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.     

Subspaces and polynomial factorizations over finite fields

Recently Niederreiter described a new method for factoring polynomials over finite fields. As with the Berlekamp technique, the method requires the construction of a linear subspace whose dimension is precisely the number of irreducible factors of the polynomial being considered. This paper explores the connection between these subspaces and gives a characterization of other subspaces having properties which are similar.     

Primitive elements in finite fields and costas arrays

In [7] Golomb made four conjectures concerning the existence of pairs of primitive elements in finite fields. In this note we resolve each of the conjectures in the affirmative. As a consequence several conjectured classes of Costas arrays do indeed exist.This author would like to thank the National Security Agency for partial support under grant agreement #MDA904-87-H-2023     

Primitive elements in finite fields and costas arrays

This author would like to thank the National Security Agency for partial support under grant agreement #MDA904-87-H-2023     

Dislocation and displacement fields of finite bodies with given boundary tractions

The traction boundary value problem for spatially finite material bodies is examined in the context of the gauge theory of dislocations. In contrast with classical theory of dislocations in infinite bodies, the boundary conditions for the dislocation fields are shown to have pronounced effects. Expansion in the load parameter that is naturally associated with the applied loading shows that dislocation effects are essentially nonlinear. If the dislocation coupling constant is of the order of the shear modulus or larger, the dislocation density tensor vanishes throughout the body in the linear engineering approximation. A sequence of well-posed linear boundary value problems are shown to provide approximate solutions to any desired degree of accuracy in the load parameter.     

Thick shell and solid finite elements with independent rotation fields

Thick shell and solid elements presented in this work are derived from variational principles employing independent rotation fields. Both elements are built on a special hierarchical interpolation and both possess six degrees of freedom per node. Performance of the elements is evaluated on a set of problems in elastostatics. However, the formulation presented herein is also suitable for transient and non-linear problems.     

A benchmark study of reduced‐strain shell finite elements: quadratic schemes

We study experimentally the accuracy and reliability of some low‐order shell finite element schemes based on modifying the standard displacement formulation by reduced‐strain expressions. We focus on quadrilateral elements with a quadratic displacement approximation. Three benchmark problems with different asymptotic behaviour in the limit of zero shell thickness is used in the experiments. Following the error analysis of a reduced‐strain scheme, we study two components of the total error, the approximation error and the consistency error. We demonstrate that the performance of the methods is both case and mesh dependent. When a bending dominated problem is solved, none of the methods studied can avoid the usual worst‐case locking effect of the approximation error on general meshes. For a membrane dominated problem the total error is typically dominated by the consistency error which often convergences slowly. Copyright © 2000 John Wiley & Sons, Ltd.     

Static analysis of Timoshenko beam resting on elastic half-plane based on the coupling of locking-free finite elements and boundary integral

Making use of a mixed variational formulation including the Green function of the soil and assuming as independent fields both the structure displacements and the contact pressure, a finite element (FE) model is derived for the static analysis of a foundation beam resting on elastic half-plane. Timoshenko beam model is adopted to describe structural foundations with low slenderness and to impose displacement compatibility between beam and half-plane without requiring the continuity of the first order derivative of the surface displacements enforced by Euler–Bernoulli beam. Numerical results are obtained by using locking-free Hermite polynomials for the Timoshenko beam and constant reaction over the soil. Foundation beams loaded by many load configurations illustrate accuracy and convergence properties of the proposed formulation. Moreover, the different behaviour of the Euler–Bernoulli and Timoshenko beam models is thoroughly discussed. Rectangular pipe loaded by a force in the upper beam exemplifies the straightforward coupling of the foundation FE with a structure described by usual FEs.     

Calculation of transient temperature fields with finite elements in space and time dimensions

A program is demonstrated which apart from linear finite elements in time also includes elements with shape functions of the second and third degree. The algorithm for discretization in the time dimension is described and, using the example of a parabolic time element, the coefficients required to form the global system are given. By various test examples the efficiency of the process is examined by comparison with the customary difference method. Generally, with finite elements in time, the solution has better stability. Comparing the time required for calculation with the accuracy of the solution it would appear that in examining problems where boundary conditions are constant in time, higher order time elements are no improvement over the linear time element. However, for the purpose of reproducing periodic processes, higher order time elements offer an advantage in that one is not limited to linear variations of the boundary conditions within the element. Thus, for example, the temperature curve for parabolic variation of the surface temperature can be reproduced with close approximation by two time elements per period and a shape function of the third degree.     

MLS‐based variable‐node elements compatible with quadratic interpolation. Part II: application for finite crack element

Two‐dimensional finite ‘crack’ elements for simulation of propagating cracks are developed using the moving least‐square (MLS) approximation. The mapping from the parental domain to the physical element domain is implicitly obtained from MLS approximation, with the shape functions and their derivatives calculated and saved only at the numerical integration points. The MLS‐based variable‐node elements are extended to construct the crack elements, which allow the discontinuity of crack faces and the crack‐tip singularity. The accuracy of the crack elements is checked by calculating the stress intensity factor under mode I loading. The crack elements turn out to be very efficient and accurate for simulating crack propagations, only with the minimal amount of element adjustment and node addition as the crack tip moves. Numerical results and comparison to the results from other works demonstrate the effectiveness and accuracy of the present scheme for the crack elements. Copyright © 2005 John Wiley & Sons, Ltd.     

