bluuebunny/crossref_metadata_2025 huggingface dataset:
A coupled analysis method for structures with independently modelled finite element subdomains
Authors: Aminpour Mohammad A. , Ransom Jonathan B. , McCleary Susan L.
Published: August 2005

A new method for analysing plate and shell structures with two or more independently modelled finite element subdomains is presented, assessed, and demonstrated. This method provides a means of coupling local and global finite element models whose nodes do not coincide along their common interface. In general, the method provides a means of coupling structural components (e.g., wing and fuselage) which may have been modelled by different analysts. In both cases, the need for transition modelling, which is often tedious and complicated, is eliminated. The coupling is accomplished through an interface for which three formulations are considered and presented. These formulations are: collocation, discrete least‐squares, and hybrid variational. Several benchmark problems are analysed and it is shown that the hybrid variational formulation provides the most accurate solutions.
Numerical modelling for scattering of waves in three‐dimensional cracked material
Authors: Han Sixiong , Ichikawa Yasuaki
Published: August 2005

In this paper, a mathematical model is proposed for the problem of the scattering of plane waves in a three‐dimensional cracked materials. Instead of obtaining closed‐form solutions as in conventional theoretical analysis methods, this approach, called the Equivalent Nodal Force (ENF) method formulates the mechanical effects of cracks as an equivalent nodal force in a numerical procedure, and physically translates cracked material into an equivalent continuous one. Several mechanical relations between waves and cracks are evident from this method. Also the results of several numerical calculations are presented and these are compared with those obtained by the conventional methods.
An efficient assumed strain element model with six DOF per node for geometrically non‐linear shells
Authors: Park Hoon C. , Cho Chahngmin , Lee Sung W.
Published: August 2005

The present paper describes an assumed strain finite element model with six degrees of freedom per node designed for geometrically non‐linear shell analysis. An important feature of the present paper is the discussion on the spurious kinematic modes and the assumed strain field in the geometrically non‐linear setting. The kinematics of deformation is described by using vector components in contrast to the conventional formulation which requires the use of trigonometric functions of rotational angles. Accordingly, converged solutions can be obtained for load or displacement increments that are much larger than possible with the conventional formulation with rotational angles. In addition, a detailed study of the spurious kinematic modes and the choice of assumed strain field reveals that the same assumed strain field can be used for both geometrically linear and non‐linear cases to alleviate element locking while maintaining kinematic stability. It is strongly recommended that the element models, described in the present paper, be used instead of the conventional shell element models that employ rotational angles.
Finite strip analysis of imperfect laminated plates under end shortening and normal pressure
Authors: Dawe D. J. , Wang S. , Lam S. S. E.
Published: August 2005

The finite strip method is developed to predict the geometrically non‐linear response of rectangular laminated plates subjected to progressive end shortening whilst initial imperfection and/or applied pressure loading are also present. The development is an extension of earlier work concerned with perfect, unpressurized laminates which have simply supported loaded ends at which lateral in‐plane expansion is either allowed freely or is prevented completely. The analysis is conducted in the contexts of classical plate theory and first‐order shear deformation plate theory, in turn. A number of applications involving both isotropic plates and laminated plates is described.
A posteriori estimation and adaptive control of the pollution error in the h‐version of the finite element method
Authors: Babuška I. , Strouboulis T. , Upadhyay C. S. , Gangaraj S. K.
Published: August 2005

In References 1 and 2 we showed that the error in the finite‐element solution has two parts, the local error and the pollution error, and we studied the effect of the pollution error on the quality of the local error‐indicators and the quality of the derivatives recovered by local post‐processing. Here we show that it is possible to construct a posteriori estimates of the pollution error in any patch of elements by employing the local error‐indicators over the mesh outside the patch. We also give an algorithm for the adaptive control of the pollution error in any patch of elements of interest.
Interior and boundary difference equations for hyperbolic differential equations
Authors: Lick Wilbert , Ziegler Kirk , Lick James
Published: August 2005

Interior and boundary difference equations are derived for several hyperbolic partial differential equations by means of an integral method. The method is applied to a simple transport equation, to waves in a compressible, isentropic fluid, and to surface waves in shallow water. Boundary conditions treated are (a) a perfectly reflecting boundary, (b) an open boundary with outgoing waves and a specified incoming wave, and (c) a partially reflecting boundary. For open boundaries, the major assumption for the algorithms to be valid is that outgoing waves can be defined, an assumption equivalent to the most general statement of Sommerfeld's radiation condition. The difference equations obtained are conservative, second‐order accurate, two time‐level, explicit, and stable (for one‐dimensional, time‐dependent problems) for cΔt/Δx ⩽ 1 where c is the wave speed, Δt is the temporal grid size, and Δx is the spatial grid size. Numerical calculations demonstrate the excellent accuracy of the procedure.
Solution of governing differential equations of vibrating cylindrical shells using B‐spline functions
Authors: Peng‐Cheng Shen , Jian‐Guo Wang
Published: August 2005

