A micromechanically motivated model for ferroelectrics combined with polygonal finite elements

Piezoelectric materials are one of the most prominent smart materials due to their strong electromechanical coupling behaviour. Ferroelectric ceramics behave like piezoelectric materials under low electrical and mechanical loads, but exhibit pronounced nonlinear response at higher loads due to microscopic domain switching. Modern smart devices consist of complex geometries that may force the ferroelectrics employed within them to experience higher fields than they were originally designed for, so that the material responds within its nonlinear region. Hence, models predicting the nonlinear effects of ferroelectrics under complex loading cases are important from the design point of view. Within standard finite element models dealing with electromechanical problems, each grain may be subdiscretized by several finite elements. This problem can be approximated or rather overcome by a polygonal finite element method, where each grain is modelled by solely one single finite element. In this contribution, a micromechanically motivated switching model for ferroelectric ceramics, as based on volume fraction concepts, is combined with polygonal finite element approach. Related representative numerical examples allow to further study and understand the nonlinear response of this material under complex loading cases. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Advanced finite element formulations for modeling thin piezoelectric structures

This contribution is concerned with mixed finite element formulations for modeling piezoelectric beam and shell structures. Due to the electromechanical coupling, specific deformation modes are joined with electric field components. In bending dominated problems incompatible approximation functions of these fields cause incorrect results. These effects occur in standard finite element formulations, where interpolation functions of lowest order are used. A mixed variational approach is introduced to overcome these problems. The mixed formulation allows for a consistent approximation of the electromechanical coupled problem. It utilizes six independent fields and could be derived from a Hu-Washizu variational principle. Displacements, rotations and the electric potential are employed as nodal degrees of freedom. According to the Timoshenko theory (beam) and the Reissner-Mindlin theory (shell), the formulations account for constant transversal shear strains. To incorporate three dimensional constitutive relations all transversal components of the electric field and the strain field are enriched by mixed finite element interpolations. Thus the complete piezoelectric coupling is appropriately captured. The common assumption of vanishing transversal stress and dielectric displacement components is enforced in an integral sense. Some numerical examples will demonstrate the capability of the presented finite element formulation. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A microcrack-based continuum damage model with application towards refractories

The present work discusses numerical results of damage evolution due to thermoshock processes at refractory ceramics. Damage patterns have been generated using a two-scale approach for brittle materials, implemented into a finite element framework. For this purpose, a cell model has been employed, incorporating cracks on a microscopic level. The impact of these discontinuities on macroscopic material properties and damage evolution is determined with the help of analytical homogenization techniques. Finally, the potential of the numerical tool is demonstrated by means of refractory bricks, being imposed by thermomechanical loading. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Shape control of piezolaminated thin-walled structures using nonlinear piezoelectric actuators

In the present paper a 2D-shell finite element model is proposed to carry out static analysis of piezolaminated composite shells by incorporating nonlinear constitutive relations in order to describe the electromechanical coupling under strong electric fields. The present shell element has 5 mechanical DOFs and 3 electrical DOFs per node. The developed composite piezolaminated shell element is employed to study the static behavior and shaping of spherical antenna reflector laminated with piezo-patches under both weak and strong electric fields. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A three-field,bi-domain based approach to the strongly coupled electromechanics of the heart

We propose a novel, unconditionally stable and fully coupled finite element method for the bidomain based approach to cardiac electromechanics. To this end, the transmembrane potential, the extracellular potential, and the displacement field are treated as independent variables such that the already coupled electrophysiology problem in the bidomain setting is further extended to the electromechanical coupling. In this multifield problem, the intrinsic coupling arises from both excitation-induced contraction of cardiac cells and the deformation-induced generation of intra-cellular currents. The respective bidomain reaction-diffusion and the momentum balance equations are recast into the corresponding weak forms through a conventional isoparametric Galerkin approach. The resultant set of non-linear residual equations is consistently linearized. The monolithic scheme is employed to avoid stability issues that may arise due to the strong coupling between excitation and deformation. The performance of the put forward framework is further assessed through three-dimensional representative electromechanical initial-boundary value problems. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite deformations of thin shells with piezoelectric sensors and actuators

We treat coupled electromechanical problem of finite deformations of piezoelectric shells with the help of the direct approach. A shell is considered as a material surface with mechanical degrees of freedom of particles and with an additional field variable, namely electric potential on the electrodes. This results both in the nonlinear system of equations of piezoelectric shells and in the appropriate numerical scheme. Application of the direct approach is preceded with the three-dimensional asymptotic analysis of a linear electromechanical problem for a non-homogeneous piezoelectric plate, which provides the constitutive relations for the nonlinear theory. As a sample problem, we present finite element analysis of deformation and local buckling of cylindrical panel, equipped with piezoelectric sensors. The latter influence the mechanical behavior and produce signals, which can be interpreted in terms of structural entities. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Electromechanics of Cardiac Tissue: A Unified Approach to the Fully Coupled Excitation-Contraction Problem

