bluuebunny/crossref_metadata_2025 huggingface dataset:
Point matching computation of transverse resonances
Authors: Bates R. H. T. , Ng F. L.
Published: August 2005

Point matching (collocation) methods are reviewed and the conditions under which they break down are examined. The detailed treatment applies to complex‐valued solutions of the two‐dimensional Helmholtz equation with either Neumann or Dirichlet boundary conditions. Point matching methods (when they are valid) can have very real computational advantages. It is shown, by a combination of analysis and computational example, how to estimate whether the cross‐section of any particular resonant system is suitable for the application of an existing point matching method. An extended point matching method is introduced to handle cases for which previous methods fail. The method is illustrated with several computational examples. The discussion is specifically concerned with electromagnetic wave guides, but the results can be applied to a wide range of physical systems.
Solution methods for eigenvalue problems in structural mechanics
Authors: Bathe Klaus‐Jürgen , Wilson Edward L.
Published: August 2005

A survey of probably the most efficient solution methods currently in use for the problems Kϕ = ω2Mϕ and KΨ = λKGΨ is presented. In the eigenvalue problems the stiffness matrices K and KG and the mass matrix M can be full or banded; the mass matrix can be diagonal with zero diagonal elements. The choice is between the well‐known QR method, a generalized Jacobi iteration, a new determinant search technique and an automated sub‐space iteration. The system size, the bandwidth and the number of required eigenvalues and eigenvectors determine which method should be used on a particular problem. The numerical advantages of each solution technique, operation counts and storage requirements are given to establish guidelines for the selection of the appropriate algorithm. A large number of typical solution times are presented.
A direct integral equation method for the potential flow about arbitrary bodies
Authors: Grodtkjær Erik
Published: August 2005

The velocity on the surface of an arbitrary three‐dimensional body in the potential flow of an ideal fluid is determined by means of an integral equation method. The equations to be solved are a pair of singular integral equations of the second kind, the unknowns being the desired velocity components. The integral equations are approximated by a set of linear equations, which are solved numerically. For ellipsoids the results are compared with the analytic solutions.
Criteria for finite element discretization of shells of revolution
Authors: Sen Subir K. , Gould Phillip L.
Published: August 2005

A method is described whereby an efficient finite element discretization scheme for a shell of revolution can be predetermined and subsequently refined for successive analyses by considering the following criteria:

Stress intensification in the vicinity of a concentrated action.

Variation in the strain energy intensity function.

Several case problems have been studied to validate the procedure.
A computer program for filter design having arbitrary magnitude specifications in the frequency domain
Authors: Jullien G. A. , Sid‐Ahmed M. A.
Published: August 2005

A general procedure is presented for the design of analogue or digital filters having arbitrary specifications in the frequency domain. The algorithm developed uses a simple technique for constraining the pole zero locations, for stability and minimum phase considerations, which is far more elegant than methods used by previous workers.1 This constraint procedure yields a unified approach to the design of both digital and analogue filters.

The method employed is that of minimizing a square‐error criterion using the Fletcher–Powell2 conjugate gradient minimization routine.

The programme is written in FORTRAN IV double precision complex arithmetic.
The orthogonal projection method for the numerical solution of a field problem
Authors: Rafalski P.
Published: August 2005

A numerical method, based on the orthogonal projection method of Hilbert space, is given. Two approximate solutions, which bound the exact solution from above and below, are described. The error of the approximate solution is calculated directly. An example of a calculation of temperature field in a nuclear reactor shell is presented.
Flow of plastic and visco‐plastic solids with special reference to extrusion and forming processes
Authors: Zienkiewicz O. C. , Godbole P. N.
Published: August 2005

When large, progressive, deformation occurs under plastic or visco‐plastic conditions, elastic deformations are negligible and the material flows in a viscous manner. The flow is non‐Newtonian and the viscosity is a function of the current strain rates. An effective numerical treatment of such non‐Newtonian flow by the finite element method is indicated here and is illustrated by examples of an extrusion process for which approximate slip line solutions are known.

The methodology is extended to the solution of transient, quasi‐static situations and coupling with thermodynamic equations is discussed.

A stream function formulation of the problem is used.
An improved transfer matrix procedure
Authors: Dawson B. , Davies M.
Published: August 2005

This paper presents an extension to the Matrix Transfer Method which gives quadratic convergence to a natural frequency from within a range on either side.

The method is described and illustrated by application to:

A torsional vibration problem, and

A lumped mass beam vibration problem.
Application of finite element methods to thermohydrodynamic lubrication
Authors: Huebner Kenneth H.
Published: August 2005

A finite element formulation is presented for the equations governing the steady thermohydrodynamic behaviour of liquid lubricated bearings. This formulation permits application of the iterative solution scheme to bearings of arbitrary geometry.