The cubic B3‐spline functions and eigenfunctions are used to obtain the approximate solution for the vibration of cylindrical shells in this paper. Unified computational schemes suited for various types of boundary conditions are formulated here. In comparison with the conventional finite elements method and finite strip method, the main features of the present method are higher accuracy, fewer unknowns, ease in programming, and economy in computer solution. The numerical results are given and compared with other numerical solutions.
Amplification of surface ground motion by an inclusion of arbitrary shape for harmonic surface line loading
Authors: Dravinski M. , Mossessian T. K.
Published: August 2005

The plane strain model for the Lamb's problem with an elastic inclusion of arbitrary shape embedded completely within an elastic half space is investigated by using an indirect boundary integral equation method for steady‐state elastodynamics. The surface of the half space is subjected to vertical or horizontal harmonic line loads. The displacement field is evaluated throughout the elastic medium so that the continuity of the displacement and traction fields along the interface between the half space and the inclusion is satisfied in a least‐square sense. The numerical results demonstrate that the presence of the inclusion may cause locally very large amplification of the surface ground motion and that the amplification pattern depends upon the frequency and the type of the input load, the impedance contrast between the half space and the inclusion, the type of the inclusion, and the location of the observation point at the surface of the half space.
A highly accurate and stable explicit finite difference solution to the convection‐dispersion equation with varying velocity and dispersion coefficients
Authors: El‐Ageli Muktar Ali , Menzie Don E.
Published: August 2005

Based on the implicit form of the finite difference analogue to the convection‐dispersion equation of variable velocity and dispersion coefficients, a highly accurate and stable explicit finite difference scheme has been developed by extending the von Rosenberg linear scheme to the varying cases. This variable velocity and dispersion coefficient scheme has been tested for the entire range of (2D/vΔ x) between zero and unity, the region where no completely satisfactory numerical method has been previously available. No oscillations or numerical dispersion were observed in any of the solutions.
Solution of general ordinary differential equations using the algebraic theory approach
Authors: Celia Michael , Herrera Ismael
Published: August 2005

The algebraic theory for numerical methods, as developed by Herrera [3–7], provides a broad theoretical framework for the development and analysis of numerical approximations. To this point, the technique has only been applied to ordinary differential equations with constant coefficients. The present work extends the theory by developing a methodology for equations with variable coefficients. Approximation of the coefficients by piecewise polynomials forms the foundation of the approach. Analysis of the method provides firm error estimates. Furthermore, the analysis points to particular procedures that produce optimal accuracy. Example calculations illustrate the computational procedure and verify the theoretical convergence rates.
Convergence of an element‐partitioned subcycling algorithm for the semi‐discrete heat equation
Authors: Hughes Thomas J. R. , Belytschko Ted , Liu Wing Kam
Published: August 2005

A convergence analysis is performed for an element‐partitioned subcycling algorithm for the semi‐discrete heat equation. It is shown that the algorithm generally attains first‐order rate‐of‐convergence.
An efficient boundary element method for a class of parabolic differential equations using discretization in time
Authors: Ingber M. S.
Published: August 2005

A new boundary element method using discretization in time is proposed to solve a class of parabolic differential equations. The method treats the term containing the time derivative as a forcing term. This necessitates the introduction of additional unknowns in the interior of the domain. At the same time, however, values for the dependent variable are determined directly in the interior. The boundary element formulation is reduced to essentially solving a Poisson equation. The accuracy and efficiency of the method are demonstrated with several examples.
The algebraic theory approach for ordinary differential equations: Highly accurate finite differences
Authors: Herrera Ismael
Published: August 2005

This article reports further developments of Herrera's algebraic theory approach to the numerical treatment of differential equations. A new solution procedure for ordinary differential equations is presented. Finite difference algorithms of 0(hr), for arbitrary “r” are developed. The method consists in constructing local approximate solutions and using them to extract information about the sought solution. Only nodal information is derived. The local approximate solutions are constructed by collocation, using polynomials of degree G. When “n” collocation points are used at each subinterval, G = n + 1and the order of accuracy is 0(h2n−1). The procedure here presented is very easy to implement. A program in which n can be chosen arbitrarily, was constructed and applied to selected examples.
Optimization of computational grids
Authors: Pardhanani Anand , Carey Graham F.
Published: August 2005

We consider the problem of mesh grading or gridr‐point redistribution using optimization techniques. Some representate forms of the objective functions are considered and the assocated scaling parameterization and nonlinear programming algorithms examined. Numerical results for representative test problems involving two‐and three‐dimensional grids are presented.
Numerical methods for structured population models: The Escalator Boxcar Train
Authors: de Roos A. M.
Published: August 2005

A numerical integration method is introduced for the class of hyperbolic partial differential equations that occur in physiologically structured population models. These equations describe the time evolution of the population density‐function over the individual state‐space Ω. Exploiting the biological interpretation of the equation, this density‐function is represented by a set of moments over a collection of subdomains in Ω, moving along the characteristics of the partial differential equation (PDE). These moments are readily interpreted as numbers of individuals in a cohort, mean individual state in a cohort, etc. The numerical method consists of approximating the differential equations.

