bluuebunny/crossref_metadata_2025 huggingface dataset:
Numerical experiments on turbulent flow using the random vortex method
Authors: Laitone Jonathan
Published: August 2005

Chorin's random vortex method is used to predict the growth of a large‐scale coherent vortex structure in the early stages of the development of turbulence in a two‐dimensional co‐flowing shear layer. The numerical algorithm has been simplified to such an extent that the numerical analysis can be performed on a microcomputer. The numerical solution exhibits the same early turbulent instabilities and vorticity pairings as found in recent flow‐visualization experiments. In addition the results are in reasonable agreement with experimental measurements of mean velocity, root mean square fluctuations and Reynolds stresses. One could thus test the shear layer sensitivity to initial conditions and the upsteam boundary conditions.
Singular element construction using a mapping technique
Authors: Tsamasphyros George
Published: August 2005

Conformal mapping regularizes (in the new plane) the singular stress field around the crack tip. So, conventional finite elements can be used in the transformed plane. The inverse mapping of the above elements gives another class of ‘mapped elements’. The crack‐tip mapped element is a singular element. Thus, using various elements (the ‘parent elements’) in the mapped plane we obtain an infinity of singular elements, the ‘product elements’. These singular elements are introduced naturally and have no defect, as many other singular elements have.
An extension of the application of transmission line modelling (TLM) to thermal diffusion to include non‐infinite heat sources
Authors: Pulko S. , Green W. A. , Johns P. B.
Published: August 2005

The modelling of heat transfer from a moving gaseous atmosphere by the TLM technique is described. For the case of very rapidly moving gas, results are compared with those derived from finite element analysis.
A grid generator based on 4‐triangles conforming mesh‐refinement algorithms
Authors: Rivara Maria‐Cecilia
Published: August 2005

A procedure to generate 2D meshes of triangles in a quite general fashion allowing for local and selective mesh‐refinement is presented and discussed. The grid generator is based on the iterative application of 4‐triangles conforming mesh‐refinement algorithms for triangulations, which are also introduced in this paper. These algorithms are modified versions of those proposed in Int. j. numer. methods eng., 20, 745–756 (1984), and they can be used for global refinement of a grid, as well as for local refinement. The grid generator works in the following way: given any initial coarse triangulation which properly defines the geometry of the problem, a set of user‐defined refinement subregions Ri, i = 1,2,…, N, and an associated set of tolerance parameters hi, an irregular and conforming final triangulation is automatically constructed in such a way that the diameter of all the triangles contained in Ri is smaller than hi, i= 1,2,…, N. Moreover, all angles in the final triangulation are greater than or equal to half the smallest angle in the initial, coarse one. The refinement is propagated only to assure the conformity and smoothness of the grid, and consequently, the number of involved nodes will be minimized. The refinement of the final mesh will be determined by the subregions Ri and the parameters Ni and will be essentially independent of the initial coarse grid. The 4‐triangles conforming mesh‐refinement algorithms are presented and their properties discussed. The implementation of these techniques is discussed and examples of the application of the grid generator are given.
On the numerical integration of a class of pressure‐dependent plasticity models
Authors: Aravas N.
Published: August 2005

An unconditionally stable algorithm for the numerical integration of elastoplastic pressure‐dependent constitutive relations is analysed in detail in this paper. The application of the method to plane stress problems, in which the out‐of‐plane strain component is not defined kinematically, is discussed. The tangent moduli resulting from this integration algorithm are obtained by consistent linearization of the elastoplastic constitutive equations. The algorithm is applied to Gurson's constitutive model, some one‐dimensional problems are solved, and comparisons with exact solutions are made. The paper closes with a numerical study of the necking of an axi‐symmetric specimen using Gurson's plasticity model to describe the constitutive behaviour of the material.
Dynamic response optimization using an active set RQP algorithm
Authors: Lim O. K. , Arora J. S.
Published: August 2005

A recently developed recursive quadratic programming (RQP) algorithm is applied to dynamic response design problems. The algorithm incorporates updated Hessians of the Lagrange function and uses an active set strategy in which only a subset of the constraints is included in the direction finding QP subproblem. Hessian updating with dynamic response constraints presents some difficulties. The primary difficulty is that the total number of constraints and the location of time points where they have to be imposed can change from iteration to iteration. This can cause inconsistencies in Hessian updating if proper numerical procedures are not used. A numerical procedure to handle the situation is developed, implemented and evaluated. Automatic restarting procedures are necessary for proper convergence of the algorithm. The new algorithm is robust as well as more efficient than the purely linear algorithm. The active set strategy plays an important role for application to dynamic response problems. The RQP algorithms that do not use such a strategy are not applicable to this class of problems.
Interactive solution to multiobjective optimization problems
Authors: Diaz Alejandro
Published: August 2005