A generalized Reynolds equation resulting from the combination of the mass and momentum conservation equations is cast into variational form and used to derive general finite element equations. The method of weighted residuals with Galerkin's criterion is used to generate finite element matrix equations for the thermal energy equation. In addition to the finite element formulation, a discussion of appropriate finite difference techniques is also given for problems without complex geometry.

As an example, the formulations are applied to obtain numerical solutions for a three‐dimensional sector thrust bearing operating in the thermohydrodynamic regime. Pressure, velocity and temperature distributions are give, and the thermohydrodynamic solutions are compared with the results of classical isothermal theory.
Torsion of axi‐symmetric solids of variable diameter—including acceleration effects
Authors: Owen D. R. J. , Zienkiewicz O. C.
Published: August 2005

A finite element method for the analysis of axisymmetric bodies in torsion is presented. The effects of an angular acceleration about the axis of symmetry is included. The accuracy of the method is tested against experimental results available for a stepped shaft and finally the problem of an accelerating shaft‐turbine disc system is considered.
Cubic spline Galerkin approximations to parabolic systems with coupled non‐linear boundary conditions
Authors: Murphy W. D.
Published: August 2005

This paper describes a practical implementation of the Galerkin finite element procedure for solving systems of parabolic partial differential equations with non‐linear boundary conditions. This technique consists of finding an approximation in the form of a finite sum of cubic B‐splines, which yield high‐order accurate results, and consequently, solutions can be developed with remarkable precision and speed up to the steady state region where conventional finite difference methods often fail. In addition, the choice of mesh width and nodal spacing can be automatically determined for a predictor‐corrector routine, thereby relieving the engineer of a great deal of ‘guess work’ that is normally characteristic of solving such problems.
Three‐dimensional elasto‐plastic finite element analysis
Authors: Owen D. R. , Salonen E. M.
Published: August 2005

Advances in technology and interest in limit state design have made the inclusion of non‐linear effects, such as elasto‐plastic behaviour, desirable in the analysis of many structures. Improvements in solution algorithms coupled with parallel developments in high speed digital computers have now made the practical solution of such problems possible. The paper presents numerical solutions to three‐dimensional elasto‐plastic problems illustrating the applicability of isoparametric elements and the order of computation times involved.
Panel flutter optimization by gradient projection
Authors: Pierson B. L.
Published: August 2005

A gradient projection optimal control algorithm incorporating conjugate gradient directions of search is described and applied to several minimum weight panel design problems subject to a flutter speed constraint. New numerical solutions are obtained for both simply‐supported and clamped homogeneous panels of infinite span for various levels of inplane loading and minimum thickness. The minimum thickness inequality constraint is enforced by a simple transformation of variables.
Thermal deformation vector for a bilinear temperature distribution in an anisotropic quadrilateral membrane element
Authors: Wallerstein D. V.
Published: August 2005

This paper presents a thermal‐structural deformation vector for an anisotropic quadrilateral membrane element. The elastic properties of the element are based on linear stress assumptions and were developed by Robinson1. A general bilinear temperature field within the element is chosen in order to obtain a thermal deformation capability for the element commensurate with its elastic capability. A pseudo shear term is developed to account for part of the thermal deformation. The element is suitable for finite element analysis of advanced aircraft structures under elevated temperature environments and is particularly well suited for analysis of regions with severe temperature gradients such as occur in the vicinity of local heat sources and sinks. An example problem is included to demonstrate the behaviour of the element.
Crack tip finite elements are unnecessary
Authors: Henshell R. D. , Shaw K. G.
Published: August 2005

It is shown that a singularity occurs in isoparametric finite elements if the mid‐side nodes are moved sufficiently from their normal position. By choosing the mid‐side node positions on standard isoparametric elements so that the singularity occurs exactly at the corner of an element it is possible to obtain quite accurate solutions to the problem of determining the stress intensity at the tip of a crack. The solutions compare favourably with those obtained using some types of special crack tip elements, but are not as accurate as those given by a crack tip element based on the hybrid principle. However, the hybrid elements are more difficult to use.
Velocity potential computation by finite differences for compressible flow in ducts
Authors: O'Carroll M. J.
Published: August 2005

Preliminary studies of computation with velocity potential are made with a view to the analysis of complex three‐dimensional flows. The methods used are applicable more generally to quasilinear elliptic problems with derivative boundary conditions on irregular domains. Second order finite difference approximations are constructed in simple form for plane ducts of general shape by using an irregular net. Derivative boundary conditions are handled quite easily. An iterative method is described which corresponds to freezing the coefficients in the quasilinear differential equation for velocity potential. The discretization is such that this is a ‘generalized Newton’ method for the non‐linear algebraic equations. Good convergence has been found in practice even when there are small supersonic zones. The discretization accuracy is tested by comparisons with the exact solution for incompressible flow between confocal hyperbolas.
Heat transfer during fluidized‐bed coating
Authors: Meyer Gunter H.
Published: August 2005

