A triangular hybrid equilibrium plate element of general degree

Hybrid stress‐based finite elements with side displacement fields have been used to generate equilibrium models having the property of equilibrium in a strong form. This paper establishes the static and kinematic characteristics of a flat triangular hybrid equilibrium element with both membrane and plate bending actions of general polynomial degree p. The principal characteristics concern the existence of hyperstatic stress fields and spurious kinematic modes. The former are shown to exist for p>3, and their significance to finite element analysis is reviewed. Knowledge of the latter is crucial to the determination of the stability of a mesh of triangular elements, and to the choice of procedure adopted for the solution of the system of equations. Both types of characteristic are dependent on p, and are established as regards their numbers and general algebraic forms. Graphical illustrations of these forms are included in the paper. Copyright © 2005 John Wiley & Sons, Ltd.     

The stability of stars of simplicial hybrid equilibrium finite elements for solid mechanics

In this paper, we define the spurious kinematic modes of hybrid equilibrium 2D and 3D simplicial elements of general degree and present the results of studies on the stability of star patches of such elements. The approach used in these studies is based on first establishing the kinematic properties of a pair of elements that share an interface. This approach is introduced by its application to star patches of hybrid equilibrium triangular plate elements for modelling membrane and plate bending problems and then generalised to 3D continua. It is then shown how the existence of spurious kinematic modes depends on the topological and geometrical properties of a patch, as well as on the degree of the polynomial approximation functions of stress and displacement. Copyright © 2015 John Wiley & Sons, Ltd.     

An efficient assumed strain element model with six DOF per node for geometrically non-linear shells

The present paper describes an assumed strain finite element model with six degrees of freedom per node designed for geometrically non-linear shell analysis. An important feature of the present paper is the discussion on the spurious kinematic modes and the assumed strain field in the geometrically non-linear setting. The kinematics of deformation is described by using vector components in contrast to the conventional formulation which requires the use of trigonometric functions of rotational angles. Accordingly, converged solutions can be obtained for load or displacement increments that are much larger than possible with the conventional formulation with rotational angles. In addition, a detailed study of the spurious kinematic modes and the choice of assumed strain field reveals that the same assumed strain field can be used for both geometrically linear and non-linear cases to alleviate element locking while maintaining kinematic stability. It is strongly recommended that the element models, described in the present paper, be used instead of the conventional shell element models that employ rotational angles.     

Elimination of spurious static modes in meshes of hybrid compatible triangular and tetrahedral finite elements

In the assumed displacement, or primal, hybrid finite element method, the requirements of continuity of displacements across the sides are regarded as constraints, imposed using Lagrange multipliers. In this paper, such a formulation for linear elasticity, in which the polynomial approximation functions are not associated with nodes, is presented. Elements with any number of sides may be easily used to create meshes with irregular vertices, when performing a non‐uniform h‐refinement. Meshes of non‐uniform degree may be easily created, when performing an hp‐refinement. The occurrence of spurious static modes in meshes of triangular elements, when compatibility is strongly enforced, is discussed. An algorithm for the automatic selection, based on the topology of a mesh of triangular elements, of the sides in which to decrease the degree of the approximation functions, in order to eliminate all these spurious modes and preserve compatibility, is presented. A similar discussion is presented for the occurrence of spurious static modes in meshes of tetrahedral elements. An algorithm, based on heuristic criteria, that succeeded in eliminating these spurious modes and preserving compatibility in all the meshes of tetrahedral elements of uniform degree that were tested, is also presented. Copyright © 2001 John Wiley & Sons, Ltd.     

On the modelling of incompressibility in linear and non‐linear elasticity with the master–slave approach

The master–slave approach is adapted to model the kinematic constraints encountered in incompressibility. The method presented here allows us to obtain discrete displacement and pressure fields for arbitrary finite element formulations that have discontinuous pressure interpolations. The resulting displacements satisfy exactly the incompressibility constraints in a weak sense, and are obtained by solving a system of equations with the minimum (independent) degrees of freedom. In linear analysis, the method reproduces the well‐known stability results for inf–sup compliant elements, and permits to compute the pressure modes (physical or spurious) when they exist. By rewriting the equilibrium equations of a hyperelastic material, the method is extended to non‐linear elasticity, while retaining the exact fulfilment of the incompressibility constraints in a weak sense. Problems with analytical solution in two and three dimensions are tested and compared with other solution methods. Copyright © 2007 John Wiley & Sons, Ltd.     

