bluuebunny/crossref_metadata_2025 huggingface dataset:
Dynamic finite element analysis of viscoelastic composite plates in the time domain
Authors: Yi Sung , Hilton Harry H.
Published: August 2005

Numerical procedures for analysing transient responses of anisotropic viscoeiastic composite plates have been developed in the time domain using variational principles and a direct time integration method. Recursion relationships have been formulated in order to reduce computer storage substantially. The method developed herein requires only two previous lime solutions to evaluate the next time solution and the necessary storage is independent of the number of the time steps. Verification studies were conducted to evaluate the accuracy and convergence of the present numerical algorithm and they show that the numerical solutions agree well with analytical ones. As examples, dynamic transient responses of viscoeiastic composite plates subjected to unit step loads were calculated using the presently developed procedure.
Spatial stability and free vibration of shear flexible thin‐walled elastic beams. I: Analytical approach
Authors: Kim Moon‐Young , Chang Sung‐Pil , Kim Sung‐Bo
Published: August 2005

An improved formulation for spatial stability and free vibration analysis of thin‐walled elastic beams is presented by applying Hellinger–Reissner principle and introducing Vlasov's assumption. It includes shear deformation effects due to flexural shear and restrained warping stress, rotary inertia effects and bendirsg–torsional coupling effects due to unsymmetric cross sections. Closed‐form solutions for determining flexural–torsional buckling loads and natural frequencies of unsymmetric simply supported beam‐columns subjected to eccentric axial force are newiy derived and also, the tangent stiffness matrix and stability functions for symmetric thin‐walled beam elements subjected to axial force are presented. In a companion paper,26 these analytic solutions are compared with the finite element solutions according to the increase of shear deformation effects.
Charge iteration: A procedure for the finite element computation of unbounded electrical fields
Authors: Aiello G. , Alfonzetti S. , Coco S.
Published: August 2005

An iterative procedure is presented for the finite element computation of unbounded electrical fields created by voltaged conductors. The procedure is based on successive evaluations of the potential on a fictitious boundary enclosing all the conductors, according to the charge lying on their surface. The convergence of the procedure to the solution of the unbounded field problem is demonstrated. Indications for the optimal placement of the fictitious boundary are provided, also taking into account the accuracy of the solution. The way in which computational efficiency can be reached is also discussed. The main advantage of this procedure lies in its simplicity of implementation in the context of a standard FE code for bounded problems, because a very limited amount of additional software is required; moreover, 2‐D and axisymmetric versions can be implemented with minor changes from a suitable 3‐D one. Examples of application are given in order to illustrate the practical use of the procedure and to validate it by comparisons with available solutions.
A modal aeroelastic finite element analysis method for advanced turbomachinery stages
Authors: Jacquet‐Richardet G. , Henry R.
Published: August 2005

This paper presents a formulation suitable for the flutter analysis of rotating bladed assemblies. The blades are modelled using the finite element method. The aeroelastic displacements, expressed in terms of travelling wave co‐ordinates, are written as a linear combination of a number of undamped modes leading to a complex eigenvalues eigenvectors problem. The associated unsteady aerodynamic model is either two or three dimensional, leading for both cases to a consistent and flexible aeroelastic analysis method.

Three numerical applications are presented which illustrate the theory and emphasize the possible effects of blade cross‐section (chord) deformation on aeroelastic damping.
A fixed grid method for the numerical solution of phase change problems
Authors: Clavier L. , Arquis E. , Caltagirone J. P. , Gobin D.
Published: August 2005

A fixed grid method using an updated iterative implicit scheme is developed to solve one‐dimensional phase change problems. The temperature field is deduced from the resolution of the governing equations whose discretization takes into account the discontinuous variation of the temperature derivative at the phase change front. At each iteration an updated position of the moving front is found from the resolution of the energy conservation at the solid‐liquid interface. The accuracy of the proposed numerical method has been checked on three test problems.
Reduced minimization of Mindlin plate
Authors: Oak‐Key Min , Yong‐Woo Kim
Published: August 2005

The use of inconsistent displacement fields for Mindlin plate elements causes unmatched coefficients to appear in shear strain interpolations. The role of the numerical integration order in relation to these unmatched coefficients, and the existence of optimal stress points are explained through the use of the reduced minimization concept. This concept makes it possible to test whether assumed displacement fields or shear strain fields result in inconsistent strain fields which contain spurious constraints. We have included applications of reduced minimization to the conventional C0‐continuous elements, which employ reduced integration, and to quadrilateral Mindlin plate elements of Hinton and Huang to demonstrate how these elements alleviate shear locking.
Flow chart for numerical coupling of dynamic problems in elastic media
Authors: Martin Vincent , Peseux Bernard
Published: August 2005