Sensitivity analysis is presented as a natural addition to interactive multiobjective optimization methods based on compromise programming. It is shown that useful information regarding trade‐offs in the objectives can be generated effectively by means of an analysis of the sensitivity of solutions to variations in preference structures. An implementation based on sequential quadrative programming is provided and examples are given for illustration.
The use and efficiency of the theorems of structural variation for finite element analysis
Authors: Topping B. H. V. , Kassim A. M. Abu
Published: August 2005

This paper describes how the theorems of structural variation may be used in finite element analysis. The efficiency of the theorems is examined and their application to structural and geotechnical problems illustrated.
A posteriori estimates of the solution error caused by discretization in the finite element, finite difference and boundary element methods
Authors: Kelly D. W. , Mills R. J. , Reizes J. A. , Miller A. D.
Published: August 2005

A theory is described which guarantees an upper and lower bound estimate of the discretization error in numerical solutions of elliptic boundary value problems. This method gives bounded global estimates of the error in the energy norm. Pointwise estimates of the error in the solution variable or its derivatives can then be obtained if the numerical solution is exhibiting pointwise monotonic convergence.

The versatility of this method is illustrated by its application to numerical solutions from finite element, finite difference and boundary element methods.
Convected systems for curved structural elements
Authors: Gattass Marcelo , Abel John F.
Published: August 2005

This paper presents a new approach to the use of the updated Lagrangian formulation with convected systems. The development of this new method is similar to the updated Lagrangian formulation presented by Bathe et al.4 The main difference is that here the tensors are written in their components in the curvilinear convected system, while Bathe, Ramm, and Wilson employ Cartesian components. A special form of the natural approach5 is also introduced in this formulation to simplify the element computations. The matrices of a two‐dimensional curved beam‐column element are derived as an example of this new formulation. Finally, a numerical evaluation is performed to evaluate critically the new curved element and the method proposed here.
On artificial strain of thin curved elements
Authors: Gattass Marcelo , Abel John F.
Published: August 2005

This paper presents an attempt to clarify further the nature of the overstiffness of thin curved structural elements. It follows the approach presented by Ashwell and co‐workers1,2 who based their study on shape functions of arch elements. Such elements display the same stiff behaviour as complex shell elements but are simpler to study and understand. An examination of several discretizations for a two‐dimensional curved beam‐column element for linear elastic analysis of arches is performed. This examination and the numerical results obtained from these approximations provide a new interpretation of the artificial strains based on the mapping variables of shape functions. Two new methods to reduce the artificial stiffness of the element are proposed.
Selective refinement: A new strategy for automatic node placement in graded triangular meshes
Authors: Frey William H.
Published: August 2005

Automating triangular finite element mesh generation involves two interrelated tasks: generatine a distribution of well‐placed nodes on the boundary and in the interior of a domain, and constructing a triangulation of these nodes. For a given distribution of nodes, the Delaunay triangulation generally provides a suitable mesh, and Watson's algorithm26 provides a flexible means of constructing it. In this paper, a new method is described for automating node placement in a Delaunay triangulation by seieclive refinement of an initial triangulation. Grading of the mesh is controlled by an explicit or implicit node spacing function. Although this paper describes the technique only in the planar context, the method generalizes to three dimensions as well.
Petrov—Galerkin method for multidimensional, time‐dependent, convective‐diffusion equations
Authors: Yu C.‐C. , Heinrich J. C.
Published: August 2005

A Petrov—Galerkin finite element method for two and three‐dimensional time dependent convection‐diffusion equations is presented in the context of two space dimensions. The method involves perturbed weighting functions in the weighted residuals formulation, that are bilinear in space and quadratic in time and depend on two parameters which are calculated according to a local analysis for the one dimensional case. The perturbations to the weighting functions can be interpreted as an added local anisotropic balancing diffusion and an added dispersion in the direction of the convective motion. The effectiveness of the method is shown through several examples involving the convective and diffusive transport of scalar distributions in known velocity fields.
A new partially bonded beam element for analysis of shear walls
Authors: Chakrabarti S. C. , Nayak G. C. , Paul D. K.
Published: August 2005

A finite element formulation for multi‐panel shear walls based on a partially bonded beam system has been proposed. Idealization of a shear wall by a partially bonded multi‐layered beam element reduces the storage and computer time considerably. The finite element model has been employed extensively for parametric study of prefabricated shear walls of different widths and heights. Different numbers of in‐situ joints in the prefabricated shear wall assembly with varying shear stiffness of joints have also been considered. The parametric study gave an insight to the structural response of the wall system. Based on the non‐linear shear stiffness characteristic of in‐situ joints determined experimentally, the nonliner behaviour of a prefabricated shear wall has been determined and the failure load has been predicted.
A new approach to the finite element modelling of beams with warping effect
Authors: Lee S. W. , Kim Y. H.
Published: August 2005

