bluuebunny/crossref_metadata_2025 huggingface dataset:
Finite deformation post‐buckling analysis involving inelasticity and contact constraints
Authors: Simo J. C. , Wriggers P. , Schweizerhof K. H. , Taylor R. L.
Published: August 2005

This paper is concerned with the numerical solution of large deflection structural problems involving finite strains, subject to contact constraints and unilateral boundary conditions, and exhibiting inelastic constitutive response. First, a three‐dimensional finite strain beam model is summarized, and its numerical implementation in the two‐dimensional case is discussed. Next, a penalty formulation for the solution of contact problems is presented and the correct expression for consistent tangent matrix is developed. Finally, basic strategies for tracing limit points are reviewed and a modification of the arc‐length method is proposed. The good performance of the procedures discussed is illustrated by means of numerical examples.
p‐adaptive boundary elements
Authors: Alarcon E. , Reverter A.
Published: August 2005

This paper presents the implementation of an adaptive philosophy to plane potential problems, using the direct boundary element method. After some considerations about the state of the art and a discussion of the standard approach features, the possibility of separately treating the modelling of variables and their interpolation through hierarchical shape functions is analysed. Then the proposed indicators and estimators are given, followed by a description of a small computer program written for an IBM PC. Finally, some examples show the kind of results to be expected.
Effect of rotary inertia on vibration of tapered beams
Authors: Gupta Arvind K.
Published: August 2005

Rotary inertia and dynamic‐correction rotary inertia matrices for a linearly tapered beam element of any cross‐sectional shape are derived in explicit form. Exact expressions for the required displacement functions are used in the derivation of these matrices. Variation of area and moment of inertia of the cross‐section along the axis of the element are exactly represented by simple functions involving shape factors. Vibration analysis results of a tapered cantilever beam of I‐section, obtained with and without the rotary inertia matrices are compared and effect of rotary inertia on vibration of tapered beams discussed.
Dynamic stress intensity factors studied by boundary integro‐differential equations
Authors: Sládek J. , Sládek V.
Published: August 2005

The boundary integro‐differential equation method is illustrated by two numerical examples concerning the study of the dynamic stress intensity factor around a penny‐shaped crack in an infinite elastic body. Harmonic and impact load on the crack surface has been considered. Applying the Laplace transform with respect to time to the governing equations of motion the problem is solved in the transformed domain by the boundary integro‐differential equations. The Laplace transformed general transient problem can be used to solve the steady‐state problem as a special case where no numerical inversion is involved.
A unified set of single step algorithms. Part 4: Backward error analysis applied to the solution of the dynamic vibration equation
Authors: Wood W. L.
Published: August 2005

This paper applies the method of backward error analysis to the numerical solution of the dynamic vibration equation by the single step algorithms proposed in References 1–3 and obtains some general rules to help with the choice of parameters.
Modelling of radiation damping in fluids by finite elements
Authors: Sharan Shailendra K.
Published: August 2005

A very efficient technique is presented to model the effects of radiation damping in the computation of added mass for the dynamic analysis of submerged structures. The structure is assumed to be surrounded by an infinite, incompressible and inviscid fluid field and the effect of the free surface is neglected. The technique is implemented in the finite element analysis of two‐dimensional problems, assuming pressure to be the nodal unknown. The implementation procedure is quite simple and the symmetrical and banded form of the matrix of coefficients remains unchanged. With the use of the proposed radiation condition, the fluid field may be truncated at a relatively very short distance from the solid—fluid interface. This results in great computational advantages. Furthermore, a guideline is suggested for the selection of the geometry and the location of the truncation boundary to enhance the computational efficiency. The effectiveness and efficiency of the technique is demonstrated by analysing several cases for different geometries of the solid—fluid interface and the truncation boundary.
Finite layer analysis of layered elastic materials using a flexibility approach. Part 2—Circular and rectangular loadings
Authors: Small J. C. , Booker J. R.
Published: August 2005

The finite layer techniques introduced in Part 1 of this paper are extended to allow the analysis of circular or general loadings applied to horizontally layered anisotropic materials. The analysis is considerably simplified through the use of Hankel transforms (circular loadings) and double Fourier transformations (general loadings).

Once again an exact flexibility matrix is found for each finite layer, and it is shown that this flexibility matrix has precisely the same form whether the loading is a strip, circular or general loading. The flexibility matrix has the advantage of not becoming singular for incompressible materials as is the case for the stiffness matrices used in a conventional finite strip analysis.