From the starting point of dynamic coupling problems in the harmonic field, such as fluid/fluid, fluid/structure, fluid/structure/fluid problems, a single procedure capable of resolving the couplage is shown when only one domain is under study. The notions of impedance transfer familiar in analytical methods in the dynamic harmonic field are introduced into numerical methods especially when transferring the structure to the fluid. The proposed work goes beyond the problems themselves and the methods currently applied io solve them (modal series, integral method, finite element method) and suggests how the ideas developed here could simplify greatly the way more complex problems might be tackled.
Sensitivity analysis of the non‐linear dynamic viscoplastic response of 2‐D structures with respect to material parameters
Authors: Kulkarni Makarand , Noor Ahmed K.
Published: August 2005

A computational procedure is presented for evaluating the sensitivity coefficients of the viscoplastic response of structures subjected to dynamic loading. A state of plane stress is assumed to exist in the structure, a velocity strain‐Cauchy stress formulation is used, and the geometric non‐linearities arising from large strains are incorporated. The Jaumann rate is used as a frame indifferent stress rate. The material model is chosen to be isothermal viscoplasticity, and an associated flow rule is used with a von Mises effective stress. The equations of motion emanating from a finite element semi‐discretization are integrated using an explicit central difference scheme with an implicit stress update. The sensitivity coefficients are evaluated using a direct differentiation approach. Since the domain of integration is the current configuration, the sensitivity coefficients of the spatial derivatives of the shape functions must be included. Numerical results are presented for a thin plate with a central circular cutout subjected to an in‐plane compressive loading. The sensitivity coefficients are generated by evaluating the derivatives of the response quantities with respect to Young's modulus, and two of the material parameters characterizing the viscoplastic response. Time histories of the response and sensitivity coefficients, and spatial distributions at selected times are presented.
A Chebyshev collocation method for computing the eigenvalues of the Laplacian
Authors: Lu Ya Yan
Published: August 2005

Chebyshev collocation techniques are developed in this paper to compute the eigenvalues of the Laplacian based on a boundary integral formulation for two‐dimensional domains with piecewise smooth boundaries. Unlike the traditional domain methods (for example, the finite element method) which discretizes the eigenfunctions on the two‐dimensional domain, only a one‐dimensional function defined on the boundary is discretized. Global expansions in terms of Chebyshev polynomials are used in each smooth piece of the boundary to solve the integral equation. Comparing with the boundary element method, this method obtains higher accuracy for a smaller discretized matrix. Finally, an efficient algorithm for generating the discretized matrix (say, n × n) is developed that requires only O(n2 log n) operations.
Calculation of flows using three‐dimensional overlapping grids and multigrid methods
Authors: Tu J. Y. , Fuchs L.
Published: August 2005

A computational methodology combining overlapping grid techniques with multigrid methods has been developed for three‐dimensional flow calculations in or around complex geometries. The computational accuracy, efficiency and capability of the present approach are investigated in this paper. The incompressible Navier–Stokes equations are discretized using a finite volume method on a semi‐staggered grid. The discrete problem is solved by a multigrid algorithm. Some numerical examples are chosen for evaluating numerical accuracy: (a) a straight pipe for which the exact solution is known; (b) curved pipes where previous experimental and numerical data are available; (c) an axisymmetric sudden expansion. The performance of the multigrid method on overlapping grids is assessed. Several cases of flows in stationary and time‐dependent complex geometries are given to demonstrate the capability and the potential of the methods that we employ.
The dual boundary element formulation for elastoplastic fracture mechanics
Authors: Leitão V. , Aliabadi M. H. , Rooke D. P.
Published: August 2005

In this paper the extension of the dual boundary element method (DBEM) to the analysis of elastoplastic fracture mechanics (EPFM) problems is presented. The dual equations of the method are the displacement and the traction boundary integral equations. When the displacement equation is applied on one of the crack surfaces and the traction equation on the other, general mixed‐mode crack problems can be solved with a single‐region formulation. In order to avoid collocation at crack tips, crack kinks and crack‐edge corners, both crack surfaces are discretized with discontinuous quadratic boundary elements. The elastoplastic behaviour is modelled through the use of an approximation for the plastic component of the strain tensor on the region expected to yield. This region is discretized with internal quadratic, quadrilateral and/or triangular cells. This formulation was implemented for two‐dimensional domains only, although there is no theoretical or numerical limitation to its application to three‐dimensional ones. A centre‐cracked plate and a slant edge‐cracked plate subjected to tensile load are analysed and the results are compared with others available in the literature. J‐type integrals are calculated.
Finite element simulation of combined buoyancy and thermocapillary driven convection in open cavities
Authors: Chippada S. , Jue T. C. , Ramaswamy B.
Published: August 2005

