Implementation of a Free-locking Composite Piezoelectric Shell Element

In this contribution, a free–locking shell element with piezoelectric ceramic layers is presented. The material law and the constitutive equations of the coupled electro–mechanical problem are derived in convective coordinates, so that the presented element accounts for geometrical nonlinearities [2]. To avoid zero–energy mode and locking effects of the thin shell structure, assumed natural strain (ANS) method [1] is implemented. Finally, some numerical examples demonstrate the ability of this element to solve linear and nonlinear problems. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Advanced Shell Element Formulations for Coupled Electromechanical Systems

With the significantly increasing applications of smart structures, piezoelectric material is widely used in branches of engineering sciences. Normally, the Finite Element Method is employed in the numerical analysis of these structures [2]. In this contribution, in order to avoid the locking effects and zero energy modes, the Assumed Natural Strain (ANS) Method [4] is implemented into four‐node piezoelectric shallow shell elements, by using the two‐field variational formulation in which displacements and electric potentials serve as independent variables and the three‐field variational formulation in which the dielectric displacement is taken as an independent variable additionally [3]. Moreover, a quadratic variation of the electric potential through the thickness direction is applied in the two‐field formulation. Numerical examples of piezoelectric sensors and actuators are presented, showing the behaviour of the shell elements by using different hybrid finite element formulations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Piezoelectric Finite Shell Element for the Geometrically Nonlinear Analysis of Sensors

A geometrically nonlinear finite element formulation to analyze piezoelectric shell structures is presented. The formulation is based on the mixed field variational functional of Hu–Washizu. Within this variational principle the independent fields are displacements, electric potential, strains, electric field, stresses and dielectric displacements. The mixed formulation allows an interpolation of the strains and the electric field through the shell thickness, which is an essential advantage when using a three dimensional material law. It is remarked that no simplification regarding the constitutive relation is assumed. The normal zero stress condition and the normal zero dielectric displacement condition are enforced by the independent resultant stress and resultant dielectric displacement fields. The shell structure is modeled by a reference surface with a four node element. Each node possesses six mechanical degrees of freedom, three displacements and three rotations, and one electrical degree of freedom, which is the difference of the electric potential through the shell thickness. The developed mixed hybrid shell element fulfills the in–plane, bending and shear patch tests, which have been adopted for coupled field problems. A numerical investigation of a smart antenna demonstrates the applicability of the piezoelectric shell element under the consideration of geometrical nonlinearity. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Static and dynamic analysis of smart cylindrical shell

Shell type components and structures are very common in many mechanical and structural systems. In smart structural applications, piezolaminated plates and shells are commonly used. In this paper a finite element formulation is presented to model the static and dynamic response of laminated composite shells containing integrated piezoelectric sensors and actuators subjected to electrical, mechanical and thermal loadings. The formulation is based on the first order shear deformation theory and Hamilton's principle. In this formulation, the mass and stiffness of the piezo-layers have been taken into account. A nine-noded degenerated shell element is implemented for the analysis. The model is validated by comparing with existing results documented in the literature. A simple negative velocity feedback control algorithm coupling the direct and converse piezoelectric effects is used to actively control the dynamic response of an integrated structure through a closed control loop. The influence of the stacking sequence and position of sensors/actuators on the response of the laminated cylindrical shell is evaluated. Numerical results show that piezoelectric sensors/actuators can be used to control the shape and vibration of laminated composite cylindrical shell.     

A finite element model for laminated piezoelectric shell structures

Thin piezoelectric laminates are used for a wide range of technical applications. A four-node piezoelectric shell element is presented to analyse such structures effectively. In case of bending dominated problems incompatible approximation functions of the electrical and mechanical fields cause incorrect results. In order to overcome this problem the finite element formulation is based on a mixed variational principle implying six independent fields: displacements, electric potential, strains, electric field, mechanical stresses and dielectric displacements. This allows for an interpolation of the strains and the electric field in thickness direction independent of the bilinear interpolation functions. A piecewise quadratic approach for the shear strains in thickness direction and the corresponding electric field is proposed for arbitrarily layered shells. Regarding coupling of electrical and mechanical fields this yields to an appropriate balance of the approximation functions. Numerical examples show more precise results in contrast to standard elements with lowest order interpolation functions. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A mixed finite element formulation for piezoelectric shells

A finite element formulation to analyze piezoelectric shell problems is presented. A reference surface of the shell is modelled with a four node element. Each node possesses six mechanical degrees of freedom, three displacements and three rotations, and one electric degree of freedom, which is the difference of the electric potential in thickness direction. The formulation is based on the mixed field variational principle of Hu-Washizu. The independent fields are displacements u , electric potential φ, strains E , electric field E , stresses S and dielectric displacements D . The mixed formulation allows an interpolation of the strains and the electric field in thickness direction. Accordingly a three-dimensional material law is incorporated in the variational formulation. It is remarked that no simplification regarding the constitutive law is assumed. The formulation allows the consideration of arbitrary constitutive relations. The normal zero stress condition and the normal zero dielectric displacement condition are enforced by the independent stress and dielectric displacement fields. They are defined as zero in thickness direction. The present shell element fulfills the important patch tests: the in-plane, bending and shear test. Some numerical examples demonstrate the applicability of the present piezoelectric shell element. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solution of a coupled problem of thermopiezoelectricity based on a geometrically exact shell element

