bluuebunny/crossref_metadata_2025 huggingface dataset:
Geometrically nonlinear formulation for the curved shell elements
Authors: Surana Karan S.
Published: August 2005

This paper presents a geometrically nonlinear formulation using total lagrangian approach for the three‐dimensional curved shell elements. The basic element geometry is constructed using the coordinates of the middle surface nodes and the mid‐surface nodal point normals. The element displacement field is described using three translations of the mid‐surface nodes and the two rotations about the local axes. The existing shell element formulations are restricted to small nodal rotations between two successive load increments. The element formulation presented here removes such restrictions. This is accomplished by retaining nonlinear nodal rotation terms in the definition of the displacement field and the consistent derivation of the element properties. The formulation presented here is very general and yet can be made specific by selecting proper nonlinear functions representing the effects of nodal rotations. The element properties are derived and presented in detail. Numerical examples are also presented to demonstrate the behaviour and the accuracy of the elements.
A note on universal matrices for triangular finite elements for the quasi‐harmonic equation
Authors: Subramanian G. , Bose C. Jeyachandra , Babu C. Ramesh
Published: August 2005

An algorithm to generate universal matrices for plane triangular finite elements for the general ‘quasi‐harmonic’ equation is presented. For every member of the triangle family three numerical universal matrices are obtained which are independent of the size, shape and ‘material’ properties of the element. Of these, two are basic and the third can be generated from one of these two. The element ‘stiffness’ matrix is conveniently generated by manipulating these two basic matrices taking into account the size, shape and material properties of the element in a simple manner.
Dynamic elasto‐plastic response of shells in an acoustic medium—EPSA code
Authors: Atkatsh Robert S. , Bieniek Maciej P. , Baron Melvin L.
Published: August 2005

A nonlinear, large deflection, elasto‐plastic finite element code (EPSA) has been developed for the analysis of shells in an acoustic medium subjected to dynamic loadings. The nonlinear equations of shells are discretized with the aid of a finite difference/finite element method based upon the principle of virtual work. The resulting system of equations contains the nodal displacements as the generalized co‐ordinates of the problem. The integration in time of the equations of motion is done explicitly via a central difference scheme.

Shell strain‐displacement relations are established by a two‐dimensional finite difference scheme. The shell constitutive equations are formulated in terms of the shell stress resultants and the shell strains and curvatures. The fluid‐structure interaction is accounted for by means of the doubly asymptotic approximation (DAA) expressed in terms of orthogonal fluid expansion functions. The analytically produced results satisfactorily reproduce available experimental data for dynamically loaded shells.
A decomposition procedure for large‐scale optimum plastic design problems
Authors: Kaneko Ikuyo , Ha Cu Duong
Published: August 2005

A decomposition procedure is proposed in this paper for solving a class of large‐scale optimum design problems for perfectly‐plastic structures under several alternative loading conditions. The conventional finite element method is used to cast the problem into a finite dimensional constrained nonlinear programming problem. Structures of practically meaningful size and complexity tend to give rise to a large number of variables and constraints in the corresponding mathematical model. The difficulty is that the state‐of‐the‐art mathematical programming theory does not provide reliable and efficient ways of solving large‐scale constrained nonlinear programming problems. The natural idea to deal with the large‐scale structural problem is somehow to decompose the problem into a collection of small‐size problems each of which represents an analysis of the behaviour of each finite element under a single loading condition. This paper proposes one such way of decomposition based on duality theory and a recently developed iterative algorithm called the proximal point algorithm.
An infinite element and a formula for numerical quadrature over an infinite interval
Authors: Pissanetzky Sergio
Published: August 2005

In this publication we present a new infinite element and discuss a formula for numerical quadrature over infinite regions. Both the element and the formula are based on the properties of a set of orthonormal functions, obtained by mapping the Legendre polynomials on the infinite interval. The element can represent any (physical) function over the entire infinite region, and the formula can integrate most integrands over the infinite interval. They are thus very convenient for implementation in a general computer program, which could then be used to solve different problems with no restrictions concerning the asymptotic behaviour of the solutions or the form of integrands to be integrated over the infinite interval. However, the element and the formula will perform at their best when used in conjunction, and for problems of electrostatic, magnetostatic, etc., potential.

The formula was compared with other existing formulae and found to be either superior or satisfactory in practically all cases, as shown by the results of many numerical experiments.