The method appears to be very efficient for this special type of PDE. It combines such desirable properties as as lack of dispersion and dissipation and a preservation of monotonicity of the density function along the characteristics with a strong relation to the biological background of the equations for these moments to arrive at a closed system of ordinary differential equations. The method also allows one to incorporate the usual type of nonlinearities that occur in physiologically structured population models. A numerical example is presented as an illustration of the application of the method.
Two different approaches for the treatment of boundary singularities
Authors: Michavila F. , Gavete L. , Díez F.
Published: August 2005

The finite element treatment of boundary singularities in elliptic problems has demanded that special techniques be developed. Many of these use some form of singular element in the neighborhood of a singular point.

For a homogenous second‐order problem, defined in a domain with a polygonal boundary, we studied three cases with different singularity orders. Some results about the accuracy of the solution are presented. Numerical results have been obtained using Akin singular triangular elements with three and six nodes and quadrilateral elements having four and eight nodes. The behavior of the gradient in each of these elements is also analyzed. However, these elements are not completely satisfactory, so that an alternative technique using curved isoparametric elements is given here. The results obtained with the two methods are compared. Conclusions about numerical accuracy of each method, the order of integration and the simplicity of application are made.
A network model for incompressible two‐fluid flow and its numerical solution
Authors: Chou S. H.
Published: August 2005

In this paper we consider an incompressible version of the two‐fluid network model proposed by Porsching (Nu. Methods Part. Diff. Eq., 1, 295–313 [1985]). The system of equations governing the model is a mixed system of differential and algebraic equations (DAEs). These DAEs are then recast, through proper transformation, into a system of ordinary differential equations on a submanifold of ℝn, for which uniqueness, existence, and stability theorems are proved. Numerical simulations are presented.
An experimental performance analysis for the rate of convergence of collocation on general domains
Authors: Mu Mo , Rice John R.
Published: August 2005

This paper presents an experimental performance analysis for the rate of convergence of collocation on general domains using a bicubic Hermite basis. Twenty domains are selected for the experiment from the population of PDEs on nonrectangular domains found in Realistic PDE Solutions for Non‐rectangular Domains (C. J. Ribbens and J. R. Rice, CSD‐TR 639, Department of Computer Sciences, Purdue University, 1986), including one rectangle for comparison. The result shows that the convergence of the ELLPACK module COLLOCATION behaves as O(h4) on all 19 of the nonrectangular domains. This set includes a large variety of nonrectangular domains (only two have reentrant corners). We conclude that, with very high probability, this collocation module has O(h4) convergence on general domains.

The experiment is made by using the Performance Evaluation System (PES) of ELLPACK, which includes the population of PDEs on nonrectangular domains. Several performance analysis tools are used to analyze the rate of convergence, the most informative is a visual examination of the convergence behavior.
A parallel domain reduction method
Authors: Brezzi Franco , Douglas Cralg C. , Marini L. Donatella
Published: August 2005

We relate a particular version of a parallel multigrid method to a domain decomposition method, showing that the parallel multigrid method reduces computation to a small portion of the domain and then extends the solution to the entire domain using the correct reflections to get the exact solution. We extend a particular example to double the parallelism in a nonobvious manner. While the techniques of this paper are applied to twodimensional problems, they can be applied to higher dimensional problems in an obvious manner.
Solution of the advective‐dispersive transport equation using a least squares collocation, Eulerian‐Lagrangian method
Authors: Bentley L. R. , Pinder G. F. , Herrera I.
Published: August 2005

Numerical solution of the advective‐dispersive transport equation is difficult when advection dominates. Difficulties arise because of the first‐order spatial derivatives which can be elminated by a local coordinate transformation to the characteristic lines of the first order hyperbolic portion of the equation. The resulting differential equation is discretized using a finite difference in time and finite elements in space employing cubic Hermite basis functions. The residuals at individual collocation points are then computed. The sum of the squares of the residuals is minimized to form the necessary set of algebraic equations. The method has performed well in one‐dimensional test problems.
The thermistor: A problem in heat and current flow
Authors: Westbrook D. R.
Published: August 2005

