bluuebunny/crossref_metadata_2025 huggingface dataset:
The method of moments and its optimization
Authors: Leventhal Stephen H.
Published: August 2005

The method of moments is a semidiscrete numerical method for solving partial differential equations. The method approximates the solution of a partial differential equation by a finite sum of products of two functions. One function in the product is an unknown function of a single variable and the other function (moment function) is a prescribed function in the remaining variables. Using variational technique we obtain a finite system of boundary value problems of ordinary differential equations for the unknown functions.

The main goal of this paper is the study of the theoretical background and numerical effectiveness of the method of moments for solving linear partial differential equations on rectangular‐like domains.

The mathematical formulation of the method together with error estimates and the theory of optimal moment functions are given. If for the one‐dimensional moment functions piecewise polynomials of degree K are used then finite element type error bounds are obtained for the approximate solution in two dimensions.

We also consider the numerical implementation of the method through the factorization method and efficient initial value methods. Several numerical examples showing the efficiency of the method are presented.
Finite element formulations for large deformation dynamic analysis
Authors: Bathe Klaus‐Jürgen , Ramm Ekkehard , Wilson Edward L.
Published: August 2005

Starting from continuum mechanics principles, finite element incremental formulations for non‐linear static and dynamic analysis are reviewed and derived. The aim in this paper is a consistent summary, comparison, and evaluation of the formulations which have been implemented in the search for the most effective procedure. The general formulations include large displacements, large strains and material non‐linearities. For specific static and dynamic analyses in this paper, elastic, hyperelastic (rubber‐like) and hypoelastic elastic‐plastic materials are considered. The numerical solution of the continuum mechanics equations is achieved using isoparametric finite element discretization. The specific matrices which need be calculated in the formulations are presented and discussed. To demonstrate the applicability and the important differences in the formulations, the solution of static and dynamic problems involving large displacements and large strains are presented.
Mathematical programming in wing theory
Authors: Bera Rajendra K.
Published: August 2005

It is shown that mathematical programming (MP) finds useful application in wing theory. This is illustrated by posing wing theoretical problems as programming problems within the framework of the linearized lifting surface theory. A great merit of programming techniques is their ability to accommodate inequality constraints in a problem. This feature allows a wing theoretician to expand into a domain of practical problems where qualitative constraints can be included into his formulations. A few numerical examples involving the Prandtl lifting line equation with equality and inequality constraints are given to show that the method works. Its extension to complex wing‐body problems poses no conceptual difficulties, but requires further numerical experimentation.
Calculation of the elastic buckling loads of thin flat reinforced plates using triangular finite elements
Authors: Allman D. J.
Published: August 2005

Details are given of a finite element scheme for the determination of the elastic buckling loads of thin flat plates of arbitrary shape under general kinematic and traction boundary conditions. The buckling loads are calculated using a triangular finite element with corner connections which is based on a previously derived variational principle combining both stresses and displacements. Numerical solutions are presented to a number of example problems, including plates with reinforcing members, and comparisons are made, where possible, with results from the finite element analyses of other authors and with exact or classical Rayleigh–Ritz solutions.
Non‐linear dynamic analysis of axisymmetric shells
Authors: Nagarajan S. , Popov E. P.
Published: August 2005

Incremental equations of motion are derived from a Lagrangian variational formulation for the large displacement elastic‐plastic and elastic‐viscoplastic dynamic analysis of deformable bodies. The material constitutive behaviour is described in terms of the symmetric Piola–Kirchhoff stress and Lagrangian strain tensors. Degenerate isoparametric elements, permitting relaxation of the Kirchhoff–Love hypothesis, are used in a finite element formulation specialized for the analysis of shells of revolution subjected to axisymmetric loading. The linearized incremental equations of motion are solved using direct integration procedures, with added accuracy obtained from application of equilibrium correction at each step. The effectiveness of the numerical techniques is illustrated by the dynamic response analyses carried out on a shallow spherical cap subjected to uniform external step pressure loadings.
A finite element method for the general solution of ordinary differential equations
Authors: Csendes Z. J.
Published: August 2005

A finite element method is developed by which it is possible to obtain the general solution of an ordinary differential equation directly. The procedure consists of approximating the differential equation with a rectangular matrix equation and of solving the latter equation by using generalized matrix inversion. It is shown in the paper that the homogeneous and inhomogeneous solutions of the two systems correspond and that the approximate solutions produced form the complete general solution of the original differential equation.
A Fortran program to generate finite difference formulas
Authors: Csendes Z. J.
Published: August 2005

