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Article
Publication date: 1 August 1995

T. Laclair and J.I. Frankel

One‐dimensional radiative heat transfer is considered in aplane‐parallel geometry for an absorbing, emitting, and linearly anisotropicscattering medium subjected to…

Abstract

One‐dimensional radiative heat transfer is considered in a plane‐parallel geometry for an absorbing, emitting, and linearly anisotropic scattering medium subjected to azimuthally symmetric incident radiation at the boundaries. The integral form of the transport equation is used throughout the analysis. This formulation leads to a system of weakly‐singular Fredholm integral equations of the second kind. The resulting unknown functions are then formally expanded in Chebyshev series. These series representations are truncated at a specified number of terms, leaving residual functions as a result of the approximation. The collocation and the Ritz‐Galerkin methods are formulated, and are expressed in terms of general orthogonality conditions applied to the residual functions. The major contribution of the present work lies in developing quantitative error estimates. Error bounds are obtained for the approximating functions by developing equations relating the residuals to the errors and applying functional norms to the resulting set of equations. The collocation and Ritz‐Galerkin methods are each applied in turn to determine the expansion coefficients of the approximating functions. The effectiveness of each method is interpreted by analyzing the errors which result from the approximations.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 5 no. 8
Type: Research Article
ISSN: 0961-5539

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Article
Publication date: 11 May 2010

Adrien Catella, Victorita Dolean and Stéphane Lanteri

The purpose of this paper is to develop a time implicit discontinuous Galerkin method for the simulation of two‐dimensional time‐domain electromagnetic wave propagation on…

Abstract

Purpose

The purpose of this paper is to develop a time implicit discontinuous Galerkin method for the simulation of two‐dimensional time‐domain electromagnetic wave propagation on non‐uniform triangular meshes.

Design/methodology/approach

The proposed method combines an arbitrary high‐order discontinuous Galerkin method for the discretization in space designed on triangular meshes, with a second‐order Cranck‐Nicolson scheme for time integration. At each time step, a multifrontal sparse LU method is used for solving the linear system resulting from the discretization of the TE Maxwell equations.

Findings

Despite the computational overhead of the solution of a linear system at each time step, the resulting implicit discontinuous Galerkin time‐domain method allows for a noticeable reduction of the computing time as compared to its explicit counterpart based on a leap‐frog time integration scheme.

Research limitations/implications

The proposed method is useful if the underlying mesh is non‐uniform or locally refined such as when dealing with complex geometric features or with heterogeneous propagation media.

Practical implications

The paper is a first step towards the development of an efficient discontinuous Galerkin method for the simulation of three‐dimensional time‐domain electromagnetic wave propagation on non‐uniform tetrahedral meshes. It yields first insights of the capabilities of implicit time stepping through a detailed numerical assessment of accuracy properties and computational performances.

Originality/value

In the field of high‐frequency computational electromagnetism, the use of implicit time stepping has so far been limited to Cartesian meshes in conjunction with the finite difference time‐domain (FDTD) method (e.g. the alternating direction implicit FDTD method). The paper is the first attempt to combine implicit time stepping with a discontinuous Galerkin discretization method designed on simplex meshes.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 29 no. 3
Type: Research Article
ISSN: 0332-1649

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Article
Publication date: 6 January 2012

S.A. Yousefi, Zahra Barikbin and Mehdi Dehghan

The purpose of this paper is to implement the Ritz‐Galerkin method in Bernstein polynomial basis to give approximation solution of a parabolic partial differential…

Abstract

Purpose

The purpose of this paper is to implement the Ritz‐Galerkin method in Bernstein polynomial basis to give approximation solution of a parabolic partial differential equation with non‐local boundary conditions.

Design/methodology/approach

The properties of Bernstein polynomial and Ritz‐Galerkin method are first presented, then the Ritz‐Galerkin method is utilized to reduce the given parabolic partial differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the new technique.

Findings

The authors applied the method presented in this paper and solved three test problems.

