζ as the objective function;

[0023] Based on the multi-phase material interpolation model and the objective function, the topological model of the piezoelectric energy harvesting structure under static load is:

[0024] find: ρ E,e , ρ P,e

[0025] min: ζ = 1 + Π S / Π E

[0026] subject to:

[0027]

[0028] V(ρ E ) / V0 ≤ θ E

[0029] V(ρ P ) / V0 ≤ θ P

[0030] 0 ≤ ρ E,e , ρ P,e ≤ 1, e = 1, …, N

[0031] Among them, the mechanical response of the structure is determined by the matrix K 11Description; Matrix K 12 and K 21 is a piezoelectric coupling matrix for coupling the structural and electrical responses; Matrix K 22 The dynamic characteristics in are determined by the inductance matrix R, resistance matrix R L , resistance matrix R R , capacitance matrix R C , dielectric matrix K φφ and the conductance matrix Z of the electrodes; F is the external excitation load; V(ρ E ) represents the total volume of the actual structural material; V(ρ P ) represents the actual volume of the piezoelectric material in the total volume; V0 is the total volume of the design domain, θ E and θ P are the given total material target volume fraction and the target volume fraction of the piezoelectric material, respectively.

[0032] In an embodiment of the present invention, the method in step S3 includes:

[0033] For the topological model of the piezoelectric energy harvesting structure under static load, assuming the piezoelectric material is ceramic PZT-5A material and the substrate material is aluminum material, balance the order of magnitude of different material property coefficients to make K 11 , K 12 , K 21 and K 22 maintained at the same order of magnitude, expressed as:

[0034]

[0035] Among them, the assembly of the element stiffness matrix K uu involves the piezoelectric elastic coefficient C pzt and the elastic coefficient C of aluminum material nonpzt , the element piezoelectric coupling matrix K uφ involves the piezoelectric coupling coefficient e pzt , the element dielectric matrix K φφ involves the dielectric coefficient κ pzt , τ is an integer and takes the value of 9.

[0036] In an embodiment of the present invention, the method of step S3 further includes:

[0037] Balance the order of magnitude of the electric potential Φ in the topological model of the piezoelectric energy harvesting structure under static load, expressed as: τ is an integer and takes the value of 9;

[0038] An equal potential constraint is applied to the upper and lower surfaces of the piezoelectric material with a silver electrode layer on the surface, and the voltage on the electrode layer is equivalent to a node voltage, which is used as the output voltage of the piezoelectric energy harvesting structure. The equipotential constraint is expressed as:

[0039]

[0040] in, is the electric potential after equipotential, the Boolean matrix B eq The dimension is N e ×N P , N e is the number of nodes, N P is the number of electrodes.

[0041] In one embodiment of the present invention, the process of solving the topological model of the piezoelectric energy harvesting structure under static load includes solving the mechanical displacement U and the equivalent potential Φ, specifically:

[0042] The charge conservation equation of the piezoelectric energy-harvesting structure q piezo +q electrode +q circuit = 0 as the target, the charge equation generated by the piezoelectric material Electrode charge equation and the charge equation in the equivalent external circuit (-ω 2 R L +iωR R +R C )Φ=q circuit , the coupling equation of the whole system of piezoelectric energy harvesting structure is obtained as follows:

[0043]

[0044] Where ω = 0, indicating the static load effect; the equivalent charge of the piezoelectric energy-harvesting structure is the charge q generated by the piezoelectric material. piezo , the charge generated on the electrode q electrode , the charge q in the equivalent external circuit circuit Three parts: Global system matrix By K 11 , K 12 , K 21 and K 22 The four matrices are combined to obtain;

[0045] The mechanical displacement U and equivalent electric potential Φ are solved by the coupling equation of the whole system of the piezoelectric energy harvesting structure.

[0046] In one embodiment of the present invention, when solving the mathematical model in step S4, the design variable ρ is also calculated. E,e and ρ P,eOptimize the derivative process, specifically as follows:

[0047] For ρ in the topological model of the piezoelectric energy harvesting structure under static load e ={ρ E,e ,ρ P,e} Under the action of a static concentrated load, calculate the objective function ζ = 1 / J η = 1 + ∏ S / ∏ E The first derivative with respect to the element density variable ρ e is:

[0048]

[0049] Expand the strain energy Π S and the electrical energy Π E respectively by the adjoint method to obtain:

[0050]

[0051] Among them, λ1 and λ2 are adjoint displacement vectors, μ1 and μ2 are adjoint electric potential vectors. Determine the two sets of adjoint variables {λ1, μ1} and {λ2, μ2} by solving the following adjoint equations:

[0052]

[0053] By solving the formulas and and substituting Π S and Π E expanded by the adjoint method, the numerical values of the adjoint variables are obtained. Then, the derivatives of the strain energy Π S and the electrical energy Π E with respect to the design variable ρ e are respectively:

[0054]

[0055] Among them, F represents the external load vector that has nothing to do with the design variable ρ e under static conditions, that is, satisfying

[0056] Construct K 11 , K 12 , K 21 , K 22 The first derivatives with respect to the design variable ρ e are:

[0057]

[0058]

[0059] Among them, B u represents the strain displacement matrix, and B φ represents the strain electric potential matrix;

[0060] According to the multi-phase material interpolation model, the derivative of the constitutive matrix of the piezoelectric material with respect to the design variable ρ e is obtained, including:

[0061] The derivatives of the elastic, piezoelectric coupling, and dielectric matrices in the constitutive matrix of the piezoelectric material with respect to the design variable ρ E,e are:

[0062]

[0063] The derivative of the constitutive matrix of the piezoelectric material with respect to the design variable ρ P,e is:

[0064]

[0065] To solve the above technical problems, the present invention provides a design system for a piezoelectric energy harvesting structure under static load, including:

[0066] Discretization module: used to discretize the design domain of the piezoelectric energy harvesting structure to be designed into N finite elements. For each finite element, a multi-phase material interpolation model representing the structural distribution of the base material, piezoelectric material, and air material in the piezoelectric energy harvesting structure is constructed, where the design domain is the two-dimensional cross-section of the piezoelectric energy harvesting structure;

[0067] Construction module: used to construct the objective function of the energy conversion efficiency of the piezoelectric energy harvesting structure under static load, and construct the topology model of the piezoelectric energy harvesting structure under static load based on the multi-phase material interpolation model and the objective function of the energy conversion efficiency;

[0068] Optimization module: used to perform parameter optimization on the topology model of the piezoelectric energy harvesting structure under static load;

[0069] Solution module: used to solve the topology model of the piezoelectric energy harvesting structure under static load after parameter optimization to obtain the designed piezoelectric energy harvesting structure.

[0070] To solve the above technical problems, the present invention provides an electronic device, including a memory, a processor, and a computer program stored on the memory and executable on the processor. When the processor executes the computer program, the steps of the design method of the piezoelectric energy harvesting structure under static load as described above are implemented.

[0071] To solve the above technical problems, the present invention provides a computer-readable storage medium, on which a computer program is stored. When the computer program is executed by a processor, the steps of the design method of the piezoelectric energy harvesting structure under static load as described above are implemented.

[0072] The above technical solution of the present invention has the following advantages compared with the prior art:

[0073] The design method of the piezoelectric energy harvesting structure under static load according to the present invention can effectively improve the energy conversion efficiency of the piezoelectric energy harvesting structure by optimally distributing the positions of the piezoelectric material and the substrate material; aiming at the convergence problem of the energy harvesting efficiency in the optimization design, the present invention takes the strain energy as a constraint condition, improves the bearing capacity of the structure, and ensures the stability of convergence; under the same material consumption and strain energy constraint, keeping the piezoelectric material fixed and optimizing the substrate, the energy conversion efficiency of the structure is 3.35%, while the energy conversion efficiency obtained by the collaborative optimization design reaches 12.44%. Description of the Drawings

[0074] In order to make the content of the present invention easier to be clearly understood, the present invention will be further described in detail below according to the specific embodiments of the present invention in conjunction with the drawings.

[0075] Figure 1 is the flowchart of the method of the present invention;

[0076] Figure 2 is a schematic diagram of the interpolation model of the piezoelectric energy harvesting structure with multiple materials in the embodiment of the present invention;

[0077] Figure 3 is a schematic diagram of the model of the piezoelectric energy harvesting structure (piezoelectric energy harvesting cantilever beam) under static load in the embodiment of the present invention;

[0078] Figure 4 is the analysis result of the energy conversion efficiency and strain energy of the piezoelectric energy harvesting structure under different volume fractions of the piezoelectric material in the embodiment of the present invention;

[0079] Figure 5 is a schematic diagram of the material distribution optimized with the maximum stiffness as the target and the optimal topological configuration of the piezoelectric energy harvesting structure in the embodiment of the present invention;

[0080] Figure 6 is an iteration diagram of the volume constraint and the objective function under the collaborative optimization of the piezoelectric and the substrate with the maximum stiffness as the target and the strain energy as the constraint in the embodiment of the present invention;

[0081] Figure 7 is a schematic diagram of the topological configuration of the piezoelectric energy harvesting structure obtained by the collaborative design of the energy conversion efficiency of the piezoelectric material and the substrate material with the energy conversion efficiency as the target and the strain energy as the constraint in the embodiment of the present invention

[0082] Figure 8It is an iterative graph of volume fraction and objective function under the collaborative optimization of piezoelectric and substrate, with the maximization of energy conversion efficiency as the goal and strain energy as the constraint, in the embodiments of the present invention. Detailed implementation manners

[0083] The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments, so that those skilled in the art can better understand the present invention and be able to implement it, but the embodiments cited are not intended to limit the present invention.