Examples are given of the behaviour of circular and general (rectangular) loadings applied to multilayered anisotropic materials. It is also shown by means of an example, that the method is extremely useful for analysis of problems involving incompressible materials.
A finite element and experimental study of rigid‐plastic compression
Authors: Houlston R. , Vickers G. W. , Anderson D. L.
Published: August 2005

This paper is concerned with the effect of material properties, boundary friction, and inertia on the flow and strain patterns in material undergoing large plane strain compression. Experimental results are presented for both the rapid compression of Plasticine and the quasi‐static compression of aluminium. In addition, a finite element code for rigid‐plastic material behaviour is discussed, and approximate procedures are described for both in‐plane and platen friction. Provision in the code was also made for strain hardening and strain rate sensitive material properties, and inertia effects. Comparison of the finite element and experimental results showed good agreement. Finite element calculations revealed that in‐plane friction had a very significant effect on the strain patterns observed in the Plasticine experiments. Finite element calculations also showed that strain hardening was of fundamental importance in determining the strain patterns that were observed to occur in the quasi‐static compression of aluminium.
Failures of local approximation in finite‐element methods
Authors: Kurtze Douglas A.
Published: August 2005

It is shown that standard finite‐element discretizations of second‐order differential equations (i.e. Galerkin and subdomain methods) using conforming linear elements may fail to approximate the original equation locally if the finite‐element grid is irregular or if subdomains are chosen improperly. This failure of local approximation can lead to spurious computational results when subdomain methods are used, but these difficulties can be averted by a judicious choice of subdomains. The conditions which the subdomains must satisfy in order for local approximation to hold are derived and used to construct an algorithm for choosing them properly. The relation of these local results to the global convergence properties of the Galerkin method is discussed.
Dynamic response of rigid strip‐foundations by a time‐domain boundary element method
Authors: Spyrakos C. C. , Beskos D. E.
Published: August 2005

The dynamic response of rigid strip‐foundations placed on or embedded in a homogeneous, isotropic, linear elastic half‐space under conditions of plane strain to either external forces or obliquely incident seismic waves of arbitrary time variation is numerically obtained. The above mixed boundary‐value problems are treated by the time domain boundary element method which is used in a step‐by‐step timewise fashion to provide the foundation response to a rectangular impulse. Numerical examples are presented in detail to demonstrate the use and importance of the proposed method. The method appears to be more advantageous than frequency domain techniques, because it provides the transient foundation response in a natural and direct way and can form the basis for extension to the non‐linear case.
Wave propagation in anisotropic layered media
Authors: Kausel Eduardo
Published: August 2005

The determination of the natural modes of wave propagation in an anisotropic layered medium requires the solution of a transcendental eigenvalue problem that is usually approached numerically with the aid of search techniques. Such computations require great effort. The method presented in this paper provides an alternate solution to this problem in terms of a quadratic eigenvalue problem involving tridiagonal matrices, for which the eigenvalues can be found with great speed and accuracy. The technique is then illustrated by means of an example involving a cross‐anisotropic Gibson solid.
A solution of the eigenproblem for undamped gyroscopic systems with the Lanczos algorithm
Authors: Bauchau O. A.
Published: August 2005

This paper presents an efficient numerical solution of the quadratic eigenproblem arising in the analysis of gyroscopic systems. Such problems are known to reduce to a generalized linear eigenproblem defined by two real non‐singular matrices, one symmetric and one skew‐symmetric. For this class of problems, the general Lanczos algorithm for unsymmetric matrices is shown to simplify considerably and yields an efficient solution of the problem. Full advantage can be taken of the sparsity of the matrices and of the specific nature of gyroscopic systems. Numerical examples are presented, which demonstrate the efficiency and accuracy of the solution procedure.
Finite element method for non‐linear forced vibrations of circular plates
Authors: Decha‐Umphai Kamolphan , Mei Chuh
Published: August 2005

Geometric non‐linearities for large amplitude free and forced vibrations of circular plates are investigated. In‐plane displacement and in‐plane inertia are included in the formulation. The finite element method is used. An harmonic force matrix for non‐linear forced vibration analysis is introduced and derived. Various out‐of‐plane and in‐plane boundary conditions are considered. The relations of amplitude and frequency ratio for different boundary conditions and various load conditions are presented.
A boundary element method for determining the effect of holes on the stress distribution around a crack
Authors: Ang W. T. , Clements D. L.
Published: August 2005

A boundary integral procedure is obtained for examining the effect of a finite number of holes on the crack tip stress intensity factors for a plane crack in a homogeneous anisotropic elastic material. Numerical results for specific examples involving a transversely isotropic material are given.
Comparison of finite element techniques for solidification problems
Authors: Dalhuijsen A. J. , Segal A.
Published: August 2005

The accuracies of the computed temperatures of a liquid in a corner region under freezing conditions are compared for various fixed‐grid finite element techniques using the analytical solution for this problem as a reference.

