A six-node finite element for plate bending

A six-node plate bending element has been developed by employing mixed formulation based on a modified Hellinger–Reissner principle and the Reissner–Mindlin plate bending theory. The numerical result indicates that, among the types of assumed independent transverse shear strains considered, a combination of 2α version with either 5α version or 6α version is free of spurious kinematic modes and leads to accurate and reliable solutions even for very thin plates.     

An accurate shear-deformable six-node triangular plate element for laminated composite structures

A six-node triangle plate/shell element is developed for the analysis of laminated composite structures. This model is formulated using Hamilton's principle along with a first-order (Reissner/Mindlin) shear deformation theory. The element is based upon an isoparametric representation along with an interdependent interpolation strategy; bicubic polynomials for the transverse displacement and biquadratic polynomials for the element geometry, in-plane displacements and rotations. The resulting element, which is evaluated using exact numerical integration, has correct rank and is free of shear ‘locking’. Numerical results are presented that validate the new element and prove its outstanding convergence capabilities in comparison to existing triangular elements using standardized test problems (elastic eigenvalue analysis, patch test, static simply supported square-plate solutions) and experimentally measured vibration data of cantilevered isotropic and composite plates.     

Hybrid-Trefftz displacement element for poroelastic media

The elastodynamic response of saturated poroelastic media is modelled approximating independently the solid and seepage displacements in the domain and the force and pressure components on the boundary of the element. The domain and boundary approximation bases are used to enforce on average the dynamic equilibrium and the displacement continuity conditions, respectively. The resulting solving system is Hermitian, except for the damping term, and its coefficients are defined by boundary integral expressions as a Trefftz basis is used to set up the domain approximation. This basis is taken from the solution set of the governing differential equation and models the free-field elastodynamic response of the medium. This option justifies the relatively high levels of performance that are illustrated with the time domain analysis of unbounded domains.     

Efficient finite element analysis of models comprised of higher order triangular elements

This paper introduces an efficient method for the finite element analysis of models comprised of higher order triangular elements. The presented method is based on the force method and benefits graph theoretical transformations. For this purpose, minimal subgraphs of predefined special patterns are selected. Self-equilibrating systems are then constructed on these subgraphs leading to sparse and banded null basis. Finally, well-structured flexibility matrices are formed for efficient finite element analysis.     

Exact finite element solutions to the Helmholtz equation

For one-, two- and three-dimensional co-ordinate systems finite element matrices for the wave or Helmholtz equation are used to produce a single difference equation holding at any point of a regular mesh. Under homogeneous Dirichlet or Neumann boundary conditions, these equations are solved exactly. The eigenfunctions are the discrete form of sine or cosine functions and the eigenvalues are shown to be in error by a term of + O(h²ⁿ) where n is the order of the polynomial approximation of the wave function. The solutions provide the means of testing computer programs against the exact solutions and allow comparison with other difference schemes.     

Strain based triangular finite element for plate bending analysis

ABSTRACT

In this work, the formulation of a new triangular finite element is presented for static and free vibration of plate bending. The developed element which contains the three essential external degrees of freedom at each of the three corner nodes is based on the Reissner/Mindlin theory and the strain-based approach. This element is based on the linear variation of the three bending strains and constant transverse shear strains. The present element performances are evaluated through several tests related to moderated thick and thin plates with various shapes where it is found to be numerically more efficient than the bilinear element.     

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.     

Construction of triangular finite element universal matrices

The various ‘universal’ matrices from which finite element matrices for triangular elements are assembled in many electromagnetics and acoustics problems, can all be derived from a basic set of three fundamental matrices. These represent, respectively, the metric of the linear manifold spanned by the triangle interpolation polynominals, the finite differentiation operator on that same manifold, and a product-embedding operator for the corresponding manifold for interpolation polynomials one order higher. Two of these have already been tabulated and published; the required method for computing the third is given in this paper, along with tables of low-order matrices.     

Hybrid-Trefftz finite element model for antisymmetric laminated composite plates using a high order shear deformation theory

International Journal of Mechanics and Materials in Design - A novel hybrid-Trefftz finite element (HTFE) has been developed for the static analysis of thick and thin antisymmetric cross-ply and...     

A refined triangular plate bending finite element

The derivation of the stiffness matrix for a refined, fully compatible triangular plate bending finite element is presented. The Kirchhoff plate bending theory is assumed. Six parameters or degrees of freedom are introduced at each of the three corner nodes resulting in an 18 degree of freedom element. This refined element is found to give better results for displacements and particularly for internal moments than any plate bending element, regardless of shape, previously reported in the literature.     

A Galerkin least-squares finite element method for the two-dimensional Helmholtz equation

In this paper a Galerkin least-squares (GLS) finite element method, in which residuals in least-squares form are added to the standard Galerkin variational equation, is developed to solve the Helmholtz equation in two dimensions. An important feature of GLS methods is the introduction of a local mesh parameter that may be designed to provide accurate solutions with relatively coarse meshes. Previous work has accomplished this for the one-dimensional Helmholtz equation using dispersion analysis. In this paper, the selection of the GLS mesh parameter for two dimensions is considered, and leads to elements that exhibit improved phase accuracy. For any given direction of wave propagation, an optimal GLS mesh parameter is determined using two-dimensional Fourier analysis. In general problems, the direction of wave propagation will not be known a priori. In this case, an optimal GLS parameter is found which reduces phase error for all possible wave vector orientations over elements. The optimal GLS parameters are derived for both consistent and lumped mass approximations. Several numerical examples are given and the results compared with those obtained from the Galerkin method. The extension of GLS to higher-order quadratic interpolations is also presented.     