[0084] Embodiment 1

[0085] Referring to Figure 1 as shown, the present invention relates to a design method of a piezoelectric energy harvesting structure under static load, including:

[0086] Step S1: Discretize the design domain of the piezoelectric energy harvesting structure to be designed into N finite elements. For each finite element, construct a multi-phase material interpolation model representing the structural distributions of the substrate material, piezoelectric material, and void material in the piezoelectric energy harvesting structure, where the design domain is the two-dimensional cross-section of the piezoelectric energy harvesting structure;

[0087] Step S2: Construct an objective function for the energy conversion efficiency of the piezoelectric energy harvesting structure under static load. Based on the multi-phase material interpolation model and the objective function of the energy conversion efficiency, construct a topology model of the piezoelectric energy harvesting structure under static load;

[0088] Step S3: Optimize the parameters of the topology model of the piezoelectric energy harvesting structure under static load;

[0089] Step S4: Solve the topology model of the piezoelectric energy harvesting structure under static load after parameter optimization to obtain the designed piezoelectric energy harvesting structure (equivalent to solving for the optimal design variables ρ E,e and ρ P,e ).

[0090] The following is a detailed introduction to this embodiment:

[0091] Please refer to Figure 1 and Figure 2 , a design method of a piezoelectric energy harvesting structure under static load provided by the present invention mainly includes the following steps:

[0092] Step S1: The topology optimization problem of discretizing the design domain into N elements (finite elements). To represent the three-phase material composed of the substrate material, piezoelectric material, and void material (i.e., the place without material), each element is described by two design variables ρ E and ρ P . The design variables take values in the interval [0, 1], and are described mathematically as: 0 ≤ ρ E,e ≤ 1 and 0 ≤ ρ P,e≤1, e = 1, 2, …, N. Design variable ρ E The design variable ρ is used to determine whether there is material in the finite element P For determining what kind of material the finite element is filled with, please refer to Figure 2 , Figure 2 Among them, the number 1 represents the piezoelectric material, the number 2 represents the substrate material, and the number 3 represents the empty material. More specifically, based on the PEMAP model extended by the SIMP method, the elastic matrix, piezoelectric coupling matrix, and dielectric matrix of the piezoelectric material are defined. Then, the mathematical expression of the multi-phase material interpolation model is:

[0093]

[0094] where C, e, and κ are the effective property matrices of the interpolated material, and C pzt and C nonpzt are the elastic matrices of the piezoelectric material and the substrate material respectively, while e pzt and κ pzt represent the piezoelectric coupling matrix and the dielectric matrix of the piezoelectric material respectively. The penalty factors p1, p2, p3, and p4 take values between 1 and 6, and the intermediate density elements are penalized to make their values tend to 0 or 1

[0095] It should be noted that according to the multi-phase material interpolation model, each element (finite element) of the piezoelectric energy harvesting structure can be composed of two materials, piezoelectric and substrate, in any proportion. When ρ E,e = 1 and ρ P,e = 1, it represents the piezoelectric material; when ρ E,e = 1 and ρ P,e = 0, it represents the substrate material. When ρ E = 0, it is the empty material (hollow filling). The design variable ρ E is used to determine the presence or absence of material, that is, the presence of an entity (piezoelectric material and substrate material) or being empty (no material); the design variable ρ P is used to determine what kind of material the unit is specifically filled with, that is, on the premise that the unit has material (ρ E,e ≠ 0), to determine whether it is filled with the solid piezoelectric material or the substrate material

[0096] Step S2: Construct the topological model of the piezoelectric energy harvesting structure under static load, specifically as follows

[0097] Construct the energy conversion efficiency of the piezoelectric energy harvesting structure under static load, expressed as

[0098]

[0099] where U T K uu U and Φ T K φφΦ represents the strain energy and the generated electrical energy of the structure, U represents the mechanical displacement, and Φ represents the equivalent electric potential. The sum of the strain energy and the electrical energy represents the work done by the external force. In this embodiment, the above J η can be further written as Π S and Π E represent the strain energy and the generated electrical energy of the structure respectively, and ∏ S +Π E The sum is the work done by the external force. Among them, U T K uu U and Φ T K φφ Φ represent the strain energy and the generated electrical energy of the structure respectively. The sum of the strain energy and the electrical energy represents the work done by the external force.

[0100] Furthermore, the problem of maximizing the energy harvesting efficiency is transformed into a minimization problem, that is, the design goal is re-expressed in the reciprocal form of the energy conversion efficiency. Therefore, the above J η can be further rewritten as: ζ = 1 / J η = 1 + Π S / Π E , and minimizing ζ is taken as the objective function. By reconstructing the objective function, maximizing the energy conversion efficiency J η is equivalent to minimizing ζ. According to the assumptions about the two design variables ρ E and ρ P , based on the multi-phase material interpolation model and the objective function ζ, and considering the constraints of the overall structural material volume fraction and the piezoelectric material volume fraction, the topological model of the piezoelectric energy harvesting structure under static load is:

[0101]

[0102] Among them, the mechanical response of the structure is described by the matrix K 11 , which changes with the layout of the material and the layout method of the structure; the matrices K 12 and K 21 are piezoelectric coupling matrices, which are used to couple the structure and the electrical response and change with the layout of the piezoelectric material and the polarization direction; the dynamic characteristics in the matrix K 22 are composed of the inductance matrix R L , the resistance matrix R R , the capacitance matrix R C , the dielectric matrix K φφ and the conductance matrix Z of the electrode in the circuit matrix of the external circuit of the piezoelectric energy harvesting structure (the external circuit for the piezoelectric energy harvesting structure in this embodiment is an RLC circuit with a series resistance R, an inductance L, and a capacitance C); the dielectric matrix K φφIt changes with the layout of the material, and the conductance matrix Z changes with the conductivity and topology of the electrodes; F is the external excitation load; V(ρ E ) represents the total volume of the actual structural material; V(ρ P ) represents the actual volume of the piezoelectric material in the total volume; V0 is the total volume of the design domain, and θ E and θ P are the given total material target volume fraction and the target volume fraction of the piezoelectric material, respectively.

