bluuebunny/crossref_metadata_2025 huggingface dataset:
Consistent force resultant distributions in displacement elements with varying sectional properties
Authors: Prathap G. , Naganarayana B. P.
Published: August 2005

Force fields computed directly from strains calculated in a displacement type finite element description of a structural element of varying sectional rigidities show extraneous oscillations. The origin of these oscillations is traced to the fact that the displacement type finite element procedure determines strains derived from the displacement field in a least squares correct sense and that force resultants computed using these strain fields and the actual sectional rigidities result in unwanted oscillations. It is necessary to introduce the concept of redistributed assumed force resultant fields that maintain a ‘consistent’ relationship to the strain fields and also are orthogonal to these strain functions. In this paper, the Hu–Washizu theorem is invoked to justify the introduction of an orthogonally correct reconstituted assumed force resultant field which will then be free of extraneous oscillations. The quadratic isoparametric tapered bar element serves to illustrate the underlying principles.

It follows that the extremely general Hu–Washizu principle is the most practical procedure of implementing an assumed force resultant, assumed strain displacement type formulation to introduce consistency and thereby remove problems associated with field‐inconsistency (such as cause locking in constrained media elasticity) and force resultant oscillations due to varying sectional properties.
Some observations on time domain and frequency domain boundary elements
Authors: Von Estorff O. , Pais A. L. , Kausel E.
Published: August 2005

The numerical solution of problems in elastodynamics involving infinite media calls for the use of discrete techniques such as the boundary element method and the finite element method. These techniques can, in turn, be formulated in the time or frequency domains, and have each relative merits and drawbacks. This paper presents a comparative study of the accuracy and limitations of three different implementations of these methods.

The problem studied is that of transient loads on the surface of homogeneous elastic halfspaces, and of finite depth strata. In each case, the response is computed first for an uninterrupted (continuous) medium, and then for a medium that includes a trench (or cavity).

Three independent computer programs were used that incorporated the following methods: (i) frequency domain boundary element method (FD‐BEM), using a discrete fundamental solution; (ii) time domain boundary element method (TD‐BEM) using an analytical fundamental solution; and (iii) a coupled time domain boundary element‐finite element model (BEM/FEM).

It is found that for convex domains (halfspace or stratum without a trench) the three independent implementations are in excellent agreement, while for non‐convex domains (trench in the path of the waves), numerical errors associated with non‐causal behaviour become evident in some cases.
The dynamic response of compliant materials to an impulsive pressure field
Authors: D'sa Joseph M. , Dalton Charles
Published: August 2005

The dynamic interaction between a compliant material and an impulsive pressure field, periodic in space and instantaneous in time, was examined as a first step at modelling the interaction between organized structures in a turbulent boundary layer and a compliant surface. The interaction, modelled two‐dimensionally, was treated dynamically by matching the pressure forces to the surface stresses in the compliant material at each instant of time. A new boundary element method was formulated to model the compliant material which was treated as a linear isotropic material, elastic in dilatation and viscoelastic (Standard) in shear. The inertial forces and viscoelastic creep stresses have been included in this formulation as transient body forces. The elastic interaction was characterized by a non‐dimensional threshold velocity, above which the elastic instabilities grew temporally and spatially in the downstream direction to produce a non‐linear breakdown of the interaction. Freestream velocities as high as 9CT (shear wave speed) were found to produce stable elastic interactions. Thinner materials produced smaller amplitude waves of higher frequencies that grew more rapidly than those in thicker materials. The stability characteristics were independent of the location of the compliant material with respect to the spatial distribution of the pressure pulse. For viscoelastic interactions, the stability curve, which serves as a bound on the types of materials capable of producing drag reduction, shows distinct regions of elastic types of interactions (Class B) and damping dominated interactions (Class A) as a function of the constants of the rheological model describing the compliant material. Class A disturbances in these interactions show slower growth or decay than Class B disturbances.
A non‐Jacobian numerical quadrature for semi‐infinite integrals
Authors: Liaw B. M. , Kamel M.
Published: August 2005

A non‐Jacobian numerical quadrature is proposed for evaluating improper integrals over a semi‐infinite range. The quadrature first transforms the semi‐infinite integration limit into a finite limit between – 1 and 1. Standard numerical integration procedures such as Gauss–Chebyshev or Gauss–Legendre schemes can then be used to obtain the integral value. Unlike traditional methods using Laguerre or Hermite polynomials, numerical results show that no specific weight function is required for the proposed quadrature to converge as long as the integral exists. The transformation also includes a scale parameter which effectively expands the numerical‐integration sampling points along its original semi‐infinite integration path. Numerical examples shows that selection of the scale parameter is rather flexible and convergence can always be achieved within a wide range of parameter values. Several applied mechanics problems are also illustrated to validate the proposed numerical scheme. The quadrature can be easily incorporated into finite element or boundary element schemes for the evaluation of semi‐infinite integrals.
Data structures and algorithms for the finite element method on a data parallel supercomputer
Authors: Johnsson S. Lennart , Mathur Kapil K.
Published: August 2005

