Hermite infinite elements and graded quadratic B-spline finite elements

Quadratic B-spline finite elements are defined for a graded mesh. Hermite infinite elements are proposed to extend the applicability of these finite elements to unbounded regions. Test problems used to compare this technique with published procedures show that the quadratic B-spline finite element solution has, as expected, lower error bounds than a linear element solution. These experiments also demonstrate that the Hermite infinite elements used to close the B-spline finite element arrays lead to error norms comparable in size with other infinite element formulations. The generation of solitary waves in a semi-infinite shallow channel by boundary forcing is modelled by the Korteweg-de Vries equation using an array of graded elements closed by a zero pole infinite element. The resulting simulation of solitary wave motion across a non-uniform mesh confirms existing work and illustrates the effectiveness of the present formulation.     

Three-dimensional hybrid-stress isoparametric quadratic displacement elements

Two 20-node quadratic displacement three-dimensional isoparametric elements are developed based on the hybrid-stress model. The elements differ in the stress interpolation used. In one case, the stress polynomials are selected to correspond approximately to the strain polynomials obtained from the displacement field, and a 57β stress field results. In the other element, complete cubic polynomials are used which are forced to satisfy the equilibrium equations and stress compatibility equations, and a 69β stress field results. Both elements possess correct rank, but only the 69β element is invariant. Results obtained using these two elements, and the corresponding 20-node assumed-displacement element, are compared and the 69β element is shown to be the better element. The 2 × 2 × 2 Gauss stations are also verified to be the optimal sampling points for these elements.     

A critical analysis of quadratic beam finite elements

A new methodology of evaluation of C⁰ beam elements is presented. It is shown that, knowing the stiffness matrix of an arbitrary type of element, it is possible to create equivalent equilibrium conditions expressed in the form of one difference equation for a regular beam discretized by these elements. The study of the convergence of one difference equation gives an interpretation of the source of troubles occurring in low-order bending elements which is more convincing than the usually applied consideration of the conditioning of element stiffness matrices. A careful examination of quadratic Mindlin elements provides a very clear explanation of the shear locking essence in the Timoshenko beam. The presented method enables one to identify errors that appear also in the reduced integrated or constrained elements. For each type of analysed quadratic element an adequate difference equation is derived and compared with the exact one. Based on this comparison a simple method of corrections is proposed that completely eliminates the errors associated with the application of C⁰ bending elements.     

Reduced modified quadratures for quadratic membrane finite elements

Reduced integration is frequently used in evaluating the element stiffness matrix of quadratically interpolated finite elements. Typical examples are the serendipity (Q8) and Lagrangian (Q9) membrane finite elements, for which a reduced 2 × 2 Gauss–Legendre integration rule is frequently used, as opposed to full 3 × 3 Gauss–Legendre integration. This ‘softens’ these element, thereby increasing accuracy, albeit at the introduction of spurious zero energy modes on the element level. This is in general not considered problematic for the ‘hourglass’ mode common to Q8 and Q9 elements, since this spurious mode is non‐communicable. The remaining two zero energy modes occurring in the Q9 element are indeed communicable. However, in topology optimization for instance, conditions may arise where the non‐communicable spurious mode associated with the elements becomes activated. To effectively suppress these modes altogether in elements employing quadratic interpolation fields, two modified quadratures are employed herein. For the Q8 and Q9 membrane elements, the respective rules are a five and an eight point rule. As compared to fully integrated elements, the new rules enhance element accuracy due to the introduction of soft, higher‐order deformation modes. A number of standard test problems reveal that element accuracy remains comparable to that of the under‐integrated counterparts. Copyright © 2004 John Wiley & Sons, Ltd.     

A quadrature rule for conforming quadratic crack tip elements

A simple quadrature rule that precisely integrates products of singular √r and quadratic shape function gradients for a two-dimensional six node triangle and a three-dimensional fifteen node prism is derived.     

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.     

Displacement oscillation in plane quadratic isoparametric elements in orthotropic situations

Long 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.     

Parallel processing of quadratic boundary elements for axisymmetric elasto-static problems

The concern of this paper is on improving the computational efficiency of boundary element methods (BEM) through the development of parallel algorithms for use on massively parallel machines. The application is on the axisymmetric elasto-static problems with quadratic boundary elements. Different ways of parallel approaches are discussed and a parallel approach suited to the BEM numerical process is developed. Numerical results from both the parallel algorithm and a serial algorithm are given in the paper to illustrate the efficiency of the parallel approach.     

Improved Y-domain magnetic film memory elements

Improved versions of a Y-domain memory element are presented. These improvements include element shape, detector configuration, element material composition, and element-substrate interface considerations. It is shown that the element can be optimized for use in a solid-state random-access memory. Using a shortened element with 60 degrees head and tail angles and an offset detector produces an output signal which approaches the ideal case.<>     

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.     

