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Article
Publication date: 4 December 2017

Milos Ivanovic, Marina Svicevic and Svetislav Savovic

The purpose of this paper is to improve the accuracy and stability of the existing solutions to 1D Stefan problem with time-dependent Dirichlet boundary conditions. The…

Abstract

Purpose

The purpose of this paper is to improve the accuracy and stability of the existing solutions to 1D Stefan problem with time-dependent Dirichlet boundary conditions. The accuracy improvement should come with respect to both temperature distribution and moving boundary location.

Design/methodology/approach

The variable space grid method based on mixed finite element/finite difference approach is applied on 1D Stefan problem with time-dependent Dirichlet boundary conditions describing melting process. The authors obtain the position of the moving boundary between two phases using finite differences, whereas finite element method is used to determine temperature distribution. In each time step, the positions of finite element nodes are updated according to the moving boundary, whereas the authors map the nodal temperatures with respect to the new mesh using interpolation techniques.

Findings

The authors found that computational results obtained by proposed approach exhibit good agreement with the exact solution. Moreover, the results for temperature distribution, moving boundary location and moving boundary speed are more accurate than those obtained by variable space grid method based on pure finite differences.

Originality/value

The authors’ approach clearly differs from the previous solutions in terms of methodology. While pure finite difference variable space grid method produces stable solution, the mixed finite element/finite difference variable space grid scheme is significantly more accurate, especially in case of high alpha. Slightly modified scheme has a potential to be applied to 2D and 3D Stefan problems.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 27 no. 12
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 11 January 2008

A. Arefmanesh and M.A. Alavi

This paper aims to develop a hybrid finite differencefinite element method and apply it to solve the three‐dimensional energy equation in non‐isothermal fluid flow past…

Abstract

Purpose

This paper aims to develop a hybrid finite differencefinite element method and apply it to solve the three‐dimensional energy equation in non‐isothermal fluid flow past over a tube.

Design/methodology/approach

To implement the hybrid scheme, the tube length is partitioned into uniform segments by choosing grid points along its length, and a plane perpendicular to the tube axis is drawn at each of the points. Subsequently, the Taylor‐Galerkin finite element technique is employed to discretize the energy equation in the planes; while the derivatives along the tube are discretized using the finite difference method.

Findings

To demonstrate the validity of the proposed numerical scheme, three‐dimensional test cases have been solved using the method. The variation of L2‐norm of the error with mesh refinement shows that the numerical solution converges to the exact solution with mesh refinement. Moreover, comparison of the computational time duration shows that the proposed method is approximately three times faster than the 3D finite element method. In the non‐isothermal fluid flow around a tube for Re=250 and Pr=0.7, the results show that the Nusselt number decreases with the increase in the tube length and, for the tube length greater than six times the tube diameter, the average Nusselt number converges to the value for the two‐dimensional case.

Originality/value

A hybrid finite differencefinite element method has been developed and applied to solve the 3D transient energy equation for different test cases. The proposed method is faster, and computationally more efficient, compared with the 3D finite element method.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 18 no. 1
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 June 2000

A. Savini

Gives introductory remarks about chapter 1 of this group of 31 papers, from ISEF 1999 Proceedings, in the methodologies for field analysis, in the electromagnetic…

1041

Abstract

Gives introductory remarks about chapter 1 of this group of 31 papers, from ISEF 1999 Proceedings, in the methodologies for field analysis, in the electromagnetic community. Observes that computer package implementation theory contributes to clarification. Discusses the areas covered by some of the papers ‐ such as artificial intelligence using fuzzy logic. Includes applications such as permanent magnets and looks at eddy current problems. States the finite element method is currently the most popular method used for field computation. Closes by pointing out the amalgam of topics.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 19 no. 2
Type: Research Article
ISSN: 0332-1649

Keywords

Article
Publication date: 19 June 2007

Mehdi Dehghan

The diffusion‐advection phenomena occur in many physical situations such as, the transport of heat in fluids, flow through porous media, the spread of contaminants in…

Abstract

Purpose

The diffusion‐advection phenomena occur in many physical situations such as, the transport of heat in fluids, flow through porous media, the spread of contaminants in fluids and as well as in many other branches of science and engineering. So it is essential to approximate the solution of these kinds of partial differential equations numerically in order to investigate the prediction of the mathematical models, as the exact solutions are usually unavailable.