Based on a 7-parameter shell model, a numerical algorithm has been developed for solving a coupled problem of thermoelectroelasticity for a laminated piezoelectric shell subjected to a thermoelectromechanical loading. As unknowns, six tangential and transverse displacements of outer surfaces and the transverse displacement of shell midsurface are chosen. This choice provides a possibility of utilizing the complete 3D constitutive equations of thermopiezoelectricity. A geometrically exact 3D hybrid piezoelectric shell element is formulated by using nonconventional analytical integration. With the help of this finite element, solutions of coupled problems of thermoelectroelasticity for laminated plates and shells with segmented and distributed piezoelectric sensors and actuators are obtained.     

A Coupled FE-BE Analysis of Smart Lightweight Structures for Active Noise and Vibration Control

Over the past years much research and development has been done in the area of active control in order to improve the acoustical and vibrational properties of thin–walled lightweight structures. An efficient technique for actively reducing the structural vibration and sound radiation is the application of smart structures. In smart structures piezoelectric materials are often used as actuators and sensors. The design of smart structures requires fast and reliable simulation tools. Therefore, the purpose of this paper is to present a coupled finite element–boundary element formulation, which enables the modeling of piezoelectric smart lightweight structures. The paper describes the theoretical background of the coupled approach in which the finite element method (FEM) is applied for the modeling of the passive vibrating shell structure as well as the surface attached piezoelectric actuators and sensors. The boundary element method (BEM) is used to characterize the corresponding sound field. In order to derive a coupled FE–BE formulation additional coupling conditions are introduced at the fluid–structure interface. Since the resulting overall model contains a large number of degrees of freedom, the mode superposition method is employed to reduce the size of the FE submodel. To validate the accuracy of the proposed approach, numerical simulations are carried out in the frequency domain and the results are compared with analytical reference solutions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A simplified formulation of a plate/shell element for geometrically nonlinear analysis of moderately large deflections with small rotations

This paper presents a total Lagrangian formulation of a plate/shell element for geometrically nonlinear analysis with the assumption of moderately large deflections but small rotations. Mallet and Marcal's nonlinear stiffness matrices [N₁] and [N₂] are derived explicitly by the procedure suggested by Rajasekaran and Murray for a four-node parallelogram isoparametric element with bilinear interpolation functions and an orthotropic constitutive relationship. The performance of the element based on this simplified formulation has been tested through a number of examples. It is concluded that this element offers potential for geometrically nonlinear analysis of large-scale practical structures.     

A finite shell element with well balanced approximation functions for piezoelectric coupling

This contribution is concerned with a piezoelectric shell formulation. The present shell element has four nodes and bilinear interpolation functions. The nodal degrees of freedom are displacements, rotations and the electric potential on top and bottom of the shell. A 3D–material law is incorporated. In case of bending dominated problems incompatible approximation functions of the electrical and mechanical fields cause incorrect results. This effect occurs in standard element formulations, where the mechanical and electrical degrees of freedom are approximated with lowest order interpolation functions. In order to overcome this problem a mixed multi–field variational approach is introduced. It allows for approximations of the electric field and the strains independent of the bilinear interpolation functions. A quadratic approach for the shear strains and the electric field is proposed through the shell thickness. This leads to well balanced approximation functions regarding coupling of electrical and mechanical fields. A numerical example illustrates the more precise results in contrast to standard elements. (© 2008 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)     

Linear and Nonlinear Finite Element Analysis of Active Composite Laminates

The paper presents a finite element concept for analysis of thin-walled active structures featuring fiber reinforced composite laminate as a passive structural material. The structure is rendered active by embedding piezoelectric material as a multifunctional material. A 9-node degenerated shell element based on the first order shear deformation theory is developed as a modelling tool capable of predicting the general behavior of the structure for controlling purposes. The von-Kármán type nonlinearities are considered. The solution strategy of the geometrical nonlinear analysis is based on the incremental approach using the updated Lagrangian formulation. Some numerical results are given to demonstrate the behavior of the element. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A flexible approach to the simulation of hybrid multibody systems

The present works deals with the incorporation of both flexible beam and shell structures into the realm of flexible multibody dynamics. Geometrically exact beam formulations based on classical Simo-Reissner kinematics are suitable for modelling beam-type flexible components in the context of finite-deformation multibody dynamics. So geometrically exact shell formulations are based on Reissner-Mindlin kinematics. In [2], a flexible framework for dealing with flexible structural elements in a multibody context is described. A specific isoparametric finite element discretization of a shell formulation leads to semi-discrete equations of motions assuming the form of differential-algebraic equations (DAEs). A compatible isoparametric formulation of beams has already been developed in [1]. The uniform DAE framework makes possible the incorporation of alternative finite element formulations. In addition to that, various time-stepping schemes such as energy-momentum methods or variational integrators can be applied. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Buckling and free vibration analysis of high speed rotating carbon nanotube reinforced cylindrical piezoelectric shell

