bluuebunny/crossref_metadata_2025 huggingface dataset:
Direct computation of Cauchy principal value integrals in advanced boundary elements
Authors: Guiggiani Massimo , Casalini Paolo
Published: August 2005

This paper presents a new method for the direct computation of strongly singular integrals existing in the Cauchy principal value sense. It can be usefully applied in the solution by the direct boundary element method of many different problems. Initially, some considerations are provided in order to summarize the state‐of‐the‐art on this issue. Then the features of the proposed method are reported. The procedure allows the direct calculation of Cauchy principal value integrals with first‐order singularity and it is applicable even in advanced boundary element methods employing high‐order elements. It requires only the use of standard Gaussian quadrature formulae plus the computation of a logarithmic term. Some examples show the effectiveness and efficiency of the procedure.
Plotting of contours in a natural way
Authors: Stelzer J. F. , Welzel R.
Published: August 2005

A simple and straightforward scheme for plotting contours is introduced. For any function the appropriate contours are traced on the paths of constant function values disclosing the information contained in the shape functions. The exactness of the contours matches that of the finite element analysis. Extreme value considerations must be carried out to decide whether contours appear in an element at all. Three co‐ordinate systems are dealt with: the local, the global and the screen co‐ordinate system, and the necessary mutual transformations. A short section deals with the treatment of contour discontinuities across element borders which frequently occur with stresses and heat fluxes. Finally, some examples are shown.
Practical experience with numerical simulations of plastic processes
Authors: Menges G. , Schwenzer C.
Published: August 2005

In recent years a number of programs have been developed for the calculation of plastic processes, but up to now the use of these programs in industrial practice has been limited to particular cases.

On the one hand, reasons for this can be found with the users who often, not knowing the theoretical principles and the restrictions of the calculations, obtain wrong results and thus generally doubt the applicability of the program.

On the other hand, in many cases calculations are only possible with low accuracy since information and data on the boundary conditions and the material behaviour are not available.

This paper describes the possibilities and limits of the numerical simulations as well as the compromises which have to be accepted.
Error estimation and adaptivity in flow formulation for forming problems
Authors: Zienkiewicz O. C. , Liu Y. C. , Huang G. C.
Published: August 2005

A very simple and reliable error estimator has recently been developed for problems of linear elasticity.1 This leads naturally toa predication of an h‐mesh refinement required for a given accuracy and with a suitable mesh generator2 for an efficient adaptive process. In this paper we extend the methodology developed previously to incompresible plastic flow of metals or polymers using the ‘flow formulation’ approach.3

The examples of application include steady state extrusion problems for which exact solutions are available and hence allow the efficiency of the error estimates to be tested as well as more complex problems of upsetting in which the mesh is updated. It is found that the estimator performs well under various circumstances and provides an economical adaptive process.
Finite‐element modelling of metal flow in three‐dimensional forming
Authors: Pillinger I. , Hartley P. , Sturgess C. E. N. , Rowe G. W.
Published: August 2005

A fully three‐dimensional elastic–plastic finite‐element program for the study of metalforming operations is described. Several applications are discussed, and the finite‐element (FE) predictions of metal flow, strain rate and temperature distributions, and the sites of initiation of ductile fracture are presented.
On the use of the boundary element method for elastoplastic, large deformation problems
Authors: Jin H. , Mattiasson K. , Runesson K. , Samuelsson A.
Published: August 2005

The boundary element method is applied to large deformation problems typical in industrial forming problems. A form of Betti's theorem relevant to an updated Lagrangian type of solution procedure is established as the basis for the boundary element formulation. Equilibrium iterations are needed in each load step. All non‐linear terms are moved to the right‐hand side and the boundary element method can be used. Cross‐anisotropy is assumed where Hill's yield criterion is adopted together with isotropic hardening. The investigation has shown that the boundary element method is applicable to this type of problem, giving good results within reasonable computer time.
Finite element simulation of hot isostatic pressing of metal powders
Authors: Abouaf M. , Chenot J. L. , Raisson G. , Bauduin P.
Published: August 2005

Rate form constitutive equations of elasto‐viscoplastic type are expressed in a manner similar to that of classical metals, but the viscoplastic part, which is no longer incompressible, is related to the rate of variation of porosity. An incremental and implicit algorithm has been implemented in a finite element program in order to simulate hot isostatic pressing of an Astroloy powder. Temperature distributions are shown to induce strong density variations in real parts during hot isostatic pressing (HIP). For one particular turbine disk, we tested the sensitivity of the final shape to different processing parameters. Computed final shapes compare well with experimental ones.
Numerical simulation of the densification behaviour of metal powder during hot isostatic pressing
Authors: Nohara Akira , Nakagawa Tomokazu , Soh Tadashi , Shinke Tohru
Published: August 2005

A numerical method is introduced which is based on metal flow theory and simulates the densification of metal powder during hot isostatic pressing.

