Quadratic Isoparametric Finite Elements Research Articles

A meshing strategy for a quadratic iso-parametric FEM in cavitation computation in nonlinear elasticity

The approximation properties of a quadratic iso-parametric finite element method for a typical cavitation problem in nonlinear elasticity in two dimensions are analyzed. More precisely, (1) the finite element interpolation errors are established in terms of the mesh parameters; (2) a mesh distribution strategy based on an error equi-distribution principle is given; (3) the convergence of finite element cavity solutions is proved. Numerical experiments show that, in fact, the optimal convergence rate can be achieved by the numerical cavity solutions.

Orientation-preservation conditions on an iso-parametric FEM in cavitation computation

The orientation-preservation condition, i.e., the Jacobian determinant of the deformation gradient $\det \nabla u$ is required to be positive, is a natural physical constraint in elasticity as well as in many other fields. It is well known that the constraint can often cause serious difficulties in both theoretical analysis and numerical computation, especially when the material is subject to large deformations. In this paper, we derive a set of sufficient and necessary conditions for the quadratic iso-parametric finite element interpolation functions of cavity solutions to be orientation preserving on a class of radially symmetric large expansion accommodating triangulations. The result provides a practical quantitative guide for meshing in the neighborhood of a cavity and shows that the orientation-preservation can be achieved with a reasonable number of total degrees of freedom by the quadratic iso-parametric finite element method.

Design variation of thin-walled composite beam cross-section properties

Purpose The purpose of this paper is to present a design sensitivity analysis continuum formulation for the cross-section properties of thin-walled laminated composite beams. These properties are expressed as integrals based on the cross-section geometry, on the warping functions for torsion, on shear bending and shear warping, and on the individual stiffness of the laminates constituting the cross-section. Design/methodology/approach In order to determine its properties, the cross-section geometry is modeled by quadratic isoparametric finite elements. For design sensitivity calculations, the cross-section is modeled throughout design elements to which the element sensitivity equations correspond. Geometrically, the design elements may coincide with the laminates that constitute the cross-section. Findings The developed formulation is based on the concept of adjoint system, which suffers a specific adjoint warping for each of the properties depending on warping. The lamina orientation and the laminate thickness are selected as design variables. Originality/value The developed formulation can be applied in a unified way to open, closed or hybrid cross-sections.

Modeling Analysis of the Wire-Drawing Operation Using the Weighted-Residual Finite Element Method

Numerical analysis of a wire drawing operation to compute the stress distribution along the blank cross-section is presented. The governing equation describing the wire drawing equation is weakened using the Bubnov-Galerkin finite element method to obtain the finite element model. The blank is descritized into a mesh of C0 quadratic isoparametric and C0 cubic finite elements. Stiffness matrices for all elements are obtained using the finite element model which were subsequently assembled by enforcing continuity of the nodal stress. Boundary conditions are applied and the resulting condensed system of equation solved for unknown nodal stresses using Gauss Seidel method. The relative performance of the C0 quadratic and C0 cubic elements are assessed. Parametric analysis is carried out to show the influence of drawing parameters on the stresses generated and drawing load. The analysis was carried out using a Visual Basic.Net program developed by the authors.The results are presented in both graphical and tabular forms.

Finite element analysis of thin-walled composite laminated beams with geometrically nonlinear behavior including warping deformation

A finite element model for structural analysis of composite laminated thin-walled beam structures, with geometrically nonlinear behavior, including torsion warping deformation, is presented. A general continuum formulation considering the updated Lagrangean procedure and a generalized displacement control method, are used to describe the deformation of the structure. The beam cross-section geometry is discretized by quadratic isoparametric finite elements to determine its bending-torsion properties. The structural discretization is performed throughout three-dimensional two-node Hermitean finite beam elements, with seven degrees-of-freedom per node. Several applications are presented, addressing the influence of lamina orientation on the structural behavior.