In this paper, buckling and free vibration behavior of a piezoelectric rotating cylindrical carbon nanotube-reinforced (CNTRC) shell is investigated. Both cases of uniform distribution (UD) and FG distribution patterns of reinforcements are studied. The accuracy of the presented model is verified with previous studies and also with those obtained by Navier analytical method. The novelty of this study is investigating the effects of critical voltage and CNT reinforcement as well as satisfying various boundary conditions implemented on the piezoelectric rotating cylindrical CNTRC shell. The governing equations and boundary conditions have been developed using Hamilton's principle and are solved with the aid of Navier and generalized differential quadrature (GDQ) methods. In this research, the buckling phenomena in the piezoelectric rotating cylindrical CNTRC shell occur as the natural frequency is equal to zero. The results show that, various types of CNT reinforcement, length to radius ratio, external voltage, angular velocity, initial hoop tension and boundary conditions play important roles on critical voltage and natural frequency of piezoelectric rotating cylindrical CNTRC shell.     

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)     

Approximation of the dual problem for error estimation in inelastic problems

In numerical simulations with the finite element method the dependency on the mesh – and for time-dependent problems on the time discretization – arises. Adaptive refinements in space (and time) based on goal-oriented error estimation [1] become more and more popular for finite element analyses to balance computational effort and accuracy of the solution. The introduction of a goal quantity of interest defines a dual problem which has to be solved to estimate the error with respect to it. Often such procedures are based on a space-time Galerkin framework for instationary problems [2]. Discretization results in systems of equations in which the unknowns are nodal values. Contrary, in current finite element implementations for path-dependent problems some quantities storing information about the path-dependence are located at the integration points of the finite elements [3], e.g. plastic strains etc. In this contribution we propose an approach – similar to [4] for sensitivity analysis – for the approximation of the dual problem which mainly maintains the structure of current finite element implementations for path-dependent problems. Here, the dual problem is introduced after discretization. A numerical example illustrates the approach. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A nonlinear extensible 4-node shell element based on continuum theory and assumed strain interpolations

Summary A quadrilateral continuum-basedC ⁰ shell element is presented, which relies on extensible director kinematics and incorporates unmodified three-dimensional constitutive models. The shell element is developed from the nonlinear enhanced assumed strain (EAS) method advocated by Sino & Armero [1] and formulated in curvilinear coordinates. Here, the EAS-expansion of the material displacement gradient leads to the local interpretation of enhanced covariant base vectors that are superposed on the compatible covariant base vectors. Two expansions of the enhanced covariant base vectors are given: first an extension of the underlying single extensible shell kinematic and second an improvement of the membrane part of the bilinear element. Furthermore, two assumed strain modifications of the compatible covariant strains are introduced such that the element performs well even in the case of very thin shells. This paper is dedicated to the memory of Juan C. Simo In honour of Professor Juan Simo who had significant collaboration with our institute and contributed important insights to our research work. This paper was solicited by the editors to be part of a volume dedicated to the memory of Juan Simo.     

An error indicator for parameter identification with stabilized mixed tetrahedrals

The identification of parameters in constitutive laws considering inhomogeneous states of stress and strain is realized by iteratively minimizing a least squares functional. In each iterative step of this optimization problem a finite element analysis is carried out which results in a significant higher numerical cost than a single finite element analysis. Consequently, an efficient discretization is required to keep the numerical cost low. To address this problem an adaptive mesh refinement is considered which is based on a posteriori error indicators [1] leading to refinements appropriate to the parameter identification problem. The goal is to apply the error indicators to the finite element method for tetrahedral elements of low order which are preferable for adaptive mesh refinements and in addition reduce computational effort. Additional stabilization terms in the element formulation [4, 6] reduce volume locking effects making the elements suitable for (nearly) incompressible material behavior. Numerical examples illustrate the progress on this work. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dual-mixed variational formulation and hp finite element method for axisymmetric shell problems in elastodynamics

A new dimensionally reduced axisymmetric shell model is presented briefly for modeling time-dependent problems. This is based on the extended version of the three-field dual-mixed variational formulation of elastostatics [1, 2] to linear elastodynamics, the independent fields of which are the non-symmetric stress tensor, the displacement- and the rotation vector. An important property of the related shell model is that the classical kinematical hypotheses regarding the deformation of the normal to the shell middle surface are not used, i.e., unmodified three-dimensional constitutive equations are applied. The computational performance of the new h- and p-version axisymmetric shell finite elements is tested through a representative cylindrical shell problems. The development presented in this paper has been motivated by the fact that efficient dual-mixed hp plate and shell finite elements were managed previously to be developed for elastostatics by [1-5]. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)