bluuebunny/crossref_metadata_2025 huggingface dataset:
Fast solution to finite element flow equations by newton iteration and modified conjugate gradient method
Authors: Gambolati Giuseppe
Published: August 2005

A modification (Irons;6 Concus et al. 1) to the conjugate gradient (CG) method by Hestenes and Stiefel5 has recently renewed the interest in this elegant technique which appears to be extraordinarily promising for large sparse systems Ax = b, where A is a symmetric positive definite matrix. A good approximation K−1 for the inverse of A is needed in the modified algorithm. For finite difference sets of equations, Meijerink and van der Vorst9 and Kershaw7 have experienced a very fast convergence with a matrix K−1 determined by the incomplete Cholesky decomposition of A.

A further acceleration of the iteration may be achieved by preliminarily processing the initial guessed solution by the Newton iterative scheme before using the modified conjugate gradient (MCG) method. The initial Newton iterations (NI) have the useful property of significantly reducing the components of the residual r0 along the eigenvectors of AK−1 associated with the eigenvalues lying in the vicinity of 1. The latter are expected to include the vast majority of the eigenvalues of AK−1. As a result the MCG method is left with a smaller number of r0 components to set to zero in a reduced dimensional eigenvector space and hence the solution is arrived at in fewer iterations. Depending on the desired final accuracy, up to 50 per cent of the MCG iterations may be equivalently replaced by an equal number of NI which are computationally faster.

This approach has been applied to the solution of finite element sets of linear equations arising from the arbitrarily irregular triangle discretization of groundwater flow domains in both steady and unsteady conditions. For diagonally dominant matrices the results emphasize the excellent performance of the MCG method which proved much faster than the first‐degree Chebyshev iteration (CHI) and required a number of iterations an order of magnitude smaller than the successive over‐relaxation technique (SOR) with optimum over‐relaxation factor.
A one‐step method for direct integration of structural dynamic equations
Authors: Brusa L. , Nigro L.
Published: August 2005

The choice on an efficient direct integration procedure for linear structural dynamic equations of motion is discussed. It is suggested that as accuracy parameter the truncation error on the exponential terms contained in the modal contributions of the exact solution be assumed. This error does not always coincide with the local truncation error. These considerations were used to design an unconditionally stable one‐step method whose accuracy is 0(h4). Numerical comparisons with some well‐known integration schemes showed the efficiency of the proposed method.
A variational principle for ideal flow over a spillway
Authors: O'carroll M. J.
Published: August 2005

A variational principle is presented which represents steady ideal flow with a free surface under gravity in terms of the stream function with Dirichlet boundary conditions. A different principle of Ikegawa and Washizu is shown to require restricted variations not consistent with the mass flow specification. The new principle is a special case of the of O'Carroll and Harrison in terms of enthalpy discontinuity between two streams. It is also reciprocal to a velocity potential functional. Finite element procedures and the determination of critical flow are discussed.
The interpolation function of a general serendipity rectangular element
Authors: Ball A. A.
Published: August 2005

The shape functions of serendipity rectangular elements had originally to be found by inspection. More recently an algorithm has been developed to formalize the construction of these shape functions. Here a simple expression is found for the interpolation function of the most general serendipity element. The shape functions can be derived immediately.
An accurate and economical eight‐node stress membrane element
Authors: Robinson J. , Ward P.
Published: August 2005

An accurate and economical eight‐node quadrilateral membrane element based on stress assumptions is presented. It is shown that this element offers distinct advantages over other available membranes.
Post‐buckling behaviour of structures using a finite element–nonlinear eigenvalue technique
Authors: Wellford L. Carter , Dib Ghassan M.
Published: August 2005

A series of finite element algorithms for the calculation of the post‐buckling behaviour of structures are proposed. The algorithms represent a finite element implementation of a variational principle for nonlinear eigenvalue problems. Incremental amplitude solution schemes are defined. Numerical results are presented for the post‐buckling behaviour of specific structures with linear and nonlinear pre‐buckling states.
Symmetricability of pressure stiffness matrices for shells with loaded free edges
Authors: Mang H. A.
Published: August 2005

For the case of free edges which are loaded, follower forces remaining normal to the middle surface of a shell throughout the deformation history do not have a load potential. In finite element analysis, this results in an unsymmetric pressure stiffness matrix. Depending on the structure of the available computer program, implementation of an equation solver permitting solution of unsymmetric simultaneous systems of algebraic equations may be a tedious task. This explains the significance of the topic of symmetricability of pressure stiffness matrices, turning out to be of special importance in the case of static buckling under the assumption of a linear prebuckling path.