Thermocapillary‐induced and buoyancy‐driven convective flows that commonly occur in crystal growth are numerically simulated using Galerkin finite element method. The physical domain comprises of a open cavity with aspect ratio one and differentially heated vertical walls. The top gas–melt interface is free to deform subject to 90° contact angle boundary conditions at the two vertical walls. The unsteady two‐dimensional Navier–Stokes equations are discretized in time using Chorin‐type splitting scheme and pressure is determined from the Poisson's equation. The free surface is taken to be resting on vertical spines and its evolution in time is determined from the kinematic free surface equation. The governing equations for heat and momentum are solved in the Arbitrary Lagrangian Eulerian frame of reference to handle the moving boundary. The influence of Grashof number, Marangoni number, Bond number, Ohnesorge number and Prandtl number on the flow field and heat transfer is investigated.
A new strategy for solving fluid‐structure problems
Authors: Sandberg Göran
Published: August 2005

A scheme for treating unsymmetrical coupled systems is outlined. Such systems occur naturally in connection with fluid–structure interaction, where an acoustic fluid is contained in an elastic structure. The discretization is performed by means of the finite element method, using displacement formulation in the structure and either pressure or displacement potential in the fluid. Based on the eigenvalues of each subdomain some simple steps give a standard eigenvalue problem. It might also be concluded that the unsymmetrical matrices have real eigenvalues.
A Galerkin least‐squares finite element method for the two‐dimensional Helmholtz equation
Authors: Thompson Lonny L. , Pinsky Peter M.
Published: August 2005

In this paper a Galerkin least‐squares (GLS) finite element method, in which residuals in least‐squares form are added to the standard Galerkin variational equation, is developed to solve the Helmholtz equation in two dimensions. An important feature of GLS methods is the introduction of a local mesh parameter that may be designed to provide accurate solutions with relatively coarse meshes. Previous work has accomplished this for the one‐dimensional Helmholtz equation using dispersion analysis. In this paper, the selection of the GLS mesh parameter for two dimensions is considered, and leads to elements that exhibit improved phase accuracy. For any given direction of wave propagation, an optimal GLS mesh parameter is determined using two‐dimensional Fourier analysis. In general problems, the direction of wave propagation will not be known a priori. In this case, an optimal GLS parameter is found which reduces phase error for all possible wave vector orientations over elements. The optimal GLS parameters are derived for both consistent and lumped mass approximations. Several numerical examples are given and the results compared with those obtained from the Galerkin method. The extension of GLS to higher‐order quadratic interpolations is also presented.
A comparison of the finite element and control volume numerical solution techniques applied to timber drying problems below the boiling point
Authors: Ferguson W. J. , Turner I. W.
Published: August 2005

Drying is a process which involves heat and mass transfer both inside the porous material, where a phase change in moisture occurs from the liquid to the gaseous state, and in the external boundary layer of the convected hot dry air, which heats the porous medium. The equations which govern this process consist of three tightly coupled, highly non‐linear partial differential equations for the unknown system variables of moisture content, temperature and pressure. Due to the inherently complex boundary conditions and intricate physical geometries in any practical drying problem, an analytical solution is not possible. In order to obtain a transient drying solution it is necessary to resort to a numerical technique. The numerical solution techniques which were employed in this research were the finite element method and the control volume method. The transient numerical results were compared and contrasted for two timber drying problems, first, at a dry bulb temperature of 50°C, and secondly, at 80°C, both cases being below the boiling point of water.
Condensation algorithms for the regular mesh substructuring
Authors: Xiao‐Lin Liu , Lam Y. C.
Published: August 2005

This paper describes and analyses the implementation of two static condensation algorithms used to exploit the regular patterns created by two schemes of regular mesh substructuring. Computer resources required by the implementation are assessed in terms of the number of multiplications, the total CPU time, the core and the disk storage requirements. It is discovered that, when implemented for the regular mesh substructuring, the newly proposed condensation algorithm is generally more cost effective than the one proposed by Han and Abel, and modified for the regular mesh substructuring by the present authors.
Large‐eddy simulations of plane impinging jets
Authors: Voke Peter R. , Gao Shian , Leslie David
Published: August 2005

The methods used for the study of the impingement of a thermally inhomogeneous plane jet on a solid plate, using large‐eddy simulation, are described. Innovative numerical methods have been introduced to produce the thermally inhomogeneous jet, to study the transmission of thermal fluctuations into the solid plate, and to prevent overshoot in the thermal computation. We have studied the impingement of water on a perspex plate in three different geometries, and also one case of the impingement of liquid sodium on a steel plate. Comparison is made with available experimental data and closure calculations.
The hybrid Laplace transform/finite element method applied to the quasi‐static and dynamic analysis of viscoelastic Timoshenko beams
Authors: Chen Tzer‐Ming
Published: August 2005

The quasi‐static and dynamic responses of a linear viscoelastic beam are solved numerically by using the hybrid Laplace transform/finite element method. In the analysis, the Timoshenko beam theory, which includes the transverse shear and rotatory inertia effect and conventional beam theory, are used to solve this problem. The temperature field is assumed to be constant and homogeneous and that the relaxation modulus has the form of the Prony series. In the hybrid method, the Laplace transform with respect to time is applied to the coupled equations and the finite element model is developed by applying Hamilton's variational principle without any integral transformation. The numerical results of quasi‐static and dynamic responses for the models of Maxwell fluid and three parameter solid types are presented and discussed.
Unit‐impulse response matrix of unbounded medium by finite‐element based forecasting
Authors: Song Chongmin , Wolf John P.
Published: August 2005