Expressions of shape functions are given for the element. A test problem was solved using the element and the formula, and the results are shown to be more accurate than those obtained when solving the same problem using infinite elements with exponential decay and Gauss‐Laguerre numerical quadrature.
A boundary element solution to elasto‐plastic torsion of solids of revolution
Authors: Zhen‐Sheng Chen
Published: August 2005

A boundary element method for the solution of problems of elastic and elasto‐plastic torsion of solids of revolution is proposed. The displacement and stress field produced by a circumferential ring load in an infinite elastic body are derived and used as the fundamental solution to establish the governing integral equations. For the elasto‐plastic case, linear boundary elements and bilinear quadrilateral internal cells are used for discretization of these equations. In order to carry out Gaussian integrations along boundary elements and over internal cells, a method ensuring sufficient accuracy for the calculation of singular integrals is proposed. Numerical results for several problems are given.
General derivative approximations for finite different schemes
Authors: Patterson J. C.
Published: August 2005

A general expression for finite difference approximations for all derivatives up to the (N − 1)th order on a finite difference mesh with N‐nodes is established. The mesh may be non‐uniform and the derivative evaluated at any of the N‐nodes. An estimate for the truncation error is obtained and shown to depend on the nonlinearity of the mesh as well as on the mesh diameter. Specific formulae for first‐ and second‐order derivatives are given as examples.
On extrapolation boundary conditions for the numerical solution of hyperbolic difference schemes
Authors: Jamieson S. M. , Sloan D. M.
Published: August 2005

An analysis is presented for the stability of an initial and boundary value difference scheme arising from the solution of a two‐dimensional hyperbolic equation by the Lax‐Wendroff method. The positive and negative characteristic speeds of the linear equation are not necessarily equal in magnitude. Supplementary boundary conditions required for the finite difference solution are obtained using extrapolation for a linear combination of the dependent variables. The analysis produces stability intervals for the parameter which describes this linear combination of variables. Numerical experiments are described which support the theoretical analysis.
A posteriori error analysis and adaptive processes in the finite element method: Part II—adaptive mesh refinement
Authors: De S. R. Gago J. P. , Kelly D. W. , Zienkiewicz O. C. , Babuska I.
Published: August 2005

This is a paper presented in two parts dealing respectively with error analysis and adaptive processes applied to finite element calculations. Part I contains the basic theory and methods of deriving error estimates for second‐order problems. Part II of the paper deals with the strategy for adaptive refinement and concentrates again on the p‐convergent methods. It is shown that an extremely high rate of convergence is reached in practical problems using such procedures. Applications to realistic stress analysis and potential problems are presented.
A finite element for the analysis of reinforced concrete structures
Authors: Gupta Ajaya K. , Akbar Habibollah
Published: August 2005

Formulation of a four‐node isoparametric element suitable for modelling cracks in reinforced concrete structures is presented. The standard isoparametric element is known to give spurious shear stresses. The conventional remedy of selective integration breaks down in a ‘cracked’ element. It is shown that the proposed formulation gives superior results as compared to both the standard isoparametric element and the conventional selectively integrated element.
Reparametrization and its application in computer‐aided geometric design
Authors: Ball A. A.
Published: August 2005

The paper develops the theory of reparametrization within the context of computer‐aided geometric design. It is established that the parametrization plays a positive role in the numerical description of curves and surfaces and it is proposed that the parametrization should be controlled, independently from the shape, via reparametrization. A general approach to reparametrization is adopted in which a function to change the parametrization is modelled in addition to the basic parametric equation. Examples are given for parametric polynomial curves, probably the most commonly used curve segments in computer‐aided geometric design. Finally it is shown how reparametrization might be applied to enhance existing surface definition procedures. A surface equation is developed based on the parametric cubic segment that can define shapes virtually independent of the parametrization.
Modelization of phase changes by fictitious‐heat flow
Authors: Roose J. , Storrer O.
Published: August 2005

This paper presents an efficient algorithm to solve phase change problems. The first part gives the definition of the problem and mentions some practical applications. The second part analyses how to deal with phase changes by means of concentrated fictitious heat flows. The principles of the method are described and compared to those of other methods. The concept of ‘the area associated with a node’ is developed and some practical values for the weight coefficients are recommended. Conclusions are given about the convergence and the stability of the algorithm, and about the improvements obtained by prescribing the temperatures and using relaxation techniques. Theoretical and practical examples illustrate the validity and efficiency of the proposed method.
Boundary integral equation method for linear porous‐elasticity with applications to fracture propagation
Authors: Cheng Alexander H‐D. , Liggett James A.
Published: August 2005

For the semi‐infinite crack propagating quasi‐statically in a porous‐elastic medium, the boundary integral equation method (BIEM), based on a reciprocal principle, is formulated for the governing equations of the Biot model of linear porous‐elasticity. The resulting numerical scheme is efficient since the discretization of the solution unknowns is required only along the axis of symmetry of a fracture. Fracture propagation criteria based on both elastic and plastic constitutive relations are investigated. Practical applications of the model are expected in the failure of overconsolidated clay, earthquake prediction and underground hydraulic fracturing for energy exploration.
Non‐iterative solution of a tridiagonal‐type matrix coupled by diagonal submatrices
Authors: Deshpande M. D.
Published: August 2005