In the finite element formulation of the problem different time‐stepping schemes are compared: the implicit Euler‐backward algorithm combined with an iterative scheme and two three‐time‐level methods—the Lees algorithm and a Dupont algorithm, which are both applied as non‐iterative schemes.

Furthermore, different methods for handling the evolution of latent heat are examined: an approximation method suggested by Lemmon and one suggested by Del Giudice, both using the enthalpy formulation as well as a fictitious heat‐flow method presented by Rolph and Bathe.

Results of calculations performed with the consistent heat‐capacity matrix are compared with those performed with a lumped heat‐capacity matrix.
A time—memory trade‐off frontwidth reduction algorithm for finite element analysis
Authors: Webb J. P. , Froncioni A.
Published: August 2005

A frontwidth reduction algorithm is presented with an execution time which may be traded against its primary memory requirement, making it possible to optimize the performance of the algorithm on a particular computer. With an amount of primary memory O(E1/2), where E is the number of elements, the execution time of the algorithm is O(E3/2), in two or three dimensions. The algorithm has two parts: first, new node‐based data structures are derived from the conventional element list, then these structures are used to reorder the elements for reduced frontwidth.
A sixth‐order multiderivative method for two beam problems
Authors: Twizell E. H. , Tirmizi S. I. A.
Published: August 2005

A sixth‐order method is developed and analysed for the numerical solution of the linear fourth‐order boundary value problem for which the boundary conditions are given in terms of functional values and (i) first‐order derivatives (the clamped‐clamped beam problem) or (ii) second‐order derivatives (the simple‐simple beam problem). The method is tested on a number of problems from the literature.
Grid optimization for the boundary element method
Authors: Ingber Marc S. , Mitra Ambar K.
Published: August 2005

The quality of solution obtained using the boundary element method (BEM) is dependent on how the boundary is discretized. This is particularly true in domains of complex geometry. A rule for grid optimization for the BEM is derived on the bases of an asymptotic measure of the boundary element error that preserves the number of elements (degrees of freedom). Three example problems are provided to show the advantages of grid optimization in terms of accuracy and cost.
Concurrent Cholesky factorization of positive definite banded Hermitian‐matrices
Authors: Utku S. , Salama M. , Melosh R. J.
Published: August 2005

First the Cholesky factorization is extended to cover uniformly partitioned banded positive definite matrices of rank n which may be real symmetric or Hermitian. Then two stratagems are given for the use of the algorithm in concurrent machines where the number of processing elements is less than required to factor the matrix in as few serial steps as possible, and where uniformly high efficiency is expected from all processing elements. Expressions are given for the efficiency factor e appearing in the speed‐up expression g = eN, and these are specialized for the N node hypercube machine as a function of partition size s, the number N of processing elements of the hypercube machine, and the cost μ of interelement transmission relative to computation. It is shown that efficiency factor e is inversely proportional to μ/s, and that e is almost independent of N when N is large and μ/s = 0. The task is completed in n/s serial steps with no limit on n. The half bandwidth b of the matrix is 2Ns.
Parallel solution of closely coupled systems
Authors: Utku S. , Salama M.
Published: August 2005

The odd—even permutation and associated unitary transformations for reordering the matrix coefficient A is employed as a means of breaking the strong seriality which is characteristic of closely coupled systems. The nested dissection technique is also reviewed, and the equivalence between reordering A and dissecting its network is established. The effect of transforming A with odd—even permutation on its topology and the topology of its Cholesky factors is discussed. This leads to the construction of directed graphs showing the computational steps required for factoring A, their precedence relationships and their sequential and concurrent assignment to the available processors. Expressions for the speed‐up and efficiency of using N processors in parallel relative to the sequential use of a single processor are derived from the directed graph. Similar expressions are also derived when the number of available processors is fewer than required.
Performance of some SQP algorithms on structural design problems
Authors: Thanedar P. B. , Arora J. S. , Tseng C. H. , Lim O. K. , Park G. J.
Published: August 2005

This paper describes the basic ideas of sequential quadratic programming (SQP) algorithms for design optimization. There are two fundamental differences between the various algorithms: (i) the difinition of the QP subproblem solved at each iteration, and (ii) the descent function used during step size determination. The performances of the algorithms can change dramatically depending on how the two steps are executed. Numerical implementation details of various computational steps are discussed. Three programs based on SQP algorithms are used to solve 17 structural design problems having 7 to 96 design variables and 10 to 1051 performance constraints besides design variable bounds. Based on the performance of these programs, efficient procedures to execute various steps of the SQP methods are determined. It is concluded that the potential constraint strategy, where only a subset of the constraints is used to define the QP subproblem, is essential for large scale engineering design applications. With this strategy the SQP methods are quite robust and have great potential for routine application in engineering design.
Gradients of behaviour constraints and reanalysis via the integrated force method
Authors: Patnaik S. N. , Gallagher Richard H.
Published: August 2005