At first, incremental equations for tracing the nonlinear load–displacement path are derived. Thereafter, the buckling condition is deduced. Then, it is demonstrated that symmetrization of the pressure stiffness matrix is admissible if the so‐obtained ‘buckling pressure’ differs ‘very little’ from the ‘buckling pressure’ resulting from an alternative symmetric ‘buckling matrix’, as is shown to be the case for a simply supported cylindrical shell with a free upper edge, subjected to hydrostatic external pressure. The alternative symmetric ‘buckling matrix’ is a consequence of deleting the virtual work term, causing the unsymmetry of the pressure stiffness matrix, in the expression for the external virtual work. A mechanical interpretation of this virtual work term is given. It is shown to be equal to the difference of virtual work of the original pressure load and of a ‘substitute pressure‐field’, of the form of a Fourier series of the former. This explains why, normally, the buckling coefficient resulting from the ‘substitute pressure‐field’ represents a good approximation to the buckling coefficient stemming from the original pressure load.
An analysis procedure for plane strain contained plastic deformation problems
Authors: Corradi L. , Gioda G.
Published: August 2005

An analysis procedure for the plane strain contained plastic deformation problem is proposed. It is shown that, when piecewise‐linear elastic–plastic laws are adopted, the plane strain problem can be formulated in terms of exactly the same variables appearing in a plane stress problem, even if the transverse stress and plastic strain components do not vanish in general. The two problems are governed by the same equilibrium and compatibility equations and the only difference is in the elastic–plastic material constitutive laws, in complete analogy with the linear elastic case.

The problem is discretized by using a compatible finite element model on which basic elastic‐plastic constitutive laws for an element in plane strain conditions can be obtained. When the assemblage is performed, the resulting set of governing relations presents a mathematical structure which enable, for solution, techniques efficiently used for truss and frame problems. Some solutions for axisymmetric tubes subjected to pressure and temperature gradients are illustrated.
On boundary conditions for the numerical solution of hyperbolic differential equations
Authors: Sloan D. M.
Published: August 2005

Various boundary methods are considered for the numerical solution of the linear wave equation. The solution in the interior region is obtained using the Lax‐Wendroff method. The theory of Gustafsson, Kreiss and Sundström is used to establish stability and the theory of Sköllermo is used to compare the accuracies of the various methods. The accuracy predictions are compared with computed results.
Modal methods in finite element fluid–structure eigenvalue problems
Authors: Daniel W. J. T.
Published: August 2005

The technique of fixed‐interface modal synthesis used to solve large structural eigenvalue problems is extended to fluid‐structure eigenvalue problems with a view to efficient solution of problems involving localized modifications. Three cases are considered: namely, those fluid or fluid‐structure vibration problems directly analogous to the structural case; the modification of the fluid model of a structure enclosing a compressible fluid; and the modification of fluid and structural models of a structure enclosing an incompressible fluid. Examples of the three cases are given, as are estimates of the computation that can be saved when using the methods to avoid resolving modified eigenvalue problems. It is also shown that the hard‐walled modes of a fluid enclosure can be used to modify the incompressible‐fluid modes of the fluid‐structure system to account for a degree of compressibility in the fluid. An example and computation estimates are given.
Performance of reduction methods for fluid–structure and acoustic eigenvalue problems
Authors: Daniel W. J. T.
Published: August 2005

A methodology is presented for measuring the effectiveness of a finite element approach when compared with a reference finite element approach with respect to accuracy and computation. The effectiveness measure is applied to the Hughes reduction method for structural eigenvalue problems and three proposed analogous methods for fluid or fluid–structure problems. Comparisons of the three methods with consistent interpolation solutions indicate that improved effectiveness can be obtained.
Optimum synthesis of a slider‐crank mechanism using geometric programming
Authors: Rao A. C.
Published: August 2005

Geometric programming is a powerful technique and its application to mechanism synthesis has not been fully explored. In this paper, slider‐crank mechanism is chosen since the objective function consists of a large number of terms while there are only two design variables. Thus the degree of difficulty is more and does not lend itself to easy application of the geometric programming. Here, modification of the objective function is suggested so that the degree of difficulty reduces to zero even when the working space constraint is imposed. This results in a near optimum solution. Application of signomial geometric programming is illustrated through a numerical example.
Two‐dimensional shape optimal design by the finite element method
Authors: Queau J. P. , Trompette Ph.
Published: August 2005

This paper is concerned with the optimal shape design of plane or axisymmetric structures in order to minimize the stress concentration factor along the boundary. This boundary is described by straight lines and circles. The structure is analysed using the finite element method, and the optimization procedure is based on an extended interior penalty function. Three example problems are reported.
More on infinite elements
Authors: Bettess Peter
Published: August 2005

The method of infinite elements is briefly reviewed. New more logical and general formulations of infinite elements are presented. The simplicity of the programming is emphasized. Results are given for elasticity and potential problems. Future extensions are discussed.
Dispersion analysis in homogeneous lakes
Authors: Salmon James R. , Liggett James A. , Gallagher Richard H.
Published: August 2005