A discretization process is described by which it is possible to generate finite difference formulas for arbitrary linear two‐dimensional partial differential equations. The process is based on a novel approach to finite difference analysis in which differential operators are approximated by rectangular matrices. In this approach, the discrete form of a complicated operator is obtained by performing simple numerical operations on elementary matrix factors. The analysis is augmented by a listing of a computer program based on the method for the automatic generation of finite difference formulas.
A comparison of time marching schemes for the transient heat conduction equation
Authors: Wood W. L. , Lewis R. W.
Published: August 2005

This paper investigates the phenomenon of ‘noise’ which is common in most time‐dependent problems. The emphasis is on the achievable accuracy that is obtained with various time‐stepping algorithms and how this can be improved if noise is artificially damped to an acceptable level. A series of experiments are made where the space domain is discretized using the finite element method and the variation with time is approximated by several finite difference methods. The conclusion is reached that the Crank–Nicolson scheme with a simple averaging process is superior to the other methods investigated.
Finite element large deflection analysis of shallow shells
Authors: Batoz Jean Louis , Chattopadhyay Arup , Dhatt Gurbachan
Published: August 2005

A finite element formulation is presented to study the non‐linear buckling of arbitrary shallow elastic thin shells with general boundary conditions and subjected to conservative pressure loading. Pre and post buckling behaviour of a large number of shallow and semi deep doubly curved shells is studied in detail. Unsymmetrical bifurcation paths of a shallow spherical shell subjected to uniform inward pressure are also investigated.
Sequential interpolation functions
Authors: Feijoo Raul A. , Bevilacqua Luiz
Published: August 2005

In the present paper it is shown how to achieve special interpolation functions that can be applied to the finite element method, least squares methods or other related numerical techniques. We call these functions sequential interpolation functions since one of the main characteristics of this technique refers to the fact that the refinement of the solution does not require the redefinition of the interpolation funtions used in a previous step, as the case for instance in the FEM.

The present theory and the FEM using an isoparametric element were compared for a plane stress problem. The results show that both have the same degree of accuracy. The choice of the sequential interpolation functions is easier than those used in the finite element formulation and the computer time was reduced by about 50 per cent.
Enumeration of networks
Authors: Thomson P. F.
Published: August 2005

The results of previous workers relating the number of internal and external vertices in a two‐dimensional network to the number of elements of one given geometry are generalized to cover elements of any geometry (not necessarily convex) and any number of edge nodes, which may be vertex nodses of adjacent elements.

It is shown that a relationship between the number of elements and number of vertices (nodes) alone exists only on a surface (plane or otherwise) and therefore cannot be extended to three‐dimensional networks.

In three‐dimensions, the number of elements with any number of edges and vertices is related to the number of internal and external edges and vertices, is related to the number of internal and external edges and vertices, enclosed by a surface which may be non‐simple, (i.e. of topological genus greater than zero).
Design of efficient stiffened shells of revolution
Authors: Majumder D. K. , Thornton W. A.
Published: August 2005

A method to produce efficient piecewise uniform stiffened shells of revolution is presented. The approach uses a first order differential equation formulation for the shell prebuckling and buckling analyses and the necessary conditions for an optimum design are derived by a variational approach. A variety of local yielding and buckling constraints and the general buckling constraint are included in the design process. The local constraints are treated by means of an interior penalty function and the general buckling load is treated by means of an exterior penalty function. This allows the general buckling constraint to be included in the design process only when it is violated. The self adjoint nature of the prebuckling and buckling formulations is used to reduce the computational effort. Results for four conical shells and one spherical shell are given.
A degenerate solid element for linear fracture analysis of plate bending and general shells
Authors: Barsoum Roshdy S.
Published: August 2005

A general curved element of arbitrary shape for both thick and thin shells is proposed for the linear fracture analysis of a through crack in a shell or a plate. The element is derived from a degenerate 20‐noded solid isoparametric element using reduced integration technique. The 1/√(r) singularity of the strains is obtained by the same procedure proposed earlier for two‐ and three‐dimensional problems,1,2 viz. by placing the mid‐side nodes near the crack at the quarter points.

Several illustrated examples ranging from classical solutions to practical problems are given to assess the accuracy of solution attainable.
Local linearization of the finite element method and its applications to compressible flows
Authors: Shen S. F. , Habashi W.
Published: August 2005

In applications of the finite element method to complicated nonlinear problems, it is proposed to simplify the operator in each element, typically by a linerization process. Local approximations can then be more easily constructed in the usual manner, and a proper assembly can be made. Such an approach would lead to an approximate global variational statement; or if the variational principle exists but is hard to handle, it should suggest an iterative scheme with a built‐in physical basis, hence a better chance to converge. The case of high subsonic flow over a circular cylinder is studied in detail by this method. With linear triangular elements, numerical results are obtained for 100 nodal unknowns in each quadrant. The convergence is extremely rapid up to a free stream Mach number of 0.42, slightly above critical. The accuracy is excellent as compared against Imai's analytical results.
Non‐linear analysis for anisotropic materials
Authors: Valliappan S. , Boonlaulohr P. , Lee I. K.
Published: August 2005

An elasto‐plastic analysis of anisotropic work‐hardening materials, using the finite element method is presented. The analysis is based on the generalized Huber‐Mises yield criterion extended by Hill for anisotropic materials. General expressions for the anisotropic parameters in the yield criterion have been derived both for initial yielding as well as subsequent yielding in the case of work‐hardening materials.