A global mapping method for improving the performance of quadratic elements

A simple method is proposed in this paper for improving the performance (accuracy and convergence) of quadratic elements. In this the physical domain of the problem, as well as the constitutive relationships, are transformed globally by implementing a suitable co-ordinate transformation, and the problem is then solved in the mapped (transformed) domain using finite elements of the mapped space following the processes of the conventional FEM. Results show that the method significantly improves the performance of quadratic elements in both two and three dimensions. The mathematical implications of mapping in this way indicate that the method can also be applied for improved performance to higher-order elements if required.     

The boundary contour method for two-dimensional linear elasticity with quadratic boundary elements

This paper presents a further development of the Boundary Contour Method (BCM) for two-dimensional linear elasticity. The new developments are: (a) explicit use of the rigid body motion solution to regularize the BCM and avoid computation of the corner tensor, (b) quadratic boundary elements compared to linear elements in previous work and (c) evaluation of stresses both inside and on the boundary of a body. This method allows boundary stress computations at regular points (i.e. at points where the boundary is locally smooth) inside boundary elements without the need of any special algorithms for the numerical evaluation of hypersingular integrals. Numerical solutions for illustrative examples are compared with analytical ones. The numerical results are uniformly accurate.     

On quadratic isoparametric transition elements for a crack normal to a bimaterial interface

The quadratic isoparametric crack‐tip elements proposed by R. E. Abdi and G. Valentin (Computers and Structures 33, 241–248) are reconsidered and a simpler method for calculating the optimal position of the side nodes proposed. Quadratic isoparametric transition elements for an r^λ−1 (0<λ<1) strain singularity are formulated. The effects of these transition elements on the accuracy of the calculated stress intensity factors are shown numerically for a crack normal to and terminating at a bimaterial interface. Finally, layered transition elements are formulated for this case and their effects studied numerically. Copyright © 1999 John Wiley & Sons, Ltd.     

Consistent multiscale analysis of heterogeneous thin plates with smoothed quadratic Hermite triangular elements

A consistent multiscale formulation is presented for the bending analysis of heterogeneous thin plate structures containing three dimensional reinforcements with in-plane periodicity. A multiscale asymptotic expansion of the displacement field is proposed to represent the in-plane periodicity, in which the microscopic and macroscopic thickness coordinates are set to be identical. This multiscale displacement expansion yields a local three dimensional unit cell problem and a global homogenized thin plate problem. The local unit cell problem is discretized with the tri-linear hexahedral elements to extract the homogenized material properties. The characteristic macroscopic deformation modes corresponding to the in-plane membrane deformations and out of plane bending deformations are discussed in detail. Thereafter the homogenized material properties are employed for the analysis of global homogenized thin plate with a smoothed quadratic Hermite triangular element formulation. The quadratic Hermite triangular element provides a complete C¹ approximation that is very desirable for thin plate modeling. Meanwhile, it corresponds to the constant strain triangle element and is able to reproduce a simple piecewise constant curvature field. Thus a unified numerical implementation for thin plate analysis can be conveniently realized using the triangular elements with discretization flexibility. The curvature smoothing operation is further introduced to improve the accuracy of the quadratic Hermite triangular element. The effectiveness of the proposed methodology is demonstrated through numerical examples.     

Improved numerical integration of thick shell finite elements

A quadratic thick shell element derived from a three-dimensional isoparametric element was first introduced by Ahmad and co-workers in 1968. This element was noted, however, to be relatively inefficient in representing bending deformations in thin shell or thin plate applications. The present paper outlines a selective integration scheme for evaluating the stiffness matrix of the element, in which each component of the strain energy is evaluated separately using a different Gaussian integration grid for each contribution. By this procedure, the excessive bending stiffness of the element, which results from the use of me quadratic interpolation functions, is avoided. The improved performance of this element, as compared with the original thick shell element, is demonstrated by analyses of a variety of thin and thick shell problems.

1 Editors' note: A similar development was outlined by O. C. Zienkiewicz and co-workers in lnt. J. num. Meth. Engng, 3 , 275–290 (1971). Some important details differ between the two papers which are thus complementary.

     

Improved conditioning of infinite elements for exterior acoustics

An optimized version of the so‐called mapped wave envelope elements, also known as Astley–Leis elements, is introduced. These elements extend to infinity in one dimension and therefore provide an approach to the simulation of exterior acoustical problems in both frequency and time domains. Their formulation is improved significantly through the proper choice of polynomial bases in the direction of radiation. In particular, certain Jacobi polynomials are identified which behave well with respect to conditioning of the system matrices. As a consequence, the size of the finite element discretization may reduce considerably without any loss of accuracy. In addition, the new polynomial bases lead to superior performance of the infinite elements in conjunction with iterative solvers. Copyright © 2003 John Wiley & Sons, Ltd.