This paper is devoted to a comparison of the relative accuracy of two‐dimensional vs. three‐dimensional finite element representations of contaminant dispersion in shallow lakes. Formulations of both types are developed, followed by numerical calculations of a hypothetical lake. The results indicate that for typical lakes a two‐dimensional dispersion analysis can be employed in the absence of a significant advective contribution. With significant advection the two‐dimensional approach is not sufficiently accurate. A two‐dimensional dispersion analysis requires approximately the same computational resources as a three‐dimensional circulation analysis.
Numerical solutions for unsteady flow in unconfined aquifers
Authors: Guvanasen V. , Volker R. E.
Published: August 2005

Two numerical methods for solving the problem of unsteady flow in unconfined aquifers are studied. They are an explicit finite difference method (FDM), and the finite element method (FEM). The FEM is further subdivided into three schemes: vertical displacement approach, explicit scheme (FEM1), normal velocity approach, explicit scheme (FEM2), and vertical displacement approach, implicit scheme (FEM3). Results from the above methods are compared with experimental results from a sand box model. Various factors affecting the accuracy and numerical stability are investigated. Results indicate that, for a similar degree of accuracy, the FEM requires less computational effort than the explicit FDM. Amongst the three FEM schemes, FEM3 appears to be most attractive as it is the most stable and economical of the three schemes compared.
Smooth interpolation of large sets of scattered data
Authors: Franke Richard , Nielson Greg
Published: August 2005

Methods for solving the following data‐fitting problems are discussed: given the data (xi, yi, fi),i = 1,…,N construct a smooth bivariate function S with the property that S(xi, yi) = fii = 1,…,N. Because the desire to fit this type of data is encountered frequently in many areas of scientific applications, an investigation of the available methods for solving this problem was undertaken. Several aspects, such as computational efficiency, fitting characteristics and ease of implementation, were analysed and compared. Within the context of a general‐purpose method for large sets of data, two of these methods emerged as being generally superior to the others. It is the purpose of this paper to describe these two methods and present examples illustrating their use and application.
Finite element analysis of earthquake‐induced sloshing in axisymmetric tanks
Authors: Aslam Mohammad
Published: August 2005

A finite element anaylysis to predict the sloshing dislacements and hydrodynamic pressures in liquid‐filled tanks subjected to earthquake ground motions is presented. Finite element equations were derived using the Galerkin formulation, and the predicted results were checked against the test data, showing a good agreement between the test and finite element results.
A ψ–ω finite element formulations for the Navier‐Stokes equations
Authors: Dhatt Gouri , Fomo Bonaventure Kamga , Bourque Claude
Published: August 2005

A new method of solving the two‐dimensional Navier–Stokes equations by using the finite element method is proposed. The flow is represented by the stream function–vorticity formulation and the no‐slip boundary conditions are explicitly introduced in the nonlinear equations. This formulation coupled with the Newton‐Raphson method enables the study of stationary flows for high Reynolds number, without any convergence problem. A number of flow problems are analysed in order to demonstrate the efficiency of the present formulation.
A program to integrate a function tabulated at unequal intervals
Authors: McNamee John Michael
Published: August 2005

A FORTRAN subroutine is presented to integrate a function given only by a set of tabular values at unequal intervals. It uses the trapezoidal rule and a generalization of Simpson's rule, obtaining a conservative error estimate by comparing the two results.
Numerical integration of structural dynamics equations including natural damping and periodic forcing terms
Authors: Wood W. L.
Published: August 2005

This paper shows that it is comparatively simple to analyse algorithms for the numerical integration of the Space discretized equations from structural dynamics when applied to \documentclass{article}\pagestyle{empty}\begin{document}$\ddot x + \mu \dot x\omega ^2 x = p{\rm e}^{ist} $\end{document}, instead of the usual \documentclass{article}\pagestyle{empty}\begin{document}$ \ddot x + \omega ^2 x = 0 $\end{document}, and suggests that this should be done in order to gain some insight into the effect with natural damping and/or a periodic forcing term. The method is illustrated on some three‐ and four‐time‐level schemes.
Finite element analysis of deformation of strain‐softening materials
Authors: Pietruszczak St. , Mróz Z.
Published: August 2005

A finite element analysis of strain‐softening materials is presented in which the shear band of prescribed thickness is assumed to exist within elements where maximal stress intensity is reached. The incremental stiffness matrix of the element is derived including shear band deformation. Examples presented in the Paper demonstrate that the load‐displacement curve and the displacement field are not sensitive to the mesh size used in the solution.
Two‐dimensional stress intensity factor computations using the boundary element method
Authors: Blandford George E. , Ingraffea Anthony R. , Liggett James A.
Published: August 2005

