bluuebunny/crossref_metadata_2025 huggingface dataset:
An integrated sequential solver for large matrix equations
Authors: Vendhan C. P. , Kapoor M. P. , Das Y. C.
Published: August 2005

An integrated sequential computer programme for the solution of large linear algebraic systems encountered in the discrete analysis of complex structures using finite element idealization is described. The programme uses an algorithm based on Gauss elimination for solution and is independent of the possible symmetric nature of the system matrix thus enhancing the applicability of the programme for systems with unsymmetric coefficient matrices. The present programme can handle large band widths and the core memory is used efficiently for all feasible problem sizes. A FORTRAN print‐out of the programme with relevant explanatory comments and a full notation list are appended.
Exponential fitting using integral equations
Authors: Moore Edward
Published: August 2005

The fitting of a function y =\documentclass{article}\pagestyle{empty}\begin{document}$ \sum\nolimits_{i = 1}^n {A_i {\rm{e}}^{\lambda ix}} $\end{document} to experimental data is considered. Integral equations are developed which have the functions \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\nolimits_{i = 1}^n {A_i {\rm{e}}^{\lambda ix}} $\end{document} as their solutions for n = 1, 2, 3, These integral equations are used to find the frequencies λi. Examples are worked out to illustrate the method. The method is shown to be capable of extension to other functions.
Semi‐analytical finite element procedure for conduction in anisotropic axisymmetric solids
Authors: Padovan Joseph
Published: August 2005

Through the use of complex series representations and the properties of adjoint differential operators, a semi‐analytical finite element procedure is developed which can analyse the steady state and transient temperature fields of thermally anisotropic axisymmetric bodies with complete circumferential properties. Due to its generality, the procedure can handle arbitrary laminate construction with possible meridional and radial variations in locally or globally thermally anisotropic materials. In this respect, the procedure developed herein supercedes the classical semi‐analytical treatment which is limited to specially orthotropic materials. Furthermore, in contrast with the classical procedure, the present treatment reveals several important effects of material anisotropy which are entirely missed by the specially orthotropic assumption.
Numerical quadrature used for radiative heat‐transfer computations
Authors: Look D. C. , Love T. J.
Published: August 2005

Physical reasoning makes it possible to use artificial parameters with numerical quadrature in order to obtain approximate solutions for the Fredholm integral equations associated with radiative heat transfer calculations. With the aid of such parameters, decisions can be made concerning the applicability and usefulness of the numerical quadrature technique.
Effect of time‐dependent material properties on dynamic response
Authors: Yamada Y. , Takabatake H. , Sato T.
Published: August 2005

By representing the time‐dependent material properties by mechanical models, i.e. the generalized Maxwell and/or Voigt (or Kelvin) models, it is possible to formulate a consistent procedure to analyse the effect of material damping on dynamic response. As a continuation of the previous work which concerned the creep and relaxation of viscoelastic materials, the present paper applies this representation to the harmonic response analysis. The harmonic solution can be obtained by decomposing the relevant forces and displacements into real and imaginary parts and it is shown that the modal analysis by an appropriate eigensolution procedure is useful for obtaining the dependence of vibration amplitude on frequencies of sinusoidal external excitations.
On the treatment of boundary conditions by the method of artificial parameters
Authors: Kao Robert
Published: August 2005

A procedure to deal with various boundary conditions in the Rayleigh‐Ritz method is discussed herein, namely, the method of artificial parameters. The method is then applied to solve a simple string problem; results obtained compare very nicely with exact solutions.
Local and global smoothing of discontinuous finite element functions using a least squares method
Authors: Hinton E. , Campbell J. S.
Published: August 2005

The concepts and potential advantages of local and global least squares smoothing of discontinuous finite element functions are introduced.

The relationship between local smoothing and the ‘reduced’ integration' technique is established.

Examples are presented to illustrate the application of the two smoothing techniques to the finite element stresses from several structural analysis problems.

The paper concludes with some practical recommendations for discontinuous finite element function smoothing.
Transient two‐dimensional heat conduction problems solved by the finite element method
Authors: Bruch John C. , Zyvoloski George
Published: August 2005

A finite element weighted residual process has been used to solve transient linear and non‐linear two‐dimensional heat conduction problems. Rectangular prisms in a space‐time domain were used as the finite elements. The weighting function was equal to the shape function defining the dependent variable approximation. The results are compared in tables with analytical, as well as other numerical data. The finite element method compared favourably with these results. It was found to be stable, convergent to the exact solution, easily programmed, and computationally fast. Finally, the method does not require constant parameters over the entire solution domain.
Finite element solution of non‐linear heat conduction problems with special reference to phase change
Authors: Comini G. , Del Guidice S. , Lewis R. W. , Zienkiewicz O. C.
Published: August 2005

The paper presents a generally applicable approach to transient heat conduction problems with non‐linear physical properties and boundary conditions. An unconditionally stable central algorithm is used which does not require iteration.

