A Method and System for Performance Analysis of Flexoelectric Materials Based on Small Sample Data Augmentation Analysis

Abstract

This invention discloses a method and system for analyzing the performance of flexoelectric materials based on small-sample data augmentation analysis, belonging to the field of flexoelectric material performance analysis technology. First, the geometry of a flexoelectric cantilever beam is analyzed, and a simplified model with parallel simulation lines of equal scaling is constructed. Then, a finite element subroutine developed based on the flexoelectric hybrid variational principle is used for simulation to determine the physical deformation and electrical signal output under different vibration characteristics, and this drives the simplified model to undergo morphological changes. A morphological-electrical signal mapping model is constructed. Finally, this model is used to predict the performance of the flexoelectric material. This invention solves the problem of stochastic-cognitive hybrid uncertainty caused by data scarcity at the micro-nano scale. By using a deformation-driven mapping mechanism to replace time-consuming simulation, it improves computational efficiency while ensuring the robustness of prediction results, providing a theoretical basis for the design of key components of robust and reliable novel micro-nano electromechanical systems.

Description

Technical Field

[0001] This invention relates to the field of flexural electrical material performance analysis technology, and in particular to a method and system for flexural electrical material performance analysis based on small sample data enhancement analysis. Background Technology

[0002] Flexural electric cantilever beams, as key building blocks of next-generation micro / nano-electromechanical systems (MEMS / NEMS), demonstrate broad application prospects in precision sensing, actuation, and energy harvesting by leveraging the universality of strain gradient-electric field coupling and significant size effects. However, the flexural electric effect at the micro / nano scale is significantly affected by fabrication process fluctuations, and the measurement of its physical properties is extremely difficult. This often results in a "small sample" environment with severely scarce data during the research and development process, leading to a complex mix of stochastic and cognitive uncertainties caused by the inherent randomness of material preparation and cognitive biases due to data scarcity. Traditional deterministic theories and high-fidelity multiphysics finite element simulations are not only computationally intensive and time-consuming, making it difficult to meet the needs of rapid design iteration, but also lack effective means to handle such mixed uncertainties and enhance analysis of limited experimental data. This results in a difficult-to-quantify consistency gap between simulation predictions and actual physical responses, severely restricting the robust design and service performance evaluation of high-performance flexural electric structures under extreme or complex operating environments. Summary of the Invention

[0003] The purpose of this invention is to provide a method and system for analyzing the performance of flexural electrical materials using small sample data augmentation analysis, which can reduce simulation overhead.

[0004] This invention discloses a method for analyzing the properties of flexural materials based on small sample data augmentation analysis, including: Step S100: Analyze the geometric structure of the flexural electric cantilever beam, and based on the analysis results, construct a set of parallel simulation lines to obtain a simplified model of the point cantilever beam. The spacing and length of the simulation lines are scaled proportionally according to the actual structural parameters of the flexural electric cantilever. Step S200: Perform finite element simulation analysis on the flexural electric cantilever beam of the preset material to determine the physical deformation and electrical signal output characteristics of the flexural electric cantilever beam under different vibration characteristics, and use the corresponding physical deformation as the driving characteristic to make the simplified model of the cantilever beam produce the corresponding bending deformation. Step S300: Based on the relationship between the morphological changes and electrical signal output characteristics of the simplified cantilever beam model, perform a unified mapping logic analysis with the same material and size, and construct a morphological-electrical signal mapping model between the morphological changes and electrical signal output characteristics based on the analysis structure. Step S400: Using the morphological electrical signal mapping model, the performance of the flexural electric cantilever to be analyzed is performed.

[0005] In some embodiments disclosed in this invention, the method for performing finite element simulation analysis on a flexural cantilever beam of a preset material includes: Step S201: Based on the flexoelectric hybrid variational principle, develop a three-dimensional hybrid finite element subroutine suitable for characterizing the coupling relationship between strain gradient and electric field intensity, and embed the subroutine into a general finite element analysis software environment to establish a numerical model of a flexoelectric cantilever beam with mechanical-electric coupling calculation capability. Step S202: Simulate the external mechanical load on the cantilever beam in the simulation environment, calculate the non-uniform deformation generated inside the material through the three-dimensional hybrid finite element, and then solve the electric polarization intensity distribution caused by the strain gradient simultaneously according to the flexural electric constitutive relation to achieve deep coupling simulation of mechanical field and electric field. Step S203: The cantilever beam model is divided using a three-fold meshing scheme from coarse to fine. The mesh convergence index is calculated using the Richardson extrapolation method. This index is used to quantitatively evaluate and eliminate the numerical discretization uncertainty in the finite element simulation, ensuring the accuracy of the extracted physical deformation morphology and electrical signal data. Step S204: Set electrical open circuit or short circuit boundary conditions according to the analysis requirements to simulate the forced vibration response of the cantilever beam under different vibration frequencies and amplitudes, thereby obtaining the deflection displacement data at the beam end and the corresponding voltage or charge output characteristics, which serve as the basic data for driving the deformation of the simplified model.

[0006] In some embodiments disclosed in this invention, the method for causing the simplified cantilever beam model to produce corresponding bending deformation includes: Step S205: Overlap the simulated cantilever beam and the simplified cantilever beam model from the finite element simulation analysis, and uniformly set several associated points on each simulation line of the simplified cantilever beam model, and map the associated points to the corresponding nodes on the simulated cantilever beam. Step S206: During the simulation of the deformation of the cantilever beam, each associated point changes accordingly, and as the position of the associated point changes, the shape of the simulation line changes accordingly. The overall shape change of all simulation lines is identified as the shape change of the simplified cantilever model.

[0007] In some embodiments disclosed in this invention, a method for performing unified mapping logic analysis based on the relationship between the morphological changes and electrical signal output characteristics of a simplified cantilever beam model includes: Step S301: Extract the shape changes of the simplified cantilever beam model frame by frame, and construct a simplified shape change sequence according to time. Determine the maximum inclination point of each simulated line in each shape change frame of the simplified shape change sequence, and construct an inclined tangent at the maximum inclination point. Step S302: Map the spatial representation of the inclined tangent in the simplified morphological representation sequence to a preset positioning space, and obtain the spatial dynamic representation of the inclined tangent by representing the change of the inclined tangent in the simplified morphological representation sequence through the preset positioning space, and simultaneously align the electrical signal output curve and the spatial dynamic representation of the inclined tangent in time. Step S303: The dynamic performance of the inclined tangent space and the electrical signal output curve are used as input data for unified mapping logic analysis.