A shear-flexible triangular finite element model for laminated composite plates

Formulation and numerical evaluation of a shear-flexible triangular laminated composite plate finite element is presented in this paper. The element has three nodes at its vertices, and displacements and rotations along with their first derivatives have been chosen as nodal degrees-of-freedom. Computation of element matrices is highly simplified by employing a shape function subroutine, and an optimal numerical integration scheme has been used to improve the performance. The element has satisfactory rate of convergence and acceptable accuracy with mesh refinement for thick as well as thin plates of both homogeneous isotropic and laminated anisotropic materials. The numerical studies also suggest that reliable prediction of the behaviour of laminated composite plates necessitates the use of higher order shear-flexible finite element models, and the proposed finite element appears to have some advantages over available elements.     

A cubic triangular finite element for flat plates with shear

This paper presents the development of a straightforward displacement type triangular finite element for bending of a flat plate with the inclusion of transverse (or lateral) shear effects. The element has twenty two degrees of freedom consisting of ten for the lateral displacement of the midplane and six for rotations of the normal to the undeformed midplane of the plate. The latter are taken as independent of the slopes of the deformed midplane in order to include deformation due to transverse shear. The element is fully conforming and may be orthotropic. At interelement boundaries, the element matches adjacent elements both with respect to lateral displacement of the midplane and the rotations of the normal. The result is an efficient ‘linear moment’ triangular element but with transverse shear deformation included. Numerical computations for a number of examples are presented. The results show the element to be more flexible than most other finite element models and agree closely with those from a numerical solution of the three dimensional elasticity equations. The results also converge to those from thin plate theory when the thickness to length ratio becomes small or when the transverse shear moduli are artificially increased.     

A conforming triangular finite element plate bending solution

A conforming plate bending solution using simple polynomial deflection functions of third-degree inside each triangular element is presented. In order to avoid normal slope discontinuities along the sides of the elements, the plate displacement parameters are subjected to ‘slope continuity conditions’ acting as constraints to the minimum potential energy problem. This is then solved by the classical method of Lagrange introducing multipliers as new auxiliary variables. If a special variational formulation of the problem is used, it can be shown that the Lagrangean multipliers are generalized stress parameters. The suggested solution is therefore basically a ‘mixed’ solution, the unknown variables of the problem being both displacement and stress parameters. Several numerical results are presented.     

A residual‐based finite element method for the Helmholtz equation

A new residual‐based finite element method for the scalar Helmholtz equation is developed. This method is obtained from the Galerkin approximation by appending terms that are proportional to residuals on element interiors and inter‐element boundaries. The inclusion of residuals on inter‐element boundaries distinguishes this method from the well‐known Galerkin least‐squares method and is crucial to the resulting accuracy of this method. In two dimensions and for regular bilinear quadrilateral finite elements, it is shown via a dispersion analysis that this method has minimal phase error. Numerical experiments are conducted to verify this claim as well as test and compare the performance of this method on unstructured meshes with other methods. It is found that even for unstructured meshes this method retains a high level of accuracy. Copyright © 2000 John Wiley & Sons, Ltd.     

An assumed stress hybrid curvilinear triangular finite element for plate bending

An assumed stress hybrid curvilinear triangular finite element is described which is based upon the Kirchhoff theory of plate bending. The derivation extends the assumed stress hybrid technique to curvilinear boundaries where the twelve connectors are related to those of an equilibrium rectilinear element and to Semiloof. The solution process demands only first derivatives of the shape functions. The element is subjected to various patch tests for constant bending, e.g. where the central element is in close approximation to a circle. All tests are passed for stress couples and vertex displacements, but values of the remaining connectors do not resemble exact results. Patch tests for rigid-body movements are passed exactly in every respect.     

A general degree semihybrid triangular compatible finite element formulation for Kirchhoff plates

We revisit compatible finite element formulations for Kirchhoff plates and propose a new general degree hybridized approach that strictly imposes C¹ continuity. These new elements are triangular and based on nodal polynomial approximation functions that only use displacement and rotation degrees of freedom for assembly, and thereby “nearly” impose C¹ continuity. This condition is then strictly enforced by adding appropriately chosen hybrid constraints and the corresponding Lagrange multipliers. Unlike all other existing approaches, this formulation allows for the definition of elements of arbitrary degree considering a single polynomial basis for each element, without using degrees of freedom associated with second-order derivatives. The convergence is compared with that of alternative approaches in terms of numbers of elements and degrees of freedom.     

Global-local finite element models for calculating fracture parameters

The finite element iterative method (FEIM) is applied to two types of problems for the evaluation of fracture parameters in laminated composite plates. The FEIM is first used to evaluate the order of power-type singularities and the angular variation in the displacement components near the crack tip at an interface between two dissimilar materials. Next, a numerically enriched global-local finite element is formulated that incorporates the asymptotic singular behavior as calculated by the FEIM in the assumed form of the displacements for calculating stress intensity factors. Examples of both approaches yield results that agree very well with existing analytic solutions.     

Oscillatory crack tip triangular elements for finite element analysis of interface cracks

In this paper a special crack tip element has been developed in which displacements and stresses have the same behaviour as those of bi‐material interface cracks with open tips. The element degenerates into a traditional triangular quarter point element in cases of homogeneous cracks. An isoparametric co‐ordinate system (ρ, t) is defined in this study, and numerical techniques using these co‐ordinates to evaluate Jacobian matrices, shape function derivatives, and element stiffness matrices are developed. Also, equations calculating the complex stress intensity factor using displacements are obtained in this study. Numerical results are in good agreement with known analytical solutions in two examples. Copyright © 2003 John Wiley & Sons, Ltd.