Several examples involving phase change (where latent heat effects are incorporated as heat capacity variations) and non‐linear radiation boundary conditions are given which show very good accuracy.

Simple triangular elements are used throughout but the formulation is generally valid and not restricted to any single type of element.
Automatic triangulation of arbitrary planar domains for the finite element method
Authors: Cavendish James C.
Published: August 2005

The object of this paper is to describe a new algorithm for the semi‐automatic triangulation of arbitrary, multiply connected planar domains. The strategy is based upon a modification of a finite element mesh genration algorithm recently developed. 1 The scheme is designed for maximum flexibility and is capable of generating meshes of triangular elements for the decomposition of virtually any multiply connected planar domain. Moreover, the desired density of elements in various regions of the problem domain is specified by the user, thus allowing him to obtain a mesh decomposition appropriate to the physical loading and/or boundary conditions of the particular problem at hand. Several examples are presented to illustrate the applicability of the algorithm. An extension of the algorithm to the triangulation of shell structures is indicated.
Projective method applied to three‐dimensional elasticity equations
Authors: Schmidt William F.
Published: August 2005

It is shown how the elasticity equations can be formulated in terms of a general procedure of abstract analysis, called the projective method, for constructing an approximate solution. This allows the many studies of convergence and uniqueness of solutions which have been made in modern mathematics to be applied to the elasticity problem. The abstract formulation also suggests an organizational scheme for generating a computer code. A computer code for hollow or solid cylinders of arbitrary cross section was developed and several numerical examples are presented. The procedure is computationally advantageous for the class of problems under consideration because of the relatively small number of equations that must be solved as compared to the finite element method. The results presented demonstrate the accuracy of the method.
A finite element analysis in polar co‐ordinates of the Saint Venant torsion problem
Authors: Heise Ulrich
Published: August 2005

A characteristic feature of the variational functionals for several boundary value problems in polar co‐ordinates is the fact that one independent variable occurs explicitly in the denominator. Therefore, the coefficients of the finite element equations for sectors of circular ring shaped elements are not constants but functions of the distance of the elements from the origin of co‐ordinates.1 We name them coefficient functions. In order to show the particular aspects of the calculations in terms of polar co‐ordinates we deal here with the solution of the torsion problem by bilinear and bicubic Hermitian interpolation.

The finite element equations are arranged according to Schaefer2 in the form of block which can easily be transformed into ‘stars’3 or molecules4,5 similar to those used in finite difference methods. The origin of co‐ordinates requires a special consideration, firstly because of the coincidence of several nodes at that point and secondly because of the divergent behaviour of some coefficient functions. It turns out to be advantageous for the numerical calculations to expand the coefficient functions in power series. Besides, the expansions are required to deduce the equations for rectangular elements by limiting processes. The twisting moments and shearing stresses calculated for several cross sections illustrate the numerical suitability of the method. The finite element values are compared partly with exact solutions and partly with experimental results obtained by a moiré method using Prandtl's soap film analogy.6 Finally it is shown how the accuracy of the finite element values can be improved by the Richardson extrapolation7.
Elastoplastic analysis of hollow spheres subjected to external pressure loading
Authors: Grant Donald A.
Published: August 2005

The elastoplastic deformation of spherical shells of uniform thickness acted upon by external pressure is investigated in this paper. A deformation theory of plasticity is combined with the classical elasticity equations and a solution is obtained by the method of successive elastic solutions. A logarithmic effective stress‐effective plastic strain diagram is used to determine the volume change for thick polypropylene spheres under the action of external pressure. These results are compared with previously published experimental volume changes in polypropylene spheres. This comparison shows that the predicted volume changes are similar to the experimental results.
Visco‐plasticity—plasticity and creep in elastic solids—a unified numerical solution approach
Authors: Zienkiewicz O. C. , Cormeau I. C.
Published: August 2005

The visco‐plastic model of material behaviour is of much practical interest in its own right and initial strain techniques for its solution have been developed and proved efficient. More important however is the fact that the visco‐plastic model can be used to generate plasticity solutions in a simple manner when stationary conditions are reached and, at the other extreme, can reproduce standard creep phenomena. Used in this sense it allows the treatment of non‐associated plasticity and strain softening situations which present difficulties in conventional plasticity approaches. A standard programme thus allows the treatment of a wide range of materially non‐linear problems.