[0008] In some embodiments disclosed in this invention, the method for performing unified mapping logic analysis includes: Step S3031: Classify the dynamic performance of the inclined tangent space into similar sets to obtain a set of similar dynamic performance of the inclined tangent space. Compare the electrical signal output curves corresponding to the set of dynamic performance of the inclined tangent space and analyze the degree of difference between the electrical signal output curves. If the degree of difference is less than or equal to a preset value, the set of similar dynamic performance of the inclined tangent space is identified as a high-precision set of similar dynamic performance of the inclined tangent space. Step S3032: The non-high-precision inclined tangent space dynamic performance similar set is divided several times to obtain several new inclined tangent space dynamic performance similar sets. Each inclined tangent space dynamic performance similar set is analyzed to see if it is a high-precision inclined tangent space dynamic performance similar set. If not, the steps of dividing and judging whether it is a high-precision inclined tangent space dynamic performance similar set are repeated until the number of inclined tangent space dynamic performances in the divided inclined tangent space dynamic performance similar sets is less than or equal to a preset value. The inclined tangent space dynamic performance similar set at this time is identified as a low-precision inclined tangent space dynamic performance similar set. Step S3033: Use all the dynamic spatial representations of the tilted tangent as comparison data for the morphological electrical signal mapping model. If the dynamic spatial representation of the tilted tangent belongs to a low-precision set of similar dynamic spatial representations, it is determined that the accuracy of the output electrical signal output features is insufficient. If the dynamic spatial representation of the tilted tangent belongs to a high-precision set of similar dynamic spatial representations, output the output curves of several electrical signals corresponding to the high-precision set of similar dynamic spatial representations.

[0009] In some embodiments disclosed in this invention, the method for equivalent classification of the spatial dynamic performance of inclined tangents includes: Step S30311: Align the spatial dynamic representation of the tilted tangents and compare the tangent point position node and tilt angle of the tilted tangents in each relative dynamic representation frame. If the distance between the tangent point positions of the tilted tangents is less than or equal to a preset value and the tilt angle is less than or equal to a preset value, then the tilted tangents are considered to match. Then determine the first matching ratio of the matching tilted tangents to all tilted tangents. If the first matching ratio is greater than or equal to a preset value, then the dynamic representation frames are considered to match. Step S30312: If the proportion of matching dynamic performance frames to the second matching proportion of all dynamic performance frames is greater than or equal to a preset value, then the dynamic performance of the tilted tangent space is considered to be matching, and the matching dynamic performance of the tilted tangent space is classified equally.

[0010] In some embodiments disclosed in this invention, the method for equivalent classification of the spatial representation of inclined tangents further includes: Step S30313: Divide and delineate the preset positioning space to obtain several preset positioning subspaces. Set spatial positioning parameters for each preset positioning subspace. Determine the preset positioning subspaces of all the tilted tangent mappings of each dynamic display frame in each tilted tangent spatial display, and record them as the preset positioning subspaces that need to be paid attention to. Step S30314: Calculate the average tilt angle of all tilted tangents to obtain the average tilt angle, determine the preset tilt angle interval to which the average tilt angle belongs, and determine the corresponding tilt parameter based on the preset tilt angle interval to which it belongs. Step S30315: Based on the preset positioning subspace that needs attention for each dynamic performance frame, determine the corresponding spatial positioning parameters, and combine them with the tilt parameters to determine the frame attribute equivalent of the dynamic performance frame. Calculate the average value of all frame attribute equivalents to obtain the performance equivalent of the tilted tangent spatial performance. Step S30316: Based on the difference between the performance equivalents, determine the inclined tangent space performance for comparison. ; Where D is the representational equivalent, and n is the number of frames in the dynamic representation of the tangent space. Let be the tilt parameter of the i-th dynamic display frame. Let N be the spatial positioning parameter of the z-th preset positioning subspace, N be the number of preset positioning subspaces mapped, L be the adjustment coefficient for the influence of the number of mapped spaces, and B be the adjustment constant for the influence of the number of mapped spaces.

[0011] In some embodiments disclosed in this invention, the method for analyzing the degree of difference between electrical signal output curves includes: Step S30121: Perform time axis alignment and baseline drift correction on the multiple electrical signal output curves corresponding to the similar set to eliminate sequence misalignment caused by sampling phase shift and sensor zero drift; Step S30122: Extract the time-domain feature vector and frequency-domain feature vector of each output curve of the aligned electrical signal. The time-domain feature vector includes the peak response amplitude, rise time constant, decay half-life and waveform symmetry coefficient. The frequency-domain feature vector obtains the energy ratio of the main resonance frequency band and the second harmonic distortion rate through short-time Fourier transform. Step S30123: Construct a multi-dimensional feature space based on the time-domain feature vector and the frequency-domain feature vector, and use a weighted fusion algorithm of dynamic time warping distance and Mahalanobis distance to calculate the feature difference degree between any two electrical signal output curves, and identify the statistical extreme value or spatial distribution variance of the feature difference degree as the degree of difference between the electrical signal output curves.

[0012] In some embodiments disclosed in this invention, a method for performing performance analysis on flexural electrical materials requiring analysis using a morphological electrical signal mapping model includes: Step S401: Convert the flexural electric cantilever to be analyzed into a simplified real-time cantilever beam model, compare the simplified real-time cantilever beam model with the simplified cantilever beam model in the morphological electrical signal mapping model, determine the simplified cantilever beam model with the same material and size, and select the high-precision inclined tangent spatial dynamic performance similar set and the low-precision inclined tangent spatial dynamic performance similar set of the simplified cantilever beam model. Step S402: The dynamic performance of the simplified real-time cantilever beam model is transformed to obtain the real-time inclined line spatial dynamic performance. The real-time inclined line spatial dynamic performance and the inclined line spatial dynamic performance in the simplified cantilever beam model are compared equally. The mutually equivalent inclined line spatial dynamic performances are selected, and it is determined whether they belong to the high-precision inclined tangent spatial dynamic performance similar set or the low-precision inclined tangent spatial dynamic performance similar set. If the compared inclined tangent spatial dynamic performance belongs to the low-precision inclined tangent spatial dynamic performance similar set, it is determined that the accuracy of the output electrical signal output feature is insufficient. If the compared inclined tangent spatial dynamic performance belongs to the high-precision inclined tangent spatial dynamic performance similar set, several electrical signal output curves corresponding to the high-precision inclined tangent spatial dynamic performance similar set are output.