The isoparametric ‘quadratic’ quadrilateral elements have been used for the analysis and the ‘initial stress technique’ has been adopted for the iterative solution of the non‐linear problems. The results of the various numerical examples have been compared with the available solutions.
An algorithm for minimal degree linear Chebyshev approximation on a discrete set
Authors: Roberts F. D. K.
Published: August 2005

An algorithm for computing a linear Chebyshev approximation to a function defined on a finite set of points is presented. The method requires the accuracy of the approximation to be specified, and determines the least degree approximation which achieves this accuracy. The algorithm is based upon the simplex method of linear programming. A FORTRAN program is supplied in the Appendix.
A simultaneous iteration algorithm for symmetric eigenvalue problems
Authors: Corr R. B. , Jennings A.
Published: August 2005

A FORTRAN IV algorithm is presented for determining sets of dominant eigenvalues and corresponding eigenvectors of symmetric matrices. It is also extended to the solution of the equations of natural vibration of a structure for which symmetric stiffness and mass matrices are available. The matrices are stored and processed in variable bandwidth form, thus enabling advantage to be gained from sparseness in the equations. Some of the procedures may also be used to solve symmetric positive definite equations such as those arising from the static analysis of structures loaded within the elastic range.
Continuously distributed dislocation model for shear‐bands in softening materials
Authors: Cleary Michael P.
Published: August 2005

The classical concept of slip‐surfaces in soil mechanics is assigned a physical significance based on frequent observations of rupture in overconsolidated soils and rock masses; propagation of a narrow shear‐band (i.e. slip‐surface) is commonly the mechanism of failure. Effectively, relative sliding occurs between adjacent material points on the surface, a concept familiar in rack mechanics; just as failure by crack propagation can occur in metals well below net section yielding, so can progressive failure by band propagation cause slope failure at nominal stresses well below the peak strength of the material. The relative sliding is simulated here, for a material response which is linear outside the band, by superposing continuously distributed dislocations; use is made of influence functions, for dislocation and point‐force singularities in a variety of materials and geometries. The method has further application in rock mechanics and metallurgy. A stress‐displacement constitutive relation is imposed on the line of the band and the resulting non‐linear singular integral equations necessitate numerical solutions; the routines are prefected on two problems for which answers are otherwise available. Extension to previously intractable plane‐strain problems is then described and results are given for a surface‐parallel shear‐band corresponding to a flake slide on a long slope.
On the numerical solution of phase change problems in transient non‐linear heat conduction
Authors: Mastanaiah K.
Published: August 2005

A finite difference solution for the transient non‐linear heat conduction with phase change in a finite slab is proposed. A two‐time level implicit method is used while Taylor's forward projection method is employed for taking into account the non‐linearities. The stability due to the boundary conditions at the moving front is verified. The numerical solution is compared with the analytical solutions and found to be in close agreement.
Artificial adjustment of element stiffness with a particular reference to plate bending
Authors: Cook Robert D.
Published: August 2005

A finite element may satisfy all convergence requirements and work well except for being too stiff or too flexible when the mesh is coarse. Here we discuss a technique for softening or stiffening such elements, provided that they have internal degrees‐of‐freedom. The technique is applied to a previously derived plate element, and the element is found to be much improved.
Comment on hybrid finite elements
Authors: Bartholomew P.
Published: August 2005

It is shown that an apparently sound selection of trial functions can give rise to hybrid elements with rank deficient stiffness matrices and unused interior fields. An account is given of modifications which overcome these problems for a particular plate bending element.
On the optimum shape of fillets in plates subjected to multiple in‐plane loading cases
Authors: Kristensen Erik Steen , Madsen Niels Flemming
Published: August 2005

The shape of a plate in plane stress is determined, such that the maximum elastic stress corresponding to given loads is minimized. The shape of the boundary is approximated by a series and optimization is carried out by solving a sequence of linearized minimum–maximum problems using linear programming. The optimization problem is extended to include multiple loading cases and geometrical constraints. The stress derivatives are found using analytical expressions for stiffnesses and stiffness derivatives of the finite elements. For the optimum design of a hole in an infinite plate under biaxial stretching the numerical result is compared with an analytical solution. As another example the method is used to optimize the edge shape of a shape of a junction in a web frame.
Computing three‐dimensional free surface flows
Authors: Larock Bruce E. , Taylor Cedric
Published: August 2005

Finite element techniques are developed for the analysis of high‐speed three‐dimensional free surface flows within the context of potential flow theory. The primary dependent variable is the velocity potential, which is approximated as a quadratic within each isoparametric element. An iterative approach is used to seek the initially unknown free surface location.