The method, in which nodal velocities are variables, takes into consideration the dependence of flow stress on the temperature, strain rate and relative density of the material. The compressibility of porous metal as well as the incompressibility of fully dense metal are also taken into account.

Porous specimens of the superalloy MERL76 were subjected to uniaxial compression tests at high temperature in order to determine the parameters of the equations on which the method is based. The densification behaviour of a turbine disk and two cylindrical components was numerically simulated and compared with experimental results.
Punch stretching of sheet metal and differential equations on manifolds
Authors: Cavendish James C. , Wenner Michael L. , Burkardt John , Hall Charles A. , Rheinboldt Werner C.
Published: August 2005

The goal of mathematical modelling of sheet metal forming processes is to provide predictive tools for use in the design of stamping processes and the selection of sheet materials. Most current approaches to finite element modelling of large deformation, elastic–plastic sheet metal forming problems use a rate form of the virtual work (equilibrium) equations, and a single‐field finite element representation of the displacement components. Called the incremental method, this approach does not produce approximations which satisfy the discrete equilibrium equations at all times, and consequently it demands small time steps to insure stability and numerical accuracy. This paper describes a variant of the mixed method in which displacements, stresses, effective strain and pressures are all given separate finite element representations. The equilibrium equations in non‐rate form are discretized to produce a system of algebraic equations which are coupled with the constitutive equations and then integrated using state‐of‐the‐art numerical software. When used to model rate sensitive sheet materials in hydrostatic bulging, plane strain punch stretching and hemispherical punch stretching, the new approach proved to be between 6 and 26 times as fast as the old incremental method.
Numerical simulation of the transport processes in a heat treatment furnace
Authors: Jaluria Y.
Published: August 2005

The numerical simulation of the heat transfer and fluid processes in a furnace is carried out using finite difference techniques. A heat treatment furnace for the annealing of steel is considered as an example and a mathematical model of the system is obtained by simplifying the equations that govern the transport in the various components that constitute the system. The resulting set of coupled equations then governs the time‐dependent temperature distribution in the furnace. Because of the complexity of the full problem, the transport processes undergone by individual components of the furnace are simulated first, using appropriate boundary conditions that decouple them from the other components. These individual simulations are considered in terms of the underlying physical mechanisms for validation. Finally, these are combined to obtain the simulation of the full, coupled problem. The temperature cycle undergone by the material is computed, along with the time‐dependent temperature distributions in other components. A comparison with experimental data indicates close agreement, lending strong support to the numerical model. The relevance of these results to the design of the overall thermal system is also outlined.
Oscillatory convection in liquid metals in a horizontal temperature gradient
Authors: Winters K. H.
Published: August 2005

When growing semiconductor crystals from the melt, the free convection induced in the liquid phase by the imposed thermal boundary conditions can become periodic in time. Recently, the onset of this oscillatory convection has been simulated directly through solution of the time‐dependent governing equations for the particular case of an imposed horizontal temperature gradient. In this paper a new approach for detecting the oscillatory instability is presented in which the transition from steady to periodic flow is identified with a Hopf bifurcation in the solution of the steady equations. The critical Grashof number and frequency are predicted by solving an extended system of steady equations that locates exactly the Hopf bifurcation point, and the variation with aspect ratio and Prandtl number of the threshold for oscillations is obtained through continuation methods. By introducing a homotopy parameter into the boundary conditions the variation of the critical Grashof number is computed as the thermal and viscous conditions on the upper surface vary.
On the numerical modelling of heat transfer during solidification processes
Authors: Salcudean M. , Abdullah Z.
Published: August 2005

An extensive amount of work has been published on the treatment of heat transfer associated with phase change. A few recent advances are discussed in this paper with some emphasis on the phase change of metals. The apparent capacity, effective capacity, enthalpy, post iterative, source based and semi‐analytical methods are discussed and relative advantages and disadvantages of each are analysed. Recent developments in nodelling the flow during pouring and natural convection with applications of two widely used convective liffusive codes, the Los Alamos MAC and the Imperial College TEACH, are presented. An alternative tream function–vorticity approach is also discussed. Applications of these methods to turbulent convection during mould filling and continuous casting are presented. Areas of interest for further research work are identified as modelling of turbulence in liquid metals, flow through mushy regions and improvement of the performance of weak methods in multidimensional problems when the ratio of latent heat to sensible heat is large.
The numerical solution of 3‐D thermoelastic problems using only Laplace operator discretization methods
Authors: Fanelli Michele A.
Published: August 2005

The application of potential theory to the solution of theory of elasticity problems harks back to an old tradition, witness the Boussinesq treatment of the half‐space point loading.