The thermistor is an electrical device that can be used as a current surge protector. The basis of its operation is the temperature‐dependent electrical conductivity that drops by 4 to 5 orders of magnitude over a temperature range of 100–200°C. In the present work the coupled heat and electrical current problems are examined with the conductivity modeled by a step function. A numerical scheme is suggested for the nonlinear coupled problem. Convergence of this scheme for some boundary conditions is demonstrated and some numerical results are given.
Scattering of elastic waves by nonaxisymmetric three‐dimensional dipping layer
Authors: Eshraghi Hossein , Dravinski Marijan
Published: August 2005

Using a boundary method, we investigated the scattering of elastic plane harmonic SH, SV, P, and Rayleigh waves by three‐dimensional nonaxisymmetric dipping layers embedded in an elastic half‐space. The valley was subjected to incident Rayleigh wave and oblique incident SH, SV, and P waves. The method utilized spherical wave functions to express the unknown scattered field. These functions satisfy the equation of motion and radiation conditions at infinity but they do not satisfy the stress‐free boundary conditions at the surface of the half‐space. The boundary and continuity conditions are imposed locally in the least‐square‐sense at several points on the layer interface and on the surface of the half‐space. A comparative study was done to examine the validity and limitations of the two‐dimensional approximations (antiplane and plane strain models) of three‐dimensional models. It is demonstrated that the two‐dimensional approximations may be inadequate to represent actual displacement field for three‐dimensional irregularities.
Convergence properties of a class of boundary element approximations to linear diffusion problems with localized nonlinear reactions
Authors: Peirce Anthony P. , Askar Attila , Rabitz Herschel
Published: August 2005

We consider a boundary element (BE) Algorithm for solving linear diffusion desorption problems with localized nonlinear reactions. The proposed BE algorithm provides an elegant representation of the effect of localized nonlinear reactions, which enables the effects of arbitrarily oriented defect structures to be incorporated into BE models without having to perform severe mesh deformations.

We propose a one‐step recursion procedure to advance the BE solution of linear diffusion localized nonlinear reaction problems and investigate its convergence properties. The separation of the linear and nonlinear effects by the boundary integral formulation enables us to consider the convergence properties of approximations to the linear terms and nonlinear terms of the boundary integral equation separately.

For the linear terms we investigate how the degree of piecewise polynomial collocation in space and the size of the spatial mesh relative to the time step affects the accumulation of errors in the one‐step recursion scheme. We develop a novel convergence analysis that combines asymptotic methods with Lax's Equivalence Theorem. We identify a dimensionless meshing parameter θ whose magnitudé governs the performance of the one‐step BE schemes. In particular, we show that piecewise constant (PWC) and piecewise linear (PWL) BE schemes are conditionally convergent, have lower asymptotic bounds placed on the size of time steps, and which display excess numerical diffusion when small time steps are used. There is no asymptotic bound on how large the tie steps can be–this allows the solution to be advanced in fewer, larger time steps. The piecewise quadratic (PWQ) BE scheme is shown to be unconditionally convergent; there is no asymptotic restriction on the relative sizes of the time and spatial meshing and no numerical diffusion. We verify the theoretical convergence properties in numerical examples. This analysis provides useful information about the appropriate degree of spatial piecewise polynomial and the meshing strategy for a given problem.

For the nonlinear terms we investigate the convergence of an explicit algorithm to advance the solution at an active site forward in time by means of Caratheodory iteration combined with piecewise linear interpolation. We consider a model problem comprising a singular nonlinear Volterra equation that represents the effect of the term in the BE formulation that is due to a single defect. We prove the convergence of the piecewise linear Caratheodory iteration algorithm to a solution of the model problem for as long as such a solution can be shown to exist. This analysis provides a theoretical justification for the use of piecewise linear Caratheodory iterates for advancing the effects of localized reactions.
A generalized finite difference method for solid mechanics
Authors: Dow John O. , Jones Michael S. , Harwood Shawn A.
Published: August 2005

Procedures are developed that improve the applicability of the finite difference method to problems in solid mechanics. This is accomplished by formulating the coefficients of the Taylor series expansion used to approximate derivative quantities in terms of physically interpretable strain gradients. Improvements realized include modeling of boundary conditions that has intuitive appeal and the use of irregular grids in a natural manner. These developments are demonstrated for the analysis of plane stress problems with traction boundary conditions. The results compare well with finite element solutions. The approach suggests further generalization of the finite difference method.
Numerical solution of the equations of compressible flow by a transport method
Authors: Rivlin J. , Kaniel S.
Published: August 2005

This paper implements a numerical solution of the equations of compressible flow, by a transport method. The model is based on a kinetic model, built for the fluid motion. It transforms the equations of motion into integral equations, solved by an analytical‐numerical method. Precomputation of tables for some canonical integrals reduces the amount of work. A boundary value problem (flow into a wedge) is treated and numerical solutions are exhibited. The paper deals with isentropic flow. It is an important special case, among the variety of problems where the assumption of constant entropy causes a negligible error.
A fourth order difference method for the one‐dimensional general quasilinear parabolic partial differential equation
Authors: Jain M. K. , Jain R. K. , Mohanty R. K.
Published: August 2005

