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Vol.:(0123456789)
Engineering with Computers (2024) 40:3233–3251
https://doi.org/10.1007/s00366-024-01969-1
ORIGINAL ARTICLE
Element‑free Galerkin analysis ofMHD duct flow problems atarbitrary
andhigh Hartmann numbers
XiaolinLi1· ShulingLi1
Received: 10 December 2023 / Accepted: 11 March 2024 / Published online: 15 April 2024
© The Author(s), under exclusive licence to Springer-Verlag London Ltd., part of Springer Nature 2024
Abstract
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
≤
10
16
, which is much larger than that in existing numerical
methods.
Keywords Meshless· Element-free Galerkin method· Numerical integration· Stabilization· MHD duct flows· High
Hartmann numbers
1 Introduction
The magnetohydrodynamics (MHD) equation is one of the
most important practical models for describing the dynam-
ics of electrically conducting fluids and their interactions
with magnetic fields. MHD duct flow problems come up
in a wide variety of applications in physics and engineer-
ing, such as blood flow measurements, nuclear reactor cool-
ing systems, and MHD flowmeters, generators, pumps and
accelerators [1–8]. However, partly owing to the coupling
of fluid dynamics and electrodynamics equations, analytical
solutions of MHD duct flow problems are available only for
some special geometries with simple boundary conditions
[1, 8, 9]. Therefore, effective numerical methods are neces-
sary and important for solving MHD duct flow problems.
In the past two decades, MHD duct flow problems have
been extensively studied by some mesh-based numerical
methods such as the finite element method (FEM) [1–3],
the finite difference method (FDM) [4–6], the boundary
element method (BEM) [7, 8] and the tailored finite point
method (TFPM) [10]. The success and accuracy of mesh-
based methods largely rely on the ability to generate and
keep sufficient quality elements throughout the entire numer-
ical simulation.
In order to reduce the difficulty in generating elements
and enhance the computational accuracy, some meshless (or
meshfree) methods have been developed. Several meshfree
point collocation method (MPCM) [11–16] have been pro-
posed for MHD duct flow problems in recent years. Galer-
kin meshless methods have been widely used due to their
excellent stability and accuracy. As one of the commonly
used Galerkin meshless methods, the element-free Galerkin
(EFG) method [17] has also contributed a great deal to solv-
ing MHD duct flow problems [18–22]. It should be pointed
out that the improved EFG method [23], the complex vari-
able EFG method [24], and the interpolating EFG method
[25] have been developed respectively by using the improved
MLS approximation, the complex variable MLS approxima-
tion and the interpolating MLS method to construct mesh-
less shape functions.
In meshless methods, shape functions are generated with
high smoothness and accuracy by scattered nodes instead of
elements. Nevertheless, due to the non-polynomial nature
* Xiaolin Li
lxlmath@163.com
1 School ofMathematical Sciences, Chongqing Normal
University, Chongqing400047, People’sRepublicofChina
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