A matrix formulation of structural analysis, known as the integrated force method (IFM) is an attempt to establish an approach in which internal forces are the unknowns. Gradients of such behaviour constraints as stress, displacement and frequency can be obtained in closed form via the IFM. Closed form reanalysis formulae can also be generated. Computational experiments reported here show that reanalysis and gradient generation can be performed efficiently by this approach.
Curved beam elements with penalty relaxation
Authors: Tessler Alexander , Spiridigliozzi Luciano
Published: August 2005

Shallowly curved beam elements, including shear deformation and rotary inertia effects, are derived from Hamilton's variational principle. Different degree polynomials, labelled ‘anisoparametric’, are used to interpolate the kinematic variables, instead of uniform interpolations as in the conventional isoparametric procedure. This approach yields a correct representation of the bending strain and, importantly, the membrane and transverse shear strains. Consequently, the severe shortcomings of the exactly integrated isoparametric elements, characterized by excessively stiff solutions in the thin regime (a phenomenon often referred to as membrane and shear locking), are overcome. Uniform (isoparametric‐like) nodal patterns are achieved by explicitly enforcing higher‐degree penalty modes in the membrane and shear strains. This procedure preserves the compatibility of the kinematic field and the capability of the element to move rigidly without straining. Exact quadratures are used on all element matrices, producing a correct rank stiffness matrix, a consistent load vector and a consistent mass matrix. The elements suffer no limitations over the entire theoretical range of the slenderness ratio. For further enhancement and, particularly, in coarse‐mesh situations, an effective relaxation of penalty constraints at the local element level is introduced. This technique ensures a well‐conditioned stiffness matrix. Although the element penalty constraints are relaxed, the corresponding global structure constraints are enforced as is required by the analytic theory. Particular attention is given to the simplest element—a two‐node, six degree‐of‐freedom beam in which all strains are constant. Solutions to static and free vibration arch and ring problems are presented, demonstrating the exceptional modelling capabilities of this element.
Beam moments by a general finite difference formulation in cable‐assisted bridge analysis
Authors: Yiu P. K. A. , Brotton D. M.
Published: August 2005

This paper presents a general finite difference equation for the determination of beam bending moments. The scope of its application to the solution of cable‐assisted bridge problems is discussed. Its derivation and relation to previous difference formulations is explained. It is found that some previous formulations can be conveniently represented by the proposed general difference equation through the variation of a general difference coefficient. The accuracy of the proposed equation is illustrated with numerical examples.
Accurate finite difference approximations for the solution of parabolic partial differential equations by semi‐discretization
Authors: Huntley E.
Published: August 2005

The advantages in formulation and use of semi‐discretized approaches to the numerical solution of initial/boundary value problems are well known. The aim of this paper is to demonstrate that it is feasible to obtain accurate results even with a coarse spatial mesh. A method is developed which produces in a simple manner matrix representations for high‐order central difference operators. Dirichlet, Neumann and mixed boundary conditions are considered, both homogeneous and non‐homogeneous. It is shown in all cases that, for linear problems at least, there is no need to use a finer mesh than that dictated by the essential frequency content of the initial function data.
Formulation of numerical procedures for dynamic analysis of spinning structures
Authors: Gupta K. K.
Published: August 2005

This paper presents the descriptions of recently developed numerical algorithms that prove to be useful for the solution of the free vibration problem of spinning structures. First, a generalized procedure for the computation of nodal centrifugal forces in a finite element owing to any specified spin rate is derived in detail. This is followed by a description of an improved eigenproblem solution procedure that proves to be economical for the free vibration analysis of spinning structures. Numerical results are also presented which indicate the efficacy of the currently developed procedures.
Experiences in non‐linear analysis of temperature fields with finite elements
Authors: Stelzer J. F. , Welzel R.
Published: August 2005

This paper describes practical experiences in dealing with non‐linearities in temperature field calculations. First the use of the load vector for representing the heat radiation on the outer surfaces is reported. However, if there are also additional non‐linearities in the conductance matrix, then it may happen that because of possible counteractions of the non‐linearities no convergence is achieved, at least with the direct iteration. Therefore it is more appropriate to put the radiation terms into the conductance matrix where the other non‐linearities, such as temperature dependent thermal conductivity and natural convection also appear. It is shown how the Newton–Raphson iteration can be applied in an easy way. In the case of temperature dependent heat sources the load vector also needs to be considered. The appropriate special measures are described.
Numerical studies of buoyancy‐driven turbulent flow in a rectangular cavity
Authors: Thompson C. P. , Wilkes N. S. , Jones I. P.
Published: August 2005