A two‐level implicit difference scheme using three spatial grid points of Crandall form of O(k2 + kh2 + h4) is obtained for solving the one‐dimensional quasilinear parabolic partial differential equation, uxx = f(x, t, u, ut, ux) with Dirichlet boundary conditions. The method, when applied to a linear convection‐diffusion problem, is shown to be unconditionally stable. The numerical results show that the proposed method produces accurate and oscillation‐free solutions.
A coupled double splitting ADI scheme for the first biharmonic using collocation
Authors: Cooper K. D. , Prenter P. M.
Published: August 2005

A new method for solving the first biharmonic equation via a double splitting into coupled systems of ordinary differential equations is presented. The first splitting reduces this fourth order partial differential equation into the coupled system uxx + uyy = v and vxx + vyy = f, where u carries both Dirichlet and Neumann boundary data and v carries no boundary data. The pair is then iteratively solved using coupled alternating direction collocation on the resulting systems of ordinary differential equations. This can also be viewed as an alternating direction method of lines for a system of partial differential equations. Although there is no separation of variables underlying the splitting, the method yields a convergent sequence of iterates for a variety of examples under a restricted range of acceleration parameters, and possesses O(h4) accuracy. Desirable features of the algorithm are discussed together with the reduction of bandwidth of the associated collocation matrices under intersticing of u and v variables. Interesting open questions are also discussed.
A probabilistic approach to numerical solution of the nonlinear diffusion equation
Authors: Korzeniowski A.
Published: August 2005

Based on the Feynman‐Kac formula an algorithm for solving the scalar reaction‐diffusion equation is given. The method reduces the approximation to a simple nonrandom averaging which does not require a simulation of Brownian motion. The rate of convergence is also established.
The h‐p version of the finite element method for parabolic equations. II. The h‐p version in time
Authors: Babuš Ivo , Janik Tadeusz
Published: August 2005

The paper is the second in the series addressing the h‐p version of the finite element method for parabolic equations. The present paper addresses the case when in both variables, the spatial and time, the h‐p version is used. Error estimation is given and numerical computations are presented.
Approximation of radial functions by piecewise polynomials on arbitrary grids
Authors: Tourigny Y. , Sanz‐Serna J. M. , Morris J. Li.
Published: August 2005

We study the approximation of radial functions in IRN, N ≥ 2, by means of Lagrange finite elements on arbitrary grids. As an application, error estimates are obtained for the finite‐element discretization of the radial Schrödinger equation. (AMS subject classification number: 65M60).
An O(h6) cubic spline interpolating procedure for harmonic functions
Authors: Papamichael N. , Soares Maria Joana
Published: August 2005

An O(h6) method for the interpolation of harmonic functions in rectangular domains is described and analyzed, The method is based on an earlier cubic spline technique [N. Papamichael and J.R. Whiteman, BIT 14, 452–459 (1974)], and makes use of recent results concerning the a posteriori correction of interpolatory cubic splines.
Parallel finite element splitting‐up method for parabolic problems
Authors: Tai Xue‐Cheng , Neittaanmäki Pekka
Published: August 2005

An efficient method for solving parabolic systems is presented. The proposed method is based on the splitting‐up principle in which the problem is reduced to a series of independent 1D problems. This enables it to be used with parallel processors. We can solve multidimensional problems by applying only the 1D method and consequently avoid the difficulties in constructing a finite element space for multidimensional problems. The method is suitable for general domains as well as rectangular domains. Every 1D subproblem is solved by applying cubic B‐splines. Several numerical examples are presented.
Construction of a novel finite‐difference scheme for a nonlinear diffusion equation
Authors: Mickens Ronald E.
Published: August 2005

A novel finite‐difference scheme is constructed for a particular nonlinear diffusion equation. This scheme does not correspond to any of those that might be expected to follow from application of the standard procedures. A major advantage is that in the appropriate limit the difference equation model reduces to schemes that reproduce exactly the behaviors of the solution to the corresponding differential equation.
Discrete least‐squares global approximations to solutions of partial differential equations
Authors: Pen Ue‐Li
Published: August 2005

Solutions of partial differential equations (PDEs) using globally nonvanishing approximating functions are discussed, and the particular case of global polynomial solutions is studied. Convergence and error bounds are examined. Examples are given and compared with analytic solutions. This method seems particularly well suited for elliptic PDEs with continuous boundary conditions and nonhomogeneous terms, even for irregular domains, offering geometric convergence rates. By providing the minimized residues, strong error indicators are obtained. This algorithm's implementation retains simplicity under a variety of applications.
Finite element approach to the Stokes problem
Authors: Chinosi C. , Comodi M. I.
Published: August 2005

