A finite element formulation for piezoelectric shell structures considering geometrical and material non‐linearities

In this paper, we present a non‐linear finite element formulation for piezoelectric shell structures. Based on a mixed multi‐field variational formulation, an electro‐mechanical coupled shell element is developed considering geometrically and materially non‐linear behavior of ferroelectric ceramics. The mixed formulation includes the independent fields of displacements, electric potential, strains, electric field, stresses, and dielectric displacements. Besides the mechanical degrees of freedom, the shell counts only one electrical degree of freedom. This is the difference in the electric potential in the thickness direction of the shell. Incorporating non‐linear kinematic assumptions, structures with large deformations and stability problems can be analyzed. According to a Reissner–Mindlin theory, the shell element accounts for constant transversal shear strains. The formulation incorporates a three‐dimensional transversal isotropic material law, thus the kinematic in the thickness direction of the shell is considered. The normal zero stress condition and the normal zero dielectric displacement condition of shells are enforced by the independent resultant stress and the resultant dielectric displacement fields. Accounting for material non‐linearities, the ferroelectric hysteresis phenomena are considered using the Preisach model. As a special aspect, the formulation includes temperature‐dependent effects and thus the change of the piezoelectric material parameters due to the temperature. This enables the element to describe temperature‐dependent hysteresis curves. Copyright © 2011 John Wiley & Sons, Ltd.     

An algorithm for optimization of non-linear shell structures

An algorithm for optimal design of non-linear shell structures is presented. The algorithm uses numerical optimization techniques and nonlinear finite element analysis to find a minimum weight structure subject to equilibrium conditions, stability constraints and displacement constraints. A barrier transformation is used to treat an apparent non-smoothness arising from posing the stability constraints in terms of the eigenvalues of the Hessian of the potential energy of the structure. A sequential quadratic programming strategy is used to solve the resulting non-linear optimization problem. Matrix sparsity in the constraint Jacobian is exploited because of the large number of variables. The usefulness of the proposed algorithm is demonstrated by minimizing the weight of a number of stiffened thin shell structures.     

An advanced finite element formulation for piezoelectric beam structures

Since many piezoelectric components are thin rod-like structures, a piezoelectric finite beam element can be utilized to analyse a wide range of piezoelectric devices effectively. The mechanical strains and the electric field are coupled by the constitutive relations. Finite element formulations using lower order functions to interpolate mechanical and electrical fields lead to unbalances within the numerical approximation. As a consequence incorrect computational results occur, especially for bending dominated problems. The present contribution proposes a concept to avoid these errors. Therefore, a mixed multi-field variational approach is introduced. The element employs the Timoshenko beam theory and considers strains throughout the width and the thickness enabling to directly use 3D constitutive relations. By means of several numerical examples it is shown that the element formulation allows to analyse piezoelectric beam structures for all typical load cases without parasitically affected results.     

A geometrically non-linear tensorial formulation of a skewed quadrilateral thin shell finite element

A 48-degree-of-freedom (d.o.f.) skewed quadrilateral thin shell finite element, including the effect of geometrical non-linearity, is formulated and appropriate numerical procedures are adopted for the development of an efficient approach for the static and dynamic analysis of general thin shell structures. The element surface is described by a variable-order polynomial in curvilinear co-ordinates. The displacement functions are described by bicubic Hermitian polynomials in curvilinear co-ordinates. The directions of the curvilinear co-ordinates at each nodal point are uniquely defined to coincide with the directions of the boundaries of the element. In the present case of a skewed quadrilateral with non-orthogonal curvilinear coordinates, the coupling terms of the metric tensor and curvature tensor of the surface no longer vanish, such as in the case of orthogonal co-ordinates. The tensor form is used in the setup of the shape functions, geometric derivatives, stiffness matrix and computer code. This allows for the treatment of shells with irregular shapes and variable curvatures. To evaluate the efficiency and accuracy of this formulation, a systematic list of examples is chosen: (i) linear and non-linear static analysis of square and rhombic plates, cylindrical and spherical shells; (ii) linear vibrations of trapezoidal flat and curved plates; (iii) large amplitude vibrations of a rhombic plate. For the square plate and cylindrical and spherical shell, shewed element meshes with various distortion angles are used to study the effect of the distortion angles on the accuracy of the results and to demonstrate the versatility of the present element. All results are compared with alternative available solutions including those obtained using regular rectangular meshes. Pinched thin cylindrical and spherical shells are studied using different skewed meshes and various Gauss integration meshes, and no membrane locking phenomenon is observed.     