Initial Failure Analysis of Laminated Composite Plates under Humid Conditions

A quadratic isoparametric finite element formulation based on the first-order shear deformation theory is presented for the initial failure analysis of a laminated composite plate exposed to moist conditions. The moisture concentration causing the failure of composite laminate has been predicted in the present study. An iteration method has been developed for the prediction of moisture concentration for the weakest ply failure of laminated composite plates. Several numerical examples are considered and the results are compared with those available in the published literature. The moisture concentration for the starting of failure has been computed for cross-ply and angle-ply laminations with various boundary conditions, taking into account the moisture independence of the materials properties.

HYGROTHERMAL EFFECTS ON THE DYNAMIC BEHAVIOR OF MULTIPLE DELAMINATED COMPOSITE PLATES AND SHELLS

A quadratic isoparametric finite element formulation based on the first order shear deformation theory is presented for the free vibration and transient response analysis of multiple delaminated doubly curved composite shells subjected to a hygrothermal environment. A simple multiple delamination model developed by the authors earlier is employed to take care of any number/size of delamination located anywhere in the laminate. The analysis takes into account the lamina material properties at elevated moisture concentration and temperature. Newmark's direct integration scheme is used to solve the dynamic equation of equilibrium at every timestep during the transient analysis. Several numerical examples are considered and the results are compared with those available in the literature. The results show a reduction in the fundamental frequency with an increase in the percentage of uniform moisture content as well as temperature for simply supported antisymmetric crossply and angleply laminates for any size of delamination considered. The central dynamic displacements and the stresses at the center of laminates are observed to increase due to the effect of moisture/temperature subjected to the suddenly applied uniform pulse loading

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.

A family of elements with λ singularity

A new finite element family which contains a λ singularity is developed. Three elements are derived; namely three-, four- and five-noded elements which are compatible with linear, quadratic and cubic standard isoparametric finite elements, respectively. The elements are tested on two different examples. In the first example, an edge crack problem is analysed using two different meshes and different integration orders. The second example is a crack perpendicular to the interface problem which is solved for different material properties and, in turn, different singularity order λ. The results of those examples illustrate the efficiency of the proposed elements.

Behaviour of paraboloid of revolution shell using cross-ply and anti-symmetric angle-ply laminates

A generalised formulation for the analysis of a doubly curved laminated composite shell using an eight-noded curved quadratic isoparametric finite element has already been presented. Based on that formulation, the behaviour of the shell characteristics of the paraboloid of revolution shell using eight types of cross-ply laminates and twelve types of anti-symmetric angle-ply laminates are presented in this paper. A detailed study of the variations of the deflections and other shell actions for the above laminates reveals that symmetric cross-ply laminates appear to be superior to anti-symmetric cross-ply laminates while 45/-45 laminates are superior to 30/-30 and 60/-60 laminates. An overall comparison between cross-ply and angle-ply laminates reveals that angle-ply laminates are more efficient than cross-ply laminates. In general, a decrease in the value of the shell actions is observed with an increase in the number of layers.

Finite element analysis of laminated composite paraboloid of revolution shells

A generalized formulation for the doubly curved laminated composite shell is attempted using eight-noded curved quadratic isoparametric finite elements with all three radii of curvature. The formulation is also applied to the isotropic material as a special case. In the present investigation, only the paraboloid of revolution is taken up for computing the deflections and stress resultants. Various parametric studies are carried out and the current results for both isotropic and laminated composite shells are compared with those available in the published literature. The shape functions are obtained from interpolation polynomial and the element stiffness matrices are formed on the basis of macromechanical analysis of laminates using the principle of minimum potential energy.

Two-dimensional off-state modelling of high-voltage semiconductor devices with floating guard rings

The paper presents novel algorithms for the off-state modelling of high-voltage devices with floating guard rings. The computation is based on quadratic isoparametric finite elements which improved accuracy in the determination of electric fields and ionisation integrals. The special boundary conditions for floating rings are considered together with an approach that automatically selects which ones to include in the calculations at a given bias. In addition, an efficient voltage-stepping technique is described for determining the breakdown voltage. Comparisons are made with measured values of floating ring potential and breakdown voltage on a variety of test devices with excellent agreement being found.

Determining invertible two and three dimensional quadratic isoparametric finite element transformations

This paper presents an algorithm determining the invertibility of any planar, triangular quadratic isoparametric finite element transformation. Extensions of the algorithm to three-dimensional isoparametric finite element transformations yield conditions which guarantee invertibility of 10-node tetrahedra and 8-node bricks.