A non‐iterative method is presented for solving a system of algebraic equations forming a banded matrix with each diagonal submatrix being tridiagonal and all other submatrices being diagonal. A recursion relation is assumed between successive elements of the solution vector. This relation is determined by equating a linearly combined form of it with the original set of equations also being linearly combined. The solution vector is determined by back‐substitution of elements in the recursion relation. Since the method takes advantage of the banded structure of the matrix, it is more efficient than the conventional methods. A FORTRAN program that incorporates the present method is included.
Pure opening mode stress‐intensity factor computation for elliptical crack fronts
Authors: Manu Corneliu
Published: August 2005

Further insight into the functional consistent displacement correlation method to calculate the stress‐intensity factor for elliptical cracks will be presented. The method will be modified to account for the actual crack face shape in pure opening mode.
An optimized solution for the diffusion equation on a non‐uniform grid
Authors: Sobey R. J.
Published: August 2005

A lowest order difference scheme has been developed for a spatially non‐uniform grid such that the integrated square error is a minimum. This algorithm accommodates grid asymmetry and maintains reasonable accuracy without recourse to higher order approximations. Numerical experiments and computations of the Fourier response factor confirm the performance of the optimized asymmetric algorithm.
Deflection of thick beams of multimodular materials
Authors: Bertm C. W. , Gordaninejad F.
Published: August 2005

A transfer‐matrix analysis is presented for determining the static behaviour of thick beams of ‘multimodular materials’ (i.e. materials which have different elastic behaviour in tension and compression, with nonlinear stress–strain curves approximated as piecewise linear, with four or more segments). To validate the transfer‐matrix method results, a closed‐form solution is also presented for cases in which the neutral‐surface location is constant along the beam axis. Numerical results for axial displacement, transverse deflection, bending slope, bending moment, transverse shear, axial force and location of neutral surface are presented for multimodular and bimodular models of unidirectional aramid cordrubber. The transfer‐matrix method results agree very well with the closed‐form solutions.
Numerical solution of a parabolic system with coupled nonlinear boundary conditions
Authors: Liu Ta‐Jo
Published: August 2005

A numerical technique has been developed to solve a system that consists of m linear parabolic differential equations with coupled nonlinear boundary conditions. Such a system may represent chemical reactions, chemical lasers and diffusion problems. An implicit finite difference scheme is adopted to discretize the problem, and the resulting system of equations is solved by a novel technique that is a modification of the cyclic odd–even reduction and factorization (CORF) algorithm. At each time level, the system of equations is first reduced to m nonlinear algebraic equations that involve only the m unknown grid points on the nonlinear boundary. Newton's method is used to determine these m unknowns, and the corresponding Jacobian matrix can be computed and updated easily. After convergence is achieved, the remaining unknowns are solved directly. The efficiency of this technique is illustrated by the numerical computations of two examples previously solved by the cubic spline Galerkin method.
An invariant eight‐node hybrid‐stress element for thin and thick multilayer laminated plates
Authors: Spilker R. L.
Published: August 2005

A hybrid‐stress eight‐node isoparametric element is developed for the analysis of thin or thick multilayer fiber‐reinforced composite plates. Transverse shear deformation effects are included by allowing for individual layer cross‐section warping for thick laminates, or alternatively, laminate non‐normal cross‐section rotations for thin to moderately thick laminates. All stress components are included and are interpolated independently within each layer. Interlayer surface traction continuity and appropriate upper/lower surface traction‐free conditions are exactly satisfied. The layer stress field is selected on the basis of earlier single‐layer element studies so that the resulting element is naturally invariant with respect to co‐ordinate translation or rotations, is non‐locking in the thin‐plate limit, and the element stiffness is of correct rank. An example for which an elasticity solution is available is used to demonstrate the element performance. Schemes for reduction of element stiffness computation time are also presented.
Analysis of multi‐layer laminates using three‐dimensional super‐elements
Authors: Jones R. , Callinan R. , Teh K. K. , Brown K. C.
Published: August 2005

Analysis of multi‐layer composite laminates using three‐dimensional finite element models often results in high computational cost. This communication reports a three‐dimensional super‐element scheme which can reduce the computational cost.
Finite element analysis of wave motion
Authors: Kawahara Mutsuto , Miwa Toshihiko
Published: August 2005

To estimate wave motion, the finite element method is presented based on the linear interpolation function and two‐step explicit numerical integration in time. For the determination of free surface position, the two‐step scheme based on the Eulerian technique is usefully employed. Travel and run‐up of solitary wave have been analysed and compared with the analytical solution. Both results are reasonably well in agreement. This method is applied to estimate the wave force on the practical breakwater to show the validity of the method.
Second derivative methods applied to implicit first‐ and second‐order systems
Authors: Addison C. A. , Gladwell I.
Published: August 2005