This article describes a formulation of the finite element method and its implementation on a data parallel computing system. The Connection Machine® system, CM‐2, has been used as the model architecture. Data structures, storage requirements, communication and parallel arithmetic complexity are analysed in detail for the cases when a processor represents an unassembled finite element and when a processor is assigned to an unassembled nodal point. Data parallel algorithms for the grid generation, the evaluation of the elemental stiffness matrices and for the iterative solution of the linear system are presented. The algorithm for evaluating the elemental stiffness matrices computes the matrix elements concurrently without communication. This concurrency is in addition to the inherent parallelism present among different finite elements. A conjugate gradient solver with diagonal pre‐conditioner is used for the solution of the resulting linear system. Results from an implementation of the three‐dimensional finite element method based on Lagrange elements are reported. For single‐precision floating‐point operations, the measured peak performance is approximately 2·4 G flops s−1for evaluating the elemental stiffness matrices and approximately 850 M flops s−1for the conjugate gradient solver. On a Connection Machine system with 16K physical processors, the time per conjugate gradient iteration for an application with 400 000 degrees of freedom is approximately 0·13 s for double‐precision floating‐point operations.
A subspace forward iteration method for solving the quadratic eigenproblem associated with the FDE formulation
Authors: Gmuer Thomas E. C.
Published: August 2005

This paper presents the development of an efficient and numerically stable algorithm for accurately computing the eigensolutions of the quadratic real symmetric eigenproblem arising in the finite dynamic element (FDE) formulation. Closely related to the subspace forward iteration method, the proposed scheme is well suited to extracting the lowest natural frequencies and associated mode shapes of large practical eigenproblems, takes full advantage of the banded configuration of the stiffness, inertia and dynamic correction matrices involved in the eigenproblem and ensures a monotonic convergence from above to the required eigenpairs. A shifting technique for convergence acceleration and an eigenpair verification scheme are also presented. Numerical examples are shown demonstrating the excellent performance of the solution procedure.
A fixed grid finite element technique for modelling phase change in steady‐state conduction–advection problems
Authors: Pardo E. , Weckman D. C.
Published: August 2005

A new fixed grid finite element technique for modelling phase‐change effects in steady‐state conduction–advection problems is described. The method is appropriate for materials that transform at a fixed critical temperature and is based on the assumption that the enthalpy experiences a jump discontinuity at this temperature, yielding a singular specific heat. This singularity is described using a Dirac delta function and the appropriate finite element matrix equations are developed for isoparametric elements. A simple first order upwinding scheme is used to avoid numerical oscillations in advection dominated problems. The utility of the method is illustrated through the solution of a one‐dimensional test problem using eight‐node hexahedral isoparametric finite elements. The technique may be applied to a wide range of finite element types and offers a number of advantages over other schemes for modelling phase‐change effects.
Causal FFT treatment applicable to singularity functions
Authors: Katukura Hiroshi , Hayashi Yasuhiro
Published: August 2005

The objective of this paper is to present a method of causal FFT treatment which is available to not only approximately time‐limited and band‐limited functions but also singularity functions appearing frequently in earthquake engineering problems such as time domain response analyses and seismic record processing. By means of the causal FFT treatment for a singularity function, it is possible to determine several causal functions. The properties of the obtained causal functions are discussed in detail. The 0 second and 0 Hz value problems of engineering importance concerning the determination of the discrete singularity values of non‐time‐limited or non‐band‐limited functions are presented. Finally, by presenting numerical examples, several problems existing in the FFT analyses of singularity functions and a method to solve the problems are explained in detail.
A Computationally viable higher‐order theory for laminated composite plates
Authors: Tessler Alexander , Saether Erik
Published: August 2005

A variational higher‐order theory involving all transverse strain and stress components is proposed for the analysis of laminated composite plates. Derived from three‐dimensional elasticity with emphasis on developing a viable computational methodology, the theory is well suited for finite element approximations as it incorporates both C0 and C−1 continuous kinematic fields and Poisson boundary conditions. From the theory, a simple three‐node stretching‐bending finite element is developed and applied to the problem of cylindrical bending of a symmetric carbon/epoxy laminate for which an exact solution is available. Both the analytic and finite element results were found to be in excellent agreement with the exact solution for a wide range of the length‐to‐thickness ratio. The proposed higher‐order theory has the same computational advantages as first‐order shear‐deformable theories. The present methodology, however, provides greater predictive capabilities, especially, for thick‐section composites.
On bounding properties of eigenvalues from linear initial FE stability analyses of thin, elastic shells with respect to stability limits from geometrically non‐linear prebuckling analyses
Authors: Mang H. A.
Published: August 2005

