Equipotential Condition Research Articles

Understanding intra-pore electrolyte resistances and distributed capacitances in carbon electrodes used in supercapacitors using a robust modeling process to separate the charge-storage occurring into macro-, meso-, and micro-pore structures

In this study, a Modified Generic Electrochemical (MGE) model is introduced for studying rough/porous carbon-based electrode materials employed in electrical double-layer capacitors (EDLC) which contain different types of surface defects. Instead of adopting the common theoretical approach involving the use of transmission line models or ad hoc equivalent circuits, a rational equivalent circuit model has been devised, which permitted us to derive the different equations for each electrochemical technique. The MGE representing the overall electrochemical behavior of EDLCs obtained using the different techniques (e.g., cyclic voltammetry, chronoamperometry, chronopotentiometry, and impedance) is formally derived based on a simple premise considering that the different inhomogeneous surface defects, which are geometrically arranged under equipotential conditions, are not electrochemically equivalent. In this sense, the presence of hierarchically distributed charge-storage processes with different time constants, accounting for the ionic accumulation at the electrode/solution interface, can be ascertained. We verified that is imperative to avoid the determination of the specific capacitances and resistances in EDLCs using a single electrochemical technique or several techniques whose model equations are based on different theoretical premises. The use of the MGE model unequivocally revealed that isolated measurements of the specific capacitance in the absence of the associated intra-pore resistances are not sufficient to ensure the presence of rapid charge storage allied with high-power characteristics. With the appropriate choice of carbon-based materials with very low intra-pore resistances, substantial progress can be made in ushering in the next generation of EDCLs with very high-power characteristics.

Multiscale modeling and simulation methods for electromagnetic and multiphysics problems

Multiscale modeling and simulation methods for electromagnetic and multiphysics problems

A hybridizable discontinuous Galerkin method for simulation of electrostatic problems with floating potential conductors

AbstractIn an electrostatic simulation, an equipotential condition with an undefined/floating potential value has to be enforced on the surface of an isolated conductor. If this conductor is charged, a nonzero charge condition is also required. While implementation of these conditions using a traditional finite element method (FEM) is not straightforward, they can be easily discretized and incorporated within a discontinuous Galerkin (DG) method. However, DG discretization results in a larger number of unknowns as compared to FEM. In this work, a hybridizable DG (HDG) method is proposed to alleviate this problem. Floating potential boundary conditions, possibly with different charge values, are introduced on surfaces of each isolated conductor and are weakly enforced in the global problem of HDG. The unknowns of the global HDG problem are those only associated with the nodes on the mesh skeleton and their number is much smaller than the total number of unknowns required by DG. Numerical examples show that the proposed method is as accurate as DG while it improves the computational efficiency significantly.

An efficient facet shell element with layerwise mechanics for coupled electromechanical response of piezolaminated smart shells

In this work, an efficient four-node facet shell element is presented for analysis of doubly curved multilayered piezoelectric shells, based on an electromechanically coupled third order zigzag theory. It is motivated by the excellent performance of the shell element developed earlier by the authors for elastic laminated shells. The laminate theory not only incorporates a layerwise description of the inplane displacements but also accounts for the layerwise normal deformation due to the d33-effect of the piezoelectric layers. The number of primary displacement variables is, however, the same as for the smeared third order theory (TOT). A quadratic variation of the electric potential across the piezoelectric layers is assumed, while satisfying the equipotential condition of electroded piezoelectric surfaces using the concept of electric nodes. The performance of the element is assessed for the stress analysis under mechanical and electric potential loads, and for the free vibration response of hybrid piezolaminated deep shells in comparison with the three dimensional piezoelasticity based analytical and finite element solutions. The comparison shows excellent accuracy of the present element for the deflection, stresses, sensory potential, electric displacement and natural frequencies of hybrid shells made of composite as well as the highly inhomogenous soft-core sandwich laminates, for which the smeared TOT is shown to produce highly erroneous results. It is also shown to yield better accuracy than the other available elements of similar computational efficiency for hybrid composite shells.