A fully implicit numerical method for linear parabolic free boundary problems with coupled and integral boundary conditions is described. The partial differential equation and the boundary conditions are time discretized with the method of lines. An auxiliary function is introduced to remove the coupled and integral boundary conditions from the resulting free boundary problem for ordinary differential equations. Once separated boundary conditions are obtained, invariant imbedding is used to solve the free boundary problem numerically. The method is illustrated by solving the heat transfer equations for the fluidized‐bed coating of a thin‐walled cylinder.
A finite element method for the axisymmetric flow computation in a turbomachine
Authors: Hirsch Ch. , Warzee G.
Published: August 2005

The equations for the meridional through‐flow in a turbomachine are formulated as a quasi‐harmonic non‐linear equation. This equation is then solved iteratively by the finite element method and in order to obtain convergence an under‐relaxation factor has to be introduced. Comparisons with experimental results in single and multistage axial compressors show very good agreement. The finite element method is thus shown to be an efficient tool for this type of problem.
Theory and examples of partial approximation in the finite element method
Authors: Kikuchi Fumio
Published: August 2005

This paper presents theory and examples of partial approximation as a modification of the displacement method in the finite element analysis. This method requires different shape functions for different terms in the potential energy expression to curtail the processes in the standard displacement method. Explanation of the theory is given by use of a simple example for Poisson's equation. It can also be effectively utilized to give mathematical foundation to some finite element models based on physical reasonings, such as Melosh's rectangular element for plate bending and beam element approximation of circular arches.
Numerical quadrature and two‐dimensional radiative heat transfer
Authors: Look D. C. , McKinney S. E.
Published: August 2005

Four methods of numerically integrating a two‐variable integral equation, which is used to describe non‐dimensional radiosity in a diffuse grey cavity, are presented. Physical reasoning is used to organize the successful method for obtaining physically plausible solutions.
Transformation of elliptic partial differential equations for solving two‐dimensional boundary value problems in fluid flow
Authors: Boadway J. D.
Published: August 2005

It is possible to transform elliptic partial differential equations to exchange the dependent with one of the independent variables. The Laplace equation for a stream function ‘Ψ’ over the X and Y co‐ordinate system, for example, can be transformed into a relationship expressing the Y position of streamlines in terms of strem function Ψ and X.

Although the resulting new partial differential equation is much more complex it is much more convenient to use in a computer. An irregular Y boundary becomes, with the new relationship, merely the boundary values assigned to the outer streamlines and the computer always need only deal with a rectangular array. The resulting answer is in the form of the position of streamlines which is the information directly required for plotting flow maps.
Reduction scheme for some structural eigenvalue problems by a variational theorem
Authors: Hughes Thomas J. R.
Published: August 2005

An accurate reduction scheme for structural eigenvalue problems is deduced from a variational theorem in which the displacement, velocity and/or momentum fields are taken to be independent.
Numerical techniques in probabilistic load flow problems
Authors: Allan R. N. , Grigg C. H. , Al‐Shakarchi M. R. G.
Published: August 2005

This paper describes numerical techniques which have been developed in order to solve the load flow problem of a power system network probabilistically instead of using normal deterministic methods. The aim in the development of these techniques has been to reduce the computational time and storage commensurate with maintaining a high degree of accuracy.
On the relationship between the finite element and finite difference methods
Authors: Gray William F. , Pinder George F.
Published: August 2005

The integral equations which arise from application of the Galerkin‐finite element scheme to the convective diffusion equation are examined to illustrate how this method represents differential equations. The formulae obtained are effectively spatial averages of standard finite difference equations written at a node. The truncation error in the finite element solution at a node is obtained for various nodal configurations.
Subspace iteration accelrated by using Chebyshev polynomials for eigenvalue problems with symmetric matrices
Authors: Yamamoto Yoshiyuki , Ohtsubo Hideomi
Published: August 2005

Bathe's algorithm of subspace iteration for the solution of the eigenvalue problem with symmetric matrices is improved by incorporating an acceleration technique using Chebyshev polynomials. This method of acceleration is particularly effective for this kind of iteration. The rate of convergence of the iteration scheme presented is considerably improved when compared with the original one, and satisfactory rates of convergence can be obtained for a wider range of eigenvalues.
Non‐linear programming—gradient projection via orthogonalization
Authors: Russell S. S.
Published: August 2005