As a first step toward developing a finite element formulation that can model coupling among extensional, bending and torsional behaviour of beams, a new method is proposed to properly represent the warping of arbitrary cross‐sections. The basic approach is to introduce a small warping displacement superimposed over flat cross‐sections of a shear‐flexible beam in a deformed configuration. Numerical tests involving simple isotropic beams undergoing a small elastic displacement demonstrate the validity of the new approach. The present approach can be extended to composite beams as well as isotropic beams experiencing a large deflection or finite rotation.
An assumed covariant strain based 9‐node shell element
Authors: Jang Junho , Pinsky Peter M.
Published: August 2005

A new shell element, which is free from serious locking problems and which does not possess hourglass modes, is proposed. Solutions obtained with this element exhibit good convergence and are satisfactorily insensitive to mesh distortion. The element also exhibits good convergence for plates with variable thickness. This element is based on the use of assumed covariant strains which are obtained from the covariant strain field defined with respect to the element natural co‐ordinate system. Only the linear version for thin shell cases is considered. The element performance is tested by application to several standard plate and shell problems. A problem involving variable thickness is also presented.
A parallel householder tridiagonalization stratagem using scattered row decomposition
Authors: Chang H. Y. , Utku S. , Salama M. , Rapp D.
Published: August 2005

Householder's method for tridiagonalizing a real symmetric matrix, a major step in evaluating eigenvalues of the matrix, is modified into a parallel algorithm for a concurrent machine of message passing type. Each processor of the concurrent machine has its own CPU, communications control and local memory. Messages are passed through connections between processors. Although the basic algorithm is inherently serial, the computations can be spread over all processors by scattering different rows of the matrix into processors, hence the term “Scattered Row Decomposition”. The steps in the serial and the parallel algorithms are identified. Expressions for efficiency and speedup are given in terms of problem and machine parameters. For a concurrent machine of ring type interconnection, a selected representative problem of large order exhibits efficiency approaching 66 per cent.
Quadratic‐spline collocation methods for two‐point boundary value problems
Authors: Houstis E. N. , Christara C. C. , Rice J. R.
Published: August 2005

A new collocation method based on quadratic splines is presented for second order two‐point boundary value problems. First, O(h4) approximations to the first and second derivative of a function are derived using a quadratic‐spline interpolant of u. Then these approximations are used to define an O(h4) perturbation of the given boundary value problem. Second, the perturbed problem is used to define a collocation approximation at interval midpoints for which an optimal O(h3‐J) global estimate for the jth derivative of the error is derived. Further, O(h4‐J) error bounds for the jth derivative are obtained for certain superconvergence points. It should be observed that standard collocation at midpoints gives O(h2‐J) bounds. Results from numerical experiments are reported that verify the theoretical behaviour of the method.
Unconditionally stable staggered solution procedure for soil‐pore fluid interaction problems
Authors: Zienkiewicz O. C. , Paul D. K. , Chan A. H. C.
Published: August 2005

This paper presents a simple procedure for modifying the dynamic coupled soil‐pore fluid equations in a manner which allows a stable and efficient staggered solution procedure to be used. The solution method is simpler than an alternative suggested by Park15 and has a direct physical interpretation. This stabilized procedure is efficient in numerical time domain analysis of soil‐pore fluid occurring in earthquake and consolidation problems. Numerical examples of one and two dimensional saturated poroelastic media demonstrate the procedure.
Resolving membrane and shear locking phenomena in curved shear‐deformable axisymmetric shell elements
Authors: Tessler Alexander , Spiridigliozzi Luciano
Published: August 2005

A simple two‐node axisymmetric shell element with shallowly curved meridian assumptions and the inclusion of shear deformation and rotary inertia is presented. The principal developments include: (a) consistent resolution of the membrane and shear related excessive stiffening (locking) via anisoparametric interpolations of the displacement variables; (b) further upgrading of strain energy by means of a shear relaxation (correction) parameter. The resulting element possesses an improved condition of the stiffness matrix, increased efficiency in explicit time integration and enhanced accuracy in coarse discretizations. Comprehensive vibration examples are carried out to assess the element performance. The numerical results demonstrate a wide applicability range with respect to element slenderness and curvature properties.
Stress and vibration analyses of anisotropic shells of revolution
Authors: Noor Ahmed K. , Peters Jeanne M.
Published: August 2005

An efficient computational strategy is presented for reducing the cost of the stress and free vibration analyses of laminated anisotropic shells of revolution. The analytical formulation is based on a form of the Sanders‐Budiansky shell theory including the effects of both the transverse shear deformation and the laminated anisotropic material response. The fundamental unknowns consist of the eight strain components, the eight stress resultants and the five generalized displacements of the shell. Each of the shell variables is expressed in terms of trigonometric functions (Fourier series) in the circumferential co‐ordinate, and a three‐field mixed finite element model is used for the discretization in the meridional direction.