On non-linear dynamics of shells: implementation of energy-momentum conserving algorithm for a finite rotation shell model

Continuum and numerical formulations for non-linear dynamics of thin shells are presented in this work. An elastodynamic shell model is developed from the three-dimensional continuum by employing standard assumptions of the first-order shear-deformation theories. Motion of the shell-director is described by a singularity-free formulation based on the rotation vector. Temporal discretization is performed by an implicit, one-step, second-order accurate, time-integration scheme. In this work, an energy and momentum conserving algorithm, which exactly preserves the fundamental constants of the shell motion and guaranties unconditional algorithmic stability, is used. It may be regarded as a modification of the standard mid-point rule. Spatial discretization is based on the four-noded isoparametric element. Particular attention is devoted to the consistent linearization of the weak form of the initial boundary value problem discretized in time and space, in order to achieve a quadratic rate of asymptotic convergence typical for the Newton–Raphson based solution procedures. An unconditionally stable time finite element formulation suitable for the long-term dynamic computations of flexible shell-like structures, which may be undergoing large displacements, large rotations and large motions is therefore obtained. A set of numerical examples is presented to illustrate the present approach and the performance of the isoparametric four-noded shell finite element in conjunction with the implicit energy and momentum conserving time-integration algorithm. © 1998 John Wiley & Sons, Ltd.     

Hybrid stress finite element model for non-linear shell problems

The present paper describes a hybrid stress finite element formulation for geometrically non-linear analysis of thin shell structures. The element properties are derived from an incremental form of Hellinger-Reissner's variational principle in which all quantities are referred to the current configuration of the shell. From this multi-field variational principle, a hybrid stress finite element model is derived using standard matrix notation. Very simple flat triangular and quadrilateral elements are employed in the present study. The resulting non-linear equations are solved by applying the load in finite increments and restoring equilibrium by Newton-Raphson iteratioin. Numerical examples presented in the paper include complete snap-through buckling of cylindrical and spherical shells. It turns out that the present procedure is computationally efficient and accurate for non-linear shell problems of high complexity.     

An investigation of a finite rotation four node assumed strain shell element

The paper presents a shell formulation based on the ‘degenerated solid approach’. The theory employs covariant strains and performs explicit integration through the shell thickness. The rigid body motion is exactly represented. The consistent tangent stiffness matrix is evaluated for the four node quadrilateral. It is shown, in the final part, that this type of element, which distinguishes itself by a very simple and easily understandable theory, gives good answers for linear as well as non-linear applications.     

A finite element formulation with stabilization matrix for geometrically non-linear shells

An assumed strain finite element formulation with a stabilization matrix is developed for analysis of geometrically non-linear problems of isotropic and laminated composite shells. The present formulation utilizes the degenerate solid shell concept and assumes an independent strain as well as displacement. The assumed independent strain field is divided into a lower order part and a higher order part. Subsequently, the lower order part is set equal to the displacement-dependent strain evaluated at the lower order integration points and the remaining higher order part leads to a stabilization matrix. The strains and the determinant of the Jacobian matrix are assumed to vary linearly in the thickness direction. This assumption allows analytical integration through thickness, independent of the number of plies. A nine-node element with a judiciously chosen set of higher order assumed strain field is developed. Numerical tests involving isotropic and composite shells undergoing large deflections demonstrate the validity of the present formulation.     

A mixed finite element formulation for non-linear analysis of plane problems

Based on the incremental non-linear theory of solid bodies and the Hellinger-Reissncr principle, a mixed updated Lagrangian formulation of the large displacement motion of solid bodies is derived, and an associated mixed finite element model is developed. The model contains the displacements and stresses as the nodal degrees of freedom. The model is used for the large deformation elasto-plastic analysis of plane problems. In solving non-linear problems, the Newton-Raphson method with arc-length control is adopted to trace the post-buckling response. The computational steps to calculate the elasto-plastic stress increments at Gauss points in the elasto-plastic analysis by the present mixed model are described in detail. Numerical results are presented and compared with those of the displacement model and existing solutions to show the accuracy of the present mixed model in the large deformation elasto-plastic analysis of plane problems.     

