bluuebunny/crossref_metadata_2025 huggingface dataset:
Solving the model of isothermal reactors with axial mixing by the differential ouadrature method
Authors: Wang Kuo‐Ming
Published: August 2005

The numerical solution of nonlinear partial differential equations plays a very important role in may areas of physics, engineering and biology. In many cases all that is desired is a moderately accurate solution at a few points whcih can be calculated rapidly.

In this work, we present a simple and direct technique, the differential quadrature method, which can be applied in various classes of partial differential equations. It produces moderately accurate results for the points of interest, generally few in number, and eliminates the difficulties of programming complex algorithms for the computer, as well as excessive use of storage and computer time. We illustrate this technique with the solution of the model of isothermal reactors with axial mixing. The results are compared with the results obtained by using the finite difference method. It is shown that differential quadrature of order 9 (9 points) provides approximately the same accurate results as the finite difference method (480 points) does.
Effects of internal finite element node placement on field problem solutions
Authors: Kraus Hal G.
Published: August 2005

Given herein, in abbreviated form, is a discussion of the effects of element internal and edge node location in the finite element‐Galerkin method. The solution dependencies on node placement of many numerical schemes are reviewed. Then, a transient linear one‐dimensional heat conduction problem, solved using the finite element‐Galerkin method, is used for illustration. Results show that, in contrast to many other numerical techniques, if the finite element‐Galerkin method is strictly adhered to with respect to spatial integrations for a given number of nodes in an element, the solution is unaffected by shifting node locations as long as the specific shape of any given element remains unaltered. The reason for this invariance becomes readily apparent upon examining the orthogonality requirements imposed by this method.
An analysis of alternatives for computing axisymmetric element stiffness matrices
Authors: Palacios Jose Antonio , Henriksen Mogens
Published: August 2005

This note compares the use of numerical and closed form integration in computation of element stiffness matrices for axisymmetric finite element analysis. Only constant strain elements are considered. Results obtained with Gaussian quadrature and closed form integrators are compared mutually and with an exact solution obtained from classical methods. The FEM global equations estimate the force‐displacement behaviour of an elastic continuum with an accuracy that depends on the integration method used. Selection of an integration order minimizing error is particularly critical in the presence of high stress gradients. Best results in the vicinity of the axis of revolution may be obtained with single‐point integration rather than higher order approximations or exact integration of the element stiffness matrix. This phenomenon and its consequences are subsequently discussed.
Direct‐iterative solution of ill‐conditioned finite element stiffness matrices
Authors: Gen‐Hua Shi
Published: August 2005

In this paper we discuss and expand the direct‐iterative method proposed originally by Wilson.1 First we introduce several simple numerical examples to illustrate the basic idea of this method before we proceed to prove the convergence of the direct‐iterative method. We then discuss the methods for selecting the transformative matrix (Q) to be used in transforming an ill‐conditioned matrix into a well‐conditioned matrix in the direct‐iterative method. There are two methods used to choose the matrix Q, namely the rigid body movement method and the imaginary element method. From examples 1‐3 we can see that the imaginary element mesh is optional, and the finite element mesh is not necessary. The imaginary element method is a generalization of the mesh refinement method development in Reference 3. Because instead of local rotation angle we only choose displacements of nodes to represent rigid body movement, the rigid body movement method is an improvement of the method in Reference 2. The advantage of these two methods is that, in order to obtain well‐conditioned matrices, only a few changes in the stiffness matrices are required even with general ill‐conditioned stiffness matrices, and then convergency is achieved rapidly under SOR iteration. Finally, the examples for computing each type of the ill‐conditioned matrix in three‐dimensional finite element analysis are presented to demonstrate the effectiveness of the direct‐iterative method in solving the large bandwidth problems.
A finite element model for plane elasticity problems using the complementary energy theorem
Authors: Vallabhan C. V. Girija , Azene Muluneh
Published: August 2005

A finite element stress analysis capability for plane elasticity problems, employing the principle of stationary complementary energy, is developed. Two models are investigated. The first is a 24 d.o.f. rectangular finite element. The second model consists of an 18 d.o.f. triangular element. In order to allow for self‐equilibrating stresses which are continuous within the element, the well‐known Airy stress function ø is used. The function ø is represented by means of quintic Hermitian polynomials within the finite element. The values of the ø function and its derivatives up to order two are used as nodal parameters. For matching the stress function with the prescribed boundary tractions, additional equations are developed considering the force and moment equilibrium equations on the boundary consistent with the assumed stress function. These additional boundary equations are incorporated into the system equations using the Lagrangian multiplier technique. Excellent results are obtained for linear elastic problems even with coarse finite element discretization. Some examples of plane elasticity problems are solved and results compared.
A warning against misuse of quarter‐point elements
Authors: Peano A. , Pasini A.
Published: August 2005