The shell response associated with a range of Fourier harmonics is approximated by a linear combination of a few global approximation vectors, which are generated at a particular value of the Fourier harmonic, within that range. The full equations of the finite element model are solved for only a single Fourier harmonic, and the response corresponding to the other Fourier harmonics is generated using a reduced system of equations with considerably fewer degrees of freedom.
Exact numerical solutions for radiative heat transfer in cylindrical furnaces
Authors: Selçuk Nevin , Tahiroglu Zeynep
Published: August 2005

Exact expressions for the distributions of the components of radiative flux density and the radiative energy source term in terms of wall and medium temperature distributions have been formulated for an emitting absorbing medium of constant properties bounded by black walls of a cylindrical enclosure. The accuracy of numerical solutions has been tested on an idealized enclosure for which exact analytical solution of the expressions is possible and shown to have six‐figure accuracy. The exact expressions have then been solved numerically for an enclosure problem based on data reported previously on a large scale experimental furnace. The principal feature of the data is highly non‐uniform temperature distributions which are typical of the conditions encountered in industrial furnaces. These data have been chosen because of their practical importance and the non‐availability of exact solutions for such data. The resulting exact solutions have been tabulated and are intended to serve in the future as standards for testing the accuracy of the approximate predictions produced using various three‐dimensional flux models in cylindrical configurations.
Curved composite boundary elements for second‐order two‐dimensional problems
Authors: Silva Paul J. , Mote C. D.
Published: August 2005

Finite elements with a curved edge often require relatively large numerical effort to form.1–4 Relatively simple triangular elements with a single curved edge are developed in this paper for second‐order, two‐dimensional problems. A convex boundary element is formed as a composite of two straight‐edged triangles and a circular sector.

Application of the convex composite boundary element results in less numerical effort for a comparable error in circular and elliptical domain test problems than the application of straight‐edged elements in all cases shown, and in most cases when compared to the curved isoparametric elements. For domains with concave boundaries, the application of straight‐edged, concave composite boundary and curved isoparametric elements give comparable accuracies and numerical efforts because of a fortuitous cancellation of error that occurs with straight‐edged elements in this case.
Non‐local and semi‐local optimal weighting functions for symmetric problems involving a small parameter
Authors: Givoli Dan
Published: August 2005

The method of optimal weighting functions for symmetric problems is described in a general form. It is based on a Petrov‐Galerkin formulation in which the best approximation property and other mathematical features are achieved for a chosen norm, different from the original ‘energy norm’ of the problem. The nonlocality of the weighting functions is shown to have only a minor effect on the efficiency of the method, although a localization scheme is also suggested. The method is applied to a one‐ and two‐dimensional singular perturbation problems, as well as to a cylindrical shell problem.
An eighteen‐node solid element for thin shell analysis
Authors: Ausserer M. F. , Lee S. W.
Published: August 2005

An eighteen‐node, three‐dimensional, solid element with 54 degrees of freedom is presented for the finite element analysis of thin plates and shells. The element is based on the Hellinger‐Reissner principle with independent strain. The assumed independent strain is divided into higher and lower terms. The stiffness matrix associated with the higher order independent strain plays the role of stabilization matrix. A modified stress‐strain relation decoupling inplane and normal strain is used to incorporate thin shell behaviour. Numerical results demonstrate that, with a properly chosen set of assumed strain, this element is effectively free of locking even for very thin plates and shells.
Damped dynamic substructures
Authors: Leung A. Y. T.
Published: August 2005

The dynamic substructure method is extended to lightly or heavily damped systems. Both internal and external dampings are considered. The damped dynamic flexibility associated with the slave co‐ordinates is first expanded in terms of the damped fixed interface natural modes and the condensed dynamic stiffness associated with the master co‐ordinates is formed subsequently. The convergence of the condensed dynamic stiffness with respect to the damped natural modes can be improved by means of the static matrices. Since the dynamic stiffness method is equivalent to the modal synthesis method, the component mode method and Kron's method, the theory presented here is readily applicable to these methods are restricted to symmetric damping matrices.
Boundary element analysis of an integral equation for three‐dmensional crack problems
Authors: Le Van A. , Peseux B.
Published: August 2005

In another paper, the authors proposed an integral equation for arbitrary shaped three‐dimensional cracks. In the present paper, a discretization of this equation using a tensor formalism is formulated. This approach has the advantage of providing the displacement discontinuity vector in the local basis which varies as a function of the point of the crack surface. This also facilitates the computation of the stress intensity factors along the crack edge. Numerical examples reported for a circular crack and a semi‐elliptical surface crack in a cylindrical bar show that one can obtain good results, using few Gaussian points and no singular elements.
Application of the boundary element method to dense dispersions
Authors: Durand Paul P. , Ungar Lyle H.
Published: August 2005

