bluuebunny/crossref_metadata_2025 huggingface dataset:
Solution of quadratic matrix equations for free vibration analysis of structures
Authors: Gupta K. K.
Published: August 2005

An efficient digital computer procedure and the related numerical algorithm are presented herein for the solution of quadratic matrix equations associated with free vibration analysis of structures. Such a procedure enables accurate and economical analysis of natural frequencies and associated modes of discretized structures.

The numerically stable algorithm is based on the Sturm sequence method, which fully exploits the banded form of associated stiffness and mass matrices. The related computer program written in FORTRAN V for the JPL UNIVAC 1108 computer proves to be substantially more accurate and economical than other existing procedures of such analysis. Numerical examples are presented for two structures: a cantilever beam and a semicircular arch.
Direct numerical solution of three‐dimensional equations containing elliptic operators
Authors: Martin E. Dale
Published: August 2005

A direct three‐dimensional elliptic solver is presented for application in a wide class of numerical methods for solving partial differential equations in physics and engineering. The derived algorithm and FORTRAN code implement Buzbee, Golub and Nielson's proposed extension of Buneman's Cyclic‐Reduction Poisson solver to three dimensions. Both a ‘most direct’ cyclic reduction and a revised method (to eliminate roundoff error difficulties) are derived. Tests on an IBM 360/67 computer, using various optional combinations of subroutines, showed significant differences in accuracy and computing time, with the optimum subroutine combination depending on mesh size.
Application of the Fast Fourier Transform to linear systems in civil engineering
Authors: O'Brien M. A. , Fitzgerald J. J. , Dillon E. C.
Published: August 2005

The Fast Fourier Transform is employed as a method of Laplace transform inversion to solve problems in the civil engineering fields of visco‐elasticity and hydrology. When these problems are accurately represented by a linear time‐invariant model, it is shown that the Fast Fourier Transform inversion procedure is often more accurate than standard convolution and inversion techniques. The speed and accuracy of solution resulting from this application of the Fast Fourier Transform is illustrated by referring to case studies solved on an IBM 1130 computer, model 3D processor with 32K memory.
Optimization of complex structures to satisfy static, dynamic and aeroelastic requirements
Authors: Rao S. S.
Published: August 2005

A structural optimization problem is considered in which the design requirements include restrictions on the strength, stability, frequency and flutter characteristics of the structure. One of the central concerns of this phase of the work has been to overcome the problems inherent in analysing the dynamic and aeroelastic behaviour of structures with many degrees of freedom. The multiweb delta wing structure under supersonic flight conditions is the model upon which this exploratory study is based. The finite element idealization, with three different kinds of elements, is used to model the wing structure. The constant stress triangular plate elements, the rectangular shear panels and the pin‐jointed bar elements are used to represent, respectively, the cover skins, webs and stringers of the wing structure. The design problem is formulated as a minimum weight optimization problem and is solved by using non‐linear programming techniques. Computationally efficient schemes are developed for the necessary derivatives of the behaviour constraints. Numerical examples are presented to illustrate the feasibility and the computational effectiveness of the method.
Restrictions in the application of automatic mesh generation schemes by ‘isoparametric’ co‐ordinates
Authors: Steinmueller G.
Published: August 2005

To obtain meshes of reasonable form there have to be made some restrictions in the application of automatic mesh generation schemes using ‘isoparametric’ co‐ordinates. In this present paper it will be shown that such restrictions exist for plane meshes. The results figured here are also valid for three‐dimensional meshes.
Buckling under combined loading of thin, flat‐walled structures by a complex finite strip method
Authors: Plank R. J. , Wittrick W. H.
Published: August 2005

A finite strip method is presented for determining the initial buckling stresses of any structure consisting of a series of thin flat isotropic plates rigidly connected together at their longitudinal edges. Each plate may be subjected to a combination of longitudinal and transverse compression, longitudinal in‐plane bending, and shear, and it is assumed that the buckling mode, of whatever type, is sinusoidal in the longitudinal direction. Due to the presence of shear, the perturbation forces and displacements which occur at the edges of component plates during buckling are out of phase, and this is accounted for by defining their magnitudes in terms of complex quantities. Stiffness matrices relating the amplitudes of these forces and displacements are derived using an approximate method based upon assumed displacement functions across the width of the plate. It is shown how the method can be used to calculate natural frequencies of prismatic structures, and finally an indication of the accuracy of the method is given together with some illustrative results.
Least square‐finite element for elasto‐static problems. Use of ‘reduced’ integration
Authors: Zienkiewicz O. C. , Owen D. R. J. , Lee K. N.
Published: August 2005

A least square based finite element algorithm is developed for some elasto‐static problems. In the formulation both stresses and displacements appear as simultaneous variables. In two dimensional (plane) analysis, parabolie isoparametric elements are used. Considerable improvement of performance is obtained with a numerical integration based on 2 × 2 Gauss point distribution over more accurate integration schemes. Reasons for this are presented.