Quarter‐point elements are used very frequently for fracture mechanics computations, because the quarter‐point technique yields the required singular interpolation without any modification to existing software. This advantage is particularly significant for three‐dimensional stress intensity factor computations because of the difficulty of implementing other techniques. However, in practical 3‐D applications, the crack front is usually curved, and this note proves that a crack front distortion leads to a negative Jacobian in the region surrounding the crack front. The numerical difficulties to be expected depend on the aspect ratio of the elements.
On a finite element model for solving Dirichlet's problem of Laplace's equation
Authors: Yi‐Heng Chen
Published: August 2005

This paper is a sequel to the author's previous paper. 1In soliving Dirichlet's problem of Laplace's equation by the finite element method, we can subdivide the original region into many elements, which in general are chosen to be triangular or rectangular. Customarily, 2,3,7 polynomial trial functions are chosen as the basis of the finite element model.

In this paper we divide the original region into rectangles and choose a particular series for the finite element model. This model is shown to be very efficient in obtaining the analytical solution of Laplace's equation over complex regions. From some computer trials it is demonstrated that this finite element model can be used to reduce the main memory storage and computer time requirements.
Some vector expansion functions for quadrilateral and triangular domains
Authors: Okon E. E.
Published: August 2005

Vector functions having a constant or zero divergence over a quadrilateral domain are derived using a parametric co‐ordinate formulation. Expansion functions with a constant curl, defined over parallelograms and triangles, which can be used in moments‐based investigations of electromagnetic wave interactions with surfaces and apertures, are then obtained. Finally, two basis functions, which have appeared in the literature and have been used to solve field problems associated with rectangular apertures and arbitrary surfaces, are identified as special cases of these expansion functions.
Round‐off error in the penalty finite element analysis of incompressible continuous media
Authors: Bar‐Yoseph Pinchas
Published: August 2005

Upper and lower bounds on the spectral condition number of the coefficient matrix are computed for several Lagrangian and serendipity elements.
Hybrid‐stress eight‐node elements for thin and thick multilayer laminated plates
Authors: Spilker R. L.
Published: August 2005

Hybrid‐stress eight‐node isoparametric multilayer plate elements are developed for the analysis of thin to thick fibre‐reinforced composite plates. Transverse shear deformation effects are incorporated by allowing a linear through‐thickness displacement variation for in‐plane displacements in each layer, with displacement continuity enforced along interlayer surfaces. An alternative, kinematically constrained through‐thickness displacement distribution is also studied, characterized by laminate reference plane displacements and non‐normal rotations. All stress components are included and are interpolated independently within each layer; interlayer surface traction continuity and laminate upper/lower surface traction‐free conditions are satisfied exactly. Alternative forms of in‐plane stress distributions are examined and the best is shown to be based on a rationale used previously for single layer hybrid‐stress plate elements to avoid locking in the thin plate limit. Selected example problems are used to demonstrate the convergence and effects of plate thickness for each of the elements, and limited stress results serve to verify the expected optimal sampling points.
Interactive computer graphic preprocessing for three‐dimensional finite element analysis
Authors: Perucchio Renato , Ingraffea Anthony R. , Abel John F.
Published: August 2005

An interactive computer graphic system has been developed for generating and editing three‐dimensional finite element meshes. The various difficulties inherent to the generation of solid three‐dimensional geometries through an effective use of computer graphic techniques are examined to explain the advantages of this system. The method is a combination of discrete transfinite mapping and cubic spline blending algorithms, used, respectively, to create plane cross‐sectional meshes, and to generate the three‐dimensional geometry by interpolaring between the cross‐sections. The communication between analyst and machine is based on a digitizing tablet and a refresh vector scope. The dynamic viewing capabilities and depth cueing of the vector display greatly enhance the perception of the mesh three‐dimensionality. Multi‐view dynamic display is also employed to facilitate node or element selection. The flexibility and usefulness of the system are illustrated with the help of three examples of practical interest.
Estimating stress intensity factors with singular components of the total finite element solution
Authors: Staab G. H.
Published: August 2005

Displacement formulated singular elements are compared to isoparametric quarter‐node elements. The total stress and displacement solution for each element is decomposed into singular and regular components for evaluating stress intensity factors. Specific relationships between nodal displacements and the singular component of stress are presented for the isoparametric element at selected locations within the element. Numerical results for KI and KII in the slant crack problem are presented. The singular components are shown to produce superior results when considering crack opening displacements.
Minimum weight design of a spherical shell under a concentrated load at the apex
Authors: Thevendran V.
Published: August 2005