A wide variety of methods have been proposed for solving implicit second‐order systems of differential equations such as arise in structural and mechanical vibration analyses. Here we discuss a class of methods, based on second derivative formulae, some of which have previously been used by Enright.1 We show them to have desirable analytic and implementation properties. We relate these methods to others recently proposed by Brusa and Nigro and by Serbin, showing that we obtain algorithms which have all the desirable properties of their methods and which can also be extended to classes of practical problems which were not considered by these other authors. Since implicit first‐order systems are also important in practice, for example in finite element analyses of heat conduction problems, we demonstrate the efficient implementation of the second derivative formulae in this case too.
The cylinder in the wind tunnel, revisited
Authors: Habashi Wagdi G. , Kotiuga Peter L.
Published: August 2005

Analytical comparative solutions are often needed to assess the accuracy of proposed elements or proposed iterative schemes for potential flow numerical calculations. Traditionally, the flow over a circular cylinder, placed in a wind tunnel, has served for such a test and the results compared to what is thought to be an exact solution. We point out here that there is no exact solution for the flow over a circular cylinder in a wind tunnel, for practical values of wind tunnel height to cylinder radius, but that an exact solution can only be found for the flow over an oval cylinder. We derive that exact solution and provide numerical comparisons, using triangular elements, with both the stream function and the velocity potential for the incompressible case. In addition, numerical results for compressible flows up to critical Mach number are presented.
The methods of Gellert and of Brusa and Nigro are Padé approximant methods
Authors: Thomas R. M. , Gladwell I.
Published: August 2005

We discuss properties of methods proposed recently by Gellert1 and by Brusa and Nigro2 for solving systems of second‐order ordinary differential equations. In both cases, we extend the derivations to obtain whole families of methods with improved stability properties (roughly equivalent to L‐stability for first‐order systems) and we show that obtaining improvement in stability properties results in a deterioration in accuracy and/or implementation properties. Also we bring out the relationship (and in some cases equivalence) of these methods to others proposed in the literature. In particular, we show that the methods proposed by the above authors are essentially equivalent to well‐known methods of long standing, namely Padé approximant methods for equivalent first‐order systems of ordinary differential equations.
Bending of bilinear quadrilateral shell elements
Authors: Morley L. S. D.
Published: August 2005

Formulae are derived for exact inextensional bending solutions for arbitrary quadrilateral shell finite elements of bilinear parametric representation. It is found that the polynomial degree of parametric representation of the rectangular components of displacement requires to be at least cubic in order to describe any inextensional bending modes.
h‐ and p‐version finite element analyses of a rhombic plate
Authors: Wang D. W. , Katz I. N. , Szabo B. A.
Published: August 2005

A simply‐supported rhombic plate with obtuse angle equal to 150 degrees is analysed by the finite element method, using both the h‐version and the newer p‐version. Results obtained using the computer code CONE (C1‐continuity) for plate bending problems are compared with theoretical predictions and with computational results reported in the literature. If accuracy in terms of the number of degrees‐of‐freedom is used as a criterion, the solutions presented here are the most efficient that have been published to date.
An iterative approach for the evaluation of delamination stresses in laminated composites
Authors: Barsoum Roshdy S. , Freese C. E.
Published: August 2005

A computational strategy for the evaluation of delamination stresses in multilayered fibre‐reinforced composites is presented. The approach uses special inter‐laminar shear elements and an iterative procedure in the calculation of out‐of‐plane stresses. Both regular and singular elements for the free edge are presented. The solutions are carried on by switching between direct matrix solution methods of small systems of equations and an iterative conjugate gradient method. The proposed discretization avoids ill‐conditioning problems, modelling difficulties and large computational cost associated with three‐dimensional analysis of delamination of holes, cut‐outs, joints and areas of complex stress states.
An improved explicit finite element method for tidal flow
Authors: Kikukawa H. , Ichikawa H.
Published: August 2005

A weighting function by which the mass matrix is made to be diagonal is deduced on the two‐dimensional simplex element. Adopting the weighted residual method with the deduced weighting function for spatial domain and the two‐step Lax–Wendroff method for time differentiation, the two‐dimensinoal tidal flow in a model basin is calculated by means of the explicit finite element method. The water mass transport integrated over one tidal period at some cross‐sections of the basin is also estimated, in order to verify the water mass conservation. It is found that the water mass is nearly conserved in the present method, although the usual lumped mass matrix technique fails to conserve.