The boundary element method (BEM) is used to find the effective conductivity of a random dispersion of disks in a second material. Small separations between inclusions give rise to difficulties not usually associated with the use of the BEM. We find that the integration point spacing on a given surface element has to be of the order of the shortest distance between disks. The spacing requirement becomes a limitation at high densities, even when the ratio of the conductivity of the disks to that of the host material is close to one. This limitation is overcome by increasing the order of the integration scheme used rather than by increasing the number of surface elements. A more conventional concern is that the surface mesh itself must be sufficiently fine to represent the temperature and flux profiles. As with all simulations of dense dispersions, periodic boundary conditions have to be used and further, because random dispersions are of interest, averages of calculations over many configurations are needed. Taking account of the above considerations we calculate the first exact results for the effective conductivity of a dispersion of cylinders at a high (0.6) density. The influence of vectorization on the performance of the code is briefly mentioned, as are other possible improvements.
A local boundary integral equation method for potential problems
Authors: Fenner R. T. , Watson J. O.
Published: August 2005

Boundary integral equation (boundary element) methods have the advantage over other commonly used numerical methods that they do not require values of the unknowns at points within the solution domain to be computed. Further benefits would be obtained if attention could be confined to information at one small part of the boundary, the particular region of interest in a given problem. A local boundary integral equation method based on a Taylor series expansion of the unknown function is developed to do this for two‐dimensional potential problems governed by Laplace's equation. Very accurate local values of the function and its derivatives can be obtained. The method should find particular application in the efficient refinement of approximate solutions obtained by other numerical techniques.
Finite‐element models of viscoelasticity and diffusion in adhesively bonded joints
Authors: Roy S. , Reddy J. N.
Published: August 2005

The paper contains a description of the finite‐element model based on Schapery's non‐linear viscoelastic constitutive equation for the adhesive, a non‐linear generalized Fickean diffusion model and an updated Lagrangian formulation of a two‐dimensional stress state, to analyse adhesively bonded joints. The finite‐element analysis program, called NOVA, is used to analyse a number of adhesive joints, and the results are compared with existing experimental results and analytical solutions.
Numerical modelling of wave‐induced currents in a breakwater situation
Authors: Yoo D.
Published: August 2005

A numerical model system is developed to describe waves and wave‐induced currents in a breakwater situation, where refraction, diffraction, reflection and wave–current interaction may all play important roles in the transformation of waves. A turbulence transport model is employed to estimate the eddy viscosity, and the friction factors associated with the major variables are clearly defined using a high level closure of combined flow bed friction. The model system is tested against a laboratory situation, and the effects of reflection are demonstrated on the wave‐induced currents as well as on the wave distribution.
Numerical modelling for tide flow and pollutant transport in Taipong Bay, China
Authors: Xiaoyong Zhan
Published: August 2005

A two‐dimensional finite difference alternating direction implicit (ADI) scheme numerical model to solve the vertically integrated form of the non‐linear hydrodynamics equations and convection–diffusion equation has been developed for the description of tide flow and pollutant transport in Taipong Bay, China. Computer software for the modelling is well designed to reduce storage and costs of computation. The computed results agree quite well with the observations available. The tidal level, tidal current, residual current and organic pollution in the bay are discussed based on the computed results and observed data. Applications for the environmental problems along the coast of the bay are also mentioned briefly.
The influence of the form of the log‐conductivity covariance on non‐Fickian dispersion in random permeability fields
Authors: Salandin Paolo , Rinaldo Andrea
Published: August 2005

The paper deals with non‐Fickian dispersion of inert solutes in random permeability fields. Attention is focused critically on the statistical characterization of the porous medium which affects pollutant dispersion in groundwater. After a brief account of recent results of stochastic theories of transport in porous media and of the fundamental indications of large‐scale field experiments, it is inferred from numerical studies that the particular choice of an analytical form of covariance of log‐conductivity has a poor influence on the overall dispersion process. In fact, different covariance structures with the same macroscale (a measure of the distance between two points beyond which the permeability ceases to be correlated) yield very similar dispersion processes. The result has a noteworthy bearing on field studies of pollutant dispersion in groundwater because it underlines the reliability of exponential correlation structures yielding analytical expression for time‐varying macrodispersion coefficients.
An introduction to the NSPCG software package
Authors: Kincaid David R. , Oppe Thomas C. , Joubert Wayne D.
Published: August 2005

NSPCG is the most recent research‐oriented software package developed as part of the ITPACK Project at the Center for Numerical Analysis of The University of Texas at Austin. It is designed to solve large sparse systems of linear equations by a variety of different non‐symmetric preconditioned conjugate gradient methods. Several different sparse data storage schemes are available for storing the coefficient matrix of the linear system. Matrices with a wide range of structures from highly structured to completely unstructured can be accommodated. To limit memory requirements, the accelerators in the package can be called directly with the matrix stored in a user‐defined sparse data storage format (for example, from a particular application) by supplying some user‐written routines for certain matrix–vector operations.