Originality/value

This is the first time that the Ritz‐Galerkin method in Bernstein polynomial basis is employed to solve the model investigated in the current paper.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 22 no. 1
Type: Research Article
ISSN: 0961-5539

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Article
Publication date: 1 June 2003

G.Y. Yu

An indirect symmetric Galerkin BEM (SGBEM) is applied to 2D potential problems in this paper. Based on the assumption that solutions from different methods should be the…

Abstract

An indirect symmetric Galerkin BEM (SGBEM) is applied to 2D potential problems in this paper. Based on the assumption that solutions from different methods should be the same, the hypersingular matrix appeared in SGBEM is approximately expressed by those matrices appeared in asymmetric Galerkin BEM (AGBEM). As only strong and weak singularities need to be solved, the problem becomes much simpler. The space derivatives of potential are expressed with a set of new meaning distributed flux, which will produce the same potential on the boundary position for Ω in the unbounded domain Ω+Ω′, so that hypersingularity will not appear for boundary points. Therefore, there is no need of C1,α for the spatial interpolation function (no Galerkin integration can be used for this purpose). Formulations for both the steady‐state and time‐domain potential problems are given. Three numerical examples are analyzed to demonstrate the effectiveness and accuracy of the proposed indirect method.

Details

Engineering Computations, vol. 20 no. 4
Type: Research Article
ISSN: 0264-4401

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Article
Publication date: 4 July 2016

Qingdong Zhang, Boyang Zhang and Xingfu Lu

The purpose of this paper is to propose a hybridization numerical method to solve the plastic deformation of metal working based on the flow function method and meshless…

Abstract

Purpose

The purpose of this paper is to propose a hybridization numerical method to solve the plastic deformation of metal working based on the flow function method and meshless method.

Design/methodology/approach

The proposed method is named as flow function-element free Galerkin (F-EFG) method. It uses the flow function as the basic unknown quantity to get the basic control equation, the compactly supported approximate function to establish a local approximate flow function by means of moving least square approximation, and the element free Galerkin (EFG) method to solve variational equation. The F-EFG method takes the upper limit method essence of flow function method, and the convergence, stability, and error characteristics of EFG method.

Findings

The steady extrusion process of the axisymmetric extrusion problems as well as the extrusion deformation law and main field variables are subjects in the modeling and simulation analysis using F-EFG method. The results show that the F-EFG method has good computational efficiency and accuracy.

Originality/value

The F-EFG method proposed in this paper has the advantages of high-solution precision of flow function method and large deformation solution of element free method. It overcomes the difficulties in global flow function establishment in flow function method and low-solution efficiency in element free method. The method is beneficial to the development of flow function method and element free method.

Details

Engineering Computations, vol. 33 no. 5
Type: Research Article
ISSN: 0264-4401

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Article
Publication date: 9 April 2019

Uddhaba Biswal, Snehashish Chakraverty and Bata Krushna Ojha

The purpose of this paper is to carry out a detailed investigation to study the natural convection of a non-Newtonian nanofluid flow between two vertical parallel plates…

Abstract

Purpose

The purpose of this paper is to carry out a detailed investigation to study the natural convection of a non-Newtonian nanofluid flow between two vertical parallel plates. In this study, sodium alginate has been taken as a base fluid and nanoparticles that added to it are copper and silver. Maxwell–Garnetts and Brinkman models are used to calculate the effective thermal conductivity and viscosity of nanofluid, respectively.

Design/methodology/approach

The authors used two methods in this study, namely, Galerkin’s method and homotopy perturbation method.

Findings

This paper investigates the velocity and temperature profile of nanofluid and the real fluid flow between two vertical parallel plates. The impacts of physical parameters such as nanofluid volume fraction and dimensionless non-Newtonian viscosity are discussed.

Originality/value

Coupled non-linear differential equations are solved for velocity and temperature. A model is proposed in such a way that the authors may get the solution of real fluid from the nanofluid by neglecting the nano term. The authors do not require a further calculation for real fluid problem.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 29 no. 6
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 1 July 2006

J. Belinha and L.M.J.S. Dinis

The aim of this paper is to extend the Element Free Galerkin method (EFGM) in order to perform the elasto‐plastic analysis of isotropic plates.

Abstract

Purpose

The aim of this paper is to extend the Element Free Galerkin method (EFGM) in order to perform the elasto‐plastic analysis of isotropic plates.