The paper discusses various applications of the new general formulation and introduces certain numerical information on solution stability.
Imperfections in axially loaded plane frames
Authors: Harung H. S. , Millar M. A. , Brotton D. M.
Published: August 2005

A finite beam‐column element computer program for the structural failure analysis of proportionally loaded skeletal plane frames is described. The program, which is based on an iterative stiffness approach, provides for non‐linear elastic or plastic‐non‐linear elastic behaviour. Particular attention is given to the incorporation into the analysis of geometrical imperfections, return of elasticity at plastic hinges and negative frame stiffness. The effect of lack‐of‐fit displacements on the load‐deflection path and failure load of frames that carry relatively large axial loads is described in general terms. The theory is developed further through the application of the computer program to two examples which show the necessity for the allowing for the effect of imperfections when designing axially loaded redundant frames.
Least squares finite element analysis of laminar boundary layer flows
Authors: Lynn Paul P.
Published: August 2005

Based on the least squares error criterion, a class of finite element is formulated for the numerical analysis of steady state viscous boundary layer flow problems. The method is essentially a discrete element‐wise minimization of square and weighted residuals which arise from the attempts in approximately satisfying boundary layer equations. An iterative linearization scheme is developed to circumvent the mathematical difficulties posed by the non‐linear boundary layer equations. It results in a process of successive least squares minimizations of residual errors arising from satisfying a set of linear differential equations. A mathematical justification for the method is presented. A major feature of the method lies in the linearization approach which renders non‐linear differential equations amenable to linear least squares finite element analysis. Another important feature rests on the proposed finite element formulation which preserves the symmetric nature of finite element matrix equations through the use of the least squares error criterion. Numerical examples of viscous flow along a flat plate are presented to demonstrate the applicability of the method as well as to illuminate discussions on the theoretical aspects of the method.
Eigenproblem solution of damped structural systems
Authors: Gupta K. K.
Published: August 2005

This article presents an efficient numerical algorithm a complete listing of the associated computer program, developed for the eigenproblem solution of discrete damped structures, including spinning ones. The numerically stable procedure is based on a combined Sturm sequence and inverse iteration technique, which fully exploits the banded form of the relevant matrices and proves to be most substantially economical when compared to similar existing softwares. Numerical results are presented for representative structures, solved by the present computer program written in FORTRAN V for the JPL UNIVAC 1108 computer.
Redundant structure concepts in finite element analyses
Authors: Meek J. L.
Published: August 2005

The paper examines two triangular finite elements used in plate bending analyses for which Lagrange multiplier techniques have been used to set up the necessary equations for the solution. These equations involve either equilibrium of the system or the inter‐element compatibility. Starting from only the notions of equilibrium and continuity as used in the analyses of redundant structures, the paper proceeds to solutions of the same problems. It is found that end results are the same when the equations obtained from the minimization techniques are reduced by Gauss‐elimination in partitioned from.
Three dimensional analysis of compressible potential flows with the finite element method
Authors: Periaux J.
Published: August 2005

Solutions of elliptic boundary value problems are obtained with variational formulations by minimization of functionals.

Conventional tools of the finite element strategy are emphasized; linear or quadratic approximations on triangular elements, curved iso‐parametric elements, Ni shape funtions, Li homogeneous co‐ordinates. High speed of computation of the final assembled matrices require the use of the Formal–FORMAC calculus. Illusrations of the finite element method are chosen in the class of the 2d–3d compressible subsonic irrotational flows in a nozzle and around one or several lifting bodies of arbitary shape.
Finite elements for the vibration of cones and cylinders
Authors: Ross C. T. F.
Published: August 2005

Elemental mass matrices have been produced for the vibration of conical and cylindrical shells, based on a semi‐analytical approach. Frequencies and modes of vibration have been compared with existing solutions and also with experimental results obtained from other sources. Good agreement has been found between theory and experiment for thin‐walled circular cylinders and cones, a cone‐cylinder combination, and a cooling tower model.

A theoretical investigation was also made on the vibration of a circular cylinder when subjected to uniform pressure.
The economical solution of elastic problem for a range of Poisson's ratio
Authors: Booker J. R. , Small J. C.
Published: August 2005

In this paper an economical method for calculating the stress and displacement fields in an elastic body for a range of Poission's ratios is given. The solution is expanded as a power series in Poission's ratio, the coefficients of the series being determined successively. The range of convergence of the solution is examined, and it is shown that the power series converges for values of Poission's ratio in the range zero to a half, provided a suitable point of expansion is chosen. Particular features of the method are firstly that only one effective inversion of the stiffness matrix, for Poission's ratio zero, is required to obtain the solution for all Poission's ratio and secondly that no special formulation for an incompressible material is required.
A comparison of Crandall and Crank–Nicolson methods for solving a transient heat conduction problem
Authors: Moore A. H. , Kaplan B. , Mitchell D. B.
Published: August 2005

