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.

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.

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.

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.

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.

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.

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.

An isogeometric approach to static and transient analysis of fluid-infiltrated porous metal foam piezoelectric nanoplates with flexoelectric effects and variable nonlocal parameters

In this work,a novel refined higher-order shear deformation plate theory is integrated with nonlocal elasticity theory for analyzing the free vibration,bending,and transient behaviors of fluid-infiltrated porous metal foam piezoelectric nanoplates resting on Pasternak elastic foundation with flexoelectric effects.Isogeometric analysis(IGA)and the Navier solution are applied to the problem.The innovation in the present study is that the influence of the in-plane variation of the nonlocal parameter on the free and forced vibration of the piezoelectric nanoplates is investigated for the first time.The nonlocal parameter and material characteristics are assumed to be material-dependent and vary gradually over the thickness of structures.Based on Hamilton’s principle,equations of motion are built,then the IGA approach combined with the Navier solution is used to analyze the static and dynamic response of the nanoplate.Lastly,we investigate the effects of the porosity coefficients,flexoelectric parameters,elastic stiffness,thickness,and variation of the nonlocal parameters on the mechanical behaviors of the rectangular and elliptical piezoelectric nanoplates.

Flexoelectricity-enhanced photovoltaic effect in trapezoid-shaped NaNbO_(3)nanotube array composites

In this work,we successfully prepared vertically aligned NaNbO_(3)nanotube(NN-NT)with trapezoidal shapes,in which the orthorhombic and monoclinic phases coexisted.According to the structure analysis,the NN-NT/epoxy composite film had excellent flexoelectric properties due to the lattice distortion caused by defects and irregular shape.The flexoelectric effect is the greatest in the vertical direction in the flexible NN-NT/epoxy composite film,and the flexoelectric coefficient()is 2.77×10^(−8)C·m^(−1),which is approximately 5-fold higher than that of the pure epoxy film.The photovoltaic current of the NN-NT/epoxy composite film increased from 39.9 to 71.8 nA·cm^(−2)in the direction of spontaneous polarization when the sample was bent upward due to the flexoelectricity-enhanced photovoltaic(FPV)effect.The flexoelectric effect of the NN-NT/epoxy composite film could modulate the photovoltaic response by increasing it by 80%or reducing it to 65%of the original value.This work provides a new idea for further exploration in efficient and lossless ferroelectric memory devices.

Variational principles and governing equations in nano-dielectrics with the flexoelectric effect

The flexoelectric effect is very strong and coupled with large strain gradients for nanoscale dielectrics. At the nanoscale, the electrostatic force cannot be ignored. In this paper, we have established the electric enthalpy variational principle for nanosized dielectrics with the strain gradient and the polarization gradient effect, as well as the effect of the electrostatic force. The complete governing equations, which include the effect of the electrostatic force, are derived from this variational principle, and based on the principle the generalized electrostatic stress is obtained, the generalized electrostatic stress contains the Maxwell stress corresponding to the polarization and strain, and stress related to the polarization gradient and strain gradient. This work provides the basis for the analysis and computations for the electromechanical problems in nanosized dielectric materials.

Flexoelectric materials and their related applications: A focused review

Flexoelectricity refers to the mechanical-electro coupling between strain gradient and electric polarization, and conversely, the electro-mechanical coupling between electric field gradient and mechanical stress. This unique effect shows a promising size effect which is usually large as the material dimension is shrunk down. Moreover, it could break the limitation of centrosymmetry, and has been found in numerous kinds of materials which cover insulators, liquid crystals, biological materials, and semiconductors. In this review, we will give a brief report about the recent discoveries in flexoelectricity, focusing on the flexoelectric materials and their applications. The theoretical developments in this field are also addressed. In the end, the perspective of flexoelectricity and some open questions which still remain unsolved are commented upon.

Bending of a Flexoelectric Semiconductor Plate

We study electromechanical responses of a flexoelectric semiconductor plate in bending under mechanical loads.A two-dimensional theory for classical bending without shear deformation is derived from the three-dimensional macroscopic theory of flexoelectric semiconductors.A simple solution is obtained for pure bending.A combination of physical and geometric parameters is introduced as a measure of the strength of the coupling between the mechanical load and the redistribution of charge carriers.A trigonometric series solution is obtained for a simply supported rectangular plate under a local normal mechanical load,which shows concentration of mobile charges and the formation of electric potential barriers near the loading area.The results are fundamental to the mechanical manipulation of charge carriers in these plates.We also analyze the buckling of a simply supported rectangular plate under in-plane compressive forces.A series of buckling loads and modes are obtained.Numerical results show that flexoelectric coupling exhibits a stiffening effect and increases the buckling load,while semiconduction weakens the flexoelectric stiffening.The distributions of mobile charges in the first few buckling modes are presented.