Finite element model based on an efficient layerwise theory for dynamics and active vibration control of smart functionally graded beams

In this work, we present a two-noded efficient finite element (FE) model incorporating the layer-wise mechanics for the dynamics and active vibration control of smart functionally graded (FG) beams. The material properties in the FG beam are assumed to vary smoothly in the thickness direction according to power law variation. The effective properties are computed using Mori-Tanaka homogenization scheme. Electric potential profile in the electroded piezoelectric layers/patches is assumed quadratic across its thickness. The equations of motion are derived using extended Hamilton’s principle. Due to the complex algebraic expressions involved in the effective properties of FG system, the inertia and stiffness parameters are computed numerically using six point Gauss integration method. To fulfill the convergence requirements for weak integration of various energy terms in the variational formulation, the transverse displacement is interpolated with Hermite interpolation function possessing C1-continuity while the inplane displacement, shear rotation and quadratic component of the electric potential are interpolated with linear Lagrangian functions of C0-continuity. The equipotential condition of the electroded piezoelectric sensors and actuators is conveniently modelled using electric node concept. The control system is designed for constant gain velocity feedback (CGVF), and optimal LQR and LQG control strategies for a reduced order model using state space approach. The control performance is studied for single-input-single-output (SISO) and multi-input-multi-output (MIMO) configurations under various excitations. The effect of material inhomogeneity on stability/instability of the closed-loop response in the CGVF control has been discussed.

Modeling Floating Potential Conductors Using Discontinuous Galerkin Method

Isolated conductors appear in various electrostatic problems. In simulations, an equipotential condition with an undefined/floating potential value is enforced on the surface of isolated conductors. In this work, a numerical scheme making use of the discontinuous Galerkin (DG) method is proposed to model such conductors in electrostatic problems. A floating-potential boundary condition, which involves the equipotential condition together with a total charge condition, is “weakly” enforced on the conductor surfaces through the numerical flux of the DG method. Compared to adaptations of the finite element method used for modeling conductors, this proposed method is more accurate, capable of imposing charge conditions, and simpler to implement. Numerical results, which demonstrate the accuracy and applicability of the proposed method, are presented.

Analytical model of mechanically excited piezoelectric unimorph beams.

A model is presented of a composite beam with one elastic and one piezoelectric layer. A reduced set of piezoelectric equations of state that has only the longitudinal components of stress and strain and the transverse components of electric field and charge density is consistently used to include the effect of piezoelectric coupling in all the equations. The equi-potential boundary conditions on the electrodes, the open-circuit condition, and the Gauss condition are satisfied. The position of the neutral axis and the dynamic equilibrium equation are derived after including the effect of piezoelectric coupling. All equations are combined to derive an equation of motion that contains only the displacement and the mechanical excitation. The solution to the equation is expressed in terms of a complete set of functions and an auxiliary function that contains the electric potential. The latter is needed to satisfy piezoelectric boundary conditions at the ends of the beam. The electric potential varies along the length of the beam and has a quadratic variation between the electrodes. Analytical expressions for displacement and potential, and numerical results at low frequencies and in the neighborhood of resonance, are presented for certain sets of boundary conditions.

Energy Absorption by the Membrane Rafts in the Modulated Electro-Hyperthermia (mEHT)

Aim: Heating by nanoparticles, which are located in the tissue to be treated, is a well-recognized method in hyperthermic oncology. Our objective is to investigate selective, nanoscopic heating without concentrating extra artificial nanoparticles. We have in silico calculation to study the heating of the transmembrane protein clusters (rafts) on cell-membrane. The transmembrane protein domains have significantly higher dielectric constant than their lipid neighborhood in the membrane. This difference causes a local gradient in the Specific Absorption Rate (SAR), which could be a factor of heating of the membranes locally, as well as exciting the receptors for various signal transduction in the cells. We suppose that this process determines the observed cellular effects of modulated electro-hyperthermia (mEHT, trade-name: oncothermia). Materials and Methods: In silico models with highly specialized software (Computer Simulation Technology (CST), Darmstadt, Germany) were performed visualizing the selectivity for the membrane domains. Local raft models were created to simulate the electromagnetic (EM) effect of a 13.56 MHz excitation between two perfect electrical conductor plates, simulating the equipotential conditions of the sides of the membrane in the vicinity of the raft. The simulations were performed with near-field (EQS) solver of CST. The electric field, current density, and electric loss density were monitored by the simulations. The applied material properties and parameters refer to the recent literature. Results: In silico models show ten times higher energy-absorption of the transmembrane domains than that of its lipid-membrane surrounding, and intra- and extracellular neighborhood. Depending on the size, orientation, and location of the membrane rafts, the value of SAR varies, but we use only two simplified models to see the absorption properties. Taking into account the characteristics of the EM field effects we showed that the selective energy-absorption increased further by the cell-cell interactions. The model-calculation could confirm the opportunity of the local membrane heating. Conclusion: Our results indicate the heating in nanoscopic range with energy-absorption by the transmembrane proteins. The heated protein-clusters (membrane rafts) are used the same way as the artificial nanoparticles, while these absorbers are natural parts of the biological system.