This contribution is concerned with a new, unified finite element approach to the fully coupled problem of cardiac electromechanics. In contrast to the existing numerical approaches suggested in the literature; to the best of our knowledge, for the first time, we propose a fully implicit, purely finite-element-based approach to the coupled problem. The system of coupled algebraic equations obtained by simultaneous linearization of non-linear weighted residual terms is solved monolithically. The put forward modular algorithmic framework leads to an unconditionally stable and geometrically flexible structure that can readily be extended towards complex ionic models of cardiac electrophysiology. The performance of the proposed approach is illustrated by the coupled electromechanical analysis of a biventricular generic heart model. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A strong discontinuity based adaptive refinement approach for the modeling of crack branching

In the current work, the physical phenomena of dynamic fracture of brittle materials involving crack growth, acceleration and consequent branching is simulated. The numerical modeling is based on the approach where the failure in the form of cracks or shear bands is modeled by a jump in the displacement field, the so called ‘strong discontinuity’. The finite element method is employed with this strong discontinuity approach where each finite element is capable of developing a strong discontinuity locally embedded into it. The focus in this work is on branching phenomena which is modeled by an adaptive refinement method by solving a new sub-boundary value problem represented by a finite element at the growing crack tip. The sub-boundary value problem is subjected to a certain kinematic constraint on the boundary in the form of a linear deformation constraint. An accurate resolution of the state of material at the branching crack tip is achieved which results in realistic dynamic fracture simulations. A comparison of resulting numerical simulations is provided with the experiment of dynamic fracture from the literature. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation of micro-cutting considering finite deformation crystal plasticity

Micro-machining processes on metalic microstructures are influenced by the crystal structure, i. e. the grain orientation. Furthermore, the chip formation underlies large deformations. To perform finite element simulations of micro-cutting processes, a large deformation material model is necessary in order to model the hyperelastic and finite plastic material behaviour. In the case of cp-titanium material with hcp-crystal structure the anisotropic behaviour must be considered by an appropriate set of slip planes and slip directions. In the present work the impact of the grain orientation on the plastic deformation is demonstrated by means of finite element simulations of a finite deformation single slip crystal plasticity model. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An adaptive approach for modeling a fiber-matrix composite with the FE2 method

In materials with a complicated microstructre [1], the macroscopic material behaviour is unknown. In this work a Fiber-Matrix composite is considered with elasto-plastic fibers. A homogenization of the microscale leads to the macroscopic material properties. In the present work, this is realized in the frame of a FE² formulation. It combines two nested finite element simulations. On the macroscale, the boundary value problem is modelled by finite elements, at each integration point a second finite element simulation on the microscale is employed to calculate the stress response and the material tangent modulus. One huge disadvantage of the approach is the high computational effort. Certainly, an accompanying homogenization is not necessary if the material behaves linear elastic. This motivates the present approach to deal with an adaptive scheme. An indicator, which makes use of the different boundary conditions (BC) of the BVP on microscale, is suggested. The homogenization with the Dirichlet BC overestimates the material tangent modulus whereas the Neumann BC underestimates the modulus [2]. The idea for an adaptive modeling is to use both of the BCs during the loading process of the macrostructure. Starting initially with the Neumann BC leads to an overestimation of the displacement response and thus the strain state of the boundary value problem on the macroscale. An accompanying homogenization is performed after the strain reaches a limit strain. Dirichlet BCs are employed for the accompanying homogenization. Some numerical examples demonstrate the capability of the presented method. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiple level finite element analysis and its applications to tidal current flow in Tokyo Bay

A finite element method for the analysis of a one level and a multiple level current flow is presented. The basic equations can be derived from the three-dimensional Navier-Stokes equations under the shallow water assumptions. The standard finite element method has been introduced using the linear interpolation function based on a triangular finite element. For each level, the finite element subdivisions are not required to be coincident. To integrate the discretized equations numerically in time, an improved two step explicit scheme is employed. The multiple level finite element method is applied to a tidal flow analysis of Tokyo Bay.The multiple level tidal flow analysis is performed at the entrance channel of Tokyo Bay. The density of water is assumed to be constant for each level. The vertical profiles of the numerical velocity are compared with those of the observed velocity. The flow directions and the order of velocity are both well in agreement with the observed data. The tidal flow pattern in Tokyo Bay has been shown to be expressed by the multiple level flow assuming that the density of seawater is levelwise constant.The numerical tidal flow computation of Tokyo Bay carried out using a one level model is compared with observed data. The one level numerical values will be used to specify the boundary conditions for the multiple level analysis. Both numerical and observed results correspond extremely well in this computation. The two dominant circulated residual flows have been computed, and they coincide with the observed facts.     

Structural Optimization with FEM-based Shakedown Analyses

In this paper, a mathematical programming formulation is presented for the structural optimization with respect to the shakedown analysis of 3-D perfectly plastic structures on basis of a finite element discretization. A new direct algorithm using plastic sensitivities is employed in solving this optimization formulation. The numerical procedure has been applied to carry out the shakedown analysis of pipe junctions under multi-loading systems. The new approach is compared to so-called derivative-free direct search methods. The computational effort of the proposed method is much lower compared to this methods.     