A comparison is made of the Crandall and Crank–Nicolson methods applied to a transient heat conduction problem with a convective boundary layer on its surface for a range of values of Nusselt modulus and Fourier modulus. A new equation is derived to handle the convective boundary condition. For reasonable accuracy requirements the Crank–Nicolson method gives superior performance with respect to accuracy.
Optimum design of stiffened cylindrical panels with constraint on the frequency in the presence of initial stresses
Authors: Patnaik S. N. , Sankaran G. V.
Published: August 2005

This paper presents the optimum design of stiffened cylindrical panels with weight as the objective function and constraints on the frequencies in the presence of initial stresses, using unconstrained minimization techniques of non‐linear mathematical programming problem. The interaction between the buckling constraints and the frequency constraints in the presence of initial stresses are included in the formulation. Loss of load carrying capacity due to imperfection and due to suddenly applied loads are included in the buckling analyses. Results for cylindrical shell are obtained by setting the panel angle to 360 degrees. The relationship between the weight of the cylinder and the panel (a segment of the cylinder) is examined.
Solution of plane elasticity problems by the displacement discontinuity method. I. Infinite body solution
Authors: Crouch S. L.
Published: August 2005

This paper is concerned with the development of a numerical procedure for solving complex boundary value problems in plane elastostatics. This procedure—the displacement discontinuity method—consists simply of placing N displacement discontinuities of unknown magnitude along the boundaries of the region to be analyzed, then setting up and solving a system of algebraic equations to find the discontinuity values that produce prescribed boundary tractions or displacements. The displacement discontinuity method is in some respects similar to integral equation or ‘influence function’ techniques, and contrasts with finite difference and finite element procedures in that approximations are made only on the boundary contours, and not in the field. The method is illustrated by comparing computed results with the analytical solutions of two boundary value problems: a circular disc subjected to diametral compression, and a circular hole in an infinite plate under a uniaxial stress field. In both cases the numerical results are in excellent agreement with the exact solutions.
Multivariable approach to the finite difference solution of elasto‐plastic problems
Authors: Brown David K.
Published: August 2005

The solutions of plane elasto‐plastic problems usually use one or two field variables—namely a stress functions or the displacements. The use of three, four or five field variables is investigated and it is concluded that the three stresses from the best basis for a multivariable approach. Attempts to solve the governing equations with an initial value technique were exhaustively tried and discarded in preference to a boundary value or elliptical technique. The problem solved to check the method is that of a hole in plane strain uniaxial tension. Effective plastic strain distributions are plotted, along with stress and strain concentration data and distributions of the mean to effective stress ratio. Comparison is made between solutions produced using incremental and deformation theories.
Frontal solution program for unsymmetric matrices
Authors: Hood P.
Published: August 2005

A frontal solution program is presented which may be used for the solution of unsymmetric matrix equations arising in certain applications of the finite element method to boundary value problems. Based on the Gaussian elimination algorithm, it has advantages over band matrix methods in that core requirements and computation times may be considerably reduced; furthermore numbering of the finite element mesh may be completed in an arbitrary manner. The program is written in FORTRAN and a glossary of terms is provided.
Finite element method applied to the supersonic flutter of circular cylindrical shells
Authors: Bismarck‐Nasr Maher N.
Published: August 2005

The application of the finite element method to the supersonic flutter of circular cylindrical shells subjected to internal pressure and axial compression is presented. A circular cylindrical shell element is used. The element stiffness, mass and initial stiffness matrices are given. The element aerodynamic matrix is derived based on a first order high Mach number approximation to the linear potential flow theory. The eigenvalue problem is solved by the QR algorithm. Numerical results are presented and these are compared with analytical solutions and experimental data.
Comparison of curve fitting techniques
Authors: Deshpande R. B. , Shankaramurthy B. K. , Revankar C. P.
Published: August 2005

Various methods of curve fitting used in solving aero‐thermodynamic problems of turbomachines have been compared. The first and second derivatives predicted by using orthogonal polynomial fits are much more accurate than those obtained by using spline fits (with different boundary conditions) and method of least squares.
Some new rational approximations to the error function
Authors: Ersoy O.
Published: August 2005

Some new rational approximations to the error function erf(x) are discussed. They have the same asymptotic behaviour as erf(x), and are more accurate than the ones previously published.
An efficient iterative method for numerical evaluation of integrals over a semi‐infinite range
Authors: Squire William
Published: August 2005