Slender piezoelectric beams with resistive-inductive electrodes - modeling and axial wave propagation

This contribution presents an extended one-dimensional theory for piezoelectric beam-type structures with non-ideal electrodes. For these types of electrodes the equipotential area condition is not satisfied. The main motivation of our research is originated from passive vibration control: when an elastic structure is covered by several piezoelectric patches that are linked via resistances and inductances, vibrational energy is efficiently dissipated if the electric network is properly designed. Assuming infinitely small piezoelectric patches that are connected by an infinite number of electrical, in particular resistive and inductive elements, one obtains the Telegrapher\'s equation for the voltage across the piezoelectric transducer. Embedding this outcome into the framework of Bernoulli-Euler, the final equations are coupled to the wave equations for the longitudinal motion of a bar and to the partial differential equations for the lateral motion of the beam. We present results for the wave propagation of a longitudinal bar for several types of electrode properties. The frequency spectra are computed (phase angle, wave number, wave speed), which point out the effect of resistive and inductive electrodes on wave characteristics. Our results show that electrical damping due to the resistivity of the electrodes is different from internal (=strain velocity dependent) or external (=velocity dependent) mechanical damping. Finally, results are presented, when the structure is excited by a harmonic single force, yielding that resistive-inductive electrodes are suitable candidates for passive vibration control that might be of great interest for practical applications in the future.

Transverse dynamics of slender piezoelectric bimorphs with resistive-inductive electrodes

This paper presents and compares a one-dimensional (1D) bending theory for piezoelectric thin beam-type structures with resistive-inductive electrodes to ANSYS three-dimensional (3D) finite element (FE) analysis. In particular, the lateral deflections and vibrations of slender piezoelectric beams are considered. The peculiarity of the piezoelectric beam model is the modeling of electrodes in such a manner that is does not fulfill the equipotential area condition. The case of ideal, perfectly conductive electrodes is a special case of our 1D model. Two-coupled partial differential equations are obtained for the lateral deflection and for the voltage distribution along the electrodes: the first one is an extended Bernoulli-Euler beam equation (second-order in time, forth order in space) and the second one the so-called Telegrapher\'s equation (second-order in time and space). Analytical results of our theory are validated by 3D electromechanically coupled FE simulations with ANSYS. A clamped-hinged beam is considered with various types of electrodes for the piezoelectric layers, which can be either resistive and/or inductive. A natural frequency analysis as well as quasi-static and dynamic simulations are performed. A good agreement between the extended beam theory and the FE results is found. Finally, the practical relevance of this type of electrodes is shown. It is found that the damping capability of properly tuned resistive or resistive-inductive electrodes exceeds the damping performance of beams, where the electrodes are simply linked to an optimized impedance.

High-order plate finite elements for smart structure analysis

This paper presents new finite elements for plate analysis of smart composite structures. Based on the sinus model, Murakami’s Zig–Zag functions are introduced in the three directions, improving the accuracy for multilayered modeling. The transverse normal stress is included allowing use of the three-dimensional constitutive law. Three different eight-node finite elements are developed using C0 approximations, each with a different number of unknown functions: 9, 11 or 12. For the piezoelectric approximation, a layer wise description is used with a cubic variation in the thickness of each layer while the potential is assumed to be constant on each elementary domain for the in-plane variation. These finite elements aims at modeling both thin and thick plates without any pathologies of the classical plate finite elements (shear and Poisson or thickness locking, spurious modes, etc.). This family is evaluated on classical piezoelectric problems of the literature and special emphasis is pointed towards the introduction of equipotential conditions.