[0013] In some embodiments disclosed in this invention, a flexural electrical material performance analysis system based on small sample data augmentation analysis is also disclosed, including: The first module is used to analyze the geometric structure of the flexural electric cantilever beam and, based on the analysis results, construct a set of parallel simulation lines to obtain a simplified model of the point cantilever beam. The spacing and length of the simulation lines are scaled proportionally according to the actual structural parameters of the flexural electric cantilever. The second module is used to perform finite element simulation analysis on a flexural electric cantilever beam of a preset material, determine the physical deformation morphology and electrical signal output characteristics of the flexural electric cantilever beam under different vibration characteristics, and use the corresponding physical deformation morphology as the driving characteristic to make the simplified model of the cantilever beam produce the corresponding bending deformation. The third module is used to perform a unified mapping logic analysis of the same material and size based on the relationship between the shape changes and electrical signal output characteristics of the simplified cantilever beam model, and to construct a shape-electric signal mapping model between the shape changes and electrical signal output characteristics based on the analysis structure. The fourth module is used to perform performance analysis on the flexural electric cantilever that needs to be analyzed using a morphological electrical signal mapping model.

[0014] This invention discloses a method and system for analyzing the performance of flexoelectric materials based on small-sample data augmentation analysis, belonging to the field of flexoelectric material performance analysis technology. First, the geometry of a flexoelectric cantilever beam is analyzed, and a simplified model with parallel simulation lines of equal scaling is constructed. Then, a finite element subroutine developed based on the flexoelectric hybrid variational principle is used for simulation to determine the physical deformation and electrical signal output under different vibration characteristics, and this drives the simplified model to undergo morphological changes. A morphological-electrical signal mapping model is constructed. Finally, this model is used to predict the performance of the flexoelectric material. This invention solves the problem of stochastic-cognitive hybrid uncertainty caused by data scarcity at the micro-nano scale. By using a deformation-driven mapping mechanism to replace time-consuming simulation, it improves computational efficiency while ensuring the robustness of the prediction results, providing a theoretical basis for the design of key components of robust and reliable novel micro-nano electromechanical systems.

[0015] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0016] Figure 1 This diagram illustrates the steps of the method for analyzing the performance of flexural electrical materials based on small sample data augmentation analysis disclosed in this embodiment of the invention. Detailed Implementation

[0017] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0018] The technical solution of the present invention will be clearly and completely described below with reference to the accompanying drawings and specific embodiments. It should be understood that the preferred embodiments described herein are only for illustration and explanation of the present invention and should not be construed as limiting the scope of protection of the present invention. Those skilled in the art can make some non-essential improvements and adjustments based on the following content of the present invention. In the present invention, unless otherwise expressly specified and limited, the technical terms used in the present invention should have the ordinary meaning understood by those skilled in the art.

[0019] Example: This invention discloses a method for analyzing the properties of flexural electrical materials based on small sample data augmentation analysis. (See reference...) Figure 1 ,include: Step S100: Analyze the geometric structure of the flexural electric cantilever beam, and based on the analysis results, construct a set of parallel simulation lines to obtain a simplified model of the point cantilever beam. The spacing and length of the simulation lines are scaled proportionally according to the actual structural parameters of the flexural electric cantilever.

[0020] The core of this step lies in achieving a physical reduction of the order of complex continuum mechanics models through geometric abstraction, addressing the modeling challenges at the microscale caused by insufficient understanding or data scarcity. Since the flexural electrical effect exhibits significant size effects, its polarization intensity is proportional to the strain gradient, and the strain gradient changes most dramatically along the thickness direction of the structure. By analytically constructing a three-dimensional cantilever beam as a set of parallel simulated lines, the continuous material hierarchy is essentially discretized into a geometric skeleton highly sensitive to strain distribution. The spacing and length of the simulated lines are scaled proportionally, accurately capturing the strain gradient characteristics that sharply increase due to the reduction in thickness in micro / nanostructures. For example, when the cantilever beam is subjected to bending, the relative displacement difference between these parallel simulated lines can intuitively reflect the local deformation rate within the material, thus providing a fundamental mathematical basis for subsequent quantification of the stochastic geometric uncertainties introduced by manufacturing process fluctuations.

[0021] Step S200: Perform finite element simulation analysis on the flexural electric cantilever beam of the preset material to determine the physical deformation morphology and electrical signal output characteristics of the flexural electric cantilever beam under different vibration characteristics, and use the corresponding physical deformation morphology as the driving characteristic to make the simplified model of the cantilever beam produce the corresponding bending deformation.

[0022] This step aims to provide realistic physical behavior criteria for the simplified model using high-fidelity numerical methods, solving the complex multi-physics coupling mapping problem between mechanical deformation and electrical signal output in flexoelectric effects. By embedding a three-dimensional hybrid finite element subroutine (UEL) developed based on the flexoelectric hybrid variational principle into the simulation environment, the non-uniform deformation and polarization intensity distribution of the cantilever beam under forced vibration can be solved simultaneously, ensuring that the obtained physical deformation characteristics conform to the constitutive relations of the material. The accurate deformation morphology obtained from the finite element simulation is used as a driving feature and loaded onto the simulation line of the simplified model, causing the simulation line to produce a nonlinear bending behavior highly consistent with the real physical model. This process realizes the energy and information transfer from "high-fidelity numerical solution" to "geometric morphological characteristics," laying a data foundation for replacing time-consuming full-scale simulation calculations in a small sample environment.

[0023] Step S300: Based on the relationship between the morphological changes and electrical signal output characteristics of the simplified cantilever beam model, perform a unified mapping logic analysis with the same material and size, and construct a morphological-electrical signal mapping model between the morphological changes and electrical signal output characteristics based on the analysis structure.

