An accuracy study for a class of rectangular isoparametric finite elements

A method is reported whereby accuracy losses, which result from the distortion of rectangular isoparametric elements, are greatly reduced or virtually eliminated. Rectangular elements subjected to either skew or curved edge distortion are considered and accuracy loss is judged by virtue of solving a cantilever beam test problem. A 16-node element with four appropriately placed internal nodes is developed and comparisons between this element and its 12-node precursor are presented. It is demonstrated that the accuracy loss is eliminated in the case of skew distortions and greatly reduced for curved edge distortion without resorting to reduced integration procedures. This development holds promise for the many applications where extremely distorted elements are essential to the problem solution.     

Cubic isoparametric rolling/traveling finite elements

In this paper, the development and formulation of a new cubic isoparametric rolling/travelling finite element is described. The element can be used to simulate the dyanamic response of steadily rolling/traveling viscoelastic structures without the necessity of numerical time integration of the governing equations. Due to its higher order shape function, the element can more accurately simulate both the inertia and viscoelastic effects associated with rolling/traveling structures than currently available lower order moving elements. To illustrate the capabilities of the new cubic element, comparisons with exact and lower order generated results are presented.     

Anisotropic interpolation error estimates for isoparametric quadrilateral finite elements

Anisotropic local interpolation error estimates are derived for quadrilateral and hexahedral Lagrangian finite elements with straight edges. These elements are allowed to have diameters with different asymptotic behaviour in different space directions. The case of affine elements (parallel-epipeds) with arbitrarily high degree of the shape functions is considered first. Then, a careful examination of the multi-linear map leads to estimates for certain classes of more general, isoparametric elements. As an application, the Galerkin finite element method for a reaction diffusion problem in a polygonal domain is considered. The boundary layers are resolved using anisotropic trapezoidal elements.     

The equation of the curved edge for isoparametric cubic finite elements

We derive the equation of the actual curve that results when a curved edge is approximated using an isopaiametric cubic finite element. This implied curve depends only on the parameters of the nodes associated with the curved side and does not depend on the shape of the element or on the basis functions used. By choosing a special set of basis functions for a nine-parameter cubic triangle we obtain an isoparametric transformation with a simple form that lends itself to elementary analysis. For triangular finite elements, both the Lagrange and Hermite interpolants are considered. A special case is described which permits an important class of cubic curves to be exactly matched very simply by an appropriate choice of mid-side nodes.     

Applications of isoparametric three-dimensional hybrid-stress finite elements with least-order stress fields

This paper investigates the factors relating to element stability, coordinate invariance and optimality in 8- and 20-noded three-dimensional brick elements in the context of hybrid-stress formulations with compatible boundary displacements and both a priori and a posteriori equilibrated assumed stresses. Symmetry group theory is used to guarantee the essential non-orthogonality of the stress and strain fields, resulting in a set of least-order selections of stable invariant stress polynomials. The performance of these elements is examined in numerical examples providing a broad range of analytical stress distributions, and the results are favourably compared to those of the displacement formulation and a stress-function based complete stress approach with regard to displacements, stresses, sampling, convergence and distortion sensitivity.     

Coordinate-free isoparametric elements

Formulations for general solid elements are presented which do not rely explicitly on an arbitrary global coordinate system, and which are independent of the dimension of the embedding space. Abstract data types corresponding to points, vectors and tensors replace matrix representations, allowing the coordinate-free nature of the derivation to be carried over into the implementation. The formulations are oriented primarily toward applications involving general geometric and material nonlinearities, iterative solution algorithms and element-by-element implementations, but traditional linear applications are considered as well.     

Affine approximation of isoparametric elements

The use of rectangular isoparametric elements in finite element analysis of second-order boundary-value problems requires evaluating integrals of rational polynomial functions. Gaussian quadrature formulas are currently the most popular method of obtaining approximations to the exact integrals. A new method is described in which the isoparametric finite element function spaces are approximated. The resulting integrals can be evaluated exactly, avoiding the computational expense of the Gaussian quadrature schemes, particularly the 27 point formula used in three-dimensional elements.     

Integration techniques for isoparametric and higher order bases on finite elements with a curved side

Efficient integration techniques are developed for a class of integrals over finite elements bounded by two straight sides and a parabolic arc. The techniques can be used to speed up the evaluation of the element matrices for both high order transformation bases and for isoparametric bases.     