The formulation is extended in the section ‘General least square formulation’ to beams and plates with a similar success of ‘reduced’ integration.
Finite elements in the solution of electromagnetic induction problems
Authors: Donea J. , Giuliani S. , Philippe A.
Published: August 2005

This paper describes a method of solving electromagnetic induction problems by means of the finite element technique. A variational equation associated with the differential equation for the vector potential is formulated and solved via the concept of finite elements. Sinusoidal driving currents and linear, isotropic, but inhomogeneous media are assumed. A typical example is included in order to illustrate the suggested solution technique.
Observations regarding assumed‐stress hybrid plate elements
Authors: Cook Robert D. , Ladkany Samaan G.
Published: August 2005

Edge displacement of assumed‐stress hybrid plate elements are commonly chosen as either linear, quadratic or cubic functions of the edge co‐ordinate. The present paper examines the effect of this choice on the acceptability, accuracy, and possible zero energy modes of elements based on a linear stress field. It is concluded that triangular elements may be preferable to quadrilaterals, and that linear edge displacements are not desirable.
Representation of singularities with isoparametric finite elements
Authors: Benzley S. E.
Published: August 2005

The development of a generalized quadrilateral finite element that includes a singular point at a corner node is presented. Inter‐element conformability is maintained so that monotone convergence is preserved. The global‐local concept of finite elements is used to formulate the complete set of equations. Examples of crack tip singularities are given.
A new triangular element for finite difference solution of axisymmetric conduction problems in cylindrical co‐ordinates
Authors: Mahata Prafulla C. , McNary Orlo
Published: August 2005

The triangular element is one of the most useful finite difference elements for the potential problems because of its versatility in fitting irregular boundary, in connecting one element type to another, and in changing grid fineness. Such triangular elements find many applications in conduction problems in cartesian co‐ordinates. This paper presents a new triangular element for the finite' difference solution of axisymmetric conduction problems in cylindrical co‐ordinates. The validity of the proposed triangular element is analyzed, and its workability is demonstrated using three selected examples. Also, an industrial application highlights the advantageous characteristics of the triangular element, and gives a comparison with known results obtained by finite element method.
Computer analysis of a rotating axial‐turbomachine blade in coupled bending‐bending‐torsion vibrations
Authors: Fu C. C.
Published: August 2005

Basic equations are derived for a comprehensive computer analysis for an equivalent lumped parameter system which simulates a pre‐twisted, rotating or non‐rotating. Timoshenko beam in coupled bending‐bending‐torsion vibrations. These equations enable one to develop straightforwardly digital computer programs for studying vibration problems related to compressor or turbine blades in turbomachinery as well as in other structural dynamics applications. The validity of the lumped parameter approach is established through a free vibration study. Good agreement between the present computer results and those obtained theoretically or experimentally by other investigators is demonstrated. In particular, it is found that the natural frequencies obtained in the free vibration study for all beams examined converge from below the exact values at a convergence rate of N−4, where N is the total number of the lumped parameters used in the computer simulation. The rapid rate of convergence shows that the mathematical model used in deriving the basic equations can adequately represent the type of beams under consideration. The known convergence rate also provides an estimate on the accuracy of the computer results, and a means to improve these results by extrapolation.
Calculation of transient temperature fields with finite elements in space and time dimensions
Authors: Köhler W. , Pittr J.
Published: August 2005

A program is demonstrated which apart from linear finite elements in time also includes elements with shape functions of the second and third degree. The algorithm for discretization in the time dimension is described and, using the example of a parabolic time element, the coefficients required to form the global system are given. By various test examples the efficiency of the process is examined by comparison with the customary difference method. Generally, with finite elements in time, the solution has better stability. Comparing the time required for calculation with the accuracy of the solution it would appear that in examining problems where boundary conditions are constant in time, higher order time elements are no improvement over the linear time element. However, for the purpose of reproducing periodic processes, higher order time elements offer an advantage in that one is not limited to linear variations of the boundary conditions within the element. Thus, for example, the temperature curve for parabolic variation of the surface temperature can be reproduced with close approximation by two time elements per period and a shape function of the third degree.
Automatic generation of the coefficient matrix of finite difference equations
Authors: Chugh A. K. , Gesund H.
Published: August 2005

A unified procedure is presented for the automatic generation of the coefficient matrix of finite difference equations. The method is applicable to the solution of both initial and boundary value problems, to any order of difference equation, and may be used with any arbitrary initial or boundary conditions.
The torsional stiffness of prismatic members with sections divisible into rectangles
Authors: Golley B. W. , Cashman J. D.
Published: August 2005

