On mixed boundary element solutions of convection-diffusion problems in three dimensions

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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COMPARISON BETWEEN GLOBAL, CLASSICAL DOMAIN DECOMPOSITION AND LOCAL, SINGLE AND DOUBLE COLLOCATION METHODS BASED ON RBF INTERPOLATION FOR SOLVING CONVECTION-DIFFUSION EQUATION

International Journal of Modern Physics C ◽

10.1142/s0129183108013102 ◽

2008 ◽

Vol 19 (11) ◽

pp. 1737-1751 ◽

Cited By ~ 3

Author(s):

GAIL GUTIERREZ ◽

WHADY FLOREZ

Keyword(s):

Diffusion Equation ◽

Domain Decomposition ◽

Mixed Boundary ◽

Domain Decomposition Method ◽

Collocation Methods ◽

Special Treatment ◽

Convection Diffusion ◽

Convection Diffusion Equation ◽

Full Domain ◽

Diffusion Problems

This work presents a performance comparison of several meshless RBF formulations for convection-diffusion equation with moderate-to-high Peclet number regimes. For the solution of convection-diffusion problems, several comparisons between global (full-domain) meshless RBF methods and mesh-based methods have been presented in the literature. However, in depth studies between new local RBF collocation methods and full-domain symmetric RBF collocation methods are not reported yet. The RBF formulations included: global symmetric method, symmetric double boundary collocation method, additive Schwarz domain decomposition method (DDM) when it is incorporated into two anterior approaches, and local single and double collocation methods. It can be found that the accuracy of solutions deteriorates as Pe increases, if no special treatment is used. From the numerical tests, it seems that the local methods, especially the derived double collocation technique incorporating PDE operator, are more effective than full domain approaches even with iterative DDM in solving moderate-to-high Pe convection-diffusion problems subject to mixed boundary conditions.

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A Boundary Element Method Formulation for Design Sensitivities in Steady-State Conduction-Convection Problems

Journal of Applied Mechanics ◽

10.1115/1.2899426 ◽

1992 ◽

Vol 59 (1) ◽

pp. 182-190 ◽

Cited By ~ 12

Author(s):

Abhijit Chandra ◽

Cho Lik Chan

Keyword(s):

Steady State ◽

Boundary Element Method ◽

Boundary Element ◽

Machining Processes ◽

Convection Diffusion ◽

Design Sensitivities ◽

Singular Kernels ◽

Method Formulation ◽

Diffusion Problems ◽

Element Method

A Boundary Element Method (BEM) formulation for the determination of design sensitivities of temperature distributions to various shape and process parameters in steady-state convection-diffusion problems is presented in this paper. The present formulation is valid for constant or piecewise-constant convective velocities. This approach is based on direct differentiation (DDA) of the relevant BEM formulation of the problem. It retains the advantages of the BEM regarding accuracy and efficiency while avoiding strongly singular kernels. The BEM formulation is also observed to avoid any false diffusion. This approach provides a new avenue toward efficient optimization of steady-state convection-diffusion problems and may be easily adapted to investigate the thermal aspects of various machining processes.

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