[0024] The principle behind this step is to construct an efficient mathematical mapping model using a limited number of simulation sample points to address the pain points of low computational efficiency and difficulty in data acquisition for flexural electrical structures at the microscale. Correlation analysis is performed on the simplified model's geometric deformation sequence (such as the curvature and tilt tangent changes of the simulated line) and the corresponding charge or voltage signals, aiming to extract the inherent electromechanical coupling response law of the material from small sample data. By performing unified mapping logic analysis within the parameter space, non-essential interferences caused by size and load fluctuations can be shielded, constructing a mathematical mapping model between morphological changes and output signals. For example, this model can identify the corresponding flexural polarization intensity distribution when the simulated line exhibits a specific bending trajectory, thereby solidifying the complex physical evolution logic into a quickly invoked functional relationship and achieving effective propagation of uncertainties between input and output.

[0025] Step S400: Using the morphological electrical signal mapping model, the performance of the flexural electric cantilever to be analyzed is performed.

[0026] This step uses the trained mapping model to comprehensively evaluate the performance of the target structure. Its core lies in using quantitative criteria to assess the accuracy of the numerical model in describing the real physical model, thus achieving robust design. After obtaining the parameters of the flexural electrical material to be analyzed, the model can quickly provide its output response distribution under the combined influence of random uncertainties (such as processing errors) or cognitive uncertainties (such as insufficient understanding).

[0027] In some embodiments disclosed in this invention, the method for performing finite element simulation analysis on a flexural cantilever beam of a preset material includes: Step S201: Based on the flexoelectric hybrid variational principle, a three-dimensional hybrid finite element subroutine suitable for characterizing the coupling relationship between strain gradient and electric field intensity is developed, and the subroutine is embedded in a general finite element analysis software environment to establish a numerical model of a flexoelectric cantilever beam with force-electric coupling calculation capability.

[0028] The principle behind this step lies in using mathematical methods to overcome the limitations of general-purpose simulation software in handling higher-order mechanical effects. Since flexoelectric effects involve the coupling of strain gradients and electric field strengths, this exceeds the computational scope of standard element libraries in traditional finite element software. Its core principle is based on the flexoelectric hybrid variational principle, deriving a mathematical constitutive model capable of simultaneously handling displacement fields and electric fields and their higher-order derivatives. By writing a three-dimensional hybrid finite element UEL subroutine and embedding it into general-purpose simulation platforms such as Abaqus, a customized "physics engine" is essentially established within the standard computational framework. This endows the model with the ability to capture the polarization caused by inversion symmetry breaking in non-centrosymmetric materials, thereby establishing a numerical logical starting point capable of accurately simulating mechanoelectric conversion behavior at the micro-nano scale.

[0029] Step S202: Simulate the external mechanical load on the cantilever beam in the simulation environment, calculate the non-uniform deformation generated inside the material through the three-dimensional hybrid finite element, and then solve the electric polarization intensity distribution caused by the strain gradient simultaneously according to the flexural electric constitutive relation, so as to realize the deep coupling simulation of mechanical field and electric field.

[0030] Step S203: The cantilever beam model is divided using a three-fold meshing scheme from coarse to fine. The mesh convergence index is calculated using the Richardson extrapolation method. This index is used to quantitatively evaluate and eliminate the numerical discretization uncertainty in the finite element simulation, ensuring the accuracy of the extracted physical deformation morphology and electrical signal data.

[0031] Step S204: Set electrical open circuit or short circuit boundary conditions according to the analysis requirements to simulate the forced vibration response of the cantilever beam under different vibration frequencies and amplitudes, thereby obtaining the deflection displacement data at the beam end and the corresponding voltage or charge output characteristics, which serve as the basic data for driving the deformation of the simplified model.

[0032] In some embodiments disclosed in this invention, the method for causing the simplified cantilever beam model to produce corresponding bending deformation includes: Step S205: Overlap the simulated cantilever beam and the simplified cantilever beam model from the finite element simulation analysis, and uniformly set several associated points on each simulation line of the simplified cantilever beam model, and map the associated points to the corresponding nodes on the simulated cantilever beam.

[0033] Step S206: During the simulation of the deformation of the cantilever beam, each associated point changes accordingly, and as the position of the associated point changes, the shape of the simulation line changes accordingly. The overall shape change of all simulation lines is identified as the shape change of the simplified cantilever model.

[0034] In some embodiments disclosed in this invention, a method for performing unified mapping logic analysis based on the relationship between the morphological changes and electrical signal output characteristics of a simplified cantilever beam model includes: Step S301: Extract the shape changes of the simplified cantilever beam model frame by frame, and construct a simplified shape change sequence according to time. Determine the maximum inclination point of each simulated line in each shape change frame of the simplified shape change sequence, and construct an inclined tangent at the maximum inclination point.

[0035] The principle behind this step lies in capturing the core physical quantity driving the flexoelectric effect—the strain gradient—through geometric means. The flexoelectric effect essentially describes the coupling relationship between non-uniform deformation (i.e., strain gradient) within a material and the polarization intensity. Since the curvature change of a cantilever beam during vibration directly determines the magnitude of the strain gradient, refining the deformation sequence frame by frame and determining the maximum inclination point of each simulation line essentially involves finding the key locations where the beam deformation is most severe and the polarization response is most significant. Constructing inclined tangents at these points simplifies the complex continuous curve deformation into quantifiable slope change characteristics, thus providing a refined geometric description for characterizing the significantly enhanced electromechanical coupling behavior within the material due to size effects.

[0036] Step S302: Map the spatial representation of the inclined tangent in the simplified morphological representation sequence to a preset positioning space, and obtain the spatial dynamic representation of the inclined tangent by representing the change of the inclined tangent in the simplified morphological representation sequence through the preset positioning space, and simultaneously align the electrical signal output curve and the spatial dynamic representation of the inclined tangent in time.