Design/methodology/approach

The EFGM shape‐function construction is briefly presented. The Newton‐Raphson method and the elasto‐plastic algorithm adapted to the EFGM, are described. Several plate bending non‐linear material problems are solved and the obtained solutions are compared with available finite element method (FEM) solutions.

Findings

The paper finds that the developed EFGM approach is a good alternative to the FEM for the solution of non‐linear problems, once the obtained results with the EFGM show a high similarity with the obtained FEM results.

Research limitations/implications

Comparing the FEM and the EFGM there are some drawbacks for the EFGM. The computational cost of the EFGM is higher, the imposition of the essential boundary conditions is more complex and there is a high sensitivity of the method in what concerns the choice of the influence domain and the choice of the weight function.

Practical implications

The knowledge that the EFGM formulation can be treated almost as the FEM formulation once the EFGM parameters are calibrated and optimized.

Originality/value

The extension of the EFGM to the elasto‐plastic analysis of isotropic plates.

Details

Engineering Computations, vol. 23 no. 5
Type: Research Article
ISSN: 0264-4401

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Article
Publication date: 4 January 2016

Mehdi Jamei and H Ghafouri

The purpose of this paper is to present a novel sequential implicit discontinuous Galerkin (DG) method for two-phase incompressible flow in porous media. It is based on…

Abstract

Purpose

The purpose of this paper is to present a novel sequential implicit discontinuous Galerkin (DG) method for two-phase incompressible flow in porous media. It is based on the wetting phase pressure-saturation formulation with Robin boundary condition (Klieber and Riviere, 2006) using H(div) velocity projection.

Design/methodology/approach

The local mass conservation and continuity of normal component of velocity across elements interfaces are enforced by a simple H(div) velocity projection in lowest order Raviart-Thomas (RT0) space. As further improvements, the authors use the weighted averages and the scaled penalties in spatial DG discretization. Moreover, the Chavent-Jaffre slope limiter, as a consistent non-oscillatory limiter, is used for saturation values to avoid the spurious oscillations.

Findings

The proposed model is verified by a pseudo 1D Buckley-Leverett problem in homogeneous media. Two homogeneous and heterogeneous quarter five-spot benchmark problems and a random permeable medium are used to show the accuracy of the method at capturing the sharp front and illustrate the impact of proposed improvements.

Research limitations/implications

The work illustrates incompressible two-phase flow behavior and the capillary pressure heterogeneity between different geological layers is assumed to be negligible.

Practical implications

The proposed model can efficiently be used for modeling of two-phase flow in secondary recovery of petroleum reservoirs and tracing the immiscible contamination in porous media.

Originality/value

The authors present an efficient sequential DG method for immiscible incompressible two-phase flow in porous media with improved performance for detection of sharp frontal interfaces and discontinuities.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 26 no. 1
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 15 February 2008

Ali Şahin, İdris Dağ and Bülent Saka

This paper seeks to develop an efficient B‐spline Galerkin scheme for solving the Fisher's equation, which is a nonlinear reaction diffusion equation describing the…

Abstract

Purpose

This paper seeks to develop an efficient B‐spline Galerkin scheme for solving the Fisher's equation, which is a nonlinear reaction diffusion equation describing the relation between the diffusion and nonlinear multiplication of a species.

Design/methodology/approach

The solution domain is partitioned into uniform mesh and, using the quartic B‐spline functions, the Galerkin method is applied to the Fisher's equation.

Findings

The method yields stable accurate solutions. Obtained results are acceptable and in unison with some earlier studies.

Originality/value

Using the uniform mesh, quartic B‐spline Galerkin method is employed for finding the numerical solutions of Fisher's equation.

Details

Kybernetes, vol. 37 no. 2
Type: Research Article
ISSN: 0368-492X

Keywords

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Article
Publication date: 1 January 1988

A.E. Kanarachos and C.N. Spentzas

Considering typical self‐adjoint and non‐self‐adjoint problems that are governed by differential equations with predominant lower order derivatives, a comparison is…

Abstract

Considering typical self‐adjoint and non‐self‐adjoint problems that are governed by differential equations with predominant lower order derivatives, a comparison is presented of their finite element solutions by Ritz, Galerkin, least square (LSQ) and discrete least square (DLSQ) methods.

Details

Engineering Computations, vol. 5 no. 1
Type: Research Article
ISSN: 0264-4401

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