A Deep Learning Approach to Shape Optimization Problems for Flexoelectric Materials Using the Isogeometric Finite Element Method

A new approach for flexoelectricmaterial shape optimization is proposed in this study.In this work,a proxymodel based on artificial neural network(ANN)is used to solve the parameter optimization and shape optimization problems.To improve the fitting ability of the neural network,we use the idea of pre-training to determine the structure of the neural network and combine different optimizers for training.The isogeometric analysis-finite element method(IGA-FEM)is used to discretize the flexural theoretical formulas and obtain samples,which helps ANN to build a proxy model from the model shape to the target value.The effectiveness of the proposed method is verified through two numerical examples of parameter optimization and one numerical example of shape optimization.

Flexoelectricity in hydroxyapatite for the enhanced piezocatalytic degradation of phenanthrene in soil

Coupling the effects of flexoelectricity with piezoelectricity has been proved to effectively harvest mechanical energy.In this study,a composition-graded core–shell structure(HAP@FAP)was prepared by surface-gradient F-doping in hydroxyapatite,which could introduce flexoelectricity by a built-in strain gradient.A flexoelectric-boosted piezoelectric response was demonstrated by piezoresponse force microscopy(PFM)characterization,showing that the piezoelectric constant of HAP@FAP was increased by 2.25 times via a lattice strain gradient induced by chemical heterogeneities derived from the unique composition-graded core-shell structure.Thus,the piezocatalytic activity of HAP@FAP for phenanthrene(PHE)degradation in soil was enhanced.This work provides a new strategy for the modification of piezoelectric catalysts for the remediation of organics-contaminated soils on industrial land.

Computational Machine Learning Representation for the Flexoelectricity Effect in Truncated Pyramid Structures

In this study,machine learning representation is introduced to evaluate the flexoelectricity effect in truncated pyramid nanostructure under compression.A Non-Uniform Rational B-spline(NURBS)based IGA formulation is employed to model the flexoelectricity.We investigate 2D system with an isotropic linear elastic material under plane strain conditions discretized by 45×30 grid of B-spline elements.Six input parameters are selected to construct a deep neural network(DNN)model.They are the Young's modulus,two dielectric permittivity constants,the longitudinal and transversal flexoelectric coefficients and the order of the shape function.The outputs of interest are the strain in the stress direction and the electric potential due flexoelectricity.The dataset are generated from the forward analysis of the flexoelectric model.80%of the dataset is used for training purpose while the remaining is used for validation by checking the mean squared error.In addition to the input and output layers,the developed DNN model is composed of four hidden layers.The results showed high predictions capabilities of the proposed method with much lower computational time in comparison to the numerical model.

Vibration analysis of piezoelectric sandwich nanobeam with flexoelectricity based on nonlocal strain gradient theory

A nonlocal strain gradient theory(NSGT) accounts for not only the nongradient nonlocal elastic stress but also the nonlocality of higher-order strain gradients,which makes it benefit from both hardening and softening effects in small-scale structures.In this study, based on the NSGT, an analytical model for the vibration behavior of a piezoelectric sandwich nanobeam is developed with consideration of flexoelectricity. The sandwich nanobeam consists of two piezoelectric sheets and a non-piezoelectric core. The governing equation of vibration of the sandwich beam is obtained by the Hamiltonian principle. The natural vibration frequency of the nanobeam is calculated for the simply supported(SS) boundary, the clamped-clamped(CC) boundary, the clamped-free(CF)boundary, and the clamped-simply supported(CS) boundary. The effects of geometric dimensions, length scale parameters, nonlocal parameters, piezoelectric constants, as well as the flexoelectric constants are discussed. The results demonstrate that both the flexoelectric and piezoelectric constants enhance the vibration frequency of the nanobeam.The nonlocal stress decreases the natural vibration frequency, while the strain gradient increases the natural vibration frequency. The natural vibration frequency based on the NSGT can be increased or decreased, depending on the value of the nonlocal parameter to length scale parameter ratio.