Localization of elastic waves in one-dimensional detuned phononic crystals with flexoelectric effect

Although forbidden band effect in perfectly periodic phononic crystals(PC)is very attractive,random disordered(i.e.detuning)phenomenon is inevitable in engineering processing,thus explor-ing the effect of detuning on the wave characteristics of PC becomes a necessity.In this study,fundamental governing equa-tions and boundary conditions are derived from the principle of virtual work.Wave characteristics and localization factor of one-dimensional(1D)detuned nano-PC are investigated based on the transfer matrix method,with flexoelectric effect duly accounted for.Subsequently,with BaTiO_(3)􀀀SrTiO_(3) nano-PC taken for illustration,forbidden band properties and localization factor of 1D elastic waves in harmonic and detuned states are systematically character-ized.It is demonstrated that localization factor can characterize the energy band structure of 1D PC perfectly.Flexoelectric effect tends to widen the width of forbidden band with increasing detuning,and detuning is linearly related to the bandwidth.The forbidden band is more sensitive to flexoelectric coefficient detuning than thickness detuning.The research results provide useful theoretical guidance for designing high-frequency nanoscaled devices with the function of filtering based on band gap effect of nano-PC.

Universal converse flexoelectricity in dielectric materials via varying electric field direction

Flexoelectricity is a symmetry independent electromechanical cou-pling phenomenon that outperforms piezoelectricity at micro and nanoscales due to its size-dependent behavior arising from gradi-ent terms in its constitutive relations.However,due to this gradient term flexoelectricity,to exhibit itself,requires specially designed geometry or material composition of the dielectric material.First of its kind,the present study put forward a novel strategy of achieving electric field gradient and thereby converse flexoelectri-city,independent of geometry and material composition of the material.The spatial variation of the electric field is established inside the dielectric material,Ba_(0.67)Sr_(0.33)TiO_(3)(BST),by manipulating electrical boundary conditions.Three unique patterns of electrode placement are suggested to achieve this spatial variation.This varying direction of electric field gives rise to electric field gradient,the prerequisite of converse flexoelectricity.A multi-physics cou-pling based theoretical framework is established to solve the flexo-electric actuation by employing isogeometric analysis(IGA).Electromechanically coupled equations of flexoelectricity are solved to obtain the electric field distribution and the resulting displace-ments thereby.The maximum displacements of 0.2 nm and 2.36 nm are obtained with patterns I and II,respectively,while pattern III can yield up to 85 nm of maximum displacement.

Static and Dynamic Analysis of a Piezoelectric Semiconductor Cantilever Under Consideration of Flexoelectricity and Strain Gradient Elasticity

In this work,the static and dynamic response of a piezoelectric semiconductor cantilever under the transverse end force with consideration of flexoelectricity and strain gradient elasticity is systematically investigated.The one-dimensional governing equations and the corresponding boundary conditions are derived based on Hamilton’s principle.After that,combining with the linearized equations for the conservation of charge,the effects of characteristic length and flexoelectric coefficient on the working performance of a ZnO nanowire are demonstrated as a numerical case,including the static mechanical and electric fields,natural frequencies,and the frequency–response characteristics at resonances.The results indicate that the flexoelectric effect has a great influence on the electric properties of the nanowire,while the strain gradient effect directly contributes to its mechanical properties.To some extent,the increase in characteristic length is equivalent to the stiffness strengthening.The qualitative results and quantitative data are beneficial for revealing the underlying physical mechanism and provide guidance for the design of piezoelectric semiconductor devices.

Magnetically induced electric potential in first-order composite beams incorporating couple stress and its flexoelectric effects

A new model of a first-order composite beam with flexoelectric and piezomagnetic layers is developed.The new model is under a transverse magnetic field and can capture the couple stress and its flexoelectric effects.The governing equations are obtained through a variational approach.To illustrate the new model,the static bending problem is analytically solved based on a Navier’s technique.The numerical results reveal that the extension,deflection,and shear deformation of the current or couple stress relevant flexoelectric model are always smaller than those of classical models at very small scale.It is also found that the electric potentials only appear with the presence of the flexoelectric effect for this non-piezoelectric composite beam model.Furthermore,various electric potential distributions can be manipulated by the particular magnetic fields,and remote/non-contact control at micro-and nano-scales can be realized by current functional composite beams.