Design method and system for piezoelectric energy harvesting structure under dynamic load effect

Abstract

The invention relates to a design method and system for a piezoelectric energy harvesting structure under the action of a dynamic load, and the method comprises the steps: S1, dispersing a design domain of the piezoelectric energy harvesting structure to be designed into N units, and for each unit, carrying out the design of the piezoelectric energy harvesting structure; a multiphase material interpolation model representing structural distribution of a substrate material, a piezoelectric material and an empty material in the piezoelectric energy harvesting structure is constructed, and a design domain is a two-dimensional cross section of the piezoelectric energy harvesting structure; s2, constructing a dynamic flexibility objective function of the piezoelectric energy harvesting structure under the action of the dynamic load and an objective function of energy conversion efficiency, and constructing a topological model of the piezoelectric energy harvesting structure under the dynamic load based on the dynamic flexibility objective function, the multiphase material interpolation model and the objective function of the energy conversion efficiency; and S3, solving the topological model of the piezoelectric energy harvesting structure under the dynamic load to obtain a designed piezoelectric energy harvesting structure. According to the invention, a piezoelectric energy harvesting structure with high energy harvesting efficiency can be designed under the action of a dynamic load.

Description

Technical Field

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

[0002] In the study of topological optimization design of piezoelectric structure static load energy harvesting efficiency, it is usually assumed that the structure is optimized under static load. However, in actual applications, piezoelectric energy harvesting structures are often continuously acted upon by electromagnetic exciters, and are mainly subjected to the periodic simple harmonic inertial force excitation generated by the exciter, which manifests as forced vibration. In this case, the structure not only needs to cope with static loads, but also needs to optimize its response under dynamic excitation to ensure efficient energy harvesting.

[0003] This periodic vibration load will affect the energy conversion efficiency of the piezoelectric material. Therefore, the optimization design needs to consider the impact of dynamic response, including the vibration mode, frequency response characteristics, and how to maximize energy conversion under different vibration conditions of the piezoelectric material. In this case, the goal of topology optimization is not only to improve the static energy conversion efficiency, but also to take into account the dynamic performance of the structure to ensure the overall performance of the piezoelectric structure under actual working conditions.

[0004] Existing research mainly focuses on the optimization of the layout of a single piezoelectric material, but the actual piezoelectric energy-harvesting structure is usually composed of a piezoelectric material and a substrate material. In addition, most of the existing dynamic structure designs consider load-independent or fixed-frequency situations, while there are relatively few studies on dynamic inertial loads and wide-frequency ranges that are closer to practical applications. In recent years, research on structural topology optimization has mainly focused on solving the optimal material distribution form under the action of external irrelevant loads. However, these studies often ignore the influence of the material's own weight on structural performance, or equate the material's own weight to a concentrated external load applied to the center of gravity of the structure. Although this treatment method has little effect on the topological optimization of general structures, it cannot be ignored for structures such as bridges that need to consider the material's own weight effect. In addition, for piezoelectric energy-harvesting structures under periodic excitation of electromagnetic exciters, they mainly bear simple harmonic inertial loads related to their own weight. Therefore, when performing structural topology optimization, it is crucial to fully consider the influence of inertial loads on the optimization target.

[0005] In summary, the existing piezoelectric energy harvesting structures designed under dynamic loads are often not combined with the actual application process, resulting in unreasonable designed piezoelectric energy harvesting structures. Summary of the invention

[0006] Therefore, the technical problem to be solved by the present invention is to overcome the problem in the prior art that the piezoelectric energy-harvesting structure designed under the action of dynamic loads is often not combined with the situation in the actual application process.

[0007] To solve the above technical problems, the present invention provides a design method for a piezoelectric energy harvesting structure under dynamic loads, including:

[0008] Step S1: Discretize the design domain of the piezoelectric energy harvesting structure to be designed into N elements. For each element, construct a multi-phase material interpolation model representing the structural distributions of the base 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;

[0009] Step S2: Construct the dynamic compliance objective function of the piezoelectric energy harvesting structure under dynamic loads and the objective function of the energy conversion efficiency. Based on the dynamic compliance objective function, the multi-phase material interpolation model, and the objective function of the energy conversion efficiency, construct the topology model of the piezoelectric energy harvesting structure under dynamic loads;

[0010] Step S3: Solve the topology model of the piezoelectric energy harvesting structure under dynamic loads to obtain the designed piezoelectric energy harvesting structure.

[0011] In an embodiment of the present invention, the method for constructing the multi-phase material interpolation model representing the structural distributions of the base material, piezoelectric material, and void material in the piezoelectric energy harvesting structure in step S1 includes:

[0012] Under the action of a harmonic inertial force, construct the multi-phase material interpolation model of the piezoelectric energy harvesting structure based on the RAMP model, expressed as:

[0013]

[0014] 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, ρ represents the density of the multi-phase material after interpolation, ρ pzt and ρ nonpzt are the mass densities of the piezoelectric material and the base material respectively, and q1, q2, q3, and q4 are penalty factors.