Design/methodology/approach

The difficulties arising in numerical solutions of the transport equation are well known. Hence, the study of transport equation continues to be an active field of research. A number of mathematicians have developed the method of time‐splitting to divide complicated time‐dependent partial differential equations into sets of simpler equations which could then be solved separately by numerical means over fractions of a time‐step. For example, they split large multi‐dimensional equations into a number of simpler one‐dimensional equations each solved separately over a fraction of the time‐step in the so‐called locally one‐dimensional (LOD) method. In the same way, the time‐splitting process can be used to subdivide an equation incorporating several physical processes into a number of simpler equations involving individual physical processes. Thus, instead of applying the one‐dimensional advection‐diffusion equation over one time‐step, it may be split into the pure advection equation and the pure diffusion equation each to be applied over half a time‐step. Known accurate computational procedures of solving the simpler diffusion and advection equations may then be used to solve the advection‐diffusion problem.

Findings

In this paper, several different computational LOD procedures were developed and discussed for solving the two‐dimensional transport equation. These schemes are based on the time‐splitting finite difference approximations.

Practical implications

The new approach is simple and effective. The results of a numerical experiment are given, and the accuracy are discussed and compared.

Originality/value

A comparison of calculations with the results of the conventional finite difference techniques demonstrates the good accuracy of the proposed approach.

Details

Kybernetes, vol. 36 no. 5/6
Type: Research Article
ISSN: 0368-492X

Keywords

Article
Publication date: 1 June 2000

K. Wiak

Discusses the 27 papers in ISEF 1999 Proceedings on the subject of electromagnetisms. States the groups of papers cover such subjects within the discipline as: induction…

Abstract

Discusses the 27 papers in ISEF 1999 Proceedings on the subject of electromagnetisms. States the groups of papers cover such subjects within the discipline as: induction machines; reluctance motors; PM motors; transformers and reactors; and special problems and applications. Debates all of these in great detail and itemizes each with greater in‐depth discussion of the various technical applications and areas. Concludes that the recommendations made should be adhered to.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 19 no. 2
Type: Research Article
ISSN: 0332-1649

Keywords

Article
Publication date: 9 July 2020

Tadeusz Sobczyk and Marcin Jaraczewski

Discrete differential operators (DDOs) of periodic functions have been examined to solve boundary-value problems. This paper aims to identify the difficulties of using…

Abstract

Purpose

Discrete differential operators (DDOs) of periodic functions have been examined to solve boundary-value problems. This paper aims to identify the difficulties of using those operators to solve ordinary nonlinear differential equations.

Design/methodology/approach

The DDOs have been applied to create the finite-difference equations and two approaches have been proposed to reduce the Gibbs effects, which arises in solutions at discontinuities on the boundaries, by adding the buffers at boundaries and applying the method of images.

Findings

An alternative method has been proposed to create finite-difference equations and an effective method to solve the boundary-value problems.

Research limitations/implications

The proposed approach can be classified as an extension of the finite-difference method based on the new formulas approximating the derivatives. This can be extended to the 2D or 3D cases with more flexible meshes.

Practical implications

Based on this publication, a unified methodology for directly solving nonlinear partial differential equations can be established.

Originality/value

New finite-difference expressions for the first- and second-order derivatives have been applied.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering , vol. 39 no. 4
Type: Research Article
ISSN: 0332-1649

Keywords

Article
Publication date: 1 February 2000

A.N. Pavlov, S.S. Sazhin, R.P. Fedorenko and M.R. Heikal

Detailed results of numerical calculations of transient, 2D incompressible flow around and in the wake of a square prism at Re = 100, 200 and 500 are presented. An…

