Towards improving the MITC6 triangular shell element

Effective triangular shell elements are of utmost interest in engineering practice, and the MITC6a element – a 6 node quadratic general shell element of the MITC family – has been shown to significantly reduce the locking phenomena arising in bending dominated behaviours. However, for some specific combinations of midsurface geometry and boundary conditions, the MITC6a element features some non-physical displacement modes with vanishing membrane strain energy. This phenomenon is thoroughly analyzed, and a remedy based on a stabilized bilinear form is proposed. Detailed numerical tests are included and the results demonstrate the good performance of the proposed method both for membrane and bending dominated problems.     

Nonlinear analysis of shells using the MITC formulation

Summary The formulation of general shell elements using the method of mixed interpolation of tensorial components (MITC) is reviewed. In particular three elements that were formulated using the MITC method are examined: the MITC4 and MITC8 that were developed for general nonlinear analysis under the restriction of small strains and the MITC4-TLH that was developed for finite strain elasto-plastic analysis of shells. In memoriam of Juan Carlos Simo.     

Optimal error bounds for the MITC4 plate bending element

Abstract One of the most popular finite element method for Reissner-Mindlin plates is the so-called MITC4 method. Unfortunately, until recently, the best error estimate available in the literature for this element needs a non-optimal and generally unrealistic regularity for the solution. Here we derive optimal error estimates for this well-known low-order finite element method, which are valid for a general family of meshes. A similar result has recently been obtained elsewhere, with the same mesh restrictions. The exposition presented here is totally different.     

On optimal stabilized MITC4 plate bending elements for accurate frequency response analysis

The mixed interpolation technique of the well-established MITC4 quadrilateral plate finite element is combined with shear and generalized least-squares stabilization methods for accurate frequency response analysis. Dispersion analysis is used to determine optimal combinations of stabilization parameters, which, for a given mesh, provide for a three-fold increase in the frequency range over which accurate solutions are obtained, thus allowing for accurate solutions at significantly lower cost. Numerical results for the forced vibration of Reissner–Mindlin plates validate the observations made from the dispersion analysis.     

MITC elements for a classical shell model

The formulation of nine-node mixed-interpolated shell elements based on a classical mathematical shell theory is presented, taking into account some fundamental considerations for the finite element analysis of shells. The elements are based on the mixed interpolation of tensorial components approach (MITC), but the assumed covariant strain fields are applied only for the membrane and shear components. Two different types of elements are considered, depending on whether or not geometric approximations are included in the formulation. The performance of the proposed elements is illustrated with a well-established test problem––the Scordelis-Lo roof.     

Electric potential approximations for an eight node plate finite element

The aim of this work is to develop a computational tool for multilayered piezoelectric plates: a low cost tool, simple to use and very efficient for both convergence velocity and accuracy, without any classical numerical pathologies. In the field of finite elements, two approaches were previously used for the mechanical part, taking into account the transverse shear stress effects and using only five unknown generalized displacements: C⁰ finite element approximation based on first-order shear deformation theories (FSDT) [Polit O, Touratier M, Lory P. A new eight-node quadrilateral shear-bending plate finite element. Int J Numer Meth Eng 1994;37:387–411] and C¹ finite element approximations using a high order shear deformation theory (HSDT) [Polit O, Touratier M. High order triangular sandwich plate finite element for linear and nonlinear analyses. Comput Meth Appl Mech Eng 2000;185:305–24]. In this article, we present the piezoelectric extension of the FSDT eight node plate finite element. The electric potential is approximated using the layerwise approach and an evaluation is proposed in order to assess the best compromise between minimum number of degrees of freedom and maximum efficiency. On one side, two kinds of finite element approximations for the electric potential with respect to the thickness coordinate are presented: a linear variation and a quadratic variation in each layer. On the other side, the in-plane variation can be quadratic or constant on the elementary domain at each interface layer. The use of a constant value reduces the number of unknown electric potentials. Furthermore, at the post-processing level, the transverse shear stresses are deduced using the equilibrium equations.Numerous tests are presented in order to evaluate the capability of these electric potential approximations to give accurate results with respect to piezoelasticity or finite element reference solutions. Finally, an adaptative composite plate is evaluated using the best compromise finite element.     

Finite element analysis of shell structures

Summary A survey of effective finite element formulations for the analysis of shell structures is presented. First, the basic requirements for shell elements are discussed, in which it is emphasized that generality and reliability are most important items. A general displacement-based formulation is then briefly reviewed. This formulation is not effective, but it is used as a starting point for developing a general and effective approach using the mixed interpolation of the tensorial components. The formulation of various MITC elements (that is, elements based on Mixed Interpolation of Tensorial Components) are presented. Theoretical results (applicable to plate analysis) and various numerical results of analyses of plates and shells are summarized. These illustrate some current capabilities and the potential for further finite element developments.     