In this paper some numerical predictions are presented for the buoyancy‐driven turbulent flow and heat transfer in a large cavity using both a prescribed eddy viscosity model and a k–ϵ model. The results from these models are in good agreement, although the calculations with the prescribed eddy viscosity model are considerably cheaper. The results also show the correct qualitative behaviour, but there are some differences between the predictions and the available experimental data. Possible reasons for these are discussed.
Least‐squares schemes for time integration of thermal problems
Authors: Kujawski J. , Wiberg N.‐E.
Published: August 2005

The paper proposes a generalized least‐squares time finite element procedure for analysis of transient field problems. It leads to a very general four parameter and two time level family of schemes. The free parameters are optimized in terms of accuracy, stability and oscillatory behaviour of the computational algorithm. In the case of linear problems an unconditionally stable scheme of fourth‐order accuracy is obtained. Special choices of the free parameters also give a number of well‐known time‐integration schemes, such as the Zienkiewicz–Lewis third‐order algorithm, the Goodrich scheme and standard schemes such as the Crank–Nicolson and backward fully implicit schemes. The main disadvantage of the higher order algorithm is a large bandwidth of the system matrix owing to the matrix terms of the form KTK. This drawback is reduced for special choices of the parameters by forming a product formula. Somewhat more computation is required than for popular schemes such as the generalized midpoint rule, depending on the prescribed variables. Numerical comparisons are made between the proposed higher order schemes and standard schemes.
Lanczos method for heat conduction analysis
Authors: Nour‐Omid Bahram
Published: August 2005

An analysis technique that uses the Lanczos vectors to construct a Reyleigh–Ritz solution for the transient heat conduction problem is presented. The Lanczos vectors are derived from the heat supply vector and incorporate all the information peculiar to a given problem. They are orthonormal with respect to the heat capacitance matrix and are generated with a minimum of computational effort. Furthermore, an error bound is derived that can be used to terminate the Lanczos algorithm as soon as the number of vectors required to obtain a desired degree of accuracy has been generated. A summary of the algorithm is presented and the accuracy and efficiency of the method is illustrated using a numerical example.
An enthalpy method for convection/diffusion phase change
Authors: Voller V. R. , Cross M. , Markatos N. C.
Published: August 2005

An enthalpy formulation for convection/diffusion phase change is developed. The essential feature of this formulation is that latent heat effects are isolated in a source term. This formulation is applicable to a general convection/diffusion phase change, i.e. it is valid in the cases of evolution of latent heat either at an isothermal temperature or over a temperature range. Before implementation of the enthalpy formulation, a technique is required to ensure that velocities predicted to be in a solid region actually take the value zero. Three alternative schemes for achieving this are presented.

The enthalpy formulation and velocity correction schemes are independent of the numerical technique. As an example of how the method can be implemented a control volume numerical discretization is chosen. This implementation is applied to two test problems: a solidification phase change in a cavity under conduction and the same phase change under conduction and natural convection. The natural convection problem is used to compare the performances of the various velocity correction schemes. The results of the problems are in good agreement with available analytical solutions and previous numerical solutions.
A simple error estimator and adaptive procedure for practical engineerng analysis
Authors: Zienkiewicz O. C. , Zhu J. Z.
Published: August 2005

A new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes.

The estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors. It can thus be combined with a full adaptive process of refinement or, more simply, provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials.

When combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable.

Estimates other than the energy norm have successfully been applied giving, for instance, a predetermined accuracy of stresses.
Evaluating eigenvalues for boundary value problems of heat conduction in rectangular and cylindrical co‐ordinate systems
Authors: Nowak A. , Białecki R. , Kurpisz K.
Published: August 2005

A method of evaluating eigenvalues for any combination of the first, second and third boundary conditions of heat conduction in a plate, a solid cylinder or a hollow cylinder is described. Some general relationships between eigenvalues are derived. Based on these relationships a universal algorithm is developed. A list of a package of computer subroutines is included.
Model‐size reduction for the analysis of symmetric structures with asymmetric boundary conditions
Authors: Noor Ahmed K. , Whitworth Sandra L.
Published: August 2005

A simple computational procedure is presented for reducing the size of the analysis model for a symmetric structure with asymmetric boundary conditions to that of the corresponding structure with symmetric boundary conditions. The procedure is based on approximating the asymmetric response of the structure by a linear combination of symmetric and antisymmetric global approximation vectors (or modes). The key elements of the procedure are (a) restructuring the governing finite element equations to delineate the contributions to the symmetric and antisymmetric components of the asymmetric response, (b) successive application of the finite element method and the classical Rayleigh–Ritz technique. The finite element method is first used to generate a few global approximation vectors (or modes). Then the amplitudes of these modes are computed by using the Rayleigh–Ritz technique.

