![Distribution of quadrature points for the RKGSI and 13-point Gauss rule in the unit triangle Ωξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{{ \xi }} $$\end{document}](https://www.researchgate.net/publication/379833715/figure/fig1/AS:11431281283929933@1729043583117/Distribution-of-quadrature-points-for-the-RKGSI-and-13-point-Gauss-rule-in-the-unit_Q320.jpg)
![Nodes and integration cells by regular and irregular discretizations in the domain Ω=0,12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega =\left( {0,1} \right) ^2$$\end{document} and quadrature points for the RKGSI and 13-point Gauss rule. The circle and the dot represent nodes and quadrature points, respectively](https://www.researchgate.net/publication/379833715/figure/fig2/AS:11431281283889850@1729043583302/Nodes-and-integration-cells-by-regular-and-irregular-discretizations-in-the-domain_Q320.jpg)
![Analytical solutions (left column), and EFG solutions without stabilization (middle column) and with stabilization (right column) of the velocity field u computed by using ε=10-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon =10^{-2}$$\end{document} (top row), ε=10-4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon =10^{-4}$$\end{document} (middle row), and ε=10-6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon =10^{-6}$$\end{document} (bottom row) for the square-channel flow](https://www.researchgate.net/publication/379833715/figure/fig4/AS:11431281283975841@1729043583854/Analytical-solutions-left-column-and-EFG-solutions-without-stabilization-middle_Q320.jpg)
![Analytical solutions (left column), and EFG solutions without stabilization (middle column) and with stabilization (right column) of the induced magnetic field b computed by using ε=10-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon =10^{-2}$$\end{document} (top row), ε=10-4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon =10^{-4}$$\end{document} (middle row), and ε=10-6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon =10^{-6}$$\end{document} (bottom row) for the square-channel flow](https://www.researchgate.net/publication/379833715/figure/fig5/AS:11431281283918268@1729043584303/Analytical-solutions-left-column-and-EFG-solutions-without-stabilization-middle_Q320.jpg)
April 2024
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18 Reads
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2 Citations
Engineering with Computers
A stabilized element-free Galerkin (EFG) method is proposed in this paper for numerical analysis of the generalized steady MHD duct flow problems at arbitrary and high Hartmann numbers up to 1016. Computational formulas of the EFG method for MHD duct flows are derived by using Nitsche’s technique to facilitate the implementation of Dirichlet boundary conditions. The reproducing kernel gradient smoothing integration technique is incorporated into the EFG method to accelerate the solution procedure impaired by Gauss quadrature rules. A stabilized Nitsche-type EFG weak formulation of MHD duct flows is devised to enhance the performance damaged by high Hartmann numbers. Several benchmark MHD duct flow problems are solved to testify the stability and the accuracy of the present EFG method. Numerical results show that the range of the Hartmann number Ha in the present EFG method is 1≤Ha≤1016, which is much larger than that in existing numerical methods.