A method of determining the torsional stiffness of a wide range of structural sections, both homogeneous and composite, is presented. Prandtl's stress function along the boundaries of component sections is approximated by finite Fourier series with coefficients chosen to minimize a functional. An example indicates that the stiffness may be determined to acceptable accuracy by the solution of very few equations.
Discrete (Legendre) orthogonal polynomials—a survey
Authors: Neuman C. P. , Schonbach D. I.
Published: August 2005

The discrete (Legendre) orthogonal polynomials, (DLOP's) are useful for approximation purposes. This set of mth degree polynomials {Pm(K, N)} are orthogonal with unity weight over a uniform discrete interval and are completely determined by the normalization Pm(O, N) = 1. The authors are employing these polynomials as assumed modes in engineering applications of weighted residual methods. Since extensive material on these discrete orthogonal polynomials, and their properties, is not readily available, this paper is designed to unify and summarize the presently available information on the DLOP's and related polynomials. In so doing, many new properties have been derived. These properties, along with sketches of their derivation, are included. Also presented are a representation of the DLOP's as a product of vectors and matrices, and an efficient computational scheme for generating these polynomials.
Variational principles and the patch test
Authors: De Veubeke B. Fraeijs
Published: August 2005

The patch test is shown to be contained in the variational formulations of the finite element methods at the assembling level, all of which require the vanishing of the virtual work of interface connection loads. By a systematic introduction of stress generating functions, attention is drawn to the fact that any given finite element model can be assembled in two different ways: either by identification of a set of boundary displacements (leading to the direct stiffness method), or by identification of a set of local stress function values (leading to the dual direct flexibility method). Looking at any conjugate couple (generalized displacement‐generalized surface traction) at an interface, one is strongly transmitted, the other weakly. Discretization of the zero virtual work condition at an interface of plate bending models, by means of Legendre polynomial expansions, allows a systematic construction of so‐called ‘non‐conforming’ elements that pass the patch test. They are in fact identified with weakly conforming, but strongly diffusive, hybrids, and the lowest degree element (quadratic) is in fact the Morley constant‐moment element. Examples are given for higher degree displacement fields.

The case of plate stretching elements can be handled by duality, the difficulties being here associated with the requirements for diffusivity. Non‐diffusive elements that pass the zero interface virtual work test can be constructed systematically and are identifiable with weakly diffusive, but strongly conforming, hybrids of the type first proposed by T. H. H. Pian.
Finite difference analysis for laminar flow heat transfer in concentric annuli with simultaneously developing hydrodynamic and thermal boundary layers
Authors: Coney J. E. R. , El‐Shaarawi M. A. I.
Published: August 2005

The results of a finite difference analysis are presented for the problem of incompressible laminar flow heat transfer in concentric annuli with simultaneously developing hydrodynamic and thermal boundary layers, the boundary conditions of one wall being isothermal and the other wall adiabatic. This corresponds to the fundamental solution of the third kind according to the four fundamental solutions classified by Reynolds, Lundberg and McCuen1.

Firstly, the hydrodynamic entry length problem, based on the boundary layer simplifications of the Navier–Stokes equations, was solved by means of an extension of the linearized finite difference scheme used previously by Bodia and Osterle2 to solve a similar problem between parallel plates. The energy equation is then solved, using the velocity profiles previously obtained, by means of an implicit finite difference technique.

The accuracy of the numerical solution was checked by comparing results for the annulus of radius ratio 0.25 with the avaiable solution of Shumway and McEligot3.
Effect of shear deformations and rotary inertia on optimum beam frequencies
Authors: Kamat M. P.
Published: August 2005

Accounting for shear deformations and rotary inertia effects, necessary condition for optimum fundamental frequency of a vibrating beam of constant volume and with a given distribution of non‐structural mass, is obtained through the calculus of variations. Minimum cross‐section of the beam is controlled by the introduction of an inequality constraint. A finite element displacement formulation is then used in an iterativve manner to arrive at the optimum fundamental frequency and the corresponding material distribution for the discretized beam models with various boundary conditions. A comparison is then made with the corresponding results of an Euler beam.
Structural optimization using the finite element method applied to a beam
Authors: Hornbuckle J. C. , Nevill G. E. , Boykin W. H.
Published: August 2005

The optimum profile of a beam is obtained by using finite elements with mathematical programming techniques. Beam profiles of preceding studies can be diplicated by the finite element method with the appropriate choice of numerical tolerance used in the convergence test.
Sampling techniques in the analysis of Power‐System transients
Authors: McCarthy T. O.
Published: August 2005

A detailed investigation of analytical approaches to the determination of transients on Power Systems has been undertaken1, the main features of which are presented in this paper. The shortcomings of conventional travelling‐wave methods, particularly where long power lines constitute complex sub‐systems, have led to a numerical solution approach based on a discrete form of the Laplace Transform which has been labelled the ‘Fast Laplace Transform’ (FLT). The method allows efficient treatment of a wide variety of linear problems including, in particular, cases involving frequency‐dependent parameters.