[0015] In an embodiment of the present invention, the dynamic compliance objective function of the piezoelectric energy harvesting structure under dynamic loads constructed in step S2 is expressed as:

[0016]

[0017] where C d is the dynamic compliance of the piezoelectric energy harvesting structure, U is the vector of the amplitude of the steady-state displacement of the structure, Φ is the vector of the amplitude of the steady-state electric potential response of the structure, and P is the vector of the amplitude of the external load.

[0018] In an embodiment of the present invention, the method for constructing the objective function of the energy conversion efficiency of the piezoelectric energy harvesting structure under dynamic load in step S2 includes:

[0019] Construct the energy conversion efficiency of the piezoelectric energy harvesting structure under a harmonic inertial force, expressed as:

[0020]

[0021] where ω is the angular frequency of the external load of the given harmonic inertial force, J η J(ω) is the energy conversion efficiency; Π S Π(ω) and Π E (ω) respectively represent the strain energy and the generated electrical energy of the piezoelectric energy harvesting structure at the angular frequency ω, and Π S (ω)+Π E (ω) is the total external input energy; and are the complex conjugate vectors of Φ and U respectively; K φφ is the global dielectric matrix, and K uu is the global stiffness matrix;

[0022] Rewrite the energy conversion efficiency J η (ω) as ζ(ω), and minimize ζ(ω) as the objective function, expressed as:

[0023] ζ(ω) = 1 / J η (ω) = 1 + Π S (ω) / Π E (ω).

[0024] In an embodiment of the present invention, constructing the topology model of the piezoelectric energy harvesting structure under dynamic load in step S2 includes:

[0025] Construct a topology optimization model for minimizing the dynamic compliance according to the dynamic compliance objective function, expressed as:

[0026] find: ρ E,e , ρ P,e

[0027]

[0028] subject to:

[0029]

[0030] 0 ≤ ρ E,e ≤ 1, e = 1, …, N

[0031] where, is the dynamic stiffness matrix, ω Pω is the angular frequency of the given simple harmonic varying external load, M is the global mass matrix, V(ρ E ) is the total volume of the control entity piezoelectric material and the substrate material, V0 is the total volume of the initial design domain, is the given lower limit of the total material target volume fraction, is the given upper limit of the total material target volume fraction;

[0032] According to the multiphase material interpolation model and the objective function ζ(ω), a topology optimization model of the energy conversion efficiency under dynamic load is constructed, expressed as:

[0033] find: ρ E,e , ρ P,e

[0034] min: ζ(ω) = 1 + Π S (ω) / Π E (ω)

[0035] subject to:

[0036]

[0037] 0 ≤ ρ E,e ≤ 1, e = 1, …, N

[0038] where, K 11 、K 12 、K 21 and K 22 are respectively expressed as:

[0039] K 11 = K uu - Mω 2 , K 12 = K uφ , K 21 = K φu , K 22 = K φφ .

[0040] In an embodiment of the present invention, when solving the topology model of the piezoelectric energy harvesting structure under dynamic load in step S3, the derivative process of the design variables ρ E,e and ρ P,e is optimized, specifically:

[0041] Calculate the first-order derivative of the objective function ζ(ω) with respect to the element density variable ρ e = {ρ E,e , ρ P,e}:

[0042]

[0043] The strain energy Π under the excitation of the simple harmonic inertial force is expanded by the adjoint method S (ω) to obtain:

[0044]

[0045] where λ1 is the adjoint displacement vector and μ1 is the adjoint electric potential vector; the expanded Π S (ω) is differentiated with respect to the e-th design variable ρ e to obtain:

[0046]

[0047] By solving the adjoint equation to determine the values of the first set of adjoint variables {λ1, μ1}:

[0048] Then is arranged as:

[0049]

[0050] By solving to determine the second set of adjoint variables {λ2, μ2}, then the electrical energy Π E (ω) with respect to the e-th design variable ρ e is differentiated to obtain:

[0051]

[0052] Construct K 11 、K 12 、K 21 、K 22 、The first-order derivative of the mass matrix F with respect to the design variable ρ e is:

[0053]

[0054] where a is the linear acceleration vector, B u is the strain-displacement matrix, B φ is the strain-electric potential matrix, and N is the shape function matrix;

[0055] 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,e is obtained and expressed as:

[0056]

[0057] In an embodiment of the present invention, when solving the dynamic compliance minimization topology optimization model in step S3, the differentiation process of the design variables ρ E,e and ρ P,e is optimized, specifically:

[0058] Construct the structural steady-state response equation of the dynamic flexibility, expressed as:

