73 research outputs found

    Discretisations and Preconditioners for Magnetohydrodynamics Models

    Full text link
    The magnetohydrodynamics (MHD) equations are generally known to be difficult to solve numerically, due to their highly nonlinear structure and the strong coupling between the electromagnetic and hydrodynamic variables, especially for high Reynolds and coupling numbers. In the first part of this work, we present a scalable augmented Lagrangian preconditioner for a finite element discretisation of the B\mathbf{B}-E\mathbf{E} formulation of the incompressible viscoresistive MHD equations. For stationary problems, our solver achieves robust performance with respect to the Reynolds and coupling numbers in two dimensions and good results in three dimensions. Our approach relies on specialised parameter-robust multigrid methods for the hydrodynamic and electromagnetic blocks. The scheme ensures exactly divergence-free approximations of both the velocity and the magnetic field up to solver tolerances. In the second part, we focus on incompressible, resistive Hall MHD models and derive structure-preserving finite element methods for these equations. We present a variational formulation of Hall MHD that enforces the magnetic Gauss's law precisely (up to solver tolerances) and prove the well-posedness of a Picard linearisation. For the transient problem, we present time discretisations that preserve the energy and magnetic and hybrid helicity precisely in the ideal limit for two types of boundary conditions. In the third part, we investigate anisothermal MHD models. We start by performing a bifurcation analysis for a magnetic Rayleigh--B\'enard problem at a high coupling number S=1,000S=1{,}000 by choosing the Rayleigh number in the range between 0 and 100,000100{,}000 as the bifurcation parameter. We study the effect of the coupling number on the bifurcation diagram and outline how we create initial guesses to obtain complex solution patterns and disconnected branches for high coupling numbers.Comment: Doctoral thesis, Mathematical Institute, University of Oxford. 174 page

    Robust stabilised finite element solvers for generalised Newtonian fluid flows

    Get PDF
    Various materials and solid-fluid mixtures of engineering and biomedical interest can be modelled as generalised Newtonian fluids, as their apparent viscosity depends locally on the flow field. Despite the particular features of such models, it is common practice to combine them with numerical techniques originally conceived for Newtonian fluids, which can bring several issues such as spurious pressure boundary layers, unsuitable natural boundary conditions and coupling terms spoiling the efficiency of nonlinear solvers and preconditioners. In this work, we present a finite element framework dealing with such issues while maintaining low computational cost and simple implementation. The building blocks of our algorithm are (i) an equal-order stabilisation method preserving consistency even for lowest-order discretisations, (ii) robust extrapolation of velocities in the time-dependent case to decouple the rheological law from the overall system, (iii) adaptive time step selection and (iv) a fast physics-based preconditioned Krylov subspace solver, to tackle the relevant range of discretisation parameters including highly varying viscosity. Selected numerical experiments are provided demonstrating the potential of our approach in terms of robustness, accuracy and efficiency for problems of practical interest

    Preconditioning for linear systems arising from discretization of the Navier-Stokes equations using isogeometric analysis

