bluuebunny/crossref_metadata_2025 huggingface dataset:
An orthogonal residual procedure for non‐linear finite element equations
Authors: Krenk Steen
Published: August 2005

A general and robust solution procedure for non‐linear finite element equations with limit points is developed. At each equilibrium iteration the magnitude of the load is adjusted such that the residual force is orthogonal to the current displacement increment from the last equilibrium state. The method implements the physical condition that the orthogonal residual force will neither increase nor decrease the magnitude of the current displacement increment vector. The orthogonality condition is formulated directly in terms of conjugate variables and therefore does not contain any scaling parameters. Passage of load and displacement limit points is discussed as well as the relation to line search, minimum residual, and are‐length methods. The method is illustrated by two examples.
Large strain constitutive modelling and computation for isotropic, creep elastoplastic damage solids
Authors: Li Xikui
Published: August 2005

A three‐dimensional fully coupled creep elastoplastic damage model at finite strain for isotropic non‐linear material is developed. The model is based on the thermodynamics of an irreversible process and the internal state variable theory. A hyperelastic form of stress–strain constitutive relation in conjunction with the multiplicative decomposition of the deformation gradient into elastic and inelastic parts is employed. The pressure‐dependent plasticity with strain hardening and the damage model with two damage internal variables are particularly considered. The rounding of stress–strain curves appearing in cycling loading is reproduced by introduction of the creep mechanism into the model. A numerical integration procedure for the coupled constitutive equations with three hierarchical phases is proposed. A consistent tangent matrix with consideration of the fully coupled effects at finite strain is derived. Numerical examples are tested to demonstrate the capability and performance of the present model at large strain.
A numerical study of mechanisms of accretion for rotating snow sleeves evolving on conductor rods
Authors: Poots G. , Skelton P. L. I.
Published: August 2005

It is the purpose of this paper to construct a mathematical/computational model for the accretion of snow on rotating snow sleeves attached to conductor rods, and to consider some of the limitations in the use of the assumption of perfect cylindrical‐sleeve growth, as currently employed to provide estimates of snow‐loading intensity.

For purposes of illustration, three rotating snow accretion regimes are considered, in which the direction of the airflow is always taken as perpendicular to the axis of the conductor rod. Firstly, a horizontal conductor rod is free to rotate and is initially bare. Rotation of the system (conductor rod and accreted snow) is initiated by snow torque, due to eccentric snow loading on the windward side of the conductor, and is driven by aerodynamic torque generated by the windflow past the evolving, rotating, asymmetrical sleeve growth. Secondly, a vertical conductor rod is free to rotate and the bare rod is given an initial small angular velocity; the rotation of the accreting system is then driven by aerodynamic torque alone. Thirdly, a horizontal rod is fixed and is initially covered by an annulus of snow, supported by a thin layer of (unfrozen) water existing between the conductor and the snow sleeve. In this case, the rotation of the snow sleeve is due to the combined snow and aerodynamic torques and is now retarded by the viscous torque due to the rotating thin film of water.

Asymptotic solutions obtained for small accretion time supplement numerical results derived from the computational models. Numerical results on these rotating non‐periodic snow accretion regimes are presented as a function of the wind speed, snow precipitation rate and diameter of the conductor rod.
Three‐dimensional parallel unstructured grid generation
Authors: Shostko Alexander , Löhner Rainald
Published: August 2005

An algorithm for the parallel generation of 3‐D unstructured grids is presented. The technique is an extension of the algorithm presented in Reference 21 for the 2‐D case. The method uses a background grid as the means to separate spatially different regions, enabling the concurrent, parallel generation of elements in different domains and interdomain regions. The parallel 3‐D grid generator was implemented and tested on the INTEL hypercube and Touchstone Delta parallel computers. The results obtained demonstrate the effectiveness of the algorithm developed. The methodology is applicable to the parallel implementation of a wide range of problems that are, in principle, scalar by nature, and do not lend themselves to SIMD parallelization.
Mesh partitioning for implicit computations via iterative domain decomposition: Impact and optimization of the subdomain aspect ratio
Authors: Farhat Charbel , Maman Nathan , Brown Gregory W.
Published: August 2005

Optimal domain decomposition methods have emerged as powerful iterative algorithms for parallel implicit computations. Their key preprocessing step is mesh partitioning, where research has focused so far on the automatic generation of load‐balanced subdomains with minimum interface nodes. In this paper, we emphasize the importance of the subdomain aspect ratio as a mesh partitioning factor, and highlight its impact on the convergence rate of an optimal domain decomposition based iterative method. We also present a fast optimization algorithm for improving the aspect ratio of existing mesh partitions, and illustrate it with several examples from fluid dynamics and structural mechanics applications. For a stiffened shell problem decomposed by the optimal Recursive Spectral Bisection scheme and solved by the FETI method, this optimization algorithm is shown to improve the solution time by a factor equal to 1·54 and to restore numerical scalability.
Moving asymptotes and active set strategy for constrained optimization design in magnetostatic problems
Authors: Mahey P. , Saldanha R. R. , Coulomb J. L.
Published: August 2005