The present explicit finite element method with the conservative form of governing equations is also investigated and the water mass conservation is found to be a little more improved in this case than in the case of non‐conservative forms. Application to a real bay indicates that the method with the conservative form of equations could be used for the estimation of tidal residual flow.
Incremental elastoplastic stiffness matrix for materials obeying the principal stress yield criterion
Authors: Fo O. J. A. Goncalves , Owen D. R. J.
Published: August 2005

In this work, the plastic stress–strain relation and the incremental elastoplastic stiffness matrix for materials obeying the principal stress yield criterion are derived in the context of the associated theory of plasticity. Particular attention is paid to the treatment of the singularity conditions associated with this yield criterion.
A unified set of single step algorithms. Part 1: General formulation and applications
Authors: Zienkiewicz O. C. , Wood W. L. , Hine N. W. , Taylor R. L.
Published: August 2005

A general algorithm for a single step time marching scheme for use in dynamic or diffusion equations is presented. This algorithm is easily programmed in its universal form for all orders of approximation and covers most of the currently used schemes as well as presenting many new possibilities. In many cases it presents a computationally advantageous form over conventional procedures—this is particularly so when compared with the Newmark algorighm and its variants.
Substructure condensation using modified decomposition
Authors: Han Tao‐Yang , Abel John F.
Published: August 2005

A new method for static condensation is presented. This method utilizes Choleski decomposition with some modification to obtain the effective stiffness matrix and the effective load vector. A comparison between this modified decomposition method and the usual matrix decomposition method shows that the former is more efficient than the latter.
Analysis of laminated composite shells using a degenerated 3‐D element
Authors: Chao W. C. , Reddy J. N.
Published: August 2005

A special three‐dimensional element based on the total Lagrangian description of the motion of a layered anisotropic composite medium is developed, validated and employed to analyse laminated anisotropic composite shells. The element contains the following features: geometric nonlinearity, dynamic (transient) behaviour and arbitrary lamination scheme and lamina properties. Numerical results of nonlinear bending, natural vibration and transient response are presented to illustrate the capabilities of the element.
A combination of the finite element and singular‐integral equation methods for the solution of the generally cracked body
Authors: Theocaris P. S. , Tsamasphyros G. , Theotokoglou E. E.
Published: August 2005

A method which combines the finite element technique and the singular‐integral equation method is presented. The association of the two methods is obtained with the help of Schwarz's alternating method (SAM). The method was applied with satisfactory results to the solution of a series of problems of a circular arc crack lying inside a finite thin plate for various lengths of the circular arc and for various dimensions of the rectangular cracked sheet.
Linear multipoint constraints applied via transformation as part of a direct stiffness assembly process
Authors: Shephard Mark S.
Published: August 2005

A procedure is presented for the application of linear multipoint constraints. The procedure employs the transformation approach for constraint application which reduces the number of equations to be solved by the number of constraints. However, the matrix operations implied by the approach are avoided by the proper construction of the constraint equations. A specific procedure for carrying out the constraint application process as part of the assembly of the global stiffness matrix is given.
Reducing relative error from the CVBEM by proper treatment of the known boundary conditions
Authors: Hromadka T. V. , Guymon G. L.
Published: August 2005

By a proper treatment of the known boundary conditions of a boundary value problem, a complex variable boundary element method (CVBEM) can be used to exactly satisfy the known nodal point boundary values. In this fashion, a numerical model can be developed which generates relative error information along the problem boundary that can be used to reduce the modelling error by either an integrated measure or a maximum relative error measure.
The complex variable boundary element method: Applications
Authors: Hromadka T. V. , Yen C. C. , Guymon G. L.
Published: August 2005

The complex variable boundary element method (CVBEM) is used to approximate several potential problems where analytical solutions are known: A modelling result produced from the CVBEM is a measure of relative error in matching the known boundary condition values of the problem. A CVBEM error‐reduction algorithm is used to reduce the relative error of the approximation by adding nodal points in boundary regions where error is large. From the‐test problems, overall error is reduced significantly by utilizing the adaptive integration algorithm.
Generalized finite difference representations of the Laplacian operator in n‐dimensional Euclidean space
Authors: Okon E. E.
Published: August 2005

Two generalized finite difference representations of the n‐dimensional Laplacian operator are obtained vectorialiy by extensions of the techniques previously developed for three‐dimensional space. Each representation is presented in a form which can readily be applied at the nodes generated in an automatic decomposition of an n‐dimensional region. The finite difference expressions for the Laplacian are applied to the n‐dimensional sphere, as well as to certain important polytopes for which detailed algebraic descriptions are provided.
Nonlinear dynamic analysis with a 48 d.o.f. curved thin shell element
Authors: Saigal Sunil , Yang T. Y.
Published: August 2005