The minimum weight design of a spherical shell under a concentrated load at the apex is presented in this paper. The usual restrictions of statical equilibrium and various yield criteria have been utilized. The finite element method is used to discretize the design parameters and the resulting nonlinear minimization problems are solved using the modified Rosenbrock's optimization method.
The optimum size of quarter‐point crack tip elements
Authors: Harrop L. P.
Published: August 2005

Ingraffea and Manu1 and Lynn and Ingraffea2 have shown that the size of the quarter‐point elements can affect the computed elastic stress intensity factor. The nature of the effect is such that, all other details remaining constant, there is a particular crack tip element size which minimizes the error in the computed stress intensity factor. Here, size of element means the radial edge length. The reasons for this size dependence are discussed below. It will be seen that the discussion is in terms of the need to simultaneously represent the singular and finite stress terms in a given problem. The discussion has relevance to other formulations of crack tip elements.
Finite element method in magnetohydrodynamic channel flow problems
Authors: Singh Bani , Lal Jia
Published: August 2005

The finite element method has been applied to the steady‐state fully developed magnetohydrodynamic channel flow of a conducting fluid in the presence of transverse magnetic field. Simple elements have been used to obtain the numerical values of velocity and induced magnetic field. To test the efficiency of the method, three different geometries, viz., rectangle, circle and triangle, are taken as the section of the pipe whose walls are non‐conducting. Comparison is made with those cases in which exact solutions are available. Apart from giving good results, the FEM makes it possible to solve the problem for a pipe with arbitrary cross‐section which was not possible by the other methods.
A finite strip method for the geometrically nonlinear analysis of plate structures
Authors: Lengyel P. , Cusens A. R.
Published: August 2005

A finite strip method is presented for the geometrically nonlinear analysis of prismatic structures under arbitrary loading. The method employs an iterative finite element approach but replaces numerical integration by an explicit formulation. A computer program has been developed and is described in the paper; a numerical example is given.
Variable upwinding and adaptive mesh refinement in convection‐diffusion
Authors: Carey G. F. , Plover T.
Published: August 2005

An analysis of the stability and oscillation properties of upwind finite element methods for convection‐diffusion is given and used to develop a variable upwinding formulation. This formulation is particularly well suited to problems with nonconstant coefficients, nonlinearity or non‐uniform meshes. We present the theoretical analysis and numerical studies for a standard steady‐state and transient model one‐dimensional convection‐diffusion problem. The variable upwind strategy can also be used to significant advantage in conjunction with adaptive mesh refinement. Numerical results for the steady‐state case in which convection dominates and the mesh is adaptively refined into a boundary layer confirm the efficacy of the scheme.
Nonlinear flow analysis in pipe networks
Authors: Ohtmer O.
Published: August 2005

Pipe networks are computed in an analogous mannner to frameworks in structural mechanics loaded only by moments. The mesh method (force method) is applied. Due to the boundary layer effects in special pipe members, the flow problem is in general nonlinear. Therefore, the Newton‐Raphson iteration procedure is used to solve the nonlinear system of equations. Using the computed flow rates in the TREE structure (determined by graph theory) as initial values, the iteration procedure converges rapidly to a user specified tolerance value. The loss coefficients of pressure for different pipe members (TUBE, VALVE, BOW, TEE, PUMP, KNEE, ±CONTR, ±DIFSR) need only be given in diagrams. These diagrams are used in digitalized form. In the back‐substitution phase with known flow rates in all members, the pressure at the joints is computed.

The main advantages of the analysis as outlined are that no initial values for the member flow rates need be known, the iteration procedure converges rapidly, and within each iteration step only small systems of linear equations need to be solved. Due to the fact that the loss coefficients of pressure need only be given in diagrams, arbitrary nonlinear networks can be analysed by the unchanged program system. A flow rate assumption may be specified in the input for a member of a mesh.

The pressures at the joints are defined as unknowns (displacement method) in References 7, 8 and 10. The flow rates in the members are defined as unknowns (force method) in Reference 9. The nonlinear system of equations is always solved by the Newton‐Raphson procedure. In Reference 10 a strategy is presented to take into account different pipe members, but in a different way from that outlined in this paper. The members BOW, TEE, KNEE, CONTR, DIFSR are not examined in Reference 10.
Predictor‐corrector methods for parabolic partial differential equations
Authors: Jacques I. B.
Published: August 2005

In this paper we extend predictor‐corrector methods, commonly used for the numerical solution of ordinary differential equations (o.d.e.s), to parabolic partial differential equations (p.d.e.s), typically of the form ut = auxx + ƒ(u, ux, x, t).