Spurious modes in two-dimensional isoparametric elements

The reasons for the presence of spurious modes in plane isoparametric elements are examined. Eigenvalue and eigenvector analyses of stiffness matrices for single elements and unconstrained and constrained meshes reveal the types of mechanisms and near-mechanisms that can be expected when reduced or selective integration techniques are used with quadrilateral elements in the analysis of plane strain problems.     

A triangular finite element for thin plates and shells

A finite element modelling technique which utilizes a triangular element with 45 degrees-of-freedom and seven-point integration has been tested for analysis of thin plate and shell structures. The element is based on the degenerate solid shell concept and the mixed formulation with assumed independent inplane and transverse shear strains. Numerical result indicates effectiveness of the present modelling technique which features combined use of elements with kinematic modes and those without kinematic modes in an attempt to eliminate both locking and spurious kinematic modes at global structural level.     

Some observations on element performance in isochoric and dilatant plastic flow

The performance of finite elements is scrutinized in isochoric and dilatant/contractant plastic flow. Standard displacement based elements, uniformly and selectively integrated elements, and elements with augmented strain rate fields are considered in plane-strain, axisymmetric and three-dimensional configurations with particular reference to the kinematic constraint imposed by dilatant/contractant plastic flow. It turns out that findings for isochoric deformations do not necessarily carry over to cases with plastic dilatancy or contraction. For the elements with augmented strain rate fields the danger of spurious mechanisms in ideal plasticity is brought out. A particular strategy which augments only the normal strain rates at the expense of a lesser improvement for the bending behaviour does not suffer from the possibility of spurious modes, while preserving the ability to accommodate dilatant/contractant plastic flow. Illustrative examples on plane-strain, axisymmetry and three-dimensional structures are included which support the above findings.     

A stabilization procedure for the quadrilateral plate element with one-point quadrature

A stabilization procedure is developed for controlling the kinematic modes of the four-node, bilinear quadrilateral element when single-point quadrature is used. These kinematic modes manifest themselves by spatial oscillations or singularity of the total stiffness. In this stabilization procedure, additional generalized strains are defined which are activated by the kinematic modes; these generalized modes are furthermore not activated by rigid body motions regardless of the shape of the quadrilateral. By using a scaling law developed in an earlier paper, the stabilization parameters are defined so they do not adversely affect the element's performance. Several problems which are subject to kinematic modes are presented to illustrate the performance of this stabilization procedure for linear problems.     

Restraining approach for the spurious kinematic modes in hybrid equilibrium element

The present paper proposes a rigorous approach for the elimination of spurious kinematic modes in hybrid equilibrium elements, for three well known mesh patches. The approach is based on the identification of the dependent equations in the set of inter-element and boundary equilibrium equations of the sides involved in the spurious kinematic mode. Then the kinematic variables related to the dependent equations are reciprocally constrained and, by application of master slave elimination method, the set of inter-element equilibrium equations is reduced to full rank. The elastic solutions produced by means of the proposed approach verify the homogeneous, the inter-element and the boundary equilibrium equations. Hybrid stress formulation is developed in a rigorous mathematical setting. The results of linear elastic analysis obtained by the proposed approach and by classical displacement based method are compared for some structural examples.     

Determination of coefficients of the crack tip asymptotic field by fractal hybrid finite elements

This paper applies the fractal finite element method (FFEM) together with 9-node Lagrangian hybrid elements to the calculation of linear elastic crack tip fields. An explicit stabilization scheme is employed to suppress the spurious kinematic modes of the sub-integrated Lagrangian element. An extensive convergence study has been conducted to examine the effects of the similarity ratio, reduced integration and the type of elements on the accuracy and stability of the numerical solutions. It is concluded that (i) a similarity ratio close to unity should be used to construct the fractal mesh, (ii) sub-integrated Lagrangian elements with occasionally unstable behaviour should not be used, and (iii) the good accuracy (with differences less than 0.5% with existing available solutions) and stability over a wide range of numerical tests support the use of fractal hybrid finite elements for determining crack tip asymptotic fields.     