Micromechanically based stochastic finite elements: length scales and anisotropy

The present stochastic finite element (SFE) study amplifies a recently developed micromechanically based approach in which two estimates (upper and lower) of the finite element stiffness matrix and of the global response need first to be calculated. These two estimates correspond, respectively, to the principles of stationary potential and complementary energy on which the SFE is based. Both estimates of the stiffness matrix are anisotropic and tend to converge towards one another only in the finite scale limit; this points to the fact that an approximating meso-scale continuum random field is neither unique nor isotropic. The SFE methodology based on this approach is implemented in a Monte Carlo sense for a conductivity (equivalently, out-of-plane elasticity) problem of a matrix-inclusion composite under mixed boundary conditions. Two versions are developed: in one an exact calculation of all the elements' stiffness matrices from the microstructures over the entire finite element mesh is carried out, while in the second one a second-order statistical characterization of the mesoscale continuum random field is used to generate these matrices.     

Determination of T-stress from finite element analysis for mode I and mixed mode I/II loading

The elastic T-stress has been recognised as a measure of constraint around the tip of a crack in contained yielding problems. A review of the literature indicates that most methods for obtaining T are confined to simple geometry and loading configurations. This paper explores direct use of finite element analysis for calculating T. It is shown that for mode I more reliable results with less mesh refinement can be achieved if crack flank nodal displacements are used. Methods are also suggested for calculating T for any mixed mode I/II loading without having to calculate stress intensity factors. There is good agreement between the results from the proposed methods and analytical results. T-stress is determined for a test configuration designed to investigate brittle and ductile fracture in mixed mode loading. It is shown that in shear loading of a cracked specimen T vanishes only when a truly antisymmetric field of deformation is provided. However this rarely happens in practice and the presence of T in shear is often inevitable. It is shown that for some cases the magnitude of T in shear is much more than that for tension. The effect of crack length is also investigated. This revised version was published online in August 2006 with corrections to the Cover Date.     

Polynomial evaluation over finite fields: new algorithms and complexity bounds

An efficient evaluation method is described for polynomials in finite fields. Its complexity is shown to be lower than that of standard techniques, when the degree of the polynomial is large enough compared to the field characteristic. Specifically, if n is the degree of the polynomiaI, the asymptotic complexity is shown to be ${O(\sqrt{n})}$ , versus O(n) of classical algorithms. Applications to the syndrome computation in the decoding of Reed-Solomon codes are highlighted.     

Generalizing the finite element method: Diffuse approximation and diffuse elements

This paper describes the new diffuse approximation method, which may be presented as a generalization of the widely used finite element approximation method. It removes some of the limitations of the finite element approximation related to the regularity of approximated functions, and to mesh generation requirements. The diffuse approximation method may be used for generating smooth approximations of functions known at given sets of points and for accurately estimating their derivatives. It is useful as well for solving partial differential equations, leading to the so called diffuse element method (DEM), which presents several advantages compared to the finite element method (FEM), specially for evaluating the derivatives of the unknown functions.     

Virtual and smoothed finite elements: A connection and its application to polygonal/polyhedral finite element methods

We show both theoretically and numerically a connection between the smoothed finite element method (SFEM) and the virtual element method and use this approach to derive stable, cheap and optimally convergent polyhedral FEM. We show that the stiffness matrix computed with one subcell SFEM is identical to the consistency term of the virtual element method, irrespective of the topology of the element, as long as the shape functions vary linearly on the boundary. Using this connection, we propose a new stable approach to strain smoothing for polygonal/polyhedral elements where, instead of using sub‐triangulations, we are able to use one single polygonal/polyhedral subcell for each element while maintaining stability. For a similar number of degrees of freedom, the proposed approach is more accurate than the conventional SFEM with triangular subcells. The time to compute the stiffness matrix scales with the in case of the conventional polygonal FEM, while it scales as in the proposed approach. The accuracy and the convergence properties of the SFEM are studied with a few benchmark problems in 2D and 3D linear elasticity. Copyright © 2015 John Wiley & Sons, Ltd.     

Modeling mixed-mode dynamic crack propagation nsing finite elements: Theory and applications

Previous work in modeling dynamic fracture has assumed the crack will propagate along predefined mesh lines (usually a straight line). In this paper we present a finite element model of mixed-mode dynamic crack propagation in which this constraint is removed. Applying linear elasto-dynamic fracture mechanics concepts, discrete cracks are allowed to propagate through the mesh in arbitrary directions. The fracture criteria used for propagation and the algorithms used for remeshing are described in detail. Important features of the implementation are the use of triangular elements with quadratic shape functions, explicit time integration, and interactive computer graphics. These combine to make the approach robust and applicable to a broad range of problems.Example analyses of straight and curving crack problems are presented. Verification problems include a stationary crack under dynamic loading and a propagating crack in an infinite body. Comparisons with experimental data are made for curving propagation in a cracked plate under biaxial loading.     

Transient field problems: Two-dimensional and three-dimensional analysis by isoparametric finite elements

The transient field problem of the type encountered in heat conduction problems is formulated in terms of the finite element process using the Galerkin approach. Curved two-dimensional and three-dimensional, isoparametric elements are used in a time-stepping solution and their advantages illustrated by means of several examples.     

A virtual crack closure-integral method (VCCM) to compute the energy release rates and stress intensity factors based on quadratic tetrahedral finite elements

This paper proposes a new virtual crack closure-integral method (VCCM) for quadratic tetrahedral finite element to compute the energy release rates/stress intensity factors. The formulations, numerical implementations and some numerical results of proposed VCCM are presented in this paper. Proposed VCCM enables us to adopt the tetrahedral finite element in 3D crack problems and us to use automatic mesh generation programs. Therefore process time to perform 3D crack analysis drastically reduces compared with the case of hexahedral elements.