A multidomain boundary element formulation for the analysis of general two‐dimensional plane strain/stress crack problems is presented. The numerical results were accurate and efficient. The analyses were performed using traction singular quater‐point boundary elements on each side of the crack tip(s) with and without transition elements. Traction singular quarter‐point boundary elements contain the correct √r displacement and 1/√r traction variations at the crack tip. Transition elements are appended to the traction singular elements to model the √r displacement variation. The 1/√r traction singularity is not represented with these elements. Current research studies for the crack propagation analysis of quasi‐static and fatigue fracture problems are discussed.
Mixed mode fracture analysis of rectilinear anisotropic plates by high order finite elements
Authors: Heppler Glenn , Hansen Jorn S.
Published: August 2005

A singular finite element is developed for direct calculation of combined modes I and II stress intensity factors for planar rectilinear anisotropic structures subject to arbitrary loading. Twelve‐node conventional elements are used in conjunction with a linear elastic fracture mechanics enrichment of the same element which is formed into a four‐element macro‐element. Example problems show this formulation to be exceptionally accurate and results are presented for a variety of modern fibre‐reinforced composites in simple mode I extension and in mixed mode I and II situations. In addition, it is shown that the meshes for accurate results are relatively coarse and thus calculations are quite economical.
Mathematical simulation of singularities for isoparametric elements of arbitrary orders
Authors: Nayfeh Adnan H. , Nassar Elsayed Abdel‐Ati M.
Published: August 2005

For the general quadrilateral isoparametric elements with 4(n−1) nodes (i.e. n nodes along each side), it is shown that the inverse square root singularity of the strain field at the crack tip can be obtained by a general but simple rule. This amounts to collapsing the quadrilateral elements into triangular elements around the crack tip and placing the (n−2) mid‐side nodes at locations γ2/(n−1)2 (γ=1,2,…, n−2) times the length of side emanating from the crack tip. These locations are measured from the crack tip. The known results for n = 3 and n = 4 are thus obtained as special cases.
A finite element solution for the two‐dimensional elastic contact problems with friction
Authors: Sachdeva T. D. , Ramakrishnan C. V.
Published: August 2005

A numerical procedure developed for solving the two‐dimensional elastic contact problems with friction is presented. This is a generalization of a procedure developed by Francavilla and Zienkiewicz to include frictional effects under proportionate loading. The method uses the flexibility matrix obtained by inversion of condensed stiffness matrix formed by eliminating all the nodes except those where contact is likely to take place and those with external forces. Compatibility of displacements for both normal and tangential directions is applied to those nodes which do not slip. However, for the nodes which slip, compatibility of displacements is applied for normal direction only and slip condition is applied in the tangential direction. The technique has been applied to several problems and very good results have been obtained. The number of iterations needed are very small.
An algorithm for programming the element matrices of doubly curved quadrilateral shell finite elements
Authors: Nelson Robert L.
Published: August 2005

This paper describes an algorithm which reduces the apparently diverse and complicated matrix calculations required to evaluate the finite element matrices of doubly curved quadrilateral shell finite elements to a few basic manipulations which are then repeated a large number of times. The algorithm can be transcribed particularly quickly into computer code. It lends itself to a structured programming approach and the flexibility of the ensuing coding is high. The accuracy of the code can be ascertained with relative ease and fault‐finding is comparatively straightforward; hence, a high degree of confidence can be placed in its execution. The element chosen to illustrate the application of the algorithm is a particularly difficult example. It is expected that the algorithm described would be useful in a wide variety of finite element computer program applications.
Geometrically nonlinear formulation for the axisymmetric shell elements
Authors: Surana Karan S.
Published: August 2005

A geometrically nonlinear formulation using total Lagrangian approach is presented for the axisymmetric shell elements. The basic element is formulated using the co‐ordinates of the mid‐surface nodes and the mid‐surface nodal point normals. An important aspect of the formulation presented here is that the restriction on the magnitude of the nodal rotations is eliminated. This is accomplished by retaining true nonlinear nodal rotation terms in the definition of the displacement field and the consistent derivation of the element properties based on this displacement field. The element properties are derived and presented in detail. Numerical examples are also presented to demonstrate the element behaviour and the accuracy.
Constraint relationships in linear and nonlinear finite element analyses
Authors: Barlow John
Published: August 2005

Finite element analysis of engineering structures commonly requires the use of inter‐nodal displacement constraints. Current algorithms for this facility—elimination, Lagrange multipliers and penalty function methods—are briefly reviewed. An alternative approach is proposed which avoids some of the problems of these methods. This technique uses solution of the unconstrained stiffness equation with an extended number of right‐hand sides. The resulting solutions are then combined to satisfy the constraints. This method has application in the synthesis of sub‐structures with incompatible boundary variables. It is particularly efficient when the constraint affects many variables but the kinematics of the constraint can be expressed in terms of few independent variables, e.g. rigid body motion of part of the structure. In common with the Lagrange method, it produces data for use in a nonlinear analysis with constraints. The formulation of ‘rigid body’ constraints in geometric nonlinearity is detailed and demonstrated.
A family of solution algorithms for nonlinear structural analysis based on relaxation equations
Authors: Park K. C.
Published: August 2005