In this paper, we present an improvement of the optimization technique of the moving asymptotes applied to a class of magnetostatic design problem. Our approach includes, among other aspects, an active set strategy which turns possible to find feasible points during the iterative process. We present some numerical results obtained on real‐life case studies where the functions are computed by a complex finite‐element routine.
A parallelized Lattice‐Gas solver for transient Navier–Stokes‐flow: Implementation and simulation results
Authors: Krafczyk M. , Rank E.
Published: August 2005

The last decade has seen the development of Lattice‐Gas (LG) schemes as a complementary if not alternative method for the simulation of moderate Reynolds‐Number Navier–Stokes flow. After a short theoretical introduction we present a detailed discussion of implementation features for a specific 2D‐LG algorithm, which runs in parallel on a workstation‐cluster, discuss simulation results and compare one of them to experimental studies. Finally, we attempt to point out present problems and perspectives of these methods.
Assumed strain formulation for the four‐node quadrilateral with improved in‐plane bending behaviour
Authors: Stolarski Henryk K. , Chen Yung‐I
Published: August 2005

A new assumed strain quadrilateral element with highly accurate in‐plane bending behaviour is presented for plane stress and plane strain analysis. The basic idea of the formulation consists in identification of various modes of deformation and then in proper modification of the strain field in some of these modes. In particular, the strain operator corresponding to the in‐plane bending modes is modified to simulate the strain field resulting from the assumptions usually made in structural mechanics. The modification of the strain field leads to the assumed strain operator on the element level. As a result, the so‐called shear and membrane locking phenomena are alleviated. The element exhibits remarkable success in bending‐ dominated problems even when severely distorted and high aspect ratio meshes are used. Another advantage of the present assumed strain element is that locking for nearly incompressible materials is also mitigated. While this assumed strain element passes the patch test only for the parallelogram shapes, the element provides convergent solutions as long as the initially general form of the element approaches a parallelogram shape with the refinement of the mesh.
Embedded representation of fracture in concrete with mixed finite elements
Authors: Lotfi Hamid R. , Shing P. Benson
Published: August 2005

As an alternative to the smeared and discrete crack representations, an embedded representation of fracture for finite element analysis of concrete structures is presented. The three‐field Hu–Washizu variational statement is extended to bodies with internal discontinuities. The extended variational statement is then utilized for formulating elements with a discontinuous displacement field. The new elements are capable of modelling different deformation modes of an internal discontinuity at the element level. The satisfactory performance of the embedded crack representation is verified by several case studies on concrete fracture.
High performance computational kernels for selected segments of a p finite element code
Authors: Barragy E. , Van De Geijn R.
Published: August 2005

A high performance implementation is presented for three kernel routines commonly found in element‐byelement preconditioned conjugate gradient finite element codes. These routines include forming the element stiffness matrices and loading vectors, or in the case of a non‐linear problem, element residual vectors; and routines for applying element matrix–vector products. The present study considers tensor product elements of arbitrary mapping in 2‐D, although the generalization to triangular elements and serendipity elements is straightforward. The implementation presented is most appropriate for high p type finite element methods, where the element matrices are relatively large and dense. This results in a set of high performance kernels for superscalar architectures, which otherwise may be memory bandwidth limited. Performance studies are presented for a representative superscalar microprocessor, the Intel i860. As these types of microprocessors are at the heart of modern workstations as well as several parallel supercomputing systems, this work is relevant across a variety of platforms. The resulting kernels yield both high performance on a variety of sequential architectures as well as a high degree of code portability through the basic linear algebra subprograms mechanism.
Stable and time‐dissipative finite element methods for the incompressible Navier–Stokes equations in advection dominated flows
Authors: Simo J. C. , Armero F. , Taylor C. A.
Published: August 2005

This paper examines a new Galerkin method with scaled bubble functions which replicates the exact artificial diffusion methods in the case of 1‐D scalar advection–diffusion and that leads to non‐oscillatory solutions as the streamline upwinding algorithms for 2‐D scalar advection–diffusion and incompressible Navier–Stokes. This method retains the satisfaction of the Babuska–Brezzi condition and, thus, leads to optimal performance in the incompressible limit. This method, when, combined with the recently proposed linear unconditionally stable algorithms of Simo and Armero (1993), yields a method for solution of the incompressible Navier–Stokes equations ideal for either diffusive or advection‐dominated flows. Examples from scalar advection–diffusion and the solution of the incompressible Navier–Stokes equations are presented.
Out‐of‐core solver for large, multi‐zone boundary element matrices
Authors: Rigby R. H. , Aliabadi M. H.
Published: August 2005