In a preceding study by the authour,1 it has been shown how plane thermoelastic problems can be solved by using only numerical methods suited to treat the Laplace operator Δ2φ, hence by introducing only one d.o.f. or every node of the discretization.

In the present paper it is proposed to show how similar techniques can be used in a three‐dimensional case. The reduction achieved in the number of nodal unknowns is from three d.o.f. to one; thus it is hoped that—notwithstanding the greater complexity of the present treatment—a computational advantage can be gained over the direct approach, at least for discretization meshes of a few thousands of nodes (which is not unusual for 3‐D domains).

The basic formulation draws from fairly classical works2–6 and differs from more recent ones (e.g. Goodier as quoted in Reference 7) insofar as the particular solutions of the thermoelastic problems do not ntail, in the present approach, any integration, but rather are expressed directly in terms of the thermal field.
Treating exceptionally non‐linear temperature fields
Authors: Stelzer J. F. , Welzel R.
Published: August 2005

By some examples it is shown how to treat non‐everyday heat transfer problems which are characterized by temperature dependent heat sources and heat radiation.
Application of the finite element iterative method to the eigenvalue problem of a crack between dissimilar media
Authors: Barsoum Roshdy S.
Published: August 2005

The finite element iterative method is applied to the eigenvalue problem of a crack between dissimilar media. In this case the transfer matrix is non‐symmetric, which leads to complex eigenvalues. The singularity obtained agrees with the analytical results of r(−1/2+ie). The method of evaluating the eigenfunctions is general and can be applied to more complex cases of material and geometry, which are frequently encountered in composite materials. A powerful method for evaluating stress intensities in dissimilar media is given in the Appendix. The method is also reduced for homogeneous media to give stress intensity factors for modes I, II and III.
An integrated approach to the synthesis of geometrically non‐linear structures
Authors: Smaoui H. , Schmit L. A.
Published: August 2005

An integrated approach to the minimum weight design of geometrically non‐linear three dimensional truss structures with geometric imperfections, subject to inequality constraints on static displacements, stresses, local buckling and cross sectional areas, is investigated. The integrated structural synthesis problem involves design and response quantities as independent variables and equilibrium equations, describing the finite element model, as equality constraints. The non‐linear structural analysis and the optimization are thus merged together into a single process. A computer program developed to compute the contraint values and analytical gradients is coupled with a generalized reduced gradient algorithm to solve the integrated problem. Numerical results for a geometrically non‐linear shallow dome example problem are presented for various types of imperfections. Furthermore, it is found that the algorithm is capable of detecting and guarding against system as well as element elastic instability using equilibrium information only, that is, without imposing system and local buckling inequality constraints.
Post‐buckling analysis of shallow shells by the field‐boundary‐element method
Authors: Zhang J. D. , Atluri S. N.
Published: August 2005

The non‐linear field‐boundary‐element technique is applied to the analysis of snap‐through phenomena in thin shallow shells. The equilibrium path is traced by using the arc‐length method and the solution strategy is discussed in detail. The results show that, as compared to the approaches based on the popular symmetric‐variational Galerkin finite element formulation, the current approach based on an unsymmetric variational Petrov–Galerkin field‐boundary‐element formulation gives a faster convergence while using fewer degrees of freedom. The illustrative numerical examples deal with post‐buckling responses of several shallow shells with different geometries.
A numerical case study of a non‐Newtonian flow problem
Authors: Webster M. F. , Süli E. E. , Morton K. W.
Published: August 2005

This paper addresses some of the theoretical aspects involved in the numerical study of non‐Newtonian flow problems. We consider the second‐order Rivlin–Erickson constitutive model due to the simple differential form that emerges for the system of equations that govern the flow when expressed in stream function–vorticity variables. This model describes slightly elastic fluids that exhibit a constant viscosity behaviour. A steady two‐dimensional flow is studied through a planar contraction geometry.