An examination of continuous evaluation procedures, which are subject to truncation difficulties and to inefficient usage of computer time, is followed by a review of frequency domain sampling methods. 4,5,11 The FLT inversion technique is described with emphasis on the efficient use of computer time.

The method is illustrated by application to a number of switching studies and comparison with travelling‐wave solutions is made. Problems involving frequency‐dependent ground impedance, even where ground resistivity varies with depth, are easily simulated.
A boundary integral solution to torsion of cylinders and solids of revolution
Authors: Battenbo H. , Banies B. H.
Published: August 2005

The boundary integral equation known as Green's Theorem has been solved to generate displacement and stress fields directly on the boundary for cylinders and solids of revolution in pure elastic torsion. The computer programs which effect the solution utilize conventional numerical methods and discrete analytical elemental integrations and are applicable to multiply‐connected shapes. The integral equation has been differentiated in each case, so that internal stresses may be generated directly, where required.

Computed results are compared with analytical solutions and with experimental evidence produced by photoelastic techniques.
The application of node‐element rules for forecasting problems in the generation of finite element meshes
Authors: Fuhring H.
Published: August 2005

The number of equations which relate the mesh parameters (number of nodes, number of elements etc.) to each other is smaller than the number of unknowns contained therein. These rules are only applicable when several parameters of a defined mesh are known, but are unsuitable for use in the particularly interesting field of mesh projection unless additional inter‐relationships can be found. Such relationships are given for uniform meshes in which besides the characteristics of the element type the ‘solidity value’ of the structure plays an important role.

The equations are derived for two‐dimensional triangular and quadrilateral elements and also for three‐dimensional tetrahedral and hexahedral elements. The fact that these relationships are also suitable for estimating the parameters of any non‐uniform mesh, discloses a wide field of application in the management of storage and in the control of programs for automatic mesh generation.
An Extended Matrix Transfer Method with an automatic root search capability
Authors: Dawson B. , Davies M.
Published: August 2005

An ‘Extended Matrix Transfer Method’ is presented which has an automatic root search capability that can be guaranteed to determine all the roots of the problem with the exception of those roots lying within prescribed tolerance bandwidths.

The power and generality of the method is demonstrated by application to a specially synthesized problem.
The calculation of branch flows in a directed network
Authors: Gay B.
Published: August 2005

This paper describes a procedure for the calculation of branch flows in a directed network in which the node‐to‐node flows are prescribed.

The node‐to‐node paths of the network are enumerated using a combinatorial process. Procedures based on the path list are used to allocate the node‐to‐node flows to the alternative paths of the network according to Wardrop's first principle. The calculation, although necessarily iterative, is rapidly convergent.

The method has been implemented as a computer program and the results obtained for a typical network are discussed.

A linear relationship between branch potential and branch flow has been used, but the method can be extended to piece‐wise linear and non‐linear relationships.
Isoparametric finite element methods for two‐dimensional transport calculations
Authors: Gerin‐Roze J. , Lesaint P.
Published: August 2005

Finite element methods for solving the two‐dimensional x–y transport equation are considered. Numerical schemes are defined and tested on some simple practical problems.
Diagonal dominance and S.O.R. performance with skew nets
Authors: O'carroll M. J.
Published: August 2005

Alternative finite difference formulations for elliptic equations on skew nets are investigated for suitability for rapid convergence of S.O.R. iterations. In dealing with the differential equation directly, diagonal dominance is a key property for convergence theorems. Mixed derivatives boundary conditions threatens this property, but formulas are presented which protect it as long as the net skewness is not too severe. Test calculations are reported showing convergence for these formulas to be as good as that of optimal S.O.R. theory, whereas other more direct formulas give a very inferior performance.
Optimal stress locations in finite element models
Authors: Barlow John
Published: August 2005

The existence of optimal points for calculating accurate stresses within finite element models is discussed. A method for locating such points is proposed and applied to several popular finite elements.
Transmission‐line matrix solution of waveguides with dielectric losses
Authors: Akhtarzad S. , Johns P. B.
Published: August 2005

The transmission‐line matrix method is a time‐domain numerical method for solving wave problems. This paper describes how a minor change in the computer program for loss‐free dielectrics extends the method to include dielectric losses. Results are given in terms of the wave impedance of a waveguide with losses, and also in terms of power decay due to losses in the dielectric of a resonant cavity.
A simple cubic displacement element for plate bending
Authors: Allman D. J.
Published: August 2005