[0059]

[0060] Derive the structural steady-state response equation of the dynamic flexibility with respect to the design variable ρ e ={ρ E,e , ρ P,e} to obtain:

[0061]

[0062] After arrangement, the derivative of the frequency response is obtained as:

[0063]

[0064] Finally, according to the formulas and obtain the derivative of the objective function C d with respect to the design variable ρ e as:

[0065]

[0066] To solve the above technical problems, the present invention provides a design system for a piezoelectric energy harvesting structure under dynamic loads, including:

[0067] Discretization module: used to discretize the design domain of the piezoelectric energy harvesting structure to be designed into N elements. For each 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;

[0068] Construction module: used to construct the dynamic flexibility objective function of the piezoelectric energy harvesting structure under dynamic loads and the objective function of the energy conversion efficiency. Based on the dynamic flexibility objective function, the multi-phase material interpolation model, and the objective function of the energy conversion efficiency, construct the topology model of the piezoelectric energy harvesting structure under dynamic loads;

[0069] Solution module: used to solve the topology model of the piezoelectric energy harvesting structure under dynamic loads 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 above design method for the piezoelectric energy harvesting structure under dynamic loads 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 for the piezoelectric energy harvesting structure under dynamic 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 for the piezoelectric energy harvesting structure under dynamic load designed by the present invention improves the energy harvesting efficiency under fixed frequency and broadband harmonic inertial load by optimizing the topological configuration of the substrate structure;

[0074] The present invention adopts the RAMP material model and proposes a continuous design strategy with gradually changing frequency to improve the convergence of the topological evolution process. The design results show that the dynamic flexibility optimization makes the fundamental characteristic frequency of the structure approach the maximum value, thus moving away from the lower excitation frequency to avoid resonance at low frequencies; under the condition that the volume fraction of the piezoelectric material remains fixed at 20%, the substrate structure is optimized in the broadband range (100 - 500 Hz), and the energy conversion efficiency of the structure is 3.19% - 4.48%, effectively improving the broadband energy harvesting efficiency. Description of the Drawings

[0075] 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.

[0076] Figure 1 is the flowchart of the method of the present invention;

[0077] Figure 2 is a schematic diagram of the piezoelectric energy harvesting structure (piezoelectric cantilever beam model) under the action of a harmonic excitation source (harmonic inertial force) in the embodiment of the present invention;

[0078] Figure 3 is the evolution process of the energy conversion efficiency iteration curve and the optimal topological configuration at an excitation frequency of 0 Hz in the embodiment of the present invention, that is, when the structure is only under its own weight;

[0079] Figure 4 is the iteration history curve graph of the dynamic flexibility and static flexibility of the substrate material structure at an excitation frequency of 0 Hz in the embodiment of the present invention, that is, when the structure is only under its own weight;

[0080] Figure 5 is the iteration curve graph of the fundamental frequency of the substrate material structure at an excitation frequency of 0 Hz in the embodiment of the present invention, that is, when the structure is only under its own weight;

[0081] Figure 6It is the evolution process of the energy conversion efficiency design and the optimal topological configuration of the substrate material at a fixed excitation frequency of 500 Hz in the embodiments of the present invention;

[0082] Figure 7 It is the iterative history curve graph of the structural dynamic flexibility and static flexibility of the substrate material at a fixed excitation frequency of 500 Hz in the embodiments of the present invention;

[0083] Figure 8 It is the iterative curve graph of the fundamental frequency of the substrate material at a fixed excitation frequency of 500 Hz in the embodiments of the present invention;

[0084] Figure 9 It is the relationship curve graph between the energy conversion efficiency and the frequency under fixed excitation frequency and broadband optimization in the embodiments of the present invention. Detailed implementation manners

[0085] 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 specific embodiments cited are not intended to limit the present invention.

[0086] Embodiment 1

[0087] Refer to Figure 1 As shown, since the piezoelectric energy harvesting structure in reality is continuously affected by the electromagnetic exciter, that is, the piezoelectric energy harvesting structure mainly bears the forced vibration of the periodic simple harmonic inertial force excitation generated by the exciter, therefore, the present invention relates to a design method for a piezoelectric energy harvesting structure under dynamic load, including:

[0088] Step S1: Discretize the design domain of the piezoelectric energy harvesting structure to be designed into N elements (finite elements). For each element, construct a multi-phase material interpolation model representing the substrate material, piezoelectric material, and air material structure distribution in the piezoelectric energy harvesting structure, where the design domain is the two-dimensional cross-section of the piezoelectric energy harvesting structure;

[0089] Step S2: Construct the dynamic flexibility objective function of the piezoelectric energy harvesting structure under dynamic load and the objective function of the energy conversion efficiency. Based on the dynamic flexibility objective function, the multi-phase material interpolation model, and the objective function of the energy conversion efficiency, construct the topological model of the piezoelectric energy harvesting structure under dynamic load;

[0090] Step S3: Solve the topological model of the piezoelectric energy harvesting structure under dynamic load to obtain the designed piezoelectric energy harvesting structure.