An auxiliary variable is introduced into the problem formulation producing a non‐linear system of differential equations comprising two elliptic equations and one hyperbolic equation. This system is discretized by finite difference methods and the resulting system of non‐linear algebraic equations is solved iteratively by successive substitutions. The simple structure of this iteration permits a convergence analysis which is presented in Section 2. This analysis is performed prior to the spatial discretization and establishes the dependence of the iteration upon the material parameters. At the discrete linearized equation level a combination of inner iterations for elliptic equations and direct marching for the hyperbolic equation is used. The stability of the marching scheme is considered in Section 4.3 and a discussion on the results is given in Section 5.
A quadrilateral finite element including vertex rotations for plane elasticity analysis
Authors: Allman D. J.
Published: August 2005

A finite element of arbitrary quadrilateral shape is presented for plane elasticity analysis. The element is derived from triangular fields of compatible quadratic displacements with vertex connectors which include rotations. All rigid body movements and constant strain states are recovered exactly. Results from several test problems demonstrate that very good numerical accuracy is obtained for both displacements and stresses, even for quite coarse finite element meshes.
Analysis of infilled frames using A coupled finite element and boundary element solution scheme
Authors: Papia Maurizio
Published: August 2005

The behaviour of infilled frames subjected to horizontal loads is analysed by an iterative numerical procedure. The stiffness of the structural system is determined with variations in geometrical and mechanical characteristics.

The analysis is carried out utilizing the boundary element method (BEM) for the infill and opportunely dividing the frame into finite elements, so as to transform the mutual interactions of the two subsystems into stresses distributed along the boundary for the infill and into nodal actions for the frame.

This makes it possible to take into account the separation arising between the two substructures when mutual tensile stresses are involved.

At first, infills without openings are considered, using BEM with constant elements for two‐dimensional problems in elasticity.

The results are compared with those obtained using the simplified equivalent pin‐jointed strut model, which is very common in the literature.

Subsequently, using an analogous procedure, panels with openings are considered. For these systems, for which no satisfactory simplified models exist, the loss of stiffness in relation to the size of the opening is evaluated.
Finite element analysis of viscoelastic solids responding to periodic disturbances
Authors: Purushothaman N. , Moore I. D. , Heaton B. S.
Published: August 2005

A finite element analysis for viscoelastic solids responding to periodic disturbances travelling at constant speed is developed. The disturbance is decomposed into harmonic components using the Discrete Fourier Series, and the viscoelastic material response is determined using the Correspondence principle. The procedure is used to solve two and three dimensional contact problems to demonstrate its effectiveness.
The specified boundary stiffness/force SBSF method for finite element subregion analysis
Authors: Jara‐Almonte C. C. , Knight C. E.
Published: August 2005

It is difficult to analyse a large, complex structure in sufficient detail to obtain accurate results everywhere. One approach is simply to refine the whole structure model in the regions of interest. Another approach is to identify a subregion of the structure and develop a separate refined model of the subregion. It is difficult to assure accuracy in this subregion model because of uncertainties in specifying boundary conditions and loading from the whole structure model solution. In the current literature, three methods other than whole model refinement have been described to deal with this problem. These are called the specified boundary displacement method, the linear constraint method and the zooming method.

This paper describes a new approach to this problem of modelling subregions. This approach uses the stiffnesses and forces from the whole model solution at the nodes on the boundary of the sub‐region model. Accurate displacement and stress solutions are obtained with this method as it takes into account the interaction between the new stiffness of the subregion and the rest of the structure. This approach is similar to substructuring; however, the equations outside the subregion are discarded rather than condensed out, which results in much less computation effort.

Examples of the application of this method to the problem of a plate with a centre hole in tensile loading are presented. The results compared favourably with the results of the same problem solved using other methods, with significant improvement in accuracy over the specified boundary displacement method. Also presented are some results from design modification of the subregion which illustrate the potential of this method for redesign application.
Linear elastic contact problems using curved elements and including dynamic friction
Authors: Pascoe S. K. , Mottershead J. E.
Published: August 2005

A finite element solution for two‐dimensional contact between elastic bodies is presented. Equations of constraint and equilibrium governing ‘sticking’ and ‘sliding’ contact are imposed at mesh contact boundaries and incorporated in the usual displacement solution routine. The method allows sliding over curved bodies and the dynamic coefficient of friction between the bodies is included in a direct (non‐iterative) solution at each loading stage.

A progressive loading approach is implemented whereby full loads are initially applied. If a change in the contact boundary condition occurs, then scaling (reduction) of loads and displacements is undertaken until the first new boundary constraint becomes just operative again. The remainder of the load is then applied with a modified stiffness matrix and the process is repeated. Finally the mesh is fully loaded and the contact boundary conditions are satisfied.

