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Exact Solution (exact + solution)

Distribution by Scientific Domains

Engineering	51%
Mathematics and Statistics	25%
Physics and Astronomy	9%
Chemistry	5%
Earth and Environmental Science	4%
2 Other Domains	6%

Distribution within Engineering

Numerical Methods & Algorithms	70%
General & Introductory Mechanical Engineering	3%
8 Other Subdomains	27%

Selected Abstracts

Notice of Plagiarism: A Single Recovery Type Curve from Theis' Exact Solution

GROUND WATER, Issue 1 2004
Article first published online: 9 OCT 200
Shortly after the September-October 2003 issue of the journal was mailed, three readers called our attention to similarities between the paper by N. Samani and M. Pasandi (2003, ?A single recovery type curve from Theis? exact solution,?Ground Water 41, no. 5: 602-607) and a paper published in 1980 by Ram G. Agarwal. Agarwal?s paper, ?A new method to account for producing time effects when drawdown type curves are used to analyze pressure buildup and other test data,? was published by the Society for Petroleum Engineers (1980, in Society of Petroleum Engineers 55th Annual Fall Technical Conference, September 2 1-24, Dallas, Texas: SPE Paper 9289). An investigation by the journal verified that the approach and some of the wording used in the two papers are identical. Dr. Samani and Mr. Pasandi acknowledge the similarity and offer an explanation and apology. [source]

Exact Solution of the Six-Vertex Model with Domain Wall Boundary Conditions: Antiferroelectric Phase

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 6 2010
Pavel Bleher
We obtain the large- n asymptotics of the partition function Zn of the six-vertex model with domain wall boundary conditions in the antiferroelectric phase region, with the weights a = sinh(, , t), b = sinh(, + t), c = sinh(2,), |t| < ,. We prove the conjecture of Zinn-Justin, that as n , ,, Zn = C,4(n,)F [1 + O(n,1)], where , and F are given by explicit expressions in , and t, and ,4(z) is the Jacobi theta function. The proof is based on the Riemann-Hilbert approach to the large- n asymptotic expansion of the underlying discrete orthogonal polynomials and on the Deift-Zhou nonlinear steepest-descent method. © 2009 Wiley Periodicals, Inc. [source]

Exact solution of position dependent mass Schrödinger equation by supersymmetric quantum mechanics

ANNALEN DER PHYSIK, Issue 11-12 2003
R. Koç
Abstract A supersymmetric technique for the solution of the effective mass Schrödinger equation is proposed. Exact solutions of the Schrödinger equation corresponding to a number of potentials are obtained. The potentials are fully isospectral with the original potentials. The conditions for the shape invariance of the potentials are discussed. [source]

Resonance behaviour of viscoelastic fluid in Poiseuille flow in the presence of a transversal magnetic field

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 8 2005
Muhammad R. Mohyuddin
Abstract The oscillatory flow of a viscoelastic fluid in a circular pipe under the influence of a transversal magnetic field is studied. Exact solutions for the axial velocity and flow rate are presented. The velocity enhancement and the resonance behaviour are analysed both numerically and asymptotically in the case of small pipe radii. Approximations for the resonance frequencies and the achievable velocity enhancements are derived. Copyright © 2005 John Wiley & Sons, Ltd. [source]

Exact solutions for a perturbed nonlinear Schrödinger equation by using Bäcklund transformations

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2009
Hassan A. Zedan
Abstract In this paper, the method of deriving the Bäcklund transformation from the Riccati form of inverse method is presented for the perturbed nonlinear Schrödinger equation (PNSE). Consequently, the exact solutions for the PNSE can be obtained by the AKNS class. The technique developed relies on the construction of the wave functions that are solutions of the associated AKNS, that is, a linear eigenvalues problem in the form of a system of partial differential equation. Moreover, we construct a new soliton solution from the old one and its wave function. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Exact solutions of space,time dependent non-linear Schrödinger equations

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2004
Hang-yu Ruan
Abstract Using a general symmetry approach we establish transformations between different non-linear space,time dependent evolution equations of Schrödinger type and their respective solutions. As a special case we study the transformation of the standard non-linear Schrödinger equation (NLS)-equation to a NLS-equation with a dispersion coefficient which decreases exponentially with increasing distance along the fiber. By this transformation we construct from well known solutions of the standard NLS-equation some new exact solutions of the NLS-equation with dispersion. Copyright 2004 John Wiley & Sons, Ltd. [source]