We describe linear multistep methods for p.d.e.s, the nonlinear algebraic equations arising from implicit formulae being solved using a corrector analogous to those used for o.d.e.s. A sufficient condition for convergence of the iteration is then derived and is found, in most cases, to be far less restrictive than that obtained from the usual method of lines approach. Numerical results are presented to investigate the necessity of this condition. They also indicate that we can accelerate convergence by reducing the time increment. This allows us to achieve convergence within a prescribed number of iterations and so to construct PCm methods corresponding to P(EC)m methods for o.d.e.s. Numerical results are also given to test the absolute stability of the Crank‐Nicolson corrector for various predictors P, and iterations, m.
Spurious reflection of elastic waves due to gradually changing finite element size
Authors: Celep Zekai , Bažant Zdeněk P.
Published: August 2005

When a homogeneous elastic continuum is modelled by a non‐uniform finite element grid, the differences in the size of adjacent finite elements can cause a spurious wave reflection which does not exist in the continuum. Extending previous studies in which the spurious wave reflection was investigated for the case of two uniform grids with a sudden element size change from one grid to the other, the present study deals with the case when the two uniform grids of different element sizes are separated by a transition zone through which the element size varies gradually, either by a geometric progression or by an arithmetic progression. The solution is carried out in complex variables and numerical plots of the results are given. When the ratio of wavelength to the largest element size is as low as 4 : 1, the spurious wave reflection is very significant, while for the ratios over 10 : 1 it is insignificant. Inserting a transition zone with gradually varying element size between the two uniform parts of the grid somewhat mitigates the phenomenon of spurious reflection, but this is significant only when the ratio of element sizes in the uniform grids is small (less than about 1·5 : 1). Varying the element size throughout the transition zone in an arithmetic progression seems slightly better than in a geometric progression. The spurious wave reflection is less pronounced for higher‐order elements, in particular for linear strain elements as compared to constant strain elements. The spurious reflection is also less severe for consistent mass than for lumped mass.
Numerical methods for partial differential equations based on power series
Authors: Small Robert D.
Published: August 2005

Stable difference schemes that approximate partial differential equations to any desired order are generated by means of power series. The method is demonstrated on the one‐dimensional heat equation and the stability is studied in detail. Stability analyses are then presented for parabolic linear equations with constant coefficients. A nonlinear example is presented which indicates the efficiency of the higher ordered schemes.
Elements for the numerical analysis of wave motion in layered strata
Authors: Tassoulas John L. , Kausel Eduardo
Published: August 2005

A technique is developed for the numerical analysis of wave motion in layered strata. Semidiscrete particular solutions satisfying inhomogeneous boundary conditions are calculated by the finite element method. These solutions are combined with semidiscrete modes of an appropriate eigenvalue problem. The boundary conditions corresponding to rigid and rough footings on a layered stratum are treated in detail. Applications are considered which demonstrate the validity of the technique.
Numerical solution of nonlinear quasi‐stationary magnetic fields
Authors: Melkes Frantis̃ek , Zlámal Milos̋
Published: August 2005

The problem of quasi‐stationary nonlinear magnetic fields is formulated in a variational form. A Galerkin finite element solution is constructed. Discretization in time is carried out by means of one‐ and two‐step linear A‐stable schemes. The resulting system of nonlinear equations is solved by Newton's method. A numerical example is given.
Finite element analysis of sinusoidal small‐amplitude vibrations in deformed viscoelastic solids. Part I: Theoretical development
Authors: Morman K. N. , Nagtegaal J. C.
Published: August 2005

A general method is presented for the isothermal mechanical analysis of incompressible material solids in which a small‐amplitude time harmonic oscillation is superposed on a static finite deformation field. Material behaviour is assumed to be of the ‘fading memory’ type describable by the finite linear viscoelasticity theory of Coleman and Noll.4 Existing finite element codes, that treat finite elasticity material behaviour, can be extended with minor modifications to treat the subject problem. A specialized form of the finite linear viscoelasticity constitutive equations proposed by Morman8 has been implemented in the MARC nonlinear finite element program for predicting the response of statically deformed elastomeric components to small amplitude vibrations. Numerical results obtained with MARC for the static force‐deflection behaviour and dynamic (complex) stiffness for a viscoelastic cylinder subject to combined axial and twisting pre‐loads, are in excellent agreement with corresponding analytical results.
Design of a recursive, shape controlling mesh generator
Authors: Bykat A.
Published: August 2005

A recursive, shape controlling triangulation method is described. The method is designed to produce a labelling which implies reduced fill in the solution of (finite element) equations assembled from such a triangulation and allows simple implementation of a nested disection algorithm for irregular domains. This approach saves a substantial amount of time usually spent on discovering a suitable relabelling of the triangulation. In addition, the matrix of the resulting system is then endowed with a recursive doubly bordered block diagonal form. This allows us to develop a recursive parallel bisection method for the solution of the system of equations.
A stream function‐vorticity finite element formulation for Navier‐Stokes equations in multi‐connected domain
Authors: Mizukami Akira
Published: August 2005