In this novel forecasting algorithm to calculate the unit‐impulse response matrix of an unbounded medium to analyse dynamic medium‐structure interaction in the time domain based on the substructure method, a finite‐element region adjacent to the structure–medium interface consisting of two or three rows only is introduced. Its exterior boundary is chosen to be similar to the interior boundary which coincides with the structure–medium interface. At similar boundaries of the unbounded medium the unit‐impulse response matrices are a function of the dimensionless time. The unit‐impulse response matrix at the exterior boundary can be forecasted from that at the interior boundary via this dimensionless time by taking advantage of the time delay of waves propagating between these two boundaries. The forecasting algorithm based solely on the finite‐element method is exact in the finite‐element sense in modelling the unbounded medium.
A parallel Monte‐Carlo finite‐element procedure for the analysis of multicomponent random media
Authors: Cruz Manuel E. , Patera Anthony T.
Published: August 2005

We present a new first‐principle framework for the prediction of effective properties and statistical correlation lengths for multicomponent random media. The methodology is based upon a variational hierarchical decomposition procedure which recasts the original multiscale problem as a sequence of three scale‐decoupled subproblems. The focus of the current paper is the computationally intensive mesoscale subproblem, which comprises: Monte‐Carlo acceptance–rejection sampling; domain generation and parallel partition based on Voronoi tesselation; parallel Delaunay mesh generation; homogenization‐theory formulation of the governing equations; finite‐element discretization; parallel iterative solution procedures; and implementation on message‐passing multicomputers, here the Intel iPSC/860 hypercube. Two (two‐dimensional) problems of practical importance are addressed: heat conduction in random fibrous composites, and creeping flow through random fibrous porous media.
A quadrilateral mixed finite element with two enhanced strain modes
Authors: Piltner R. , Taylor R. L.
Published: August 2005

An improved plane strain/stress element is derived using a Hu–Washizu variational formulation with bilinear displacement interpolation, seven strain and stress terms, and two enhanced strain modes. The number of unknowns of the four‐node element is increased from eight to ten degrees of freedom. For linear and non‐linear applications, the two unknowns associated with the enhanced strain terms can be eliminated by static condensation so that eight displacement degrees of freedom remain for the proposed element, which is denoted by QE2. The excellent performance of the proposed element is demonstrated using several linear and non‐linear examples.
A cubic‐spline boundary integral method for two‐dimensional free‐surface flow problems
Authors: Sen Debabrata
Published: August 2005

A cubic‐spline boundary‐integral‐equation method is presented for solution of free‐surface potential problems in two dimensions. In this scheme, it is possible to enforce a condition of conitnuous velocity through a geometric corner point in addition to the condition of continuous potential. Only the latter condition is possible in a linear‐element version of the model. A detailed description of the method is provided. Test computations concerning a propagating linear wave show a remarkable improvement in the performance of the method compared to the linear and constant‐element versions, both of which are prevalent in free‐surface hydrodynamics applications. What is particularly emphasized is the level of accuracy that can be achieved from the presented method both globally and locally at and near the corner.
A continuum‐based shell theory for non‐linear applications
Authors: Parisch H.
Published: August 2005

The paper introduces a non‐linear shell theory, which provides a complete three‐dimensional state of stress. Since the theory is derived from simple three‐dimensional continuum mechanics, it is very easy to understand. As an example, the theory is applied to quadrilateral shell elements, which provide only displacement degrees of freedom located at nodes on the outer surfaces and one degree of freedom at the middle surface. It is proposed to eliminate this degree of freedom on element level, so that the elements have the same layout as the equivalent brick elements, but have a better behaviour in bending, have stress resultants and are cheaper with respect to computational effort. The advantages with respect to implementation in a finite element program, as well as in special applications, are obvious. However, well‐known conditioning problems in thin shell applications must be expected. Therefore emphasis is put on this issue in the example problems. It is shown that the elements can give acceptable answers in engineering applications and offer a potential for material non‐linear applications, which will be considered in a forthcoming paper.
Variational principles and membrane finite elements with drilling rotations for geometrically non‐linear elasticity
Authors: Ibrahimbegović Adnan , Frey François
Published: August 2005

Variational principles for geometrically non‐linear continuum with independent rotation field are constructed based on the polar factorization theorem. Their regularized forms are next discussed suitable for the finite element implementation. The considerations are specialized to a two‐dimensional membrane problem, and standard isoparametric interpolations are used in order to construct membrane elements with drilling rotations. The elements are evaluated on a set of problems in geometrically non‐linear elastostatics.
Dual finite element analysis for plasticity–friction torsion of composite bar
Authors: Wieckowski Zdzislaw
Published: August 2005