In this paper an out‐of‐core solver is developed for a three‐dimensional elastostatic boundary element program. This program includes multi‐regions which enables a large model to be divided into a number of homogeneous sub‐models, each with their own material properties. The system matrix produced by multi‐regions is sparse, blocked and unsymmetric in character and so reduces disc space and solution time. The solver presented in this paper utilizes efficiently the structure of this matrix by holding only non‐zero parts of this matrix in a sequential, unformatted direct access file, reading in as much of the file as possible into the working space, performing Gauss elimination, then writing part of the matrix back to file. The maximum utilization of working space is particularly important on vector machines as vector activity is maximized whilst record read/write is minimized. The effect of multi‐regions on CPU is demonstrated on both the CONVEX and CRAY machines.
Reproducing kernel particle methods for structural dynamics
Authors: Liu Wing Kam , Jun Sukky , Li Shaofan , Adee Jonathan , Belytschko Ted
Published: August 2005

This paper explores a Reproducing Kernel Particle Method (RKPM) which incorporates several attractive features. The emphasis is away from classical mesh generated elements in favour of a mesh free system which only requires a set of nodes or particles in space. Using a Gaussian function or a cubic spline function, flexible window functions are implemented to provide refinement in the solution process. It also creates the ability to analyse a specific frequency range in dynamic problems reducing the computer time required. This advantage is achieved through an increase in the critical time step when the frequency range is low and a large window is used. The stability of the window function as well as the critical time step formula are investigated to provide insight into RKPMs. The predictions of the theories are confirmed through numerical experiments by performing reconstructions of given functions and solving elastic and elastic–plastic one‐dimensional (1‐D) bar problems for both small and large deformation as well as three 2‐D large deformation non‐linear elastic problems. Numerical and theoretical results show the proposed reproducing kernel interpolation functions satisfy the consistency conditions and the critical time step prediction; furthermore, the RKPM provides better stability than Smooth Particle Hydrodynamics (SPH) methods. In contrast with what has been reported in SPH literature, we do not find any tensile instability with RKPMs.
Asymptotic post‐buckling analysis of rectangular plates by HC finite elements
Authors: Lanzo Antonio D. , Garcea Giovanni , Casciaro Raffaele
Published: August 2005

The present paper extends the finite element perturbation approach already presented for pin‐jointed and framed structures15 to rectangular thin plates. Koiter's asymptotic strategy2 is coupled with a High‐Continuity finite element discretization of the plate.22 The consistency of the discrete model is discussed from the kinematical and numerical points of view and several numerical tests are reported. It appears that use of the HC elements makes the perturbation algorithm insensitive to the locking phenomenon occurring in the evaluation of the postbuckling behaviour. It also allows the use of very fine discretization meshes at low computational cost.
Analysis of symmetric domains in advanced applications with the boundary element method
Authors: Kaljević Igor , Saigal Sunil
Published: August 2005

The strategy in the boundary element method for the analysis of symmetric domains that does not require the modelling of contours corresponding to the axes of symmetry is investigated for a number of advanced applications. These applications include: treatment of domain loadings, two‐dimensional time domain transient elastodynamics, and the analysis of probabilistic problems in elastostatics with a random geometric configuration. Both symmetric as well as arbitrary loadings acting on the symmetric objects are considered. A number of case studies are presented to provide comparisons of computer memory and CPU time requirements for the analyses of the entire object versus the analyses of only the symmetric portion of the object. The numerical data presented clearly demonstrates the advantages of developing efficient symmetric formulations for advanced applications.
Singular boundary elements for the analysis of cracks in plane strain
Authors: Watson J. O.
Published: August 2005

Straight and curved cracks are modelled by direct formulation boundary elements, of geometry defined by Hermitian cubic shape functions. Displacement and traction are interpolated by the Hermitian functions, supplemented by singular functions which multiply stress intensity factors corresponding to the dominant modes of crack opening in which displacement is proportional to the square root of distance r from the crack tip, and subdominant modes in which it is proportional to r1·5. The singular functions extend over many boundary elements on each crack face.