In this paper we present a stream function‐vorticity finite element formulation for Navier‐Stokes equations which does not require an iterative procedure for satisfying the boundary conditions, and show that this formulation is of great advantage in solving the flow problems in a multi‐connected domain.
Accurate Petrov‐Galerkin methods for transient convective diffusion problems
Authors: Dick Erik
Published: August 2005

The Petrov‐Galerkin method is applied to transient convective diffusion problems with convection dominance. Weighting functions are used which differ from the shape functions by both symmetric and non‐symmetric modifications. It is proved that the symmetric modifications largely influence the accuracy. By a one‐dimensional and a two‐dimensional example it is shown how to choose this modification to maximize accuracy.
Fast modal response method for structures
Authors: Leung Y. T.
Published: August 2005

Regardless of their simplicity, all structures have an infinite number of degrees‐of‐freedom (d.o.f.) when subjected to dynamic loading. The usual finite element method reduces the infinite d.o.f. system to a model with a limited d.o.f. while capturing the significant physical behaviour. The modal analysis reduces the number of d.o.f. further to a limited number of modal co‐ordinates. However, accurate results comparable to the original finite element model may not be possible unless higher modes are included. The present paper is to recommend a response analysis which makes use of both the natural modes and the mass and stiffness matrices of the system to improve the convergence with respect to the number of modes. While the effects of lower modes are analysed similar to the modal analysis, the effects of higher modes are included in the system matrices and the information for higher modes is not needed.
A two‐dimensional finite element idealization for thermo‐elastic deflection in beams
Authors: Abouaf Marc , Chenot Jean‐Loup , Marcelin Jean‐Luc
Published: August 2005

This paper addresses the problem of thermo‐elastic deflection in beams using a two‐dimensional finite elemement idealization of the cross‐section. Bending effects are introduced in a gerneralized plane strain formulation assuming that constant curvature dominates the flexural deformation. The method leads to a plane strain calculation which is very interesting from the computational point of view; thermal deendence of the physical properties can be taken into account, which would be inconceivable in a three‐dimensional problem.
High‐accuracy product integration methods for the Abel equation, and their application to a problem in scattering theory
Authors: Cameron R. F. , McKee S.
Published: August 2005

In this paper we present orsder three, four and five backward product integration methods for the generalized Abel equation

equation image

We illustrate their use by silving the small angle deflection problem, and show computationally that these methods are convergent for all ϵ ϵ(0,1). The coefficients of order three and four methods are given in Applenidx I.
Optimum synthesis of linkages with autocorrelation concept (modified least squares technique)
Authors: Lapalikar S. R. , Rao A. C.
Published: August 2005

Correlation is a measure of similarity between two functions. When two functions are exactly similar and equal, the correlation between them is maximum. The correlation concept can be used for system design, e.g. the four‐bar mechanism, the slider crank mechanism, etc. The motion generated by four‐bar linkage can be expressed as a function of link dimensions, input and output angles. The product of generated function and desired function gives correlation. The best possible design would be when the deviation of actual correlation from its maximum value (i.e. 100 per cent correlation) is least. This concept, in fact, leads to a modified least squares approach and based on this an objective function is formulated and minimized to give link dimensions. The results obtained are better than the least squares technique, and this is illustrated through different numerical examples.
Numerically efficient procedures for dynamic contact problems
Authors: Madsen Nels
Published: August 2005

The solution of dynamic contact (elastic impact) problems is complicated by the changing nature of the contact area. If a finite element approach is used, the system matrices vary with the contact area. If the problem is properly formulated, such changes are rank one. Rank one changes produce easily determinable changes in the LDLT decompositions of the system matrices. This renders a class of dynamic contact problems soluble with the same accuracy and computational effort as is associated with the solution of any dynamic problem using finite element procedures. Consideration of an example problem involving the impact of spring–mass systems confirms this claim.
A complex variable boundary element method: Development
Authors: Hromadka T. V. , Guymon G. L.
Published: August 2005

A generalized boundary integral equation method for the solution of the Laplace equation is developed based on the Cauchy integral theorem for analytical complex variable functions. Although the approach is complicated by the utilization of complex variable theory, the resulting model involves direct integration along straight‐line boundary segments (elements) rather than using quadrature formulae that are required in current real variable boundary element formulations. Previously published boundary integral equation methods based on the Cauchy integral theorem are shown to be a subset of the generalized model theory developed in this paper.
The numerical solution of two generalizations of the neumann moving interface problem into two dimensions
Authors: Schulze H. J. , Howarth J. A. , Poots G.
Published: August 2005