The quasi‐static problem of torsion of an elastic–plastic, prismatic, composite bar is considered in the paper. The phenomenon of slip on the interfaces between the components of the bar is taken into account. The elastic–plastic behaviour of the material is described by the Prandtl‐Reuss constitutive relation. The slip on the interface is governed by the Coulomb friction law—it is assumed that there is no cohesion between components of the bar. The stresses normal to the interfaces are considered to be caused by shrinkage of the matrix of the bar or by external forces acting perpendicularly to its longitudinal axis. The problem is set in the dual variational forms and solved with the help of the finite element method. Two approximate kinematically and statically admissible solutions are obtained. The stress function is used for calculation of the second one. The iterative algorithms solving the problem and some numerical results are presented in the paper.
Improved assumed‐stress hybrid shell element with drilling degrees of freedom for linear stress, buckling and free vibration analyses
Authors: Rengarajan Govind , Aminpour Mohammad A. , Knight Norman F.
Published: August 2005

An improved 4‐node quadrilateral assumed‐stress hybrid shell element with drilling degrees of freedom is presented. The formulation is based on Hellinger–Reissner variational principle and the shape functions are formulated directly for the 4‐node element. The element has 12 membrane degrees of freedom and 12 bending degrees of freedom. It has 9 independent stress parameters to describe the membrane stress resultant field and 13 independent stress parameters to describe the moment and transverse shear stress resultant field. The formulation encompasses linear stress, linear buckling and linear free vibration problems. The element is validated with standard test cases and is shown to be robust. Numerical results are presented for linear stress, buckling, and free vibration analyses.
An element‐by‐element strategy for non‐linear shape optimization
Authors: Kohli H. S. , Carey G. F.
Published: August 2005

A strategy for the efficient solution of non‐linear shape optimization problems is developed. This strategy employs an integrated element‐by‐element approach to the solution of the governing partial differential equations, and, more particularly, to the computation of the necessary gradients of the objective function and constraints using an adjoint formulation. This proves to be a very efficient strategy and also is relatively easy to implement, because the local effect of design changes can be exploited. The method is tested with an application involving the design of the shape of electromagnet poles in order to obtain a desired field in the interpolar region.
On the numerical solution of non‐linear reservoir flow models with gravity
Authors: Hansen Ragnhild , Espedal Magne S.
Published: August 2005

Based on a two‐phase fluid model for immiscible displacement in a porous medium, we develop and analyse numerical solution techniques for certain non‐linear phenomena. Two different solution strategies for the treatment of gravity effects, which are non‐trivial to model by existing solution techniques and may be of great influence in many practical flow situations, are presented. The solution procedures are based on an operator‐splitting technique, combining the modified method of characteristics with finite element techniques and adaptive grid refinement.
A class of generalized mid‐point algorithms for the Gurson–Tvergaard material model
Authors: Zhang Z. L. , Niemi E.
Published: August 2005

We investigate the generalized mid‐point algorithms for the integration of elastoplastic constitutive equations for the pressure‐dependent Gurson–Tvergaard yield model. By exact linearization of the algorithms and decomposition of the stresses into hydrostatic and deviatoric parts, a formula for explicitly calculating the consistent tangent moduli with the generalized mid‐point algorithms is derived for the Gurson–Tvergaard model. The generalized mid‐point algorithms, together with the consistent tangent moduli, have been implemented into ABAQUS via the user material subroutine. An analytical solution of the Gurson–Tvergaard model for the plane strain tension case is given and the performances of the generalized mid‐point algorithms have been assessed for plane strain tension and hydrostatic tension problems and compared with the exact solutions. We find that, in the two problems considered, the generalized mid‐point algorithms give reasonably good accuracy even for the case using very large time increment steps, with the true mid‐point algorithm (α = 0·5) the most accurate one. Considering the extra non‐symmetrical property of the consistent tangent moduli of the algorithms with α < 1, the Euler backward algorithm (α = 1) is, perhaps, the best choice.
Numerical analysis of singularities in two‐dimensions part 1: Computation of eigenpairs
Authors: Yosibash Zohar , Szabó Barna
Published: August 2005

A numerical method is described for the computation of eigenpairs which characterize the exact solution of linear second‐order elliptic partial differential equations in two dimensions in the vicinity of singular points. The singularities may be caused by re‐entrant corners and abrupt changes in boundary conditions or material properties.

Such singularities are of great interest from the point of view of failure initiation: The eigenpairs characterize the straining modes and their amplitudes quantify the amount of energy residing in particular straining modes. For this reason, failure theories directly or indirectly involve the eigenpairs and their amplitudes.