There is consensus in the literature that the eigenvalues of smallest absolute value from linear initial stability analyses of thin, elastic shells by the finite element method (FEM) do not posses bounding properties with respect to corresponding stability limites from geometrically non‐linear stability analyses. A ‘linear initial stability analysis’ by the FEM representes the first step of an ‘accompanying linear stability analysis’ by this method. In this paper, two modes of such stability analyses of thin, elastic shells will be presented. It will be proved that, for mode 1, in contrast to mode 2, bounding properties of eigenvalues of smallest absolute value with respect to corresponding stability limits from geometrically non‐linear stability analyses, in fact, do exist. Moreover, bounding properties of such eigenvalues from mode 1 relative to corresponding eigenvalues from mode 2 will be shown to exist. The existence of these properties is important from the standpoint of engineering practice.
On FEM‐BEM coupling for fluid‐structure interaction analyses in the time domain
Authors: Estorff O. Von , Antes H.
Published: August 2005

The investigation of coupled fluid‐structure systems, subjected to dynamic loads, calls for the use of discrete techniques such as the finite element method (FEM) and the boundary element method (BEM). This paper reports on the development of a FEM‐BEM coupling procedure, where finite elements are used to model the linear elastic structure, while the adjacent fluid is represented by boundary elements. The new approach is formulated entirely in the time domain to be extensible to non‐linear problems.

Various numerical calculations are presented which clearly demonstrate the applicability and the merits of the hybrid method. The obtained results are in excellent agreement with solutions stemming from a boundary element analysis which uses subdomain techniques.
A continuum finite element for single‐set jointed media
Authors: Murakami H. , Impelluso T. , Hegemier G. A.
Published: August 2005

In response to the need for an advanced computational model for wave propagation in jointed‐rock media a new finite element for jointed media with a single set of regularly spaced joints is developed. The element is a numerical implementation of the higher‐order homogenization model recently proposed by Murakami and Hegemier. Due to the dispersive effects induced by regularly spaced joints, wave phenomena in jointed media are altered significantly. Therefore, in order to improve the interpretation of seismograms for accurate source identification, it is necessary to develop a higher‐order continuum element. The accuracy and efficiency of the new element is investigated by applying it to wave‐guide and wave‐normal problems of a jointed half‐space and by comparing the wave response with that of DYNA2D. The analyses by DYNA2D discretize explicitly the details of the joint microstructure, and are adopted as numerically exact measures for the assessment of the proposed finite element; good correlations were obtained. The validation study also confirmed the importance of wave dispersion for non‐linear as well as linear joint responses. Finally, as a more practical application of the proposed element, the problem of a jointed full‐space with a cylindrical cavity pressurized by step and pulse loadings was solved. Velocities at several observation points were compared with the numerically exact results of DYNA2D. Similar analyses carried out for elastic isotropic media predicted totally different velocity responses and confirmed the need for the proposed element.
Transient thermoelasticity and other body force effects in boundary element shape sensitivity analysis
Authors: Kane James H. , Kumar B. L. Keshava , Stabinsky Mark
Published: August 2005

Several new formulations for structural design sensitivity analysis (DSA) of elastic solids are presented that consistently account for steady and transient thermal loading, along with centrifugal and gravity loading. The approach employs implicit differentiation of the discretized boundary integral equations and facilitates the production of accurate and efficient computational procedures that obviate the need for matrix factorizations in the DSA step. For centrifugal, gravity and general thermal loading, a new DSA concept employing a particular integral technique is developed. For centrifugal, gravity and steady thermal effects, a surface integration approach is also presented, and the two methods compared. Alternative strategies for modelling the behaviour of temperature distributions of evolving shapes undergoing shape optimization and the ramifications of these strategies in the DSA process are also discussed. It is shown that, for certain classes of thermoelastic problems, shape optimization can be performed without the need to incorporate thermal analysis or DSA into the overall process. A thermoelastic surface stress sensitivity analysis formulation is presented that does not require additional numerical integration. A new thermoelastic sharp corner traction component consistent sensitivity analysis formulation is also presented. This technique is shown to work for any initial stress distribution. Numerical sensitivity results are presented for many problems with analytical solutions.
Hierarchic models for laminated composites
Authors: Babuška Ivo , Szabó Barna A. , Actis Ricardo L.
Published: August 2005