[0091] The following is a detailed introduction to this embodiment:

[0092] Please refer to Figure 1 and Figure 2 A design method for a piezoelectric energy harvesting structure under dynamic load provided by the present invention mainly includes the following steps:

[0093] Step S1: For the dynamic response topology optimization model under the action of the simple harmonic inertial force, the RAMP model is used to interpolate the density and constitutive matrix of the multi-phase materials (substrate material, piezoelectric material, and void material). The multi-phase material interpolation model can be written as:

[0094]

[0095] 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 substrate material respectively. e pzt and κ pzt represent the piezoelectric coupling matrix and the dielectric matrix of the piezoelectric material respectively. ρ represents the density of the multi-phase material after interpolation. ρ pzt and ρ nonpzt are the mass densities of the piezoelectric material and the substrate material respectively. q1, q2, q3, and q4 are penalty factors, and their values are generally between 5 and 120 to push the design variables to extreme values close to 0 or 1. In the material interpolation model, the physical meanings of the design variable values of ρ E,e and ρ P,e are as follows:

[0096]

[0097] where ρ E,e is used to determine whether the material is filled in element e. If it is filled, it is the piezoelectric material or the substrate material or a mixture of both; if not, it is the void material. If there is no void material in element e, then ρ P,e is used to distinguish the piezoelectric material and the substrate material in this element.

[0098] Step S2: Construct the dynamic compliance objective function of the piezoelectric energy harvesting structure under the action of dynamic loads, which is specifically as follows:

[0099] This embodiment considers the optimization problem of the dynamic compliance response of a linear elastic structure under a given simple harmonic external excitation. The structural response under the simple harmonic inertial force consists of two parts: the transient response and the steady-state response. Due to the existence of damping, the transient response part of the structural response decays rapidly with time. Therefore, this embodiment mainly considers the steady-state response part during optimization. The following takes the undamped case as an example for illustration. The dynamic compliance is defined as the integral over the entire design domain of the product of the amplitude of the external load and the amplitude of the steady-state response of the structural displacement at the load application point.

[0100] Specifically, in this embodiment, according to the full-system coupling equations of the piezoelectric energy harvesting structure (corresponding to formulas (3) and (4)), the steady-state response equation of the structure after finite element discretization (corresponding to formula (5)) is obtained as:

[0101]

[0102] Among them, the global system matrix is assembled from four sub-module matrices K 11 , K 12 , K 21 and K 22 . The mechanical response of the structure is described by the matrix K 11 , which varies with the layout of the material and the layout of the structure. The matrices K 12 and K 21 are piezoelectric coupling matrices used to couple the structural and electrical responses and vary 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 , resistance matrix R R , capacitance matrix R C , dielectric matrix K φφ and the conductance matrix Z of the electrodes in the circuit matrix of the external circuit of the piezoelectric energy harvesting structure (in this embodiment, the external circuit of the piezoelectric energy harvesting structure is an RLC circuit with a series resistance R, inductance L, and capacitance C). Among them, the dielectric matrix K φφ varies with the layout of the material, and the conductance matrix Z varies with the conductivity of the electrodes and the topology. φφ is defined as the dynamic stiffness matrix, ω is the angular frequency of the given harmonic-varying external load, where ω P = 2πf P . When the external excitation is a harmonic-varying external load, the external load can be expressed in complex form as f(t) = Fe P , the corresponding steady-state structural displacement response is u(t) = Ue iωt i ωt , and the steady-state electrical potential response of the structure is φ(t) = Φe iωt . Among them, F is the amplitude vector of the external load after finite element discretization, U is the amplitude vector of the steady-state structural displacement response, and Φ is the amplitude vector of the steady-state electrical potential response of the structure. For the vibrational piezoelectric energy harvesting structure, the structure only bears inertial loads and there is no external charge input. Therefore, according to formula (5), the dynamic flexibility C d d T T can be further expressed as the vector product of the external load amplitude vector P and the steady-state structural displacement amplitude vector {U, Φ}:

[0103]

[0104] It should be noted that when the external load frequency is high, the dynamic stiffness matrix K d may be non-positive definite. At this time, P T {U, Φ} TIt may be negative, and minimizing it will lead to failure (tending to negative infinity). For this reason, the absolute value sign is introduced into the expression of dynamic flexibility for correction.

[0105] It should be noted that the piezoelectric energy harvesting structure has an external circuit to solve 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 , and the final coupling equation of the whole system of the piezoelectric energy harvesting structure (i.e., formula (3)) is obtained.