A nodal collocation scheme is used, in which additional boundary integral equations are obtained by differentiation of the equation obtained from Betti's theorem. The hypersingular kernels of the equations so derived are integrated by consideration of trial displacement fields of subdomains lying to either side of the crack. Examples are shown of the analysis of buried and edge cracks, to demonstrate the effects of modelling subdominant modes and extending singular shape functions over many elements.
A conforming quadrilateral plate bending element revisited
Authors: Liu Ine‐Wei
Published: August 2005

A new formulation of a 16‐degree‐of‐freedom conforming quadrilateral plate bending element using the Cartesian co‐ordinates is presented. The displacement function w is taken as a slight modification of bicubic polynomial in the Cartesian co‐ordinates. Use of this expression in the construction of finite element representations is expected to be useful specially for microcomputers. The simple stiffness matrix subroutine, coded in FORTRAN 77, for the general quadrilateral plate bending element is listed in the Appendix.
Wave envelope examples for progressive waves
Authors: Bettess Peter , Chadwick Edmund
Published: August 2005

The wave envelope method, due to Astley,1 is applied to some simple propagating wave problems governed by the Helmholtz equation and similar equations. It is shown that the method yields solutions which for some of the problems are exact and for the others are approximate and complex. In these cases however, when the number of elements is increased the trial function becomes a better approximation.
A plasticity model and algorithm for mode‐I cracking in concrete
Authors: Feenstra Peter H. , De Borst René
Published: August 2005

A class of plasticity models which utilize Rankine's (principal stress) yield locus is formulated to simulate cracking in concrete and rock under monotonic loading conditions. The formulation encompasses isotropic and kinematic hardening/softening rules, and incremental (flow theory) as well as total (deformation theory) formats are considered. An Euler backward algorithm is used to integrate the stresses and internal variables over a finite loading step and an explicit expression is derived for a consistently linearized tangent stiffness matrix associated with the Euler backward scheme. Particular attention is paid to the corner regime, that is when the two major principal stresses become equal. A detailed comparison has been made of the proposed plasticity‐based crack formulations and the traditional fixed and rotating smeared‐crack models for a homogeneously stressed sample under a non‐proportional loading path. A comparison between the flow‐theory‐based plasticity crack models and experimental data has been made for a Single Edge Notched plain concrete specimen under mixed‐mode loading conditions.
Numerical methods for evaluation of service life of gear
Authors: Abers̆ek B. , Flas̆ker J.
Published: August 2005

The service life of mechanical parts, of which the gears are most important, is influenced to the greatest extent by defects occurring as cracks. In case of cracks the most important stage is the initiation and propagation of short fatigue cracks; further, the propagation of these crack up to their critical and/or permissible length also must be taken into account. As these are quite different areas we have introduced into calculations a completely new factor, i.e. the gear stress intensity factor Z. On the basis of the factor Z we have developed appropriate models, e.g. the programe STATFAG which ensures calculation of the service life of gears with and without cracks.
Automatic three‐dimensional finite element mesh generation via modified ray casting
Authors: Yeker Cengiz , Zeid Ibrahim
Published: August 2005

Three‐dimensional (3‐D) finite element mesh generation has been the target of automation due to the complexities associated with generating and visualizing the mesh. A fully automatic 3‐D mesh generation method is developed. The method is capable of meshing CSG solid models. It is based on modifying the classical ray‐casting technique to meet the requirements of mesh generation. The modifications include the utilization of the element size in the casting process, the utilization of 3‐D space box enclosures, and the casting of ray segments (rays with finite length). The method begins by casting ray segments into the solid. Based on the intersections between the segments and the solid boundary, the solid is discretized into cells arranged in a structure. The cell structure stores neighbourhood relations between its cells. Each cell is meshed with valid finite elements. Mesh continuity between cells is achieved via the neighbourhood relations. The last step is to process the boundary elements to represent closely the boundary. The method has been tested and applied to a number of solid models. Sample examples are presented.
A novel finite element formulation for frictionless contact problems
Authors: Papadopoulos Panayiotis , Jones Reese E. , Solberg Jerome M.
Published: August 2005

This article advocates a new methodology for the finite element solution of contact problems involving bodies that may undergo finite motions and deformations. The analysis is based on a decomposition of the two‐body contact problem into two simultaneous sub‐problems, and results naturally in geometrically unbiased discretization of the contacting surfaces. A proposed two‐dimensional contact element is specifically designed to unconditionally allow for exact transmission of constant normal traction through interacting surfaces.
A linear programming formulation for incremental contact analysis
Authors: Šimunović Srdan , Saigal Sunil
Published: August 2005

A novel algorithm for the analysis of contact problems in elasticity has been presented in this paper. The algorithm is based on the boundary element method and a direct approximation of the contact complementarity conditions using linear programming. An incremental loading scheme has been developed to ensure an accurate approximation of the deformation path that the object experiences during the process of contact. Several numerical examples have been analysed to illustrate the validity of the proposed formulations.
Single and multiobjective structural optimization in discrete‐continuous variables using simulated annealing
Authors: Bennage W. A. , Dhingra A. K.
Published: August 2005