Exact solution of position dependent mass Schrödinger equation by supersymmetric quantum mechanics

Exact solutions to the time-dependent supersymmetric Jaynes-Cummings model and the Chiao-Wu model

ANNALEN DER PHYSIK, Issue 3 2003
J.-Q. Shen
Abstract The present paper obtains the exact solutions to the time-dependent supersymmetric two-level multiphoton Jaynes-Cummings model and the Chiao-Wu model that describes the propagation of a photon inside an optical fiber. On the basis of the fact that the two-level multiphoton Jaynes-Cummings model possesses a supersymmetric structure, an invariant is constructed in terms of the supersymmetric generators by working in the sub-Hilbert-space corresponding to a particular eigenvalue of the conserved supersymmetric generators (i.e., the time-independent invariant). By constructing the effective Hamiltonian that describes the interaction of the photon with the medium of the optical fiber, it is further verified that the particular solution to the Schrödinger equation is the eigenfunction of the second-quantized momentum operator of photons field. This, therefore, means that the explicit expression (rather than the hidden form that involves the chronological product) for the time-evolution operator of wave function is obtained by means of the invariant theories. [source]

A lattice Boltzmann method for solute transport

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 8 2009
Jian Guo Zhou
Abstract A lattice Boltzmann method is developed for solute transport. Proper expressions for the local equilibrium distribution functions enable the method to be formulated on rectangular lattice with the same simple procedure as that on a square lattice. This provides an additional advantage over a lattice Boltzmann method on a square lattice for problems characterized by dominant phenomenon in one direction and relatively weak in another such as solute transport in shear flow over a narrow channel, where the problems can efficiently be approached with fine and coarse meshes, respectively, resulting in more efficient algorithm. The stability conditions are also described. The proposed method on a square lattice is naturally recovered when a square lattice is used. It is verified by solving four tests and compared with the analytical/exact solutions. They are in good agreement, demonstrating that the method is simple, accurate and robust for solute transport. Copyright © 2008 John Wiley & Sons, Ltd. [source]

A unified formulation of the piecewise exact method for inelastic seismic demand analysis including the P -delta effect

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, Issue 6 2003
M. N. Ayd
Abstract The non-linear analysis of single-degree-of-freedom (SDOF) systems provides the essential background information for both strength-based design and displacement-based evaluation/design methodologies through the development of the inelastic response spectra. The recursive solution procedure called the piecewise exact method, which is efficiently used for the response analysis of linear SDOF systems, is re-formulated in this paper in a unified format to analyse the non-linear SDOF systems with multi-linear hysteresis models. The unified formulation is also capable of handling the P-delta effect, which generally involves the negative post-yield stiffness of the hysteresis loops. The attractiveness of the method lies in the fact that it provides the exact solution when the loading time history is composed of piecewise linear segments, a condition that is perfectly satisfied for the earthquake excitation. Based on simple recursive relationships given for positive, negative and zero effective stiffnesses, the unified form of the piecewise exact method proves to be an extremely powerful and probably the best tool for the SDOF inelastic time-history and response spectrum analysis including the P-delta effect. A number of examples are presented to demonstrate the implementation of the method. Copyright © 2003 John Wiley & Sons, Ltd. [source]

Analytic Element Modeling of Embedded Multiaquifer Domains

GROUND WATER, Issue 1 2006
Mark Bakker
An analytic element approach is presented for the modeling of multiaquifer domains embedded in a single-aquifer model. The inside of each domain may consist of an arbitrary number of aquifers separated by leaky layers. The analytic element solution is obtained through a combination of existing single-aquifer and multiaquifer analytic elements and allows for the analytic computation of head and leakage at any point in the aquifer. Along the boundary of an embedded multiaquifer domain, the normal flux is continuous everywhere; continuity of head across the boundary is met exactly at collocations points and approximately, but very accurately, in between. The analytic element solution compares well with an existing exact solution. A hypothetical example with a river intersecting two embedded domains illustrates the practical application of the proposed approach. [source]

Notice of Plagiarism: A Single Recovery Type Curve from Theis' Exact Solution

He's homotopy perturbation method for two-dimensional heat conduction equation: Comparison with finite element method