[0106] Step S2: construct an objective function of the energy conversion efficiency of the piezoelectric energy harvesting structure under dynamic load, as follows:

[0107] The energy conversion efficiency of the piezoelectric energy-harvesting structure under simple harmonic inertial force is expressed as:

[0108]

[0109] Where ω is the angular frequency of the external load of the given simple harmonic inertial force, and the energy conversion efficiency J η (ω) corresponds to the energy conversion efficiency under the excitation frequency ω. S (ω) and Π E (ω) represent the strain energy and electric energy generated by the piezoelectric energy-harvesting structure at the angular frequency ω, Π S (ω)+Π E (ω) is the total input energy from the outside world, and are the complex conjugate vectors of Φ and U respectively; K φφ is the global dielectric matrix, K uu is the global stiffness matrix. The objective function of this embodiment is to transform the maximization problem into a minimization problem, that is, to re-express the design target as the inverse form of the energy conversion efficiency (i.e., formula (7)), and the dynamic load energy conversion efficiency is further rewritten as:

[0110] ζ(ω)=1 / J η (ω)=1+Π S (ω) / Π E (ω) (8)

[0111] After reconstructing the objective function, the energy conversion efficiency J of the piezoelectric energy harvester will be maximized. η$(ω)$ is equivalent to minimizing $ζ(ω)$.

[0112] In step S2, the construction of the topological model of the piezoelectric energy harvesting structure under dynamic load includes two models. First, a topology optimization model for minimizing dynamic compliance is constructed according to the dynamic compliance objective function. Second, a topology optimization model for energy conversion efficiency under dynamic load is constructed according to the multi-phase material interpolation model and the objective function $ζ(ω)$.

[0113] (1) A topology optimization model for minimizing dynamic compliance is constructed according to the dynamic compliance objective function (ignoring damping and external resistance), expressed as:

[0114]

[0115] Among them, the material usage is controlled by two volume constraints. In the constraint conditions, $V(ρ$ E ) is the total volume of the controlled solid piezoelectric material and the substrate material. $V_0$ is the total volume of the initial design domain. is the given lower limit of the total material target volume fraction. is the given upper limit of the total material target volume fraction. is the dynamic stiffness matrix, $ω$ P is the angular frequency of the given harmonically varying external load, and $M$ is the global mass matrix.

[0116] According to the multi-phase material interpolation model and the objective function $ζ(ω)$, and considering the volume constraint, a topology optimization model for energy conversion efficiency under dynamic load is constructed, expressed as:

[0117]

[0118] Among them, $K$ 11 , $K$ 12 , $K$ 21 and $K$ 22 are four sub-module matrices. In the case of ignoring damping and external resistance, the four sub-module matrices $K$ 11 , $K$ 12 , $K$ 21 and $K$ 22 can be simplified as:

[0119] $K$ 11 = $K$ uu - $Mω$ 2 , $K$ 12 = $K$ uφ , $K$ 21 = $K$ φu , $K$ 22 = $K$ φφ (11)

[0120] It should be noted that in the process of solving the above two models, considering problems such as complexity, the piezoelectric material is fixed, and only the optimization of the substrate material is carried out. That is, in this embodiment, ρ P,e is taken as a fixed value, and only ρ E,e needs to be solved during the actual solution.

[0121] Since optimizing the energy conversion efficiency of the piezoelectric energy harvesting structure is a singularity problem, in the optimization process, it often tends to sacrifice the stiffness of the structure to meet the objective function of energy conversion efficiency. This will weaken the load-bearing performance of the structure and it is difficult to ensure convergence. Therefore, to solve this problem, the static compliance constraint is introduced into the optimization model, that is, the following constraint can also be added in formula (10):

[0122]

[0123] where is the given static compliance constraint value, and the selection of its numerical value can refer to the static compliance result obtained by minimizing the dynamic compliance.

[0124] Step S3: When solving the topology optimization model for minimizing the dynamic compliance, the solution process is also optimized, including optimizing the derivative process of the design variables ρ E,e and ρ P,e (i.e., sensitivity optimization, and the sensitivity includes ), specifically as follows:

[0125] First, in the case of no damping and external resistance, the derivative of the structural steady-state response equation in formula (5) with respect to the design variable ρ e ={ρ E,e ,ρ P,e} is obtained as:

[0126]

[0127] Assume that the external excitation frequency ω P is not equal to any natural frequency of the structure. Then, after sorting, the derivative of the frequency response is obtained as:

[0128]

[0129] Finally, according to formula (13) and the derivative of the objective function C d with respect to the design variable ρ e is obtained as:

[0130]

[0131] Step S3: When solving the topology model of the piezoelectric energy harvesting structure under dynamic load, the solution process is also optimized, including the design variable ρ E,eand ρ P,e Optimize the derivative process of (i.e., sensitivity optimization, where sensitivity includes and in the following formula), as follows:

[0132] Under the action of a simple harmonic inertial load, the first derivative of the objective function ζ(ω) with respect to the element density variable ρ e ={ρ E,e , ρ P,e} is written as:

[0133]

[0134] ω≠0 indicates the action of a dynamic load (i.e., a simple harmonic inertial force). Under the excitation of a simple harmonic inertial force, the strain energy ∏ E (ω) and the electrical energy ∏ E (ω) are expressed as the time-average values, i.e.:

[0135]

[0136] The specific form of the adjoint equation of the strain energy under the excitation of a simple harmonic inertial force is as follows:

[0137]

[0138] where λ1 is the adjoint displacement vector and μ1 is the adjoint electric potential vector. Differentiate formula (18) with respect to the e-th design variable ρ e , and its specific form is as follows:

[0139]

[0140] By solving the following adjoint equation (formula (20)), the values of the first set of adjoint variables {λ1, μ1} can be determined, and its specific form is as follows:

[0141]

[0142] Subsequently, formula (19) is further arranged as:

[0143]

[0144] Similarly, by solving the second set of adjoint variables {λ2, μ2}, its specific form is as follows:

[0145]

[0146] By solving the equation, the derivative of the electrical energy Π E (ω) with respect to the e-th design variable ρ e is:

[0147]

[0148] Construct K 11 , K 12 , K 21 , K 22 , mass matrix F, and design-related simple harmonic inertial loads with respect to design variables ρ e The first-order derivative of is expressed as follows:

[0149]

[0150]

[0151] Wherein, a is a linear acceleration vector, which refers to the acceleration due to gravity in this embodiment, and the negative sign represents that the acceleration is in the opposite direction along the z-axis. u is the strain displacement matrix, B φ is the strain potential matrix, and N is the shape function matrix. According to the RAMP multiphase material interpolation model, the material constitutive matrix is ​​obtained with respect to the design variable ρ E,e The specific form of the derivative is:

[0152]

[0153] By combining the strain energy derivative formula and the electric energy derivative formula, the first-order derivative formula of the objective function with respect to the unit density variable is solved respectively, and the first-order sensitivity information of the objective function with respect to the design variable can be obtained.

[0154] When solving, substitute formula (29) into formula (24)-(28), and then substitute the results of formula (24)-(28) into formula (23). At the same time, substitute the accompanying variables {λ2,μ2} obtained by formula (22) into formula (23), and finally solve formula (23) to obtain the derivative of electric energy with respect to the design variable. Similarly, substitute the accompanying variables {λ1,μ1} obtained by formula (20) into formula (21), and at the same time, substitute formula (29) into formula (24)-(28), and then substitute the results of formula (24)-(28) into formula (21), and finally solve formula (21) to obtain the derivative of strain energy with respect to the design variable.

[0155] Step S3 also includes filtering technology: different filtering methods are used according to different optimization targets and types. When the objective function is to maximize the energy conversion efficiency and the piezoelectric material and the substrate material are optimized at the same time, density filtering is used. For the remaining optimization targets, sensitivity filtering is used. It should be noted that sensitivity filtering and density filtering are both used to prevent numerical calculations from being unstable and causing calculation failures.

[0156] Step S3 also includes updating the design variables: When calculating the new density design variables, the sensitivity information filtered by the filtering technique is used 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 .

[0157] After completing the above steps S1 to S3, check whether the objective function and the 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 times. If the convergence criterion is met, an image binarization algorithm that satisfies the volume constraint is used to obtain a clear structural contour and display the distribution of the piezoelectric material and the substrate material. Otherwise, the updated design variables are used as the initial values of step S2, and a new round of loop iteration is performed.

[0158] Please refer to Figure 3 、 Figure 4 and Figure 5 . In the embodiment of the present invention, at an excitation frequency of 0 Hz, that is, the energy conversion efficiency design of the substrate material under the action of its own weight only, that is, the static flexibility of the structure is equal to the dynamic flexibility. Figure 3 Under the action of the self-weight load, the substrate material is mainly distributed at the root and the free end of the cantilever beam. After 100 steps of iteration, the objective function does not converge. Figure 4 In

[0159] Please refer to Figure 6 、 Figure 7 and Figure 8 . In the embodiment of the present invention, it is the iteration curve of the energy conversion efficiency of the substrate material at an excitation frequency of 500 Hz. As the iteration progresses, after 50 iterations, the energy conversion efficiency tends to be stable and finally is 4.41%. There is a peak in the dynamic flexibility curve. The reason for this peak is that when iterating to the 17th to 21st times, the materials connected at the fixed end and the free end of the piezoelectric energy harvesting structure gradually separate and finally break, resulting in the oscillation of the dynamic flexibility.