This paper addresses the problem of determining the eigenpairs numerically on the basis of the Steklov formulation. Numerical results are presented for several cases. Importantly, the method is applicable to three‐dimensional cases.
Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation
Authors: Ihlenburg Frank , Babuška Ivo
Published: August 2005

When applying numerical methods for the computation of stationary waves from the Helmholtz equation, one obtains ‘numerical waves’ that are dispersive also in non‐dispersive media. The numerical wave displays a phase velocity that depends on the parameter k of the Helmholtz equation. In dispersion analysis, the phase difference between the exact and the numerical solutions is investigated. In this paper, the authors' recent result on the phase difference for one‐dimensional problems is numerically evaluated and discussed in the context of other work directed to this topic. It is then shown that previous error estimates in H1‐norm are of nondispersive character but hold for medium or high wavenumber on extremely refined mesh only. On the other hand, recently proven error estimates for constant resolution contain a pollution term. With certain assumptions on the exact solution, this term is of the order of the phase difference. Thus a link is established between the results of dispersion analysis and the results of numerical analysis. Throughout the paper, the presentation and discussion of theoretical results is accompanied by numerical evaluation of several model problems. Special attention is given to the performance of the Galerkin method with a higher order of polynomial approximation p(h‐p‐version).
A new triangular and tetrahedral basis for high‐order (hp) finite element methods
Authors: Sherwin Spencer J. , Karniadakis George Em
Published: August 2005

In this paper we describe the foundations of a new hierarchical modal basis suitable for high‐order (hp) finite element discretizations on unstructured meshes. It is based on a generalized tensor product of mixed‐weight Jacobi polynomials. The generalized tensor product property leads to a low operation count with the use of sum factorization techniques. Variable p‐order expansions in each element are readily implemented which is a crucial property for efficient adaptive discretizations. Numerical examples demonstrate the exponential convergence for smooth solutions and the ability of this formulation to handle easily very complex two‐ and three‐dmensional computational domains employing standard meshes.
New numerical approach in the analysis of a thin wire radiating over a lossy half‐space
Authors: Poljak Dragan
Published: August 2005

The finite element formulation for a half‐space electric field integral equation is described. Sommerfeld integrals, appearing in the kernel of the integral equation are calculated by means of exponential approximations. This approach shows advantages over the usual techniques. Obtained results are compared with other results available.
hp‐version discontinuous Galerkin methods for hyperbolic conservation laws: A parallel adaptive strategy
Authors: Bey Kim S. , Patra Abani , Oden J. Tinsley
Published: August 2005

This paper describes a parallel algorithm based on discontinuous hp‐finite element approximations of linear, scalar, hyperbolic conservation laws. The paper focuses on the development of an effective parallel adaptive strategy for such problems. Numerical experiments suggest that these techniques are highly parallelizable and exponentially convergent, thereby yielding efficiency many times superior to conventional schemes for hyperbolic problems.
A consistent penalty method for contact between a deforming viscoplastic workpiece and a rigid tool
Authors: Dawson Paul R. , Boyce Donald E. , Eggert Geoffrey M. , Beaudoin Armand J.
Published: August 2005

A methodology is presented for enforcing a contact constraint between a viscoplastic workpiece and the tooling which deforms it. The algorithm is based on a consistent penalty approach wherein an elemental discretization permits elimination of contact tractions prior to solution for the velocity field. When written in rate form, the new algorithm allows contact constraints to be embedded in a viscoplastic formulation without altering the structure of the global matrix equation. Examples are presented to illustrate its use in metal forming simulations.
A new 3D finite element for adaptive h‐refinement in 1‐irregular meshes
Authors: Morton D. J. , Tyler J. M. , Dorroh J. R.
Published: August 2005

A new finite element, viable for use in the three‐dimensional simulation of transient physical processes with sharply varying solutions, is presented. The element is intended to function in adaptive h‐refinement schemes as a versatile transition between regions of different refinement levels, ensuring interelement continuity by constructing a piecewise linear solution at the element boundaries, and retaining all degrees of freedom in the solution phase. Construction of the element shape functions is described, and a numerical example is presented which illustrates the advantages of using such an element in an adaptive refinement problem. The new element can be used in moving‐front problems, such as those found in reservoir engineering and groundwater flow applications.
Optimization of truss topology using tabu search
Authors: Bennage W. A. , Dhingra A. K.
Published: August 2005

A design procedure for integrating topological considerations in the framework of structural optimization is presented. The proposed approach is capable of considering multiple load conditions, stress, displacement and local/global buckling constraints, and multiple objective functions in the problem formulation. Further, since the proposed method permits members to be added to or deleted from an existing topology and the topology is not defined by member areas, the difficulty of not being able to reach singular optima is also avoided. These objectives are accomplished using a discrete optimization procedure which uses 0–1 topological variables to optimize alternate designs. Since the topological variables are discrete in nature and the member cross‐sections are assumed to be continuous, the topological optimization problem has mixed discrete‐continuous variables. This non‐linear programming problem is solved using a memory‐based combinatorial optimization technique known as tabu search. Numerical results obtained using tabu search for single and multiobjective topological optimization of truss structures are presented. To model the multiple objective functions in the problem formulation, a cooperative game theoretic approach is used. The results indicate that the optimum topologies obtained using tabu search compare favourably, and in some instances, outperform the results obtained using the ground–structure approach. However, this improvement occurs at the expense of a significant increase in computational burden owing to the fact that the proposed approach necessitates that the geometry of each trial topology be optimized.
An analysis of alternating‐direction methods for parabolic equations
Authors: Celia Michael A. , Plnder George F.
Published: August 2005