HEAT TRANSFER - ASIAN RESEARCH (FORMERLY HEAT TRANSFER-JAPANESE RESEARCH), Issue 4 2010
M. Jalaal
Abstract Heat conduction appears in almost all natural and industrial processes. In the current study, a two-dimensional heat conduction equation with different complex Dirichlet boundary conditions has been studied. An analytical solution for the temperature distribution and gradient is derived using the homotopy perturbation method (HPM). Unlike most of previous studies in the field of analytical solution with homotopy-based methods which investigate the ODEs, we focus on the partial differential equation (PDE). Employing the Taylor series, the gained series has been converted to an exact expression describing the temperature distribution in the computational domain. Problems were also solved numerically employing the finite element method (FEM). Analytical and numerical results were compared with each other and excellent agreement was obtained. The present investigation shows the effectiveness of the HPM for the solution of PDEs and represents an exact solution for a practical problem. The mathematical procedure proves that the present mathematical method is much simpler than other analytical techniques due to using a combination of homotopy analysis and classic perturbation method. The current mathematical solution can be used in further analytical and numerical surveys as well as related natural and industrial applications even with complex boundary conditions as a simple accurate technique. © 2010 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/htj.20292 [source]

Human and machine effects in a just-in-time scheduling problem

HUMAN FACTORS AND ERGONOMICS IN MANUFACTURING & SERVICE INDUSTRIES, Issue 4 2009
Tamer Eren
In this article, single-machine scheduling problems with learning effects of setup and removal times and deterioration effects of processing time are considered. The objective function of the problem is minimization of the weighted sum of total earliness and total tardiness. To get an exact solution to the problem, a mathematical programming model is proposed. Also the model is tested on an example. To the best of our knowledge, no work exists in the literature that considers the problem presented in this article. © 2009 Wiley Periodicals, Inc. [source]

On the solution of the nonlinear Korteweg,de Vries equation by the homotopy perturbation method

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 12 2009
Ahmet Yildirim
Abstract In this paper, the homotopy perturbation method is used to implement the nonlinear Korteweg,de Vries equation. The analytical solution of the equation is calculated in the form of a convergent power series with easily computable components. A suitable choice of an initial solution can lead to the needed exact solution by a few iterations. Copyright © 2008 John Wiley & Sons, Ltd. [source]

A variational multiscale model for the advection,diffusion,reaction equation

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 7 2009
Guillaume Houzeaux
Abstract The variational multiscale (VMS) method sets a general framework for stabilization methods. By splitting the exact solution into coarse (grid) and fine (subgrid) scales, one can obtain a system of two equations for these unknowns. The grid scale equation is solved using the Galerkin method and contains an additional term involving the subgrid scale. At this stage, several options are usually considered to deal with the subgrid scale equation: this includes the choice of the space where the subgrid scale would be defined as well as the simplifications leading to compute the subgrid scale analytically or numerically. The present study proposes to develop a two-scale variational method for the advection,diffusion,reaction equation. On the one hand, a family of weak forms are obtained by integrating by parts a fraction of the advection term. On the other hand, the solution of the subgrid scale equation is found using the following. First, a two-scale variational method is applied to the one-dimensional problem. Then, a series of approximations are assumed to solve the subgrid space equation analytically. This allows to devise expressions for the ,stabilization parameter' ,, in the context of VMS (two-scale) method. The proposed method is equivalent to the traditional Green's method used in the literature to solve residual-free bubbles, although it offers another point of view, as the strong form of the subgrid scale equation is solved explicitly. In addition, the authors apply the methodology to high-order elements, namely quadratic and cubic elements. The proposed model consists in assuming that the subgrid scale vanishes also on interior nodes of the element and applying the strategy used for linear element in the segment between these interior nodes. The proposed scheme is compared with existing ones through the solution of a one-dimensional numerical example for linear, quadratic and cubic elements. In addition, the mesh convergence is checked for high-order elements through the solution of an exact solution in two dimensions. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Time 2D fundamental solution for saturated porous media with incompressible fluid

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 3 2005
Behrouz Gatmiri
Abstract The derivation of analytical transient two-dimensional fundamental solution for porous media saturated with incompressible fluid in u-p formulation is discussed in detail. First, the explicit Laplace transform solution in terms of solid displacements and fluid pressure are obtained. Then, the closed-form time-dependent fundamental solution is derived by the analytical inversion of the Laplace transform solution. Finally, a set of numerical results is presented to investigate the accuracy of the proposed solution. It is shown that this solution can be considered as a good approximation of exact solution, especially for the long time. Copyright © 2004 John Wiley & Sons, Ltd. [source]