A simple error estimator for size and distortion of 2D isoparametric finite elements

The shape of elements in the finite element analysis may be the most important of many factors which induce a discretizing error. In particular, the efficiency of adaptive refinement analysis depends on the shape of the elements, and so estimating the quality of element shape is requisite during the adaptive analysis being performed. Unfortunately, most posterior error estimates can not evaluate the shape error of an element, so that some difficulties remain in the application of an adaptive analysis. For this purpose, an error estimator which can separately evaluate size error and distortion error from Zienkiewicz-Zhu's error estimator is presented for bilinear and quadratic isoparametric finite elements. As deduced from the results of numerical experiments, the suggeted estimator gives a reasonable evaluation of error due to element shape as well as discretizing error.     

Basis for isoparametric stress elements

The family of the so-called ‘isoparametric strain (displacement) elements’ is restricted to membranes and solids. The reason for this restriction has led to the development of a new family based on stress assumptions; these elements will be referred to as ‘isoparametric stress elements’. This family contains plates and solids but no membranes. The omission of a particular element in each family is consistent with the plate-membrane analogies. The basic flexibility matrix of an isoparametric stress element is singular since the zero stress state is directly included. The rank technique is adopted to automatically extract the zero stress modes such that the element can be completely interchangeable between any finite element system. The theory for stress assumed isoparametric “quadrilateral” plate bending elements with curved boundaries is given. A brief presentation of the theory for isoparametric stress solid elements is also included.     

Doubly-curved variable-thickness isoparametric heterogeneous finite element

This paper describes an element streamlined for the analysis of doubly-curved, variable-thickness structural components and illustrates its effective application to vibration and static problems. The element is isoparametric, doubly-curved, thin-shell and triangular with variable thickness and accounts for anisotropic, inhomogeneous elastic material behavior. The element has six nodes (three corner and three mid-side) with five degrees-of-freedom (DOF) per node—three translations and two rotations. Quadratic isoparametric interpolation polynomials are used to express the element geometry and displacement variables in terms of corresponding nodal variables.     

Mixed isoparametric elements for saint-venant torsion

Mixed isoparametric elements are presented for the Saint-Venant torsion problem of laminated and anisotropic bars. Both triangular and quadrilateral elements are considered. The “generalized” element stiffness matrix is obtained by using a modified form of the Hellinger-Reissner mixed variational principle. Group-theoretic techniques are used in conjunction with computerized symbolic integration to obtain analytic expressions for the stiffness coefficients. The accuracy of the mixed isoparametric elements developed is demonstrated by means of numerical examples, and their advantages over commonly used stress and displacement elements are discussed.     

On the accuracy of nodal stress computation in plane elasticity using finite volumes and finite elements

This paper addresses accuracy issues of nodal stresses in both finite elements and finite volumes. Strategies to evaluate nodal stresses for finite elements are also briefly discussed. Emphasis is placed on the application of finite volumes to solid mechanics and an overview of recent advancements is presented. Numerical simulation of test problems shows that accuracy of nodal stresses evaluated using finite elements is strongly dependent on the recovery strategy. In all cases, stress recovery at boundary nodal points renders large errors for all non-patch schemes. Contrastingly, finite volumes provide an accurate computation of nodal stresses, including at domain boundaries.     

A simple isoparametric three-node shell finite element

A three-node isoparametric shell finite element including membrane and bending effects is proposed. The element is based on the degenerated solid approach and uses an assumed strain method to avoid shear locking. An intermediate convected covariant frame is used in order to construct the modified shear strain interpolation matrix. Validation tests show that shear locking is avoided and that a reduced integration procedure can be used without any loss of accuracy which is useful for the numerical efficiency.     

Design sensitivity analysis with isoparametric shell elements

This paper deals with design sensitivity calculation by the direct differentiation method for isoparametric curved shell elements. Sensitivity parameters include geometric variables which influence the size and the shape of a structure, as well as the shell thickness. The influence of design variables, therefore, may be separated into two distinct contributions. The parametric mapping within an element, as well as the influence of geometric variables on the orientation of an element in space, is accounted for by the sensitivity calculation of geometric variables, and efficient formulations of sensitivity calculation are derived for the element stiffness, the geometric stiffness and the mass matrices. The methods presented here are applied to the sensitivity calculations of displacement, stress, buckling stress and natural frequency of typical basic examples such as a square plate and a cylindrical shell. The numerical results are compared with the theoretical solutions and finite difference values.     

