Two-level finite element method with a stabilizing subgrid for the incompressible MHD equations

We consider the Galerkin finite element method (FEM) for the incompressible magnetohydrodynamic (MHD) equations in two dimension. The domain is discretized into a set of regular triangular elements and the finite-dimensional spaces employed consist of piecewise continuous linear interpolants enriched with the residual-free bubble functions. To find the bubble part of the solution, a two-level FEM with a stabilizing subgrid of a single node is described and its application to the MHD equations is displayed. Numerical approximations employing the proposed algorithm are presented for three benchmark problems including the MHD cavity flow and the MHD flow over a step. The results show that the proper choice of the subgrid node is crucial to get stable and accurate numerical approximations consistent with the physical configuration of the problem at a cheap computational cost. Furthermore, the approximate Solutions obtained show the well-known characteristics of the MHD flow. Copyright (C) 2009 John Wiley & Sons, Ltd

Stabilized finite element approximation of the incompressible MHD equations

No es frecuente encontrar un campo donde dos ramas principales de la Física estén involucradas. La Magnetohidrodinámica es uno de tales campos debido a que involucra a la Mecánica de Fluidos y al Electromagnetismo. Aun cuando puede parecer que esas dos ramas de la Física tienen poco en común, comparten similitudes en las ecuaciones que gobiernan los fenómenos involucrados en ellas. Las ecuaciones de Navier-Stokes y las ecuaciones de Maxwell, ambas en la raíz de la Magnetohidrodinámica, tienen una condición de divergencia nula y es esta condición de divergencia nula sobre la velocidad del fluido y el campo magnético lo que origina algunos de los problemas numéricos que surgen en la modelación de los fenómenos donde el flujo de fluidos y los campos magnéticos están acoplados.El principal objetivo de este trabajo es desarrollar un algoritmo eficiente para la resolución mediante elementos finitos de las ecuaciones de la Magnetohidrodinámica de fluidos incompresibles.Para lograr esta meta, los conceptos básicos y las características de la Magnetohidrodinámica se presentan en una breve introducción informal.A continuación, se da una revisión completa de las ecuaciones de gobierno de la Magnetohidrodinámica, comenzando con las ecuaciones de Navier-Stokes y las ecuaciones de Maxwell. Se discute la aproximación que da origen a las ecuaciones de la Magnetohidrodinámica y finalmente se presentan las ecuaciones de la Magnetohidrodinámica.Una vez que las ecuaciones de gobierno de la Magnetohidrodinámica han sido definidas, se presentan los esquemas numéricos desarrollados, empezando con la linealización de las ecuaciones originales, la formulación estabilizada y finalmente el esquema numérico propuesto. En esta etapa se presenta una prueba de convergencia.Finalmente, se presentan los ejemplos numéricos desarrollados durante este trabajo.Estos ejemplos pueden dividirse en dos grupos: ejemplos numéricos de comparación y ejemplos de internes tecnológico. Dentro del primer grupo están incluidas simulaciones del flujo de Hartmann y del flujo sobre un escalón. El segundo grupo incluye simulaciones del flujo en una tobera de inyección de colada continua y el proceso Czochralski de crecimiento de cristales.It is not frequent to find a field where two major branches of Physics are involved. Magnetohydrodynamics is one of such fields because it involves Fluid Mechanics and Electromagnetism. Although those two branches of Physics can seem to have little in common, they share similarities in the equations that govern the phenomena involved. The Navier-Stokes equations and the Maxwell equations, both at the root of Magnetohydrodynamics, have a divergence free condition and it is this divergence free condition over the velocity of the fluid and the magnetic field what gives origin to some of the numerical problems that appear when approximating the equations that model the phenomena where fluids flow and magnetic fields are coupled.The main objective of this work is to develop an efficient finite element algorithm for the incompressible Magnetohydrodynamics equations.In order to achieve this goal the basic concepts and characteristics of Magnetohydrodynamics are presented in a brief and informal introduction.Next, a full review of the governing equations of Magnetohydrodynamics is given, staring from the Navier-Stokes equations and the Maxwell equations. The MHD approximation is discussed at this stage and the proper Magnetohydrodynamics equations for incompressible fluid are reviewed.Once the governing equations have been defined, the numerical schemes developed are presented, starting with the linearization of the original equations, the stabilization formulations and finally the numerical scheme proposed. A convergence test is shown at this stage.Finally, the numerical examples performed while this work was developed are presented. These examples can be divided in two groups: numerical benchmarks and numerical examples of technological interest. In the first group, the numerical simulations for the Hartmann flow and the flow over a step are included. The second group includes the simulation of the clogging in a continuous casting nozzle and Czochralski crystal growth process.Postprint (published version