[0103] During the topology optimization iteration process, in order to optimize the energy conversion efficiency, the optimized piezoelectric energy harvesting structure will have large deformations in the area where the load is applied, resulting in excessive distortion of the elements, and then deteriorating the load-bearing performance of the piezoelectric energy harvesting structure. Therefore, in this embodiment, considering that the piezoelectric energy harvesting structure has a certain load-bearing capacity, the strain energy constraint is introduced into formula (3):

[0104]

[0105] Among them, is the constraint value of the strain energy. Numerical design shows that the additional consideration of the strain energy constraint will also accelerate the convergence speed of the optimization.

[0106] Step S3: For the topology model of the piezoelectric energy harvesting structure under static load, before solving the displacement U and equivalent electric potential Φ of the structure by finite element method, it needs to be preprocessed. The piezoelectric energy harvesting structure is mainly composed of two materials, piezoelectric and substrate. Among them, the piezoelectric material involves three material properties: piezoelectric elastic coefficient, piezoelectric coupling coefficient, and dielectric coefficient, while the substrate material is the material elastic coefficient property. There are different orders of magnitude among various material properties, which makes the assembled global stiffness matrix ill-conditioned ( ill-conditioned), that is, the numerical difference between the maximum and minimum values is too large, resulting in instability or difficult convergence in the output response solution process, and then causing a large error in the calculation results.

[0107] (1) Process the ill-conditioned stiffness matrix. Through algebraic operations, balance the orders of magnitude of the mechanical displacement and electric potential involved in the electromechanical problem, scale the elastic matrix and dielectric constant matrix, and finally make the entire assembled global stiffness matrix maintain the same order of magnitude. Taking the piezoelectric ceramic PZT-5A material and the substrate aluminum material as an example, the variable dimension method is used to make K 11 、K 12 、K 21 and K 22 maintain the same order of magnitude. The orders of magnitude of the material property coefficients are as follows:

[0108]

[0109] Among them, the element stiffness matrix K uu The assembly involves the piezoelectric elastic coefficient C pzt and the elastic coefficient C of the aluminum material nonpzt , the element piezoelectric coupling matrix K uφ involves the piezoelectric coupling coefficient e pzt , the element dielectric matrix K φφ involves the dielectric coefficient κ pzt . Among them, τ is an arbitrary integer.

[0110] This embodiment balances the order of magnitude of the electric potential Φ in the topological model of the piezoelectric energy harvesting structure under static load, expressed as:

[0111] Through the above algebraic balancing operation, the order of magnitude of the mechanical displacement and the electric potential involved in the electromechanical problem can be balanced. The positive integer used to balance the order of magnitude should satisfy the following conditions The order of the global material property matrix can be obtained as 10 1 , eliminating the numerical instability in the finite element solution caused by the ill-conditioned matrix of the stiffness matrix involved in the topological optimization model.

[0112] (2) Since the surface of the actual piezoelectric material is often plated with a silver electrode layer, the potential can be made equal on the surface of this electrode layer. To numerically simulate the equipotential electrode layer, the equipotential boundary condition is forced to be applied to the surface of the piezoelectric material. In the equipotential constraint, the silver electrode layer is replaced by applying an equal potential constraint on the upper and lower surfaces of the piezoelectric material, that is, all the nodes on the electrode have the same voltage value, and the voltage on the electrode layer can be equivalent to a node voltage. Mathematically, considering a perfectly conducting electrode layer, the equipotential boundary condition can be achieved by applying a Boolean matrix in the following form:

[0113]

[0114] Among them, is the electric potential after equipotential, and the Boolean matrix B eq has a dimension of N e ×N P . Among them, N e is the number of nodes, and N P is the number of electrodes. The Boolean matrix B eq imposes the equipotential boundary condition on Φ.

[0115] (3) Solve for the output mechanical displacement U and equivalent electric potential Φ

[0116] The mechanical displacement U and equivalent electric potential Φ are used to indicate the increase or decrease of the final required design variables ρ E and ρ P , and the specific solution is as follows:

[0117] After completing the above parameter preprocessing steps, since the piezoelectric energy harvesting structure has an external circuit (as described above), this embodiment uses the system charge conservation equation q piezo +q electrode +q circuit = 0 as the target, the charge equation generated by the piezoelectric material Electrode charge equation and the charge equation in the equivalent external circuit (-ω 2 R L +iωR R +R C )Φ=q circuit , the final coupling equation of the whole system of piezoelectric energy harvesting structure is obtained as:

[0118]

[0119] Among them, the four submodules of the stiffness matrix can be further expanded as follows:

[0120]

[0121] Where ω = 0 represents the static load action, and the equivalent charge of the piezoelectric energy-harvesting structure is the charge q generated by the piezoelectric material. piezo , the charge generated on the electrode q electrode , and the charge q in the equivalent external circuit circuit It consists of three parts. Global system matrix By K 11 , K 12 , K 21 and K 22 Four submodule matrices are obtained by assembly.

