A robust solver for the finite element approximation of stationary incompressible MHD equations in 3D

Li, Lingxiao; Zheng, Weiying

doi:10.1016/j.jcp.2017.09.025

Cited by 31 publications

(26 citation statements)

References 45 publications

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“…The magnetic field lines are not significantly altered due to the relatively low magnetic Reynolds number. The streamtraces on the z =0.5 plane are also provided in Figure A and the results are in excellent agreement with the result of Li and Zheng in their Figure 1. The present streamtraces are also compared with those of the two‐dimensional numerical simulation in Figure B and the computed u −velocity vector components are also provided for both two and three dimensions at the cavity vertical centerline in Figure C.…”

Section: Numerical Resultssupporting

confidence: 87%

“…Although the computed velocity profiles are quite similar with each other, the braking effect of the magnetic field in three dimensions is significantly disappeared. The reason is that there is a counter‐rotating vortex pair at x =0 plane as seen in Figure 1A in the work of Li and Zheng due to a relatively low aspect ratio of 1:1 and these vortices induce an upward velocity on the z =0 midplane. Since the velocity values are rather small in the lower part of the cavity, the small induced velocity can significantly alter the streamtraces in this region.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…In recent years, the motion of an electrically conducting fluid within a three‐dimensional lid‐driven cavity has been extensively investigated in the literature . In here, the calculations are carried out for a lid‐driven cubic cavity with a horizontally applied magnetic field, B =(1,0,0) ⊤ corresponding to the work of Li and Zheng due to its relatively high Hartmann number. Therefore, nondimensional numbers are set to Re =100, Re m =1, and N =100.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…It may be noted that the ratio between the off‐diagonal entries in the B 12 and B 21 blocks is proportional to

H a^{2} false/ {R e_{m}}^{2}

and its large values adversely effect the system condition number. In the literature, the several block preconditioning techniques are proposed . In here, to remove the zero block in the original system, an upper triangular right preconditioner is used as follows:

([], \begin{matrix} center center center center \\ array B_{11} & array B_{12} & array B_{13} & array 0 \\ array B_{21} & array B_{22} & array 0 & array B_{24} \\ array B_{31} & array 0 & array 0 & array 0 \\ array 0 & array B_{42} & array 0 & array 0 \end{matrix}) ([], \begin{matrix} center center center center \\ array I & array 0 & array - B_{13} & array 0 \\ array 0 & array I & array 0 & array - B_{24} \\ array 0 & array 0 & array I & array 0 \\ array 0 & array 0 & array 0 & array I \end{matrix}) ([], \begin{matrix} center \\ array r^{n + 1} \\ array s^{n + 1} \\ array P^{n + 1} \\ array q^{n + 1} \end{matrix}) = ([], \begin{matrix} center \\ array d_{1} \\ array d_{2} \\ array 0 \\ array 0, \end{matrix})

which leads to

([], \begin{matrix} center center center center \\ array B_{11} & array B_{12} & array - B_{11} B_{13} + B_{13} & array - B_{12} \end{matrix})

…”

Section: Mathematical and Numerical Formulationmentioning

confidence: 99%

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A face‐based monolithic approach for the incompressible magnetohydrodynamics equations

Ata

Şahin

2019

Numerical Methods in Fluids

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Summary A novel numerical algorithm has been developed to solve the incompressible resistive magnetohydrodynamics equations in a fully coupled form. The numerical method is based on the face‐centered unstructured finite volume approximation, where the velocity and magnetic field vector components are defined at the center of edges/faces; meanwhile, the pressure term is defined at element centroid. In order to enforce a divergence‐free magnetic field, the gradient of a scalar Lagrange multiplier is introduced into the induction equation. A special attention will be given to satisfy the continuity equation and the Gauss' law for magnetism within each element and the summation of the equations can be exactly reduced to the domain boundary. The first modification to the original algorithm involves the evaluation of the convective fluxes over the two neighboring elements, where the discrete continuity equations are exactly satisfied. The second modification is based on the neglecting electric field term from the Lorentz force in two dimensions. The resulting large‐scale algebraic linear equations are solved in a fully coupled manner using the one‐ and two‐level restricted additive Schwarz preconditioners to avoid any time step restrictions forced by stability requirements. The spatial convergence of the algorithm is confirmed by solving the Hartmann flow, and then the algorithm is applied to the classical lid‐driven cavity and backward facing step benchmark problems in two and three dimensions. The lid‐driven cavity flow calculations at relatively high Stuart numbers indicate the perfect braking effect of the magnetic field in two dimensions.

