Alternative integration formulae for triangular finite elements

Some new formulae are presented for numerical integration over triangular regions. These formulae, which are triangularly symmetric, involve four, six, and seven integration points, respectively. They were derived by determining co-ordinates and weight factors for which polynomial functions of up to a certain degree are integrated exactly. The effectiveness of the formulae is demonstrated by applying them to the integration of five non-polynomial functions, two of which are singular within the region of integration. Results obtained with the new formulae are compared with results obtained with other, comparable formulae. It is seen that the new formulae are more accurate than some existing ones with the same numbers of points.     

On the use of isoparametric finite elements in linear fracture mechanics

Quadratic isoparametric elements which embody the inverse square root singularity are used in the calculation of stress intensity factors of elastic fracture mechanics. Examples of the plane eight noded isoparametric element show that it has the same singularity as other special crack tip elements, and still includes the constant strain and rigid body motion modes. Application to three-dimensional analysis is also explored. Stress intensity factors are calculated for mechanical and thermal loads for a number of plane strain and three-dimensional problems.     

A drafting program for isoparametric finite elements

Considerable effort has been invested lately in the application of isoparametric finite elements for numerical solution of a wide range of applied mechanics problems. In fact, several general purpose computer programs are now available which are based upon such finite elements. In the present paper, a new application of the isoparametric finite element concept is introduced which significantly extends its usefulness for many practical structural configurations. In this application, final working or architectural drawings of the structure are made from the same (or similar) finite element model as was utilized in a structural integrity analysis. The hardware necessary to produce such drawing, a computer driven plotter or automated drafting machine, is available commercially or through most data centres, and the software concepts required are described herein.     

Direct Gauss quadrature formulae for logarithmic singularities on isoparametric elements

Weakly singular logarithmic integrals occur in two-dimensional (2D) BEM problems when the integration is over an element which includes the current source node. For boundary elements with curved geometry such as the quadratic and cubic element then numerical integration must be used. Recently developed direct Gauss quadrature schemes which implicitly consider the integral to include a sum of singular and non-singular terms appear to be superior to conventional schemes which represent these separate terms explicitly. This paper discusses these direct quadratures and introduces new Gauss schemes which integrate exactly the logarithmic singularities on any one of the 3 nodes of a quadratic element using a single formula. This new quadrature may also be used for linear and constant elements. Similar quadrature formulae for the 4 singularities on a cubic element are also discussed. This new approach is both accurate and simple, reducing the size of the computer program and allowing the use of the same quadrature for several isoparametric types.     

Flexible finite-difference stencils from isoparametric finite elements

A new finite-difference technique for the numerical solution of boundary value problems for partial differential equations in two space variables is described. Isoparametric finite elements are used in a finite-difference context to derive difference approximations to space derivatives on a locally curvilinear grid. The result is a generalization of a classical finite-difference stencil which is ‘flexible’ in that it is adaptable to variable meshes, such as those arising from regions with curved boundaries. Numerical results presented for a test problem (potential flow past a circle) using a 9 × 10 grid agree with the analytic solution to within one per cent.     

Representation of singularities with isoparametric finite elements

The development of a generalized quadrilateral finite element that includes a singular point at a corner node is presented. Inter-element conformability is maintained so that monotone convergence is preserved. The global-local concept of finite elements is used to formulate the complete set of equations. Examples of crack tip singularities are given.     

Complete quadratic isoparametric finite elements in fracture mechanics analysis

For about a decade, each and every researcher who used isoparametric quadratic elements to solve fracture mechanics problems employed only incomplete versions of these elements. Although complete formulations of isoparametric elements are almost as available as incomplete ones in general-purpose finite element computer programs, no attempt to use complete formulations has been seen yet. The purpose of this paper is to show how the complete quadratic isoparametric elements can be employed in the field of fracture mechanics.     

Analysis of free surface flows using isoparametric finite elements

A theory for free surface flow using variable geometry finite elements is numerically modelled using isoparametric finite elements, for the first time. The resulting nonlinear equations are solved using the Newton-Raphson method. Because a functional is used as the basis of the formulation, the resulting equations are symmetrical. The realistic problem of flow over a Crump weir is solved for a large range of discharges.     

Behaviour of isoparametric quadrilateral family of Lagrangian fluid finite elements

This paper presents the formulation of both the consistent and inconsistent four‐, eight‐ and nine‐noded isoparametric quadrilateral fluid finite elements that are based on Lagrangian frame of reference. The mesh locking phenomenon due to simultaneous enforcement of twin constraints, namely the incompressibility and irrotationality constraints, is studied in detail. The study shows that the characteristic of the locked fluid elements is that it always generates numerous spurious acoustic (volume change) modes upon the enforcement of rotational constraints. That is, the rotational constraints change the character of certain volume change modes. The study further reinforces the necessity of rotational constraints in not only identifying the spurious pressure modes, but also in reducing the computational effort for determining the eigenvalues and eigenvectors. It is found that all fully integrated inconsistent models exhibit locking behaviour. However, the inconsistent eight‐ and nine‐noded elements, integrated with full integration of volumetric stiffness and one point integration of the rotational stiffness matrices, gives excellent performance, although they do not pass the inf–sup test. The four‐ and nine‐noded consistent models are found to give locking free performance while their eight‐noded counterpart exhibited locking behaviour. The study shows that only consistent nine‐noded element models pass the inf–sup test. The utility of these elements in the coupled fluid–structure interaction problem is also demonstrated. Copyright © 2002 John Wiley & Sons, Ltd.     