We present numerical solutions for the following Stefan problems. The half‐space z > 0 is initially filled with liquid at its fusion temperature. The boundary z = 0, taken to be the x axis, is maintained at a constant temperature, less than the fusion temperature, for x < 0. For x > 0, the first problem considers the case of an insulated boundary, and the second problem considers the case of the boundary maintained at the fusion temperature. This gives rise to a solid‐liquid interface curved in the (x, z) plane.
Optimal control finite element approximation for penalty variational formulation of three‐dimensional Navier–Stokes problem
Authors: Kai‐Tai Li , Ai‐Xiang Huang , Yi‐Chen Ma , Du Li , Zhi‐Xing Liu
Published: August 2005

An optimal control finite element approximation of the penalization type for the solution of incompressible viscous Navier–Stokes equations is presented. This paper has proved the convergence of its minimizing sequence and the existence of the solution for the optimal control problem corresponding to the penalty variational problem of the Navier–Stokes equations. Based on this method and the conjugate gradient algorithm, the program for solving the three‐dimensional Navier–Stokes problem has been developed, and some numerical results have also been provided.
Compact analytic expressions of two‐dimensional finite difference forms
Authors: Reali M. , Rangogni R. , Pennati V.
Published: August 2005

In this paper a straightforward derivation of one‐ and two‐dimensional finite difference forms for general cartesian networks is given. General analytic compact expressions up to third order for first derivatives are specifically derived. General cartesian networks with locally telescoping subnetworks are also introduced and the basic problem of approximating derivative boundary conditions is clarified. The applicability of these general finite difference forms is shown by solving numerically the Laplace problem with mixed Dirichlet–Neumann boundary conditions for an elliptic domain.
A comment on the effectiveness of uniform finite element refinements
Authors: Yunus Shah M. , Shephard Mark S.
Published: August 2005

Numerical results using uniform h‐refinement approaches to the solution of a pair of elasticity problems with singularities are presented and compared to recently published results based on the uniform p‐refinement approaches.
A shear‐flexible triangular finite element model for laminated composite plates
Authors: Lakshminarayana H. V. , Murthy S. Sridhara
Published: August 2005

Formulation and numerical evaluation of a shear‐flexible triangular laminated composite plate finite element is presented in this paper. The element has three nodes at its vertices, and displacements and rotations along with their first derivatives have been chosen as nodal degrees‐of‐freedom. Computation of element matrices is highly simplified by employing a shape function subroutine, and an optimal numerical integration scheme has been used to improve the performance. The element has satisfactory rate of convergence and acceptable accuracy with mesh refinement for thick as well as thin plates of both homogeneous isotropic and laminated anisotropic materials. The numerical studies also suggest that reliable prediction of the behaviour of laminated composite plates necessitates the use of higher order shear‐flexible finite element models, and the proposed finite element appears to have some advantages over available elements.
Algorithms for refining triangular grids suitable for adaptive and multigrid techniques
Authors: Rivara M. Cecilia
Published: August 2005

Two general algorithms for refining triangular computational meshes based on the bisection of triangles by the longest side are presented and discussed. The algorithms can be applied globally or locally for selective refinement of any conforming triangulation and always generate a new conforming triangulation after a finite number of interactions even when locally used. The algorithms also ensure that all angles in subsequent refined triangulations are greater than or equal to half the smallest angle in the original triangulation; the shape regularity of all triangles is maintained and the transition between small and large triangles is smooth in a natural way. Proofs of the above properties are presented. The second algorithm is a simpler, improved version of the first which retains most of the properties of the latter. The algorithms can be used either for constructing irregular computational meshes or for locally refining any given triangulation. In this sense they can be adequately combined with adaptive and/or multigrid techniques for solving finite element systems. Examples of the application of the algorithms are given and two possible generalizations are pointed out.
Hellinger–Reissner principles for plate and shell finite elements
Authors: Morley L. S. D.
Published: August 2005

The Hellinger–Reissner generalized variational principle is used to deduce further principles directed especially towards formulation of high integrity finite elements for plates and curved shells in polynomial parametric representation. Assumed stresses in these principles are derived from displacements and supplemented with stresses derived from stress functions.
A C0 triangular plate element with one‐point quadrature
Authors: Belytschko T. , Stolarski H. , Carpenter N.
Published: August 2005

A triangular plate element of the Mindlin type, which uses linear fields for rotations and transverse deflections, is developed. Success is achieved, in contrast to previous failures with this element, by decomposing the displacements into well‐defined bending and shear modes which are associated with bending and shear energies, respectively. The element only requires a single quadrature point, which is very attractive for nonlinear analysis. Results are presented for a variety of plate problems; the accuracy is similar to the 4‐node bilinear Mindlin plate and its rate of convergence is approximately of order h2.
Surface integral finite element hybrid (SIFEH) method for fracture mechanics
Authors: Annigeri Balkrishna S. , Cleary Michael P.
Published: August 2005