Three-Dimensional Isogeometric Analysis of Flexoelectricity with MATLAB Implementation

Flexoelectricity is a general electromechanical phenomenon where the electric polarization exhibits a linear dependency to the gradient of mechanical strain and vice versa.The truncated pyramid compression test is among the most common setups to estimate the flexoelectric effect.We present a three-dimensional isogeometric formulation of flexoelectricity with its MATLAB implementation for a truncated pyramid setup.Besides educational purposes,this paper presents a precise computational model to illustrate how the localization of strain gradients around pyramidal boundary shapes contributes in generation of electrical energy.The MATLAB code is supposed to help learners in the Isogeometric Analysis and Finite Elements Methods community to learn how to solve a fully coupled problem,which requires higher order approximations,numerically.The complete MATLAB code which is available as source code distributed under a BSD-style license,is provided in the part of Supplementary Materials of the paper.

Determining the sum of flexoelectric coefficients in nematic liquid crystals by the capacitance method

A detailed theoretical analysis of determining the sum of flexoelectric coefficients in nematic liquid crystals using the capacitance method is given. In the strong anchoring parallel aligned nematic （PAN） and hybrid aligned nematic （HAN） cells, the dependences of the capacitance on the sum of flexoelectric coefficients and the applied voltage are obtained by numerical simulations, and the distributions of the director and the electric potential for different applied voltages and flexoelectric coefficients are also given. Based on this theoretical analysis, we propose an experimental design for measuring the capacitance of a liquid crystal cell using the improved precision LCR meter E4980A （Agilent）. Through comparing the experimental data with the simulated results, the sum of flexoeletric coefficients can be determined.

Flexoelectric-Induced Voltage Shift in Hybrid Aligned Nematic Liquid Crystal Cell

Flexoelectric-induced voltage shift in a weak anchoring hybrid aligned nematic fiquid crystai cell is investigated theoretically. Based on the elastic theory of liquid crystal and the variation method, the equations for the bulk and the boundary of the cell are derived. By computer simulation, the dependence of the shift voltage on the sum of the ttexoelectric coefficients and the anchoring energy strength is obtained. As a result, a novel method to determine the sum of the flexoelectric coefficients by measuring the shift voltage is put forward.

Size-dependent electromechanical properties in piezoelectric superlattices due to flexoelectric effect

Piezoelectric superlattice is a potential component for nanoelectromechanical systems. Due to the strong nonlocal effect such as flexoelectric effect at interfaces, classical piezoelectric theory is unable to accurately describe the electromechanical response of piezoelectric superlattice at nanoscale scale. Based on the previous nonlocal thermodynamics theory with flexoelectric effect Liu et al. （2016）, the size- dependent electromechanical properties of piezoelectric superlattices made of BaTiO3 （BTO） and PbTiO3 （PRO） layers are investigated systematically in the present work. Giant strain gradient is found near the interface between BTO and PTO layers, which leads to the significant enhancement of polarization in the superlattice due to the flexoelectric effect. For the piezoelectric BTO-PTO superlattices with different unit- cell sizes, the thickness of interface with nontrivial strain gradient is almost constant. The influence of strain gradient at the interface becomes significant when the size of superlattice decreases, As a result, a strong size dependence of electromechanical properties is predicted for the piezoelectric BTO-PTO superlattices, In particular, for the superlattices with a specific thickness ratio of BTO and PTO layers, the piezoelectric response can be several times larger than that of bulk structure. The present work demonstrates a practical wast to design the piezoelectric superlattices with high piezoelectric coefficient by using the nonlocal effect at nanoscale.

