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Stabilized Finite Element Formulations for Solving Incompressible Magnetohydrodynamics

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Stabilized Finite Element Formulations for Solving Incompressible Magnetohydrodynamics UNIVERSITAT POLITECNICA` DE CATALUNYA PhD Thesis STABILIZED FINITE ELEMENT FORMULATIONS FOR SOLVING INCOMPRESSIBLE MAGNETOHYDRODYNAMICS by Ramon Planas Badenas Barcelona, September 2013 Stabilized finite element formulations for solving incompressible magnetohydrodynamics Author: Ramon Planas Advisors: Santiago Badia, Ramon Codina Escola T`ecnicaSuperior d'Enginyers de Camins, Canals i Ports Universitat Polit`ecnicade Catalunya September 2013 Curs acadèmic: Acta de qualificació de tesi doctoral Nom i cognoms Programa de doctorat Unitat estructural responsable del programa Resolució del Tribunal Reunit el Tribunal designat a l'efecte, el doctorand / la doctoranda exposa el tema de la seva tesi doctoral titulada __________________________________________________________________________________________ _________________________________________________________________________________________. Acabada la lectura i després de donar resposta a les qüestions formulades pels membres titulars del tribunal, aquest atorga la qualificació: APTA/E NO APTA/E (Nom, cognoms i signatura) (Nom, cognoms i signatura) President/a Secretari/ària (Nom, cognoms i signatura) (Nom, cognoms i signatura) (Nom, cognoms i signatura) Vocal Vocal Vocal ______________________, _______ d'/de __________________ de _______________ El resultat de l’escrutini dels vots emesos pels membres titulars del tribunal, efectuat per l’Escola de Doctorat, a instància de la Comissió de Doctorat de la UPC, atorga la MENCIÓ CUM LAUDE: SÍ NO (Nom, cognoms i signatura) (Nom, cognoms i signatura) Presidenta de la Comissió de Doctorat Secretària de la Comissió de Doctorat Barcelona, _______ d'/de ____________________ de _________ Acknowledgements I would like to thank my co-advisors, Santi Badia and Ramon Codina, for their support, advice and never-ending patience to solve my doubts. It's been a great experience to work with them, who have such a vast knowledge of this field. I also really appreciate the advice, help and discussions with my co-workers during these years, Javier Principe, Alberto Mart´ın,Rub´enOt´ın,Elisabet Mas de les Valls, Oriol Colom´es,Alba Hierro, Marc Olm, Joan Baiges, Vladimir Jazarevic, and the rest of people at CIMNE and RMEE-UPC. The financial support received from the Universitat Polit`ecnica de Catalunya (UPC) and from the Col.legi d'Enginyers de Camins, Canals i Ports de Catalunya is gratefully acknowledged. 5 Abstract Magnetohydrodynamics (MHD) is the physics branch that studies electrically conduct- ing fluids under external magnetic fields. This thesis deals with the numerical approx- imation using stabilized finite element methods of two different formulations to model incompressible MHD, namely the resistive and inductionless MHD problems. Further, the linear systems of equations resulting from the application of these discrete formula- tions to simulate real cases are typically ill-conditioned and can have as many as 106-109 degrees of freedom. An efficient and scalable solver strategy is mandatory in these cases. On one hand, a new stabilized finite element formulation for the approximation of the resistive magnetohydrodynamics equations has been proposed. The novelty of this formulation with respect to existing ones is that it always converges to the physical so- lution, even when it is singular, which has been proved through a detailed stability and convergence analysis of the formulation. Moreover, it is inferred from the convergence analysis 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. Finally, different operator splitting schemes have been proposed for solving the transient incom- pressible resistive MHD system that are unconditionally stable. Two levels of splitting have been considered. On the first level, the segregation of the Lagrange multipliers, the fluid pressure and the magnetic pseudo-pressure, from the vectorial fields computation is achieved. On the second level, the fluid velocity and induction fields are also decoupled. This way, the fully coupled indefinite multiphysics system is transformed into smaller uncoupled one-physics problems. On the other hand, a stabilized formulation to solve the inductionless magnetohy- drodynamic problem using the finite element method is presented. The inductionless MHD problem models the flow of an electrically charged fluid under the influence of an external magnetic field where the magnetic field induced in the fluid by the currents is negligible with respect to the external one. This system of partial differential equations is strongly coupled and highly nonlinear for real cases of interest. Therefore, solving the multiphysics linear systems of equations resulting from the discretization of these equa- tions with finite element methods is a very challenging task which requires efficient and scalable preconditioners. A new family of recursive block LU preconditioners has been designed to improve the convergence of iterative solvers for this problem. These precon- ditioners are obtained after splitting the fully coupled matrix into one-physics problems for every variable (velocity, pressure, current density and electric potential) that can be optimally solved, e.g. using preconditioned domain decomposition algorithms. Further- more, these ideas have been extended for developing recursive block LU preconditioners for the thermally coupled inductionless MHD problem. 