Algorithms for determining invertible two‐ and three‐dimensional quadratic isoparametric finite element transformations

AbstractThis paper presents an algorithm which determines the invertibility of any planar, triangular quadratic isoparametric finite element transformation. Extensions of the algorithm to three‐dimensional isoparametric finite element transformations yield conditions which guarantee invertibility of 10‐node tetrahedra and 8‐node bricks. The mathematical basis for the algorithm focuses on the Jacobian as a continuous function defined over a compact set where the Jacobian attains a maximum and a minimum value. The algorithm then determines whether these values are of opposite sign.

An algorithm for determining invertible quadratic isoparametric finite element transformations

This paper derives an algorithm which determines the invertibility of arbitrary two-dimensional quadratic isoparametric finite element transformations. Theorems verifying the algorithm and guiding the construction of invertible transformations are proven.

Application of finite element analysis techniques for predicting crack propagation in lugs

The plane elastic problem corresponding to a through radial crack emanating from the internal boundary of a symmetrical lug is considered. A pin bearing pressure distribution was developed by utilizing photoelastic test data and differs considerably from the usually assumed uniform or cosine pressure distributions. The stress intensity factors at the crack tip were obtained by using recently derived quadratic isoparametric finite elements which embody the inverse square root singularity. Fatique crack growth tests of 17 aluminium, titanium and steel lugs were utilized to verify stress intensity factor solutions.

Displacement oscillation in plane quadratic isoparametric elements in orthotropic situations

AbstractLong thin test specimens of orthotropic material are modelledby quadratic isoparametric finite elements in plane strain. Displacement oscillations occur down the long free edge of themodel with both reduced (2 × 2) and full (3 × 3) integration, themagnitude of the oscillations being affected by element aspectratio and the elastic constants of the material. The rate ofdiffusion of stress as predicted by the classical shear‐lag theory, being dependent on the material constants, is reflectedin the displacement oscillations near the load in 3 × 3integration. In 2 × 2 integration, however, this effect is swampedby the build‐up of oscillations away from the restraints in alowenergy deformation mode which becomes dominant as the elementaspect ration is increased. These latter voscillations arepredicted by the illconditioning of the structural stiffnessmatrix measured by a solution diagonal decay factor.

Three-Dimensional Versus Axisymmetric Finite-Element Analysis of a Cylindrical Vessel Inlet Nozzle Subject to Internal Pressure—A Comparative Study

This paper presents a comparative study between a three-dimensional and an axisymmetric finite-element analysis of a reactor pressure-vessel inlet nozzle subject to internal pressure. A quarter-symmetric section of the nozzle is modeled with a three-dimensional quadratic isoparametric finite element. This comparative study proves that the axisymmetric analysis is unconservative if based upon common axisymmetric modeling techniques. This inadequacy, for the PWR vessel inlet nozzle studied herein, can be offset by a modification of the modeling techniques, i.e., if the value of the radius of the equivalent spherical vessel is taken as 3.2 instead of, say, 2. The results of the three-dimensional finite-element analysis are also compared with those of a photo-elastic stress analysis and with the stress indices indicated by the ASME Section III Code. These additional comparisons, based upon a continuous distribution of hoop and tangential stress indices in both the transverse and longitudinal planes, shows good agreement between the three-dimensional finite-element and photoelastic analyses. The ASME Section III stress indices are found to be relatively conservative.

Stress Analysis of Adhesive Bonded Tubular Lap Joints

Abstract The stresses in adhesive bonded tubular lap joints, subjected to axial and torsional loads, have been analysed using axisymmetric quadratic isoparametric finite elements. In the axial load case, the results are compared with a previously published closed-form solution and in the torsional case the results are compared with a closed-form solution presented here. The influences on the stress distributions of an adhesive fillet and of partial tapering of the adherends are also investigated, and an extension to the range of validity pf Goland and Reissner's second criterion is proposed.

Further application of quadratic isoparametric finite elements to linear fracture mechanics of plate bending and general shells

Further application of quadratic isoparametric finite elements to linear fracture mechanics of plate bending and general shells