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

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

On an unconditionally convergent stabilized finite element approximation of resistive magnetohydrodynamics

In this work, we propose a new stabilized finite element formulation for the approximation of the resistive magnetohydrodynamics equations. The novelty of this formulation with respect to existing ones is the fact that it always converges to the physical solution, even for singular ones. We have performed a detailed stability and convergence analysis of the formulation in a simplified setting. From the convergence analysis, we infer that a particular type of meshes with a macro-element structure is needed, which can be easily obtained after a straight modification of any original mesh. A detailed set of numerical experiments have been performed in order to validate our approach.Peer ReviewedPreprin

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

Analysis of an unconditionally convergent stabilized finite element formulation for incompressible magnetohydrodynamics

In this work, we analyze a recently proposed stabilized finite element formulation for the approximation of the resistive magnetohydrodynamics equations. The novelty of this formulation with respect to existing ones is the fact that it always converges to the physical solution, even when it is singular. We have performed a detailed stability and convergence analysis of the formulation in a simplified setting. From the convergence analysis, we infer that a particular type of meshes with a macro-element structure is needed, which can be easily obtained after a straight modification of any original mesh.Preprin

Disseny i implementació d'un prototip de placa electrònica amb protocol de comunicació MODBUS per a aplicació industrial

El PFC té com a objectiu el disseny i posterior implementació d'un prototip de placa electrònica industrial, capaç de capturar i gestionar dades de qualsevol procés automatitzat mitjançant protocol de comunicació Modbus. Particularment, el nostre projecte permet fer una adquisició de dades de tots els possibles sensors i actuadors que poden formar part d'un procés industrial i transmetre-les a un PC per comunicació RS-485. Posteriorment, es tracten tot el seguit de dades, es verifiquen i gestionen a un correcte ús automatitzat. El desenvolupament del projecte, comprèn el disseny del prototip, la insolació de la placa de circuit imprès, muntatge del prototip, el software de programació de la placa en un entorn automatitzat amb protocol Modbus i una interfície gràfica perquè l'usuari pugui interactuar amb la placa. Tot el mòdul d'entrades analògiques i entrades / sortides digitals està controlat per un microcontrolador PIC 18F4620

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