A procedure for numerical quadrature over a semi‐infinite range is described which does not require storing weights or nodes. It is applicable to a wide variety of integrands including oscillatory functions and forms with integrable singularites. A computer program and numerical examples are presented.
A Block–Stodola eigensolution technique for large algebraic systems with non‐symmetrical matrices
Authors: Dong Stanley B.
Published: August 2005

A Block–Stodola eigensolution method is presented for large algebraic eigensystems of the form AU = λBU where A is real but non‐symmetric. The steps in this method parallel those of a previous technique for the case when both A and B were real and symmetric. The essence of the technique is simultaneous iteration using a group of trial vectors instead of only one vector as is the case in the classical Stodola–Vianello iteration method. The problem is then transformed into a subspace where a direct solution of the reduced algebraic eigenvalue problem is sought. The main advantage is the significant reduction of computational effort in extracting a subset of eigenvalues and corresponding eigenvectors. Theorems from linear algebra serve to underlie the basis of the present technique. Complex eigendata that emerge during iteration can be handled without doubling the size of the problem. Higher order eigenvalue problems are reducible to first order form for which this technique is applicable. The treatment of the quadratic eigenvalue problem illustrates the details of this extension.
Finite element analysis of anisotropic plates
Authors: Noor Ahmed K. , Mathers Michael D.
Published: August 2005

Two aspects of the finite element analysis of mid‐plane symmetrically laminated anisotropic plates are considered in this paper. The first pertains to exploiting the symmetries exhibited by anisotropic plates in their analysis. The second aspect pertains to the effects of anisotropy and shear deformation on the accuracy and convergence of shear‐flexible displacement finite element models. Numerical results are presented which show the effects of increasing the order of approximating polynomials and of using derivatives of generalized displacements as nodal parameters.
Integration over the unit triangle
Authors: Akin J. E.
Published: August 2005

An integration formula for polynomial expressions in terms of the unit triangle co‐ordinates is presented.
Extended numerical integration method for triangular surfaces
Authors: Lague Gerald , Baldur Roman
Published: August 2005

The paper includes a technique for numerical integration over the surface of a triangle. The mathematical transformation which is described, overcomes the difficulties associated with the derivation of new quadrature coefficients and yet leads to results which are accurate.

An application to an axisymmetric finite element method with triangular elements is also included.
Automatic generation of stiffness matrices for finite element analysis
Authors: Cecchi Maria Morandi , Lami Carlo
Published: August 2005

Procedures for the automatic generation of symbolic stiffness matrices are presented that greatly reduce the amount of work involved in the construction of finite element stiffness matrices. The related system of programs is presented which completely automates the matrix generation and an example is given.
Optimum design of laminated fibre composite plates
Authors: Schmit L. A. , Farshi B.
Published: August 2005

A method is presented for the minimum weight optimum design of laminated fibre composite plates, subject to multiple inplane loading conditions, which includes stiffness, strength and elastic stability constraints. The buckling analysis is based on an equivalent orthotropic plate approach leading to two uncoupled eigenproblems per load condition. Overall computational efficiency is achieved by using constraint deletion techniques in conjunction with Taylor series approximations for the constraints retained. The optimization algorithm used, namely the method of inscribed hyperspheres, is a sequence of linear programs technique which exhibits rapid convergence in this application. Several example problems are given to demonstrate that the method presented offers an efficient and practical optimum design procedure for the fundamental and recurring problem treated.
A parametric cubic modelling system for general solids of composite material
Authors: Stanton E. L. , Crain L. M. , Neu T. F.
Published: August 2005

A new approach to modelling solids that are anisotropic and heterogeneous is presented with applications to structures of composite material. A parametric cubic modelling system is presented for lines, surfaces, volumes, and physical data that uses construction‐in‐context to generate numerical data. This system automates the construction of discrete element models and can reduce input data requirements by more than an order of magnitude. A tricubic isoparametric discrete element is presented that does not require displacement derivatives to define connectivity. This element is capable of exact displacement and strain continuity over a surface while permitting strain discontinuities at heterogeneous material interfaces. The shape of an element can be any hexahedron, pentahedron, or tetrahedron and the material properties are allowed to vary over the volume. Evaluation of modelling error with respect to closed‐form solutions for curved geometries indicate a single element can model up to 90‐degree segments with stresses accurate to 1 per cent. Applications of the system to composite structures are presented for interlaminar edge effects and attachment stresses in a sandwich panel.
Determination of eigenvalues of large structural systems in an arbitrarily specified range
Authors: Kaul Maharaj K.
Published: August 2005