A large deformation,rotation-free,isogeometric shell

Conventional finite shell element formulations use rotational degrees of freedom to describe the motion of the fiber in the Reissner–Mindlin shear deformable shell theory, resulting in an element with five or six degrees of freedom per node. These additional degrees of freedom are frequently the source of convergence difficulties in implicit structural analyses, and, unless the rotational inertias are scaled, control the time step size in explicit analyses. Structural formulations that are based on only the translational degrees of freedom are therefore attractive. Although rotation-free formulations using C⁰ basis functions are possible, they are complicated in comparison to their C¹ counterparts. A C^k-continuous, k ? 1, NURBS-based isogeometric shell for large deformations formulated without rotational degrees of freedom is presented here. The effect of different choices for defining the shell normal vector is demonstrated using a simple eigenvalue problem, and a simple lifting operator is shown to provide the most accurate solution. Higher order elements are commonly regarded as inefficient for large deformation analyses, but a traditional shell benchmark problem demonstrates the contrary for isogeometric analysis. The rapid convergence of the quadratic element is demonstrated for the NUMISHEET S-rail benchmark metal stamping problem.     

A comparison of the linear,quadratic and cubic mindlin strip elements for the analysis of thick and thin plates

The behaviour of the linear, quadratic and cubic elements of the Mindlin plate strip family for thick and very thin plate analysis is investigated in this paper. Selective integration techniques are used to ensure the good behaviour of the elements when dealing with thin plates. Numerical results showing the convergence and accuracy of the elements for the analysis of plates of a wide range of thicknesses are given. The general performance of the three elements is discussed in detail. In particular, the linear element with a single integration point seems to be the best value strip element for practical purposes.     

A Kirchhoff-mode method for C0 bilinear and serendipity plate elements

A new formulation for the C⁰ quadrilateral and Serendipity plate elements is presented. It is an extension of the mode-decomposition approach that has been successfully applied in evaluating the C⁰ triangular linear plate element. In contrast to the triangular element, the concept of an equivalent discrete Kirchhoff configuration is utilized. The elements are applied to several examples and the performance of these elements is shown to be quite good.     

Isoparametric linear bending element and one-point integration

An isoparametric linear plate bending element is introduced and its use for the analysis of beams, square and rhombic plates is examined. One-point integration for the generation of an element stiffness matrix has been carried out. When compared with available solutions, the agreement of the results have been excellent. Isoparamctric linear elements have been found to be more economical than isoparametric quadratic elements at least for beams and plates with straight edges.     

Measuring convergence of mixed finite element discretizations: an application to shell structures

We consider the problem of assessing the convergence of mixed-formulated finite elements. When displacement-based formulations are considered, convergence measures of finite element solutions to the exact solution of the mathematical problem are well known. However when mixed formulations are considered, there is no well-established method to measure the convergence of the finite element solution. We first review a number of approaches that have been employed and discuss their limitations. After having stated the properties that an ideal error measure would possess, we introduce a new physics-based procedure. The new proposed error measure can be used for many different types of mixed formulations and physical problems. We illustrate its use in an assessment of the performance of the MITC family of shell elements.     

Accurate force evaluation with a simple,bi-linear,plate bending element

A great deal of attention has been given to the development of simple C ° continuous plate and shell elements based on the shear flexible theories for application to thick plates, sandwich or cellular plates and transversely isotropic or laminated plates. After considerable experimentation using unconventional approaches such as reduced integration, selective integration, mixed methods using discontinuous force fields, etc., it has been possible to develop simple displacement-type elements which can be reliably used. The stress recovery at nodes from such elements is often unreliable as the nodes are usually the points where strains or stresses are least accurate in the element domain. Further, nodal values can reflect severe oscillations at some difficult corner or edge conditions. In this paper, we focus attention on the optimal stress recovery from such an element. This is done after an interpretation of the displacement method as a procedure that obtains strains over the finite-element domain in a least-squares accurate fashion. If a shear flexible element is field-consistent, there are optimal locations at which bending moments and shear forces are accurate in a least-squares sense. These points are identified for the present element and used to study stresses in typical plate problems. Another difficulty faced is the rapid variation of twisting moments at free edges and corners of shear flexible plates and its influence on the shear forces at that edge. A related source of difficulty is the distinction made in Kirchhoff theory between shear forces and the effective shear reactions of that theory. The present study is seen to give accurate enough shear force and twisting moment predictions to allow one to draw the severe conclusion that the use of the Kirchhoff shear reaction at edges in classical plate theory is an ambiguous and unnecessary one and can be avoided. The findings confirm a recent suggestion that it may be more appropriate to have three (as introduced originally by Poisson) instead of the two boundary conditions (as modified by Kirchhoff) usually applied on the edge of a thin plate, especially if that edge is unsupported.     

On convergence of nonconforming convex quadrilateral finite elements AGQ6

The 4-node quadrilateral membrane elements AGQ6-I and II are two novel incompatible models formulated by the Quadrilateral Area Coordinate Method (QACM). In this paper, the sufficient conditions for their convergence are established. It is further shown theoretically that the convergence in the energy norm is linear, and the convergence in the L² norm is quadratic provided that certain geometric conditions are met requiring asymptotically parallelograms meshes to be used. The necessity of conditions is also discussed. The results of numerical examples completely confirm the theoretical findings.     