We approximate the Stokes problem by using a finite element method. This method utilizes the approach of Kleiser–Schumann, in which a boundary condition for the pressure is implicitly defined by a condition on the velocity. We consider a suitable splitting of the unknowns that allows one to reduce the Stokes problem to a cascade of classical Dirichlet problems and to a boundary integral equation.
Numerical simulation of the turbulent boundary layer equations via a Runge‐Kutta algorithm
Authors: Campo Antonio , Schuler Carlos
Published: August 2005

This paper addresses a hybrid computational procedure for the step‐by‐step calculation of momentum transfer in turbulent boundary layer flows along flat plates. The proposed procedure relies on a modified method of lines wherein transversal discretizations are carried out by a “control volume” being infinitesimal in the streamwise direction and finite in the transversal direction of the fluid flow. Using mixing length theory and coarse intervals in the transversal direction, the resulting system of ordinary differential equations of first order may be readily integrated on a personal computer utilizing a fourth‐order Runge‐Kutta algorithm. In general, a maximum number of sixteen lines is necessary at the trailing edge of the flat plate for a typical calculation. As a consequence, computing time and storage for each run were very small when compared to other finite‐difference methods. Furthermore, to validate the hybrid procedure involving the method of lines and control volumes (MOLCV), comparisons with experimental data have been done in terms of both velocity distributions and local skin friction coefficients. For all cases tested, the proposed methodology predicts the growth of the boundary layer of gases correctly.
Modified methods of fundamental solutions for harmonic and biharmonic problems with boundary singularities
Authors: Karageorghis Andreas
Published: August 2005

A modified version of the method of fundamental solutions which incorporates the singular behavior of the problem under consideration is introduced. The method is tested on potential and biharmonic problems and its performance is compared to the performance of the standard method of fundamental solutions.
Projection methods for the numerical solution of non‐self‐adjoint elliptic partial differential equations
Authors: Kamath C. , Weeratunga S.
Published: August 2005

We compare the relative performances of two iterative schemes based on projection techniques for the solution of large sparse nonsymmetric systems of linear equations, encountered in the numerical solution of partial differential equations. The Block–Symmetric Successive Over‐Relaxation (Block‐SSOR) method and the Symmetric–Kaczmarz method are derived from the simplest of projection methods, that is, the Kaczmarz method. These methods are then accelerated using the conjugate gradient method, in order to improve their convergence. We study their behavior on various test problems and comment on the conditions under which one method would be better than the other. We show that while the conjugate‐gradient‐accelerated Block‐SSOR method is more amenable to implementation on vector and parallel computers, the conjugate‐gradient accelerated Symmetric–Kaczmarz method provides a viable alternative for use on a scalar machine.
Comparison of S and V cycles in multigrid method for linear elliptic equations with variable coefficients
Authors: Lyengar Satteluri R. K. , Goyal Anu
Published: August 2005

We implement and compare the Sawtooth (S) and V cycles in multigrid method (mgd) for the Poisson equation and for the fourth‐order 13‐point methods derived by Krishnaiah et al. Defect correction in the mgd context is also studied. Computations reveal that the S cycle is much cheaper compared to the V cycle in most of the situations. Defect correction is much cheaper than the direct application of the 13‐point formula, giving almost the fourth‐order accurate results.
Computation of viscous or nonviscous conservation law by domain decomposition based on asymptotic analysis
Authors: Bourgeat A. , Garbey M.
Published: August 2005

An accurate and efficient numerical method has been developed for a nonlinear diffusion convection‐dominated problem. The scheme combines asymptotic methods with usual solution techniques for hyperbolic problems. After having localized shock or corner layers and rescaling, first terms of the inner expansion are computed. Using the same concepts gives a method to compute a very accurate solution of the nonlinear conservation law. Because our numerical scheme is based on a uniform approximation throughout the domain, the shock is localized very accurately and there is practically no smearing out. Numerical computations are presented. Another novel feature is the ability to break down the problem according to subdomains of different local behavior, based on asymptotic analysis, which may make it feasible to do computations with different processors.
Elliptic partial differential equation and optimal control
Authors: Wan‐xie Zhong , Xiang‐Xiang Zhong
Published: August 2005

The theory of optimal control and the semianalytical method of elliptic partial differential equation (PDE) in a prismatic domain are mutually simulated issues. The simulation of discrete‐time linear quadratic (LQ) control with the substructural chain problem in static structural analysis is given first. From the minimum potential energy variational principle of substructural chain, the generalized variational principle with two kinds of variables and the dual equations are derived. The simulation relation is then recognized by comparing the variational principle and dual equations of the LQ control theory. The simulation between elliptic PDE in the prismatic domain and continuous‐time LQ control is established in the same way, and the interval energy is naturally introduced, as in the case of substructural chain. The assembling and condensation equations can help one to derive the differential equations of the submatrices of potential energy and mixed energy. The well known Riccati equation is one of them. The interval assembling and condensation algorithm can be used to solve the Riccati equation. Some numerical examples are given to illustrate the method.
A fully Sinc‐Galerkin method for Euler–Bernoulli beam models
Authors: Smith Ralph C. , Bowers Kenneth L. , Lund John
Published: August 2005

