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Mai-Duy, Nam; Thanh Tran-Cong,. "A stable and accurate control-volume technique based on integrated radial basis function networks for fluid-flow problems.(conference paper)." Australian Journal of Mechanical Engineering. The Institution of Engineers, Australia. 2011. HighBeam Research. 19 Sep. 2018 <https://www.highbeam.com>.
Mai-Duy, Nam; Thanh Tran-Cong,. "A stable and accurate control-volume technique based on integrated radial basis function networks for fluid-flow problems.(conference paper)." Australian Journal of Mechanical Engineering. 2011. HighBeam Research. (September 19, 2018). https://www.highbeam.com/doc/1G1-272246455.html
Mai-Duy, Nam; Thanh Tran-Cong,. "A stable and accurate control-volume technique based on integrated radial basis function networks for fluid-flow problems.(conference paper)." Australian Journal of Mechanical Engineering. The Institution of Engineers, Australia. 2011. Retrieved September 19, 2018 from HighBeam Research: https://www.highbeam.com/doc/1G1-272246455.html
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1 INTRODUCTION
The control-volume (CV) formulation (Patankar, 1980) provides an effective way to simulate fluid flows numerically. Radial basis function networks (RBFNs) are regarded as a powerful approximation tool (Fasshauer, 2007). In this paper, the CV formulation is employed with integrated RBFNs (IRBFNs) (Mai-Duy & Tran-Cong, 2001; 2007). The diffusion term is approximated using local 1D-IRBFNs, while the convection term is treated by a high-order upwind scheme incorporating global 1D-IRBFNs with the deferred-correction (DC) strategy. It is noted that the DC strategy, which was proposed by Khosla & Rubin (1974), casts the convected face value as the upstream value and the streamwise correction term. The proposed technique achieves both good accuracy and stability properties. The remainder of the paper is organised as follows. Brief reviews of the control-volume formulation and 1D-IRBFNs are given in sections 2 and 3, respectively. Section 4 describes the proposed technique. In section 5, two examples--heat transfer and fluid flow--are presented to demonstrate attractiveness of the present implementation. Section 6 concludes the paper.
2 CONTROL-VOLUME FORMULATION
Consider the following convection-diffusion equation:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
where [phi] is the field variable; t is the time; [rho] is the density; [??] is the convection velocity vector; [kappa] is the diffusion coefficient; R is the source term; [??] is the position vector; and [OMEGA] is the domain of interest. By directly integrating equation (1) over a control volume of the grid point P, [[OMEGA].sub.p], the following equation is obtained:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
Applying the Gauss divergence theorem to equation (2) results in:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
where [[GAMMA].sub.p] is the boundaries of [[OMEGA].sub.pi] [??] is the unit outward vector normal to [[GAMMA].sub.p]; and d[GAMMA].sub.p] is a differential element of [[GAMMA].sub.p]. The governing differential equation (1) is thus transformed into a CV form equations (2)/(3). It is noted that no approximation is made at this stage.
3 ONE-DIMENSIONAL IRBFNS
The basic idea of the integral RBF scheme is to decompose the highest-order derivatives of [phi] in a given differential equation (eg. second-order derivatives for the convection-diffusion equation (1)) into RBFs. Consider a univariate function [phi]x). The present 1D-IRBF scheme starts with:
[d.sup.2][phi](x)/d[x.sup.2] = [N.summation over (i=1)] [w.sub.i][I.sup.(2).sub.i] (x) (4)
where N is the number of RBFs; [{w.sub.i]}.sup.N.sub.i=1] is the set of network weights; and {[I.sup.(2).sub.i](x)}.sup.N.sub.i=1] is the set of RBFs. Expressions for the first-order derivative and function itself are then obtained through integration:
d[phi](x)/dx = [N.summation over (i=1)][w.sub.i][I.sup.(1).sub.i](x)+[c.sub.1] (5)
[phi](x)=[N.summation over (i=1)][w.sub.i][I.sup.(0).sub.i](x)+[c.sub.1]x+[c.sub.2] (6)
where [I.sup.(i).sub.i](x) = [integral][I.sup.(2).sub.i](x)dx, [I.sup.(0).sub.i](x)= [integral][I.sup.(1).sub.i] (x)dx and ([c.sub.1], [c.sub.2]) are the constants of integration. Evaluation of equations (4) to (6) at a set of collocation points [{[x.sub.i]}.sup.N.sub.i=1] leads to:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
in which [d. …
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