Interior point optimization and limit analysis: an application

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 10 2003
Joseph Pastor
Abstract The well-known problem of the height limit of a Tresca or von Mises vertical slope of height h, subjected to the action of gravity stems naturally from Limit Analysis theory under the plane strain condition. Although the exact solution to this problem remains unknown, this paper aims to give new precise bounds using both the static and kinematic approaches and an Interior Point optimizer code. The constituent material is a homogeneous isotropic soil of weight per unit volume ,. It obeys the Tresca or von Mises criterion characterized by C cohesion. We show that the loading parameter to be optimized, ,h/C, is found to be between 3.767 and 3.782, and finally, using a recent result of Lyamin and Sloan (Int. J. Numer. Meth. Engng. 2002; 55: 573), between 3.772 and 3.782. The proposed methods, combined with an Interior Point optimization code, prove that linearizing the problem remains efficient, and both rigorous and global: this point is the main objective of the present paper. Copyright © 2003 John Wiley & Sons, Ltd. [source]

On the stability and convergence of a Galerkin reduced order model (ROM) of compressible flow with solid wall and far-field boundary treatment,

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 10 2010
I. Kalashnikova
Abstract A reduced order model (ROM) based on the proper orthogonal decomposition (POD)/Galerkin projection method is proposed as an alternative discretization of the linearized compressible Euler equations. It is shown that the numerical stability of the ROM is intimately tied to the choice of inner product used to define the Galerkin projection. For the linearized compressible Euler equations, a symmetry transformation motivates the construction of a weighted L2 inner product that guarantees certain stability bounds satisfied by the ROM. Sufficient conditions for well-posedness and stability of the present Galerkin projection method applied to a general linear hyperbolic initial boundary value problem (IBVP) are stated and proven. Well-posed and stable far-field and solid wall boundary conditions are formulated for the linearized compressible Euler ROM using these more general results. A convergence analysis employing a stable penalty-like formulation of the boundary conditions reveals that the ROM solution converges to the exact solution with refinement of both the numerical solution used to generate the ROM and of the POD basis. An a priori error estimate for the computed ROM solution is derived, and examined using a numerical test case. Published in 2010 by John Wiley & Sons, Ltd. [source]

Error estimation in a stochastic finite element method in electrokinetics

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2010
S. Clénet
Abstract Input data to a numerical model are not necessarily well known. Uncertainties may exist both in material properties and in the geometry of the device. They can be due, for instance, to ageing or imperfections in the manufacturing process. Input data can be modelled as random variables leading to a stochastic model. In electromagnetism, this leads to solution of a stochastic partial differential equation system. The solution can be approximated by a linear combination of basis functions rising from the tensorial product of the basis functions used to discretize the space (nodal shape function for example) and basis functions used to discretize the random dimension (a polynomial chaos expansion for example). Some methods (SSFEM, collocation) have been proposed in the literature to calculate such approximation. The issue is then how to compare the different approaches in an objective way. One solution is to use an appropriate a posteriori numerical error estimator. In this paper, we present an error estimator based on the constitutive relation error in electrokinetics, which allows the calculation of the distance between an average solution and the unknown exact solution. The method of calculation of the error is detailed in this paper from two solutions that satisfy the two equilibrium equations. In an example, we compare two different approximations (Legendre and Hermite polynomial chaos expansions) for the random dimension using the proposed error estimator. In addition, we show how to choose the appropriate order for the polynomial chaos expansion for the proposed error estimator. Copyright © 2009 John Wiley & Sons, Ltd. [source]

An exponentially fitted method for solving Burgers' equation

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 6 2009
Turgut Özi
Abstract In this paper, an exponentially fitted method is used to numerically solve the one-dimensional Burgers' equation. The performance of the method is tested on the model involving moderately large Reynolds numbers. The obtained numerical results show that the method is efficient, stable and reliable for solving Burgers' equation accurately even involving high Reynolds numbers for which the exact solution fails. Copyright © 2009 John Wiley & Sons, Ltd. [source]

Shape optimization of piezoelectric devices using an enriched meshfree method

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2009
C. W. Liu
Abstract We present an enriched reproducing kernel particle method for shape sensitivity analysis and shape optimization of two-dimensional electromechanical domains. This meshfree method incorporates enrichment functions for better representation of discontinuous electromechanical fields across internal boundaries. We use cubic splines for delineating the geometry of internal/external domain boundaries; and the nodal coordinates and slopes of these splines at their control points become the design parameters. This approach enables smooth manipulations of bi-material interfaces and external boundaries during the optimization process. It also enables the calculation of displacement and electric-potential field sensitivities with respect to the design parameters through direct differentiation, for which we adopt the classical material derivative approach. We verify this implementation of sensitivity calculations against an exact solution to a variant of Lamé's problem, and also, finite-difference approximations. We follow a sequential quadratic programming approach to minimize the cost function; and demonstrate the utility of the overall technique through a model problem that involves the shape optimization of a piezoelectric fan. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Dispersion analysis of the meshfree radial point interpolation method for the Helmholtz equation