A multivariable optimization technique based on the Monte‐Carlo method used in statistical mechanics studies of condensed systems is adapted for solving single and multiobjective structural optimization problems. This procedure, known as simulated annealing, draws an analogy between energy minimization in physical systems and objective function minimization in structural systems. The search for a minimum is simulated by a relaxation of the statistical mechanical system where a probabilistic acceptance criterion is used to accept or reject candidate designs. To model the multiple objective functions in the problem formulation, a cooperative game theoretic approach is used. Numerical results obtained using three different annealing strategies for the single and multiobjective design of structures with discrete‐continuous variables are presented. The influence of cooling schedule parameters on the optimum solutions obtained is discussed. Simulation results indicate that, in several instances, the optimum solutions obtained using simulated annealing outperform the optimum solutions obtained using some gradient‐based and discrete optimization techniques. The results also indicate that simulated annealing has substantial potential for additional applications in optimization, especially for problems with mixed discrete‐continuous variables.
A0 – Stable family of single step methods for semidiscretized parabolic problems
Authors: Niemi Jarmo
Published: August 2005

A general family of single‐step methods for use in first‐order ODEs is presented. This family of methods, called SS3‐methods, embraces generalized trapezoidal methods, SS21‐methods, α‐methods and many other well‐known methods, thus providing a good basis for comparison of these methods. Corresponding parameter values of SS3 methods for these methods are presented. The analysis of SS3‐methods is restricted to linear and symmetric systems. Stability, convergence, accuracy, numerical dissipation and overshoot of the methods are considered. The conditions for algorithm parameters to exhibit good characteristics in these respects are given. It is shown that the optimal subfamily of SS3‐methods coincides with α‐methods.
Dynamic analysis of thin‐walled and variable open section beams with shear flexibility
Authors: Ambrosini Ricardo Daniel , Riera Jorge Daniel , Danesi Rodolfo Francisco
Published: August 2005

The equations of motion of thin‐walled beams with open cross‐section, considering the effects of shear flexibility and rotatory inertia in the stress resultants, as well as variable cross‐sectional properties are presented using a state variables approach. These equations are based on Vlasov's theory of thin‐walled beams, which is modified to include the effects indicated above. The resulting equations are used in the determination of the natural frequencies of an open cross‐section prestressed concrete beam. Comparisons with other formulations are also presented employing the waves dispersion equation.
Equivalence of the driving elastic forces in flexible multibody systems
Authors: Gofron M. , Shabana A. A.
Published: August 2005

When the driving joint forces, determined using the inverse dynamics procedure, are applied in the feedforward control of mechanical systems, discrepancies between the specified and the actual motion are observed. In some recent publications, these discrepancies were attributed to the wave phenomenon. It is shown in this investigation that the solution of the inverse dynamics of flexible mechanical systems defines two types of driving forces which can be classified as driving joint forces and driving elastic forces. The driving joint forces which depend on the deformation of the flexible bodies define the torque and the actuator forces which must be applied at the joints. The driving elastic forces are associated with the deformation degrees of freedom, and therefore, there is no gaurantee that an algorithm that ignores these driving elastic forces will converge and achieve the desired solution. It is the objective of this investigation to examine the nature of the driving elastic forces in the solution of the inverse dynamics problem, and demonstrate that the driving elastic forces associated with two different sets of vibration modes which produce the same physical displacements are basically the same and they differ only by a co‐ordinate transformation. The effect of the selection of the deformable body co‐ordinate system on these forces is also examined numerically using a slider crank mechanism with a flexible connecting rod.
Dynamic analysis and drilling degrees of freedom
Authors: Hughes Thomas J. R. , Masud Arif , Harari Isaac
Published: August 2005

Finite element methods for dynamic analysis employing elements with drilling degrees of freedom are presented. The formulation is based on a variational principle in which displacements and rotations are interpolated independently. The issue of zero masses corresponding to rotational degrees of freedom is addressed and techniques for defining consistent and lumped rotational mass matrices are presented.
An effective data parallel self‐starting explicit methodology for computational structural dynamics on the connection machine CM‐5
Authors: Namburu Raju R. , Turner David A. , Tamma Kumar K.
Published: August 2005

This paper discusses the implementation aspects and our experiences towards a data parallel explicit self‐starting finite element transient methodology with emphasis on the Connection Machine (CM‐5) for linear and non‐linear computational structural dynamic applications involving structured and unstructured grids. The parallel implementation criteria that influence the efficiency of an algorithm include the amount of communication, communication routing, and load balancing. To provide simplicity, high level of accuracy, and to retain the generality of the finite element implementation for both linear and non‐linear transient explicit problems on a data parallel computer which permit optimum amount of communications, we implemented the present self‐starting dynamic formulations (in comparison to the traditional approaches) based on nodal displacements, nodal velocities, and elemental stresses on the CM‐5. Data parallel language CMFortran is employed with virtual processor constructs and with:SERIAL and:PARALLEL layout directives for arrays. The communications via the present approach involve only one gather operation (extraction of element nodal displacements or velocities from global displacement vector) and one scatter operation (dispersion of element forces onto global force vector) for each time step. These gather and scatter operations are implemented using the Connection Machine Scientific Software Library communication primitives for both structured and unstructured finite element meshes. The implementation aspects of the present self‐starting formulations for linear and elastoplastic applications on serial and data parallel machines are discussed. Numerical test models for linear and non‐linear one‐dimensional applications and a two‐dimensional unstructured finite element mesh are then illustrated and their performance studies are discussed.
On the calculation of boundary stresses with the Somigliana stress identity
Authors: Fiedler Christian
Published: August 2005