The principles governing the formulation of hierarchic models for laminated composites are discussed. The essential features of the hierarchic models described herein are: (a) the exact solutions corresponding to the hierarchic sequence of models converge to the exact solution of the corresponding problem of elasticity for a fixed laminate thickness, and (b) the exact solution of each model converges to the same limit as the exact solution of the corresponding problem of elasticity with respect to the laminate thickness approaching zero. Hierarchic models make the computation of any engineering data possible to an arbitrary level of precision within the framework of the theory of elasticity. Examples are presented.
An adaptive h‐r boundary element algorithm for the laplace equation
Authors: Sun W. , Zamani N. G.
Published: August 2005

In this paper, the combination of the h‐method (mesh refinement) and the r‐method (mesh redistribution) is employed to solve the Laplace equation using the boundary element procedure. The key in this approach is to derive on upper bound for the residual associated with the boundary element solution and minimize this bound with respect to an unknown grading function. The latter part is achieved by employing techniques of the calculus of variations.
A hybrid finite element formulation of the linear biphasic equations for hydrated soft tissue
Authors: Vermilyea Mark E. , Spilker Robert L.
Published: August 2005

A hybrid finite element method has been developed for application to the linear biphasic model of soft tissues. The biphasic model assumes that hydrated soft tissue is a mixture of two incompressible, immiscible phases, one solid and one fluid, and employs mixture theory to derive governing equations for its mechanical behaviour. These equations are time dependent, involving both fluid and solid velocities and solid displacement, and will be solved by spatial finite element and temporal finite difference approximation. The first step in the derivation of this hybrid method is application of a finite difference rule to the solid phase, thus obtaining equations with only velocities at discrete times as primary variables. A weighted residual statement of the temporally discretized governing equations, employing C° continuous interpolations of the solid and fluid phase velocities and discontinuous interpolations of the pore pressure and elastic stress, is then derived. The stress and pressure functions are chosen so that the total momentum equation of the mixture is satisfied; they are jointly referred to as an equilibrated stress and pressure field. The corresponding weighting functions are chosen to satisfy a relationship analogous to this equilibrium relation. The resulting matrix equations are symmetric. As an illustration of the hybrid biphasic formulation, six‐noded triangular elements with complete linear, several incomplete quadratic, and complete quadratic stress and pressure fields in element local co‐ordinates are developed for two dimensional analysis and tested against analytical solutions and a mixed‐penalty finite element formulation of the same equations. The hybrid method is found to be robust and produce excellent results; preferred elements are identified on the basis of these results.
Time domain boundary element method for dynamic stress intensity factor computations
Authors: Dominguez J. , Gallego R.
Published: August 2005

This paper presents a procedure for transient dynamic stress intensity factor computations using traction singular quarter‐point boundary elements in combination with the direct time domain formulation of the Boundary Element Method. The stress intensity factors are computed directly from the traction nodal values at the crack tip. Several examples of finite cracks in finite domains under mode‐I and mixed mode dynamic loading conditions are presented. The computed stress intensity factors are represented versus time and compared with those obtained by other authors using different methods. The agreement is very good. The results are reliable and little mesh dependent. These facts allow for the analysis of dynamic crack problems with simple boundary discretizations. The versatile procedure presented can be easily applied to problems with complex geometry which include one or several cracks.
Finite element calculation of energy release rate for 2‐D rubbery material problems with non‐conservative crack surface tractions
Authors: Chang J. H. , Becker E. B.
Published: August 2005

The energy release rate G for 2‐D rubbery material problems with non‐conservative crack surface tractions is calculated by a modified version of virtual crack extension method with finite element solutions. The formulation is demonstrated to be ‘patch‐independent’ and therefore a complicated finite element model around the crack tip is not required.
A numerical technique for the analysis of cracks subjected to normal compressive stresses
Authors: Scavia Claudio
Published: August 2005

The purpose of the present paper is the development of a numerical method for the evaluation of the Stress Intensity Factor in mode II at the tips of closed cracks subjected to compressive normal stresses and monotonically increasing shear stresses. A hybrid Boundary Element Technique, originally developed for the analysis of open cracks, has been suitably modified. The Stress Intensity Factor is evaluated directly on the basis of the displacement discontinuities computed at the crack tip element, by taking into account the development of frictional resistance on the crack surfaces. A limited number (10‐15) of displacement discontinuity elements is found to be sufficient for an accurate evaluation of the Stress Intensity Factor. Some basic aspects of the mechanical behaviour of closed cracks are also investigated on the basis of the proposed method.
Delaunay's mesh of a convex polyhedron in dimension d. application to arbitrary polyhedra
Authors: George P. L. , Hermeline F.
Published: August 2005

This paper presents a method for creating a Delaunay triangulation connected to a set of specified points. The theoretical aspect is recalled for an arbitrary dimension and the method is discussed in order to derive a practical approach, valid for dimensions 2 and 3, which is simple, robust and well adapted to computation. Convex polyhedral and arbitrary polyhedral situations are introduced.
Derivation of a new analytical solution for a general two‐dimensional finite‐part integral applicable in fracture mechanics
Authors: Mayrhofer K. , Fischer F. D.
Published: August 2005

