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1 – 10 of over 1000The paper gives the description of boundary element method(BEM) with subdomains for the solution ofconvection—diffusion equations with variable coefficients and Burgers’equations…
Abstract
The paper gives the description of boundary element method (BEM) with subdomains for the solution of convection—diffusion equations with variable coefficients and Burgers’ equations. At first, the whole domain is discretized into K subdomains, in which linearization of equations by representing convective velocity by the sum of constant and variable parts is carried out. Then using fundamental solutions for convection—diffusion linear equations for each subdomain the boundary integral equation (in which the part of the convective term with the constant convective velocity is not included into the pseudo‐body force) is formulated. Only part of the convective term with the variable velocity, which is, as a rule, more than one order less than convective velocity constant part contribution, is left as the pseudo‐source. On the one hand, this does not disturb the numerical BEM—algorithm stability and, on the other hand, this leads to significant improvement in the accuracy of solution. The global matrix, similar to the case of finite element method, has block band structure whereas its width depends only on the numeration order of nodes and subdomains. It is noted, that in comparison with the direct boundary element method the number of global matrix non‐zero elements is not proportional to the square of the number of nodes, but only to the total number of nodal points. This allows us to use the BEM for the solution of problems with very fine space discretization. The proposed BEM with subdomains technique has been used for the numerical solution of one‐dimensional linear steady‐state convective—diffusion problem with variable coefficients and one‐dimensional non‐linear Burgers’ equation for which exact analytical solutions are available. It made it possible to find out the BEM correctness according to both time and space. High precision of the numerical method is noted. The good point of the BEM is the high iteration convergence, which is disturbed neither by high Reynolds numbers nor by the presence of negative velocity zones.
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Karen L. Ricciardi and Stephen H. Brill
The Hermite collocation method of discretization can be used to determine highly accurate solutions to the steady‐state one‐dimensional convection‐diffusion equation (which can be…
Abstract
Purpose
The Hermite collocation method of discretization can be used to determine highly accurate solutions to the steady‐state one‐dimensional convection‐diffusion equation (which can be used to model the transport of contaminants dissolved in groundwater). This accuracy is dependent upon sufficient refinement of the finite‐element mesh as well as applying upstream or downstream weighting to the convective term through the determination of collocation locations which meet specified constraints. Owing to an increase in computational intensity of the application of the method of collocation associated with increases in the mesh refinement, minimal mesh refinement is sought. Very often this optimization problem is the one where the feasible region is not connected and as such requires a specialized optimization search technique. This paper aims to focus on this method.
Design/methodology/approach
An original hybrid method that utilizes a specialized adaptive genetic algorithm followed by a hill‐climbing approach is used to search for the optimal mesh refinement for a number of models differentiated by their velocity fields. The adaptive genetic algorithm is used to determine a mesh refinement that is close to a locally optimal mesh refinement. Following the adaptive genetic algorithm, a hill‐climbing approach is used to determine a local optimal feasible mesh refinement.
Findings
In all cases the optimal mesh refinements determined with this hybrid method are equally optimal to, or a significant improvement over, mesh refinements determined through direct search methods.
Research limitations
Further extensions of this work could include the application of the mesh refinement technique presented in this paper to non‐steady‐state problems with time‐dependent coefficients with multi‐dimensional velocity fields.
Originality/value
The present work applies an original hybrid optimization technique to obtain highly accurate solutions using the method of Hermite collocation with minimal mesh refinement.
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Salam Adel Al-Bayati and Luiz C. Wrobel
The purpose of this paper is to describe an extension of the boundary element method (BEM) and the dual reciprocity boundary element method (DRBEM) formulations developed for one…
Abstract
Purpose
The purpose of this paper is to describe an extension of the boundary element method (BEM) and the dual reciprocity boundary element method (DRBEM) formulations developed for one- and two-dimensional steady-state problems, to analyse transient convection–diffusion problems associated with first-order chemical reaction.
Design/methodology/approach
The mathematical modelling has used a dual reciprocity approximation to transform the domain integrals arising in the transient equation into equivalent boundary integrals. The integral representation formula for the corresponding problem is obtained from the Green’s second identity, using the fundamental solution of the corresponding steady-state equation with constant coefficients. The finite difference method is used to simulate the time evolution procedure for solving the resulting system of equations. Three different radial basis functions have been successfully implemented to increase the accuracy of the solution and improving the rate of convergence.
Findings
The numerical results obtained demonstrate the excellent agreement with the analytical solutions to establish the validity of the proposed approach and to confirm its efficiency.
Originality/value
Finally, the proposed BEM and DRBEM numerical solutions have not displayed any artificial diffusion, oscillatory behaviour or damping of the wave front, as appears in other different numerical methods.
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L.C. WROBEL and D.B. DE FIGUEIREDO
This paper presents a boundary element formulation for transient convection‐diffusion problems employing the fundamental solution of the corresponding steady‐state equation with…
Abstract
This paper presents a boundary element formulation for transient convection‐diffusion problems employing the fundamental solution of the corresponding steady‐state equation with constant coefficients and a dual reciprocity approximation. The formulation allows the mathematical problem to be described in terms of boundary values only. Numerical results show that the BEM does not present oscillations or damping of the wave front as appear in other numerical techniques.
