Formulation of a thin shell finite element with continuity C 0 and convected material frame notion

 This paper presents a formulation for a new family of thin shell finite elements. The element is formulated by using a convected material frame notion which offers an interesting framework to take into account large transformations. Bending behaviour is calculated from the Love–Kirchhoff assumptions and from a finite difference technique between adjacent elements. We therefore called this element SFE for semi-finite-element. This method allows us to keep C ⁰ continuity without introducing other variables than the 3 classical displacements, which reduces computational time. In this paper, a full formulation of this element is described more precisely. It takes into account the coupling effect between both membrane and bending behaviour. Various sample solutions that illustrate the effectiveness of the element in linear and nonlinear analysis are presented, with some sheet metal forming examples. Received 10 January 2001     

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 quasi-incompressible and quasi-inextensible element formulation for transversely isotropic materials

The contribution presents a new finite element formulation for quasi-inextensible and quasi-incompressible finite hyperelastic behavior of transeversely isotropic materials and addresses its computational aspects. The material formulation is presented in purely Eulerian setting and based on the additive decomposition of the free energy function into isotropic and anisotropic parts, where the former is further decomposed into isochoric and volumetric parts. For the quasi-incompressible response, the Q1P0 element formulation is outlined briefly, where the pressure-type Lagrange multiplier and its conjugate enter the variational formulation as an extended set of variables. Using the similar argumentation, an extended Hu-Washizu–type mixed variational potential is introduced, where the volume averaged fiber stretch and fiber stress are additional field variables. Within this context, the resulting Euler-Lagrange equations and the element formulation resulting from the extended variational principle are derived. The numerical implementation exploits the underlying variational structure, leading to a canonical symmetric structure. The efficiency of the proposed approached is demonstrated through representative boundary value problems. The superiority of the proposed element formulation over the standard Q1 and Q1P0 element formulation is studied through convergence analyses. The proposed finite element formulation is modular and exhibits very robust performance for fiber reinforced elastomers in the inextensibility limit.     

The moment redistribution effect of the C0 structural elements formulation

A new formulation was proposed recently for the removal of the shear and membrane locking mechanisms from the C⁰ structural elements. The performance of the new formulation was shown to be excellent in many cases of beam, plate and shell element applications, completely eliminating all locking problems. However, this formulation has its own problems. The potential introduction of softening effects (yielding softer models) and a rotational zero energy mode describe the problematic behaviour of the new formulation in cases of C⁰ plate and shell element applications. Analysis of this behaviour reveals some interesting aspects of the classical finite element formulation and allows for a better insight into the overall behaviour of the C⁰ structural elements. As a result of the present analysis, a modification of new formulation, remedying its problematic behaviour, will appear soon.     

Isogeometric FEM‐BEM coupled structural‐acoustic analysis of shells using subdivision surfaces

We introduce a coupled finite and boundary element formulation for acoustic scattering analysis over thin‐shell structures. A triangular Loop subdivision surface discretisation is used for both geometry and analysis fields. The Kirchhoff‐Love shell equation is discretised with the finite element method and the Helmholtz equation for the acoustic field with the boundary element method. The use of the boundary element formulation allows the elegant handling of infinite domains and precludes the need for volumetric meshing. In the present work, the subdivision control meshes for the shell displacements and the acoustic pressures have the same resolution. The corresponding smooth subdivision basis functions have the C¹ continuity property required for the Kirchhoff‐Love formulation and are highly efficient for the acoustic field computations. We verify the proposed isogeometric formulation through a closed‐form solution of acoustic scattering over a thin‐shell sphere. Furthermore, we demonstrate the ability of the proposed approach to handle complex geometries with arbitrary topology that provides an integrated isogeometric design and analysis workflow for coupled structural‐acoustic analysis of shells.     

Alternate hybrid stress finite element models

Alternate hybrid stress finite element models in which the internal equilibrium equations are satisfied on the average only, while the equilibrium equations along the interelement boundaries and the static boundary conditions are adhered to exactly a priori, are developed. The variational principle and the corresponding finite element formulation, which allows the standard direct stiffness method of structural analysis to be used, are discussed. Triangular elements for a moderately thick plate and a doubly-curved shallow thin shell are developed. Kinematic displacement modes, convergence criteria and bounds for the direct flexibility-influence coefficient are examined.     