The main entry point into the package is through a single subroutine call. The various methods are accessed by using a particular naming convention for the first two parameters that in turn selects a preconditioner, an accelerator and a data storage scheme. Some of the basic preconditioners that are available are Jacobi, Incomplete LU Decomposition and Symmetric Successive Overrelaxation as well as block preconditioners. The user can select from a large collection of accelerators such as Conjugate Gradient (CG), Chebyshev (SI, for semi‐iterative), Generalized Minimal Residual (GMRES), Biconjugate Gradient Squared (BCGS) and many others. The package is modular so that almost any accelerator can be used with almost any preconditioner. One of the main purposes for the development of the package was to provide a common modular structure for research on iterative methods. The entire package is written in Fortran 77 with vectorization in mind for applications on supercomputers.
On a class of preconditioned iterative methods on parallel computers
Authors: Axelsson O. , Carey G. , Lindskog G.
Published: August 2005

The standard implementations of iterative solvers for finite element and finite difference methods frequently use a diagonal (Jacobi) preconditioner, particularly for element‐by‐element schemes. However, for such methods the actual order of the condition number with respect to mesh size is not reduced by the preconditioner. In the present paper we describe an iterative method where, in addition, the condition number is reduced by an order of magnitude. Moreover, the scheme may also be implemented as an element‐by‐element method. The method uses a generalized SSOR preconditioner and a wave front or multi‐frontal ordering of the mesh nodes. For a general irregular finite element mesh a striped irregular wave front ordering may be used. The performance of the method as well as various iterative acceleration techniques for a parallel computer are examined in the numerical studies.
Numerical analysis of two‐level finite difference schemes for unsteady diffusion–convection problems
Authors: Rigal Alain
Published: August 2005

This paper develops, for linear diffusion–convection model problems, the numerical analysis of some two‐time‐level second order finite difference schemes: modified centred, upstream weighted, Samarskii explicit and implicit schemes. The stability and related properties—dissipation, amplitude and phase errors, numerical diffusion and dispersion—are studied through Fourier analysis. The behaviour of the numerical solution is analysed from the matrix properties of the schemes with special emphasis on parabolicity and positivity. These analyses allow a comparison of the properties of every scheme, the main criterion being the efficiency of the scheme for strongly convective problems.
An operator‐splitting algorithm for two‐dimensional convection–dispersion–reaction problems
Authors: Ding Daoyang , Liu Philip L.‐F.
Published: August 2005

An operator‐splitting algorithm for the two‐dimensional convection–dispersion–reaction equation is developed. The flow domain is discretized into triangular elements which are fixed in time. The governing equation is split into three successive initial value problems: a pure convection problem, a pure dispersion problem and a pure reaction problem. For the pure convection problem, solutions are found by the method of characteristics. The solution algorithm involves tracing the characteristic lines backwards in time from a vertex of an element to an interior point. A cubic polynomial is used to interpolate the concentration and its derivatives on an element. For the pure dispersion problem, an explicit finite element algorithm is employed. Analytical solutions are obtained for the pure reaction problem. The treatment of the boundary conditions is also discussed. Several numerical examples are presented. Numerical results agree well with analytical solutions. Because cubic polynomials are used in the interpolation, very little numerical damping and oscillation are introduced, even for the pure convection problem.
A general finite element for beams or beam‐columns with or without an elastic foundation
Authors: Sirosh S. N. , Ghali A. , Razaqpur A. G.
Published: August 2005

A general finite element is derived for beams or beam‐columns with or without a continuous Winkler type elastic foundation. The need to discretize members into shorter elements for convergence towards an ‘exact’ solution is eliminated by employing in the derivation of the element exact shape functions obtained from the equation of the elastic line. Inter‐nodal values of deflections, bending moments and shear forces are obtained using the exact shape functions and trigonometric series. The effect of heavy compressive or tensile axial forces on bending stiffness is treated as a linear problem by considering the axial force as a constant parameter affecting the stiffness. FORTRAN subroutines to compute the stiffness matrix, equivalent nodal forces, deflected shape, bending moments and shear forces are provided and verified by an example.
Numerical differentiation for use in integrating unified constitutive equations
Authors: Henshall G. A. , Tanaka T. G. , Miller A. K. , Lowe T. C.
Published: August 2005

The NONSS method for numerically integrating time‐dependent constitutive equations requires evaluation of the partial derivatives of the constitutive equations. A numerical method for evaluating these partial derivatives has been developed, thus avoiding the cumbersome task of analytically re‐deriving the partial derivatives when improvements to the constitutive model are evaluated. This method is based upon the adaptive finite‐difference algorithm published by Stepleman and Winarsky. Modifications of their algorithm are described that allow its efficient use within an extended version of NONSS, called NONSS‐ND.