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2009
Christina Wenterodt
Abstract When numerical methods such as the finite element method (FEM) are used to solve the Helmholtz equation, the solutions suffer from the so-called pollution effect which leads to inaccurate results, especially for high wave numbers. The main reason for this is that the wave number of the numerical solution disagrees with the wave number of the exact solution, which is known as dispersion. In order to obtain admissible results a very high element resolution is necessary and increased computational time and memory capacity are the consequences. In this paper a meshfree method, namely the radial point interpolation method (RPIM), is investigated with respect to the pollution effect in the 2D-case. It is shown that this methodology is able to reduce the dispersion significantly. Two modifications of the RPIM, namely one with polynomial reproduction and another one with a problem-dependent sine/cosine basis, are also described and tested. Numerical experiments are carried out to demonstrate the advantages of the method compared with the FEM. For identical discretizations, the RPIM yields considerably better results than the FEM. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Generalization of robustness test procedure for error estimators.

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2005
Part I: formulation for patches near kinked boundaries
Abstract In this part of paper we shall extend the formulation proposed by Babu,ka and co-workers for robustness patch test, for quality assessment of error estimators, to more general cases of patch locations especially in three-dimensional problems. This is performed first by finding an asymptotic finite element solution at interior parts of a problem with assumed smooth exact solution and then adding a correction part to obtain the solution near a kinked boundary irrespective of other boundary conditions at far ends of the domain. It has been shown that the solution corresponding to the correction part may be obtained in a spectral form by assuming a suitable proportionality relation between the nodal values of a mesh with repeatable pattern of macro-patches. Having found the asymptotic finite element solution, the performance of error estimators may be examined. Although in this paper we focus on the asymptotic behaviour of error estimators, the method described in this part may be used to obtain finite element solution for two/three-dimensional unbounded heat/elasticity problems with homogeneous differential equations. Some numerical results are presented to show the validity and performance of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd. [source]

Smart element method I. The Zienkiewicz,Zhu feedback

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 10 2005
Shaofan Li
Abstract A new error control finite element formulation is developed and implemented based on the variational multiscale method, the inclusion theory in homogenization, and the Zienkiewicz,Zhu error estimator. By synthesizing variational multiscale method in computational mechanics, the equivalent eigenstrain principle in micromechanics, and the Zienkiewicz,Zhu error estimator in the finite element method (FEM), the new finite element formulation can automatically detect and subsequently homogenize its own discretization errors in a self-adaptive and a self-adjusting manner. It is the first finite element formulation that combines an optimal feedback mechanism and a precisely defined homogenization procedure to reduce its own discretization errors and hence to control numerical pollutions. The paper focuses on the following two issues: (1) how to combine a multiscale method with the existing finite element error estimate criterion through a feedback mechanism, and (2) convergence study. It has been shown that by combining the proposed variational multiscale homogenization method with the Zienkiewicz,Zhu error estimator a clear improvement can be made on the coarse scale computation. It is also shown that when the finite element mesh is refined, the solution obtained by the variational eigenstrain multiscale method will converge to the exact solution. Copyright © 2004 John Wiley & Sons, Ltd. [source]

Hybrid-stabilized solid-shell model of laminated composite piezoelectric structures under non-linear distribution of electric potential through thickness

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 10 2003
Lin Quan Yao
Abstract Eighteen-node solid-shell finite element models have been developed for the analysis of laminated composite plate/shell structures embedded with piezoelectric actuators and sensors. The explicit hybrid stabilization method is employed to formulate stabilization vectors for the uniformly reduced integrated 18-node three-dimensional composite solid element. Unlike conventional piezoelectric elements, the concept of the electric nodes introduced in this paper can effectively eliminate the burden of constraining the equality of the electric potential for the nodes lying on the same electrode. Furthermore, the non-linear distribution of electric potential in the piezoelectric layer is expressed by introducing internal electric potential, which not only can simplify modelling but also obtains the same as the exact solution. Copyright © 2003 John Wiley & Sons, Ltd. [source]