Analysis of the second Piola-Kirchhoff stress in nonlinear thick and thin structures using exact geometry SaS solid-shell elements

This paper presents the finite rotation exact geometry four-node solid-shell element using the sampling surfaces (SaS) method. The SaS formulation is based on choosing inside the shell N SaS parallel to the middle surface to introduce the displacements of these surfaces as basic shell unknowns. Such choice of unknowns with the consequent use of Lagrange polynomials of degree N–1 in the through-thickness distributions of displacements, strains and stresses leads to a robust higher-order shell formulation. The SaS are located at only Chebyshev polynomial nodes that make possible to minimize uniformly the error due to Lagrange interpolation. The proposed hybrid-mixed four-node solid-shell element is based on the Hu-Washizu variational principle and is completely free of shear and membrane locking. The tangent stiffness matrix is evaluated through efficient 3D analytical integration and its explicit form is given. As a result, the proposed exact geometry solid-shell element exhibits a superior performance in the case of coarse meshes and allows the use of load increments, which are much larger than possible with existing displacement-based solid-shell elements.     

Finite elements for an optimal control problem governed by a parabolic equation

A stationary variational formulation of the necessary conditions for optimality is derived for an optimal control problem governed by a parabolic equation and mixed boundary conditions. Then a mixed finite element model with elements in space and time is utilized to solve a simple numerical example whose analytical and finite difference solutions are given elsewhere. Numerical results show that the proposed method with C^° continuity elements constitutes a powerful numerical technique for solution of optimal control problems of distributed parameter systems.     

Convergence of curved shell elements based on assumed covariant strain interpolations

A C⁰ 9-node shell element based on assumed interpolations of covariant strain components defined with respect to the element natural co-ordinate system has recently been proposed. In this formulation, the covariant strains are obtained directly from the Cartesian strains by tensor transformation without any need to compute laminar co-ordinate based strains. In the present work, the interpolated covariant strains used in this element are analysed to determine their satisfaction of the basic requirements for successful strain interpolation. These basic requirements are stated as invariance to rigid body motions and ability to represent constant and linear strain states. In the finite element formulation, the weak form of momentum balance is expressed in terms of covariant strains and contravariant stresses. The corresponding elasticity tensor is a function of the components of the metric tensor associated with the element natural co-ordinate system. The invariance properties of the metric tensor in the context of the finite element approximation are also discussed.     

Finite element Lagrange multiplier formulation for size-dependent skew-symmetric couple-stress planar elasticity

We develop a variational principle based on recent advances in couple-stress theory and the introduction of an engineering mean curvature vector as energy conjugate to the couple stresses. This new variational formulation provides a base for developing a couple-stress finite element approach. By considering the total potential energy functional to be not only a function of displacement, but of an independent rotation as well, we avoid the necessity to maintain C¹ continuity in the finite element method that we develop here. The result is a mixed formulation, which uses Lagrange multipliers to constrain the rotation field to be compatible with the displacement field. Interestingly, this formulation has the noteworthy advantage that the Lagrange multipliers can be shown to be equal to the skew-symmetric part of the force-stress, which otherwise would be cumbersome to calculate. Creating a new consistent couple-stress finite element formulation from this variational principle is then a matter of discretizing the variational statement and using appropriate mixed isoparametric elements to represent the domain of interest. Finally, problems of a hole in a plate with finite dimensions, the planar deformation of a ring, and the transverse deflection of a cantilever are explored using this finite element formulation to show some of the interesting effects of couple stress. Where possible, results are compared to existing solutions to validate the formulation developed here.     