Early attempts to construct a triangular finite element for plate bending problems from a compatible cubic displacement field are not entirely satisfactory. The present paper shows how an accurate plate element can be achieved using independent cubic polynomial assumptions for the internal and boundary displacements in conjunction with a modified potential energy principle. This approach yields a simple algebraic formulation with favourable connection quantities at the element vertices which will appeal to practical users of the conventional finite element displacement method. Moreover, in Appendix I it is shown that the cubic element is identical to a previous hybrid stress element with linear internal bending and twisting moments and cubic boundary displacements. The stresses obtained from the former hybrid finite element solution therefore satisfy the strain compatibility conditions exactly. This remarkable result has an important significance in the theory of hybrid finite elements.
Mathematical model and boundary conditions in stress analysis relating to steam turbine rotors, under transient operating conditions
Authors: Di Pasquantonio F. , Macchi A.
Published: August 2005

An appropriate mathematical model for calculation of thermoelastic stress in steam turbine rotors, under transient operating conditions, is developed in the framework of so called uncoupled quasi‐static theory. The numerical analysis has been carried out by means of the finite element code ANGUS, the accuracy of which has been evaluated by comparing it with an analytical solution given by one of the authors in Reference 2. It has been shown that, with an appropriate choice of thermal and elastic boundary conditions, it is possible to carry out stress transient analysis considering only a segment of the hottest zone of high or intermediate pressure body of rotors. The paper includes a comparison study of the influence of two types of thermal boundary conditions in the labyrinth glands.
An iterative penalty function method in structural analysis
Authors: Salonen E‐M.
Published: August 2005

An iterative solution process combined with the penalty function method employed in constrained variational priciples is described. As an example of the use of the method an axi‐symmetric plate with the complementary energy formulation is considered.
Steady and unsteady finite element analysis of incompressible viscous fluid
Authors: Kawahara Mutsuto , Yoshimura Nobutoshi , Nakagawa Katsuya , Ohsaka Hajime
Published: August 2005

Finite element procedures and related illustrative numerical examples for incompressible viscous fluid motion are discussed in this paper. The steady flow problem is solved by the Newton–Raphson method and the perturbation method. By numerical examples, it can be shown that the combined use of the Newton–Raphson method and perturbation method is suitable. For the analysis of unsteady flow, the perturbation method is employed. Assuming that the basic flow is known, unsteady flow is calculated by accumulating the solution of the linearized equation in which the boundary values are varied by small amounts. Steady flows of temperature dependent free convection are also discretized and analyzed by the same procedure as the conventional finite element Galerkin method. For shape functions, quadratic polynomials are used for velocity and temperature, and linear polynomials for pressure. It is to be noted that the selections of shape functions and solution method are the keys to the analysis of highly non‐linear fluid flow problems such as those discussed in this paper.
An ADI procedure suitable for virtual storage systems
Authors: Wang H. H.
Published: August 2005

The Peaceman–Rachford ADI technique is a well‐known iterative method for solving large systems of linear equations arising from finite difference methods of solution of partial differntial equations. The present note demonstrates how the algorithm can be conveniently rearranged for computation in a virtual storage system with paging. Timing results are given for a sample problem typical of those encountered in nuclear diffusion computation.
An automatic node relabelling scheme for minimizing a matrix or network bandwidth
Authors: Akhras Georges , Dhatt Gouri
Published: August 2005

An automatic algorithm is presented for reducing the bandwidth of symmetric sparse connectivity matrices. The procedure takes an arbitrary input sequence and creates an optimal bandwidth for complex network systems by renumbering the node labels. Thus, considerable improvement for computer storage and time is achieved in the solution of the systems.

A large number of examples is presented to demonstrate the reliability and the efficiency of the method.
The analysis of rotational shells using a curved ring element and the mixed variational formulation
Authors: Barony S. Y. , Tottenham H.
Published: August 2005

A finite element for the analysis of rotational shells using a mixed variational formulation is proposed. The element has arbitrary curvature in the meridional plane, its geometry being given by the values of the axial and radial co‐ordinates, slope and meridional curvature at the nodal circles. A linear variation in the thickness was included in the formulation as was an orthotropic material. Test cases of flat plates and spherical shells indicate that a high degree of accuracy is obtainable with relatively small number of elements.
A finite element Galerkin formulation and its numerical performance
Authors: Salaam U. , Sandhu R. S.
Published: August 2005

A finite element Galerkin formulation admitting non‐homogenous boundary conditions and discontinuous approximants is derived from a variational principle. The procedure results in matrix equations explicitly allowing for jump discontinuties across interelement boundaries. As an illustration, a specialization of the procedure similar to a reduced integration technique is used to solve several problems in elastostatics. The improvement in accuracy in comparison with conventional methods is remarkable.
Comparative study of two and three level ADI methods for parabolic equations with a mixed derivative
Authors: Iyengar Satteluri R. K. , Jain M. K.
Published: August 2005