[0037] The core principle of this step is to establish a consistent correlation between the evolution of geometric deformation and the electrical output response in both time and space, thereby solving the synchronization problem in multi-source data fusion. By mapping the spatial representation of the inclined tangent to a pre-defined positioning space, the geometric features at different amplitudes and times can be normalized, eliminating interference from absolute displacement and focusing on the dynamic evolution of the tangent slope. Synchronous time-axis alignment ensures that the observed "cause" (the dynamic representation of the geometric shape) and the resulting "effect" (the electrical signal output curve) are logically perfectly matched. This spatiotemporal alignment principle provides a standardized data framework for subsequent quantization to address cognitive uncertainties caused by insufficient understanding or testing errors.

[0038] Step S303: The dynamic performance of the inclined tangent space and the electrical signal output curve are used as input data for unified mapping logic analysis.

[0039] The principle behind this step is to utilize cross-dimensional correlation analysis to reveal the intrinsic mapping law between the evolution of geometric features and the physical polarization response, aiming to establish a rapid performance prediction logic that does not rely on full-scale simulation. Using the spatial dynamics of the inclined tangent and the synchronously aligned electrical signal output as input data essentially involves modeling the causal relationship between "morphology and performance." Through unified mapping logic analysis, the interactive effects of random uncertainties (such as manufacturing process fluctuations) and cognitive uncertainties (such as data scarcity) on output performance can be separated, thereby extracting universal material characteristic laws that transcend size limitations from small sample data. This principle not only improves the computational efficiency of uncertainty propagation analysis but also provides core algorithmic support for the robust design and model validation metrics of flexural electric cantilever beam structures.

[0040] In some embodiments disclosed in this invention, the method for performing unified mapping logic analysis includes: Step S3031: Classify the dynamic performance of the inclined tangent space into similar sets, compare the electrical signal output curves corresponding to the set, and analyze the degree of difference between the electrical signal output curves. If the degree of difference is less than or equal to a preset value, the set of similar performances of the inclined tangent space is identified as a high-precision set of similar performances of the inclined tangent space.

[0041] Because the flexural electrical effect is significantly affected by fabrication errors and experimental noise at the micro- and nano-scale, the same deformation mode may not necessarily produce completely consistent electrical signals. By classifying the "spatial dynamic behavior of inclined tangents" into equivalent categories, the essence is to find typical physical states in structural deformation; if the corresponding set of electrical signal output curves have small differences (less than a preset value), it proves that the geometric feature can stably and reliably induce specific polarization charges. This screening mechanism effectively identifies data segments less affected by "random uncertainty" and defines them as a "high-precision similarity set," providing a high-quality sample benchmark for subsequently constructing a high-confidence performance prediction function.

[0042] Step S3032: The non-high-precision inclined tangent space dynamic performance similar set is divided several times to obtain several new inclined tangent space dynamic performance similar sets. Each inclined tangent space dynamic performance similar set is analyzed to see if it is a high-precision inclined tangent space dynamic performance similar set. If not, the steps of dividing and judging whether it is a high-precision inclined tangent space dynamic performance similar set are repeated until the number of inclined tangent space dynamic performances in the divided inclined tangent space dynamic performance similar sets is less than or equal to a preset value. The inclined tangent space dynamic performance similar set at this time is identified as a low-precision inclined tangent space dynamic performance similar set.

[0043] The principle behind this step is to address "cognitive uncertainty" in the data through recursive segmentation, specifically by deeply mining mapping inaccuracies caused by data scarcity or insufficient understanding of physical mechanisms. When the electrical signals corresponding to certain geometric deformation features exhibit significant dispersion, it indicates the presence of severe random-cognitive mixed interference in that region. By segmenting these "non-high-precision sets" multiple times, the method attempts to find local consistency within a smaller parameter neighborhood, similar to finding a narrower boundary distribution in an uncertainty probability box. If the dataset still fails the accuracy check after being segmented to its minimum size, it is defined as a "low-precision similarity set," which objectively records the "vulnerable area" or "high-risk area" of the model's predictive ability, reflecting a scientific approach to the limited availability of physical sample information at the microscale.

[0044] Step S3033: Use all the dynamic spatial representations of the tilted tangent as comparison data for the morphological electrical signal mapping model. If the dynamic spatial representation of the tilted tangent belongs to a low-precision set of similar dynamic spatial representations, it is determined that the accuracy of the output electrical signal output features is insufficient. If the dynamic spatial representation of the tilted tangent belongs to a high-precision set of similar dynamic spatial representations, output the output curves of several electrical signals corresponding to the high-precision set of similar dynamic spatial representations.

[0045] The principle behind this step is to establish a "performance prediction firewall" based on feature comparison to ensure the robustness of the final output flexural electrical performance indicators. Taking the measured or analyzed geometric dynamic performance as input, the model no longer blindly provides a fixed value, but first determines its "accuracy trust domain" through comparison logic. If a low-accuracy set is matched, the system will proactively warn of "insufficient accuracy" based on model confirmation metrics, avoiding potential failure risks due to cognitive bias; if a high-accuracy set is matched, a verified stable signal curve is output. This output logic based on similar set comparison transforms the complex force-electric coupling law into a quantifiable and verifiable decision-making process, providing core technical support for the development of highly reliable micro-sensors or actuators.

[0046] In some embodiments disclosed in this invention, the method for equivalent classification of the spatial dynamic performance of inclined tangents includes: Step S30311: Align the spatial dynamic representation of the tilted tangents and compare the tangent point position node and tilt angle of the tilted tangents in each relative dynamic representation frame. If the distance between the tangent point positions of the tilted tangents is less than or equal to a preset value and the tilt angle is less than or equal to a preset value, then the tilted tangents are considered to match. Then determine the first matching ratio of the matching tilted tangents to all tilted tangents. If the first matching ratio is greater than or equal to a preset value, then the dynamic representation frames are considered to match.

[0047] Step S30312: If the proportion of matching dynamic performance frames to the second matching proportion of all dynamic performance frames is greater than or equal to a preset value, then the dynamic performance of the tilted tangent space is considered to be matching, and the matching dynamic performance of the tilted tangent space is classified equally.