An energy–momentum consistent method for transient simulations with mixed finite elements developed in the framework of geometrically exact shells

In this paper, a mixed variational formulation for the development of energy–momentum consistent (EMC) time‐stepping schemes is proposed. The approach accommodates mixed finite elements based on a Hu–Washizu‐type variational formulation in terms of displacements, Green–Lagrangian strains, and conjugated stresses. The proposed discretization in time of the mixed variational formulation under consideration yields an EMC scheme in a natural way. The newly developed methodology is applied to a high‐performance mixed shell finite element. The previously observed robustness of the mixed finite element formulation in equilibrium iterations extends to the transient regime because of the EMC discretization in time. Copyright © 2016 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.     

A higher-order facet quadrilateral composite shell element

A C⁰ finite element formulation of flat faceted element based on a higher-order displacement model is presented for the analysis of general, thin-to-thick, fibre reinforced composite laminated plates and shells. This theory incorporates a realistic non-linear variation of displacements through the shell thickness, and eliminates the use of shear correction coefficients. The discrete element chosen is a nine-noded quadrilateral with five and nine degrees of freedom per node. A comparison of results is also made with the 2-D thin classical and 3-D exact analytical results, and finite element solutions with 9-noded first-order element. © 1997 John Wiley & Sons, Ltd.     

Non‐linear exact geometry 12‐node solid‐shell element with three translational degrees of freedom per node

This paper presents the finite rotation exact geometry (EG) 12‐node solid‐shell element with 36 displacement degrees of freedom. The term ‘EG’ reflects the fact that coefficients of the first and second fundamental forms of the reference surface and Christoffel symbols are taken exactly at each element node. The finite element formulation developed is based on the 9‐parameter shell model by employing a new concept of sampling surfaces (S‐surfaces) inside the shell body. We introduce three S‐surfaces, namely, bottom, middle and top, and choose nine displacements of these surfaces as fundamental shell unknowns. Such choice allows one to represent the finite rotation higher order EG solid‐shell element formulation in a very compact form and to derive the strain–displacement relationships, which are objective, that is, invariant under arbitrarily large rigid‐body shell motions in convected curvilinear coordinates. The tangent stiffness matrix is evaluated by using 3D analytical integration and the explicit presentation of this matrix is given. The latter is unusual for the non‐linear EG shell element formulation. Copyright © 2011 John Wiley & Sons, Ltd.     

Simple and efficient shear flexible two-node arch/beam and four-node cylindrical shell/plate finite elements

Several simple and accurate C° two-node arch/beam and four-node cylindrical shell/plate finite elements are presented in this paper. The formulation used here is based on the refined theory of thick cylindrical shells and the quasi-conforming element technique. Unlike most C° elements, the element stiffness matrix presented here is given explicitly. In spite of their simplicity, these C° finite elements posseses linear bending strains and are free from the deficiencies existing in curved C° elements such as shear and membrane locking, spurious kinematic modes and numerical ill-conditioning. These finite elements are valid not only for thick/thin beams and plates, but also for arches/straight beams and cylindrical shells/plates. Furthermore, these C° elements can automatically reduce to the corresponding C¹ beam and plate elements and give the C° beam element obtained by the reduced integration as a special case. Several numerical examples indicate that the simple two-node arch/beam and four-node cylindrical shell/plate elements given in this paper are superior to the existing C° elements with the same element degrees of freedom. Only the formulation of the rectangular cylindrical shell and plate element is presented in this paper. The formulation of an arbitrarily quadrilateral plate element will be presented in a follow-up paper³².     

FREE VIBRATION ANALYSIS OF KIRCHHOFF PLATES RESTING ON ELASTIC FOUNDATION BY MIXED FINITE ELEMENT FORMULATION BASED ON GÂTEAUX DIFFERENTIAL

The main objective of the present work is to give the systematic way for derivation of Kirchhoff plate-elastic foundation interaction by mixed-type formulation using the Gâteaux differential instead of well-known variational principles of Hellinger–Reissner and Hu–Washizu. Foundation is a Pasternak foundation, and as a special case if shear layer is neglected, it converges to Winkler foundation in the formulation. Uniform variation of the thickness of the plate is also included into the mixed finite element formulation of the plate element PLTVE4 which is an isoparametric C⁰ class conforming element discretization. In the dynamic analysis, the problem reduces to solution of the standard eigenvalue problem and the mixed element is based upon a consistent mass matrix formulation. The element has four nodes and at each node transverse displacement two bending and one torsional moment is the basic unknowns. Proper geometric and dynamic boundary conditions corresponding to the plate and the foundation is given by the functional. Performance of the element for bending and free vibration analysis is verified with a good accuracy on the numerical examples and analytical solutions present in the literature. © 1997 by John Wiley & Sons, Ltd.     