An exact expression is derived for the general finite‐part integral
equation image
over an inclined ellipticaldomain Ω. r denotes the distance of a point in Ω to the singular point \documentclass{article}\pagestyle{empty}\begin{document}$\left({x,y} \right).f = x_{^0 }^i y_0^j \sqrt {Z\left({x_{0,} y_0 }\right)}$\end{document} is a general function of the Cartesian co‐ordinates x0,y0. The boundary of the region Ω represents the equation Z(x0, y0)=O. These integrals appear during the numerical solution of plane crack problems in three‐dimensional elasticity where they are the dominant part of a hypersingular integral equation. The availability of exact expressions for the integrals with arbitrary integers i and j will increase the accuracy of the numerical results and, simultaneously, lead to quicker numerical results.

The considered finite‐part integral can be expressed in closed form as function of complete elliptical integrals or Gauss hypergeometric functions, respectively.

Formuias for special cases and some i, j values and their numerical verification are given in Appendices II and III.
The s‐version of the finite element method for multilayer laminates
Authors: Fish J. , Markolefas S.
Published: August 2005

A methodology has been developed to accurately resolve the stress field in the vicinity of free edges as well as the overall response of laminated plates without significantly affecting the computational cost. This is accomplished by enriching a set of classical smooth interpolants throughout the thickness direction with C° continuous displacement interpolants (piecewise continuous strain field) in the regions where the most critical behaviour is anticipated. C° continuity of the displacement field is maintained by imposing homogeneous boundary conditions on the superimposed field in the portion of the boundary which is not contained within the boundary of the problem. Numerical experiments for both cylindrical bending and uniform extension of cross‐ply laminates are presented to validate the present formulation.
Fast decomposition of matrices generated by the boundary element method
Authors: Rezayat Mohsen
Published: August 2005

A new method to reduce the solution time of matrices generated by the Boundary Element Method is presented here. The method involves converting the fully populated system into a banded system by lumping certain coefficients of the matrix into fictitious nodes and then constraining these nodes to accurately represent each coefficient. The major advantages of lumping over the substructuring method are that lumping can be applied to arbitrarily shaped geometries and infinite‐domain problems and that it preserves the diagonal‐dominance of the matrix. It is shown here that the proposed algorithm reduces the rate of increase of solution time t of an n‐degree‐of‐freedom problem from t ∝ n3 to t ∝ n2. Although the algorithm is for thermal problems, its extension to mechanical problems is straightforward. The procedure can easily be incorporated into existing boundary‐element‐based packages.
An optimal method for the solution of the constrained eigenvalue/response problem for large structures comprising rigidly‐connected substructures
Authors: Garvey S. D. , Penny J. E. T.
Published: August 2005

Dynamic substructuring or modal synthesis is established as a powerful means by which the computational effort required to perform dynamic analyses of large structures comprising many degrees of freedom can be reduced. For the case where the substructures are rigidly connected a method of synthesis is proposed which gives very substantial improvements in efficiency over existing methods both in terms of the number of floating‐point operations which must take place and the on‐board computer memory required. The proposed method makes use of a stable co‐ordinate transformation which reduces the mass matrix for the constrained composite‐system to the identity matrix and makes the stiffness matrix banded. The bandwidth is exactly equal to twice the total number of constraints plus 1. This is usually very small in comparison with the total number of degrees of freedom in the constrained composite‐system. The proposed method is ideal for combining a number of substructures into a single new substructure in an efficient way without necessarily disposing of any degrees of freedom. Approximate formulae are presented which give the ratios of the number of operations required for established methods compared with the number needed for the proposed method. These formulae determine when it is appropriate to use the method proposed here.
Tetrahedral elements for fluid flow
Authors: Bertrand F. H. , Gadbois M. R. , Tanguy P. A.
Published: August 2005

Several old and new tetrahedral elements are tested and compared to some classical hexahedral elements for the solution of three‐dimensional incompressible fluid flow problems. The proposed elements are created by adding velocity degrees of freedom to the basic trilinear velocity, constant pressure tetrahedron so as to satisfy the Brezzi condition. Numerical tests carried out for an artificial divergence‐free flow problem and the lid‐driven cubic cavity up to a Reynolds number of 400 are used to compare the proposed elements and assess their accuracy.
The dual boundary element method: Effective implementation for crack problems
Authors: Portela A. , Aliabadi M. H. , Rooke D. P.
Published: August 2005