For a large structural system of small bandwidth a technique combining linear interpolation on the characteristic polynomial and suppression of its previously determined roots by deflation can be used to determine its eigenvalues. The eigenvalues, however, have to be found in order, beginning from either the lowest (or the highest) to obtain monotonic convergence to the polynomial roots. A method which eliminates this restriction is presented in this paper. If eigenvalues greater than a are desired, the method, in essence, consists of modifying the deflation procedure to suppress all the eigenvalues smaller than a without actually determining them. Some consequences of the procedure developed in this paper are also presented and it is shown that the standard deflation technique is a special case of this procedure. The method has been applied to a wide range of problems with success and considerable saving in computation time.
A finite element technique for non‐deterministic thermal deformation analyses including temperature dependent material properties
Authors: Case William R. , Walston William H.
Published: August 2005

A technique utilizing the finite element displacement method is developed for the static analysis of structures subjected to non‐deterministic thermal loading in which the material properties, assumed isotropic, are temperature dependent. Matrix equations are developed for the first two statistical moments of the displacements using a third order series expansion for the displacements in terms of the random temperatures.

Sample problems are included to demonstrate the range of applicability of the third order series solutions. These solutions are compared with results from Monte Carlo analyses and also, for some problems, with solutions obtained by numerically integrating equations for the statistical properties of the displacements. In general, it is shown that the effect of temperature dependent material properties can have a significant effect on the covariances of the displacements.
A finite element free boundary formulation for the problem of multiphase heat conduction
Authors: Wellford L. Carter , Ayer Robert M.
Published: August 2005

The problem of heat conduction in a multiphase medium is a free boundary problem in which the free boundary is the phase transition line. The solution to this problem is highly irregular in that the temperature gradient is discontinuous at the free boundary. Two‐dimensional, discontinuous, space–time finite elements are introduced to obtain a finite element, free boundary formulation for this problem. The jump in heat flux and position of the phase transition line become dependent variables in the finite element model. The resulting models give accurate results while allowing the calculation of free boundary problems with a fixed finite element mesh.
Static stability of plates using fully conforming element
Authors: Abbas B. A. H. , Thomas J.
Published: August 2005

In this paper the geometric stiffness matrices of a 12 degree‐of‐freedom rectangular element based on a fully compatible displacement field are developed for the first time and used to solve the elastic stability problem of rectangular plates under combined in‐plane forces.

The rapid convergence of buckling solutions based on the derived geometric stiffness matrices is demonstrated for plates with various boundary and loading conditions. Comparison of results shows that the present conforming element gives in all cases better results than the well known 12 degree‐of‐freedom non‐conforming rectangular element.
An alternate finite difference operator for plates of variable stiffness
Authors: Bradley W. A.
Published: August 2005

A finite difference approximation for the equilibrium equation of a thin, laterally loaded, elastic plate of variable stiffness is developed; it uses plate stiffnesses at node points and at points midway between the nodes of the rectangular grids. Displacement and moment results using this operator and an alternate one are compared.
Control of Crank‐Nicolson noise in the numerical solution of the heat conduction equation
Authors: Wood W. L.
Published: August 2005

The stiff equations resulting from the finite element discretization of the heat conduction equation are very susceptible to Crank‐Nicolson noise. This paper shows that the forcing function can be turned on (or the surface temperature raised to a steady value) in such a way as to make the noise produced by it considerably reduced by making it proportional to 1–e −βt.
An energy basis for mesh refinement of structural continua
Authors: Melosh R. J. , Marcal P. V.
Published: August 2005

This paper proposes an energy measure of discretization error and examines its use for finite element mesh refinement in the analysis of structural continua.

The measure is based on the strain energy contributions by the admissible displacement response modes of an element. An element energy differential is obtained by separating the energy contribution due to the higher displacement modes. This measure is suitable for use with all element types based on the direct stiffness method.

The paper presents results of membrane, plate and shell analyses using the measure. It compares sequences of analysis with successively improved meshes to explore the quality of the measure. It concludes that the element energy difference provides a quantitative measure of the efficiency of a given mesh and a qualitative measure which is useful for selecting further mesh refinements, when necessary.
Periodic Galerkin finite element method of unsteady periodic flow of viscous fluid
Authors: Kawahara Mutsuto
Published: August 2005

Finite element analysis of unsteady flow of two‐dimensional incompressible viscous fluid is presented employing stream function as a field variable. The Galerkin approach is applied in the procedure of discretizing in time. Using trigonometric series as the interpolation function, the non‐linear algebraic simultaneous equation system is derived. Several numerical examples show the adaptability of the formulation.
Efficient numerical treatment of periodic systems with application to stability problems
Authors: Friedmann P. , Hammond C. E. , Woo Tze‐Hsin
Published: August 2005