Large scale non‐linear programming problems with non‐linear constraints are of practical concern in the areas of process control and system design. Although advances in computer technology address these problems, more efficient algorithms are needed. An algorithm currently used is Rosen's Gradient Projection Method. This note presents an alternative technique of performing the steps for finding the projected gradient used in Rosen's method, a technique that gives equivalent results with fewer computations.
A review of least‐squares methods for solving partial differential equations
Authors: Eason Ernest D.
Published: August 2005

Continuous (integral) and discrete (point‐matching) least‐squares methods are presented for linear and non‐linear problems in boundary‐value, eigenvalue, and initial‐value form. The history is traced, and important theoretical and practical results are summarized. A comprehensive sample of the literature is presented, indexed to show type of application, version of least squares used, and results of comparison studies. The advantages of least‐squares methods are discussed, including convenience in formulation and error evaluation, generality of mixed and local (finite element) versions, and performance that is competitive with other methods.
A variable metric algorithm for constrained minimization based on an augmented Lagrangian
Authors: Gruver W. A. , Engersbach N. H.
Published: August 2005

This paper concerns the implementation of a recent idea, attributed to Hestenes and Powell, based on solving the equality constrained finite dimensional minimization problem

via the unconstrained problem

where ƒ is a non‐linear functional, g is a non‐linear mapping into Rp, K is a prescribed matrix of penalty constants and λ is the Lagrange multiplier. The computational algorithm is based on restoring active constraints to first order and adjusting x in the remaining necessary conditions by gradient projection. The minimization is performed by the variable metric rank‐two BGFS update with linear search by cubic interpolation. Computational results using the algorithm include two problems of minimum fuel trajectory optimization—two impulse rendezvous with Comet Encke and three impulse constrained positioning of a geostationary satellite.
A modification of the Monte Carlo method for steady heat conduction problems
Authors: Zinsmeister G. E. , Pan S. S.
Published: August 2005

The unique capability of calculating single points by Monte Carlo Methods is used in an attempt to develop an efficient Monte Carlo method for whole field calculations of heat conduction problems. Rather than calculate the whole field by Monte Carlo, only select points around the boundaries of inscribed figures of standard shapes are are so calculated. Interior points of the inscribed figures are then calculated by analytical (non‐numerical) techniques. The procedure is described and shown to be more efficient than previously published Monte Carlo modifications but not as efficient as the Gauss–Seidel method. This new procedure will most likely be competitive for subregion calculations rather than for whole field calculations (for which standard numerical procedures are most efficient) or for single point calculations (for which other Monte Carlo methods are most efficient).
Derivatives and their error bounds in second order problems by finite element
Authors: Mote C. D. , Young R. C.
Published: August 2005

A previous paper by the authors presented a finite element technique for computing accurate and bounded solutions at arbitrary points in second order mixed boundary value problems. The technique is extended herein to the computation of accurate and bounded derivatives at arbitrary interior points. Numerical results are presented to illustrate both the strengths and limitations to the method.
Optimum structural design with stability constraints
Authors: Khot N. S. , Venkayya V. B. , Berke L.
Published: August 2005

This paper presents an optimization method based on optimality criterion for minimum weight of structures with stability requirements. A recurrence relation is derived and the method is explained in the context of the displacement method of finite element analysis. The incipient buckling of the structure is determined by a linear eigenvalue solution. The method is programmed for trusses and frames. Illustrative problems are given to show the applicability of the method of design of structures with a large number of design variables.
A new method of integration over polynomal finite element boundaries
Authors: Rodin E. Y.
Published: August 2005

In this note we are developing integration techniques for triangular regions in which one side is curved. Specifically, we are considering (a) general polynomial boundaries and (b) various combinations of elliptical boundaries. Integration is performed over all of these explicitly.

One definite and considerable improvement achieved is the introduction of a new method of integration over polynomial type boundaries. This new method is not only exact for polynomials or other analytic functions, but also reduces the computing time considerably; for example, if for a sixth order polynomial the number of operations necessary (using previously available methods) to perform the integration was n, the method introduced here requires a number proportional to n1/6. Furthermore, coding time is also reduced considerably. It may be of value to mention that the availability of the results presented here could obviate the need for the use of isoparametric elements in some regions with curved boundaries.