A mixed-variable continuously deforming finite element method for parabolic evolution problems. Part III: Numerical implementation and computational results

In Part I of this paper,¹ the conceptual framework of a rate variational least squares formulation of a continuously deforming mixed-variable finite element method was presented for solving a single evolution equation. In Part II² a system of ordinary differential equations with respect to time was derived for solving a system of three coupled evolution equations by the deforming grid mixed-variable least squares rate variational finite element method. The system of evolution equations describes the coupled heat flow, fluid flow and trace species transport in porous media under conditions when the flow velocities and constituent phase transitions induce sharp fronts in the solution domain. In this paper, we present the method we have adopted to integrate with respect to time the resulting spatially discretized system of non-linear ordinary differential equations. Next, we present computational results obtained using the code in which this deforming mixed finite element method was implemented. Because several features of the formulation are novel and have not been previously attempted, the problems were selected to exercise these features with the objective of demonstrating that the formulation is correct and that the numerical procedures adopted converge to the correct solutions.     

Vibration of axisymmetric composite piezoelectric shells coupled with internal fluid

This paper presents the theoretical and finite element formulations of piezoelectric composite shells of revolution filled with compressible fluid. The originality of this work lies (i) in the development of a variational formulation for the fully coupled fluid/piezoelectric structure system, and (ii) in the finite element implementation of an inexpensive and accurate axisymmetric adaptive laminated conical shell element. Various modal results are presented in order to validate and illustrate the efficiency of the proposed fluid–structure finite element formulation. Copyright © 2007 John Wiley & Sons, Ltd.     

Approximation of Cahn–Hilliard diffuse interface models using parallel adaptive mesh refinement and coarsening with C1 elements

A variational formulation and C¹ finite element scheme with adaptive mesh refinement and coarsening are developed for phase‐separation processes described by the Cahn–Hilliard diffuse interface model of transport in a mixture or alloy. The adaptive scheme is guided by a Laplacian jump indicator based on the corresponding term arising from the weak formulation of the fourth‐order non‐linear problem, and is implemented in a parallel solution framework. It is then applied to resolve complex evolving interfacial solution behavior for 2D and 3D simulations of the classic spinodal decomposition problem from a random initial mixture and to other phase‐transformation applications of interest. Simulation results and adaptive performance are discussed. The scheme permits efficient, robust multiscale resolution and interface characterization. Copyright © 2008 John Wiley & Sons, Ltd.     

Exact geometry four-node solid-shell element for stress analysis of functionally graded shell structures via advanced SaS formulation

Abstract

The exact geometry four-node solid-shell element formulation using the sampling surfaces (SaS) method is developed. The SaS formulation is based on choosing inside the shell N not equally spaced SaS parallel to the middle surface in order to introduce the displacements of these surfaces as basic shell unknowns. Such choice of unknowns with the use of Lagrange basis polynomials of degree N???1 in the through-thickness interpolations of displacements, strains, stresses and material properties leads to a very compact form of the SaS shell formulation. The SaS are located at Chebyshev polynomial nodes that make possible to minimize uniformly the error due to Lagrange interpolation. To implement efficient 3D analytical integration, the extended assumed natural strain method is employed. As a result, the proposed hybrid-mixed solid-shell element exhibits a superior performance in the case of coarse meshes. To circumvent shear and membrane locking, the assumed stress and strain approximations are utilized in the framework of the mixed Hu-Washizu variational formulation. It can be recommended for the 3D stress analysis of thick and thin doubly-curved functionally graded shells because the SaS formulation with only Chebyshev polynomial nodes allows the obtaining of numerical solutions, which asymptotically approach the 3D solutions of elasticity as the number of SaS tends to infinity.