[0122] According to the above process, the coupling equation of the whole system of piezoelectric energy harvesting structure (Formula (7)) is used to solve the mechanical displacement U and equivalent potential Φ, and according to the formula Restore the equivalent potential Φ to the actual output voltage.

[0123] Step S4: When solving the piezoelectric energy harvesting structure topology model under static load after parameter optimization, the ρ e ={ρ E,e ,ρ P,e The derivation process of} is optimized (i.e., sensitivity filtering), as follows:

[0124] Step S41: Based on the formula ζ=1 / J η =1+Π S / Π E It can be obtained that under the action of static concentrated load, the objective function is related to the unit density variable ρe The first derivative of

[0125]

[0126] By introducing the adjoint method, the strain energy Π S and the electric energy Π E can be expanded respectively as follows:

[0127]

[0128] where λ1 and λ2 are adjoint displacement vectors, μ1 and μ2 are adjoint electric potential vectors, and two sets of adjoint variables {λ1, μ1} and {λ2, μ2} are determined by solving the following adjoint equations:

[0129]

[0130] By solving formulas (11) and (12) and substituting the numerical values of the adjoint variables obtained from the formulas (9) and (10) expanded by the adjoint method, the strain energy Π S and the electric energy Π E with respect to the design variable ρ e are respectively:

[0131]

[0132] where F represents the external load vector independent of the design variable ρ e under static conditions, that is, satisfying Under static conditions, only the electromechanical coupling characteristics of the structure are considered, and the mass, damping, electrode conduction, and external circuit of the structure are ignored. By introducing the integral expression of the stiffness matrix, the first derivatives of the stiffness matrices in the four simplified sub-modules with respect to the design variable are as follows:

[0133]

[0134] where B u represents the strain-displacement matrix, and B φ represents the strain-electric potential matrix;

[0135] According to the multi-phase material interpolation model, the derivative of the constitutive matrix of the piezoelectric material in the above formula with respect to the design variable ρ e can be further obtained, where

[0136] the derivatives of the elastic, piezoelectric coupling, and dielectric matrices in the constitutive matrix of the piezoelectric material with respect to the design variable ρ E,e can be written as:

[0137]

[0138] Similarly, the derivative of the constitutive matrix of the piezoelectric material with respect to the design variable ρ P,e can be written as:

[0139]

[0140] Finally, combining the derivative of the constitutive matrix of the piezoelectric material with respect to the design variable ρ e , substitute the formula

[0141] into the formula to obtain the first-order sensitivity information of the objective function with respect to the two design variables ρ E,e and ρ P,e .

[0142] During the solution, substitute formulas (19) and (20) into formulas (15)-(18) respectively, then substitute formulas (15)-(18) into formulas (13)-(14). At the same time, substitute the adjoint variables {λ1, μ1} and {λ2, μ2} obtained from formulas (11) and (12) into formulas (13)-(14), and finally solve for the mechanical displacement U and the equivalent electric potential Φ through formulas (13)-(14).

[0143] It should be noted that the reason for optimizing the topological model of the piezoelectric energy harvesting structure under static load (i.e., formula (3)) during the solution process after parameter optimization in this embodiment is that formulas (19) and (20) are not obtained by simple differentiation, so the above optimization of the solution process is carried out.

[0144] Furthermore, this embodiment adopts a filtering technique: different filtering methods are adopted according to different optimization objectives and types. When the objective function is to maximize the energy conversion efficiency and both the piezoelectric material and the substrate material are optimized, density filtering is adopted. For the remaining optimization objectives, sensitivity filtering is adopted. It should be noted that both sensitivity filtering and density filtering are used to prevent numerical calculation instability, resulting in inability to calculate.

[0145] Furthermore, this embodiment updates the design variables: when calculating the new density design variables, the filtered sensitivity information is adopted instead of the sensitivity of the original objective function. According to the sensitivity information and the energy constraint conditions, the MMA algorithm is used to update the design variables ρ E and ρ P .

[0146] After completing the above steps S1 to S4, check whether the objective function and constraints converge. The design variables stop when the numerical values in two consecutive iterations are less than 1% or the number of iterations is greater than 100. If the convergence criterion is met, an image binarization algorithm that satisfies the volume constraint is used to obtain the clear contour of the piezoelectric energy harvesting structure, showing the distribution of the piezoelectric material and the substrate material. Otherwise, use the design variables ρ E and ρ P as the initial values and perform a new round of cyclic iteration.

[0147] Please refer to Figure 3 , the reference domain of this embodiment is the two-dimensional cross-section of the piezoelectric energy harvesting structure with a rectangular cantilever beam, with PZT-5A piezoelectric ceramics on the upper layer and an elastic substrate layer on the lower layer.