For completeness, explicit integration formulas for elliptic shaped regions are also listed.
A hybrid element method for steady linearized free‐surface flows
Authors: Mei C. C. , Chen H. S.
Published: August 2005

It is first shown that the two‐dimensional linearized ship wave problem can be recast as the sum of a radiation and a diffraction problem for simple harmonic waves. Each problem can be solved by a hybrid element method (HEM) where conventional finite elements are used near the body and analytical solutions are used in the remaining infinite regions (super‐elements). Variational principles which incorporate the matching conditions between regular and super‐elements as natural conditions are derived. Numerical examples are presented. The theoretical aspects for extending the above ideas to a three‐dimensional ship wave problems are also described.
PFVIBAT—a computer program for plane frame vibration analysis by an exact method
Authors: Åkesson Bengt Å.
Published: August 2005

The described program PFVIBAT uses the exact displacement method to perform free and forced vibration analysis entirely within the differential equation theory of beams thus avoiding assumed modes and lumped masses. The frame may contain rigid bodies. Clamped, hinged, guided and rolling connections are allowed for. Consideration of rotatory inertia, shear deformation and second‐order bending moments and shear forces as caused by static axial load is optional. Eigenfrequencies and modal masses are calculated with an accuracy that may be specified. Displacement and moment modes are plotted. Transient vibrations are studied.
The generation of elements with singularities
Authors: Akin J. E.
Published: August 2005

A procedure is given for generating two‐dimensional conforming singularity elements from standard conforming elements. Three new elements with 0(r‐p) derivative singularities are introduced. The technique is based on the use of elements defined by numerically integrated shape functions. Special quadrature rules are suggested for triangular elements. The degeneration of these elements to crack tip elements and the direct evaluation near the singularity of element quantities, such as the stress intensity factors, is discussed.
Numerical computations of three‐dimensional thin panel structures
Authors: Růžičková H. , Ženíšek A.
Published: August 2005

A finite element model for analysing static problems of three‐dimensional thin panel constructions is presented. The potential energy of stretching of each panel is represented by the classical Lagrange formulation of two‐dimensional elasticity and in the finite element discretization piecewise linear u, v displacements are used. The potential energy of bending of each panel is represented in the finite element discretization by Hellan and Herrmann's model, i.e. piecewise constant moments and piecewise linear w‐displacements are used. The physical appropriateness of the model is verified for different assemblages of panels, and in all cases good numerical results are obtained.
The elasto‐plastic analysis of plates using hybrid finite elements
Authors: Barnard A. J. , Sharman P. W.
Published: August 2005

An elasto‐plastic plate‐bending analysis is developed based on the finite element method and utilizing an assumed‐stress hybrid (Pian) element. Plasticity is incorporated using the Prandtl–Reuss relations and isotropic hardening theory together with an incremental residual force technique. Several example solutions are included to demonstrate the ability of this approach in elasto‐plastic plate‐bending problems.
Convergence problems in the asymptotic expansions for the Bessel functions of complex argument
Authors: Kausel Eduardo
Published: August 2005

The convergence of Hankel's asymptotic expansions for the Bessel functions of large complex argument are discussed, and a method is proposed to avoid numerical problems derived from their implementation in computer programs. Whiel the error tolerance cannot in all cases be controlled by the user, excellent results can indeed be obtained with application of the proposed method.
Numerical stability of dynamic relaxation analysis of non‐linear structures
Authors: Cassell A. C. , Hobbs R. E.
Published: August 2005

The estimation of the parameters (‘fictitious densities’) which control the convergence and numerical stability of a non‐linear Dynamic Relaxation solution is described. The optimal values of these parameters vary during the iterative solution and they are predicted from the Gerschgörin bounds, that is rowsums of the stiffness matrix, which are divided into constant and variable parts for computational convenience. The procedure is illustrated by reference to the analysis of an axially loaded beam on a non‐uniform elastic foundation.
Simplex elements of C0 continuity with varying polynomial degrees
Authors: Tong Pin
Published: August 2005

A hierarchical approach is used to construct the simplex elements which permits easy merging of elements of different orders. The explicit interpolation functions and the algorithm of using such elements are presented.
Infinite elements
Authors: Bettess Peter
Published: August 2005

A series of shape functions analogous to Lagrange polynomials, but including an exponential decay term, are suggested as appropriate for elements which extend to infinity. The shape functions are applied to a number of examples, mainly of viscous flow, where the domain extends to infinity. Both explicitly integrated and numerically integrated elements are described. The technique is applicable to many problems where the domain is supposed to extend to infinity.
Large displacement, transient analysis of space frames
Authors: Belytschko T. , Schwer L. , Klein M. J.
Published: August 2005

A formulation is presented for the transient analysis of space frames in large displacement, small strain problems. For purposes of treating arbitrarily large rotations, node orientations are described by unit vectors and deformable elements are treated by a co‐rotational (rigid‐convected) description. Deformable elements may be connected either to nodes directly or through rigid bodies. The equations of motion are integrated by an explicit procedure. Sample results are presented on the snap‐through of an arch‐type structure and an idealization of a vehicle‐barrier impact.
A unidimensional convergence test for matrix iterative processes applied to Strapdown Navigation
Authors: Bar‐Itzhack Itzhack Y.
Published: August 2005