The present paper is concerned with the effective numerical implementation of the two‐dimensional dual boundary element method, for linear elastic crack problems. The dual equations of the method are the displacement and the traction boundary integral equations. When the displacement equation is applied on one of the crack surfaces and the traction equation on the other, general mixed‐mode crack problems can be solved with a single‐region formulation. Both crack surfaces are discretized with discontinuous quadratic boundary elements; this strategy not only automatically satisfies the necessary conditions for the existence of the finite‐part integrals, which occur naturally, but also circumvents the problem of collocation at crack tips, crack kinks and crack‐edge corners. Examples of geometries with edge, and embedded crack are analysed with the present method. Highly accurate results are obtained, when the stress intensity factor is evaluated with the J‐integral technique. The accuracy and efficiency of the implementation described herein make this formulation ideal for the study of crack growth problems under mixed‐mode conditions.
Memory and operations count scaling of coupled finite‐element and boundary‐element systems of equations
Authors: Paulsen Keith D. , Liu Weiping
Published: August 2005

Memory and operations count scaling for the solution of coupled finite‐element and boundary‐element systems of equations is considered. Three algebraic approaches for solving the hybrid set of equations are studied using both direct and iterative matrix methods. Results show that once the matrix solution technique is chosen, a single hybrid algebra emerges as a clear favourite. Interestingly, the most computationally attractive hybrid algebra under assumptions of direct solution becomes the least desirable approach when iterative methods are applied. The analysis has been carried out using a range of mesh‐dependent parameters which have been empirically derived through practical experience; however, the scaling expressions presented are valid for any mesh once these critical parameters have been determined.
The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity
Authors: Zienkiewicz O. C. , Zhu J. Z.
Published: August 2005

In this second part of the paper, the issue of a posteriori error estimation is discussed. In particular, we derive a theorem showing the dependence of the effectivity index for the Zienkiewicz–Zhu error estimator on the convergence rate of the recovered solution. This shows that with superconvergent recovery the effectivity index tends asymptotically to unity. The superconvergent recovery technique developed in the first part of the paper1 is the used in the computation of the Zienkiewicz–Zhu error estimator to demonstrate accurate estimation of the exact error attainable. Numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient (stress) error estimation. Several examples of the performance of the error estimator in adaptive mesh refinement are also presented.
On the three‐dimensional vibration analysis of simultaneously skewed and twisted cantilevered parallelepipeds
Authors: McGee O. G.
Published: August 2005

The natural frequencies of simultaneously skewed and twisted cantilevered parallelepipeds are determined using the Ritz method. The present work is the first known three‐dimensional study of the problem. Assumed displacement functions are in the form of algebraic polynomials which satisfy the fixed face conditions exactly, and which are mathematically complete. Reasonably accurate natural frequencies are calculated for low aspect ratio, thick parallelepipeds having arbitrary degrees of twist and skewness. Detailed numerical studies show that a three‐dimensional analysis is essential in monitoring complicated coupled‐mode sensitivities in the variation of non‐dimensional natural frequencies with increasing skew and twist angles and thickness ratio. Results obtained using the present method are compared with those obtained in a detailed three‐dimensional finite element analysis, where the suitability and limitations of s‐version solid element discretizations are clearly brought out.
An assessment of four‐noded plate finite elements based on a generalized third‐order theory
Authors: Averill R. C. , Reddy J. N.
Published: August 2005

Plate finite elements based on the generalized third‐order theory of Reddy and the first‐order shear deformation theory are analysed and compared on the basis of thick and thin plate modelling behaviour, distortion sensitivity, overall accuracy, reliability and efficiency. In particular, several four‐noded Reddy‐type elements and the nine‐noded Lagrangian and heterosis (Mindlin‐type) plate elements are analysed to assess their behaviour in bending, vibration and stability of isotropic and laminated composite plates. A four‐noded Reddy‐type element is identified which is free of all spurious stiffness and zero energy modes, computationally efficient, and suitable for use in any general‐purpose finite element program.
A finite element formulation for highly anisotropic incompressible elastic solids
Authors: Brádaigh Conchúr M. Ó , Pipes R. Byron
Published: August 2005

A numerical approach is developed for the solution of problems of materials with extremely strong directions. Small deformations of a transversely isotropic linear elastic solid, reinforced by a single family of inextensible fibres, are considered. The kinematic constraint equations of incompressibility and inextensibility in the fibre direction lead to the appearance of an arbitrary hydrostatic pressure and an arbitrary tension stress in the constitutive equations. A Galerkin approach is used to discretize the virtual work and weak form of the constraint equations. Independent interpolation of the displacement, pressure and tension fields leads to a mixed system of equations, with characteristic zero‐diagonal terms. The assumption of plane stress conditions in the plane of the fibres results in a simplified displacement‐tension formulation, analogous to the primitive‐variable formulation of Stokes flow. A mixed penalty approximation is then employed to solve for displacement and tension stress fields. Computations are carried out using a biquadratic displacement element with discontinuous bilinear tension stress interpolation. The formulation is used to solve a number of simple beam problems and the results compared to closed‐form solutions.
On development of a finite dynamic element and solution of associated eigenproblem by a block Lanczos procedure
Authors: Gupta K. K. , Lawson C. L. , Ahmadi A. R.
Published: August 2005