A high‐speed fluid jet issuing through an orifice plate into a transvese gravity field is examined to illustrate features of the approach. Convergence of the solutions is very good at a Froude number of 3‐53 but deteriorates as F nears unity.
Numerical solution of Fredholm integral equations with a logarithmic singularity
Authors: Sastry S. S.
Published: August 2005

Generalized quadrature is used for the numerical solution of two Fredholm integral equations which occur in electrostatics and aerodynamics.
Definition of stochastic models during soil compaction
Authors: Drakatos Panagiotis A.
Published: August 2005

Employing experiments, Correlation Analysis, the method of ‘least square estimation’, and a special computational program, the problem of similarity between the variables of the soil‐machine system during soil compaction is examined. The solution, based on the probability theory of ‘Analysis of Variance’, consists in developing the stochastic model which defines the variables and their interactions for different systems.

In this particular case of soil compaction by cylinders the theorem of Vaschy‐Buckingham or Pi terms is used to define the mononyms (terms) of similarity.

The stochastic model, operant in every case, consists of an ellipsoid characteristic of the particular cylinder and its variables, so that greatest soil compaction at a given depth is achievable with a probability of 95 per cent at a level of significance of α = 0.05.
The solution of the three‐dimensional Navier‐Stokes equations using a new finite difference approach
Authors: Roscoe D. F.
Published: August 2005

A numerical method for integrating the three‐dimensional Navier–Stokes equations for a viscous, incompressible, laminar duct flow is given. A new method is applied to the derivation of stable, and second order accurate difference representations to the momentum transport and diffusion terms. Numerical examples are given.
Sensitivity of inelastic response to numerical integration of strain energy
Authors: Kamat M. P.
Published: August 2005

The exact solution to the quasi‐static, inelastic response of a cantilever beam of rectangular cross‐section subjected to a bending moment at the tip is obtained. The material of the beam is assumed to be linearly elastic‐linearly strain‐hardening. This solution is then compared with three different numerical solutions of the same problem obtained by minimizing the total potential energy using Gaussian quadratures of two different orders and a Newton‐Cotes scheme for integrating the strain energy of deformation. Significant differences between the exact dissipative strain energy and its numerical counterpart are emphasized. The consequence of this on the non‐linear transient responses of a beam with solid cross‐section and that of a thin‐walled beam on elastic supports under impulsive loads are examined.
Integration of triangular finite elements containing corrective rational functions
Authors: Katz I. Norman
Published: August 2005

Rational corrective functions are often supplemented to complete polynomials in order to create conforming triangular finite elements for plate bending. Although such elements containing complete polynomials of degree 3 or 4 have long been known, conforming triangular elements for plate bending, which contain complete polynomials of arbitrary degree p ⩽ 5 and new corrective rational functions, have been given only recently.

Integration of rational functions usually requires numerical quadrature. It is shown here that these new corrective rational functions fall into two classes. For those in one class, explicit closed form integration formulas are available. For those in the other class, integration can be performed very efficiently by summing series whose general term is O(kd) for large k, where d ⩽ 5. These formulas also apply to the older third and fourth order elements.
A numerical method for the solution of certain classes of nonlinear Volterra integro‐differential and integral equations
Authors: Prosperetti Andrea
Published: August 2005

A direct numerical method is presented for the solution of classes of non‐linear, singular Volterra integral and integro‐differential equations of the types
equation image
where y is the unknown and the Fs and Gs are given but to a large extent arbitrary functions of their arguments. The adaptation of the method to other equations, in particular with different kernels, is immediate.
Structural instability and natural vibration analysis of thin arbitrary shells by use of the Semiloof element
Authors: Martins R. A. F. , Owen D. R. J.
Published: August 2005

At present the Semiloof element is probably one of the most efficient available for the solution of thin shells of arbitrary geometry. Experience in static situations indicates that accurate results can be obtained for non‐trivial geometric and loading configurations with relatively coarse meshes. Variable thickness shells or discontinuous thicknesses can be accommodated and no difficulties are encountered in modelling sharp corners or multiple junctions in structures. The element is basically non‐conforming but some measure of C1 continuity is provided by the introduction of normal rotation variables at Loof nodes on the element boundary. This paper examines the element behaviour when applied to elastic instability and vibration situations. An eigenvalue solution scheme based on Sturm sequences is presented which does not require the usual elimination of a percentage of the total nodal variables by static condensation. Finally the method is assessed by the solution of several numerical examples.
Numerical acceleration of thermal regenerator simulations
Authors: Willmott A. J. , Kulakowski B.
Published: August 2005

The packing of a thermal regenerator provides intermediate storage capacity for the heat transferred from a hot to a cold fluid and is an intrinsic part of this type of heat exchanger. The thermal capacity of the packing imposes an inertia upon the regenerative process which prevents instantaneous response to changes in operating conditions. A regenerator which possesses a large ‘thermal inertia’ will be very slow to reach cyclic equilibrium.