The developments of an existing 48 degrees‐of‐freedom (d.o.f.), curved, quadrilateral, thin shell element, for materially and geometrically nonlinear static analysis of shell structures, are extended for the study of dynamic responses of nonlinear shells. The variable‐order polynomial representations of the shell surface and the non‐axisymmetric definition of the shell boundaries allow the study of the dynamic behaviour of a class of shell structures more general than those treated by using flat plate elements and elements with assumptions of axisymmetry. The equations of motion are based on a Lagrangian frame of reference. A combination of step‐by‐step and iterative procedures is used for the solution of nonlinear equations. The incremental equations of motion are linearized for computation purposes, and an algorithm for numerical integration based on Newmark's generalized operator for dynamic analysis, using optional iteration, is adopted. The flow theory of plasticity is used in the inelastic range, and perfectly plastic or isotropic strain hardening materials are considered. The spread of plastic zones in the thickness direction is treated by using a layered model. Numerical examples presented include the dynamic analyses of a square plate, a circular annulus, a cylindrical panel and a spherical cap. Comparisons with existing solutions demonstrate the validity and accuracy of the present developments.
A continuous plasticity version of the critical state model
Authors: Naylor D. J.
Published: August 2005

A modification to a conventional critical state model is described which introduces some plasticity in the stress region which is normally fully elastic. The effect is to blur the otherwise sharp distinction between elastic and yielding states. The modification is conceptually simple and does not require any additional material parameters. The model was developed for the dual purpose of providing a more robust constitutive law for finite element applications and to provide a more realistic representation of the soil. Its potential is demonstrated by simulating the response of a plane strain sample tested over a range of loading and unloading stress paths. The results are in accordance with expectations from physical tests. Further, they show that the model has some potential for at least the first few cycles of cyclic loading. A finite element boundary value analysis of an excavation is also described. This shows how the model copes with the severe test of stress paths which unload while yielding. Finally, the model is used to match stress‐strain curves obtained from large‐scale laboratory tests on a compacted clay. Close agreement with three of the five tests was obtained using a single set of material parameters.
Approximate force method reanalysis techniques in structural optimization
Authors: Ding Huiliang , Gallagher Richard H.
Published: August 2005

Based on the size variable‐independence of the equilibrium equations in matrix format, a rational reduced basis reanalysis and an approximate reanalysis method are presented. Because the redundancy of structures is often low, and the decomposed coefficient matrix is known, these techniques can be advantageous in the structural optimization process. Several truss structures are studied for the purpose of numerical illustration.
Processing nonlinear multipoint constraints in the finite element method
Authors: Narayanaswamy O. S.
Published: August 2005

This paper reviews and evaluates two methods for processing nonlinear constraints. Both are based on the Lagrange multiplier technique. However, one of them, the constraint forces method, is often more efficient than the commonly used method of elimination. The two methods are evaluated by applying them to an example problem and comparing corresponding solution times. Guidelines are given for selecting the more efficient method for a given constraint problem.
An alternative explicit formulation for the DKT plate‐bending element
Authors: Jeyachandrabose C. , Kirkhope J. , Babu C. Ramesh
Published: August 2005

The stiffness matrix for the DKT plate‐bending element is formulated explicitly in a global co‐ordinate system. This approach avoids transformations of stiffness, and elasticity properties for anisotropic materials, from local to global co‐ordinates, which were required in previous formulations. A FORTRAN listing of the algorithm is appended for potential users.
A superconvergence result for the natural extrapolation formula for the numerical determination of stress intensity factors
Authors: Ioakimidis N. I.
Published: August 2005

The best approach for the numerical determination of stress intensity factors at crack tips in plane and antiplane elasticity problems is frequently the numerical solution of the corresponding Cauchy‐type singular integral equation by the Gauss–Chebyshev method, followed by the application of the natural extrapolation formula for the numerical determination of the stress intensity factors. It is shown here that this approach converges for Hölder‐continuous and discontinuous (with jump discontinuities) loading distributions along the crack (or cracks) and that in all cases the rate of convergence is greater than that believed up to now. This superconvergence result is based on a theorem on the numerical equivalence of the Gauss–Chebyshev direct method to a relevant indirect method for the numerical solution of Cauchy‐type singular integral equations, also proved here. Numerical results in various crack problems corroborate the theoretical ones.
A new mesh generation scheme for arbitrary planar domains
Authors: Lo S. H.
Published: August 2005