The investigation of the convergence properties of matrix iterative processes usually involves test matrices of high order. This fact may prohibit an analytic approach to the problem. In this paper a method is presented which converts the multidimensional test procedure into a scalar one. The method is presented in conjunction with the problem of matrix orthogonalization which exists in Strapdown Inertial Navigation. Three examples are presented in which the convergence of matrix orthogonalization techniques is investigated. The examples demonstrate the use of the unidimensional convergence test in determining the order of the processes and in finding sufficient conditions for convergence. Numerical results are presented.
Technical note: The application of computers to viscosity measurement in non‐newtonian fluids
Authors: Townsend P.
Published: August 2005

A numerical method is described for the determination of viscosity‐shear rate dependence in non‐Newtonian fluids from simple pipe flow data. Results are presented to illustrate the accuracy of the method.
Compact representation of triangular finite elements for Poisson's equation
Authors: Tsipouras Paul
Published: August 2005

The energy expression associated with Poisson's equation is written, for triangular domains, in a form invariant to translation and rotation. The element matrices derived from this depend only on the intrinsic parameters of the element, say the three sides. Considerable savings in storage can be achieved by storing the global stiffness matrix in terms of these element matrices.
An attenuation‐free upwind difference formula for analogue computation of charge flow in semiconductors
Authors: O'Carroll M. J. , Foster K. , McKay C. M.
Published: August 2005

The first order partial differential equations of charge flow are commonly differenced in the space co‐ordinate, with a view to analogue computation of the resulting ordinary differential equations in time. Defects of orthodox difference formulae are identified. In particular: (a) reversed waves and stability problems arise from central differences and (b) first order errors and a ‘pseudo‐viscous’ dispersion effect with accompanying false attenuation results from upwind differences. These difficulties are particularly serious in view of the coarse differencing required for practical analogue computation. A much superior formula is presented which is entirely free of attenuation and is easily implemented. Various other desirable features are reported.
Computable finite element error bounds for Poisson's equation
Authors: Barnhill R. R. , Brown J. H. , McQueen N. , Mitchell A. R.
Published: August 2005

Theoretical error bounds of the form
equation image
are often available for finite element solutions U of elliptic problems. In this form the estimates give the order of convergence of the method but are of little practical value for estimating the size of the error because the magnitudes of the constant K and the theoretical solution u are unknown. An exception occurs in the case of the equation ∂2u/∂x2 + ∂2u/∂y2 + f = 0 in a rectangle where the Ritz–Galerkin finite element solution involves piecewise linears over a regular triangular grid. In this case where α = 1 and
equation image
Barnhill and Gregory1 have obtained the theoretical value 0·93√2 for K. In this note calculations are carried out for a variety of problems and the quantity K* = ∥u – U∥E/h∥f∥L2 measured and compared with K. The values of K* obtained fit into a well defined pattern from which we conclude that the theoretical constant K is of the correct order of magnitude.
A heuristic for root isolation
Authors: Jones Bush
Published: August 2005

A heuristic for isolating the real roots of an equation involving a continuous function over a specified region is presented. Root isolation strategies are well known for polynomial and rational functions; however, there have been no really effective root isolation strategies for transcendental functions. For a transcendental function and a given interval, there has been no way to decide the existence of roots within the interval unless the function changed sign at the endpoints or unless evaluations were performed within the interval. This paper presents a new decision procedure using only interval endpoints. The methodology developed does not guarantee to detect real roots in an interval; however, the technique will effectively indicate real roots in a given interval unless the behaviour of the function is quite erratic.
Flexible finite‐difference stencils from isoparametric finite elements
Authors: Frey William H.
Published: August 2005

A new finite‐difference technique for the numerical solution of boundary value problems for partial differential equations in two space variables is described. Isoparametric finite elements are used in a finite‐difference context to derive difference approximations to space derivatives on a locally curvilinear grid. The result is a generalization of a classical finite‐difference stencil which is ‘flexible’ in that it is adaptable to variable meshes, such as those arising from regions with curved boundaries. Numerical results presented for a test problem (potential flow past a circle) using a 9 × 10 grid agree with the analytic solution to within one per cent.
Stiffness matrix for a beam element including transverse shear and axial force effects
Authors: Chugh Ashok K.
Published: August 2005

A simple and direct procedure is presented for the formulation of an element stiffness matrix on element co‐ordinates for a beam member and a beam‐column member including shear deflections. The resulting stiffness matrices are compared with those obtained using the alternative formulation in terms of member flexibilities: The relative effects of axial force and shear force on the stiffness coefficients are presented. The critical buckling loads, considering the effects of shear force, are computed and compared with those available in the literature. Only prismatic members are considered.
An advanced boundary integral equation method for three‐dimensional thermoelasticity
Authors: Rizzo F. J. , Shippy D. J.
Published: August 2005