A discrete Kirchoff plate bending element with loof nodes

The majority of existing flat shell finite elements suffer from the deficiencies of displacement incompatibility, singularity when the elements are coplanar at a node, inability to model intersections and low-order membrane strain representation. In this paper, a plate bending element, labeled DKL (for Discrete Kirchoff element with Loof nodes), with the same nodal configuration as a triangular Semiloff plate element, but not formulated through the isoparametric concept is presented. This element when superposed with the linear strain triangle results in a faceted shell element free from the abovementioned deficiencies. Various numerical examples are tested using this plate element so as to demonstrate its reliability, accuracy and convergence characteristics.     

Finite element free vibration analysis of eccentrically stiffened plates

A new finite element model is proposed for free vibration analysis of eccentrically stiffened plates. The formulation allows the placement of any number of arbitrarily oriented stiffeners within a plate element without disturbing their individual properties. A plate-bending element consistent with the Reissner-Mindlin thick plate theory is employed to model the behaviour of the plating. A stiffener element, consistent with the plate element, is introduced to model the contributions of the stiffeners. The applied plate-bending and stiffener elements are based on mixed interpolation of tensorial components (MITC), to avoid spurious shear locking and to guarantee good convergence behaviour. Several numerical examples using both uniform and distorted meshes are given to demonstrate the excellent predictive capability of this approach.     

Triangular element for analysis of perforated plates under inplane and transverse loads

A C⁰-type triangular element formulation in orthogonal curvilinear co-ordinates has been developed, based on assumptions of transverse inextensibility and constant shear angle through thickness for analysis of perforated plates subjected to inplane and transverse loads. The assumed quadratic displacement potential energy approach is utilized in obtaining an element stiffness matrix and consistent load vector, which are numerically integrated. Numerical results have been obtained using a straight-sided triangular version, which behaves like a subparametric element, for stretching and bending analyses of perforated plates.     

Some results on the curved plate bending problem solved with non-conforming finite elements

Convergence properties are studied for two non-conforming finite elements of degree two and three for the plate bending problem, which were introduced by Morley and Fraeijs de Veubeke. Unlike some other non-conforming elements, they turn out to be very suitable for the case of curved boundaries since success in the patchtest is guaranteed without any restrictions on the shape of the plate. Two kind of support conditions are examined: for the clamped plate we prove that for both elements no kind of curved elements are needed to attain the same rates of convergence derived byLascaux & Lesaint for the polygonal case. In the case of the simply supported plate, the conclusion is the same for Morley's element. For Fraeijs de Veubeke's triangle however this is only possible with the use of second degree curved elements. In addition to the above results, an indication is given for the optimal choice of Poisson's coefficient to be used in the variational formulation for clamped plates and numerical examples are shown. Work supported by Catholic University of Rio de Janeiro.     

Evaluation of some multilayered, shear-deformable plate elements

Recently, the author has developed an improved bidimensional transverse shear deformation theory for multilayered anisotropic plates which accounts for piecewise linear distribution across the thickness of the in-plane displacements u and v, and allows the contact conditions at the interfaces between the layers to be satisfied.

Based on this refined theory and on a Mindlin's-type transverse shear deformation plate theory developed by Whitney and Pagano J. appl. Mech. 37, 1031 (1970), several triangular and quadrilateral multilayered anisotropic plate elements which include extension, bending and transverse shear deformation states have been developed by making use of the displacement formulation in conjunction with the principle of virtual work.

In order to show the accuracy and the relative merits of the developed finite elements, results are presented for the sample problems of the bending and free undamped vibrations of a three-layered, symmetric cross-ply square plate that is simply-supported on all edges.

Excepting the conventional triangular and quadrilateral elements with linear shape functions, fast convergence to the respective analytical solutions for the global response (transverse displacements and fundamental flexural frequencies) is observed for all the elements tested. The rectangular finite element developed on the basis of the refined plate theory proposed by the author is also very efficient to model the warpage of the cross-section and to predict accurate values of the flexural stress at the interfaces. The finite elements developed on the basis of the Whitney and Pagano theory fail in this respect.     

A priori error estimation of a four-node Reissner–Mindlin plate element for elasto-dynamics

The explicit finite element method for transient dynamics of linear elasticity by Reissner–Mindlin plate model is introduced. For clamped rectangular plate, the a priori error estimates are derived for the four-node Bathe–Dvorkin element. For fixed thickness, the convergence rates of deflection, rotation, and their velocities, measured both in H¹-norm and L²-norm, can possibly all be optimal under certain conditions. In some cases, the numerical examples show that the convergence rate stays optimal for a certain range of thickness. In other cases, however, the deterioration in rate of convergence and even locking may occur to the velocity terms.