The computation of boundary stresses by Boundary Element Method (BEM) is usually performed either by expressing the boundary tractions in a local co‐ordinate system, calculating the remaining stresses by shape function differentiation and inserting into Hooke's law or recently also by solving the hypersingular integral equation for the stresses. While direct solution of the hypersingular integral equation, the so‐called Somigliana stress identity, has been shown to be more reliable, the interpretation and numerical treatment of the hypersingularity causes a number of problems. In this paper, the limiting procedure in taking the load point to the boundary is carried out by leaving the boundary smooth and the contributions of all different types of singularities to the boundary integral equation are studied in detail. The hypersingular integral in the arising boundary integral equation is then reduced to a strongly singular one by considering a traction free rigid body motion. For the numerical treatment, an algorithm for multidimensional Cauchy Principal Value (CPV) integrals is extended that is applicable for the calculation of boundary stresses. Moreover, the shape of the surrounding of the singular point is studied in detail. A numerical example of elastostatics confirms the validity of the proposed method.
The least‐squares finite element method for low‐mach‐number compressible viscous flows
Authors: Yu Sheng‐Tao , Jiang Bo‐Nan , Liu Nan‐Suey , Wu Jie
Published: August 2005

The present paper reports the development of the Least‐Squares Finite Element Method (LSFEM) for simulating compressible viscous flows at low Mach numbers in which the incompressible flows pose as an extreme. The conventional approach requires special treatments for low‐speed flows calculations: finite difference and finite volume methods are based on the use of the staggered grid or the preconditioning technique, and finite element methods rely on the mixed method and the operator‐splitting method. In this paper, however, we show that such a difficulty does not exist for the LSFEM and no special treatment is needed. The LSFEM always leads to a symmetric, positive‐definite matrix through which the compressible flow equations can be effectively solved. Two numerical examples are included to demonstrate the method: driven cavity flows at various Reynolds numbers and buoyancy‐driven flows with significant density variation. Both examples are calculated by using full compressible flow equations.
Anisotropic modelling and numerical simulation of brittle damage in concrete
Authors: Govindjee Sanjay , Kay Gregory J. , Simo Juan C.
Published: August 2005

A framework for damage mechanics of brittle solids is presented and exploited in the design and numerical implementation of an anisotropic model for the tensile failure of concrete. The key feature exploited in the analysis is the hypothesis of maximum dissipation, which specifies a unique damage rule for the elastic moduli of the solid once a failure surface is specified. A complete algorithmic treatment of the resulting model is given which renders a useful tool for large‐scale inelastic finite element calculations. A rather simple three‐surface failure model for concrete, containing essentially no adjustable parameters, is shown to produce results in remarkably good agreement with sample experimental data.
A Fourier analysis and dynamic optimization of the Petrov–Galerkin finite element method
Authors: Carrano C. S. , Yeh Gourtsyh
Published: August 2005

A Fourier analysis of the linear and quadratic N + 1 and N + 2 Petrov–Galerkin finite element methods applied to the one‐dimensional transient convective‐diffusion equation is performed. The results show that a priori optimization of the N + 1 method is not possible because dissipative errors are introduced as dispersive errors are reduced (any optimization is subjective). However, a priori optimization of the N + 2 Petrov–Galerkin method is possible because the reduction of dispersion errors can be accomplished without the addition of artificial dissipation.

The Spectrally Weighted Average Phase Error Method (SWAPEM) for the optimization of the N + 2 Petrov–Galerkin method is introduced, in which the N + 2 weighting parameter is chosen at each time step to minimize the integral over wave number of the phase error of Fourier modes, weighted by the frequency content of the global solution at the previous time step (obtained via FFT). The method is dynamic, and general in that the dependence of the weighting parameter on the solution waveform is accounted for. Optimal values predicted by the method are in excellent agreement with those suggested by the numerical experimentation of others. Simulations of the pure convective transport of a Gaussian plume and a triangle wave are discussed to illustrate the effectiveness of the method.
Arbitrary Lagrangian–Eulerian finite element analysis of strain localization in transient problems
Authors: Pijaudier‐Cabot Gilles , Bodé Laurent , Huerta Antonio
Published: August 2005