Two efficient numerical methods for dealing with the stability of linear periodic systems are presented. Both methods combine the use of multivariable Floquet–Liapunov theory with an efficient numerical scheme for computing the transition matrix at the end of one period. The numerical properties of these methods are illustrated by applying them to the simple parametric excitation problem of a fixed end column. The practical value of these methods is shown by applying them to some helicopter rotor blade aeroelastic and structural dynamics problems. It is concluded that these methods are numerically efficient, general and practical for dealing with the stability of large periodic systems.
A mathematical programming method for the inelastic analysis of reinforced concrete frames
Authors: Kaneko Ikuyo
Published: August 2005

A mathematical programming method is presented for solving problems of (i) determining the moments and rotations and (ii) determining the safety factor of reinforced concrete frames. The proposed method is essentially a reformulation of those developed by De Donato and Maier. However, the reformulation results in a new mathematical programming model for which an efficient algorithm is developed. The advantage of the proposed method is mainly computational; in fact it requires roughly one half computer time and storage space as compared with those De Donato and Maier employ to solve the same problem.
Finite element method applied to the flutter of two parallel elastically coupled flat plates
Authors: Bismarck‐Nasr Maher N.
Published: August 2005

The application of the finite element method to the supersonic flutter of two parallel elastically coupled flat plates subjected to a pre‐stress state of in‐plane loadings is presented. Numerical results are given and these are compared with previous analytical solutions. It is shown that, for the coupled plates, the first spanwise modes are not necessarily the critical modes for the flutter boundaries. Previous analytical results do not exhibit such a trend since only the first spanwise modes were considered.
Analysis of power type singularities using finite elements
Authors: Tracey Dennis M. , Cook Thomas S.
Published: August 2005

A finite element formulation is described for problems with solution functions known to have local rλ variation (s), 0<λ<1, and thus singular gradients. Special 3‐node triangular elements encircle the singularity and focus to share a common node at the singular point. The shape function of each triangle has the appropriate r λ mode and a smooth angular mode expressed in element natural co‐ordinates. As with standard elements, the unknowns are the nodal values of the function. Even if the precise angular form of the asymptotic solution is known, the formulation makes no attempt to embed it, but instead piecewise approximates it. This allows assembly of the element coefficient matrix using standard procedures without nodeless variables and bandwidth complications.

The conditions of continuity, low order solution capability, and accurate numerical integration of the singularity element are discussed with a view towards establishing the general range of applicability of the formulation. Numerical applications to the elastic fracture mechanics problems of composite bondline cracking and crack branching are discussed.
Computer program for solution of large, sparse, unsymmetric systems of linear equations
Authors: Gupta S. K. , Tanji K. K.
Published: August 2005

A computer program is presented for in‐core solution of a large, sparse, unsymmetric, unbanded system of linear equations. The program employs two partially packed arrays (one for storing non‐zero elements, and one for column identifications). Used as the pivotal row is the row with the minimum number of non‐zero elements. To avoid instability, the pivot is the largest absolute element in the pivotal row. The method was tested on a system of equations encountered in field application of the three‐dimensional Galerkin finite element solution of flow and mass transport through porous media. Performance is compared with that of available alternatives.
The steady state response of geometrically non‐linear shallow arches
Authors: Hitchings D. , Ward P.
Published: August 2005

The non‐linear steady state response of structures with curvature is investigated through the expository example of a shallow circular arch. A consistent mass finite element formulation is employed to derive the governing non‐linear differential equations.

These equations are solved by assuming a single mode expansion reducing the governing equations to the single degree‐of‐freedom Duffing's equation with a quadratic term. The non‐symmetric amplitude‐frequency curve is derived and compared with results previously obtained by direct integration of the equations of motion.
Diffraction and refraction of surface waves using finite and infinite elements
Authors: Bettess P. , Zienkiewicz O. C.
Published: August 2005

The wave problem is introduced and a derivation of Berkhoff's surface wave theory is outlined. Appropriate boundary conditions are described, for finite and infinite boundaries. These equations are then presented in a variational form, which is used as a basis for finite and infinite elements. The elements are used to solve a wide range of unbounded surface wave problems. Comparisons are given with other methods. It is concluded that infinite elements are a competitive method for the solution of such problems.
A simple explicit and unconditionally stable numerical routine for the solution of the diffusion equation
Authors: Johns Peter B.
Published: August 2005

This paper describes a simple explicit and unconditionally stable numerical routine for the solution of the diffusion equation using a transmission‐line modelling (TLM) method. The paper also shows that the explicit finite difference routine and the implicit Crank–Nicolson routine may be expressed as the exact solution of certain transmission‐line models. Using these models a technique for comparing the accuracy and stability of numerical routines is developed and a detailed comparison of the new TLM methods and the well established methods is made.
More about ‘artificial’ softening of finite elements
Authors: Cook Robert D.
Published: August 2005