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

Variational Multiscale Modeling and Memory Effects in Turbulent Flow Simulations

Effective computational models of multiscale problems have to account for the impact of unresolved physics on the resolved scales. This dissertation advances our fundamental understanding of multiscale models and develops a mathematically rigorous closure modeling framework by combining the Mori-Zwanzig (MZ) formalism of Statistical Mechanics with the variational multiscale (VMS) method. This approach leverages scale-separation projectors as well as phase-space projectors to provide a systematic modeling approach that is applicable to complex non-linear partial differential equations. %The MZ-VMS framework is investigated in the context of turbulent flows. Spectral as well as continuous and discontinuous finite element methods are considered. The MZ-VMS framework leads to a closure term that is non-local in time and appears as a convolution or memory integral. The resulting non-Markovian system is used as a starting point for model development. Several new insights are uncovered: It is shown that unresolved scales lead to memory effects that are driven by an orthogonal projection of the coarse-scale residual and, in the case of finite elements, inter-element jumps. Connections between MZ-based methods, artificial viscosity, and VMS models are explored. The MZ-VMS framework is investigated in the context of turbulent flows. Large eddy simulations of Burgers' equation, turbulent flows, and magnetohydrodynamic turbulence using spectral and discontinuous Galerkin methods are explored. In the spectral method case, we show that MZ-VMS models lead to substantial improvements in the prediction of coarse-grained quantities of interest. Applications to discontinuous Galerkin methods show that modern flux schemes can inherently capture memory effects, and that it is possible to guarantee non-linear stability and conservation via the MZ-VMS approach. We conclude by demonstrating how ideas from MZ-VMS can be adapted for shock-capturing and filtering methods.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/145847/1/parish_1.pd

Approximation of the thermally coupled MHD problem using a stabilized finite element method

A numerical formulation to solve the MHD problem with thermal coupling is presented in full detail. The distinctive feature of the method is the design of the stabilization terms, which serve several purposes. First, convective dominated flows in the Navier–Stokes and the heat equation can be dealt with. Second, there is no restriction in the choice of the interpolation spaces of all the variables and, finally, flows highly coupled with the magnetic field can be accounted for. Different aspects related to the design of the final fully discrete and linearized algorithm are also discussed

Stabilized finite element approximation of the stationary magneto-hydrodynamics equations

In this work we present a stabilized finite element method for the stationary magneto-hydrodynamic equations based on a simple algebraic version of the subgrid scale variational concept. The linearization that yields a well posed linear problem is first identified, and for this linear problem the stabilization method is designed. The key point is the correct behavior of the stabilization parameters on which the formulation depends. It is shown that their expression can be obtained only on the basis of having a correct error estimate. For the stabilization parameters chosen, a stability estimate is proved in detail, as well as the convergence of the numerical solution to the continuous one. The method is then extended to nonlinear problems and its performance checked through numerical experiments

Mathematical modeling of the dissolution process of silicon into germanium melt

Numerical simulations were carried out to study the thermosolutal and flow structures observed in the dissolution experiments of silicon into a germanium melt. The dissolution experiments utilized a material configuration similar to that used in the Liquid Phase Diffusion (LPD) and Melt-Replenishment Czochralski (Cz) crystal growth systems. In the present model, the computational domain was assumed axisymmetric. Governing equations of the liquid phase (Si-Ge mixture), namely the equations of conservation of mass, momentum balance, energy balance, and solute (species) transport balance were solved using the Stabilized Finite Element Methods (ST-GLS for fluid flow, SUPG for heat and solute transport). Measured concentration profiles and dissolution height from the samples processed with and without the application of magnetic field show that the amount of silicon transported into the melt is slightly higher in the samples processed under magnetic field, and there is a difference in dissolution interface shape indicating a change in the flow structure during the dissolution process. The present mathematical model predicts this difference in the flow structure. In the absence of magnetic field, a flat stable interface is observed. In the presence of an applied field, however, the dissolution interface remains flat in the center but curves back into the source material near the edge of the wall. This indicates a far higher dissolution rate at the edge of the silicon source.We gratefully acknowledge the financial support provided by the Canadian Space Agency (CSA), Canada Research Chairs (CRC) Program, and the Natural Sciences and Engineering Research Council (NSERC) of Canada.Publisher's Versio