A mathematical model of a regenerator which simulates cycle to cycle behaviour will be similarly slow to converge. The process of numerical acceleration discussed in this paper can be employed with advantage to overcome the thermal inertia and reach the cyclic equilibrium situation in fewer cycles, thereby substantially reducing the calculation time.

The paper describes a numerical acceleration method (based on Aitken's Δ2 method) and gives details of parameter ranges for which it is suitable.
Numerical limit analysis of perforated plates
Authors: Winnicki Lech , Kwieciński Marek , Kleiber Michał
Published: August 2005

Full limit analysis is preseted of a thin plate under plane stress, perforated with circular holes arranged in a regular penetration pattern. The material is assumed to be elastic–perfectly plastic and to obey the Huber–Mises yield condition together with its associated flow law. The finite element tangential stiffness method is employed. Triangular constant‐strain elements are used, the load being generated by means of constant edge displacements at the periphery of a suitably selected subregion. Load‐displacement diagrams are obtained for various edge displacement programs and then an interaction curve is constructed for the perforated plate considered. Each computational step is associated with a certain plastic zone which develops until the limit state configuration is reached. The plate is provided with equilateral triangular configuration of holes with arbitrary cutout coefficients. The interaction curves are plotted against the Huber–Mises yield condition for the plate without openings.
The application of a variational finite element method to problems in fluid dynamics
Authors: Usuki S.
Published: August 2005

The finite element method is applied to the analysis of a two‐dimensional steady flow around a box girder in uniform flow. The local potential principle for an incompressible viscous fluid flow is introduced as a basis for establishing finite element models and a determination is made as to whether or not approximate solutions obtained by the use of the principle give upper or lower bounds of the solution. The boundaries are set so as to be appropriately far from the girder and the velocities are specified at the upstream, upper boundary and the lower boundary and the total stresses are specified at the downstream boundary. The system of non‐linear equations is solved by the standard Newton–Raphson method. Integrating the pressures and viscous stresses acting on the surface of the girder, the coefficients of drag, lift and pitching moment are numerically obtained for some Reynolds numbers. The wind tunnel test has been the only means to determine these coefficients for a body with irregular shape. The finite element method presents a new powerful procedure to determine these coefficients.
The indirect design of fluid channels using a global variational method with trial functions not satisfying the prescribed boundary conditions
Authors: Davies B. , Hendry J. A.
Published: August 2005

The indirect design problem for compressible fluid flow in a channel determines the physical shape of the channel from the prescribed flow on the channel boundary. In this paper, a global variational method for the solution of this problem is described.

The non‐linear equations are solved by iteration; at each step of this iteration the solution to the equations (which are now linear) is found by a global variational method. The variational method differs from conventional methods in that the trial function does not satisfy the prescribed boundary conditions, but the method reproduces the conditions on the boundary.

Results are presented for various physical quantities associated with the channel.

Some comments are made on the extensions of these methods to supersonic flow and to the realistic cascade design problem.
Static analysis of diaphragm‐supported cylindrical shells using a curved finite strip
Authors: Dawe D. J.
Published: August 2005

Two versions are presented of a tranversely‐curved finite strip for the static analysis of circular cylindrical shells. The strip is a refined displacement model with sixteen transverse degrees‐of‐freedom arranged either at two or at four nodal lines. The analysis is specialized for cylinders with diaphragm supports at the ends—via the employment of simple trigonometric series in the longitudinal direction—but extension to other end conditions is straightforward in principle. Detailed results of applications of the curved strip are presented.
The computation of some slipline field solutions for asymmetric extrusion
Authors: Das N. S. , Chitkara N. R. , Collins I. F.
Published: August 2005

Two families of slipline field solutions for extrusion through unsymmetrical square dies, which have hitherto only been suggested qualitatively are computed and their ranges of validity determined. The solutions are of ‘indirect type’ and have been obtained using a technique by which the integral operators are approximated by matrices. The determination of the slipline angles and other field parameters corresponding to any given geometry requires the solution of a set of non‐linear algebraic equations. This is accomplished using a non‐linear optimization procedure.
Error in eccentric beam formulation
Authors: Gupta Ajaya K. , Ma Paul S.
Published: August 2005