This paper describes a new algorithm to generate interior nodes within any arbitrary multi‐connected regions. The boundary nodes and the interior nodes are then linked up to form the best possible triangular elements by a completely revised technique in an efficient and stable manner. Owing to the generality of the central generation program, the global domain is allowed to be divided into as many irregular subdomains as desired, in order to model closely the actual physical situation. Moreover, the boundaries of the sub‐domains are updated from time to time when necessary to include the possibilities of progressive refinement around a sharp corner, generating radiating mesh from a prescribed node, generating mesh between two circular arcs, etc. Despite its flexibility and capabilities, data for triangulation have been kept to a minimum by a logical input module; no connectivity information between subregions is needed, and common boundaries are defined once only. All these features have contributed to a powerful method to generate 3‐node or 6‐node triangular element meshes of great variety within the most irregular heterogeneous regions.
Calculation of approximate weight functions in fracture mechanics by FEM
Authors: Morawietz P. , Mattheck C. , Munz D.
Published: August 2005

A method is presented for the calculation of weight functions used in fracture mechanics to determine stress intensity factors of cracks loaded by stress gradients. The reference solution for the stress intensity factor and for the reference crack opening displacement field is computed numerically by use of finite elements. The accuracy of the method is checked by comparison with well‐known solutions from the literature.
Sensitivity calculations for iteratively solved problems
Authors: Haftka Raphael T.
Published: August 2005

The calculation of sensitivity derivatives of solutions of iteratively solved systems of algebraic equations is investigated. A modified finite difference procedure is presented which improves the accuracy of the calculated derivatives. The procedure is demonstrated for a simple algebraic example as well as an element‐by element preconditioned conjugate gradient iterative solution technique applied to truss examples.
Near‐minimal matrix profiles and wavefronts for testing nodal resequencing algorithms
Authors: Armstrong Bruce A.
Published: August 2005

The speeds of algorithms that are specifically designed to solve sparse matrix equations depend on the ordering of the unknowns. Because it is difficult to know what a good ordering is, many resequencing algorithms have been developed to reorder the equations in a manner that minimizes the execution time of the solver being used. There is no theoretical way of evaluating resequencing algorithms, but four widely used algorithms (Cuthill–Mckee, Gibbs–Poole–Stockmeyer, Levy, Gibbs–King) have been compared with one another on the basis of their performance on a set of benchmark test problems. This paper reports what we believe to be are minimal or near‐minimal matrix profiles and wavefronts for the benchmark problems. Comparisons of the minimal results with those produced by the widely used resequencing algorithms show that they produce profiles typically a few tens of per cent greater than minimal, but 50 per cent to 100 per cent greater on two problem types. The algorithm that produced the near‐minimal results used a simulated annealing technique, and is far too slow for general use.
Petrov–Galerkin finite element methods with a hinged test space for singularly perturbed problems
Authors: Fitzsimons C. , Miller J. J. H. , O'riordan E.
Published: August 2005

In this paper we introduce finite element methods of Petrov–Galerkin type for the approximate solution of two‐point boundary‐value problems for singularly perturbed, second‐order, ordinary, linear differential equations. We write down Petrov–Galerkin methods on a uniform mesh which have asymptotic error estimates, as the mesh size tends to zero, whose magnitude is independent of the singular perturbation parameter. This is in marked contrast to standard finite element methods which do not possess such a property on a uniform mesh. For these, typically, the error on a fixed uniform mesh blows up as the singular perturbation parameter tends to zero.

This robust behaviour of these Petrov–Galerkin methods for singularly perturbed problems is achieved by choosing trial spaces of standard piecewise polynomial type, while the test spaces consist of hinged piecewise polynomials.

We consider self‐adjoint and non‐self‐adjoint two‐point boundary‐value problems with Dirichlet boundary conditions. We define hinged test spaces for both types of problem. We then introduce a number of sample problems and we present numerical solutions of these sample problems using a Petrov–Galerkin method with the appropriate hinged test space.
Curved beam elements to model noncircular coil shapes for tokamak reactor
Authors: Shah V. N. , Marshall N. H.
Published: August 2005

The detailed stress analysis of the toroidal field superconducting coil is needed to ensure its structural integrity, which includes conductor breakage, insulation breakage, sliding between two turns, or support structure failure. Such detailed‐analysis requires that each of the superconducting coil turns be modelled separately. In this paper, a curved beam finite element,13 having constant radius of curvature, is considered to model a turn of a noncircular toroidal field coil. Four different coil shapes subjected to electromagnetic body forces due to toroidal magnetic fields are analysed to assess the finite element models: circular, elliptic, Princeton‐D8 and compound shape9. The comparison of analytical and finite element analysis results for resultant forces, tensions and displacements suggest that the proposed approach may be extended to the detailed stress analysis of the noncircular toroidal field superconducting coil.
Elastic–viscoplastic, large deformation formulations of a curved quadrilateral thin shell element
Authors: Saigal Sunil , Yang T. Y.
Published: August 2005

