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Article

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

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Article

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

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Article

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…

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

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Article

C.P.T. GROTH and J.J. GOTTLIEB

Partially‐decoupled upwind‐based total‐variation‐diminishing (TVD) finitedifference schemes for the solution of the conservation laws governing two‐dimensional…

Abstract

Partially‐decoupled upwind‐based total‐variation‐diminishing (TVD) finitedifference schemes for the solution of the conservation laws governing two‐dimensional non‐equilibrium vibrationally relaxing and chemically reacting flows of thermally‐perfect gaseous mixtures are presented. In these methods, a novel partially‐decoupled flux‐difference splitting approach is adopted. The fluid conservation laws and species concentration and vibrational energy equations are decoupled by means of a frozen flow approximation. The resulting partially‐decoupled gas‐dynamic and thermodynamic subsystems are then solved alternately in a lagged manner within a time marching procedure, thereby providing explicit coupling between the two equation sets. Both time‐split semi‐implicit and factored implicit flux‐limited TVD upwind schemes are described. The semi‐implicit formulation is more appropriate for unsteady applications whereas the factored implicit form is useful for obtaining steady‐state solutions. Extensions of Roe's approximate Riemann solvers, giving the eigenvalues and eigenvectors of the fully coupled systems, are used to evaluate the numerical flux functions. Additional modifications to the Riemann solutions are also described which ensure that the approximate solutions are not aphysical. The proposed partially‐decoupled methods are shown to have several computational advantages over chemistry‐split and fully coupled techniques. Furthermore, numerical results for single, complex, and double Mach reflection flows, as well as corner‐expansion and blunt‐body flows, using a five‐species four‐temperature model for air demonstrate the capabilities of the methods.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 3 no. 6
Type: Research Article
ISSN: 0961-5539

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Article

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

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Article

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

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Article

G.K. Despotis and S. Tsangaris

The extrudate swell phenomenon is analysed by solving, simultaneously,the Navier‐Stokes equations along with the continuity equation bymeans of a finite volume method. In…

Abstract

The extrudate swell phenomenon is analysed by solving, simultaneously, the Navier‐Stokes equations along with the continuity equation by means of a finite volume method. In this work, the planar jet flows of incompressible viscous Newtonian and power‐law fluids for Reynolds numbers as high as 75 are simulated. The method uses the velocity components and pressure as the primitive variables and employs an unstructured triangular grid and triangular or polygonal control volume for each separate variable. The numerical results show good agreement with previously reported experimental and numerical results. Shear thickening results in an increase in swelling ratio, while the introduction of surface tension results in a describes in swelling ratio.

Details

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

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Article

Zbigniew Mańko

While calculating internal forces of a structure resulting from temperature it is necessary to know thermal conduction and what goes hand in hand to determine temperature…

Abstract

While calculating internal forces of a structure resulting from temperature it is necessary to know thermal conduction and what goes hand in hand to determine temperature distribution at various points of the analysed structures. Finite strip method (FSM) is very suitable for the analysis of thermal conduction, heating, heat and temperature distribution in engineering structures, especially rectangular of identical edge conditions. The paper presents several examples of FSM application for the analysis of conduction and heat and temperature distribution for various types of engineering structures which can appear, among others, while welding several joined elements with welds made at specified speed as linear and point welds. Bars, shields, square and rectangular plates, steel orthotropic plates, steel and combined girders (steel‐concrete), box girders subject to various loads connected with heat and temperature (loaded with temperature, non‐uniformly heated surface). The obtained results may be useful in engineering practice for determining actual temperature and load capacity in individual elements of the construction.

Details

Engineering Computations, vol. 4 no. 1
Type: Research Article
ISSN: 0264-4401

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Article

B.S. MANJUNATHA and T. KANT

This paper attempts to evaluate the transverse stresses that are generated within the interface between two layers of laminated composite and sandwich laminates by using C

Abstract

This paper attempts to evaluate the transverse stresses that are generated within the interface between two layers of laminated composite and sandwich laminates by using Cfinite element formulation of higher‐order theories. These theories do not require the use of a fictitious shear correction coefficient which is usually associated with the first‐order Reissner‐Mindlin theory. The in‐plane stresses are evaluated by using constitutive relations. The transverse stresses are evaluated through the use of equilibrium equations. The integration of the equilibrium equations is attempted through forward and central direct finite difference techniques and a new approach, named as, an exact surface fitting method. Sixteen and nine‐noded quadrilateral Lagrangian elements are used. The numerical results obtained by the present approaches in general and the exact surface fitting method in particular, show excellent agreement with available elasticity solutions. New results for symmetric sandwich laminates are also presented for future comparisons.

Details

Engineering Computations, vol. 10 no. 6
Type: Research Article
ISSN: 0264-4401

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Article

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

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