Solution modification of a piezoelectric bimorph cantilever under loads

Most of the previous investigations on the static performances of piezoelectric bimorph cantilevers are based on the elementary theory of piezo-elasticity. In fact, one fundamental problem has not been solved yet from the view of the theory of piezo-elasticity, that is, a piezoelectric bimorph cantilever covered fully with electrodes on the upper and lower surfaces and subjected to a transversely concentrated load at the free end of the beam. The major obstacle to find the solution is the equipotential conditions on both the upper and lower surfaces of the cantilever. In this article, by establishing suitable boundary conditions and introducing an appropriate Airy stress function, the analytical solutions of a piezoelectric bimorph cantilever under three different loading conditions have been obtained. Numerical analysis is also conducted and the results are consistent with the present theoretical predictions. This investigation amends the corresponding works given by other investigations based on the elementary theory. It is found that the amendments to the end deflections of a PZT-4 piezoelectric bimorph cantilever caused by an end shear force, an end moment, or a voltage difference between the upper and lower surfaces are about 10.46%, 12.78%, and 3.52%, respectively. In addition, it is also found that these amendments depend just on the material parameters, have nothing to do with the geometric parameters of the bimorph cantilever. The Airy stress function proposed in this article can also be used to study other different piezoelectric cantilevers including multilayer piezoelectric cantilevers under corresponding loads.

Experimental assessment of the piezoelectric transverse d15shear sensing mechanism

The piezoelectric transverse d15 shear sensing mechanism is firstly assessed experimentally for a cantilever smart sandwich plate made of a piezoceramic axially poled patched core and glass fiber reinforced polymer composite faces. Different electrical connections are tested for the assessment of the sensor performance under a varying amplitude harmonic (at 24 Hz) force. Also, the dynamic response of the smart sandwich composite structure is monitored using different acquisition devices. The obtained experimentally sensed voltages are compared to those resulting from the benchmark three-dimensional piezoelectric coupled finite element simulations using a commercial code where realistic features, like equipotential conditions on the patches\' electrodes and mechanical updating of the clamp, are considered. Numerically, it is found that the stiffness of the clamp, which is much softer than the ideal one, has an enormous influence on the sensed voltage of its adjacent patch; therefore, sensing with the patch on the free side would be more advantageous for a cantilever configuration. Apart from confirming the latter result, the plate benchmark experimental assessment showed that the parallel connection of its two oppositely poled patches has a moderate performance but better than the clamp side patch acting as an individual sensor.

Free vibration analysis of moderately thick asymmetric piezoelectric adaptive cantilever beams using the distributed transfer function approach

A semi-analytical distributed transfer function (DTF) approach is proposed for the free-vibration analysis of moderately thick cantilever beams with a single surface-bonded piezoelectric patch. The asymmetric piezoelectric adaptive structure is decomposed into three segments; the first and third segments are bare beam parts before and after the patch, while the second segment contains the beam part with attached piezoelectric patch bonded to its upper surface. The theoretical formulation assumes first-order shear deformation kinematics and linear electric potential through the patch thickness with an electrode equipotential physical condition, and uses the extended Hamilton׳s principle to derive the equations of motion and electromechanical boundary conditions. The latter, together with the continuity and equilibrium conditions at the segments interfaces, are then transformed into a first-order state space equation that is solved using the DTF approach. The electrodes of the piezoelectric patch are considered either in short-circuit (SC) or open-circuit (OC); this leads to two free-vibration problems to be solved for the corresponding SC and OC frequencies, from which the Electro-Mechanical Coupling Coefficient (EMCC) is post-treated. Four benchmarks from the open literature are simulated in order to validate the proposed approach. Very satisfactory correlations are obtained for all examples with maximum errors less thank 5 percent in all results. For future reference, an additional benchmark is proposed to assess the influence of the patch-to-composite host width ratio on the effective modal EMCC. It was found that the latter is mode-dependent (as expected) and decreases with increasing the former.

Coupled efficient layerwise finite element modeling for active vibration control of smart composite and sandwich shallow shells

The active vibration suppression of smart composite and sandwich shallow shells equipped with distributed monolithic piezoelectric and piezo-fiber reinforced composite (PFRC) sensors and actuators is studied using a four-node quadrilateral shallow shell element based on a fully coupled, accurate and efficient layerwise (zigzag) theory. The shell element uses the concept of electric nodes to satisfy the equipotential condition of electroded sensor surfaces without making any approximations or averaging. The effective electromechanical properties of the PFRC laminas are computed using a coupled three-dimensional iso-field micromechanical model. Both classical (constant gain velocity feedback (CGVF)) and optimal (linear quadratic Gaussian (LQG)) control strategies are studied. A truly collocated actuator–sensor arrangement is proposed and shown to remove the instability in CGVF control of shallow shells with conventionally collocated actuators and sensors. The effects of piezoelectric fiber orientation and volume fraction ratio of PFRC, and the radius of curvature and span to thickness ratio of the shell on the control performance, are studied. It is shown that the LQG control not only suppresses the transient vibration under step/impulse excitations, but also eliminates the beating phenomena under harmonic excitation when the forcing frequency is close to the natural frequency of the system.