A family of hierarchical algorithms for nonlinear structural equations are presented. The algorithms are based on the Davidenko‐Branin type homotopy and shown to yield consistent hierarchical perturbation equations. The algorithms appear to be particularly suitable to problems involving bifurcation and limit point calculations. An important by‐product of the algorithms is that they provide a systematic and economical means for computing the stepsize at each iteration stage when a Newton‐like method is employed to solve the systems of equations. Some sample problems are provided to illustrate the characteristics of the algorithms.
Integration of exponential expressions in natural co‐ordinates over finite elements
Authors: Mohsen M. F. N.
Published: August 2005

Expressions involving products of exponential functions and polynomials are encountered in the finite element formulation of the diffusion‐convection equation. Integrals of such expressions over finite elements are necessary to derive the element matrices. The derivation of the formula for such integrations is presented.
A remark on the direct numerical determination of stress intensity factors at crack tips
Authors: Ioakimidis N. I.
Published: August 2005

A numerical method for the direct determination of stress intensity factors at crack tips from the numerical solution of the corresponding singular integral equations is proposed. This method is based on the Gauss‐Chebyshev method for the numerical solution of singular integral equations and is shown to be equivalent to the Lobatto‐Chebyshev method for the numerical solution of the same class of equations.
Mixed models and reduced/selective integration displacement models for nonlinear shell analysis
Authors: Noor Ahmed K. , Andersen C. M.
Published: August 2005

Simple mixed models are developed for the geometrically nonlinear analysis of shells. A total Lagrangian description of the shell deformation is used, and the analytical formulation is based on a form of the nonlinear shallow shell theory with the effects of transverse shear deformation and bending‐extensional coupling included. The fundamental unknowns consist of eight stress resultants and five generalized displacements of the shell, and the element characteristic arrays are obtained by using the Hellinger‐Reissner mixed variational principle. The polynomial interpolation (or shape) functions used in approximating the stress resultants are, in general, of different degree than those used for approximating the generalized displacements. The stress resultants are discontinuous at the element boundaries and are eliminated on the element level.

The equivalence and ‘near‐equivalence’ between the mixed models developed herein and displacement models based on reduced/selective integration of both transverse shear and extensional energy terms is discussed. The use of reduction methods in conjunction with the mixed models is outlined and the advantages of mixed models over displacement models are delineated. Analytic expressions are derived for the rigid‐body and spurious (or zero energy) models for the various mixed models and their equivalent displacement models. Also, the advantages of mixed models over equivalent displacement models are outlined. Numerical results are presented to demonstrate the high accuracy and effectiveness of the mixed models developed, and to compare their performance with other mixed models reported in the literature.
A family of hybrid plate elements
Authors: Tong Pin
Published: August 2005

Hybrid stress elements are known to provide accurate results for analyses of plate bending; in particular, for the prediction of moment distribution. The construction of hybrid stiffness matrix requires numerical inversion of a moment matrix, evaluation of some relatively complicated boundary integrals and several matrix transformations. Each of these operations can be time‐consuming, and the matrix inversion can result in a loss of numerical accuracy.

This paper devises methods to explicitly invert the moment matrices. We find that for triangular elements the inverse is independent of the element shape and is only inversely proportional to its size. We also use a novel set of displacement variables, which greatly simplifies the boundary integration. The displacement variables are chosen in a hierarchical form so that lower order elements can be determined by straightforward reduction of excess terms in a higher order element. Except for the nodal displacements at the vertices, the present approach involves only variables at the midpoints of the sides of an element.
On computational procedures for the force method
Authors: Kaneko I. , Lawo M. , Thierauf G.
Published: August 2005

It is known that the matrix force method has certain advantages over the displacement method for a class of structural problems. It is also known that the force method, when carried out by the conventional Gauss‐Jordan procedure, tends to fill in the problem data, making the method unattractive for large size, sparse problems. This poor fill‐in property, however, is not necessarily inherent to the method, and the sparsity may be maintained if one uses what we call the Turn‐Back LU Procedure. The purpose of this paper is two‐fold. First, it is shown that there exist some close relationships between the force method and the least squares problem, and that many existing algebraic procedures to perform the force method can be regarded as applications/extensions of certain well‐known matrix factorization schemes for the least squares problem. Secondly, it is demonstrated that these algebraic procedures for the force method can be unified form the matrix factorization viewpoint. Included in this unification is the Turn‐Back LU Procedure, which was originally proposed by Topçu in his thesis.8 It is explained why this procedure tends to produce sparse and banded ‘self‐stress’ and flexibility matrices with small band width. Some computational results are presented to demonstrate the superiority of the Turn‐Back LU Procedure over the other schemes considered in this paper.
An integral equation method for the diffraction of oblique waves by an infinite cylinder
Authors: Liu Philip L‐F. , Abbaspour Madjid
Published: August 2005