An elastic–viscoplastic constitutive law is implemented in the formulations of a curved quadrilateral element based on thin shell theory. The element surface is defined using variable‐order polynomials allowing representation of a wide range of shell geometries. The large displacement effects are included by using an incremental stiffness procedure together with the Lagrangian mode of description of motion. The time rate effects may be treated by using explicit, semi‐explicit or implicit time marching schemes. A scheme for automatic calculation of time step is used. By allowing steady‐state conditions to be reached, plasticity solutions are obtained. Both perfect plasticity and strain hardening are considered. The Von Mises yield rule and the associated flow rule are used to determine plastic deformation. The spread of plasticity in the thickness direction is allowed by using a layered model. The procedures developed are applied to solve elastic‐plastic, small and large displacement problems of thin plates and shells and the results obtained are compared with existing solutions to illustrate the validity and accuracy of the present developments.
An improved solution procedure for creep problems
Authors: Heaps C. W. , Mansfield L.
Published: August 2005

A new simpler solution procedure is presented for the finite element analysis of creep problems. The creep strains are eliminated as computation variables. At each time step, a system is solved for the stresses and velocities alone.
An implicit time integration scheme for inelastic constitutive equations with internal state variables
Authors: Cordts Detlef , Kollmann Franz G.
Published: August 2005

In a broad class of inelastic constitutive models for the deformation of metals the inelastic strain rates are functions of the current state of stress and internal state variables only. All known models are in some regions of application mathematically stiff and therefore difficult to integrate. The unconditionally stable implicit Euler rule is used for integration. It leads to a system of highly nonlinear algebraic equations which have to be solved by an iterative process. The general Newton‐Raphson method, which converges under very broad conditions, requires repeated solution of the finite element system and is infeasible for large inelastic problems. But for the inelastic strains and internal state variables the Jacobian can be computed analytically and therefore the NRI can be used. For the stresses the Jacobian cannot be computed analytically and therefore the accelerated Jacobi iteration is used. A new method for computing the relaxation parameter is introduced which increases the rate of convergence significantly. The new algorithm is applied on Hart's model. A comparison with prior computations using an approximation is made.
Hybrid stress reduced‐Mindlin elements for thin multilayer plates
Authors: Spilker R. L. , Jakobs D. M.
Published: August 2005

A hybrid‐stress formulation of isoparametric elements for the analysis of thin multilayer laminated composite plates is presented. The element displacement behaviour is characterized by laminate reference surface inplane and transverse displacements and laminate non‐normal cross‐section rotations; as a result, simple Co interpolation of displacement and rotation can be used, and the number of degrees‐of‐freedom is independent of the number of layers. All components of stress are included and are related to a set of laminate stress parameters, the number of which is independent of the number of layers. Attention is restricted here to thin laminates, for which it is shown that the contributions of transverse shear stress and transverse normal stress to the internal complementary energy can be neglected. As a result of this reduction, a modified stiffness‐formation algorithm can be used which provides a significant improvement in computational efficiency. The formulation presented is used to develop an 8‐node isoparametric thin multilayer plate element. The resulting element is naturally invariant, of correct rank, and non‐locking in the thin plate limit.
Exact eigenvalue calculations for structures with rotationally periodic substructures
Authors: Williams F. W.
Published: August 2005

The theory presented enables rotationally periodic (i.e. cyclically symmetric) three‐dimensional substructures to be included when using existing algorithms to ensure that no eigenvalues are missed when the stiffness matrix method of structural analysis is used, where the eigenvalues are the natural frequencies of undamped free vibration analyses or the critical load factors of buckling problems. A substructure can be connected in any required way to a parent structure which shares its rotational periodicity, or can be connected by nodes at each end of its axis of periodicity to any parent structure, i.e. the parent structure need not be periodic. The theory uses complex arithmetic, involves only one of the rotationally repeating portions of the substructure, allows nodes and members to coincide with the axis of rotational periodicity, permits efficient multi‐level use of rotationally periodic substructures, and gives ‘exact’ eigenvalues if the member equations used are those obtained by solving appropriate differential equations. The competitiveness of the method is illustrated by approximate predictions of computation times and savings for two structures which contain rotationally periodic substructures.
Solution of two‐point boundary value problems using splines
Authors: Tewarson R. P. , Zhang Yin
Published: August 2005

Cubic spline on quintic spline interpolations are used to approximate the derivative terms for use in two high order numerical schemes for the solution of two‐point boundary value ordinary differential equations. The use of splines leads to solutions that reflect the smoothness of the slopes of the differential equations. Results of some computational experiments are given.
A field consistent three‐noded quadratic curved axisymmetric shell element
Authors: Prathap G. , Babu C. Ramesh
Published: August 2005

A curved three‐noded quadratic isoparametric axisymmetric thick shell element is developed. Field‐consistency interpretations allow various configurations of the element to be designed so as to satisfy specific problem needs. Typical applications demonstrate the versatility and accuracy of this element in its different problem‐specific forms.