An efficient finite element with layerwise mechanics for smart piezoelectric composite and sandwich shallow shells

In this work, we present a new efficient four-node finite element for shallow multilayered piezoelectric shells, considering layerwise mechanics and electromechanical coupling. The laminate mechanics is based on the zigzag theory that has only seven kinematic degrees of freedom per node. The normal deformation of the piezoelectric layers under the electric field is accounted for without introducing any additional deflection variables. A consistent quadratic variation of the electric potential across the piezoelectric layers with the provision of satisfying the equipotential condition of electroded surfaces is adopted. The performance of the new element is demonstrated for the static response under mechanical and electric potential loads, and for free vibration response of smart shells under different boundary conditions. The predictions are found to be very close to the three dimensional piezoelasticity solutions for hybrid shells made of not only single-material composite substrates, but also sandwich substrates with a soft core for which the equivalent single layer (ESL) theories perform very badly.

A nonlinear efficient layerwise finite element model for smart piezolaminated composites under strong applied electric field

In this work, we present an electromechanically coupled efficient layerwise finite element model for the static response of piezoelectric laminated composite and sandwich plates, considering the nonlinear behavior of piezoelectric materials under strong electric field. The nonlinear model is developed consistently using a variational principle, considering a rotationally invariant second order nonlinear constitutive relationship, and full electromechanical coupling. In the piezoelectric layer, the electric potential is approximated to have a quadratic variation across the thickness, as observed from exact three dimensional solutions, and the equipotential condition of electroded piezoelectric surfaces is modeled using the novel concept of an electric node. The results predicted by the nonlinear model compare very well with the experimental data available in the literature. The effect of the piezoelectric nonlinearity on the static response and deflection/stress control is studied for piezoelectric bimorph as well as hybrid laminated plates with isotropic, angle-ply composite and sandwich substrates. For high electric fields, the difference between the nonlinear and linear predictions is large, and cannot be neglected. The error in the prediction of the smeared counterpart of the present theory with the same number of primary displacement unknowns is also examined.

Advanced materials and structures response prediction and monitoring

Advanced materials and structures response prediction and monitoring

Semi-analytical free-vibration analysis of piezoelectric adaptive beams using the distributed transfer function approach

SUMMARY A distributed transfer function (DTF) approach is proposed for the free-vibration analysis of moderately thick cantilever beams with symmetrically bonded piezoelectric patches. The piezoelectric adaptive structure is decomposed into three sections; the first and third sections contain only the bare parts of the beam before and after the patches, whereas the second section contains the beam part equipped with the piezoelectric patches bonded to its upper and lower surfaces. The analytical formulation assumes first-order shear deformation kinematics and linear electric potential through each patch thickness and uses the extended Hamilton's principle to derive the equations of motion and electromechanical boundary conditions. The latter are then transformed into a first-order state space equation using the DTF approach in order to get the short-circuit and open-circuit free-vibration problems that consider the equipotential physical conditions on the electroded upper and lower surfaces of the piezoelectric patches. The implemented DTF approach is validated by running three benchmarks from open literature. The comparison of the obtained results with those provided by three-dimensional finite elements using a commercial software shows reasonable correlation. Copyright © 2011 John Wiley & Sons, Ltd.

Stability and free vibration analysis of thick piezoelectric composite plates using spline finite strip method

In the present study, a spline finite strip with higher-order shear deformation is formulated for stability and free vibration analysis of piezoelectric composite plates. At each knot, the electric potentials on the surfaces and middle plane of each piezoelectric layer are taken as nodal degrees of freedom. However, if a continuous electrode is installed on the surface of the layer, the electric potential on the electrode is changed to structural degree of freedom, so that the equipotential condition on the electrode is automatically satisfied. The analysis can be conducted based on Reddy's third-order shear deformation theory, Touratier's “Sine” model, Afaq's exponential model or Cho's higher-order zigzag laminate theory. Consequently, the shear correction coefficients are not required in the analysis, and an improved accuracy for thick plates over the first-order shear deformation theory is achieved at only little extra computational cost. The numerical results obtained based on different shear deformation theories are presented in comparison with the three-dimensional solutions. The effects of length-to-thickness ratio, fiber orientation, boundary conditions and electrical conditions on the natural frequency and critical buckling load of piezoelectric composite plates are investigated through numerical examples.