ADI difference schemes of two and three levels for the parabolic equations with a mixed derivative have been derived. Using the stability analysis and the principal part of the truncation error, a difference formula with minimum truncation error has been found. Comparative study of the two and three level formulas has been made through two examples, one with constant coefficients and another with variable coefficients.
Higher order difference formulas for a fourth order parabolic partial differential equation
Authors: Jain M. K. , Iyengar S. R. K. , Lone A. G.
Published: August 2005

In this paper, we have derived some new higher order difference formulas for the solution of a fourth order parabolic partial differential equation governing transverse vibrations of a uniform flexible beam in one and two space dimensions using Richtmyer's approach and a direct approach. Two examples illustrate the utility of the new difference methods.
Finite element methods for second order differential equations with significant first derivatives
Authors: Christie I. , Griffiths D. F. , Mitchell A. R. , Zienkiewicz O. C.
Published: August 2005

Galerkin finite element methods based on symmetric pyramid basis functions give poor accuracy when applied to second order elliptic equations with large coefficients of the first order terms. This is particularly so when the mesh size is such that oscillations are present in the numerical solution. In the present note asymmetric linear and quadratic basis functions are introduced and shown to overcome this difficulty in an appropriate two point boundary value problem. In particular symmetric quadratic basis functions are oscillation free and highly accurate for a working range of mesh sizes.
Displacement incrementation in non‐linear structural analysis by the self‐correcting method
Authors: Haisler Walter E. , Stricklin James A. , Key John E.
Published: August 2005

The prediction of non‐linear structural behaviour by the finite element method wherein buckling does not occur has received considerable attention and, with it, reasonable success has been achieved. However, the post‐buckling problem has been less actively pursued probably because of the inherent numerical difficulties encountered. This Technical Note reviews very briefly the numerical methods currently being used for pre‐ and post‐buckling analysis and presents a self‐correcting approach based on load and displacement incrementation which is shown to be efficient, reliable, and easy to program. Numerical solutions are presented which demonstrate the effectiveness of the method.
A critical assessment of the Simplified Hybrid Displacement Method
Authors: Mang Herbert A. , Gallagher Richard H.
Published: August 2005

The Simplified Hybrid Displacement Method, an attractive modification of one of the principles of stationary value of modified potential energy, is examined critically and compared to another form of such variational principles and to the classical principle of minimum of potential energy. The three methods are studied from a theoretical vantage point and through the medium of numerical examples. A non‐conforming triangular finite plate bending element serves as the vehicle to solve a number of simple problems, utilizing the classical principle of minimum potential energy, a principle of stationary value of modified potential energy and the Simplified Hybrid Displacement Method.

The attraction of this method derives from the fact that no additional independent unknown parameters—Lagrangian multipliers from a mathematical standpoint—enter into the modified energy functional, as is usually the case with hybrid methods.

It is demonstrated numcrically that the Simplified Hybrid Displacement Method may become unstable. It is also shown that the stationary values of the modified energy functional for this method may be inconsistent with the corresponding minima of the classical potential energy. Furthermore, it is demonstrated that the Simplified Hybrid Displacement method is, in general, incapable of ‘restoring’ interelement slope continuity in the sense of a principle of stationary value of modified potential energy, that is, at least in an ‘integral sense’.
Mixed explicit‐implicit iterative finite element scheme for diffusion‐type problems: I. Theory
Authors: Neuman S. P. , Narasimhan T. N.
Published: August 2005

A Galerkin finite element formulation of diffusion processes based on a diagonal capacity matrix is analysed from the standpoint of local stability and convergence. The theoretical analysis assumes that the conductance matrix is locally diagonally dominant, and it is shown that one can always construct a finite element network of linear triangles satisfying this condition. Time derivatives are replaced by finite differences, leading to a mixed explicit‐implicit system of algebraic equations which can be efficiently solved by a point iterative technique. In this work the accelerated point iterative method is adopted and is shown to converge when the conductance matrix is locally diagonally dominant. Several examples are included in Part II of this paper to demonstrate the efficiency of the new approach.
Mixed explicit‐implicit iterative finite element scheme for diffusion‐type problems: II. Solution strategy and examples
Authors: Narasimhan T. N. , Neuman S. P. , Edwards A. L.
Published: August 2005