An iterative defect-correction type meshless method for acoustics

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 15 2003
V. Lacroix
Abstract Accurate numerical simulation of acoustic wave propagation is still an open problem, particularly for medium frequencies. We have thus formulated a new numerical method better suited to the acoustical problem: the element-free Galerkin method (EFGM) improved by appropriate basis functions computed by a defect correction approach. One of the EFGM advantages is that the shape functions are customizable. Indeed, we can construct the basis of the approximation with terms that are suited to the problem which has to be solved. Acoustical problems, in cavities , with boundary T, are governed by the Helmholtz equation completed with appropriate boundary conditions. As the pressure p(x,y) is a complex variable, it can always be expressed as a function of cos,(x,y) and sin,(x,y) where ,(x,y) is the phase of the wave in each point (x,y). If the exact distribution ,(x,y) of the phase is known and if a meshless basis {1, cos,(x,y), sin, (x,y) } is used, then the exact solution of the acoustic problem can be obtained. Obviously, in real-life cases, the distribution of the phase is unknown. The aim of our work is to resolve, as a first step, the acoustic problem by using a polynomial basis to obtain a first approximation of the pressure field p(x,y). As a second step, from p(x,y) we compute the distribution of the phase ,(x,y) and we introduce it in the meshless basis in order to compute a second approximated pressure field p(x,y). From p(x,y), a new distribution of the phase is computed in order to obtain a third approximated pressure field and so on until a convergence criterion, concerning the pressure or the phase, is obtained. So, an iterative defect-correction type meshless method has been developed to compute the pressure field in ,. This work will show the efficiency of this meshless method in terms of accuracy and in terms of computational time. We will also compare the performance of this method with the classical finite element method. Copyright © 2003 John Wiley & Sons, Ltd. [source]

Numerical inclusion methods of solutions for variational inequalities

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2002
C. S. Ryoo
Abstract We consider a numerical method that enables us to verify the existence of solutions for variational inequalities. This method is based on the infinite dimensional fixed point theorems and explicit error estimates for finite element approximations. Using the finite element approximations and explicit a priori error estimates, we present an effective verification procedure that through numerical computation generates a set which includes the exact solution. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Orthogonality of modal bases in hp finite element models

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2007
V. Prabhakar
Abstract In this paper, we exploit orthogonality of modal bases (SIAM J. Sci. Comput. 1999; 20:1671,1695) used in hp finite element models. We calculate entries of coefficient matrix analytically without using any numerical integration, which can be computationally very expensive. We use properties of Jacobi polynomials and recast the entries of the coefficient matrix so that they can be evaluated analytically. We implement this in the context of the least-squares finite element model although this procedure can be used in other finite element formulations. In this paper, we only develop analytical expressions for rectangular elements. Spectral convergence of the L2 least-squares functional is verified using exact solution of Kovasznay flow. Numerical results for transient flow over a backward-facing step are also presented. We also solve steady flow past a circular cylinder and show the reduction in computational cost using expressions developed herein. Copyright © 2007 John Wiley & Sons, Ltd. [source]

Development of a class of multiple time-stepping schemes for convection,diffusion equations in two dimensions

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2006
R. K. Lin
Abstract In this paper we present a class of semi-discretization finite difference schemes for solving the transient convection,diffusion equation in two dimensions. The distinct feature of these scheme developments is to transform the unsteady convection,diffusion (CD) equation to the inhomogeneous steady convection,diffusion-reaction (CDR) equation after using different time-stepping schemes for the time derivative term. For the sake of saving memory, the alternating direction implicit scheme of Peaceman and Rachford is employed so that all calculations can be carried out within the one-dimensional framework. For the sake of increasing accuracy, the exact solution for the one-dimensional CDR equation is employed in the development of each scheme. Therefore, the numerical error is attributed primarily to the temporal approximation for the one-dimensional problem. Development of the proposed time-stepping schemes is rooted in the Taylor series expansion. All higher-order time derivatives are replaced with spatial derivatives through use of the model differential equation under investigation. Spatial derivatives with orders higher than two are not taken into account for retaining the linear production term in the convection,diffusion-reaction differential system. The proposed schemes with second, third and fourth temporal accuracy orders have been theoretically explored by conducting Fourier and dispersion analyses and numerically validated by solving three test problems with analytic solutions. Copyright © 2006 John Wiley & Sons, Ltd. [source]