Sample problems include comparison of the finite element results with classical solutions from the literature.
Two dimensional combined convection in vertical parallel plate ducts, including situations of flow reversal
Authors: Ingham D. B. , Keen D. J. , Heggs P. J.
Published: August 2005

Numerical solutions are presented for the problem of steady laminar combined convection flows in vertical parallel plate ducts. Axial diffusion is neglected in the analysis and the resulting governing equations, which are of a parabolic nature, are expressed in an implicit finite difference scheme using a vorticity‐stream function formulation and are solved using a marching technique. A constant wall temperature boundary condition is used and investigations are restricted to the case Pr = 0.72. A large range of values of the ratio Gr/Re is considered, −300 ≤ Gr/Re ≤ 70, and comparisons are made with the case of pure forced convection. For large values of the ratio |Gr/Re| reverse flow occurs in the duct. A modification to the standard marching technique is introduced and complete solutions are achieved for these situations for the first time. Results are presented in terms of velocity profiles, Nusselt numbers, friction factors and temperature distributions.
Field‐ and edge‐consistency synthesis of A 4‐noded quadrilateral plate bending element
Authors: Prathap G. , Somashekar B. R.
Published: August 2005

In this paper, we demonstrate the use of two conceptual principles, the field‐consistency requirement and the edge‐consistency requirement, as the basis for deriving a 4‐noded quadrilateral plate bending element based on Mindlin plate theory using Jacobean transformations only. The derivation is now free of the use of such devices as strain‐interpolation points and Hrennik off strain reference lines, etc., which have been the basis for many recent formulations of this element. The shear strain constraints are now consistently defined within the element domain, and ‘tangential’ shear strains are consistently matched at element boundaries so that there is no locking even under extreme distortion—e.g. even when two nodes are collapsed so that the quadrilateral becomes a triangle. Numerical experiments show that this synthesis produces an element that should be identical to other recent formulations of this element based on tensorial transformations or on shear constraint condensation on the edges, but now given a more complete and formal logical basis.
Reanalysis procedure for large structural systems
Authors: Noor Ahmed K. , Whitworth Sandra L.
Published: August 2005

An efficient procedure is presented for repetitive analysis of structures, with large numbers of degrees of freedom and design variables, as they are progressively modified during the automated optimum design process. The three key elements of the procedure are: (a) lumping of the large number of design variables into a single tracing parameter; (b) operator splitting or additive decomposition of the different arrays in the governing finite element equations of the modified structure into the corresponding arrays of the original structure plus correction terms; and (c) application of a reduction method through the successive use of the finite element method and the classical Bubnov‐Galerkin technique. The reanalysis procedure is applied to the linear static and free vibration problems of framed structures. Changes in both the sizing and shape (configuration) design variables are considered. For static problems the similarities between the proposed procedure and the preconditioned conjugate gradient technique are identified and are exploited to provide a physical meaning for the preconditioned residual vectors. The effectiveness of the proposed procedure is demonstrated by means of numerical examples.
Machine locking of degenerated thin shell elements
Authors: Briassoulis Demetres
Published: August 2005

The degenerated shell element is one of the most efficient elements for analysing shell structures. However, it is known to result in rather stiff models when used in thin element applications. The phenomena associated with this behaviour are known as locking phenomena.

This paper analyses the machine locking mechanism developed in thin to very thin Lagrangian and serendipity elements. The machine related locking phenomenon is distinguished from the shear and membrane locking phenomena. A remedy for the pure machine locking problem is developed for the two elements. The proposed remedy is based on the technique of the modified transverse shear modulus. It is also extended to control shear locking. The proposed technique is shown to completely eliminate machine locking. Also, it is shown to effectively alleviate stiffening effects due to the presence of spurious shear strain.
A shear‐deformable two‐layer plate element with interlayer slip
Authors: Latham C. T. , Toledano A. , Murakami H. , Seible F.
Published: August 2005

A new finite element for two‐layer plates with built‐in interlayer slip is developed. The new plate element is based on a new plate theory formulated using a new variational principle due to Reissner. The accuracy of the new element is investigated by applying it to the problem of a two‐layer plate with a linear slip law in cylindrical bending for which an exact elasticity solution exists. The comparison of the inplane response for several different values of interlayer shear stiffness and layer thickness ratios shows that the new element gives accurate results. The cylindrical bending of a two‐layer plate with bilinear interlayer slip law is investigated with the new element and the results obtained are compared with a plane strain analysis. Good agreement is obtained for in‐plane normal stresses and displacements. To further assess the simulation capabilities of the proposed element, a rectangular plate under concentrated load is analysed and the growth of slip failure regions is shown.
Analysis of superplastic metal forming by a finite element method
Authors: Chandra N.
Published: August 2005

A finite element velocity method for analysing the superplastic sheet metal forming process is presented. This method is developed from the principle of virtual work and is based on the use of isoparametric continuum elements. The large inelastic deformation of the superplastic material is modelled as the behaviour of an incompressible non‐linear viscous flow material. The contact and friction problem is solved by using the compatibility load step method, which is an extension of an earlier work. The finite element method is applied to selected problems to illustrate the applicability of the solution procedure.
An error‐based approach to complementary formulations of static field solutions
Authors: Rikabi J. , Bryant C. F. , Freeman E. M.
Published: August 2005

The constitutive relationship between fields in electromagnetic media is cast in error form. For well‐posed static problems, the error splits systematically into complementary functionals. Error minimization provides a universal variational principle that generates complementary and dual solution formulations. The error, which is wholly attributed to the numerical constraints in approximate solutions, is always defined and computable locally and globally.
Understanding and improving the reduced integration of Mindlin shell elements
Authors: Kamoulakos Argiris
Published: August 2005

The reason for looking in some depth at locking of Mindlin shell elements is introduced.