A fully Sinc‐Galerkin method in both space and time is presented for fourth‐order time‐dependent partial differential equations with fixed and cantilever boundary conditions. The sine discretizations for the second‐order temporal problem and the fourth‐order spatial problems are presented. Alternate formulations for variable parameter fourth‐order problems are given, which prove to be especially useful when applying the forward techniques of this article to parameter recovery problems. The discrete system that corresponds to the time‐dependent partial differential equations of interest are then formulated. Computational issues are discussed and an accurate and efficient algorithm for solving the resulting matrix system is outlined. Numerical results that highlight the method are given for problems with both analytic and singular solutions as well as fixed and cantilever boundary conditions.
Projection functions for particle‐grid methods
Authors: Bagtzoglou Amvrossios C. , Tompson Andrew F. B. , Dougherty David E.
Published: August 2005

Random walk particle methods (RWPM) can be used in operator splitting schemes to simulate reactive solute transport in porous media. Projection functions are used to transfer particle location and mass information to concentrations at selected spatial points. Because of the stochastic nature of RWPM, concentration estimates made from particle distributions include a “noisy” error component. In some cases of reactive or density‐dependent flows, this type of error may be propagated forward in time. It can be reduced by using larger numbers of particles or by using different projection functions. The effects of using different projection functions or numbers of particles in different flow regimes or dimensions are explored using concentration solutions for a set of one‐, two‐, and three‐dimensional nonreactive test problems. Resulting solutions are compared with analytic results and classical random walk error estimates. A piecewise linear projection function provides a reasonable improvement in accuracy over the more convenient box methods at a modest increase in cost. The support of the projection functions should be O(Δx) to avoid excessive smearing. Multidimensional projection functions may be advantageously formed by products of different one‐dimensional projection functions.
Nodal adjoint method for multigroup diffusion in heterogeneous slab reactors
Authors: Haidar Nassar H. S.
Published: August 2005

A variational adjoint nodal method is proposed to yield an approximation theory for the eigenvalues of the multigroup neutron diffusion boundary value problem of a heterogeneous quasicritical one‐dimensional slab reactor. Semianalytical eigensolutions are constructed for the associated whole‐reactor group nodal fluxes. The method appears to be mathematically more consistent, computationally more straightforward, and practically more convenient than alternative nodal or finite element schemes. © 1992 John Wiley & Sons, Inc.
Positive solutions and dynamics of a finite difference reaction–diffusion system
Authors: Pao C. V.
Published: August 2005

We investigate the dynamics and methods of computation for some nonlinear finite difference systems that are the discretized equations of a time‐dependent and a steady‐state reaction–diffusion problem. The formulation of the discrete equations for the time‐dependent problem is based on the implicit method for parabolic equations, and the computational algorithm is based on the method of monotone iterations using upper and lower solutions as the initial iterations. The monotone iterative method yields improved upper and lower bounds of the solution in each iteration, and the sequence of iterations converges monotonically to a solution for both the time‐dependent and the steady‐state problems. An important consequence of this method is that it leads to a bifurcation point that determines the dynamic behavior of the time‐dependent problem in relation to the corresponding steady‐state problem. This bifurcation point also determines whether the steady‐state problem has one or two non‐negative solutions, and is explicitly given in terms of the physical parameters of the system and the type of boundary conditions. Numerical results are presented for both the time‐dependent and the steady‐state problems under various boundary conditions, including a test problem with known analytical solution. These numerical results exhibit the predicted dynamic behavior of the time‐dependent solution given by the theoretical analysis. Also discussed are the numerical stability of the computational algorithm and the convergence of the finite difference solution to the corresponding continuous solution of the reaction–diffusion problem. © 1993 John Wiley & Sons, Inc.
Numerical grid generator based on Moser's deformation method
Authors: Liao Guojun , Pan Tsorng‐Whay , Su Jianzhong
Published: August 2005

A deformation method of grid generation has been recently proposed. The method is based on Moser's deformation scheme in his study of the volume elements of a Riemannian manifold. In this paper, this deformation method is improved so that the grid adaption takes place according to the physical coordinates. Some numerical results are included to demonstrate the method. © 1994 John Wiley & Sons, Inc.
Eulerian‐Lagrangian method of cells based on localized adjoint method
Authors: Herrera G. , Herrera I.
Published: August 2005