An effective surface integral and finite element hybrid (SIFEH) method has been developed to model fracture problems in finite plane domains. This hybridization by (incrementally) linear superposition combines the best features of both component methods. Finite elements are used to model the finite domain (and eventually nonlinearity), while continuous distributions of dislocations (resulting in surface integral equations) are used to model the fracture (i.e. displacement discontinuity). This method has been implemented in a computer program and results of representative problems are presented: these compare very well with known solutions and they demonstrate the computational advantages of SIFEH over other numerical methods (including the individual components).
Optimization for vibration isolation
Authors: Nack Wayne V.
Published: August 2005

An almost linear optimization problem of importance in vibration isolation has been identified and algorithms were developed to minimize the forced vibrational response of structural systems. The constraints can be either displacements or accelerations. These algorithms have been studied for transient response, frequency response and stationary random using the direct dynamic solution. Multiple response points and loading conditions may be used.
Path independent integrals for computing stress intensity factors at sharp notches in elastic plates
Authors: Sinclair G. B. , Okajima M. , Griffin J. H.
Published: August 2005

A set of path independent integrals is constructed for the calculation of the generalized stress intensity factors occurring in elastic plates having sharp re‐entrant corners or notches with stress‐free faces and subjected to Mode I, II or III type loading. The Mode I integral is then demonstrated to enjoy a reasonable degree of numerical path independence in a finite element analysis of a test problem having an exact solution. Finally, this integral is used on the same problem in conjunction with a regularizing, finite element, procedure or superposition method. The results indicate that sufficiently accurate estimates of these stress intensity factors for engineering purposes can be achieved with little computational effort.
An efficient algorithm to generate maximum entropy distributions
Authors: Basu P. C. , Templeman A. B.
Published: August 2005

The paper describes an algorithm for generating maximum entropy distributions for probabilistic data. The central moments of the data form constraint equations developed from Jaynes' formalism, which are solved by mathematical programming. Criteria are presented for selecting starting points and scaling parameters, upon which the accuracy and efficiency of the algorithm depends. Results are given of tests in which maximum entropy distributions are generated from the moment information of numerous analytical distributions. They show that four moments are generally required to produce good agreement and that maximum entropy distributions can represent most data populations very well. The notation used in the paper is defined in Appendix I.
A model of binary alloy solidification
Authors: Wilson D. G. , Solomon A. D. , Alexiades V.
Published: August 2005

A linear model for the solidification of a dilute binary alloy is presented. In this model the solidus and liquidus curves are linear. As a consequence internal energy depends linearly upon temperature and concentration. The formulation is a generalization of the well‐known enthalpy method to treat a phase change problem involving coupled heat and mass transfer. Both analytic and numerical formulations are given. Results from the latter are presented and compared with an explicit solution of Rubinstein for a Stefan‐like problem posed in a semi‐infinite slab. Some remarks on the behaviour of the explicit solution are given.
An interpolation method for an irregular net of nodes
Authors: Liszka Tadeusz
Published: August 2005

A local interpolation method for an irregular mesh of nodal points is proposed. The method is based on a Taylor expansion of the unknown function combined with the minimization of errors. Some numerical tests as well as a computer program are presented. Applicability and stability of the method are shown. By the appropirate definition of weighting coefficients, this method may be viewed as an interpolation or approximation in the sense of minimum deviation from given values. Applications in finite element and finite difference methods are shown.
Numerical evaluation of the quarter‐point crack tip element
Authors: Saouma Victor E. , Schwemmer David
Published: August 2005

This paper attempts to answer two commonly raised questions during the preparation of a finite element mesh, for the linear elastic fracture analysis of cracked structure: how to set up the finite element mesh around the crack tip, and what level of accuracy is to be expected from such a modelling.

Two test problems, with known analytical expressions for their stress intensity factors, are analysed by the finite element method using the isoparametric quadratic singular element. The modified parameters were the order of integration, aspect ratio, number of elements surrounding the crack tip, use of transition elements, the singular element length over the total crack length, the symmetry of the mesh around the crack tip. Based on these analyses, a data base is created and various plots produced. The results are interpreted, the accuracy evaluated and recommendations drawn.