Stabilization of the zero-energy modes of under-integrated isoparametric finite elements

The coefficient matrices of isoparametrically distorted finite elements are frequently computed by numerical integration formulas with a lower degree of exactness than required by the degree of the integrand. This may result in a rank deficiency of the element matrices which causes an oscillatory instability in the solution. To avoid this effect the element matrices may be stabilised, i. e. their rank may be raised artificially. A new approach is suggested which approximates exact integration without the computational expense usually connected with it. For this purpose, a new family of integration formulas is introduced which is based on the standard Gauss product formulas extended by derivatives of the integrand. For the stabilization, these derivatives can be approximated very easily. The procedure generates a stabilization matrix which, when added to the under-integrated matrix, produces the correct rank. If e. g. shear locking is a problem, the stabilization matrix may be scaled down. Two- and three-dimensional Lagrange elements in second and fourth order problems and Mindlin plate elements are presented together with the results of computations and numerical experiments.     

The 3‐D elastodynamic boundary element method: semi‐analytical integration for linear isoparametric triangular elements

A semi‐analytical integration scheme is described in this paper which is designed to reduce the errors incurred when integrals with singular integrands are evaluated numerically. This new scheme can be applied to linear triangular elements for use in steady‐state elastodynamic BEM problems and is particularly useful for predicting displacement to high accuracy, close to surfaces for a spectrum of frequencies. The scheme involves the application of Taylor expansions to formulate the integrals into two parts. One part is regular and is evaluated numerically and the other part is singular but sufficiently simple to enable its transforma tion into a line integral. The line integral is solved numerically using Gauss–Legendre quadrature. This approach caters for all the integral types that appear in steady‐state elastodynamic boundary elements but, in particular, no special treatment is required for the evaluation of the Cauchy principal value singular integrals. Numerical tests are performed on a simple test‐problem for which a known analytical solution exists. The results obtained using the semi‐analytical approach are shown to be considerably more accurate than those obtained using standard quadrature methods. Copyright © 1999 John Wiley & Sons, Ltd.     

The performance of isoparametric finite elements in stress intensity factor determination

Analysis of a compact compression specimen used for fracture toughness evaluation of cementitious materials is carried out by the finite element method using isoparametric elements. Both triangular and rectangular elements were used with those surrounding the crack tip being of the quarter point type. Solutions were obtained for different mesh subdivisions and convergenece curves for the stress intensity factor were obtained by several methods based on extrapolation and energy techniques. It is found that monotonic convergence was obtained for all cases considered. Employing uniformly graded rectangular element representations converged solutions for the stress intensity factor (assuming a 1 percent convergence criterion) were obtained by the energy methods using a total of 720 degrees of freedom for solving half the structure.Tests on modified 100 mm cubes with symmetrical notches were conducted to determine the fracture toughness. The fracture toughness was calculated from the stress intensity factor and the maximum load obtained from the tests which were conducted in a stiff Instron testing machine. The fracture toughness is found to be independent of the size of the notch.     

Application of singular quadratic distorted isoparametric elements in linear fracture mechanics

The understanding of the performance of the quarter-point and transition elements is of considerable importance as these singular elements are widely used in linear elastic fracture mechanics (LEFM) analyses. However, a number of issues remain unresolved although numerous investigations into their performance have been conducted. In particular is the question of optimum quarter-point element and transition element size. This study examines several aspects in relation to the size effect by performing a large number of numerical analyses on several standard problems. Interpretation of the numerical results was aided by the use of two concepts, the ‘zones of dominance’ and ‘zones of representation’. This study proposes a means of explaining the errors in stress intensity factor computation through the interaction between both types of zones. Consequently, a number of new and significant observations were made regarding the singular elements' performance. This study concludes with some recommendations for the application of these elements.     

Shifted integration technique in one-dimensional plastic collapse analysis using linear and cubic finite elements

The present study is concerned with the physical explanations of the linear and the cubic finite elements for beams and axisymmetric shells through comparisons of their strain energy approximations with those of the Rigid Bodies-Spring Models which are discrete elements suitable for plastic collapse analysis using the concepts of plastic hinges and hinge lines. The established conditions for the equivalence between these two modellings, which are given as the relations between the locations of the numerical integration points and those of the occurrence of plastic hinges, can be conveniently used in the economical plastic collapse analysis of framed structures and axisymmetric shells where the locations of plastic hinge formations are controlled by the movement of numerical integration points. Some numerical results are shown in order to prove numerically the obtained relations and to verify the validity of the proposed shifting technique of numerical integration points, which is identified as ‘the shifted integration technique’ in the present paper.     

Quadratic isoparametric elements as transition elements

The use of the quadratic isoparametric elements as transition elements is presented. It is shown that good estimates for the stress-intensity factor solution can be obtained over a wide range of the quarter-point element length over the crack-length ratio, provided that the aspect ratio between the transition element and the quarter-point element is held within certain limits.