[0148] Please refer to Figure 4 , in the embodiment of the present invention, as the thickness of the PZT-5A layer increases, the energy conversion rate also increases; however, as the thickness of the piezoelectric material increases, the strain energy gradually decreases.

[0149] Please refer to Figure 5 , Figure 6 and Figure 7 , Figure 5 shows the optimization result of maximizing the stiffness of the piezoelectric energy harvesting structure, corresponding to the minimum of the strain energy Π S ; Figure 6 the solid line with square dots in shows the convergence curve of the objective function (structural stiffness or energy conversion efficiency), and the solid lines with circular dots and triangular dots are the convergence curves of the structural material volume ratio. The final objective function and volume ratio tend to be stable, that is, both converge; Figure 7 shows the finally optimized piezoelectric energy harvesting structure.

[0150] Please refer to Figure 8 , in this embodiment, the global sensitivity surface and the corresponding sensitivity threshold plane of the last iteration of the cantilever beam example are output as igs files. Open this file with CAD software and directly perform an intersection operation to obtain the contour curve of the optimized structure of the cantilever beam recognizable by the CAD system. The contour curve can be directly stretched to obtain a CAD solid model, which is convenient for further design and manufacturing. For the topology optimization of three-dimensional structures, it can be directly applied. When outputting the CAD model of the optimized structure, the boundary surface of the three-dimensional optimized structure needs to be discretized into triangular patches and then output as an stl file. Subsequently, open this file with CAD software and directly perform solid filling to obtain the CAD solid model of the corresponding three-dimensional optimized structure.

[0151] By optimally allocating the positions of piezoelectric materials and substrate materials, the present invention can effectively improve the energy conversion efficiency of piezoelectric energy harvesting structures. Aiming at the convergence problem of the energy harvesting efficiency in the optimization design, the strain energy is used as a constraint condition, which improves the bearing capacity of the structure and ensures the stability of convergence. Under the same material usage and strain energy constraint, with the piezoelectric material fixed, the optimized substrate results in an energy conversion efficiency of 3.35% for the structure, while the energy conversion efficiency obtained by the collaborative optimization design reaches 12.44%. For the design of the energy harvesting efficiency of piezoelectric energy harvesting structures, the collaborative optimization design can maximize the energy conversion efficiency.

[0152] Embodiment 2

[0153] This embodiment provides a design system for a piezoelectric energy harvesting structure under static load, including:

[0154] Discrete module: used to discretize the design domain of the piezoelectric energy harvesting structure to be designed into N finite elements. For each finite element, a multi-phase material interpolation model representing the structural distribution of the substrate material, piezoelectric material, and void material in the piezoelectric energy harvesting structure is constructed, where the design domain is the two-dimensional cross-section of the piezoelectric energy harvesting structure;

[0155] Construction module: used to construct the objective function of the energy conversion efficiency of the piezoelectric energy harvesting structure under static load, and construct the topological model of the piezoelectric energy harvesting structure under static load based on the multi-phase material interpolation model and the objective function of the energy conversion efficiency;

[0156] Optimization module: used to perform parameter optimization on the topological model of the piezoelectric energy harvesting structure under static load;

[0157] Solution module: used to solve the topological model of the piezoelectric energy harvesting structure under static load after parameter optimization to obtain the designed piezoelectric energy harvesting structure.

[0158] Embodiment 3

[0159] This embodiment provides an electronic device, including a memory, a processor, and a computer program stored on the memory and executable on the processor. When the processor executes the computer program, the steps of the design method of the piezoelectric energy harvesting structure under static load described in Embodiment 1 are implemented.

[0160] Embodiment 4

[0161] This embodiment provides a computer-readable storage medium, on which a computer program is stored. When the computer program is executed by a processor, the steps of the design method of the piezoelectric energy harvesting structure under static load described in Embodiment 1 are implemented.

[0162] Those skilled in the art should understand that the embodiments of the present application can be provided as methods, systems, or computer program products. Therefore, the present application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Moreover, the present application can take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code. The solutions in the embodiments of the present application can be implemented in various computer languages. For example, object-oriented programming languages such as Java and interpreted scripting languages such as JavaScript, etc.

[0163] The present application is described with reference to the flowcharts and / or block diagrams of methods, apparatuses (systems), and computer program products according to the embodiments of the present application. It should be understood that each flow and / or block in the flowchart and / or block diagram, as well as the combination of flows and / or blocks in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to the processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing devices to generate a machine, such that the instructions executed by the processor of the computer or other programmable data processing devices generate means for implementing the functions specified in Figure 1 one or more of the flows Figure 1 or blocks or combinations of blocks.

[0164] These computer program instructions can also be stored in a computer-readable memory that can direct a computer or other programmable data processing device to work in a specific manner, such that the instructions stored in the computer-readable memory generate a manufactured article including instruction means that implement the functions specified in Figure 1 one or more of the flows Figure 1 or blocks or combinations of blocks.