Contrary to previous reports, it is found that the computed stress intensity factor (SIF) remains within engineering accuracy (10 per cent) throughout a large range of l/a (singular element length over crack length) for problems with a uniform non‐singular stress distribution ahead of the crack tip (i.e. double edge notch), and l/a should be less than 0·1 for problems with a non‐singular stress gradient (i.e three‐point bend). Also, it is found that the best results are achieved by using at least four singular elements around the crack tip, with their internal angles around 45 degrees, and a reduced (2 × 2) numerical integration.
Structural design sensitivity analysis with general boundary conditions: Static problem
Authors: Hsieh C. C. , Arora J. S.
Published: August 2005

This paper presents and analyses methods for design sensitivity analysis of static response for structural systems when general boundary conditions are used during the analysis phase. The design sensitivity analysis methods where the boundary conditions are used to reduce the equilibrium equation have been presented earlier. In such procedures, the reduced stiffness matrix is non‐singular. This fact is used to show differentiability of the displacement vector. When reduction procedures are not used, the global structural stiffness matrix may be singular. In such a case, differentiability of the displacement vector may be questionable. However, it is shown that the displacement vector is still differentiable. Several forms of the structural design sensitivity equations are presented. Although these equations are derived with Lagrange multiplier treatment of general boundary conditions, it is shown that they are applicable with any treatment of general boundary conditions. For example, they can also be used with co‐ordinate reduction schemes. Therefore, design sensitivity analysis can be carried out with any existing finite element analysis computer code. Moreover, the procedures presented can be used to calculate sensitivity of a force of constraint can that is desired. The formulations are kept quite general such that design dependent boundary conditions can be routinely treated. An example is used to illustrate the procedures.
Rational approach for assumed stress finite elements
Authors: Pian T. H. H. , Sumihara K.
Published: August 2005

A new method for the formulation of hybrid elements by the Hellinger‐Reissner principle is established by expanding the essential terms of the assumed stresses as complete polynomials in the natural coordinates of the element. The equilibrium conditions are imposed in a variational sense through the internal displacements which are also expanded in the natural co‐ordinates. The resulting element possesses all the ideal qualities, i.e. it is invariant, it is less sensitive to geometric distortion, it contains a minimum number of stress parameters and it provides accurate stress calculations. For the formulation of a 4‐node plane stress element, a small perturbation method is used to determine the equilibrium constraint equations. The element has been proved to be always rank sufficient.
The numerical treatment of laplace transforms: I. The Koizumi inversion method
Authors: Squire William
Published: August 2005

An inversion formula based on expansion in a Fourier sine series, proposed by Koizumi in 1935, is examined. It is found to give results of moderate accuracy with relatively little work.
On the computational significance of the strain space formulation of plasticity theory
Authors: Moss W. C.
Published: August 2005

The strain space and traditional stress space formulations of plasticity theory are compared. We show how plasticity theory has been implemented numerically to solve static and dynamic problems, and conclude that the numerical loading criteria that have been used for two decades are identical to the strain space loading criteria proposed by Naghdi and co‐workers.
An effective finite element approach for calculation of potential distribution in a multi‐dielectric cable
Authors: Sanavullah M. Y. , Dharmalingam K.
Published: August 2005

The radial symmetry of the cable results in circular finite elements to evaluate the potential distribution in a multi‐dielectric cable. Owing to this, a two dimensional analysis used in conjunction with triangular finite elements is reduced to one‐dimensional analysis. This method therefore suggests an alternative proposal to calculate the potential distribution in a multi‐dielectric cable with better accuracy.
An automated design scheme: Weight minimization of a single hulled ship compartment
Authors: Chowdhury M. , Caldwell J. B.
Published: August 2005

This paper describes the stages of development of an optimum structural design scheme. Any attempt to generate the optimum scantlings of a structure of even moderate complexity usually poses nonlinear programming problems for which there exist no universally accepted solution method. The structure considered here is a complete compartment of a single‐hulled ship acted upon by multiple load cases. The elastic responses under four different load types are obtained by a quasi finite element analysis routine developed specifically for this purpose. Sequential linear programming (SLP) with appropriate artifices to achieve rapid convergence has been successfully employed as an optimization routine to minimize the object function, which in this case was the total weight of the major structural members. The test problem has 490 basic degrees‐of‐freedom, 21 free design variables and more than 100 mixed constraints, the majority of which are nonlinear implicit stress functions. The method is then applied to a number of cases representative of small warship designs. Suggestions for future extensions and general conclusions are included at the end of the paper.
Use of minimum surface theory to aid finite element contour plotting
Authors: Steven G. P.
Published: August 2005

Contour plots of finite element solutions, especially for derivatives, often are discarded due to poor plotting algorithms. This note shows how a smoothing scheme based on a minimum surface theory can significantly improve the quality of the graphical results. The scheme proposed is compared with many other smoothing schemes, including least squares, on a simple problem.
Optimal shape remodelling of linearly elastic plates using finite element methods
Authors: Na Moon‐Soo , Kikuchi Noboru , Taylor John E.
Published: August 2005

The optimality conditions for the optimal shape remodelling of linearly elastic plates are obtained by introducing the total variation of a function defined on a variable domain, although the variation of a function has been taken on a fixed domain in most literature on calculus of variations. Using these optimality conditions, a solution scheme involving an iterative algorithm is proposed, together with several numerical examples.