Non‐local models guaranty that finite element computations on strain softening materials remain sound up to failure from a theoretical and computational viewpoint. The non‐locality prevents strain localization with zero global dissipation of energy, and consequently finite element calculations converge upon mesh refinements to non‐zero width localization zones. One of the major drawbacks of these models is that the element size needed in order to capture the localization zone must be smaller than the internal length. Hence, the total number of degrees of freedom becomes rapidly prohibitive for most engineering applications and there is an obvious need for mesh adaptivity. This paper deals with the application of the arbitrary Lagrangian–Eulerian (ALE) formulation, well known in hydrodynamics and fluid–structure interaction problems, to transient strain localization in a non‐local damageable material. It is shown that the ALE formulation which is employed in large boundary motion problems can also be well suited for non‐linear transient analysis of softening materials where localization bands appear. The remeshing strategy is based on the equidistribution of an indicator that quantifies the interelement jump of a selected state variable. Two well known one‐dimensional examples illustrate the capabilities of this technique: the first one deals with localization due to a propagating wave in a bar, and the second one studies the wave propagation in a hollow sphere.
A family of methods for the wave equation in one‐ and two‐space dimensions
Authors: Twizell E. H. , Tirmizi S. I. A.
Published: August 2005

A family of finite‐difference methods is developed for the numerical solution of the simple wave equation. Local truncation errors are calculated for each member of the family and each is analyzed for stability. The concepts of A0 stability and L0 stability, well used in the literature on other types of partial differential equation, are discussed in relation to second‐order hyperbolic equations. The numerical methods arc extended to cover two‐dimensional wave equations and the methods developed in this article are tested on three problems from the literature.
Finite‐element stream function solutions of transonic rotational internal and external flows
Authors: Habashi Wagdi G. , Hafez Mohamed M.
Published: August 2005

The finite‐element method is applied to the stream function formulation of transonic flows. Numerical dissipation, necessary for the calculation of mixed flows with shocks, is introduced via the artificial compressibility method. The classical problem of double‐valuedness of the mass flux versus Mach number is resolved by direct integration of the vorticity equation. Solutions are obtained for isolated airfoils, the blade‐to‐blade cascade equation, as well as the radial equilibrium equation governing the hub‐to‐shroud through flow in turbomachinery.
Principles of electricity and economics in fluid dynamics
Authors: Gustafson Karl
Published: August 2005

We present essential connections between electrical field decomposition and network theory, economic decentralization theory, and the work of Kron on decomposition of systems for computational fluid dynamics.
Monotonicity and finite element methods for hyperbolic initial value problems
Authors: Hofmann Wolf
Published: August 2005

A well known theórem about super‐ and subfunctions for the solution of hyperbolic initial value problems constructs differentiable functions as upper and lower bounds (see Walter [1], 21 XIII). The proof can be done by transforming the differential equation problem into a set of integral equations, using the monotonicity‐properties of the arising integral operators. This proof needs an integral representation for twice differentiable functions. It is shown that this proceeding can be generalized to get upper and lower bounds in terms of finite element functions. To do this, we give an integral representation for continuous, piecewise differentiable functions, including the discontinuities of their derivatives. Then the generalization of the classical proof yields interface conditions for the finite element functions. Finally, it is demonstrated how to realize numerically these conditions.
A multidomain spectral approximation of elliptic equations
Authors: Funaro Daniele
Published: August 2005

A spectral approximation for the Poisson equation defined on Ω = ]−1, 1[×] −1,1[ is studied. The domain Ω is decomposed into two rectangular regions and the equation is collocated at the Legendre nodes in each domain. On the common boundary of the two subdomains, suitable conditions are imposed in order to obtain a unique solution from the resulting linear system. Different values of the discretization parameters are allowed in each rectangle. We prove the stability of the scheme and give convergence estimates. The rate of convergence in a single subdomain, depends only on the regularity of the exact solution therein. An efficient preconditioning matrix is proposed.
On least‐squares approximations to compressible flow problems
Authors: Chen Tsu‐Fen
Published: August 2005

A direct finite‐element method for computer solutions of compressible potential flow problems is presented. An analysis of least‐squares approximation is given, including optimal order estimates for piecewise polynomial approximation spaces. The model problem considered is that of potential flow past a cylinder. Numerical results for the model problem are given for a shock‐free subsonic case.
An explicit finite difference solution to the convection‐dispersion equation
Authors: von Rosenberg D. U.
Published: August 2005

An explicit finite difference equation has been development for the solution of the convection‐dispersion equation. This equation has been over the entire range of 2D/vΔx between zero and one, region where no completely satisfactory method has been previously available. No oscillations or numerical dispersion were observed in any of the solutions.
An explicit finite difference scheme based on the modified method of characteristics for solving convection‐diffusion problems in one space dimension
Authors: Thomaidis George , Zygourakis Kyriacos , Wheeler Mary F.
Published: August 2005