[0160] Please refer to Figure 9 . For the relationship curve between the energy conversion efficiency and the frequency under the fixed excitation frequency and the broadband optimization in the embodiment of the present invention. Among them, for the broadband optimization, the objective functions of three excitation frequencies of 100 Hz, 200 Hz, and 500 Hz are weighted respectively. From Figure 9 , it can be seen that under the frequency excitation of 200 Hz, the highest energy conversion efficiency achieved by the single-frequency optimization of 200 Hz is 3.70%. Followed by the broadband optimization, and finally the 500 Hz frequency.

[0161] Embodiment 2

[0162] This embodiment provides a design system for a piezoelectric energy harvesting structure under static load, including:

[0163] Discrete module: used to discretize the design domain of the piezoelectric energy harvesting structure to be designed into N units. For each unit, a multiphase material interpolation model representing the distribution of substrate material, piezoelectric material, and air material structure in the piezoelectric energy harvesting structure is constructed, where the design domain is the two-dimensional cross-section of the piezoelectric energy harvesting structure;

[0164] Construction module: used to construct the dynamic compliance objective function of the piezoelectric energy harvesting structure under dynamic load and the objective function of energy conversion efficiency, and based on the dynamic compliance objective function, the objective function of energy conversion efficiency, and the multiphase material interpolation model, construct the topology model of the piezoelectric energy harvesting structure under dynamic load;

[0165] Solution module: used to solve the topology model of the piezoelectric energy harvesting structure under dynamic load to obtain the designed piezoelectric energy harvesting structure.

[0166] Embodiment 3

[0167] 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.

[0168] Embodiment 4

[0169] 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.

[0170] Those skilled in the art should understand that the embodiments of the present application can be provided as a method, a system, or a computer program product. Therefore, the present application can adopt the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware aspects. Moreover, the present application can adopt 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.

[0171] This application is described with reference to the flowcharts and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the present application. It should be understood that each flow and / or block in the flowchart and / or block diagram can be implemented by computer program instructions, and the combination of flows and / or blocks in the flowchart and / or block diagram can also 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, so that the instructions executed by the processor of the computer or other programmable data processing devices generate a means for implementing the functions specified in one or more of the flows Figure 1 one or more of the flows and / or blocks Figure 1 or a means for implementing the functions specified in one or more of the blocks.

[0172] 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, so that the instructions stored in the computer-readable memory generate a manufactured article including an instruction means, and the instruction means implements the functions specified in one or more of the flows Figure 1 one or more of the flows and / or blocks Figure 1 or a means for implementing the functions specified in one or more of the blocks.

[0173] These computer program instructions can also be loaded onto a computer or other programmable data processing device, so that a series of operation steps are executed on the computer or other programmable device to generate a computer-implemented process, and thus the instructions executed on the computer or other programmable device provide steps for implementing the functions specified in one or more of the flows Figure 1 one or more of the flows and / or blocks Figure 1 or a means for implementing the functions specified in one or more of the blocks.

[0174] 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.

[0175] 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 variations can be made based on the above description. It is not necessary and impossible to list all the implementation manners here. And the obvious changes or variations derived therefrom are still within the protection scope of the present invention.

Claims

1. A design method for a piezoelectric energy harvesting structure under dynamic load, characterized in that: include: Step S1: discretize the design domain of the piezoelectric energy harvesting structure to be designed into N units, and for each unit, 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 a dynamic compliance objective function and an energy conversion efficiency objective function of the piezoelectric energy harvesting structure under dynamic load, and constructing a topological model of the piezoelectric energy harvesting structure under dynamic load based on the dynamic compliance objective function, the multiphase material interpolation model and the energy conversion efficiency objective function respectively; Step S3: solving the topological model of the piezoelectric energy harvesting structure under dynamic load to obtain the designed piezoelectric energy harvesting structure.

2. The design method for a piezoelectric energy harvesting structure under dynamic 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: Under the action of simple harmonic inertial force, the multiphase material interpolation model of piezoelectric energy harvesting structure is constructed based on RAMP model, which is 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, ρ denotes the density of the multiphase material after interpolation, and ρ pzt and ρ nonpzt are the mass densities of piezoelectric material and substrate material respectively, and q1, q2, q3 and q4 are penalty factors.

3. The design method for a piezoelectric energy harvesting structure under dynamic load according to claim 2, characterized in that: In step S2, the dynamic compliance objective function of the piezoelectric energy harvesting structure under the action of dynamic load is constructed, which is expressed as: Among them, C d is the dynamic compliance of the piezoelectric energy-harvesting structure, U is the corresponding amplitude vector of the structural steady-state displacement, Φ is the amplitude vector of the structural steady-state potential response, and P is the amplitude vector of the external load.