The features of an advanced numerical solution capability for boundary value problems of linear, homogeneous, isotropic, steady‐state thermoelasticity theory are outlined. The influence on the stress field of thermal gradient, or comparable mechanical body force, is shown to depend on surface integrals only. Hence discretization for numerical purposes is confined to body surfaces. Several problems are solved, and verification of numerical procedures is obtained by comparison with accepted results from the literature.
Lagrange polynomials of a complex variable for two‐dimensional interpolation
Authors: Bates R. H. T. , Milner M. O.
Published: August 2005

We recognize that established techniques permit two‐dimensional interpolation to be accomplished efficiently as well as accurately when the grid of points, on which the data is available, is regular. Existing methods suitable for irregular grids are computationally protracted. We show that by using Lagrange polynomials of a complex variable we can interpolate, almost as conveniently as in one dimension, from an irregular grid onto particular points lying on parallel lines. Standard one‐dimensional interpolation schemes can then be used to complete the two‐dimensional interpolation. We discuss how, in any particular instance, the order of the Lagrange polynomials is chosen, and we present the results of a computational test of our method.
Some criteria for numerically integrated matrices and quadrature formulas for triangles
Authors: Laursen M. E. , Gellert M.
Published: August 2005

For a wide calss of finite element matrices integrated numerically rather than exactly, a definable number of sampling points is found to be sufficient for keeping their theoretical properties unchanged. A systematic criterion limiting the number of possible point configurations for numerical quadrature formulas on triangles is established. Some new high order formulas are presented. Tables containing optional formulas with respect to minimum number of sampling points and required degrees of accuracy are given. They are arranged so as to assist with selection of suitable quadrature formulas for finite element computer programming.
Blending interpolants in the finite element method
Authors: Marshall J. A. , Mitchell A. R.
Published: August 2005

Blending function interpolants on rectangular elements are used to construct an overall interpolant which exactly matches function and normal derivative on the perimeter of a rectangular region. This overall interpolant is incorporated into the Ritz‐Galerkin version of the finite element method and a test problem involving the biharmonic equation is solved. The numerical results obtained demonstrate the considerable increase in accuracy of the exact boundary methods as compared with the usual method using interpolated boundary conditions. A similar investigation of second order elliptic problems with Dirichlet boundary conditions where the rectangular region is divided up into triangular elements yields the perhaps surprising result that blended interpolants on triangular elements do not necessarily improve the accuracy of the Ritz‐Galekin version of the Finite Element Method.
Modification to the Suhara‐Fukuda method of network generation
Authors: Shaw R. D. , Pitchen R. G.
Published: August 2005

This note describes improvements to the Suhara‐Fukuda method for generating a triangular finite element network. These are based on the use of a node generation procedure which distributes nodes is a regular pattern across the network area and is sufficiently flexible to permit any variation in element density. Because of the regular node distribution, poorly shaped elements will not arise and so the procedure for connecting the nodes to form elements can be simplified. Examples of networks generaed in this way are given.
Periodic Galerkin finite element method of tidal flow
Authors: Kawahara Mutsuto , Hasegawa Kenichi
Published: August 2005

This paper presents the finite element method of the analysis of tidal flow. Assuming that tidal flow is periodic, the Galerkin approach is employed as the numerical integration procedure in time using a trigonometric function as the interpolation function. The present method has shown to be suitable for computation especially from the point of computing time and numerical stability.
The solution of sparse linear equations by the conjugate gradient method
Authors: Jennings A. , Malik G. M.
Published: August 2005

The convergence properties of the conjugate gradient method are discussed in relation to relaxation methods and Chebyshev accelerated Jacobi iteration when applied to the solution of large sets of linear equations which have a sparse, symmetric and positive definite coefficient matrix. The conclusion is reached that its convergence rate is unlikely to be much worse than these methods, and may be considerably better. The conjugate gradient method may either be applied to the basic unscaled or scaled equations or alternatively to various transformed equations. Preconditioning, block elimination and partial elimination methods of transforming equations are considered, and some comparative tests given for six problems.
Finite difference approximation of boundary conditions along irregular boundaries
Authors: Hunt Bruce
Published: August 2005

The approximation of normal derivatives along a curved boundary becomes a major difficulty in obtaining finite difference solutions for irregular regions. A number of authors (for‐example, Allen,1 Fox,2 Greenspan3 and Parker and Ma4) have suggested various solutions to this problem. However, most of these suggested methods are fairly cumbersome to use since they require the use of different formulas for various combinations of mesh and boundary geometries. This note shows a different way to approximate a quite general boundary condition along curved boundaries. The method uses a uniformly spaced mesh, is simple, general, relatively easy to program and appears to give accurate results.
Construction of triangular finite element universal matrices
Authors: Silvester P.
Published: August 2005