When the stiffness matrix of a beam element is transformed linearly to account for eccentricity with respect to the plate it is attached to, an incompatibility in the axial displacement field is introduced. The nature of the error so introduced is discussed.
On curved finite elements for the analysis of circular arches
Authors: Yamada Y. , Ezawa Y.
Published: August 2005

This paper proposes a straightforward criterion to warrant the displacement functions being used in the finite element approximation of circular arches. The criterion was established by studying the natural shape function, i.e. the exact solution of the deformed shape, of the circular arch element. The exact stiffness matrix [K]exact is derived from the natural shape and is confirmed to be the inverse of the well‐known flexibility matrix [F]exact in the curved beam theory. The present paper compares the inverse [K]−1 of the stiffness matrix derived from the assumed displacement function with the [F]exact. It is shown that the procedure also guarantees the implicit inclusion of rigid‐body modes in the pertinent stiffness matrix [K]. Case studies on typical approximate displacement functions assure the appropriateness as well as the ease of application of the proposed method.
A finite element for 3‐D prestressed cablenets
Authors: Gambhir M. L. , de V. Batchelor Barrington
Published: August 2005

A finite element method is presented for the analysis of prestressed cablenets. The method is based on representing the prestressed cablenet as a series of finite length curved elements. The large displacement formulation used, enables the evaluation of the static and dynamic response of 3‐D cablenets. The development is general and the mathematical basis is explained at length. It is shown by means of comparison functions that for absolute continuity at nodes, a cubic displacement field is sufficient for the prediction of the first frequency of shallow nets. For globally deep networks the accuracy can be increased by employing a quintic displacement field for the normal component of displacement.

The application of the proposed model to cable structures of shallow and also of deep global geometry, is presented. A variety of edge boundary shapes are employed in order to illustrate the versatility of the large displacement formulation. In all the example problems studied, gravity load has been taken as the initial load condition in calculating the equilibrium configuration. The stiffness and consistent mass matrices associated with the equation of motion are derived using Hamilton's variational principle.
A block iterative finite element algorithm for numerical solution of the steady–state, compressible Navier–Stokes equations
Authors: Cooke Charlie H. , Blanchard Doris K.
Published: August 2005

An iterative method for numerically solving the time independent Navier–Stokes equations for viscous compressible flows is presented. The method is based upon partial application of the Gauss–Seidel principle in block form to the systems of the non‐linear algebraic equations which arise in construction of finite element (Galerkin) models approximating solutions of fluid dynamic problems. The C0‐cubic element on triangles is employed for function approximation. Computational results for a free shear flow at Re = 1000 indicate significant achievement of economy in iterative convergence rate over finite element and finite difference models which employ the customary time dependent equations and symptotic time marching procedure to steady solution. Numerical results are in excellent agreement with those obtained for the same test problem employing time marching finite element and finite difference solution techniques.
Certain properties of numerical schemes used to solve the blade‐to‐blade flow problem
Authors: Deshpande R. B.
Published: August 2005

The compressible potential flow on a blade‐to‐blade surface of a turbomachine is analysed using the Thomas algorithm. The numerical properties of resulting non‐linear simultaneous equations are studied with respect to grid aspect ratios and convergence. The heuristic approach has established which are the important factors that affect the flow solution in typical blade‐to‐blade configurations.
The visco‐plastic approach to problems of plasticity and creep involving geometric non‐linear effects
Authors: Kanchi M. B. , Zienkiewicz O. C. , Owen D. R. J.
Published: August 2005

The visco‐plastic model is used in the context of large displacement (geometrically non‐linear) analysis. As the process now involves in numerical computation an updating of relevant stiffness matrices, formulation which is stable and computtionally more efficient. The applications shown demonstrate the applicability of the process to large displacement elasto‐plastic shell analysis as well as to problems of creep buckling.
Elastoplastic thick shell analysis by viscoplastic solid finite elements
Authors: Cormeau I.
Published: August 2005

Shell‐like assumptions introduced in a three‐dimensional isoparametric finite element provide a kinematic description of smooth thick shell structures. Elastoplastic solutions are obtained via the stationary response of an elastic/viscoplastic model. Special attention is devoted to the probem of numerical integration across the depth of the shell, as well as the applicability of the method to thin shell analysis.
A conforming crack tip element with quadratic variation in the singular fields
Authors: Stern Morris , Becker Eric B.
Published: August 2005

Shape functions are given for a six node triangular element in which the √rdisplacement filed varies quadratically in the circumferential direction. Using a convenient co‐ordinate system the stiffness development is sketched and the integrations re done exactly in the radial co‐ordinate direction. Brief numerical results are presented. Shape functions for a 15 node three‐dimensional element are given explicitly. The shape functions provide both constant strain capability and conformability with quadratic isoparametric elements.
A drafting program for isoparametric finite elements
Authors: Hofmeister Lawrence D.
Published: August 2005