Nonlinear shell analysis via mixed isoparametric elements

Mixed isoparametric elements are presented for the geometrically nonlinear analysis of laminated composite shells. The analytical formulation is based on a form of the nonlinear shallow shell theory with the effects of shear deformation, material anisotropy and bending-extensional coupling included. The fundamental unknowns consist of the 13 stress resultants and generalized displacements of the shell. The generalized stiffness matrix is obtained by using a modified form of the Hellinger-Reissner mixed variational principle. Both triangular and quadrilateral elements are considered. The accuracy of the mixed isoparametric elements developed is demonstrated by means of numerical examples and their advantages over commonly-used displacement elements are discussed. Also, computational procedures are presented for the efficient evaluation of the elemental matrices and for overcoming the difficulties associated with the large, sparse system of equations of the mixed models thus making them competitive with displacement models.     

A general isoparametric finite element program SDRC SUPERB

SDRC SUPERB is a general purpose finite element program that performs linear static, dynamic and steady state heat conduction analyses of structures made of isotropic and/or orthotropic elastic materials having temperature dependent properties. The finite element library of SUPERB contains isoparametric plane stress, plane strain, flat plate, curved shell, solid type curved shell and solid elements in addition to conventional beam and spring elements. Linear, quadratic and cubic interpolation functions are available for all isoparametric elements. Independent parameters such as displacements and temperatures are obtained from SUPERB using the stiffness method of analysis. The remaining dependent parameters, such as stresses and strains, are evaluated at element gauss points and extrapolated to nodal locations. Averaged values are given as final output. The graphic capabilities of SUPERB consists of geometry and distorted geometry plotting, and stress, strain and temperature contouring. Contours are plotted at user defined cutting planes for solids and at top, middle or bottom surfaces for plate and shell types of structures.In the first part of this paper, the program capabilities of SUPERB are summarized. Extrapolation techniques used for determining dependent nodal parameters and for contour plotting are explained in the second part of the paper. Behavior of standard, wedge and transition type isoparametric elements and the effect of interpolation function orders on accuracy are discussed in the third part. The results of several illustrative problems are included.     

Mixed isoparametric finite element models of laminated composite shells

Mixed shear-flexible isoparametric elements are presented for the stress and free vibration analysis of laminated composite shallow shells. Both triangular and quadrilateral elements are considered. The “generalized” element stiffness, consistent mass, and consistent load coefficients are obtained by using a modified form of the Hellinger-Reissner mixed variational principle. Group-theoretic techniques are used in conjunction with computerized symbolic integration to obtain analytic expressions for the stiffness, mass and load coefficients. A procedure is outlined for efficiently handling the resulting system of algebraic equations.The accuracy of the mixed isoparametric elements developed is demonstrated by means of numerical examples, and their advantages over commonly used displacement elements are discussed.     

Evaluation of integrals for a ten-node isoparametric tetrahedral finite element

The use of isoparametric finite elemts in solving three-dimensional problems typically requires the numerical evaluation of a large number of integrals over individual element domains. The evaluation of these integrals by numerical quadrature, which is the traditional approach, can be computationally expensive. For certain problems the present study provides a more efficient method for evaluation of the needed integrals. For these problems some or all of the desired integrals can be evaluated as linear combinations of basic integrals whose integrands are either (i) products of shape (interpolation) functions or (ii) a derivative of a shape function times a product of one or more shape functions. Basic integrals of these two types (when written in terms of local coordinate systems) have integrands which are polynomial both in the variables of integration and in the nodal coordinates and, thus, can be expressed as linear combinations (with rational number coefficients) of a set of polynomial functions of the nodal coordinates. Group theoretic techniques can be employed to select appropriate sets of polynomial functions for use in these expansions and to reduce substantially the number of basic integrals which need to be explicitly evaluated.

The details for the approach have been worked out for a ten-node isoparametric tetrahedral element through the use of MACSYMA, a computer system for algebraic manipulation. The symmetry group for this element has order 24. The basic integrals of types (i) and (ii) are expressed as linear combinations of 20 and 26 terms, respectively. The special case of a straight-edged tetrahedral element with mid-edge nodes is also discussed.     

Direct evaluation of reaction forces and moments using isoparametric elements

The use of isoparametric finite elements for plates, shells, solids and other structures is by now widespread. The quadratic family of elements, in general, gives the optimum results. Very good displacements are obtained and these are used to evaluate stresses (or stress resultants). When it comes to reactions at the boundaries, the generalized nodal forces are often treated as useless since it is usually not obvious how to relate them to the distributed boundary forces and moments. They usually show sharp variation and even reversal of sign between neighboring nodes. In this paper it is shown how to derive the distributed reaction forces and moments directly from the corresponding generalized forces. Several numerical examples are given to show the excellent agreement with exact and other known solutions.