A simple example of straight beam locking is examined in detail.

The observations about reduced integration that emerge are subsequently unified with some other facts the writer has come across.

A simple curved beam element is then formulated and examined under inextensional bending. The performance of this element under reduced integration is predicted for both the ‘undistorted’ and ‘distorted’ configurations (see text) and for a selective‐very reduced integration scheme.

The previous predictions are then verified through the performance of an 8 node Mindlin shell element in a suitable cylindrical shell test.

The paper finally concludes with a set of important observations that so easily could be drawn from such a simple analysis.
A new boundary element formulation for two‐ and three‐dimensional thermoelasticity using particular integrals
Authors: Henry Donald P. , Banerjee Prasanta K.
Published: August 2005

New two‐ and three‐dimensional boundary element formulations are developed for steady‐state and transient uncoupled thermoelasticity. These new procedures differ from previous work in that additional surface or volume integration is not required to incorporate thermal loads in the analysis. Instead, thermal body forces are introduced in the boundary element system via particular integrals.

The present formulation is implemented in a general purpose, multi‐region system, and examples are presented to demonstrate the accuracy and versatility of the method.
Sizing design sensitivity analysis of non‐linear structural systems. Part II: Numerical method
Authors: Santos José L. T. , Choi Kyung K.
Published: August 2005

A numerical method is presented to carry out sizing design sensitivity calculations outside established finite element analysis codes, using postprocessing data only. Geometric as well as material non‐linearities are treated. To demonstrate the accuracy of the proposed method, numerical results are presented for structural systems with linear elastic material, large displacements, large rotations and small strains. A distributed parameter approach to structural design sensitivity analysis is used to retain the continuum elasticity formulation throughout the derivation of design sensitivity results. Using this approach and an adjoint variable method, design sensitivity computations are carried out. For structural performance functionals stress and displacement are considered.

It is shown that computations can be performed with the same computational effort as for sizing design sensitivity analysis of linear structural systems. Accurate design sensitivity results are obtained for both linear and non‐linear structural systems without the uncertainty of numerical accuracy and high cost associated with the selection of finite difference perturbations. Also, the method does not require differentiation of element stiffness and mass matrices in conventional finite element models.
Modified trapezoidal quadratures and accurate potential evaluation for numerical solutions of S.I.E's on the unit circle
Authors: Theotokoglou E. N.
Published: August 2005

Plane elasticity problems that can be reduced to singular integral equations over the unit circle are considered, the method of using a dislocation density function ω(σ) is adopted. First, the trapezoidal approximation is modified for the evaluation of the contour integral ∮ y [ω(σ)/(σ – z)]dσ at the complex field point z; extra correction terms for known singularities of ω(z) can easily be determined from the error expression. Based on the quadratures obtained, expressions for the accurate evaluation of the potentials are derived using their analyticity properties. The numerical techniques proposed are used for the solution of the problem of an infinite plate weakened by a circular hole and subjected to a dislocation.
A method for modelling microstructural material discontinuities in a finite element analysis
Authors: Accorsi M. L.
Published: August 2005

A new method is proposed for studying the effects of various microstructural material discontinuities, within a body, in a finite element analysis. The material discontinuities are accounted for by introducing a transformation strain in those regions. This formulation leads to two matrix equations; the first corresponds to the finite element analysis of the body without material discontinuities, and the second accounts for the microstructure. An important feature of the new method is that the first equation is solved only once, then the second equation can be solved repetitively for different microstructures. Thus, it is possible to study the effect of different microstructures within the body without reanalysing the entire body. It is expected that this method will be particularly useful in materials research to study the mechanisms that occur in materials at the microstructural level.