4. The design method for a piezoelectric energy harvesting structure under dynamic load according to claim 3, characterized in that: The method for constructing the objective function of the energy conversion efficiency of the piezoelectric energy harvesting structure under the action of the dynamic load in step S2 includes: The energy conversion efficiency of the piezoelectric energy-harvesting structure under simple harmonic inertial force is expressed as: Where, ω is the angular frequency of the external load of the given simple harmonic inertial force, J η (ω) is the energy conversion efficiency; S (ω) and Π E (ω) represent the strain energy and electric energy generated by the piezoelectric energy-harvesting structure at the angular frequency ω, Π S (ω)+Π E (ω) is the total input energy from the outside world; and are the complex conjugate vectors of Φ and U respectively; K φφ is the global dielectric matrix, K uu is the global stiffness matrix; The energy conversion efficiency J η (ω) is rewritten as ζ(ω), and minimizing ζ(ω) is taken as the objective function, which is expressed as: ζ(ω)=1 / J η (ω)=1+Π S (w) / P E (oh)。 5. The design method for a piezoelectric energy harvesting structure under dynamic load according to claim 4, characterized in that: The step S2 constructs a topological model of a piezoelectric energy harvesting structure under dynamic load, including: According to the dynamic flexibility objective function, a dynamic flexibility minimization topology optimization model is constructed, which is expressed as: find:r E,e ,r P,e subject to: 0≤ρ E,e ≤1,e=1,…,N in, is the dynamic stiffness matrix, ω P is the angular frequency of the given harmonically varying external load, M is the global mass matrix, V(ρ E ) is the total volume of the control solid piezoelectric material and the substrate material, V0 is the total volume of the initial design domain, is the lower limit of the target volume fraction of the given overall material, is the upper limit of the target volume fraction of the given overall material; According to the multiphase material interpolation model and the objective function ζ(ω), a topology optimization model of energy conversion efficiency under dynamic load is constructed, which is expressed as: find:r E,e ,r P,e min:ζ(ω)=1+Π S (w) / P E (oh) subject to: 0≤ρ E,e ≤1,e=1,…,N Among them, K 11 , K 12 , K 21 and K 22 Respectively expressed as: K 11 =K uu -Mω 2 ,K 12 =K uφ ,K 21 =K φu ,K 22 =K φφ 。 6. The design method for a piezoelectric energy harvesting structure under dynamic load according to claim 5, characterized in that: In step S3, when solving the topological model of the piezoelectric energy harvesting structure under dynamic load, the design variable ρ E,e and ρ P,e The derivation process is optimized, specifically: Calculate the objective function ζ(ω) with respect to the cell density variable ρ e ={ρ E,e ,ρ P,e The first derivative of} is: The strain energy ∏ is calculated by the adjoint method under the excitation of simple harmonic inertial force S (ω) is expanded to obtain: Where λ1 is the associated displacement vector, μ1 is the associated potential vector; the expanded Π S (ω) for the e-th design variable ρ e The derivative is: By solving the adjoint equation To determine the value of the first set of adjoint variables {λ1,μ1}: Then Arranged as: By solving To determine the second set of accompanying variables {λ2,μ2}, the electric energy ∏ E (ω) For the e-th design variable ρ e The derivative is: Construct K 11 , K 12 , K 21 , K 22 , the mass matrix F with respect to the design variable ρ e The first-order derivative of is: Where a is the linear acceleration vector, B u is the strain displacement matrix, B φ is the strain potential matrix, N is the shape function 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,e The derivative of is expressed as:

7. The design method for a piezoelectric energy harvesting structure under dynamic load according to claim 1, characterized in that: In step S3, when solving the dynamic flexibility minimization topology optimization model, the design variable ρ E,e and ρ P,e The derivation process is optimized, specifically: The structural steady-state response equation of dynamic flexibility is constructed as follows: The structural steady-state response equation of dynamic flexibility is relative to the design variable ρ e ={ρ E,e ,ρ P,e } Taking the derivative, we get: The derivative of the frequency response is: Finally, according to the formula and Obtain the objective function C d Relative to the design variable ρ e The derivative of is:

8. A design system for a piezoelectric energy-harvesting structure under dynamic load, characterized in that: include: Discrete module: used to discretize the design domain of the piezoelectric energy harvesting structure to be designed into N units. For each unit, 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 dynamic compliance objective function and energy conversion efficiency objective function of the piezoelectric energy harvesting structure under dynamic load. The topological model of the piezoelectric energy harvesting structure under dynamic load is constructed based on the dynamic compliance objective function, the multiphase material interpolation model and the energy conversion efficiency objective function respectively. Solution module: used to solve the topological model of the piezoelectric energy harvesting structure under dynamic load 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 dynamic 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 dynamic loads as claimed in any one of claims 1 to 7 are implemented.

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