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Marcela B. Goldschmit and Eduardo N. Dvorkin
A generalized Galerkin technique originally developed by Donea,Belytschko and Smolinski for solving the steady convection—diffusionequation using elements with quadratic…
Abstract
A generalized Galerkin technique originally developed by Donea, Belytschko and Smolinski for solving the steady convection—diffusion equation using elements with quadratic interpolation has been modified to extend its application to the case of geometrically distorted 1D and 2D elements. The numerical results indicate that the modified scheme gives accurate results and presents a rather small sensitivity to element distortions.
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Nur Husnina Saadun, Nurul Aini Jaafar, Md Faisal Md Basir, Ali Anqi and Mohammad Reza Safaei
The purpose of this study is to solve convective diffusion equation analytically by considering appropriate boundary conditions and using the Taylor-Aris method to determine the…
Abstract
Purpose
The purpose of this study is to solve convective diffusion equation analytically by considering appropriate boundary conditions and using the Taylor-Aris method to determine the solute concentration, the effective and relative axial diffusivities.
Design/methodology/approach
>An analysis has been conducted on how body acceleration affects the dispersion of a solute in blood flow, which is known as a Bingham fluid, within an artery. To solve the system of differential equations analytically while validating the target boundary conditions, the blood velocity is obtained.
Findings
The blood velocity is impacted by the presence of body acceleration, as well as the yield stress associated with Casson fluid and as such, the process of dispersing the solute is distracted. It graphically illustrates how the blood velocity and the process of solute dispersion are affected by various factors, including the amplitude and lead angle of body acceleration, the yield stress, the gradient of pressure and the Peclet number.
Originality/value
It is witnessed that the blood velocity, the solute concentration and also the effective and relative axial diffusivities experience a drop when either of the amplitude, lead angle or the yield stress rises.
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Some existing finite difference methods for the numerical solution of convection dominated diffusion equations are compared. Two semi‐implicit methods, Lagrangian based and…
Abstract
Some existing finite difference methods for the numerical solution of convection dominated diffusion equations are compared. Two semi‐implicit methods, Lagrangian based and applied on an Eulerian grid system, are then derived and discussed. The new methods are demonstrated to be transportive and unconditionally stable. Moreover, the artificial diffusion and the spurious oscillations of these methods are also analysed and compared. Extensions to n‐space variables and to non‐linear equations are indicated, along with various applications.
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…
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.
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Matjaž Ramšak and Leopold Škerget
This paper aims to develop a multidomain boundary element method (BEM) for modeling 2D complex turbulent thermal flow using low Reynolds two‐equation turbulence models.
Abstract
Purpose
This paper aims to develop a multidomain boundary element method (BEM) for modeling 2D complex turbulent thermal flow using low Reynolds two‐equation turbulence models.
Design/methodology/approach
The integral boundary domain equations are discretised using mixed boundary elements and a multidomain method also known as a subdomain technique. The resulting system matrix is an overdetermined, sparse block banded and solved using a fast iterative linear least squares solver.
Findings
The simulation of a turbulent flow over a backward step is in excellent agreement with the finite volume method using the same turbulent model. A grid consisting of over 100,000 elements could be solved in the order of a few minutes using a 3.0 Ghz P4 and 1 GB memory indicating good efficiency.
Originality/value
The paper shows, for the first time, that the BEM is applicable to thermal flows using k‐ε.
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Nagesh Babu Balam and Akhilesh Gupta
Modelling accurately the transient behaviour of natural convection flow in enclosures been a challenging task because of a variety of numerical errors which have limited achieving…
Abstract
Purpose
Modelling accurately the transient behaviour of natural convection flow in enclosures been a challenging task because of a variety of numerical errors which have limited achieving the higher order temporal accuracy. A fourth-order accurate finite difference method in both space and time is proposed to overcome these numerical errors and accurately model the transient behaviour of natural convection flow in enclosures using vorticity–streamfunction formulation.
Design/methodology/approach
Fourth-order wide stencil formula with appropriate one-sided difference extrapolation technique near the boundary is used for spatial discretisation, and classical fourth-order Runge–Kutta scheme is applied for transient term discretisation. The proposed method is applied on two transient case studies, i.e. convection–diffusion of a Gaussian Pulse and Taylor Vortex flow having analytical solution.
Findings
Error magnitude comparison and rate of convergence analysis of the proposed method with these analytical solutions establish fourth-order accuracy and prove the ability of the proposed method to truly capture the transient behaviour of incompressible flow. Also, to test the transient natural convection flow behaviour, the algorithm is tested on differentially heated square cavity at high Rayleigh number in the range of 103-108, followed by studying the transient periodic behaviour in a differentially heated vertical cavity of aspect ratio 8:1. An excellent comparison is obtained with standard benchmark results.
Research limitations/implications
The developed method is applied on 2D enclosures; however, the present methodology can be extended to 3D enclosures using velocity–vorticity formulations which shall be explored in future.
Originality/value
The proposed methodology to achieve fourth-order accurate transient simulation of natural convection flows is novel, to the best of the authors’ knowledge. Stable fourth-order vorticity boundary conditions are derived for boundary and external boundary regions. The selected case studies for comparison demonstrate not only the fourth-order accuracy but also the considerable reduction in error magnitude by increasing the temporal accuracy. Also, this study provides novel benchmark results at five different locations within the differentially heated vertical cavity of aspect ratio 8:1 for future comparison studies.
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