Considerable effort has been invested lately in the application of isoparametric finite elements for numerical solution of a wide range of applied mechanics problems. In fact, several general purpose computer programs are now available which are based upon such finite elements. In the present paper, a new application of the isoparametric finite element concept is introduced which significantly extends its usefulness for many practical structural configurations. In this application, final working or architectural drawings of the structure are made from the same (or similar) finite element model as was utilized in a structural integrity analysis. The hardware necessary to produce such drawing, a computer driven plotter or automated drafting machine, is available commercially or through most data centres, and the software concepts required are described herein.
p‐convergent finite element approximations in fracture mechanics
Authors: Szabo B. A. , Mehta A. K.
Published: August 2005

The strain energy release rate (G) converges rapidly in finite element approximations in which the finite element mesh is fixed and the order of polynomial displacement interpolations (p) is increased. Numerical experiments indicate that the error inG is very closely estimated, even for small pand very coarse finite element meshes, by an expression of the form k (NDF)‐1 in which k is a mesh dependent constant and NDF is the number of degrees‐of‐freedom. The method provides for very efficient and accurate computation of G without the use of special techniques.
Ritz solutions for axisymmetric temperature and stress fields in finite elastic cylinders
Authors: Tauchert Theodore R.
Published: August 2005

Variational formulations are given for the stationary heat conduction and thermodelasticity problems of an orthotropic, non‐homogeneous, finite length, cylinder. It is demonstrated that a single‐element approach based upon Ritz approximations and the method of Lagrange multipliers offers and efficient means for computing the temperature and thermal stress distributions.
Application of direct variational method to the solution of mixed boundary value problems
Authors: Goldstein R. V. , Entov V. M. , Zazovski A. F.
Published: August 2005

A direct variational method has been considered for solving mixed boundary value problems for linear elliptical equations. The main idea of the method consists in searching the unknown solution in a class of functions satisfying the differential equation and some of the boundary conditions. The functional to be minimized may be expressed in terms of the boundary values of the unknown solution, and thus the dimension of the variational problem is reduced. Reduction of the functional defined on the domain to a functional on its boundary corresponds to the common in the elsaticity theory of application of Clapeyron's theorem for the transformation of functionals of minimum elastic potential energy and complementary work. The reduced problem may be solved by the Ritz direct method. Some examples from the elasticity and seepage theories have been given to demonstrate the effectiveness of the method. Among the problems that have been solved by the method we may mention the plane elastic problem for a crack in a strip, three‐dimensional elastic problems for a plane rectangular crack in an infinite elastic medium and the problem of a rectangular die in sliding contact with an elastic half‐space (numerical solutions exhibit correct asymptotic behaviour in the neighbourhood of corners of cracks or die), and the plane and axisymmetric seepage problems.
Development of a finite dynamic element for free vibration analysis of two‐dimensional structures
Authors: Gupta K. K.
Published: August 2005

The paper develops an efficient free‐vibration analysis procedure of two‐dimensional structures. This is achieved by employing a discretization technique based on a recently developed concept of finite dynamic elements, involving higher order dynamic correction terms in the associated stiffness and inertia matrices. A plane rectangular dynamic element is developed in detail. Numerical solution results of free‐vibration analysis presented herein clearly indicate that these dynamic element combined with a suitable quadratic matrix eigenproblem solution technique effect a most economical and efficient solution for such an analysis when compared with the usual finite element method.
Solution techniques for non−linear finite element problems
Authors: Bergan P. G. , Horrigmoe G. , Bråkeland B. , Søreide T. H.
Published: August 2005

The paper presents a classification of mathematical commonly encountered in connection with solution of non−linear finite element problems. The principal methods for numerical solution of the non−linear equations are surveyed and discussed. Special emphasis is placed upon the description of an automatic load incrementation procedure with equilibrium iterations. It is shown how this algorithm can be adapted for solving problems involving instabilities, snap−through and snap−back. A simple scalar quantity denoted the current stiffness parameter is suggested; this parameter is used to characterize the overall behaviour of non−linear problems. It can also be used as a steering parameter in the solution process. The use of the present technique is illustrated by several examples.
A simple flat triangular shell element revisited
Authors: Olson Mervyn D. , Bearden Terrence W.
Published: August 2005