7 Resum La magnetohidrodin`amica(MHD) ´esla branca de la F´ısicaque estudia el moviment de fluids el`ectricament conductors que es troben sotmesos a camps magn`eticsexterns. Aquesta tesi tracta de l'aproximaci´onum`ericaamb m`etodes d'element finits estabilitzats de dues formulacions per modelar el problema de la MHD incompressible, com s´onla MHD resistiva i la MHD sense inducci´o. A m´esa m´es,els sistemes lineals d'equacions que resulten de l’aplicaci´od'aquestes formulacions discretes per a simular casos reals solen ser mal condicionats i poden arribar a comprendre entre 106-109 graus de llibertat. La ressoluci´od'aquests sistemes lineals d'equacions necessita obligat`oriament una estrat`egia eficient i escalable. Per una banda, s'ha proposat una nova formulaci´oestabilitzada d'elements finits per a l'aproximaci´ode les equacions de la MHD resistiva. La novetat d'aquesta formulaci´o resideix en el fet que sempre convergeix a la soluci´of´ısicadel problema, fins i tot quan ´es singular, cosa que s'ha demostrat a trav´esde les an`alisisd'estabilitat i converg`enciadel m`etode. A m´es,l’an`aliside converg`encia mostra la necessitat de fer servir un tipus par- ticular de malles amb una estructura de macro-element, que es poden obtenir f`acilment a partir de qualsevol malla original. Finalment, s'han proposat diferents esquemes de segregaci´oincondicionalment estables per a resoldre el problema de la MHD resistiva transit`oria.S'han considerat dos nivells de segregaci´o.El primer nivell permet la segre- gaci´odels multiplicadors de Lagrange, la pressi´oi la pseudo-pressi´omagn`etica,del c`alcul dels camps vectorials. En el segon nivell, es desacobla el c`alculdels camps vectorials, la velocitat i l’inducci´omagn`etica.D'aquesta manera, el sistema de multif´ısicatotalment acoblat es transforma en problemes d'una f´ısicadesacoblats i m´espetits. D'altra banda, tamb´es'ha presentat una formulaci´oestabilitzada per al problema de la MHD sense inducci´o.Aquest problema permet modelar el flux d'un fluid carregat el`ectricament sota l'efecte d'un camp magn`eticextern on el camp magn`eticindu¨ıtal fluid pels corrents ´esnegligible respecte del camp magn`eticextern. Aquest sistema d'equacions ´esfortament acoblat i altament no lineal. Llavors, resoldre els sistemes d'equacions lineals que resulten de la discretitzaci´oamb m`etodes d'elements finits d'aquestes equacions ´esun gran repte que necessita precondicionadors eficients i escalables. S'ha desenvolupat una nova fam´ıliade precondicionadors LU per blocs recursius per millorar la converg`encia dels m`etodes iteratius per a resoldre aquest problema. Aquests precondicionadors permeten la segregaci´ode la matriu totalment acoblada en problemes d'una f´ısicaper a cada una de les variables del problema (velocitat, pressi´o,densitat de corrent i potencial el`ectric)que es poden resoldre de forma `optima,per exemple, fent servir algorismes de descomposici´o de domini precondicionats. A m´esa m´es,aquestes idees s'han ext`esper a desenvolupar precondicionadors LU per blocs recursius per al problema de la MHD sense inducci´o amb acoblament t`ermic. 8 Contents 1 Introduction 13 2 Resistive MHD problem 16 2.1 Introduction . 16 2.1.1 State-of-the-art . 16 2.1.2 Motivation of the work . 17 2.2 Problem statement . 18 2.2.1 The strong form . 18 2.2.2 The weak form . 20 2.3 Some finite element approximations . 21 2.4 Time discretization and linearization . 23 2.5 A stabilized FE formulation suitable for singular magnetic solutions . 24 2.6 Numerical experimentation . 27 2.6.1 Convergence to singular solutions. Case 2D . 27 2.6.2 Convergence to singular solutions. Extension to the 3D case . 29 2.6.3 Classical MHD problems with analytical solution. Shercliff's case 35 2.6.4 Classical MHD problems with analytical solution. Hunt's case . 37 2.6.5 Clogging of nozzles in steel casting processes. 38 2.7 Conclusions . 40 3 Analysis of the stabilized formulation for the resistive MHD problem 42 3.1 Introduction . 42 3.2 Problem statement . 43 3.2.1 The strong form . 43 3.2.2 The weak form . 44 3.3 A stabilized FE formulation suitable for singular magnetic solutions . 46 3.4 Stability analysis . 48 3.5 Convergence analysis . 53 3.6 Some comments on the nonlinear analysis . 56 3.7 Numerical experimentation . 57 3.8 Conclusions . 59 4 Operator splitting solvers for the resistive MHD problem 66 4.1 Introduction . 66 4.2 Problem statement . 69 4.2.1 Continuous problem . 69 9 4.2.2 Weak form . 69 4.2.3 Galerkin finite element approximation and time integration . 71 4.3 Linearization and semi-implicit algorithms . 74 4.4 Term-by-term stabilized finite element formulation . 75 4.5 Operator splitting techniques .
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