Finite differences are combined with the modified method of characteristics to develop an explicit scheme for solving convection‐dominated convection‐diffusion problems in one spatial dimension. Error analysis shows that the new algorithm is stable under a mild stability criterion. Problems with known analytical solutions are used to test the algorithm and demonstrate its convergence. Numerical solutions are free of numerical dispersion, undershoot and overshoot. The algorithm is easy to implement and requires small computational times for the test problems considered.
On the numerical analysis of analytical nodal methods
Authors: Hennart J. P.
Published: August 2005

This article presents the basic numerical analysis of the analytical nodal methods, which were originally developed in the late 1970s in relation with static and dynamic nuclear reactor calculations but are actually applicable to the numerical solution of partial differential equations (PDEs) in general, over fairly regular domains. The basic idea consists in “transverse integrating” the original PDE over all the variables minus one, leading to sets of 1D equations which are then solved in an “analytical” way, using fundamentals as well as particular solutions of the corresponding 1D operators. After examining the existing analytical methods in a critical way, we propose a more satisfactory extended analytical formalism. Superconvergence results finally lead us to useful conclusions with respect to the choice of a particular scheme.
On the potential of spectral methods to solve problems in non‐Newtonian fluid mechanics
Authors: Phillips Timothy N.
Published: August 2005

Spectral techniques for solving problems in non‐Newtonian fluid mechanics are introduced. Following the work of Coleman (J. Non‐Newtonian Fluid Mech.; 15, 227–238 [1984]), the governing equations for the creeping flow of a co‐rotational Maxwell fluid are written in terms of the Airy stress function and a stream function. This ensures that the continuity and momentum equations are automatically satisfied. The choice of trial functions for solving a one‐dimensional model problem using spectral methods is discussed. Methods for treating unbounded domains and accurately representing reentrant boundary singularities within the spectral context are also considered.
A boundary integral approach to Stefan problems with many phase changes
Authors: Coleman C. J.
Published: August 2005

This paper considers a boundary integral approach to Stefan problems that have multiple phase changes and satisfy the Laplace equation in each phase. It is shown that, by introducing artificial phase changes, the effects of moderate diffusivity can be incorporated. The paper includes the results from several computer simulations.
Least squares collocation solution of differential equations on irregularly shaped domains using orthogonal meshes
Authors: Laible J. P. , Pinder G. F.
Published: August 2005

The least squares collocation (LESCO) method has been formulated to solve differential equations defined over irregular domains using a more convenient orthogonal computational mesh. The LESCO method is described in detail for second‐order boundary value problems and applied to the time‐dependent diffusion and advection‐diffusion equations defined over two‐dimensional irregular domains. Particular attention is given to the proper procedure for applying boundary conditions. Accuracy, convergence, and consistency are examined. For cubic elements with arbitrary location of collocation points, the convergence rate is between 3rd and 4th order. The major advantages of this method are reduced input data requirements, a more robust procedure for forming the equations, positive definite matrices, and flexibility in distrbuting errors.
The h‐p version of the finite element method for parabolic equations. Part I. The p‐version in time
Authors: Babuska Ivo , Janik Tadeusz
Published: August 2005

The paper is the first in the series addressing the h‐p version of the finite element method for parabolic equations. The h‐p version is applied to both time and space variables. The present paper addresses the case when in time the p‐version with one single time element is used. Error estimation is given and numerical computations are presented.
Finite elements for transonic potential flows
Authors: Berger H. , Warnecke G. , Wendland W.
Published: August 2005

A finite element method is used for the computation of entropy solutions to the transonic full potential equation. Physically correct solutions with sharp and correctly placed shocks were obtained. (AMS (MOS) 1980 Mathematics subject classifications: 65N30, 76N15, 35M05, 76H05, 49D10, 35A40, 35L67.)
The p‐version of the finite element method for domains with corners and for infinite domains
Authors: Babuška Ivo , Oh Hae‐Soo
Published: August 2005

A special approach to deal with elliptic problems with singularities is introduced. It is shown that this approach, to be called an auxiliary mapping technique, in the framework of the p‐version of the finite element method yields an exponential rate of convergence. It is also shown that this technique can deal with elliptic problems on unbounded domains in R2 as well. (AMS(MOS) subject classifications: Primary, 65N30, 65N15.)
Viscosity‐splitting scheme for the Navier‐Stokes equations
Authors: Lung‐an Ying
Published: August 2005

A viscosity‐splitting scheme for the initial boundary value problems of the Navier‐Stokes equations is considered. In the scheme, the Stokes equation is solved in conjunction with a nonhomogeneous boundary condition which connects the tangent flow with a no‐slip flow. Convergence is proved.