The 18 degree‐of‐freedom flat triangular shell element is reformulated by combining the well‐known bending triangle with a plane stress triangle incorporating in‐plane rotations at each vertex. Both elements are displacement formulated. The plane stress element's displacement interpolation is incomplete and hence convergence to exact solutions is precluded. Comprehensive test results are presented for several types of problem including plane stress, thin shells and folded plates. The results indicate that the element does produce rapidly convergent answers. However these answers are not the correct ones, although they may be acceptable engineering approximations in many applications. Further, the element seems to provide reasonably good results even for relatively coarse element grids.
Dynamics of rotationally periodic structures
Authors: Thomas D. L.
Published: August 2005

The paper considers the finite element analysis of the free, undamped and the forced damped vibrations of rotationally periodic structures. Associated with every natural frequency (except for those for which the deflection is the same at corresponding points on every substructure) there are a pair of orthogonal mode shapes, with eigenvectors {u} and {ü}. The complex vector {z}={u}+i{ü} is also an eigenvector of the equations of motion, and represents a rotating normal mode. The deflection of one substructure has the same amplitude as, and a constant phase difference from the deflection of the preceding substructure. It is therefore possible to analysse the complete structure by considering only one substructure, and applying appropriate complex constraints at its boundary with the following substructure, so as to and applying appropriate complex constrainsts at its boundary with the following substructure, so as to impose this phase difference. The method has been implemented in a computer program and is illustrated by analyses of an alternator end winding, a cooling tower with legs, and a wheel of turbine blades.

For forced vibration, it is shown that any arbitrary oscillatory force can be decomposed into a series of rotating forces. For any one of these rotating components, there is a fixed relationship between the amplitude and phase of the force acting on one substructure, and that acting on an adjoining substructure. This relationship, which does not involve any approximation, can be used to enable a series of calculations of the response of one substructure to be performed instead of one on the whole structure. A series of calculations on an individual substructure normally requires much less computer time and storage than a single calculation on the complete structure.
A theoretically improved and easily implemented version of the Ahmad Thick Shell element
Authors: Talha M. A.
Published: August 2005

An implementation of the Ahmad Thick Shell element using vector manipulation, reduced integration and the incorporation of the missing term in the approximating polynomial is presented. The first two aspects are merely applications. The theory is presented elsewhere6,7,11,18 and will not be repeated for the sake of conciseness. The third aspect was achieved by adding to the original nodal configuration a central node having only one degree‐of‐freedom. That node is then eliminated to preserve the original number of degrees‐of‐freedom.

The Ahmad thick shell element is adequately presented in finite element literature.1‐3 The co‐ordinate definition and the displacement field are retained and will not be dealt with henceforth. However, a different concept of the strain definition due to Irons is used and the bending terms are included in the modulus matrix.

The element like all second generation isoparametric elements is easily implemented through a shape function subroutine suitable for quadrilaterals. It reproduces exactly rigid body motion for any combination of elements of any geometry even for elements withcurved sides and variable thickness. Although Ahmad's thick shell element passed the patch test for unequal parallelograms but not for quadrilaterals the version presented passes that test for quadrilaterals.

In spite of its credentials, the new version denoted by A has a major setback. the presence of one too many spurious mechanism is reported.
A simple and efficient finite element for general shell analysis
Authors: Kanok‐nukulchai Worsak
Published: August 2005

A simple, efficient and versatile finite element is introduced for shell applications. The element is developed based on a degeneration concept, in which the displacements and rotations of the shell mid‐surface are independent variables. Bilinear functions are employed in conjunction with a reduced integration for the transverse shear energy. Several examples are tested to demonstrate the effectiveness and versatility of the element. The numerical results indicate that the shell element performs accurately for both thick and thin shell situations.
A shock capturing application of the finite element method
Authors: Cooke Charlie H. , Blanchard D. K.
Published: August 2005

The finite element method is employed for the calculation in Cartesian co‐ordinates of uniform flow on a rectangular region which encounters an embedded oblique shock with known turning angle. A modified form of the code so developed is used to obtain preliminary results concerning finite element performance in the calculation of viscous recirculating flow with embedded weak shocks.
A hybrid integral‐equation method for steady two‐dimensional ship waves
Authors: Yeung Ronald W. , Bouger Yann C.
Published: August 2005

This paper presents a novel integral‐equation technique for solving the steady‐state wave‐resistance problem. The free‐surface condition is linearized, but the body condition is satisfied exactly. An integral relation describing the flow inside an arbitrarily truncated internal region is first obtained by applying Green's Theorem, using only the simple source function for an infinite fluid. The internal flow is next matched with eigen expansions in the upstream and downstream outer regions. The radiation condition can be satisfied exactly simply by a proper choice of the solution form in these outer regions. The method is applied to investigate flows about both lifting and non‐lifting two‐dimensional bodies. Agreement with existing results is excellent. The present formulation provides a simple yet rational basis for tackling the practical three‐dimensional ship‐wave problem, in which past workers have encountered considerable difficulties using a complicated free‐surface Green function.