[0165] These computer program instructions can also be loaded onto a computer or other programmable data processing device, such that a series of operation steps are performed on the computer or other programmable device to generate a computer-implemented process, so that the instructions executed on the computer or other programmable device provide steps for implementing the functions specified in Figure 1 one or more of the flows Figure 1 or blocks or combinations of blocks.

[0166] Although the preferred embodiments of the present application have been described, those skilled in the art can make additional changes and modifications to these embodiments once they know the basic creative concepts. Therefore, the appended claims are intended to be construed as including the preferred embodiments and all changes and modifications falling within the scope of the present application.

[0167] Obviously, the above embodiments are merely examples for clear illustration and are not limitations on the implementation manners. For those of ordinary skill in the art, other different forms of changes or modifications can be made based on the above description. It is not necessary and impossible to enumerate all implementation manners here. And the obvious changes or modifications derived therefrom are still within the protection scope of the present invention.

Claims

1. A design method for a piezoelectric energy harvesting structure under static load, characterized in that: include: Step S1: discretize the design domain of the piezoelectric energy harvesting structure to be designed into N finite elements, and for each finite element, construct a multiphase material interpolation model representing the distribution of substrate material, piezoelectric material and empty material structure in the piezoelectric energy harvesting structure, wherein the design domain is a two-dimensional cross section of the piezoelectric energy harvesting structure; Step S2: constructing an objective function of the energy conversion efficiency of the piezoelectric energy harvesting structure under static load, and constructing a topological model of the piezoelectric energy harvesting structure under static load based on the multiphase material interpolation model and the objective function of the energy conversion efficiency; Step S3: Optimizing parameters of the topological model of the piezoelectric energy harvesting structure under static load; Step S4: solving the topological model of the piezoelectric energy harvesting structure under static load after parameter optimization to obtain the designed piezoelectric energy harvesting structure.

2. The design method of a piezoelectric energy harvesting structure under static load according to claim 1, characterized in that: The method for constructing a multiphase material interpolation model representing the distribution of substrate material, piezoelectric material and empty material structure in the piezoelectric energy harvesting structure in step S1 includes: Each finite element of the piezoelectric energy harvesting structure is transformed into E and ρ P It means that the design variable ρ E Used to determine whether the finite element has material, design variable ρ P Used to determine what material the finite element is filled with, and 0≤ρ E,e ≤1、0≤ρ P,e ≤1, e=1,2,…,N; Combined with the design variable ρ E and ρ P Construct the multiphase material interpolation model for each finite element, expressed as: Among them, C, e and κ are the effective property matrices of the interpolated material, C pzt and C nonpzt are the elastic matrices of the piezoelectric material and the substrate material, respectively, and e pzt and κ pzt denote the piezoelectric coupling matrix and dielectric matrix of the piezoelectric material respectively, and p1, p2, p3 and p4 are penalty factors.

3. The design method of a piezoelectric energy harvesting structure under static load according to claim 2, characterized in that: The method of step S2 comprises: The energy conversion efficiency of the piezoelectric energy harvesting structure under static load is expressed as: Among them, U T K uu U and φ T K φφ φ represents the strain energy and the electric energy generated by the piezoelectric energy-harvesting structure, respectively. The sum of the strain energy and the electric energy represents the work done by the external force. U represents the mechanical displacement, and Φ represents the equivalent electric potential. J η Rewritten as: Π S and Π E represent the strain energy and electric energy generated by the piezoelectric energy-harvesting structure, respectively, corresponding to U T K uu U and Φ T K φφ φ; The problem of maximizing energy capture efficiency is transformed into a minimization problem, then J η Rewrite it as: ζ=1 / J η =1+Π S / Π E , maximize the energy conversion efficiency J η It is equivalent to minimizing ζ, and minimizing ζ is used as the objective function; Based on the multiphase material interpolation model and objective function, the topological model of the piezoelectric energy harvesting structure under static load is: find:r E,e ,r P,e min:ζ=1+Π S / P E subject to: V(r E ) / V0≤θ E V(r P ) / V0≤θ P 0≤ρ E,e ,r P,e ≤1,e=1,…,N Among them, the mechanical response of the structure is given by the matrix K 11 Description; Matrix K 12 and K 21 is the piezoelectric coupling matrix, used to couple the structure and electrical response; the matrix K 22 The dynamic characteristics of the piezoelectric energy harvesting structure are determined by the inductance matrix R in the circuit matrix of the external circuit. L , resistance matrix R R , capacitance matrix R C , dielectric matrix K φφ and the electrode conductivity matrix Z; F is the external excitation load; V(ρ E ) represents the total volume of the actual structural material; V(ρ P ) represents the actual volume of the piezoelectric material in the total volume; V0 is the total volume of the design domain, θ E and θ P are the target volume fraction of the total material and the target volume fraction of the piezoelectric material, respectively.