A new efficient mixed formulation for thin shell finite element models

A nine node shell element is developed by a new and more efficient mixed formulation. The new shell element formulation is based on the Hellinger–Reissner principle with independent strain and the concept of a degenerate solid shell. The new formulation is made more efficient in terms of computing time than the conventional mixed formulation by dividing the assumed strain fields into a lower order part and a higher order part. Numerical results demonstrate that the present nine node element is free of locking even for very thin plates and shells and is also kinematically stable. In fact the stiffness matrix associated with the higher order assumed strain plays the role of a stabilization matrix.     

A triangular finite shell element based on a fully nonlinear shell formulation

This work presents a fully nonlinear six-parameter (3 displacements and 3 rotations) shell model for finite deformations together with a triangular shell finite element for the solution of the resulting static boundary value problem. Our approach defines energetically conjugated generalized cross-sectional stresses and strains, incorporating first-order shear deformations for an inextensible shell director (no thickness change). Finite rotations are treated by the Euler–Rodrigues formula in a very convenient way, and alternative parameterizations are also discussed herein. Condensation of the three-dimensional finite strain constitutive equations is performed by applying a mathematically consistent plane stress condition, which does not destroy the symmetry of the linearized weak form. The results are general and can be easily extended to inelastic shells once a stress integration scheme within a time step is at hand. A special displacement-based triangular shell element with 6 nodes is furthermore introduced. The element has a nonconforming linear rotation field and a compatible quadratic interpolation scheme for the displacements. Locking is not observed as the performance of the element is assessed by several numerical examples, which also illustrate the robustness of our formulation. We believe that the combination of reliable triangular shell elements with powerful mesh generators is an excellent tool for nonlinear finite element analysis.Fellowship funding from FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) and CNPq (Conselho Nacional de Pesquisa), together with the material support and stimulating discussions in IBNM (Institut für Baumechanik und Numerische Mechanik), are gratefully acknowledged in this work.     

Mixed finite element formulation for geometrically non-linear analysis of shells of revolution

A Finite Element formulation for the axisymmetrical analysis of rotational shells with geometric non-linearity is developed, using a mixed finite element model of a curved rotational shell type, the two global components of displacement (u_r, u_z) and the two stress couples (M_r, M_tH) being the free and independent unknowns of the problem. The prescribed geometry at every nodal circle comprises the co-ordinates and meridional slope angle and curvature. Newton-Raphson Iteration is used in solving the non-linear system of equations. Circular plates and a spherical cap are used as examples to test the formulation; good results were achieved which are presented graphically in comparison to the analytical solution.     

A consistent finite element formulation for non-linear dynamic analysis of planar beam

A co-rotational finite element formulation for the dynamic analysis of planar Euler beam is presented. Both the internal nodal forces due to deformation and the inertia nodal forces are systematically derived by consistent linearization of the fully geometrically non-linear beam theory using the d'Alembert principle and the virtual work principle. Due to the consideration of the exact kinematics of Euler beam, some velocity coupling terms are obtained in the inertia nodal forces. An incremental-iterative method based on the Newmark direct integration method and the Newton–Raphson method is employed here for the solution of the non-linear dynamic equilibrium equations. Numerical examples are presented to investigate the effect of the velocity coupling terms on the dynamic response of the beam structures.     

Hyperbolic paraboloid shell analysis via mixed finite element formulation

An isoparametric rectangular mixed finite element is developed for the analysis of hypars. The theory of shallow thin hyperbolic paraboloid shells is based on Kirchhoff–Love's hypothesis and a new functional is obtained using the Gâteaux differential. This functional is written in operator form and is shown to be a potential. Proper dynamic and geometric boundary conditions are obtained. Applying variational methods to this functional, the HYP9 finite element matrix is obtained in an explicit form. Since only first-order derivatives occur in the functional, linear shape functions are used and a C⁰ conforming shell element is presented. Variation of the thickness is also included into the formulation without spoiling the simplicity. The formulation is applicable to any boundary and loading condition. The HYP9 element has four nodes with nine Degrees Of Freedom (DOF) per node—three displacements, three inplane forces and two bending, one torsional moment (4 × 9). The performance of this simple, and elegant shell element, is verified by applying it to some test problems existing in the literature. Since the element matrix is obtained explicitly, there is an important save of computer time.     