Influences of surface and flexoelectric polarization on the effective anchoring energy in nematic liquid crystal

The physical effects on surface and flexoelectric polarization in a weak anchoring nematic liquid crystal cell are investigated systematically. We derive the analytic expressions of two effective anchoring energies for lower and upper substrates respectively as well as their effective anchoring strengths and corresponding tilt angles of effective easy direction.All of these quantities are relevant to the magnitudes of both two polarizations and the applied voltage U. Based on these expressions, the variations of effective anchoring strength and the tilt angle with the applied voltage are calculated for the fixed values of two polarizations. For an original weak anchoring hybrid aligned nematic cell, it may be equivalent to a planar cell for a small value of U and has a threshold voltage. The variation of reduced threshold voltage with reduced surface polarization strength is also calculated. The role of surface polarization is important without the adsorptive ions considered.

Nonlinear thickness-shear vibration of an infinite piezoelectric plate with flexoelectricity based on the method of multiple scales

This paper presents a nonlinear thickness-shear vibration model for onedimensional infinite piezoelectric plate with flexoelectricity and geometric nonlinearity.The constitutive equations with flexoelectricity and governing equations are derived from the Gibbs energy density function and variational principle.The displacement adopted here is assumed to be antisymmetric through the thickness due to the thickness-shear vibration mode.Only the shear strain gradient through the thickness is considered in the present model.With geometric nonlinearity,the governing equations are converted into differential equations as the function of time by the Galerkin method.The method of multiple scales is employed to obtain the solution to the nonlinear governing equation with first order approximation.Numerical results show that the nonlinear thickness-shear vibration of piezoelectric plate is size dependent,and the flexoelectric effect has significant influence on the nonlinear thickness-shear vibration frequencies of micro-size thin plates.The geometric nonlinearity also affects the thickness-shear vibration frequencies greatly.The results show that flexoelectricity and geometric nonlinearity cannot be ignored in design of accurate high-frequency piezoelectric devices.

Theoretical analysis of the influence of flexoelectric effect on the defect site in nematic inversion walls

Based on the experimental phenomena of flexoelectric response at defect sites in nematic inversion walls conducted by Kumar et al., we gave the theoretical analysis using the Frank elastic theory. When a direct-current electric field normal to the plane of the substrate is applied to the parallel aligned nematic liquid crystal cell with weak anchoring, the rotation of 4-1 defects in the narrow inversion walls can be exhibited. The free energy of liquid crystal molecules around the ＋1 and - 1 defect sites in the nematic inversion walls under the electric field was formulated and the electric-field-driven structural changes at the defect site characterized by polar and azimuthal angles of the local director were simulated. The results reveal that the deviation of azimuthal angle induced by flexoelectric effect are consistent with the switching of extinction brushes at the - 1 and -1 defects obtained in the experiment conducted by Kumar et al.

Prediction of flexoelectricity in BaTiO_(3) using molecular dynamics simulations

Flexoelectric effect, referring to the strain gradient induced polarization, widely exists in dielectric materials, but its molecular dynamics has not been studied so much so far. In this work, the radial distribution function of BaTiO_(3) and the phase transition temperatures have been investigated, and the results show that the core-shell potential model is effective and the structure of BaTiO_(3) is stable in a temperature range of 10 K–150 K. Molecular dynamics simulated hysteresis loops of BaTiO_(3) show that anisotropy can play an important role in the coercive field. Based on the rational simulation process,the effects of cantilever beam bent angle and fixed length on the polarization are analyzed. It is found that the small bent angle of the curved cantilever beam can give a proportional relationship with a fixed end length and a non-linear relationship is presented when the bent angle is much larger. The prediction of flexoelectric coefficient in BaTiO_(3) is 18.5 nC/m. This work provides a computational framework for the study of flexoelectric effect by using molecular dynamics.