[0048] In some embodiments disclosed in this invention, the method for equivalent classification of the spatial representation of inclined tangents further includes: Step S30313: Divide and delineate the preset positioning space to obtain several preset positioning subspaces. Set spatial positioning parameters for each preset positioning subspace. Determine the preset positioning subspaces of all the tilted tangent mappings of each dynamic display frame in each tilted tangent spatial display, and record them as the preset positioning subspaces that need to be paid attention to. Step S30314: Calculate the average tilt angle of all tilted tangents to obtain the average tilt angle, determine the preset tilt angle interval to which the average tilt angle belongs, and determine the corresponding tilt parameter based on the preset tilt angle interval to which it belongs. Step S30315: Based on the preset positioning subspace that needs attention for each dynamic performance frame, determine the corresponding spatial positioning parameters, and combine them with the tilt parameters to determine the frame attribute equivalent of the dynamic performance frame. Calculate the average value of all frame attribute equivalents to obtain the performance equivalent of the tilted tangent spatial performance. Step S30316: Based on the difference between the performance equivalents, determine the inclined tangent space performance for comparison. ; Where D is the representational equivalent, and n is the number of frames in the dynamic representation of the tangent space. Let be the tilt parameter of the i-th dynamic display frame. Let N be the spatial positioning parameter of the z-th preset positioning subspace, N be the number of preset positioning subspaces mapped, L be the adjustment coefficient for the influence of the number of mapped spaces, and B be the adjustment constant for the influence of the number of mapped spaces.

[0049] L: The amount of mapping space affects the adjustment coefficient; used to adjust the space complexity (N and ...). The weight of the impact on the final equivalent value.

[0050] B: The amount of mapping space affects the adjustment constant; used to correct or shift the output range of the exponential function.

[0051] In some embodiments disclosed in this invention, the method for analyzing the degree of difference between electrical signal output curves includes: Step S30121: Time axis alignment and baseline drift correction are performed on the multiple electrical signal output curves corresponding to the similar set to eliminate sequence misalignment caused by sampling phase shift and sensor zero drift.

[0052] Step S30122: Extract the time-domain feature vector and frequency-domain feature vector of each output curve of the aligned electrical signal. The time-domain feature vector includes the peak response amplitude, rise time constant, decay half-life and waveform symmetry coefficient. The frequency-domain feature vector obtains the energy ratio of the main resonance frequency band and the second harmonic distortion rate through short-time Fourier transform.

[0053] Step S30123: Construct a multi-dimensional feature space based on the time-domain feature vector and the frequency-domain feature vector, and use a weighted fusion algorithm of dynamic time warping distance and Mahalanobis distance to calculate the feature difference degree between any two electrical signal output curves, and identify the statistical extreme value or spatial distribution variance of the feature difference degree as the degree of difference between the electrical signal output curves.

[0054] In some embodiments disclosed in this invention, a method for performing performance analysis on flexural electrical materials requiring analysis using a morphological electrical signal mapping model includes: Step S401: Convert the flexural electric cantilever to be analyzed into a simplified real-time cantilever beam model, compare the simplified real-time cantilever beam model with the simplified cantilever beam model in the morphological electrical signal mapping model, determine the simplified cantilever beam model with the same material and size, and select the high-precision inclined tangent spatial dynamic performance similar set and the low-precision inclined tangent spatial dynamic performance similar set of the simplified cantilever beam model.

[0055] Step S402: The dynamic performance of the simplified real-time cantilever beam model is transformed to obtain the real-time inclined line spatial dynamic performance. The real-time inclined line spatial dynamic performance and the inclined line spatial dynamic performance in the simplified cantilever beam model are compared equally. The mutually equivalent inclined line spatial dynamic performances are selected, and it is determined whether they belong to the high-precision inclined tangent spatial dynamic performance similar set or the low-precision inclined tangent spatial dynamic performance similar set. If the compared inclined tangent spatial dynamic performance belongs to the low-precision inclined tangent spatial dynamic performance similar set, it is determined that the accuracy of the output electrical signal output feature is insufficient. If the compared inclined tangent spatial dynamic performance belongs to the high-precision inclined tangent spatial dynamic performance similar set, several electrical signal output curves corresponding to the high-precision inclined tangent spatial dynamic performance similar set are output.

[0056] In some embodiments disclosed in this invention, a flexural electrical material performance analysis system based on small sample data augmentation analysis is also disclosed, including: The first module is used to analyze the geometric structure of the flexural electric cantilever beam and, based on the analysis results, construct a set of parallel simulation lines to obtain a simplified model of the point cantilever beam. The spacing and length of the simulation lines are scaled proportionally according to the actual structural parameters of the flexural electric cantilever. The second module is used to perform finite element simulation analysis on a flexural electric cantilever beam of a preset material, determine the physical deformation morphology and electrical signal output characteristics of the flexural electric cantilever beam under different vibration characteristics, and use the corresponding physical deformation morphology as the driving characteristic to make the simplified model of the cantilever beam produce the corresponding bending deformation. The third module is used to perform a unified mapping logic analysis of the same material and size based on the relationship between the shape changes and electrical signal output characteristics of the simplified cantilever beam model, and to construct a shape-electric signal mapping model between the shape changes and electrical signal output characteristics based on the analysis structure. The fourth module is used to perform performance analysis on the flexural electric cantilever that needs to be analyzed using a morphological electrical signal mapping model.

[0057] This invention discloses a method and system for analyzing the performance of flexoelectric materials based on small-sample data augmentation analysis, belonging to the field of flexoelectric material performance analysis technology. First, the geometry of a flexoelectric cantilever beam is analyzed, and a simplified model with parallel simulation lines of equal scaling is constructed. Then, a finite element subroutine developed based on the flexoelectric hybrid variational principle is used for simulation to determine the physical deformation and electrical signal output under different vibration characteristics, and this drives the simplified model to undergo morphological changes. A morphological-electrical signal mapping model is constructed. Finally, this model is used to predict the performance of the flexoelectric material. This invention solves the problem of stochastic-cognitive hybrid uncertainty caused by data scarcity at the micro-nano scale. By using a deformation-driven mapping mechanism to replace time-consuming simulation, it improves computational efficiency while ensuring the robustness of the prediction results, providing a theoretical basis for the design of key components of robust and reliable novel micro-nano electromechanical systems.

[0058] Through the above description of the embodiments, those skilled in the art can clearly understand that the present invention can be implemented in hardware or by means of software plus necessary general-purpose hardware platforms. Based on this understanding, the technical solution of the present invention can be embodied in the form of a software product, which can be stored in a non-volatile storage medium (such as a CD-ROM, USB flash drive, external hard drive, etc.) and includes several instructions to cause a computer device (such as a personal computer, server, or network device, etc.) to execute the methods described in the various embodiments of the present invention.