A hybrid integral equation method is formulated to study the diffraction of oblique waves by an infinite cylinder. The water depth and the geometry of the floating cylinder are assumed to be uniform in the y‐direction (one of the horizontal axes). Numerical discretization and integrations are performed in the vertical plane. Analytical solutions are used in far fields such that radiation boundary conditions are satisfied. Numerical results are obtained for the case of wave scattering by a floating rectangular cylinder in a constant water depth. The accuracy and efficiency of present method are compared with those obtained by other numerical techniques.
Errors in natural frequency calculations using eigenvalue economization
Authors: Thomas D. L.
Published: August 2005

The eigenvalue economization process is an efficient way of reducing the size of eigenvalue problems to manageable proportions, at the expense of introducing approximations. The use of automatic criteria for selecting the slave variables to be eliminated has considerably eased the use of this technique recently. However, it is necessary to ensure that unacceptable errors are not introduced by the condensation process. Until recently, there has been little available to guide the user in estimating errors except past experience. In this paper, the errors introduced in the condensation process are considered theoretically, using an algebraic approach. This enables an absolute error bound to be stated, together with an approximate bound which can be very easily evaluated. Examples are given of the use of the approximate bound.
Block‐indexed solution of very large linear equation systems with symmetric dbbf coefficient matrix
Authors: Stabrowski M. M.
Published: August 2005

The finite element solution of various partial differential equations leads to large linear equation systems, which may have a DBBF (double bounded band form) coefficient matrix, if periodicity conditions are taken into account. An efficient Fortran program solving such equation systems, developed for Control Data computers from the Cyber 70/170 series, utilizes the original block‐indexed approach to reduce input‐output time in communication with disc memory. Program performance is presented in some detail and quantitative comparison with other possible algorithms is carried out. An appendix lists the complete program.
A simple yet accurate finite element procedure for computing stress intensity factors
Authors: Sinclair G. B. , Mullan D.
Published: August 2005

A new finite element procedure for calculating stress intensity factors in elastic crack problems is developed. In common with a number of other approaches in the literature, the procedure combines the analytical singular fields present in a problem with a finite element treatment of the residual regular problem. What distinguishes the procedure is the use of path‐independent integrals to balance the analytical and numerical contributions. A set of test problems with exact solutions is analysed and demonstrates that the procedure is readily implemented and can accurately evaluate stress intensity factors with a modest amount of computational effort. The application of two competing methods to the test problems further demonstrates that the new procedure is markedly superior in both its initial accuracy and its rate of convergence. The paper concludes with two additional illustrations of the procedure as applied to the single‐edge crack and the centre crack; these also yield accurate results for little computational effort.
A strategy for the solution of soil dynamic problems involving plasticity by transform
Authors: Matthees W.
Published: August 2005

A closed solution for dynamic problems with the transform technique is possible for linear behaviour only. Therefore, a strategy for the solution involving nonlinear material behaviour has been developed. Herewith, an iterative equalization is made possible by the repeated transform of the differential equations given through the initial strain method.
On some unconditionally stable, higher order methods for the numerical solution of the structural dynamics equations
Authors: Dougalis Vassilios A. , Serbin Steven M.
Published: August 2005

Third‐ and fourth‐order accurate Nørsett rational approximations to the exponential and associated semi‐implicit Runge–Kutta methods are used for the construction of efficient, accurate and unconditionally stable schemes for the direct numerical integration of the linear, nonhomogeneous, second‐order equations of structural dynamics.
A finite boundary method for fluid flow field computations
Authors: Ponnuswamy V.
Published: August 2005

A powerful new finite boundary concept of seeking field solution, at few selected regions in the solution domain, is introduced. Also a highly economical finite boundary method (FBM) which would greatly reduce the size of the coefficient matrix of the resulting system of simultaneous algebraic equations, requiring lesser computer memory and lesser computing time, is developed. Fluid flow fields governed by the basic elliptic partial differential equations—the Laplace's and the Poisson's equations—in two independent variables are mainly considered. The computational merits of the FBM are shown by solving, as an example, a simple representative flow problem, and the relevant computational finite boundary formulae are given in tabular form. The formulae are numerically derived based on a generalized method presented here. The added feature of the FBM is that it proves to be equally economical even when the solution is sought in the entire flow domain. The problem of steady‐state viscous flows governed by the Navier‐Stokes equations, the system of two simultaneous partial differential equations—the Poisson's equation and the vorticity transport equation—makes the FBM doubly economical. The possibility of developing an efficient hybrid computational algorithm, for curved problem boundaries, in conjunction with the finite element method, is discussed. The extension of FBM to transient, non‐elliptic problems and to three‐dimensional problem fields is also indicated. The FBM has been discussed in a more detailed manner so as to clearly bring out the advantages of the new finite boundary concept.
A family of methods with three‐term recursion formulae
Authors: Papadrakakis M.
Published: August 2005