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Section: Numerical Resultssupporting

confidence: 87%

Section: Numerical Resultsmentioning

confidence: 99%

Section: Numerical Resultsmentioning

confidence: 99%

“…It may be noted that the ratio between the off‐diagonal entries in the B 12 and B 21 blocks is proportional to

H a^{2} false/ {R e_{m}}^{2}

([], \begin{matrix} center center center center \\ array B_{11} & array B_{12} & array B_{13} & array 0 \\ array B_{21} & array B_{22} & array 0 & array B_{24} \\ array B_{31} & array 0 & array 0 & array 0 \\ array 0 & array B_{42} & array 0 & array 0 \end{matrix}) ([], \begin{matrix} center center center center \\ array I & array 0 & array - B_{13} & array 0 \\ array 0 & array I & array 0 & array - B_{24} \\ array 0 & array 0 & array I & array 0 \\ array 0 & array 0 & array 0 & array I \end{matrix}) ([], \begin{matrix} center \\ array r^{n + 1} \\ array s^{n + 1} \\ array P^{n + 1} \\ array q^{n + 1} \end{matrix}) = ([], \begin{matrix} center \\ array d_{1} \\ array d_{2} \\ array 0 \\ array 0, \end{matrix})

which leads to

([], \begin{matrix} center center center center \\ array B_{11} & array B_{12} & array - B_{11} B_{13} + B_{13} & array - B_{12} \end{matrix})

…”

Section: Mathematical and Numerical Formulationmentioning

confidence: 99%

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A face‐based monolithic approach for the incompressible magnetohydrodynamics equations

Ata

Şahin

2019

Numerical Methods in Fluids

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“…The system is closed with some appropriate set of boundary conditions dependent on the problem. Preconditioners for this formulation were introduced in [32,62,63]. In [32,63], block-triangular preconditioners for the system are proposed and analyzed, while [62] proposes a block-structured approximate inverse preconditioner.…”

mentioning

confidence: 99%

Monolithic Multigrid Methods for Two-Dimensional Resistive Magnetohydrodynamics

Adler¹,

Benson²,

Cyr³

et al. 2016

SIAM J. Sci. Comput.

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The magnetohydrodynamics (MHD) equations model a wide range of plasma physics applications and are characterized by a nonlinear system of partial differential equations that strongly couples a charged fluid with the evolution of electromagnetic fields. After discretization and linearization, the resulting system of equations is generally difficult to solve due to the coupling between variables, and the heterogeneous coefficients induced by the linearization process. In this paper, we investigate multigrid preconditioners for this system based on specialized relaxation schemes that properly address the system structure and coupling. Three extensions of Vanka relaxation are proposed and applied to problems with up to 170 million degrees of freedom and fluid and magnetic Reynolds numbers up to 400 for stationary problems and up to 20,000 for time-dependent problems.

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Stabilization in 3‐D FEM and solution of the MHD equations

Aydın

ERDOĞAN

2023

Math Methods in App Sciences

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In this study, we consider the numerical solution of convection-diffusion typed equations defined in 3-D domain using the finite element method (FEM) with the stabilized version in order to fill the gap in literature on this area in the sense of different applications. The formulation processed is performed step by step. For this purpose, we started with FEM formulation of the 3-D Laplace equation.Then, the obtained formulation is extended to the convection-diffusion equation with standard Galerkin FEM. In order to solve the stability problems for the convection dominated case, the most popular stabilized formulation known as the streamline upwind Petrov-Galerkin (SUPG) type stabilization method is considered. As a further step, obtained stabilized formulation is used in the solutions of the MHD equations by transforming the coupled system of differential equations to the decoupled convection-diffusion type equations defined on different 3-D domains. For the comparison purpose, MHD equations are also formulated with finite volume method (FVM). Obtained numerical results are displayed in terms of a table and figures as the 2-D slices of the 3-D plots in order to visualize the effect and accuracy of the proposed numerical scheme over the different test problems.

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A robust solver for the finite element approximation of stationary incompressible MHD equations in 3D

Cited by 31 publications

References 45 publications

A face‐based monolithic approach for the incompressible magnetohydrodynamics equations

A face‐based monolithic approach for the incompressible magnetohydrodynamics equations

Monolithic Multigrid Methods for Two-Dimensional Resistive Magnetohydrodynamics

Stabilization in 3‐D FEM and solution of the MHD equations

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