Finite element-based approach to modeling heterogeneous objects

Advances in materials and rapid prototyping technology, makes developing rapid-prototype heterogeneous parts possible. This paper proposes a finite element-based approach for modeling heterogeneous parts to facilitate rapid prototyping. In this approach, traditional geometrical and topological data are blended with materials information in the CAD model. It builds on a CAD system to create parts’ geometry which is then subdivided into a set of tetrahedral finite elements. With this, the design problem of material distribution in the whole body can be handled at the finite element level. The material composition of each point inside a finite element is formulated in terms of four control nodes of the finite element and a blending function. Moreover, design rules for modeling heterogeneous parts are introduced in the paper, making the modeling solution more robust.     

Space–time DG methods for the coupled electro–mechanical activation of the human heart

We consider the coupled system of time–dependent nonlinear partial differential equations modeling the electromechanical response of human heart tissue. Instead of time–stepping schemes we use a discontinuous Galerkin finite element method in the space–time domain to be able to resolve the solution in space and time simultaneously. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A laminate-based framework for switching and microstructure evolution in polycrystalline ferroelectrics

This contribution deals with the numerical modelling of polycrystalline ferroelectric materials considering a sequential laminate-based approach established for tetragonal single-crystal ferroelectrics. The particular model [1] is considered and extended to predict the material behaviour of poly-crystal tetragonal ferroelectric ceramics. The derived laminate-based model is implemented in a finite element environment to simulate the time-dependent domain evolution and switching response of a bulk polycrystalline ferroelectric ceramic. © 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A multilevel correction method for Stokes eigenvalue problems and its applications

In this paper, a new multilevel correction scheme is proposed to solve Stokes eigenvalue problems by the finite element method. This new scheme contains a series of correction steps, and the accuracy of eigenpair approximation can be improved after each step. In each correction step, we only need to solve a Stokes problem on the corresponding fine finite element space and a Stokes eigenvalue problem on the coarsest finite element space. This correction scheme can improve the efficiency of solving Stokes eigenvalue problems by the finite element method. As applications of this multilevel correction method, a multigrid method and an adaptive finite element technique are introduced for Stokes eigenvalue problems. Some numerical results are given to validate our schemes. Copyright © 2013 John Wiley & Sons, Ltd.     

Flexoelectric and surface effects on the electromechanical behavior of graphene-based nanobeams

In this novel work, the electromechanical behavior of graphene-based nanocomposite (GNC) beams with flexoelectric and surface effects were investigated using size-dependent Euler-Bernoulli theory, linear piezoelectricity and Galerkin's weighted residual method along with modified strength of materials and finite element (FE) approaches. In addition, analytical and FE models were developed to study the static response of flexoelectric GNC nanobeams with various boundary conditions: cantilever, simply-supported and clamped-clamped. The developed models predict that the effective piezoelectric coefficients of GNC are responsible for the actuation capability of a graphene layer in the transverse direction due to the applied field in its axial direction and the predictions by both the models are found to be in good agreement. Results reveal that the flexoelectric and surface effects on the static response of GNC nanobeams are significant and should be taken into account. The electromechanical response of GNC nanobeams can be tailored to achieve the required coupled electromechanical characteristics of a vast range of NEMS using various boundary conditions and thickness of nanobeam as well as volume fraction of graphene. Our fundamental study sheds a light on the possibility of developing high-performance and lightweight graphene-based NEMS such as nanosensors, nanogenerators and nanoresonators using non-piezoelectric graphene.     

A 1D constitutive law for hysteresis effects in piezoceramics

This paper is concerned with a macroscopic constitutive law for domain switching effects, which occur in piezoelectric ceramics. The thermodynamical framework of the law is based on two scalar valued functions: the Helmholtz free energy and a switching surface. In common usage, the remanent polarization and the remanent strain are employed as internal variables. The novel aspect of the present work is to introduce an irreversible electric field, which serves besides the irreversible strain as internal variable. The irreversible electric field has only theoretical meaning, but leads to advantages within the finite element implementation, where displacement and the electric potential are the nodal degrees of freedoms. A common assumption is a one-to-one relation between the irreversible strain and the polarization. This simplification is not employed in the present paper. To accomplish enough space for the polarization, resulting from an applied electric field, the irreversible strains are additively split and a special hardening function is introduced. This balances the amount of space and the domain switching due to polarization. The constitutive model reproduces the ferroelastic and the ferroelectric hysteresis as well as the butterfly hysteresis for piezoelectric ceramics and it accounts for the mechanical depolarization effect. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A finite element model of localized deformation in frictional materials taking a strong discontinuity approach

A finite element model of localized deformation in frictional materials taking a strong discontinuity approach is presented. A rate-independent, non-associated, strain-softening Drucker–Prager plasticity model is formulated in the context of strong discontinuities and implemented along with an enhanced quadrilateral element within the framework of an assumed enhanced strain finite element method. For simple model problems such as uniform compression, the strong discontinuity approach has been shown to lead to mesh-independent finite element solutions when localized deformation is present. In this paper, a finite element analysis of localized deformation occurring in a more complex model problem of slope stability is conducted in a nearly mesh-independent manner. The effect of dilatancy on the orientation of slip lines is demonstrated for the slope stability problem.