A state‐of‐the‐art survey of a family of vector iterative methods with three‐term recursion formulae is presented. The methods can be classified under two broad categories. The Chebyshev semi‐iterative type of methods and the conjugate gradient type of methods. Relationships between the iteration parameters of the methods are established and computational techniques for estimating these parameters are presented. Numerical tests confirm the superiority of the conjugate gradient method for linear problems and the Chebyshev semi‐iterative methods for nonlinear problems.
Fourier series solution for a rectangular thick plate with free edges on an elastic foundation
Authors: Henwood D. J. , Whiteman J. R. , Yettram A. L.
Published: August 2005

A Fourier series solution is presented for a system of first‐order partial differential equations which describe the linear elastic behaviour of a thick rectangular plate resting on an elastic foundation and carrying an arbitrary transverse load. The lateral edges of the plate are unstressed. A central step in the method for solving the system of equations is to combine a complementary function with a particular solution of the system in order to satisfy the boundary conditions. The complementary function is the sum of two series. The terms of the first series are products of a Fourier term in one space variable with the solution of an eigenvalue problem in the other space variable. The second series is similar and comes from reversing the roles of the space variables.
The potential integral for a linear distribution over a triangular domain
Authors: Okon Ephraim E. , Harrington Roger F.
Published: August 2005

In this paper an exact expression is derived for the potential at an arbitrary point due to a linear distribution of source over a triangular domain. The integral which gives the potential is evaluated by means of an extension of an analytical procedure previously evolved for a constant distribution. Numerical values of the potential are obtained for several field points by evaluating the above analytical expression, and are compared with those calculated by using triangle quadratures.
Limitations and possible extensions to a nonlinear finite element shell‐of‐revolution model based on Reissner's shell theory
Authors: Cook W. A.
Published: August 2005

A finite element nonlinear shell model has been developed using Reissner's symmetrically loaded shell‐of‐revolution theory. This model is capable of modelling nonlinear strains, large rotations and nonlinear materials. This paper discusses the limitations of this model by deriving the basic equations and describing the simplifying assumptions used in the theory and the model. Possible extensions of the model are discussed also.
Block overrelaxation methods for non‐rectangular co‐ordinates
Authors: Evans D. J. , Biggins M. J.
Published: August 2005

In this paper, the application of successive block overrelaxation methods to obtain the deflection of a skew plate are compared, although the techniques herein are applicable to a wider range of problems. An algorithm for the direct solution of the submatrices comprising the two‐line blocks of the skew grid is presented and the results are compared with the block (line and peripheral) overrelaxation methods.
A finite element model for thin‐walled members
Authors: Gellin S. , Lee G. C. , Chern J. H.
Published: August 2005

Thin‐walled members subject to generalized loading may be modelled by using a finite element approach with conventional stress and strain parameters. A solution method for a class of problems that would use these elements is described. A numberical example of an elastic beam is given.
An iterative method for finite dimensional structural optimization problems with repeated eigenvalues
Authors: Choi Kyung K. , Haug Edward J. , Seong Hwal G.
Published: August 2005

A numerical method for solving finite dimensional structural optimization problems with constraints on natural frequency and on buckling is formulated. The method treats nondifferentiable repeated eigenvalues that have been shown to systematically arise in structural optimization. Recent results on differentiability of eigenvalues are used to develop a generalized gradient projection method for structural optimization. The algorithm is shown to overcome technical difficulties associated with nondifferentiability of repeated eigenvalues. The method is used to solve buckling and vibration optimization problems in which repeated eigenvalues occur.
2‐D multiplane finite element technique for solving a class of 3‐D problems
Authors: Pack Steven C. , Mandel James A.
Published: August 2005

A finite element technique, for efficient solution of a class of 3‐D elasticity problems, is presented. In this method, standard 2‐D finite elements are used along with a ‘connector’ element. An element, previously used to model material interfaces, is shown to provide the properties for use as a ‘connector’ element, if input variables are redefined. The accuracy of the technique is illustrated with a sample solution.
The application of the theorems of structural variation to finite element problems
Authors: Topping B. H. V.
Published: August 2005