The localized adjoint method, when applied using an Eulerian‐Lagrangian frame, has been quite successful in treating advection‐dominated transport. The resulting methodology is known as ELLAM. In previous work, bilinear functions were used as test functions. In this paper, local constant functions are used instead, leading to procedures which are appealing because, in addition to other advantages of ELLAM methods, they ensure local mass conservation, are easy to apply and can be combined without difficulty with existing solute‐transport codes which are based on finite volumes. In addition, the procedures for deriving the algorithms presented here are used as an illustration of a general methodology for treating numerically partial differential equations, which is advocated by the authors. Such methodology consists in identifying the information about the sought solution which is contained in the approximate one and then using this insight to choose the interpolation procedure to be applied. © 1994 John Wiley & Sons, Inc.
A note on the least squares solution of the groundwater flow equation
Authors: Bentley L. R.
Published: August 2005

The least squares finite element method is a member of the weighted residuals class of numerical methods for solving partial differential equations. The least squares finite element method is applied to the groundwater flow equation. Space is discretized with a C1 continuous trial function and parameters are approximated with a C0 bilinear basis. Solutions for problems containing parameters with large localized spatial gradients are characterized by errors that are propagated throughout the entire domain. Second‐order spatial convergence is observed, and extreme mesh refinement is required to match Galerkin and mixed least squares finite element results. Temporal discretization should be kept separate from the least squares spatial discretization. © 1994 John Wiley & Sons, Inc.
A best finite‐difference scheme for the fisher equation
Authors: Mickens Ronald E.
Published: August 2005

A new class of finite‐difference schemes is constructed for the Fisher partial differential equation. These schemes are constructed according to the nonstandard modeling rules formulated by Mickens. They have the property that, in the appropriate limits, the discrete models obtained are either “exact” or “best” finite‐difference schemes for corresponding differential equation. Consequently, the elementary numerical instabilities will not occur. © 1994 John Wiley & Sons, Inc.
On the numerical solution of axisymmetric domain optimization problems by dual finite element method
Authors: Hlaváček Ivan , Mäkinen Raino
Published: August 2005

Shape optimization of an axisymmetric three‐dimensional domain with an elliptic boundary value state problem is solved. Since the cost functional is given in terms of the cogradient of the solution, a dual finite element method based on the minimum of complementary energy principle is used. © 1994 John Wiley & Sons, Inc.
Pointwise superconvergence of the gradient for the linear tetrahedral element
Authors: Goodsell George
Published: August 2005

We consider the finite element approximation to the solution of a self‐adjoint, second‐order elliptic boundary value problem in three dimensions over a fully uniform mesh of piecewise linear tetrahedral elements. Although the resulting approximation to the gradient is optimal for functions from the approximating space, it is, however, only O(h). We show how this can be improved by the recovery, from the finite element solution, of an approximation to the gradient, which is pointwise of a higher order of accuracy than that of the gradient of the finite element approximation. This approximation, termed a recovered gradient function, is, thus, superconvergent. The major task of our analysis is the establishing of an (almost) constant bound on the W seminorm of the finite element approximation to a smoothed derivative Green's function. © 1994 John Wiley & Sons, Inc.
A mixed problem for electrostatic potential in semiconductors
Authors: Sacco Riccardo
Published: August 2005

In this article we deal with the solution in Ω ⊂ R2 of the quasi linear equation −Δu = f(x, y, u(x, y)) subject to mixed boundary data and representing Gauss' law in a semiconductor device, where u and f are, respectively, the electrostatic potential and the space charge density after a suitable scaling. In the following we consider the associated variational problem of finding in a suitable subspace of H1(Ω) the minimum of the functional \documentclass{article}\pagestyle{empty}\begin{document}$ J(u)\, = \,\int {_\Omega } (\frac{1}{2}\left| {\nabla u\left| {^2 \, - \,{\cal F}(x,y,u)\,d\Omega,} \right.} \right. $\end{document}, where \documentclass{article}\pagestyle{empty}\begin{document}$ {\cal F}(x,y,u)\, = \,\int f (x,y,\xi)\,d\xi, $\end{document} and we prove existence and uniqueness of a weak solution according to the technique of Convex Analysis. The numerical study is then carried on by a piecewise linear finite element approximation, which is proved to converge in the H1‐norm to the exact solution of the variational problem; some numerical examples are also included. © 1994 John Wiley & Sons, Inc.
Difference approximations for wave equations via finite elements. I. Construction and performance
Authors: Card Curtis L. , Allen Myron B.
Published: August 2005

Many practical applications of wave equations involve media in which there are interfaces, or discontinuities in material properties. The accurate numerical representation of these interfaces is important in mathematical models. One can develop generalizations of standard finite‐difference methods that accommodate sharp interfaces by modifying a straightforward finite‐element approach. In two space dimensions, these methods yield explicit, 5‐point or 9‐point difference schemes that accurately capture reflection, transmission, and refraction at interfaces. The approach also extends readily to the simulation of waves in elastic media. A companion article presents an error analysis for the approach. © 1995 John Wiley & Sons, Inc.