A free-formulation-based flat shell element for non-linear analysis of thin composite structures

A flat shell element based on the free-formulation finite element concept is developed for analysing geometrically non-linear thin composite shells. A corotational form of the updated Lagrangian formulation is utilized. Numerical results for typical validation problems are presented in order to demonstrate the accuracy and validity of this element. These results are obtained by solving the incremental equilibrium equations through the cylindrical arc-length method.     

Three-dimensional finite element for analysing thin plate/shell structures

A three-dimensional (3-D) hexahedron finite element is presented for the analysis of thin plate/shell structures. The element employs an explicit algebraic definition of six uniform (continuum) strains, six rigid body modes and classical Lagrange-Germain-Kirchhoff thin plate bending modes. Nine additional stiffness factors are used to control higher-order hourglass modes. The element may be used for plate/shell analyses where the flat plate assumptions are appropriate. Also it can easily be adapted to form transition elements to lower order 2-D elements, or to higher-order 3-D continuum elements. The stiffness matrix satisfies the geometric isotropy requirement, passes the patch test, and gives essentially identical response to either applied transverse corner forces or to twisting moments applied on the corner, a requirement of Kirchhoff's corner conditions for a classical thin plate. Several examples are presented to demonstrate the performance of this finite element.     

A variational formulation for finite elasticity with independent rotation and Biot-axial fields

The fundamentals of the geometrically nonlinear mechanics of the three-dimensional elastic continuum are derived, starting from a general variational framework established for the polar model and passing through a constitutive definition of the non-polar medium itself. A constrained variational setting follows, having as unknown vector fields the displacement, the rotation vector and the axial of the Biot stress. It embraces both the rotational equilibrium and the characterization of the rotation as Euler-Lagrange equations. These conditions can then be satisfied in a weak sense within discrete approximations. It is also shown that the classical approach of the non-polar continuum can be accomodated as a particular case of the present formulation. A consistent linearization is then proposed and a simple solid finite element developed to test the computational viability of the formulation. A few examples assess the capability of the element to represent large three-dimensional rotations. Communicated by S. N. Atluri, 2 August 1996     

Direct finite element computation of non-linear modal coupling coefficients for reduced-order shell models

We propose a direct method for computing modal coupling coefficients—due to geometrically nonlinear effects—for thin shells vibrating at large amplitude and discretized by a finite element (FE) procedure. These coupling coefficients arise when considering a discrete expansion of the unknown displacement onto the eigenmodes of the linear operator. The evolution problem is thus projected onto the eigenmodes basis and expressed as an assembly of oscillators with quadratic and cubic nonlinearities. The nonlinear coupling coefficients are directly derived from the FE formulation, with specificities pertaining to the shell elements considered, namely, here elements of the “Mixed Interpolation of Tensorial Components” family. Therefore, the computation of coupling coefficients, combined with an adequate selection of the significant eigenmodes, allows the derivation of effective reduced-order models for computing—with a continuation procedure —the stable and unstable vibratory states of any vibrating shell, up to large amplitudes. The procedure is illustrated on a hyperbolic paraboloid panel. Bifurcation diagrams in free and forced vibrations are obtained. Comparisons with direct time simulations of the full FE model are given. Finally, the computed coefficients are used for a maximal reduction based on asymptotic nonlinear normal modes, and we find that the most important part of the dynamics can be predicted with a single oscillator equation.     

Thick shell and solid finite elements with independent rotation fields

Thick shell and solid elements presented in this work are derived from variational principles employing independent rotation fields. Both elements are built on a special hierarchical interpolation and both possess six degrees of freedom per node. Performance of the elements is evaluated on a set of problems in elastostatics. However, the formulation presented herein is also suitable for transient and non-linear problems.