In Part I1 of this paper we have established local stability and convergence criteria for the mixed explicit‐implicit finite element scheme and have shown that the proposed iterative method converges under certain conditions. Part II describes various practical aspects of the solution strategy such as convergence criteria for terminating the iterations, automatic control of time step size, reclassification of nodes from explicit to implicit during execution, estimation of time derivatives, and automatic adjustment of the implicit weight factor. Several examples are included to demonstrate certain aspects of the theory and illustrate the capabilities of the new approach.
A finite element collocation method for the exact control of a parabolic problem
Authors: Harley P. J. , Mitchell A. R.
Published: August 2005

A finite element method is given for the problem of exact control of a linear parabolic equation. The basis functions consist of piecewise bicubic polynomials and the differential equation is satisfied at Gaussian collocation points within each element. The overdetermined system of equations obtained is solved by the method of least squares, and a convergence argument is given for the complete procedure. Numerical results are given for two problems of boundary control.
The coupling of the finite element method and boundary solution procedures
Authors: Zienkiewicz O. C. , Kelly D. W. , Bettess P.
Published: August 2005

The finite element method is now recognized as a general approximation process which is applicable to a variety of engineering problems—structural mechanics being only one of these. Boundary solution procedures have been introduced as an independent alternative which at times is more economical and possesses certain merits. In this survey of the field we show how such procedures can be utilized in conventional FEM context.
A homework exercise in finite elements
Authors: Strang Gilbert
Published: August 2005

We suggest an exercise which illustrates many of the fundamental ideas of finite elements, and is nevertheless easily programmed and quickly solved. It is governed by a partial differential equation on a circular region, but by maintaining symmetry the stiffness matrix becomes small and tridiagonal. The accuracy of linear elements can be studied in detail, and a straightforward extension leads to the eigenvalue problem, elements of higher degree, numerical integration, isoparametric elements, and a simple example in plasticity.
Numerical Fourier analysis in the range of higher frequencies
Authors: Ury J. F.
Published: August 2005

A novel method is proposed for the numerical computation of Fourier coefficients in the range of higher frequencies for which hitherto no practical method was available. The accuracy of the results obtained by application of this method is discussed, and bounds of a correction term are determined accordingly. An example from the field of Diesel engine research is given, and the computed spectrum is compared with the spectrum recorded by an electronic harmonic analyzer.
A stability investigation of cable suspended pipelines
Authors: Contri Lorenzo , Schrefler Bernardo
Published: August 2005

A postbuckling problem of continuous suspension bridges which are used for carrying pipelines is investigated. These are continuous over several bridge towers with pin connections at their base. A geometrically non‐linear analysis using the finite element method is performed to study the postbuckling behaviour under increasing loads (for instance by filling up the pipelines with fluid).

Two examples are presented which investigate the aspect ratios of constructed continuous suspension bridges.
On computer‐aided analytic element analysis and the similarities of tetrahedron elements
Authors: Pedersen Pauli
Published: August 2005

A complete analytical analysis of three‐dimensional, quadratic displacement tetrahedron elements is presented. The results show that the element stiffness can readily be obtained from the much simpler stiffness matrix of the linear displacement element. Thus, the task of obtaining the matrix of the quadratic displacement element is, in effect, reduced to finding that of the linear displacement element.

The analysis has been made possible only by using a symbolic manipulation language (PL/I–FORMAC) on a computer. A rather general program for computer‐aided basic element analysis is presented. It seems that the extremely effective tool of symbolic manipulation has hardly been used until now in mechanics.
Solution of non‐linear boundary value problems by discrete least squares
Authors: Eason E. D. , Mote C. D.
Published: August 2005

This paper presents a new discrete (point‐matching) least squares method for solving general, non‐linear boundary value problems. The technique uses a mixed formulation and minimizes the sum of squared residuals by Powell's (1965) algorithm. The present method is compared with finite difference and weighted residual finite element methods in the examples and discussion. It shows considerable promise as a general differential equation solver for both large and small computer systems.
Error analysis of penalty function techniques for constraint definition in linear algebraic systems
Authors: Felippa Carlos A.
Published: August 2005

The penalty function approach has been recently formalized as a general technique for adjoining constraint conditions to algebraic equation systems resulting from variational discretization of boundary value problems by finite difference or finite element techniques. This paper studies the numerical behaviour of the penalty function method for the special case of individual equation constraints imposed on a symmetric system of linear algebraic equations. Constraint representation and computational roundoff error components are distinguished and asymptotically characterized in terms of the penalty function weight coefficients. On the basis of this study, practical rules for the automatic assignment of values to those coefficients within the linear equation solver are proposed. Numerical problems encountered in the case of more general constraints are briefly discussed, and procedures for circumventing such difficulties are suggested.
The numerical solution from measurement data of linear integral equations of the first kind
Authors: Radziuk J.
Published: August 2005