A GENERAL FORMULATION OF EQUILIBRIUM MACRO-ELEMENTS WITH CONTROL OF SPURIOUS KINEMATIC MODES: THE EXORCISM OF AN OLD CURSE

This paper illustrates a method whereby a family of robust equilibrium elements can be formulated in a general manner. The effects of spurious kinematic modes, present to some extent in all primitive equilibrium elements, are eliminated by judicious assembly into macro-equilibrium elements. These macroelements are formulated with sufficient generality so as to retain the polynomial degree of the stress field as a variable. Such a family of macro-elements is a new development, and results for polynomials of degree greater than two have not been seen before. The quality of results for macro-equilibrium elements with varying degrees of polynomial is demonstrated by numerical examples.     

Pure equilibrium tetrahedral finite elements for global error estimation by dual analysis

This study presents a general procedure of creating pure equilibrium tetrahedral finite elements for use under the elastostatic hypothesis. These pure equilibrium elements are of the Fraeijs de Veubeke type and the degree of the polynomial approximation functions of their internal stress field is the parameter generating this new elements family. The spurious kinematic modes (SKM), inherent in the equilibrium approach, are eliminated at the element level by converting each tetrahedron into a super‐element defined as an assembly of four tetrahedral primitive elements. A mathematical discussion on the number of SKM of the primitive elements as well as their elimination by the super‐element technique has been carried out. The development of first and second degree elements is presented here in detail and their efficiency in the frame of global error estimation by dual analysis is emphasized by two numerical applications. The main attribute of the error estimation by dual analysis is that it provides an upper bound on the global discretization error. Copyright © 2009 John Wiley & Sons, Ltd.     

On the suppression of zero energy deformation modes

Based on the Hellinger-Reissner principle and the deformation energy due to assumed stresses and displacements, the problem of the kinematic deformation modes in assumed stress hybrid/mixed finite elements has been examined. Basic schemes are developed for the choice of assumed stress terms that will suppress all kinematic deformation modes. Quadrilateral membrane and axisymmetric elements, and three-dimensional hexahedral elements, are used to illustrate the suggested procedure.     

Suppression of spurious intermediate frequency modes in under‐integrated elements by combined stiffness/viscous stabilization

Solutions employing perturbation stiffness or viscous hourglass control with one‐point quadrature finite elements often exhibit spurious modes in the intermediate frequency range. These spurious frequencies are demonstrated in several examples and their origin is explained. Then it is shown that by critically damping the hourglass modes, these spurious mid‐range frequency modes can be suppressed. Estimates of the hourglass frequency and damping coefficients are provided for the plane 4‐node quadrilateral and a 4‐node shell element. Results are presented that show almost complete annihilation of spurious intermediate frequency modes for both linear and non‐linear problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Qualitative errors in laminated composite plate models

The presence of ‘qualitative’ errors, i.e. errors that misrepresent the sign or direction of deformation in a problem, is identified in laminated composite plate elements. The errors are found to be due to the presence of parasitic shear terms introduced when an element is formed using incompatible polynomials. Although these modelling errors can be removed from four-node elements with reduced order Gauss quadrature integration, they cannot be removed from eight-node elements by this method. However, procedures based on a physically interpretable notation are available to remove these errors from individual elements with any number of nodes. The procedures for correcting the laminated composite plate elements can be used to remove parasitic shear and spurious zero energy modes from other types of elements. Three examples are used to demonstrate the developments.     

FINITE ELEMENT SOLUTION OF INCOMPRESSIBLE FLUID–STRUCTURE VIBRATION PROBLEMS

In this paper we solve an eigenvalue problem arising from the computation of the vibrations of a coupled system, incompressible fluid – elastic structure, in absence of external forces. We use displacement variables for both the solid and the fluid but the fluid displacements are written as curls of a stream function. Classical linear triangular finite elements are used for the solid displacements and for the stream function in the fluid. The kinematic transmission conditions at the fluid–solid interface are taken into account in a weak sense by means of a Lagrange multiplier. The method does not present spurious or circulation modes for non-zero frequencies. Numerical results are given for some test cases. © 1997 by John Wiley & Sons, Ltd.     

An assessment of four-noded plate finite elements based on a generalized third-order theory

Plate finite elements based on the generalized third-order theory of Reddy and the first-order shear deformation theory are analysed and compared on the basis of thick and thin plate modelling behaviour, distortion sensitivity, overall accuracy, reliability and efficiency. In particular, several four-noded Reddy-type elements and the nine-noded Lagrangian and heterosis (Mindlin-type) plate elements are analysed to assess their behaviour in bending, vibration and stability of isotropic and laminated composite plates. A four-noded Reddy-type element is identified which is free of all spurious stiffness and zero energy modes, computationally efficient, and suitable for use in any general-purpose finite element program.