The paper first presents the details of the development of a new six‐noded plane triangular finite dynamic element. A block Lanczos algorithm is developed next for the accurate and efficient solution of the quadratic matrix eigenvalue problem associated with the finite dynamic element formulation. The resulting computer program fully exploits matrix sparsity inherent in such a discretization and proves to be most efficient for the extraction of the usually required first few roots and vectors, including repeated ones. Most importantly, the present eigenproblem solution effort is shown to be comparable to that of the corresponding finite element analysis, thereby rendering the associated dynamic element method rather attractive owing to superior convergence characteristics of such elements, presented herein.
Local error indicator in finite element analysis of Laplacian fields based on the green integration formula
Authors: Adamiak Kazimierz
Published: August 2005

The paper presents a simple and practical method for the local error assessment In the finite element solution of fields in linear media governed by the Laplace equation. The local error is estimated for each node or element of the mesh model by applying the Green integration formula, with the error distribution being used for the adaptive mesh refinement. Several alternatives of the local error estimation algorithm are compared and numerical examples are given for a typical ‘L’‐section potential problem. These confirm the efficiency of the proposed method.
Sensitivity analysis for non‐linear constrained elastostatic systems
Authors: Tortorelli Daniel A.
Published: August 2005

Adjoint and direct differentiation methods are used to formulate design sensitivities for non‐linear constrained elastostatic systems. Variations of a general response functional are expressed in explicit form with respect to all design field variations, including shape. The response functional is expressed over the undeformed configuration; however, the displacement gradient is included so that field quantities that are defined over the deformed configuration, such as Cauchy stress, may be modelled. The finite element method is used to evaluate the sensitivities in an example problem.
p‐Version plate and curved shell element for geometrically non‐linear analysis
Authors: Sorem Robert M. , Surana Karan S.
Published: August 2005

This paper presents a p‐version geometrically non‐linear formulation based on the total Lagrangian approach for a nine node three dimensional curved shell element. The element geometry is defined by the coordinates of the nodes located on its middle surface and nodal vectors describing the bottom and top surfaces of the element. The element displacement approximation can be of arbitrary and different polynomial orders in the plane of the element and in the transverse direction. The element approximation functions and the corresponding nodal variables are derived from the Lagrange family of interpolation functions. The resulting approximation functions and the nodal variables are hierarchical and the element displacement approximation ensures C° continuity.

The element properties are established using the principle of virtual work and the hierarchical element approximation. In formulating the properties of the element complete three dimensional stresses and strains are considered, hence the element is equally effective for very thin as well as extremely thick shells and plates.

Incremental equations of equilibrium are derived and solved using the standard Newton–Raphson method. The total load is divided into increments, and for each increment of load, equilibrium iterations are performed until each component of the residuals is within a preset tolerance. Numerical examples are presented to show the accuracy, efficiency and advantages of the present formulation. The results obtained from the present formulation are compared with those available in the literature.
A finite element formulation with stabilization matrix for geometrically non‐linear shells
Authors: Kim J. H. , Lee S. W.
Published: August 2005

An assumed strain finite element formulation with a stabilization matrix is developed for analysis of geometrically non‐linear problems of isotropic and laminated composite shells. The present formulation utilizes the degenerate solid shell concept and assumes an independent strain as well as displacement. The assumed independent strain field is divided into a lower order part and a higher order part. Subsequently, the lower order part is set equal to the displacement‐dependent strain evaluated at the lower order integration points and the remaining higher order part leads to a stabilization matrix. The strains and the determinant of the Jacobian matrix are assumed to vary linearly in the thickness direction. This assumption allows analytical integration through thickness, independent of the number of plies. A nine‐node element with a judiciously chosen set of higher order assumed strain field is developed. Numerical tests involving isotropic and composite shells undergoing large deflections demonstrate the validity of the present formulation.
Element by element a posteriori error estimation of the finite element analysis for three‐dimensional elastic problems
Authors: Ohtsubo Hideomi , Kitamura Mitsuru
Published: August 2005