Static Pull-In Analysis of a Composite Laminated Nano-beam with Flexoelectric Effect

Based on the new modified couple stress theory and considering the flexoelectric effect of the piezoelectric layers,the Euler Bernoulli nano-beam model of composite laminated materials driven by electrostatically fixed supports at both ends is established. The nonlinear differential governing equations and boundary conditions are derived by the Hamilton principle. The generalized differential quadrature method(GDQM) and the Newton Raphson method are used to numerically solve the differential governing equations. The influence of flexoelectric effect on the static and the dynamic pull-in characteristics of laminated nano-beams is analyzed. The results of the numerical calculation are in a good agreement with those in the literature when the model degenerated into a nanobeam model without flexoelectric effect. The stacking sequence,length scale parameter l and piezoelectric layer applied voltage V_(p) of the composite will affect the pull-in voltage,frequency and time-domain response of the structure. Given that the flexoelectric effect will reduce the pull-in voltage and dimensionless natural frequency of the structure,the maximum dimensionless displacement at the midpoint of the beam and the period of time-domain response should be increased.

Strain Gradient Finite Element Formulation of Flexoelectricity in Ferroelectric Material Based on Phase-Field Method

Flexoelectricity is a two-way coupling effect between the strain gradient and electric field that exists in all dielectrics,regardless of point group symmetry.However,the high-order derivatives of displacements involved in the strain gradient pose challenges in solving electromechanical coupling problems incorporating the flexoelectric effect.In this study,we formulate a phase-field model for ferroelectric materials considering the flexoelectric effect.A four-node quadrilateral element with 20 degrees of freedom is constructed without introducing high-order shape functions.The microstructure evolution of domains is described by an independent order parameter,namely the spontaneous polarization governed by the time-dependent Ginzburg–Landau theory.The model is developed based on a thermodynamic framework,in which a set of microforces is introduced to construct the constitutive relation and evolution equation.For the flexoelectric part of electric enthalpy,the strain gradient is determined by interpolating the mechanical strain at the node via the values of Gaussian integration points in the isoparametric space.The model is shown to be capable of reproducing the classic analytical solution of dielectric materials incorporating the flexoelectric contribution.The model is verified by duplicating some typical phenomena in flexoelectricity in cylindrical tubes and truncated pyramids.A comparison is made between the polarization distribution in dielectrics and ferroelectrics.The model can reproduce the solution to the boundary value problem of the cylindrical flexoelectric tube,and demonstrate domain twisting at domain walls in ferroelectrics considering the flexoelectric effect.

Domain Engineering the Flexoelectric Response of BaTiO_(3)–SrTiO_(3) Binary Films:A Phase Field Study

The flexoelectric effect describes the interplay between material polarization and strain gradient.While much research has focused on control strategies for flexoelectricity,the impact of ferroelectric domain structure on the flexoelectric effect remains elusive.Through phase field simulations,we conduct a comparative study on the flexoelectric response of two types of BaTiO_(3)–SrTiO_(3) binary films:BaTiO_(3)–SrTiO_(3) multilayers and(Ba,Sr)TiO_(3) solid solutions.Our findings reveal that the flexoelectric response of these two systems,at the same Ba/Sr ratio,differs not only in magnitude but also in other aspects like temperature dependence,linearity,and hysteresis,due to their different ferroelectric domain structures and evolution dynamics.Our study deepens our current understanding of flexoelectricity in ferroelectrics and suggests“domain engineering”as a potential way to tailor this effect.

Enhanced Flexoelectricity in Barium Titanate-Cellulose Composite Thin Films

Biopolymers,the potential flexoelectric materials,are environment-friendly,degradable,lightweight,cost-effective,and possess remarkable processing properties catering to the requirements of advanced devices.However,the flexoelectric coefficient of biopolymers is normally much weaker than that of ceramic materials,limiting their potential applications for designing high-performance green electromechanical coupling devices.To improve the flexoelectric response in biopolymers,we composited barium titanate(BTO)with 2,2,6,6-tetramethylpiperidine-1-oxyl-oxidized cellulose nanofibrils(TOCNF)to enhance the flexoelectric response of TOCNF.Owing to the high permittivity and flexoelectric effect of BTO,the relative dielectric constant and flexoelectric coefficient of 33.3 wt%BTO-TOCNF films reached 30.94@1 kHz and 50.05±1.88 nC/m@1 Hz,which were almost 172 times and 27 times higher than those of TOCNF,respectively.The composite thin film contains high dielectric constant and flexoelectric coefficient,as well as excellent flexibility.Our study provided a straightforward and efficient method for improving the flexoelectric effect of biopolymers,and demonstrated its great potential applications in flexoelectric-based devices.