Using the MATMOD constitutive equations, the capabilities of NONSS‐ND are demonstrated by comparing the results of a wide variety of deformation simulations with those of NONSS, for which the derivatives are computed from analytical solutions. The numerical differentiation within NONSS‐ND is shown to be sufficiently accurate for use in integrating time‐dependent constitutive equations, but results in a 30–100 per cent increase in computation time. Multi‐element structural calculations are therefore found to be uneconomical, but NONSS‐ND is well suited for testing alternative constitutive models owing to its accuracy and its general applicability to highly non‐linear systems of differential equations.
Computation of thermal stresses in quasistatic non‐stationary thermoelasticity using boundary elements
Authors: Sládek V. , Sládek J.
Published: August 2005

In this paper we present the boundary integral formulation for the solution of quasi‐static problems of two‐dimensional uncoupled thermoelasticity. A detailed scheme for the numerical solution of the boundary integral equations and computation of internal stresses by regularized integral representation is developed in time formulation. The method is tested in a numerical example.
Duality and symmetry in mixed integral methods of elastostatics
Authors: De Freitas J. A. Teixeira
Published: August 2005

Mixed formulations for integral methods of elastostatic analysis of structures are presented. Both finite and boundary element approaches are considered. A governing system exhibiting symmetry is obtained in both cases by preserving duality in the descriptions of equilibrium and compatibility and reciprocity in the elastic constitutive relations. The theorems of virtual forces and displacements are interpreted as the energetic representation of static–kinematic duality. The quadratic programs associated with the governing system are identified as the theorems of minimum potential energy and co‐energy. Mathematical programming theory is used to establish conditions for the existence and uniqueness of solutions.
Numerical solution of linear two‐point boundary problems via the fundamental‐matrix method
Authors: Asfar O. R. , Hussein A. M.
Published: August 2005

A numerical method is presented for the‐solution of linear systems of differential equations with initial‐value or two‐point boundary conditions. For y′(x) = A(x)y(x) + f(x) the domain of interest [a,b] is divided into an appropriate number L of subintervals. The coefficient matrix A(x) is replaced by its value Ak at a point xk within the Kth subinterval, thus replacing the original system by the L discretized systems yk(x) = Akyk(x) + fk(x), k = 1,2,…, L. The fundamental matrix solution Φk(x, xk) over each subinterval is found by computing the eigenvalues and eigenvectors of each Ak. By matching the solutions yk(x) at the L – 1 equispaced grid points defining the limits of the subintervals and the boundary conditions, the two‐point problem is reduced to solving a system of linear algebraic equations for the matching constants characterizing the different yk(x). The values of y1(a) and yL(b) are used to calculate the missing boundary conditions. For initial‐value problems this method is equivalent to a one‐step method for generating approximate solutions. By means of a coordinate transformation, as in the multiple shooting method,1 the method becomes particularly suitable for stiff systems of linear ordinary differential equations. Five examples are discussed to illustrate the viability of the method.
Infinite boundary elements
Authors: Beer G. , Watson J. O.
Published: August 2005

Special types of boundary elements are discussed which can be used for the modelling of surfaces which extend to infinity. The theoretical background and details of implementation are discussed. On test examples it is shown that the elements perform extremely well even for cases in which they are located close to the area of interest. A practical application of the use of the elements for the modelling of mining excavations is given.
Boundary integration and interpolation procedures for plate bending
Authors: Abdel‐Akher Ahmed , Hartley Gilbert A.
Published: August 2005

In the direct boundary element method (DBEM) for thin plate bending analysis, the integration of the kernel‐shape‐function products can lead to complications due to a combination of factors. The first is the existence of a singularity at the source point, leading to the singular kernel, which increases in strength as the integral equation is differentiated once for rotation, again for moment and yet again for shear. The second factor arises from the use of piecewise interpolation polynomials.