An a posteriori error estimation method for finite element solutions for three‐dimensional elastic problems is presented based on the theory developed by the authors for two‐dimensional problems.1 The error is estimated for the finite element solutions obtained using three‐dimensional 8‐node elements with a linear interpolation function in an arbitrary hexahedron. The method is successfully applied to three‐dimensional elastic problems. In order to decrease computing time and memory use, the error is estimated element by element. The major difficulty in the element‐wise error estimation technique is satisfying the self‐equilibrium condition of applied forces, especially in three‐dimensional problems. These forces are mainly due to traction discontinuity on the element boundaries. The difficulty is circumvented by employing an element‐wise optimal procedure. It is also shown that a very accurate stress solution can be obtained by adding estimated error to the original finite element solutions.
The use of dynamic data structures in finite element applications
Authors: Hawken D. M. , Townsend P. , Webster M. F.
Published: August 2005

The derivation of more efficient algorithms for finite element applications is not the primary goal of this article, but rather the efficient implementation of existing algorithms using dynamic data structures. In some cases the algorithms may be far from optimal, but serve to illustrate the advantages of dynamic data structures. In the first section, a tree data structure termed a PATRICIA tree is described and is shown to be suitable for storing data associated with finite element meshes. The manner in which the tree is constructed guarantees to provide rapid data retrieval times, competitive with those associated with static array data structures, whilst providing the added advantages of a dynamic data structure. The second section introduces a list of lists (LOL) data structure which is used to produce an efficient implementation of an example of a bandwidth reduction algorithm. Furthermore, the LOL data structure is also shown to be appropriate for implementing a novel method for extracting the domain boundary, with its orientation, from a given finite element connectivity matrix.
Arbitrary Lagrangian‐Eulerian finite element analysis of large deformation in contacting bodies
Authors: Ghosh Somnath
Published: August 2005

The primary aim of this research is to implement the arbitrary Lagrangian–Eulerian finite element description to model large deformation of interacting bodies. A pure Lagrangian description is often incapable of representing their contact with sufficient precision. In this work, the arbitrary Lagrangian‐Eulerian (ALE) description has been extended to the surface nodes of interacting bodies to overcome these limitations. A penalty formulation has been implemented to account for the impenetrability constraint. The materials have been characterized as elastic‐viscoplastic. Implicit integration schemes have been used at the global and constitutive levels for'improved stability.
A finite element for variable order singularities based on the displacement formulation
Authors: Maiti S. K.
Published: August 2005

An element for analysis of variable order one point singularities in plane problems has been proposed. The element meets all the convergence requirements, viz. the rigid body mode, the constant strain condition and the interelement compatibility. It can be easily incorporated in any package based on the displacement formulation. Four examples from fracture mechanics are presented to demonstrate its performance. The examples involve mechanical or thermal loadings. The accuracy of results in all the examples is very good.
Unconditionally convergent partitioned solution procedure for dynamic coupled mechanical systems
Authors: Saetta Anna V. , Vitaliani Renato V.
Published: August 2005

A new partitioned solution procedure for the direct time integration of second order coupled‐field systems is presented and it is applied to the problem of soil–pore fluid interaction. The necessary convergence analysis for the iterative method is carried out and the fulfilment of the convergence condition is achieved with the introduction of two suitable auxiliary matrices in the basic equations governing the dynamic phenomena. Numerical examples are presented to demonstrate the effectiveness and the versatility of the proposed solution procedure.
The problem of the selection of an a posteriori error indicator based on smoothening techniques
Authors: Babuška Ivo M. , Rodriguez Rodolfo
Published: August 2005

This paper addresses the problem of assessing the quality of an a posteriori error estimate of a finite element solution. An error estimate based on local L2‐projections is analysed in the case of translation‐invariant meshes. It is shown that for general meshes this technique does not lead to an asymptotically exact estimator. The problem is analysed in detail in the one‐dimensional setting. It is shown that an asymptotically exact estimator is not the optimal one when the solution is not sufficiently smooth. An optimal estimator for adaptively constructed meshes is given. Finally, a general mathematical framework for the quality assessment of estimators is introduced.
An augmented Lagrangian method for discrete large‐slip contact problems
Authors: Heegaard J.‐H. , Curnier A.
Published: August 2005

An augmented Lagrangian formulation is proposed for large‐slip frictionless contact problems between deformable discretized bodies in two dimensions. Starting from a finite element discretization of the two bodies, a node‐on‐facet element is defined. A non‐linear gap vector and its first variation are derived in terms of the nodal displacements. The relevant action and reaction principle is stated. The gap distance is then related to the conjugate pressure by a (multivalued non‐differentiable) unilateral contact law. The resulting inequality constrained minimization problem is transformed into an unconstrained saddle point problem using an augmented Lagrangian function. Large slip over several facets is possible and the effects of target convexity or concavity are investigated. A generalized Newton method is used to solve the resulting piecewise differentiable equations necessary for equilibrium and contact. The proper tangent (Jacobian) matrices are calculated. The primal (displacements) and dual (contact forces) unknowns are simultaneously updated at each iteration.