A tracing parameter is introduced which identifies all the contributions to the antisymmetric response. The global approximation vectors are selected to be the solution corresponding to a zero value of the tracing parameter and the various‐order derivatives of the solution with respect to this parameter, evaluated at zero value of the parameter. The size of the analysis model used in generating the global approximation vectors is identical to that of the corresponding structure with symmetric boundary conditions.

The effectiveness of the computational procedure is demonstrated by means of numerical examples of linear static problems of shells, and its potential for solving non‐linear problems is discussed.
Time‐domain analysis of infinite fluid vibration
Authors: Sharan S. K.
Published: August 2005

A damper technique is proposed to model the effects of radiation damping in the finite element analysis of hydrodynamic pressures generated due to the vibration of a structure submerged in a compressible fluid. The proposed damper is suitable for a time‐domain analysis. It is assumed that the fluid is inviscid and its vibration is of small amplitude. The structure–fluid system is assumed to be two‐dimensional, and the fluid domain is considered to be unbounded along the direction of structural vibration. In the finite element analysis, pressure is assumed to be the nodal unknown, and the discretized equations of motion are solved by using a direct integration procedure. As a result of the analysis of several cases, the proposed damper is found to be very effective and efficient for a wide range of the period of excitation. The only exception is the case when the period of excitation is near the natural period of vibration of the fluid domain. For this case, the pressure becomes infinitely large and there is no requirement for any damper. The implementation of the proposed damper in the finite element analysis requires practically no extra computational effort. With the use of the suggested technique, an unbounded fluid domain may be truncated at a relatively very short distance from the structure, as compared to the distance required when the Sommerfeld damper is used. As a result of these advantages of the proposed technique, the cost of computation is greatly reduced.
An automatic design optimization procedure to minimize fillet bending stresses in involute spur gears
Authors: Andrews J. D. , Hearn E. J.
Published: August 2005

This paper presents the development of an automatic design algorithm for gears. The criterion on which the design performance is assessed is the maximum tensile stress induced in the gear tooth fillet under service conditions. Starting from an initial set of design parameters the stresses in the gear tooth fillet are calculated. The maximum value of the stress is then expressed as a function of the design variables. By minimizing this function, the objective function, subject to both equality and inequality constraints a new set of design parameters, is produced. Iterative application of the analysis and minimization stages forms a sequence of non‐linear optimization problems which converges to yield the optimal design.

Finite element techniques employed to calculate accurately the stresses in the gear tooth are again used to compute the design derivatives. This process is very economical, owing to its efficient reuse of the factorized stiffness matrix.

The algorithm is illustrated by its application to a spur gear tooth of involute profile. Both linear and non‐linear forms of the objective function are used and a comparison made between the solutions obtained.
Some improvements of the finite element solution for one‐dimensional problems
Authors: Gatica Gabriel N.
Published: August 2005

New algorithms which improve the finite element solution on a fixed mesh of the domain are developed. For this, objective functions which depend on a positive parameter to be determined are proposed. These functions lead to criteria which combine in several ways the residual of the solution, the approximation in the Euclidean norm of the value of the solution at the nodes, and the approximations in the energy and Hk norms. Some aspects of optimality are studied and, finally, numerical results are given.
A new element used in the non‐orthogonal boundary plate bending theory—an arbitrarily quadrilateral element
Authors: Ming Pan Guang , Fa Luo Song
Published: August 2005

This paper develops an arbitrarily quadrilateral element to analyse bending problems of plates with non‐orthogonal boundaries. A second‐order Jacobian matrix for the co‐ordinate transformation and an explicit form of its inverse matrix are described in detail. A shape function matrix [N] for the plate element of arbitrarily quadrilateral configuration, an equivalent load vector {R}, a strain matrix [B] and element stiffness matrix are given. Finally, four illustrated examples are given and the results of computation are compared with those from other analytical methods.
Robust, geometrically based, automatic two‐dimensional mesh generation
Authors: Baehmann Peggy L. , Wittchen Scott L. , Shephard Mark S. , Grice Kurt R. , Yerry Mark A.
Published: August 2005

A technical description of the algorithms employed in the modified quadtree mesh generator is given. Although the basis of the mesh generator is the same as the original version developed by Yerry and Shephard,1,2 the actual algorithms on which it is built have been entirely changed for the purpose of ensuring the robustness of the technique. As demonstrated in the paper the algorithmic changes made do ensure the robustness of the approach, but introduce additional algorithmic difficulties, the solutions of which are also presented. In addition to examples showing the capability of the mesh generator, the linear computational growth rate of the mesh generator is demonstrated.
Sparse quasi‐Newton LDU updates
Authors: Tewarson R. P. , Zhang Yin
Published: August 2005

Numerical solution of a given non‐linear algebraic system of equations by a quasi‐Newton type method requires updating the approximation to the Jacobian at each step. Two methods for large sparse systems are described. The approximation for the Jacobian is factored into an LDU form at the first step, then all the subsequent updates are made to L, D and U. Computational evidence is given exhibiting the relative efficiency of these methods over those currently available in the published literature.
The effect of compatibility conditions on optimal design of flexural structures
Authors: Kirsch U.
Published: August 2005

In most structural optimization problems the time‐consuming implicit analysis must be repeated many times during the solution process. To alleviate this difficulty, the implicit analysis equations are often neglected or replaced by explicit approximations.