The transformation strain formulation is reviewed, and the matrix equations for the new method are derived. Several numerical examples are presented to illustrate the versatility of the new method.
A fractal patch test
Authors: Belytschko Ted , Lasry David
Published: August 2005

Element consistency is generally checked using the patch test on an element patch of finite size. This condition may in certain cases be too restrictive, and disqualifies elements that appear to be convergent. A method termed ‘fractal patch test’ is presented, in which the patch size is maintained constant while the distorted mesh is refined. Examples are given for four‐node quadrilateral elements used in plane stress and strain analysis, and for plate bending elements.
Geometric non‐linear substructuring for dynamics of flexible mechanical systems
Authors: Wu Shih‐Chin , Haug Edward J.
Published: August 2005

A substructure synthesis formulation is presented that permits use of established flexible multibody dynamic analysis computer codes to account for structural geometric non‐linear effects. Large relative displacement is permitted between points within bodies that undergo small strain elastic deformation. Components are divided into substructures, on each of which the theory of linear elasticity relative to a body reference frame is adequate to describe deformation and its coupling with system motion. Normal vibration and static correction deformation modes are used to account for elastic deformation within each substructure. Compatibility conditions are derived and imposed as constraint equations at boundary points between substructures. System equations of motion that include geometric non‐linear effects of large rotation, in terms of generalized co‐ordinates of a reference frame for each substructure and a set of deformation modes that are defined within the substructure, are assembled. The method is implemented in an industry standard flexible multibody dynamics code, with minimal modification. Use of the formulation is illustrated on the classical problem of a spinning beam with geometric stiffening and on a space structure that experiences large deformation.
Prediction of incidence effects on oscillating airfoil aerodynamics by a locally analytical method
Authors: Chiang Hsiao‐Wei D. , Fleeter Sanford
Published: August 2005

A complete mathematical model is formulated to analyse the effects of mean flow incidence angle on the unsteady aerodynamics of an oscillating airfoil in an incompressible flow field. A velocity potential formulation is utilized. The steady flow is independent of the unsteady flow field. However, the unsteady flow is coupled to the steady flow field through the boundary conditions on the oscillating airfoil. The numerical solution technique for both the steady and unsteady flow fields is based on a locally analytical method. In this method, analytical solutions are incorporated into the numerical technique, with the discrete algebraic equations which represent the differential flow field equations obtained from analytic solutions in individual local computational grid elements. This flow model and locally analytic numerical solution method are then verified through the excellent correlation obtained with the Theodorsen oscillating flat plate and Sears transverse gust classical solutions. The effects of mean flow incidence on the steady and oscillating airfoil aerodynamics are then investigated.
Paradox lost: Natural boundary conditions in the Ritz–Galerkin method
Authors: Storch Joel , Strang Gilbert
Published: August 2005

We present a simple calculation which seems at first to yield erroneous results when natural boundary conditions are not enforced. Our example is the static deformation of a cantilever beam under uniform loading. The corresponding variational principle is established and two applications of the Rayleigh–Ritz–Galerkin method are made.

The deflection is expanded in a series of cosines. The energy is rendered stationary subject to all boundary conditions, geometric and natural (they are imposed through Lagrange multipliers). The resulting function is exactly the Fourier cosine expansion of the true deflection–which is a fourth‐degree polynomial.

The deflection is again expanded in a cosine series but now the natural boundary conditions are ignored. The resulting series does not converge to the true solution.

Owing to the simplicity of the coefficients, exact summations are possible and the discrepancy is clear. The disagreement is large, and the object of the paper is to decide which method is right.
An assessment of flow oriented schemes for reducing ‘false diffusion’
Authors: Patel M. K. , Cross M. , Markatos N. C.
Published: August 2005

An important limiting factor in the accurate modelling of fluid flow problems is the numerical representation of the convection terms in the Navier‐Stokes equations. This paper reviews several approaches used to approximate the convection terms and reduce the so‐called false‐diffusion errors, within the context of finite‐difference and finite‐volume methods. Numerical errors are characterized as those due to discretization of the differential terms and those due to the influence of the multidimensional nature of the flow. Necessary criteria are identified which a numerical scheme must satisfy, if it is to be a candidate, at least in terms of accuracy and practicality, for the successful solution of the Navier‐Stokes equations. One of the criteria is the need of the scheme to account explicitly for the multidimensionality of the flow in the transport of scalar variables. All schemes except Raithby's SKEW approximation are deficient in this respect. However, the SKEW scheme does not satisfy some of the other criteria and does not always perform well. A new scheme called CUPID (Corner UPwInDing), which is based on the ideas of the SKEW scheme, yet obeys more of the criteria identified above, is described. The scheme is tested on a series of discriminating test problems which, the authors contend, demonstrate its potential for practical use in solving accurately the Navier‐Stokes equations.
Structural dynamics analysis using an unsymmetric block Lanczos algorithm
Authors: Kim Hyoung M. , Craig Roy R.
Published: August 2005