Alternating‐direction solution procedures for parabolic partial differential equations can be developed using finite‐difference, finite‐element, and collocation approximations in space. Each of these methods derives from a common alternating‐direction formulation. Furthermore, each method leads to an O[(Δt)2] error which is in addition to the discretization error associated with standard multidimensional solutions. However, when dealing with equations having spatially varying coefficients, some alternating‐direction formulations lead to yet other errors which are O(Δt). These latter errors, and thus the accuracy of the method, depend on the structure of the mass matrix associated with the approximating method.
A network model for two‐fluid, two‐phase flow
Authors: Porsching T. A.
Published: August 2005

A systematic derivation is given of two‐fluid conservation laws on an arbitrary network. These laws are shown to conserve the mass and total energy of the mixture in the network. Numerical simulations are presented to illustrate their use.
Stability of penalty finite‐element methods for nonconforming problems
Authors: Carey Graham F. , Utku Mehmet
Published: August 2005

Penalty methods have been proposed as a viable method for enforcing interelement continuity constraints on nonconforming elements. Particularly for fourth‐order problems in which C1‐continuity leads to elements of high degree or complex composite elements, the use of penalty methods to enforce the C1‐continuity constraint appears promising. In this study we demonstrate equivalence of the finite‐element penalty method to a hybrid method and provide a stability analysis which implies that the penalty method is stable only if reduced integration of a certain order is used. Numerical experiments confirm that the penalty method fails if this condition is not met.
A family of numerical methods for diffusion and reaction–diffusion equations
Authors: Twizell E. H. , Khaliq A. Q. M.
Published: August 2005

A family of methods is developed for the numerical solution of second‐order parabolic partial differential equations in one space dimension. The methods are second‐, third‐, or fourth‐order accurate in time; five of them are seen to be L0‐stable in the sense of Gourlay and Morris, while the sixth is seen to be A0‐stable, The methods are tested on four problems from the literature, three diffusion problems and one reaction–diffusion problem.
The effects of symmetric/asymmetric boundary conditions on the flow of an internally heated fluid
Authors: Carlucci L. N. , Cheung I.
Published: August 2005

The two‐dimensional flow of an internally heated fluid in a circular vessel has been investigated by using a finite‐control‐volume numerical model. It has been found that, when symmetry constraints are imposed along the vertical midplane of the vessel, two distinct flow patterns can he predicted for the same operating conditions, a jet‐momentum‐dominated pattern and buoyancy‐dominated pattern. These patterns occur in an operating region where momentum and buoyancy forces are of comparable magnitude. However, when the symmetry constraints are removed and the full vessel cross section is modeled, only the buoyancy‐dominated pattern is observed. Results for the two cases are described and possible reasons for the differences in behavior are discussed.
Fourth‐order accurate one‐step integration methods with large imaginary stability limits
Authors: Kinnmark Ingemar P. E. , Gray William G.
Published: August 2005

One‐step integration methods of fourth‐order accuracy using an odd number of function evaluations K, to solve dy/dt = A · y, are proposed. These methods have an imaginary stability limit \documentclass{article}\pagestyle{empty}\begin{document}$ S_{1\;} = \sqrt {(K - 1)^2 - 4} $\end{document}. In the case K = 5 the Kutta‐Merson method results.
Variational grid generation
Authors: Steinberg Stanly , Roache Patrick J.
Published: August 2005

Recently, variational methods have been used to numerically generate grids on geometometric objects such as plane regions, volumes, and surfaces. This article presents a new method of determining variational problems that can be used to control such properties of the grid as the spacing of the points, area or volume of the cells, and the angles between the grid lines. The methods are applied to curves, surfaces, and volumes in three‐dimensional space; then segments, plane curves, and plane regions appear as special cases of the general discussion. The methods used here are simpler and clearer and provide more direct control over the grid than methods that appear elsewhere. The methods are applicable to any simply connected region or any region that can be made simply connected by inserting artificial boundaries. The methods also generalize easily to solution‐adaptive methods.