The problem considered in this paper is the solution of the Volterra integral equation of the first kind for the input function where we are given the impulse response or kernel and discrete, noise contaminated, observed values of the output function. The present method, introduced here, is based on assuming a multipolynomial form for the input function. Using the integral equation and the least squares method, the coefficients of all the terms in these polynomials are found simultaneously, in order to avoid bias toward any subset of the data. The solution is obtained in such a way that successive polynomials are continuous in function value and derivative at the points of juncture. Thus a smoothed estimate of the input function is generated. The development of the algorithm is presented and applied to two illustrative problems. Possible improvements and extensions of the method are also discussed.
An alternating direction method for multi‐dimensional parabolic free surface problems
Authors: Meyer Gunter H.
Published: August 2005

A locally one‐dimensional method (fractional step or alternating direction method) is used to approximate multi‐dimensional free surface problems for the diffusion equation by a sequence of one‐dimensional problems. These are solved with the method of lines and invariant imbedding.
FINITE: An approach to structural mechanics systems
Authors: Lopez L. A.
Published: August 2005

FINITE is a general purpose structural mechanics system which runs under the POLO II supervisor. It is unique in that it provides users with a very flexible recursive structural modeling capability. Engineers can communicate with FINITE through a problem‐oriented language similar to STRUDL. Significant savings in computer and data entry times can be achieved when its recursive substructuring and condensation features are used appropriately. The structure of FINITE permits engineers to enter new elements with a minimal knowledge of the system operation. Nonsymmetric elements, hybrid elements, and elements with different degrees of freedom at the various nodes are all permissible. Various forms of linear constraints may be applied to the generalized displacements of a structure.

The system runs under the POLO II supervisor and derives many of its attributes from the unique data base management facilities of this system. Among them are a limited working set size in a virtual memory environment and automatic restart capabilities.
Elasto‐plastic finite element analysis
Authors: Kalev I. , Gluck J.
Published: August 2005

An elasto‐plastic finite element small displacement procedure to predict the behaviour of structures under cyclic loads is presented. The solution combines incremental procedure with correction for equilibrium and iterative scheme. Applications are made for a rectangular strip and a notched bar under cyclic axial loads. Flow theory of plasticity with either isotropic hardening or kinematic linear hardening is used.
An isoparametric Hermitian element for the solution of field problems
Authors: Frind Emil O.
Published: August 2005

A two‐dimensional isoparametric Hermitian element is investigated for use in potential field problems with irregular geometry. Nodal degrees‐of‐freedom are the potential, the gradients in the directions normal and tangential to the element sides, and the twist. In applying the isoparametric principle a simple approximation is used which in general guarantees C1 continuity only at the element corners. The present version is restricted to 90‐degree interior angles.

The element is applied in the Galerkin solution of the steady‐state continuity equation and several examples are presented. It is found that in cases where the potential field is simple but the element is strongly distorted, gradients obtained from the Hermite are not as accurate as those from the Lagrange‐type cubic element having the same number of degrees‐of‐freedom. When the potential field is complex and the distortion is small, the Hermitian gradients are comparable to those of the Lagrange cubic; with increasing distortion they are comparable at the boundaries but less so in the interior. The errors in the potentials themselves are very similar, both elements giving a high degree of accuracy.

Since all of its degrees‐of‐freedom are shared by four elements in the interior of a grid, the Hermite element is approximately 20 per cent more efficient than the Lagrange cubic for all but the smallest grids. Also, since the Hermite uses fewer nodes, data handling is greatly reduced. The element appears to be attractive at least in the area of groundwater flow.
Numerical analysis of flow with or without wake past a symmetric two‐dimensional profile without incidence
Authors: Bourgat J. F. , Duvaut G.
Published: August 2005

Formulated in the hodograph plane, the bi‐dimensional flow of a perfect fluid past a symmetric body leads to an elliptic variational inequality. By this way we show the existence of a one‐parameter family of solutions. Furthermore the study gives us a simple and effective method for calculating the solutions. The results obtained with some typical profiles are given and compared with previous works.
Optimization methods applied to hydrodynamic bearing design
Authors: Dowson D. , Blount G. N. , Ashton J. N.
Published: August 2005

Optimization techniques are applied to the design of both hydrodynamic journal and tilting pad thrust bearings. Fletcher's adaptation of Davidon's gradient search method is further modified to deal with non‐linear constraints in journal bearing design. A form of the Sequential Simplex is used for tilting pad thrust bearings. Penalty functions and variable transformations are used in both cases. Composite objective functions are included and the effect of weighting on each component of the composite function is investigated.
Ways to improve the bending response of finite elements
Authors: Cook Robert D.
Published: August 2005

Four new ways to relieve the excessive bending stiffness of linear elements are described, and incorporated into plane elements. All satisfy the ‘patch test’. The new elements are compared with existing elements in test problems. Application to three‐dimensional problems is proposed.