Assuming the force method analysis formulation, optimal solutions of two problems are studied and compared:

The equilibrium linear programme (ELP), where only explicit equilibrium conditions are considered in the analysis model

The non‐linear programme (NLP), where the implicit compatibility conditions are included to obtain a complete formulation.

The effect of geometrical parameters on optimal solutions of both problems is investigated and geometries with particular properties are identified. The main observations that have been made are as follows:

Multiple optima of the ELP have been obtained for geometries where transitions in the set of active constraints at the ELP optimum occur. The NLP optimum for such a geometry might be included in the set of ELP optima.

Identical optimal solutions of the two problems have been found for geometries where transitions in the set of active constraints at the NLP optimum occur.

The explicit ELP can be solved instead of the implicit NLP for the two classes of transition geometries.

The relationships between optimal force distributions and objective function values for the two problems are studied and examples where compatibility conditions can be neglected are illustrated.
An adaptive boundary element method
Authors: Han P. S. , Olson M. D.
Published: August 2005

This paper explores the concept of moving singularities in the boundary element analysis. The singularities are placed on an auxiliary boundary which is located outside the domain of the problem and are allowed to move as part of the solution process. This results in a highly adaptive but non‐linear method. Examples involving the two‐ and three‐dimensional Laplace's equations are solved. Excellent agreement with exact solutions is obtained using a minimal number of singularities. Also, the trajectories of the singularity motion are plotted. The behaviour seen here is that, as the solution approaches convergence, the singularities exhibit a general trend of moving away from the domain of the problem.
LU decomposition of matrices with augmented dense constraints
Authors: Thomas P. D. , Brown R. A.
Published: August 2005

Sparse matrices composed of a central band and augmented dense rows and columns are becoming prevalent in the numerical solution of a large class of boundary and initial‐value problems. A Fortran Subroutine ARROW is presented for the LU decomposition and solution of linear equation systems with such a structure. The computational speed of the program is compared in MFLOPS (millions of floating point operations per second) to the LINPACK benchmark for the solution of a dense linear system and is found to be of comparable speed on both supercomputers and minicomputers. Use of the Basic Linear Algebra Subroutines (BLAS) available on most machines significantly enhances the speed of ARROW.
The ‘effective‐stress‐function’ algorithm for thermo‐elasto‐plasticity and creep
Authors: Kojić Miloš , Bathe Klaus‐Jürgen
Published: August 2005

An algorithm for stable and accurate computations of stresses in finite element thermo‐elastic‐plastic and creep analysis of metals is presented. The effective‐stress‐function algorithm solves the governing equations of the inelastic constitutive behaviour by calculating the zero of the appropriate effective‐stress‐function: a functional relationship which involves as unknown only the effective stress. The derivation of the effective‐stress‐function for thermo‐elasto‐plasticity conditions, including creep, for 2‐D and 3‐D analysis is presented, and the algorithmic steps of the stress solution are discussed. For use in the stiffness matrix a tangent material stress–strain relationship is evaluated consistent with the effective‐stress‐function algorithm. The solution of some demonstrative problems shows the effectiveness of the solution procedure.
Making the boundary element method less sensitive to changes or errors in the input data
Authors: Vable Madhukar
Published: August 2005

To solve a problem by the boundary element method requires a solution of an integral equation. By discretizing the boundary, the integral equation is reduced to a set of linear algebraic equations. If the matrix in algebraic equation is not diagonal dominant or more precisely, poorly conditioned, then the accuracy of the numerical solution becomes very sensitive to small changes in the input data. Small errors in the input data or changes in the mesh description can change the solution drastically.