Abstract

Detailed results of numerical calculations of transient, 2D incompressible flow around and in the wake of a square prism at Re = 100, 200 and 500 are presented. An implicit finitedifference operator‐splitting method, a version of the known SIMPLEC‐like method on a staggered grid, is described. Appropriate theoretical results are presented. The method has second‐order accuracy in space, conserving mass, momentum and kinetic energy. A new modification of the multigrid method is employed to solve the elliptic pressure problem. Calculations are performed on a sequence of spatial grids with up to 401 × 321 grid points, at sequentially halved time steps to ensure grid‐independent results. Three types of flow are shown to exist at Re = 500: a steady‐state unstable flow and two which are transient, fully periodic and asymmetric about the centre line but mirror symmetric to each other. Discrete frequency spectra of drag and lift coefficients are presented.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 10 no. 1
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 April 1993

KEVIN AMARATUNGA and JOHN R. WILLIAMS

We describe how wavelets may be used to solve partial differential equations. These problems are currently solved by techniques such as finite differences, finite elements…

Abstract

We describe how wavelets may be used to solve partial differential equations. These problems are currently solved by techniques such as finite differences, finite elements and multigrid. The wavelet method, however, offers several advantages over traditional methods. Wavelets have the ability to represent functions at different levels of resolution, thereby providing a logical means of developing a hierarchy of solutions. Furthermore, compactly supported wavelets (such as those due to Daubechies) are localized in space, which means that the solution can be refined in regions of high gradient, e.g. stress concentrations, without having to regenerate the mesh for the entire problem. To demonstrate the wavelet technique, we consider Poisson's equation in two dimensions. By comparison with a simple finite difference solution to this problem with periodic boundary conditions we show how a wavelet technique may be efficiently developed. Dirichlet boundary conditions are then imposed, using the capacitance matrix method described by Proskurowski and Widlund and others. The convergence of the wavelet solutions are examined and they are found to compare extremely favourably to the finite difference solutions. Preliminary investigations also indicate that the wavelet technique is a strong contender to the finite element method.

Details

Engineering Computations, vol. 10 no. 4
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 1 January 2014

Andrzej Demenko and Jan Sykulski

Numerical three-dimensional formulations using vector potential A have been examined for magnetic fields, with emphasis on the finite difference (FDM) and edge element…

Abstract

Purpose

Numerical three-dimensional formulations using vector potential A have been examined for magnetic fields, with emphasis on the finite difference (FDM) and edge element (EEM) methods, with the view to establish common features. The paper aims to discuss these issues.

Design/methodology/approach

It has been shown that for hexahedral elements the FDM equations may be presented in the form similar to the EEM equations, providing the products of the nodal potentials and distances between the nodes are used as unknowns in FDM, instead of the usual nodal potentials.

Findings

The analogy between the FDM and the EEM approach has been established.

Originality/value

It has been demonstrated, following from this and previous publications, that analogy exists between all fundamental methods of field solutions relying on space discretisation. This is helpful in terms of classification of the methods and aids the understanding of physical processes involved.

Details

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, vol. 33 no. 1/2
Type: Research Article
ISSN: 0332-1649

Keywords

Article
Publication date: 1 February 1995

C.I. Goldstein

This paper is concerned with the numerical solution ofmulti‐dimensional convection dominated convection‐diffusionproblems. These problems are characterized by a large…

Abstract

This paper is concerned with the numerical solution of multi‐dimensional convection dominated convection‐diffusion problems. These problems are characterized by a large parameter, K, multiplying the convection terms. The goal of this work is the development and analysis of effective preconditioners for iteratively solving the large system of linear equations arising from various finite element and finite difference discretizations with grid size h. When centered finite difference schemes and standard Galerkin finite element methods are used, h must be related to K by the stability constraint, Kh ≤ C0, where the constant C0 is sufficiently small. A class of preconditioners is developed that significantly reduces the condition number for large K and small h. Furthermore, these preconditioners are inexpensive to implement and well suited for parallel computation. It is shown that under suitable assumptions, the number of iterations remains bounded as h ↓0 with K fixed and, at worst, grows slowly as K ↓ ∞. Numerical results are presented illustrating the theory. It is also shown how to apply the theoretical results to more general convection‐diffusion problems and alternative discretizations (including streamline diffusion methods) that remain stable as Kh ↓ ∞.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 5 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

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