Towards a scalable fully-implicit fully-coupled resistive MHD formulation with stabilized FE methods

Shadid, John N.; Pawlowski, Roger P.; Banks, Jeffrey W.; Chacòn, Luis; Lin, Paul; Tuminaro, Raymond S.

doi:10.1016/j.jcp.2010.06.018

Cited by 88 publications

(126 citation statements)

References 115 publications

(182 reference statements)

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“…In the context of transient systems these methods are commonly used in operator splitting techniques, for steady state solves a fixed point iteration serves to couple the system (see e.g. [2], and the references in [23]). …”

Section: Abstract Magnetohydrodynamics Iterative Methods Precondimentioning

confidence: 99%

“…For realistic applications where the systems are very large, effective preconditioning of iterative methods is required for efficiency. Some recently developed solvers for fully coupled MHD formulations include a coupled AMG preconditioned Newton-Krylov method for a vector potential formulation [23], a multigrid preconditioned Newton-Krylov method for a parabolic reformulation of the MHD equations [1], and a block preconditioned Newton-Krylov method for a vector potential formulation [4].…”

Section: Phillips Elman Cyr Shadid and Pawlowskimentioning

confidence: 99%

“…Many strategies exist for incorporating the solenoidal condition (2.1d) into the other three equations to ensure the solvability of the system. These methods include exact penalty [11,14], Lagrange multiplier [2,22], vector potential [17,23], and divergence cleaning [5] formulations. For this work, we will use an exact penalty formulation in which the solenoidal condition is weakly enforced in the weak form.…”

Section: Phillips Elman Cyr Shadid and Pawlowskimentioning

confidence: 99%

See 2 more Smart Citations

A Block Preconditioner for an Exact Penalty Formulation for Stationary MHD

Phillips¹,

Elman²,

Cyr³

et al. 2014

SIAM J. Sci. Comput.

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Abstract. The magnetohydrodynamics (MHD) equations are used to model the flow of electrically conducting fluids in such applications as liquid metals and plasmas. This system of non-self adjoint, nonlinear PDEs couples the Navier-Stokes equations for fluids and Maxwell's equations for electromagnetics. There has been recent interest in fully coupled solvers for the MHD system because they allow for fast steady-state solutions that do not require pseudo-time stepping. When the fully coupled system is discretized, the strong coupling can make the resulting algebraic systems difficult to solve, requiring effective preconditioning of iterative methods for efficiency. In this work, we consider a finite element discretization of an exact penalty formulation for the stationary MHD equations. This formulation has the benefit of implicitly enforcing the divergence free condition on the magnetic field without requiring a Lagrange multiplier. We consider extending block preconditioning techniques developed for the Navier-Stokes equations to the full MHD system. We analyze operators arising in block decompositions from a continuous perspective and apply arguments based on the existence of approximate commutators to develop new preconditioners that account for the physical coupling. This results in a family of parameterized block preconditioners for both Picard and Newton linearizations. We develop an automated method for choosing the relevant parameters and demonstrate the robustness of these preconditioners for a range of the physical non-dimensional parameters and with respect to mesh refinement. Key words. magnetohydrodynamics, iterative methods, preconditionersAMS subject classifications. 76W05, 65F08, 65M221. Introduction. The magnetohydrodynamics (MHD) model describes the flow of electrically conducting fluids in the presence of magnetic fields. A principal application of MHD is the modeling of plasma physics, ranging from plasma confinement for thermonuclear fusion to astrophysical plasma dynamics [13]. MHD is also used to model the flow of liquid metals, for instance in magnetic pumps, liquid metal blankets in fusion reactor concepts, and aluminum electrolysis [19]. The model consists of a non-self-adjoint, nonlinear system of partial differential equations (PDEs) that couple the Navier-Stokes equations for fluid flow to a reduced set of Maxwell's equations for electromagnetics. Because multiple physical processes are represented in the model, the PDEs can span over a range of length-and time-scales, making the equations difficult to solve and requiring a robust, accurate means of approximating the solution. Decoupled solution methods which solve the fluid and magnetic systems separately and possibly couple the systems by an outer iteration have been commonly employed as solvers for the transient and steady MHD systems. In the context of transient systems these methods are commonly used in operator splitting techniques, for steady state solves a fixed point iteration serves to couple the system (see e.g. [2], and th...

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Section: Abstract Magnetohydrodynamics Iterative Methods Precondimentioning

confidence: 99%

Section: Phillips Elman Cyr Shadid and Pawlowskimentioning

confidence: 99%

Section: Phillips Elman Cyr Shadid and Pawlowskimentioning

confidence: 99%

See 1 more Smart Citation

A Block Preconditioner for an Exact Penalty Formulation for Stationary MHD

Phillips¹,

Elman²,

Cyr³

et al. 2014

SIAM J. Sci. Comput.