This paper describes the way in which an unsymmetric block Lanczos algorithm can be employed for approximate eigensolutions or dynamic response solutions for large systems having arbitrary damping and/or repeated (or closely spaced) eigenvalues. To reduce a square, unsymmetric matrix which may have repeated eigenvalues to block‐tridiagonal form, an unsymmetric block Lanczos algorithm is developed. Right and left Lanczos vectors, generated by the reduction process, are employed to define the reduced‐order model based on a discrete analytical model of general linear, time‐invariant dynamic systems. A special type of starting vector is also introduced, which automatically includes the static displacement. This reduced‐order model is employed to compute approximate eigenvalues/vectors and dynamic responses of the original system. To verify the proposed algorithm, examples of an 8‐DOF beam‐rotor system are provided for eigensolutions and dynamic responses due to step and random external forces. These examples showed promising results, indicating that the proposed algorithm is worthy of further study.
The theorems of geometric variation for finite element analysis
Authors: Topping B. H. V. , Kassim A. M. Abu
Published: August 2005

This paper illustrates the use of the theorems of geometric variation for the reanalysis of finite element structures. The formulations presented here form the basis for an efficient computer implementation.
Evaluation of the constant strain triangle with drilling rotations
Authors: Allman D. J.
Published: August 2005

The performance of the constant strain triangle with added ‘drilling’ rotations is evaluated by solving a variety of plane test problems. Numerical results confirm that the new element is always superior to the standard constant strain triangle and that it is of similar accuracy to the best available alternative triangular elements of other authors. It is, however, less accurate than some rectangular elements which include drilling rotations.
Some recent developments in numerical modelling of fracture toughness in brittle matrix composites
Authors: Hom C. L. , Mataga P. A. , McMeeking R. M.
Published: August 2005

Finite element analysis was used to study the fracture toughening of a ceramic by a stress induced dilatant transformation of second phase particles. The finite element method was based on a continuum theory which modelled the composite as subcritical material. Transient crack growth was simulated in the finite element mesh by a nodal release technique. The crack's remote tensile opening load was adjusted to maintain the near‐tip energy release rate at the level necessary for crack advance. The transformation zone surrounding the crack developed as the crack propagated through the composite. Resistance curves were computed from the analysis; and the results show that during crack advance maximum toughness is achieved before a steady state is reached.

The toughening effect of a crack‐bridging ductile phase in a brittle material may be predicted if ligament deformation is characterized. A plastically deforming ligament constrained by surrounding elastic matrix material is modelled using finite elements and the relevant toughness enhancement information extracted. Comparison is made to model experiments as well as to toughness measured for technologically important materials. The results suggest that debonding along the interface between the ligament and the matrix may enhance the toughening effect of a ductile phase.
Homogenization and pre‐integration techniques for multilayer composite and sandwich finite element models
Authors: Laschet G. , Jeusette J. P. , Beckers P.
Published: August 2005

The multilayer degenerated 3‐D shell element, for which the stiffness matrix is calculated by numerical integration ply by ply, gives accurate results even for moderately thick composite panels, but needs an important linear and non‐linear generation time when the number of plies is high (N > 10). In order to reduce this time some pre‐integration and homogenization techniques are investigated. Different pre‐integration methods have been reviewed and discussed. Then, two new homogenization methods, based on a series expansion of an equivalent Hooke matrix over the thickness, are presented. Next, simple numerical validation tests of these new methods are discussed, indicating necessary further investigations. Finally, a general discussion about the properties of the pre‐integration and homogenization techniques tends to define their respective application domain.
On refined computational models of composite laminates
Authors: Reddy J. N.
Published: August 2005

Finite element models of the continuum‐based theories and two‐dimensional plate/shell theories used in the analysis of composite laminates are reviewed. The classical and shear deformation theories up to the third‐order are presented in a single theory. Results of linear and non‐linear bending, natural vibration and stability of composite laminates are presented for various boundary conditions and lamination schemes. Computational modelling issues related to composite laminates, such as locking, symmetry considerations, boundary conditions, and geometric non‐linearity effects on displacements, buckling loads and frequencies are discussed. It is shown that the use of quarter plate models can introduce significant errors into the solution of certain laminates, the non‐linear effects are important even at small ratio of the transverse deflection to the thickness of antisymmetric laminates with pinned edges, and that the conventional eigenvalue approach for the determination of buckling loads of composite laminates can be overly conservative.
Analysis of viscoelastic structural elements in the frequency domain
Authors: De Wilde W. P. , Cardon A. H.
Published: August 2005

The paper presents a formulation in the frequency domain for the viscoelastic material behaviour of structural elements. This approach is equally valid for deterministic forces, but also in the case of probabilistic descriptions of those forces. The method takes advantage of readily available experimental data and shows how to introduce them in e.g. finite element formulations.