[0059] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A method for analyzing the properties of flexural electrical materials based on small sample data augmentation analysis, characterized in that, include: Step S100: Analyze the geometric structure of the flexural electric cantilever beam, and based on the analysis results, construct a set of parallel simulation lines to obtain a simplified model of the point cantilever beam. The spacing and length of the simulation lines are scaled proportionally according to the actual structural parameters of the flexural electric cantilever. Step S200: Perform finite element simulation analysis on the flexural electric cantilever beam of the preset material to determine the physical deformation and electrical signal output characteristics of the flexural electric cantilever beam under different vibration characteristics, and use the corresponding physical deformation as the driving characteristic to make the simplified model of the cantilever beam produce the corresponding bending deformation. Step S300: Based on the relationship between the morphological changes and electrical signal output characteristics of the simplified cantilever beam model, perform a unified mapping logic analysis with the same material and size, and construct a morphological-electrical signal mapping model between the morphological changes and electrical signal output characteristics based on the analysis structure. Step S400: Using the morphological electrical signal mapping model, the performance of the flexural electric cantilever to be analyzed is performed.

2. The method for analyzing the performance of flexural electrical materials based on small sample data augmentation analysis according to claim 1, characterized in that, Methods for finite element simulation analysis of flexural cantilever beams of pre-defined materials include: Step S201: Based on the flexoelectric hybrid variational principle, develop a three-dimensional hybrid finite element subroutine suitable for characterizing the coupling relationship between strain gradient and electric field intensity, and embed the subroutine into a general finite element analysis software environment to establish a numerical model of a flexoelectric cantilever beam with mechanical-electric coupling calculation capability. Step S202: Simulate the external mechanical load on the cantilever beam in the simulation environment, calculate the non-uniform deformation generated inside the material through the three-dimensional hybrid finite element, and then solve the electric polarization intensity distribution caused by the strain gradient simultaneously according to the flexural electric constitutive relation to achieve deep coupling simulation of mechanical field and electric field. Step S203: The cantilever beam model is divided using a three-fold meshing scheme from coarse to fine. The mesh convergence index is calculated using the Richardson extrapolation method. This index is used to quantitatively evaluate and eliminate the numerical discretization uncertainty in the finite element simulation, ensuring the accuracy of the extracted physical deformation morphology and electrical signal data. Step S204: Set electrical open circuit or short circuit boundary conditions according to the analysis requirements to simulate the forced vibration response of the cantilever beam under different vibration frequencies and amplitudes, thereby obtaining the deflection displacement data at the beam end and the corresponding voltage or charge output characteristics, which serve as the basic data for driving the deformation of the simplified model.

3. The method for analyzing the performance of flexural electrical materials based on small sample data augmentation analysis according to claim 1, characterized in that, Methods for generating corresponding bending deformations in a simplified model of a cantilever beam include: Step S205: Overlap the simulated cantilever beam and the simplified cantilever beam model from the finite element simulation analysis, and uniformly set several associated points on each simulation line of the simplified cantilever beam model, and map the associated points to the corresponding nodes on the simulated cantilever beam. Step S206: During the simulation of the deformation of the cantilever beam, each associated point changes accordingly, and as the position of the associated point changes, the shape of the simulation line changes accordingly. The overall shape change of all simulation lines is identified as the shape change of the simplified cantilever model.

4. The method for analyzing the performance of flexural electrical materials based on small sample data augmentation analysis according to claim 1, characterized in that, Based on the relationship between the morphological changes and electrical signal output characteristics of a simplified cantilever beam model, methods for unified mapping logic analysis of beams of the same material and size include: Step S301: Extract the shape changes of the simplified cantilever beam model frame by frame, and construct a simplified shape change sequence according to time. Determine the maximum inclination point of each simulated line in each shape change frame of the simplified shape change sequence, and construct an inclined tangent at the maximum inclination point. Step S302: Map the spatial representation of the inclined tangent in the simplified morphological representation sequence to a preset positioning space, and obtain the spatial dynamic representation of the inclined tangent by representing the change of the inclined tangent in the simplified morphological representation sequence through the preset positioning space, and simultaneously align the electrical signal output curve and the spatial dynamic representation of the inclined tangent in time. Step S303: The dynamic performance of the inclined tangent space and the electrical signal output curve are used as input data for unified mapping logic analysis.

5. The method for analyzing the performance of flexural electrical materials based on small sample data augmentation analysis according to claim 4, characterized in that, Methods for performing unified mapping logic analysis include: Step S3031: Classify the dynamic performance of the inclined tangent space into similar sets to obtain a set of similar dynamic performance of the inclined tangent space. Compare the electrical signal output curves corresponding to the set of dynamic performance of the inclined tangent space and analyze the degree of difference between the electrical signal output curves. If the degree of difference is less than or equal to a preset value, the set of similar dynamic performance of the inclined tangent space is identified as a high-precision set of similar dynamic performance of the inclined tangent space. Step S3032: The non-high-precision inclined tangent space dynamic performance similar set is divided several times to obtain several new inclined tangent space dynamic performance similar sets. Each inclined tangent space dynamic performance similar set is analyzed to see if it is a high-precision inclined tangent space dynamic performance similar set. If not, the steps of dividing and judging whether it is a high-precision inclined tangent space dynamic performance similar set are repeated until the number of inclined tangent space dynamic performances in the divided inclined tangent space dynamic performance similar sets is less than or equal to a preset value. The inclined tangent space dynamic performance similar set at this time is identified as a low-precision inclined tangent space dynamic performance similar set. Step S3033: Use all the dynamic spatial representations of the tilted tangent as comparison data for the morphological electrical signal mapping model. If the dynamic spatial representation of the tilted tangent belongs to a low-precision set of similar dynamic spatial representations, it is determined that the accuracy of the output electrical signal output features is insufficient. If the dynamic spatial representation of the tilted tangent belongs to a high-precision set of similar dynamic spatial representations, output the output curves of several electrical signals corresponding to the high-precision set of similar dynamic spatial representations.

