45 research outputs found
Approximation of the inductionless MHD problem using a stabilized finite element method
In this work, we present a stabilized formulation to solve the inductionless magnetohydrodynamic (MHD) problem using the finite element (FE) method. The MHD problem couples the Navier–Stokes equations and a Darcy-type system for the electric potential via Lorentz’s force in the momentum equation of the Navier–Stokes equations and the currents generated by the moving fluid in Ohm’s law. The key feature of the FE formulation resides in the design of the stabilization terms, which serve several purposes. First, the formulation is suitable for convection dominated flows. Second, there is no need to use interpolation spaces constrained to a compatibility condition in both sub-problems and therefore, equal-order interpolation spaces can be used for all the unknowns. Finally, this formulation leads to a coupled linear system; this monolithic approach is effective, since the coupling can be dealt by effective preconditioning and iterative solvers that allows to deal with high Hartmann numbers
Block recursive LU preconditioners for the thermally coupled incompressible inductionless MHD problem
The thermally coupled incompressible inductionless magnetohydrodynamics (MHD) problem models the ow of an electrically charged fuid under the in uence of an external electromagnetic eld with thermal coupling. This system of partial di erential 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 di erent 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 e cient 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 fexible and general software design for the code implementation of this type of preconditioners.Preprin
Model problems in magneto-hydrodynamics: individual numerical challenges and coupling possibilities
In this work we discuss two model problems appearing in magneto-hydrodynamics
(MHD), namely, the so called full MHD problem and the inductionless MHD
problem. The first involves as unknowns the fluid velocity and pressure, the magnetic (induction)
field and a pseudo-pressure introduced to impose the zero-divergence restriction
of this last unknown. The building blocks of this model are the Stokes problem for the
velocity and the pressure and the Maxwell problem for the magnetic field and pseudopressure.
We discuss the numerical challenges of the approximation of these two model
problems having in mind the need to couple them in the full problem, where additional
coupling terms appear. The second model we consider is the inductionless MHD approximation.
Instead of the magnetic induction and pseudo-pressure, the magnetic unknowns
are now the current density and the electric potential. The building blocks are the Stokes
problem for the fluid and the Darcy problem (in primal form) for the current density and
electric potential. We discuss also the numerical challenges involved in the approximation
of this last problem, particularly considering that it has to be coupled to the former.
Once the building blocks have been analysed independently, the possibilities of dealing
with the fully coupled problems are discussed. Iterative schemes that can be shown to be
stable are presented in the stationary case, showing that a segregated solution for the flow
and the magnetic problem is not possible. Most of the results presented are elaborated
independently in previous works. Our objective in this paper is to present the different
problems with a unified perspective
Model problems in magneto-hydrodynamics: individual numerical challenges and coupling possibilities
In this work we discuss two model problems appearing in magneto-hydrodynamics (MHD), namely, the so called full MHD problem and the inductionless MHD problem. The first involves as unknowns the fluid velocity and pressure, the magnetic (induction) fi eld and a pseudo-pressure introduced to impose the zero-divergence restriction of this last unknown. The building blocks of this model are the Stokes problem for the velocity and the pressure and the Maxwell problem for the magnetic field and pseudopressure. We discuss the numerical challenges of the approximation of these two model problems having in mind the need to couple them in the full problem, where additional coupling terms appear. The second model we consider is the inductionless MHD approximation. Instead of the magnetic induction and pseudo-pressure, the magnetic unknowns are now the current density and the electric potential. The building blocks are the Stokes problem for the fluid and the Darcy problem (in primal form) for the current density and
electric potential. We discuss also the numerical challenges involved in the approximation of this last problem, particularly considering that it has to be coupled to the former. Once the building blocks have been analysed independently, the possibilities of dealing with the fully coupled problems are discussed. Iterative schemes that can be shown to be stable are presented in the stationary case, showing that a segregated solution for the flow
and the magnetic problem is not possible. Most of the results presented are elaborated
independently in previous works. Our objective in this paper is to present the di fferent problems with a unifi ed perspective.Postprint (published version
A full divergence-free of high order virtual finite element method to approximation of stationary inductionless magnetohydrodynamic equations on polygonal meshes
In this present paper we consider a full divergence-free of high order
virtual finite element algorithm to approximate the stationary inductionless
magnetohydrodynamic model on polygonal meshes. More precisely, we choice
appropriate virtual spaces and necessary degrees of freedom for velocity and
current density to guarantee that their final discrete formats are both
pointwise divergence-free. Moreover, we hope to achieve higher approximation
accuracy at higher "polynomial" orders k_{1} \geq 2, k_{2} \geq 1, while the
full divergence-free property has always been satisfied. And then we processed
rigorous error analysis to show that the proposed method is stable and
convergent. Several numerical tests are presented, confirming the theoretical
predictions
On stabilized finite element methods based on the Scott-Zhang projector: circumventing the inf-sup condition for the Stokes problem
In this work we propose a stabilized nite element method that permits us to circumvent discrete inf-sup conditions, e.g. allowing equal order interpolation. The type of method we propose belongs to the family of symmetric stabilization techniques, which are based on the introduction of additional terms that penalize the di erence between some quantities, i.e. the pressure gradient in the Stokes problem, and their nite element projections. The key feature of the formulation we propose is the de nition of the projection to be used, a non-standard Scott-Zhang projector that is well-de ned for L1() functions. The resulting method has some appealing features: the projector is local and nested meshes or enriched spaces are not required
A monolithic approach for the incompressible magnetohydrodynamics equations
A numerical algorithm has been developed to solve the incompressible magnetohydrodynamics (MHD) equations in a fully coupled form. The numerical approach is based on the side
centered finite volume approximation where the velocity and magnetic filed vector components are
defined at the center of edges/faces, meanwhile the pressure term is defined at the element
centroid. In order to enforce a divergence free magnetic field, a magnetic pressure is introduced
to the induction equation. The resulting large-scale algebraic linear equations are solved using a
one-level restricted additive Schwarz preconditioner with a block-incomplete factorization within
each partitioned sub-domains. The parallel implementation of the present fully coupled
unstructured MHD solver is based on the PETSc library for improving the effi- ciency of the
parallel algorithm. The numerical algorithm is validated for 2D lid-driven cavity
flows and backward step problems for both conducting and insulating walls