4. The design method of a piezoelectric energy harvesting structure under static load according to claim 3, characterized in that: The method in step S3 includes: For the piezoelectric energy harvesting structure topology model under static load, the piezoelectric material is assumed to be ceramic PZT-5A material and the substrate material is aluminum material. The magnitude of the coefficients of different material properties is balanced to make K 11 , K 12 , K 21 and K 22 The four matrices are maintained at the same order of magnitude and are expressed as: Among them, the element stiffness matrix K uu Assembly involves the piezoelectric elastic coefficient C pzt and the elastic coefficient of aluminum material C nonpzt , unit piezoelectric coupling matrix K uφ Involving the piezoelectric coupling coefficient e pzt , unit dielectric matrix K φφ Involving dielectric constant κ pzt , τ is an integer and its value is 9.

5. The design method of a piezoelectric energy harvesting structure under static load according to claim 3, characterized in that: The method of step S3 further includes: The magnitude of the electric potential Φ in the topological model of the piezoelectric energy harvesting structure under static load is balanced and expressed as: τ is an integer and its value is 9; An equal potential constraint is applied to the upper and lower surfaces of the piezoelectric material with a silver electrode layer on the surface, and the voltage on the electrode layer is equivalent to a node voltage, which is used as the output voltage of the piezoelectric energy harvesting structure. The equipotential constraint is expressed as: in, is the electric potential after equipotential, the Boolean matrix B eq The dimension is N e ×N P , N e is the number of nodes, N P is the number of electrodes.

6. The design method of a piezoelectric energy harvesting structure under static load according to claim 2, characterized in that: The solution process of the piezoelectric energy harvesting structure topological model under static load includes solving the mechanical displacement U and the equivalent potential Φ, specifically: The charge conservation equation of the piezoelectric energy-harvesting structure q piezo +q electrode +q circuit = 0 as the target, the charge equation generated by the piezoelectric material Electrode charge equation and the charge equation in the equivalent external circuit (-ω 2 R L +iωR R +R C )Φ=q circuit , the coupling equation of the whole system of piezoelectric energy harvesting structure is obtained as follows: Where ω = 0, indicating the static load effect; the equivalent charge of the piezoelectric energy-harvesting structure is the charge q generated by the piezoelectric material. piezo , the charge generated on the electrode q electrode , the charge q in the equivalent external circuit circuit Three parts: Global system matrix By K 11 , K 12 , K 21 and K 22 The four matrices are combined to obtain; The mechanical displacement U and equivalent electric potential Φ are solved by the coupling equation of the whole system of the piezoelectric energy harvesting structure.

7. The design method of a piezoelectric energy harvesting structure under static load according to claim 4, characterized in that: When solving the mathematical model in step S4, the design variable ρ is also calculated. E,e and ρ P,e The derivation process is optimized, specifically: The ρ in the topological model of piezoelectric energy harvesting structure under static load e ={ρ E,e ,ρ P,e }, under the action of static concentrated load, calculate the objective function ζ=1 / J η =1+Π S / Π E About the design variable ρ e The first-order derivative of is: The strain energy Π S and electrical energy E Expand them separately to get: Among them, λ1 and λ2 are the accompanying displacement vectors, μ1 and μ2 are the accompanying potential vectors, and the two sets of accompanying variables {λ1,μ1} and {λ2,μ2} are determined by solving the following accompanying equations: By solving the formula and And substitute the Π expanded by the adjoint method S and Π E , and obtain the value of the accompanying variable, then the strain energy Π under static conditions S and electrical energy E About the design variable ρ e The derivatives of are: Where F represents the static condition that is independent of the design variable ρ e The external load vector of Construct K 11 , K 12 , K 21 , K 22 About the design variable ρ e The first-order derivative of is: Among them, B u represents the strain displacement matrix, B φ represents the strain potential matrix; According to the multiphase material interpolation model, the constitutive matrix of the piezoelectric material is obtained with respect to the design variable ρ e The derivatives of include: Elasticity, piezoelectric coupling and dielectric matrices in the constitutive matrix of piezoelectric materials with respect to the design variable ρ E,e The derivative of is: The constitutive matrix of the piezoelectric material with respect to the design variable ρ P,e The derivative of is:

8. A design system for a piezoelectric energy-harvesting structure under static load, characterized in that: include: Discrete module: used to discretize the design domain of the piezoelectric energy harvesting structure to be designed into N finite elements. For each finite element, a multiphase material interpolation model representing the distribution of the substrate material, piezoelectric material and empty material structure in the piezoelectric energy harvesting structure is constructed, wherein the design domain is a two-dimensional cross-section of the piezoelectric energy harvesting structure. Construction module: used to construct the objective function of the energy conversion efficiency of the piezoelectric energy harvesting structure under static load. Based on the multiphase material interpolation model and the objective function of energy conversion efficiency, the topological model of the piezoelectric energy harvesting structure under static load is constructed. Optimization module: used to optimize the parameters of the topological model of the piezoelectric energy harvesting structure under static load; Solution module: used to solve the topological model of the piezoelectric energy harvesting structure under static load after parameter optimization to obtain the designed piezoelectric energy harvesting structure.

9. An electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein: When the processor executes the computer program, the steps of the method for designing a piezoelectric energy harvesting structure under static load as claimed in any one of claims 1 to 7 are implemented.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the steps of the method for designing a piezoelectric energy harvesting structure under static load as claimed in any one of claims 1 to 7 are implemented.

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