An important ingredient in our method is the notion of a reference grid. A reference grid is defined on a region that is simpler, but analogous to, the geometric object on which a grid is desired. Variational methods are then used to transfer the reference grid to the geometric object. This gives simple and precise control of the local properties of the grid.
Exact solutions to difference equation models of Burgers' equation
Authors: Mickens Ronald E.
Published: August 2005

This article gives exact solutions to several difference equation models of Burgers' equation. The particular cases considered correspond to the diffusion‐free, nonlinear steady‐state and linear steady‐state situations.
Boundary element solution of the two‐dimensional groundwater flow equation with stochastic free surface boundary condition
Authors: Serrano Sergio E. , Unny T. E.
Published: August 2005

An innovative approach to the approximate solution of stochastic partial differential equations in groundwater flow is presented. The method uses a formulation of the Ito's lemma in Hilbert spaces to derive partial differential equations satisfying the moments of the solution process. Since the moments equations are deterministic, they could be solved by any analytical or numerical method existing in the literature. This permits the analysis and solution of stochastic partial differential equations occurring in two‐dimensional or three‐dimensional domains of any geometrical shape. The method is tested for the first time in the present paper through a practical application in a sandy phreatic aquifer at the Chalk River Nuclear Laboratories, Ontario, Canada. The equation solved is the two‐dimensional LaPlace equation with a dynamic, randomly perturbed, free surface boundary condition. The moments equations are derived and solved by using the boundary integral equation method. A comparison is made with a previous analytical solution obtained by applying the randomly forced one‐dimensional Boussinesq equation, and some observations on modeling procedures are given.
Discontinuities across element interfaces in the global element method
Authors: Cook R. , Delves L. M. , Phillips C.
Published: August 2005

In this article we extend the global element method formalism (Delves and Hall, 1979) for two‐dimensional partial differential equations to admit the possibility of solution discontinuities across element interfaces. This extension has particular relevance to problems involving sinks and sources, such as occur in oil well or groundwater management.
Efficient boundary element methods for three‐dimensional exterior viscous flows
Authors: Hebeker Friecrich‐Karl
Published: August 2005

An efficient numerical algorithm for partial differential equations in complicated three‐dimensional (3‐D) geometries is developed in case of exterior viscous flows. The algorithm essentially consists of a boundary element method, where the resulting algebraic system is solved with a multigrid procedure. Our investigation covers and exact mathematical foundation, a detailed analysis of the discretization error, and some numerical tests with reasonable physical examples.
A Petrov‐Galerkin method and boundary approximations for hyperbolic initial boundary‐value problems
Authors: Jamieson S. M. , Sloan D. M.
Published: August 2005

The Petrov‐Galerkin method is used to construct a semidiscrete approximation for hyperbolic systems of equations in one space dimension. Stability is analyzed for the Cauchy problem and for an initial and boundary‐value problem with positive and negative characeristic speeds of unequal magnitude. Numerical experiments are conducted on a linear problem to support the stability analysis and to compare the accuracies of various boundary approximations. Experiments on a nonlinear problem check the viability of the best boundary method.
An extension of the one‐dimensional von Rosenberg finite difference scheme to multidimensional problems of the convection dispersion equation
Authors: El‐Ageli Muktar Ali , Chierici Gianluigi
Published: August 2005

The Crank Nicolson implicit form of the finite difference analogue of the convection‐dispersion equation does not yield stable solutions when the dispersion coefficient of the medium D is too small compared with the injected fluid velocity V—i.e, 0 < (R = 2D/V ΔX) < 1. Von Rosenberg developed a simple, highly accurate and stable finite difference scheme for the one‐dimensional case of constant velocity and dispersion coefficient for 0 < R < 1. El‐Ageli extended von Rosenberg's method to the multilayered case of varying velocity and dispersion coefficient. This paper goes a step further and extends von Rosenberg's scheme to the linear multidimensional system with constant velocities and dispersion coeffcients when: 0 < (Rx, Ry,…,) < 1. When this new scheme is compared with the conventional Crank‐Nicolson method, it is found that even though there are some restrictions to the spacial grid spacings in relation to the time step, the computational effort is much less and the results obtained are stable.
On the finite element approximation of incompressible flows of an electrically conducting fluid
Authors: Peterson Janet S.
Published: August 2005

We consider the finite element approximation of incompressible flows field of an electrically conducing fluid in the presence of a magnetic where it is assumed that this field is prescribed. A weak form is chosen that is similar in some respects to a weak form used by many authors for the Navier‐Stokes equations. Existence and uniqueness results are presented for the weak problem. A finite element Algorithm is given for the approximate solution of the weak problem and error estimates are derived.
Asymptotic stability of the numerical solution for an integrodifferential reaction‐diffusion system
Authors: Galeone L. , Mastroserio C. , Montrone M.
Published: August 2005

This paper presents the study of the numerical solution of a reaction‐diffusion system involving a reaction term of integral type arising from biological models. By means of a monotone approach we introduce upper and lower solutions and then we show the existence and the asymptotic behavior of nonnegative numerical solutions. To this end, we require the positivity of the numerical scheme and so we can use some properties of positive and M‐matrices. Finally we give some sufficient conditions to verify the asymptotic stability of the numerical solution.