The various ‘universal’ matrices from which finite element matrices for triangular elements are assembled in many electromagnetics and acoustics problems, can all be derived from a basic set of three fundamental matrices. These represent, respectively, the metric of the linear manifold spanned by the triangle interpolation polynominals, the finite differentiation operator on that same manifold, and a product‐embedding operator for the corresponding manifold for interpolation polynomials one order higher. Two of these have already been tabulated and published; the required method for computing the third is given in this paper, along with tables of low‐order matrices.
A partitioned parallel Runge‐Kutta method for weakly coupled ordinary differential equations
Authors: Mattione R. P. , Katz I. N. , Franklin M. A.
Published: August 2005

In many systems it is known a priori, that some states are ony weakly coupled with others. If such systems are solved on parallel processor it is possible to partition the states in such a way that one set of states is assigned to one processor and the other set of weakly coupled states to another. Computations can then be done in parallel. Because of the weak coupling it may not be necessary for information from the two sets to be communicated to each other for many integration steps. This can result in significant cost savings. In this paper, a partitined Runge—Kutta scheme is formulated for use on weakly coupleld systems of ordinary differential equations. An error expression is derived which provides a means for predicting the number of step sizes over which the partitioned formulas can be used in terms of a prespecified error tolerance. Numerical examples are presented both to verify the error expressions and to compare solutions using partitioned and unpartitioned schemes.
A starting algorithm for the numerical simulation of two‐phase flow problems
Authors: Lewis R. W. , White I. R. , Wood W. L.
Published: August 2005

The diferential equations governing two‐phase flow through a porous medium are derived and the space discretization is carried out by the Galerkin finite element method. An examination of the eigenvalues of the resulting system follows in order to decide on a suitable stability criterion to be satisfied by the time‐stepping algorithm. A class of time‐stepping schemes is introduced and their behaviour when applied to a specific problem is analyzed. The aim is to achieve an adequate modelling of the water‐oil interface and to preserve reasonable values of the capillary pressure. A technique to clustering the mesh around the interface is used to improved the modelling, and a boundary condition based on the relative mobilities of the two fiuids is imposed to enable the program to calculate the water‐oil ratio after breakthrough. The effect of lumping the mass matrix is then examined with a view to reducting the surge in the values obtained for the capillary pressure. Finally an optimum method is suggested for use as a starting algorithm for more sophisticated time‐stepping schemes.
An advanced structural analysis/synthesis capability—ACCESS 2
Authors: Schmit L. A. , Miura H.
Published: August 2005

An advanced automated design procedure for minimum weight design of structures (ACCESS 2) is reported. Design variable linking, constraint deletion, and explicit constraint approximation are used to effectively combine approach to structral synthesis is extended to problems involving fibre composite structure, thermal effects and natural frequency constraints in addition to the usual static stress and displacement limitations. Sample results illustrating these new features are given.
Analysis of some dispersion corrected numerical schemes for solution of the transport equation
Authors: Van Genuchten Martinus T H. , Gray William G.
Published: August 2005

In the last decade or so finite element techniques have been applied with increased frequency to contaminant transport problems. Whereas most of the attention has focused on finite element approximations of spatial derivatives, standard finite difference techniques are generally used for approximation of the time derivative. Such an approach yields a scheme which is at best second order correct in time. In this study several higher order approximations of the time derivative are developed and analyzed using a finite difference approximation, and Galerkin‐type finite element approximations in conjunction with several sets of basis functions. Results obtained with the different schemes exhibit significant improvements in the numerical solution of the convective‐dispersive equation.
On least squares approximations to indefinite problems of the mixed type
Authors: Fix George J. , Gunzburger Max D.
Published: August 2005

A least squares method is presented for computing approximate solutions of indefinite partial differential equations of the mixed type such as those that arise in connection with transonic flutter analysis. The mehod retains the advantages of finite difference schemes namely simplicity and sparsity of the resulting matrix system. However, it offers some great advantages over finite difference schemes. First, the method is insensitive to the value of the forcing frequency i.e., the resulting matrix system is always symmetric and positive definite,. As a result, iterative methods may be successfully employed to solve the matrix system, thus taking full advantage of the sparsity. Furthermore, the method is insensitive to the type of the partial differential equation, i.e., the computational algorithm is the same in elliptic and hyperbolic regions. In this work the method is formulated and numerical results for model problems are presented. Some theoretical aspects of least squares approximations are also discussed.