In the first part of the DBEM analysis where the unspecified boundary values are computed, the numerical procedures are fairly straightforward. It is found in this paper that special care must be taken in calculating internal actions at domain points. The problems which can be encountered are demonstrated, and a modified boundary interpolation procedure is described. Examples are given which show the superiority of the recommended approach.
Stress intensity factors by enriched mixed finite elements
Authors: Heyliger P. R. , Kriz R. D.
Published: August 2005

An enriched finite element model for linear elastic fracture mechanics is developed for a mixed variational statement. The independent approximations for the displacement and stress components are enriched by adding the near‐field analytic expressions for a cracked body to the polynomial approximations of a conventional element. This allows for an accurate representation of the stress and displacement fields near the crack tip and also results in the direct calculation of the appropriate stress intensity factors. The accuracy of this formulation is demonstrated through several numerical examples.
Non‐linear steady state vibration of frames by finite element method
Authors: Leung A. Y. T. , Fung T. C.
Published: August 2005

A finite element method based on the virtual work principle to determine the steady state response of frams in free or forced periodic vibration is introduced. The axial and flexural deformations are coupled by mean of the induced axial force along the element. The spatial discretization of the deformations is achieved by the usual finite element method and the time discretization by Fourier coefficients of the nodal displacements. No unconventional element matrices are needed. After applying the harmonic balance method, a set of non‐linear algebraic equations of the Fourier coefficients is obtained. These equations are solved by the Newtonian iteration method in terms of the Fourier coefficient increments. Nodal damping can easily be included by a diagonal damping matrix. The direct numerical determination of the Fourier coefficient increments is difficult owing to the presence of peaks, loops and discontinuities of slope along the amplitude‐frequency response curves. Parametric construction of the response curves using the phase difference between the response and excitation is recommended to provide more points during the rapid change of the phase (i.e. at resonance). For undamped natural vibration, the method of selective coefficients adopted.

Numerical examples on the Duffing equation, a hinged–hinged beam, a clamped–hinged beam, a ring and a frame are given. For reasonably accurate results, it is shown that the number of finite elements must be sufficient to predict at least the linear mode at the frequency of interest and the number of harmones considered must satisfy the conditions of completeness and balanceability, which are discussed in detail.
Adaptive time‐step control in creep analysis
Authors: Mahnken Rolf , Stein Erwin
Published: August 2005

Concerning the integration in time of constitutive relations for Norton‐type creep problems two families of algorithms are examined, namely the generalized trapezoidal rule and the generalized midpoint rule. Based on a mathematical concept, the accuracy properties are derived for both stresses and strains. Since the choice of the step‐size is crucial for the accuracy of the numerical results, an adaptive step‐size procedure is carried out by means of error estimators. Specifically, attention is paid to features arising in the context of a finite element displacement method. Finally, two numerical examples are given to illustrate the effect of the step‐size control on the accuracy.
A frontal technique for vector computers
Authors: Brusa Luisa , Riccio Francesca
Published: August 2005

At the kernel of most scientific computations lies the solution of linear equations which, in finite element codes, is often performed by the frontal method. This method, like many algorithms for sparse matrices, is usually implemented with extensive use of indirect addressing which scarcely benefits from vectorization. As a consequence, present application codes only partially exploit the innovative features of supercomputer architectures. This paper describes a new version of the frontal technique which is designed for CRAY‐like computers and is well suited to vector processing. Numerical tests have shown that the proposed algorithm is effective in improving the efficiency of finite element solutions on a vector machine.
A mixed finite element formulation for Reissner–Mindlin plates based on the use of bubble functions
Authors: Pinsky Peter M. , Jasti Raja V.
Published: August 2005

A new finite element formulation for Reissner–Mindlin plates, based on the Hellinger–Reissner variational principle, is proposed in which the displacement field is additively decomposed into two parts: a part associated with standard interpolation of nodal degrees of freedom and a part associated with a set of independent bubble functions expressed in terms of generalized parameters. The formulation employs independent approximations for every stress component so that the stress field does not a priori satisfy the homogeneous equilibrium equations. It is shown that the bubble functions provide additional variational constraints on the stress field, resulting in optimal accuracy for the mixed formulation, and also eliminating shear locking in the thin limit. A 4‐node and a 9‐node element are described in detail. Both elements pass the patch test for mixed elements described by Zienkiewicz et al.1 and are stable in the sense of the Babuška–Brezzi condition. Numerical results indicate that the elements are accurate in displacements and stresses, including transverse shear, and insensitive to mesh distortion.
A coarse/fine preconditioner for very ill‐conditioned finite element problems
Authors: Farhat Charbel , Sobh Nahil
Published: August 2005

We consider the problem of applying the conjugate gradient method to solve‐ill‐conditioned large algebraic systems of equations resulting from the finite element discretization of some three‐dimensional boundary value problems. We present an effective preconditioner for such systems based on a multigrid technique. We assess its performance with examples borrowed from large flexible aerospace structures.
Mixed‐interpolated elements for Reissner–Mindlin plates
Authors: Brezzi Franco , Bathe Klaus‐Jürgen , Fortin Michel
Published: August 2005

We present in this paper a procedure to establish Reissner–Mindlin plate bending elements. The procedure is based on the idea to combine known results on the approximation of Stokes problems with known results on the approximation of elliptic problems. The proposed elements satisfy the mathematical conditions of stability and convergence, and some of them promise to provide efficient elements for practical solutions.