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show abstract

“…The control volume algorithm is used in (Al-Najem et al, 1998;Sarris et al, 2005;Kandaswamy et al, 2008;Sheikhzadeh et al, 2011) to solve the two dimensional transient MHD equations with alternating direct implicit procedure (ADI). Finite difference method and finite element method are developed in (Borghi et al, 1996;Borghi et al, 2004;Verardi and Cardoso 1998;Verardi et al, 2001;Verardi et al, 2002;Shadid et al, 2010) for the solution of two-dimensional steady state electrodynamic problem in magnetohydrodynamic flows. A mathematical model describing the dynamics of magnetic field influence on a conducting liquid in a square cavity is presented in (Krzeminski et al, 2000) such that biharmonic mathematical model is used with stream function and the magnetic potential.…”

Section: Introductionmentioning

confidence: 99%

Magnetic Field Effect on Natural Convection Flow with Internal Heat Generation using Fast  –  Method

Taghikhani¹

2015

JAFM

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The magnetic field effect on laminar natural convection flow is investigated in a filled enclosure with internal heat generation using two-dimensional numerical simulation. The enclosure is heated by a uniform volumetric heat density and walls have constant temperature. A fixed magnetic field is applied to the enclosure. The dimensionless governing equations are solved numerically for the stream function, vorticity and temperature using finite difference method for various Rayleigh (Ra) and Hartmann (Ha) numbers in MATLAB software. The stream function equation is solved using fast Poisson's equation solver on a rectangular grid (POICALC function in MATLAB), voricity and temperature equations are solved using red-black Gauss-Seidel and bi-conjugate gradient stabilized (BiCGSTAB) methods respectively. The results show that the strength of the magnetic field has significant effects on the flow and temperature fields. For the square cavity, the maximum temperature reduces with increasing Ra number. It is also observed that at low Ra number, location of the maximum temperature is at the centre of the cavity and it shifts upwards with increase in Ra number. Circulation inside the enclosure and therefore the convection becomes stronger as the Ra number increases while the magnetic field suppresses the convective flow and the heat transfer rate. The ratio of the Lorentz force to the buoyancy force (Ha 2 /Ra) is as an index to compare the contribution of natural convection and magnetic field strength on heat transfer.

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“…There exist several approaches for preconditioning this type of problems. One approach that has been extensively used for preconditioning large-scale multi-physics problems is the algebraic multigrid (AMG) algorithm [26,34]. This technique is very efficient for Laplacian-type problems but suffers for indefinite and nonsymmetric problems.…”

Section: Introductionmentioning

confidence: 99%

Block recursive LU preconditioners for the thermally coupled incompressible inductionless MHD problem

Badia

Martı́n

Planas

2014

Journal of Computational Physics

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The thermally coupled incompressible inductionless magnetohydrodynamics (MHD) problem models the flow of an electrically charged fluid under the influence of an external electromagnetic field with thermal coupling. This system of partial differential equations is strongly coupled and highly nonlinear for real cases of interest. Therefore, fully implicit time integration schemes are very desirable in order to capture the different physical scales of the problem at hand. However, solving the multiphysics linear systems of equations resulting from such algorithms is a very challenging task which requires efficient and scalable preconditioners. In this work, a new family of recursive block LU preconditioners is designed and tested for solving the thermally coupled inductionless MHD equations. These preconditioners are obtained after splitting the fully coupled matrix into one-physics problems for every variable (velocity, pressure, current density, electric potential and temperature) that can be optimally solved, e.g., using preconditioned domain decomposition algorithms. The main idea is to arrange the original matrix into an (arbitrary) 2 × 2 block matrix, and consider a LU preconditioner obtained by approximating the corresponding Schur complement. For every one of the diagonal blocks in the LU preconditioner, if it involves more than one type of unknown, we proceed the same way in a recursive fashion. This approach is stated in an abstract way, and can be straightforwardly applied to other multiphysics problems. Further, we precisely explain a flexible and general software design for the code implementation of this type of preconditioners.

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Towards a scalable fully-implicit fully-coupled resistive MHD formulation with stabilized FE methods

Cited by 88 publications

References 115 publications

A Block Preconditioner for an Exact Penalty Formulation for Stationary MHD

A Block Preconditioner for an Exact Penalty Formulation for Stationary MHD

Magnetic Field Effect on Natural Convection Flow with Internal Heat Generation using Fast  –  Method

Block recursive LU preconditioners for the thermally coupled incompressible inductionless MHD problem

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