6. The method for analyzing the performance of flexural electrical materials based on small sample data augmentation analysis according to claim 5, characterized in that, Methods for classifying the spatial dynamics of inclined tangents into equivalent categories include: Step S30311: Align the spatial dynamic representation of the tilted tangents and compare the tangent point position node and tilt angle of the tilted tangents in each relative dynamic representation frame. If the distance between the tangent point positions of the tilted tangents is less than or equal to a preset value and the tilt angle is less than or equal to a preset value, then the tilted tangents are considered to match. Then determine the first matching ratio of the matching tilted tangents to all tilted tangents. If the first matching ratio is greater than or equal to a preset value, then the dynamic representation frames are considered to match. Step S30312: If the proportion of matching dynamic performance frames to the second matching proportion of all dynamic performance frames is greater than or equal to a preset value, then the dynamic performance of the tilted tangent space is considered to be matching, and the matching dynamic performance of the tilted tangent space is classified equally.

7. The method for analyzing the performance of flexural electrical materials based on small sample data augmentation analysis according to claim 6, characterized in that, Methods for classifying the spatial representation of oblique tangents into equivalent categories also include: Step S30313: Divide and delineate the preset positioning space to obtain several preset positioning subspaces. Set spatial positioning parameters for each preset positioning subspace. Determine the preset positioning subspaces of all the tilted tangent mappings of each dynamic display frame in each tilted tangent spatial display, and record them as the preset positioning subspaces that need to be paid attention to. Step S30314: Calculate the average tilt angle of all tilted tangents to obtain the average tilt angle, determine the preset tilt angle interval to which the average tilt angle belongs, and determine the corresponding tilt parameter based on the preset tilt angle interval to which it belongs. Step S30315: Based on the preset positioning subspace that needs attention for each dynamic performance frame, determine the corresponding spatial positioning parameters, and combine them with the tilt parameters to determine the frame attribute equivalent of the dynamic performance frame. Calculate the average value of all frame attribute equivalents to obtain the performance equivalent of the tilted tangent spatial performance. Step S30316: Based on the difference between the performance equivalents, determine the inclined tangent space performance for comparison. ； Where D is the representational equivalent, and n is the number of frames in the dynamic representation of the tangent space. Let be the tilt parameter of the i-th dynamic display frame. Let N be the spatial positioning parameter of the z-th preset positioning subspace, N be the number of preset positioning subspaces mapped, L be the adjustment coefficient for the influence of the number of mapped spaces, and B be the adjustment constant for the influence of the number of mapped spaces.

8. The method for analyzing the performance of flexural electrical materials based on small sample data augmentation analysis according to claim 5, characterized in that, Methods for analyzing the degree of difference between electrical signal output curves include: Step S30121: Perform time axis alignment and baseline drift correction on the multiple electrical signal output curves corresponding to the similar set to eliminate sequence misalignment caused by sampling phase shift and sensor zero drift; Step S30122: Extract the time-domain feature vector and frequency-domain feature vector of each output curve of the aligned electrical signal. The time-domain feature vector includes the peak response amplitude, rise time constant, decay half-life and waveform symmetry coefficient. The frequency-domain feature vector obtains the energy ratio of the main resonance frequency band and the second harmonic distortion rate through short-time Fourier transform. Step S30123: Construct a multi-dimensional feature space based on the time-domain feature vector and the frequency-domain feature vector, and use a weighted fusion algorithm of dynamic time warping distance and Mahalanobis distance to calculate the feature difference degree between any two electrical signal output curves, and identify the statistical extreme value or spatial distribution variance of the feature difference degree as the degree of difference between the electrical signal output curves.

9. The method for analyzing the performance of flexural electrical materials based on small sample data augmentation analysis according to claim 5, characterized in that, Methods for analyzing the performance of flexural materials using morphological electrical signal mapping models include: Step S401: Convert the flexural electric cantilever to be analyzed into a simplified real-time cantilever beam model, compare the simplified real-time cantilever beam model with the simplified cantilever beam model in the morphological electrical signal mapping model, determine the simplified cantilever beam model with the same material and size, and select the high-precision inclined tangent spatial dynamic performance similar set and the low-precision inclined tangent spatial dynamic performance similar set of the simplified cantilever beam model. Step S402: The dynamic performance of the simplified real-time cantilever beam model is transformed to obtain the real-time inclined line spatial dynamic performance. The real-time inclined line spatial dynamic performance and the inclined line spatial dynamic performance in the simplified cantilever beam model are compared equally. The mutually equivalent inclined line spatial dynamic performances are selected, and it is determined whether they belong to the high-precision inclined tangent spatial dynamic performance similar set or the low-precision inclined tangent spatial dynamic performance similar set. If the compared inclined tangent spatial dynamic performance belongs to the low-precision inclined tangent spatial dynamic performance similar set, it is determined that the accuracy of the output electrical signal output feature is insufficient. If the compared inclined tangent spatial dynamic performance belongs to the high-precision inclined tangent spatial dynamic performance similar set, several electrical signal output curves corresponding to the high-precision inclined tangent spatial dynamic performance similar set are output.

10. A flexural electrical material performance analysis system based on small sample data augmentation analysis, characterized in that, The method for performing the flexural electrical material performance analysis according to any one of claims 1-9 includes: The first module is used to analyze the geometric structure of the flexural electric cantilever beam and, based on the analysis results, construct a set of parallel simulation lines to obtain a simplified model of the point cantilever beam. The spacing and length of the simulation lines are scaled proportionally according to the actual structural parameters of the flexural electric cantilever. The second module is used to perform finite element simulation analysis on a flexural electric cantilever beam of a preset material, determine the physical deformation morphology and electrical signal output characteristics of the flexural electric cantilever beam under different vibration characteristics, and use the corresponding physical deformation morphology as the driving characteristic to make the simplified model of the cantilever beam produce the corresponding bending deformation. The third module is used to perform a unified mapping logic analysis of the same material and size based on the relationship between the shape changes and electrical signal output characteristics of the simplified cantilever beam model, and to construct a shape-electric signal mapping model between the shape changes and electrical signal output characteristics based on the analysis structure. The fourth module is used to perform performance analysis on the flexural electric cantilever that needs to be analyzed using a morphological electrical signal mapping model.

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