Bending Analysis of Circular Piezoelectric Semiconductor Plates Incorporating Flexoelectricity

Based on the three-dimensional(3D)basic equations of piezoelectric semiconductors(PSs),we establish a two-dimensional(2D)deformation-polarization-carrier coupling bending model for PS structures,taking flexoelectricity into consideration.The analytical solutions to classical flexure of a clamped circular PS thin plate are derived.With the derived analytical model,we numerically investigate the distributions of electromechanical fields and the concentration of electrons in the circular PS thin plate under an upward concentrated force.The effect of flexoelectricity on the multi-field coupling responses of the circular PS plate is studied.The obtained results provide theoretical guidance for the design of novel PS devices.

Giant Flexoelectric Effect in Snapping Surfaces Enhanced by Graded Stiffness

Flexoelectricity is present in nonuniformly deformed dielectric materials and has size-dependent properties, making it useful for microelectromechanical systems. Flexoelectricity is small compared to piezoelectricity;therefore, producing a large-scale flexoelectric effect is of great interest. In this paper, we explore a way to enhance the flexoelectric effect by utilizing the snap-through instability and a stiffness gradient present along the length of a curved dielectric plate. To analyze the effect of stiffness profiles on the plate, we employ numerical parameter continuation. Our analysis reveals a nonlinear relationship between the effective electromechanical coupling coefficient and the gradient of Young’s modulus. Moreover, we demonstrate that the quadratic profile is more advantageous than the linear profile. For a dielectric plate with a quadratic profile and a modulus gradient of − 0.9, the effective coefficient can reach as high as 15.74 pC/N, which is over three times the conventional coupling coefficient of piezoelectric material. This paper contributes to our understanding of the amplification of flexoelectric effects by harnessing snapping surfaces and stiffness gradient design.

Isogeometric Analysis (IGA)-Based Topology Optimization for 3D Flexoelectric Structures

This paper presents isogeometric analysis(IGA)-based topology optimization for electrical performance of three-dimensional(3D)flexoelectric structures.IGA is employed to provide{C}^{1}continuity in shape function,which is required in treating high-order electromechanical coupling equations.To improve the computational efficiency in treating 3D problems,the redundant degrees of freedom removal technique is introduced.Regularization treatments are also implemented to avoid the numerical singularity induced by flexoelectricity.Both virtual loads and strain energy constraints are taken into consideration to prevent unexpected structural disconnection.Numerical examples and experiments on optimized structures demonstrate that the flexoelectric performance can be effectively improved using the proposed approach.

Flexoelectricity in oxide thin films

Flexoelectric effect describes the electromechanical coupling between the strain gradient and its internal polarization in all dielectrics.Despite this universality,the resulting flexoelectric field remains small at the macroscopic level.However,in nanosystems,the size-dependent effect of flexoelectricity becomes increasingly significant,leading to a notable flexoelectric field that can strongly influence the material’s physical properties.This review aims to explore the flexoelectric effect specifically at the nanoscale.We achieve this by examining strain gradients generated through two distinct methods:internal inhomogeneous strain and external stimulation.In addition,advanced synthesis techniques are utilized to enhance the properties and functionalities associated with flexoelectricity.Furthermore,we delve into other coupled phenomena observed in thin films,including the coupling and utilization of flexomagnetic and flexophotovoltaic effects.This review presents the latest advancements in these areas and highlights their role in driving further breakthroughs in the field of flexoelectricity.