In this paper a scheme is described which improves the condition of the matrix. Furthermore, it also reduces the sensitivity of the condition of the matrix to changes in the mesh description. The ideas described are applicable to any boundary element formulation. However, the numerical examples are from two‐dimensional elastostatics solved by the indirect version of the boundary element method.
On the condition ‘h normal to the wall’ in magnetic field problems
Authors: Bossavit Alaln
Published: August 2005

In finite element computations of magnetic fields, the boundary condition n × h = 0 is usually enforced on walls made of a material ofgreat permeability. We purport here to question the validity of this approximation. It is shown how this condition stems from a suitable passage to the limit (when the ratio of permeabilities μ0/μ tends to zero), provided a condition of topological nature (‘no loops’) is fulfilled. What to do when this condition does not hold is discussed.
Triangular finite element for analysis of thick laminated shells
Authors: Seide Paul , Chaudhuri Reaz A.
Published: August 2005

The development of a general curved triangular element based on an assumed displacement potential energy approach is presented for the analysis of arbitrarily laminated thick shells. The associated laminated shell theory assumes transverse inextensibility and layerwise constant shear angle. The present element is a quadratic triangle of C0‐type in the curvilinear co‐ordinate plane, which is then mapped onto a curved surface. Convergence of transverse displacement, moments, stresses and the effect of two Gauss quadrature schemes also form a part of the investigation. Examples of two laminated shell problems demonstrate the accuracy and efficiency of the present element. Comparison of the present LCST (layerwise constant shear‐angle theory) based solutions, with those based on the CST (constant shear‐angle theory) clearly demonstrates the superiority of the former over the latter, especially in the prediction of the distribution of the surface‐parallel displacements and stresses through the laminate thickness.
CONDIF: A modified central‐difference scheme for convective flows
Authors: Runchal Akshai K.
Published: August 2005

For most practical purposes, the central‐difference scheme (CDS) would be ideal only if it were unconditionally stable. It is a simple and second‐order scheme which is easy to implement. It does not introduce any second‐order ‘diffusion’ like truncation error. However, for grid Peclet numbers larger than 2, the CDS leads to over‐ and under‐shoots and is unstable. This paper presents a method, called CONDIF, which eliminates this undesirable feature of the CDS. It modifies the CDS by introducing a controlled amount of numerical diffusion based on the local gradients. The numerical diffusion can be adjusted to be negligibly low for most problems.

CONDIF has been used to solve a number of test problems which have been widely used for comparative study of numerical schemes in the published literature. For all these problems the CONDIF results are significantly more accurate than those obtained from the hybrid scheme when the Peclet number is very high (→∞) and the flow is at large angles (→45 degrees) to the grid. In general the computational effort for CONDIF is comparable (within 20 per cent) to that for the hybrid scheme. However, in one instance the rate of convergence was found to be significantly slower.
Effective hybrid hierarchical element generator for vibration analysis of plates
Authors: Alaylioglu Ayse , Alaylioglu H.
Published: August 2005

Cost effective high precision hybrid elements are presented in a hierarchical form for vibration analysis of plates. The elements are constructed by introducing computer installed hierarchical element generator algorithms which take advantage of the exact analytical treatment of the energy integrals automatically. Additional benefits accrue from: (i) the careful selection of assumed stresses and displacements satisfying finite element requirements, and maintaining the correct stiffness rank and invariance; (ii) the ability to reduce computational costs associated with various operations and the independence with respect to element geometries, which serve to build up a methodology to handle the generation of finite elements in any general purpose finite element system efficiently. Extensive numerical studies are made to predict the natural frequencies and harmonic response of plates with different boundary conditions. Accuracy of the solutions is assessed by comparison with the well established results. The noteworthy aspects of the applications are that the proposed elements are faster than those cited previously, and show a more favourable comparison to the analytical solutions for a wider range of vibration modes by maintaining rapid convergence characteristics.
The space–time Sinc‐Gallerkin method for parabolic problems
Authors: Lewis David L. , Lund John , Bowers Kenneth L.
Published: August 2005

This paper contains the formulation of a space–time Sinc‐Galerkin method for the numerical solution of the parabolic partial differential equation in one space dimension. The space–time adjective means that the Galerkin technique is employed simultaneously in time and space. Salient features of the method include: exponential rate of convergence, ease of assembly of the discrete system, a global approximation and the ability to handle singular problems. Two methods of solution for the discrete system are offered and numerical results for test problems, selected from the literature, are included.
Elastic strain effects during steady‐state rolling of visco‐elastic slabs
Authors: Shimazaki Yoji , Shiojima Takeo
Published: August 2005

The effect of elastic strains in the steady‐state visco‐elastic fluid flow under the rolling process is analysed quantitatively. Spurious oscillations in the stress deviator and pressure fields are precluded and the rate of convergence is improved when the Petrov–Galerkin method, which is based on a least‐square method, is introduced into the finite element equation for the constitutive law.
Isotropic plate elements with shear and normal strain deformations
Authors: Voyiadjis George Z. , Pecquet Robert W.
Published: August 2005

Plate‐bending finite elements are developed for thin and moderately thick plates. The element takes into account the influence of the transverse normal strain and the transverse shear effects. For the computation of the stresses, an additional plate element is developed to determine the inplane displacements and forces. The combined effects of the plate‐bending and inplane elements are used for the complete solution of the problem. Several examples are solved and compared with their corresponding exact solutions.