A method is proposed for making overly stiff elements more flexible. The central idea is quite simple and is illustrated by application to a sample element, whose behaviour is greatly improved. The effort needed to soften an element is very small if its stiffness matrix is naturally produced with reference to generalized co‐ordinates before a final transformation to nodal displacement degrees‐of‐freedom.
The influence of the choice of connectors in the finite element method
Authors: Sander G. , Beckers P.
Published: August 2005

The patch test is interpreted from a variational point of view. This leads to the conclusion that non‐conforming (or non‐stress diffusive) displacement (stress) models can be considered as special hybrid models. A simple choice of mixed connectors is proposed that satisfy a priori the patch test for any assumed displacement or stress field. Finally some numerical examples are given of finite elements that do not pass the patch test but still exhibit convergence in practical applications.
Numerical solution of the point contact problem using the finite element method
Authors: Oh K. P. , Rohde S. M.
Published: August 2005

The elastohydrodynamic lubrication problem, in which the lubricant pressure and film thickness are sensitive to surface deformation, is solved by using a finite element procedure and the Newton method. The numerical procedure is applied to the point contact problem, in which a thin lubricant film is maintained between two balls loaded together by a high load under conditions of pure rolling. The present analysis shows that pressure spikes are formed near the outlet region, a result which has been found in the line contact problem and which has been conjectured in the present problem.
On the Zienkiewicz four‐time‐level scheme for the numerical integration of vibration problems
Authors: Wood W. L.
Published: August 2005

The Zienkiewicz Weighted Residual Method produces a very general (three‐parameter) four‐time‐level scheme for the numerical integration of vibration problems. This paper obtains the values of these parameters which minimize the incremental phase error and gives a general method for testing the stability and artificial damping with any other choice of the parameters.
A simple and efficient element for axisymmetric shells
Authors: Zienkiewicz O. C. , Bauer J. , Morgan K. , Onate E.
Published: August 2005

A two noded, straight element which includes shear deformation effects is presented and shown to be extremely efficient in the analysis of axisymmetric shells. A single point of numerical integration is essential for its success when applied to thin shells where the results compare favourably with those achieved with more complex curved elements.
Application of the finite element technique combined with the collocation method to low supersonic lifting surface problems
Authors: Hashimoto M. , Washizu K.
Published: August 2005

In the present paper, a finite element technique combined with the collocation method is formulated to obtain numerical solutions of integral equations which determine the airload distributions acting on a wing oscillating sinusoidally in a uniform supersonic flow. This technique, which is formulated by the simultaneous use of two integral equations, is shown to be appropriate especially for the low supersonic regions, i.e., for 1 < M < 1·3, where M is the Mach number of the uniform flow. Although the formulation is made for a sinusoidally oscillating wing, it is obvious that the corresponding formulation for a stationary wing can be derived as a special case of that for the oscillating wing. Problems of determining the airload distributions for stationary as well as sinusoidally oscillating wings are treated, and it is shown that the accuracy of numerical results obtained by the present finite element technique is encouraging.
Derivation of general algebraic constraint conditions for ‘weak’ C1 continuity for thin shells
Authors: Mang Herbert A.
Published: August 2005

Employing C0 conforming thin shell elements, a derivation of general algebraic equations for enforcing C1 interelement continuity in a ‘weak’ (that is, integral) sense is presented. ‘Restoration’ of strict C1 continuity is treated as a special case of the proposed concept. The cónstraint equations do not depend on the metric of the given shell. While, for smooth shells, this is automatically the case, if strict C1 continuity is ‘restored’, the constraint equations usually depend on the metric of the shell, if only ‘weak’ C1 continuity is enforced. The independence of the proposed constraint equations of the metric of the shell facilitates the computer implementation of the proposed approach. It is demonstrated that linear dependencies among the constraint equations can easily be detected and a priori be eliminated. It is also shown that, in certain cases, it is very easy to switch from an (intrinsically) element‐interface‐oriented concept to an element‐oriented technique of generating constraint equations with the help of the digital computer. The latter mode offers computational advantages, if an element‐oriented mode of solving the global system of algebraic equations (equilibrium and constraint equations), such as Irons' wave front technique, is adopted.
Geometrically non‐linear analysis—A correlation of finite element notations
Authors: Wood R. D. , Schrefler B.
Published: August 2005

A correlation is geven between two notations currently used in the formulation of geometrically non‐linear analysis in a Lagrangian co‐ordinate system. The first referred to as the B‐notation results in the linear elastic, initial displacement and initial stress matrices, whilst the second, here called the N‐notation features matrices that repeat in the total potential energy, equilibrium and incremental equilibrium equations. An explicit example of the correlation is presented for a pin‐jointed bar element.