In this paper the application of the theorems of structural variation to finite element problems is presented and a simplified procedure for constant strain triangular elements is developed. Applications for these techniques are discussed.
Constraint equations from rigid elements including initial strains
Authors: Robinson John , Haggenmacher Gernot W.
Published: August 2005

In many commercial finite element method systems, the formulation of constraint equations from rigid elements is based entirely on rigid body mechanics. However, when initial (thermal) strains exist, a more general approach is required for rigidizing the element. This new approach also opens up a more versatile generation scheme of constraint equations for rigid or partially rigid elements.
Dynamic (transient) analysis of layered anisotropic composite‐material plates
Authors: Reddy J. N.
Published: August 2005

A shear‐flexible finite element is employed to investigate the transient response of isotropic, orthotropic and layered anisotropic composite plates. Numerical convergence and stability of the element is established using Newmark's direct integration technique. Numerical results for deflections and stresses are presented for rectangular plates under various boundary conditions and loadings. The parametric effects of the time step, finite element mesh, lamination scheme and orthotropy on the transient response are investigated. The present results agree very closely with the results available in the literature for isotropic plates, and the results for composite plates should serve as bench marks for future comparisons by other investigators.
Visioplasticity solution of deformation problems: Data handling procedures
Authors: Moore C. , Vickers G. W. , Dwivedi S. N.
Published: August 2005

Procedures for smoothing experimentally determined displacement fields are presented such that dynamic, non‐steady‐state metalworking processes can be analysed using the visioplasticity method. The procedure establishes a rectangular grid in which strain, strain rate and stress values may optionally be obtained at all points on the grid. The condition of material continuity is automatically satisfied by constraints during polynomial smoothing in position and time, using the Gram‐Schmidt orthogonalization technique. Examples are given of quasi‐static and dynamic plane strain upsetting operations.
On the inclusion principle for the hierarchical finite element method
Authors: Meirovitch L. , Baruh H.
Published: August 2005

The inclusion principle provides a qualitative characterization of the eigenvalues of a matrix. The principle has been shown to apply to systems described by a single Hermitian matrix, the most important of which being the real symmetric matrix. Self‐adjoint distributed systems, when discretized by either the classical Rayleigh‐Ritz method or by the finite element method, lead to algebraic eigenvalue problems described in terms of two real symmetric matrices. The algebraic eigenvalues problem derived by the classical Rayleigh‐Ritz method possesses the embedding feature required by the inclusion principle, but that derived by the finite element method in general does not. This paper demonstrates that the inclusion principle can be extended to discretized systems derived by the hierarchical finite element method.
Three‐dimensional finite element analysis of cracked thick plates in bending
Authors: Alwar R. S. , Nambissan K. N. Ramachandran
Published: August 2005

By using the finite element technique, stress intensity factors have been obtained for finite rectangular plates and the results have been given for various h/a, W/a and L/W ratios. By using a three‐dimensional isoparametric element, the problem has been considered as a three‐dimensional one and the variation of stress intensity factor across the plate thickness has been found to be nonlinear.
Finite element analysis of elastic‐plastic layered soil using discrete Fourier series expansion
Authors: Runesson Kenneth , Booker John R.
Published: August 2005

A method is developed which basically concerns how to reduce the necessary storage and cost of solving a set of algebraic equations arising from applying the initial stress technique to the analysis of a uniformly layered elastic‐plastic soil. Advantage is taken of the geometrical symmetry of the problem and the consequent symmetry of the stiffness matrix. The nodal values are expanded in a discrete Fourier series so that a two‐dimensional problem is reduced to a number of one‐dimensional problems.
Coupled vibrations of beams—an exact dynamic element stiffness matrix
Authors: Friberg P. O.
Published: August 2005

A uniform linearly elastic beam element with non‐coinciding centres of geometry, shear and mass is studied under stationary harmonic end excitation. The Euler‐Bernoulli‐Saint Venant theory is applied. Thus the effect of warping is not taken into account. The frequency‐dependent 12 × 12 element stiffness matrix is established by use of an exact method. The translational and rotational displacement functions are represented as sums (real) of complex exponential terms where the complex exponents are numerically found. Built‐up structures containing beam elements of the described type can be analysed with ease and certainty using existing library subroutines.
Elasto‐plastic analysis of anisotropic plates and shells by the Semiloof element
Authors: Owen D. R. J. , Figueiras J. A.
Published: August 2005

An elasto‐plastic analysis of thin plates and shells by means of the finite element displacement method is considered for anisotropic materials, using the Semiloof curved shell element. The elasto‐plastic analysis is based on the Huber‐Mises yield criterion extended by Hill for anisotropic materials. The yield function is generalized by introducing anisotropic parameters of plasticity which are updated during the material strain hardening history.

Numerical examples are presented and compared with available solutions. The effects of anisotropy on these solutions are also discussed. The practical problem of an aerofoil blade is also considered and some results presented.