    Get PDF
    Tato práce se zabývá iteračním řešením sedlobodových soustav lineárních algebraických rovnic získaných diskretizací Navierových--Stokesových rovnic pro nestlačitelné proudění pomocí isogemetrické analýzy (IgA). Konkrétně se zaměřuje na předpodmiňovače pro krylovovské metody. Jedním z cílů práce je prozkoumat efektivitu moderních blokových předpodmiňovačů pro různé isogeometrické diskretizace, tj. pro B-spline bázové funkce různého stupně a spojitosti, a poskytnout přehled o jejich chování v závislosti na různých parametrech úlohy. Hlavním cílem je na základě této studie navrhnout vhodný přístup k řešení těchto soustav s případnými úpravami, které by zlepšily vlastnosti dané metody pro soustavy získané isogeometrickou analýzou. Práce má dvě části. V první části jsou představeny úlohy pro nestlačitelné vazké proudění a metoda diskretizace pomocí isogeometrické analýzy. Dále uvádíme podrobný přehled metod řešení sedlobodových soustav lineárních rovnic, ve kterém se zaměřujeme především na blokové předpodmiňovače. Druhá část je věnována numerickým experimentům. Provádíme srovnání vybraných předpodmiňovačů pro několik stacionárních a nestacionarních úloh ve dvou a třech dimenzích. Zvláštní pozornost je věnována aproximaci matice hmotnosti, jejíž volba se ukazuje být v kontextu IgA důležitá, a okrajovým podmínkám pro PCD předpodmiňovač. Navrhujeme vhodnou kombinaci varianty PCD, okrajových podmínek a jejich škálování, abychom získali efektivní předpodmiňovač, který je robustní vzhledem k stupni a spojitosti diskretizace. V mnoha případech se tato volba ukazuje jako nejefektivnější z uvažovaných metod.ObhájenoThis doctoral thesis deals with iterative solution of the saddle-point linear systems obtained from discretization of the incompressible Navier--Stokes equations using the isogeometric analysis (IgA) approach. Specifically, it is focused on preconditioners for Krylov subspace methods. One of the goals of the thesis is to investigate the performance of the state-of-the-art block preconditioners for various IgA discretizations, i.e., for B-spline discretization bases of varying polynomial degree and interelement continuity, and provide an overview of their behavior depending on different problem parameters. The main goal is, based on the this study, to propose suitable solution approach to the considered linear systems with possible modifications that would improve the performance for IgA discretizations in particular. The thesis is basically divided into two parts. In the first part we introduce the mathematical model of incompressible viscous flow and the isogeometric analysis discretization method. Then we provide a detailed overview of the solution techniques for saddle-point linear systems, especially aimed at the family of block preconditioners. The second part is devoted to numerical experiments. We present a comparison of the selected preconditioners for several steady-state and time-dependent test problems in two and three dimensions. A particular attention is devoted to mass matrix approximation within the preconditioners, which appears to be important in the context of IgA, and to the boundary conditions for the pressure convection--diffusion (PCD) preconditioner. A suitable combination of PCD variant, boundary conditions and their appropriate scaling is proposed, leading to an effective preconditioner which is robust with respect to the discretization degree and continuity. In many cases, this choice of preconditioner proves to be the most efficient among all considered methods

    Applications of a finite-volume algorithm for incompressible MHD problems

    Get PDF
    We present the theory, algorithms and implementation of a parallel finite-volume algorithm for the solution of the incompressible magnetohydrodynamic (MHD) equations using unstructured grids that are applicable for a wide variety of geometries. Our method implements a mixed Adams-Bashforth/Crank-Nicolson scheme for the nonlinear terms in the MHD equations and we prove that it is stable independent of the time step. To ensure that the solenoidal condition is met for the magnetic field, we use a method whereby a pseudo-pressure is introduced into the induction equation; since we are concerned with incompressible flows, the resulting Poisson equation for the pseudo-pressure is solved alongside the equivalent Poisson problem for the velocity field. We validate our code in a variety of geometries including periodic boxes, spheres, spherical shells, spheroids and ellipsoids; for the finite geometries we implement the so-called ferromagnetic or pseudo-vacuum boundary conditions appropriate for a surrounding medium with infinite magnetic permeability. This implies that the magnetic field must be purely perpendicular to the boundary. We present a number of comparisons against previous results and against analytical solutions, which verify the code's accuracy. This documents the code's reliability as a prelude to its use in more difficult problems. We finally present a new simple drifting solution for thermal convection in a spherical shell that successfully sustains a magnetic field of simple geometry. By dint of its rapid stabilization from the given initial conditions, we deem it suitable as a benchmark against which other self-consistent dynamo codes can be tested

    Incompressible lagrangian fluid flow with thermal coupling

    Get PDF
    A method is presented for the solution of an incompressible viscous fluid flow with heat transfer and solidification using a fully Lagrangian description of the motion. The originality of this method consists in assembling various concepts and techniques which appear naturally due to the Lagrangian formulation. First of all, the Navier-Stokes equations of motion coupled with the Boussinesq approximation must be reformulated in the Lagrangian framework, whereas they have been mostly derived in an Eulerian context. Secondly, the Lagrangian formulation implies to follow the material particles during their motion, which means to convect the mesh in the case of the Finite Element Method (FEM), the spatial discretisation method chosen in this work. This provokes various difficulties for the mesh generation, mainly in three dimensions, whereas it eliminates the classical numerical difficulty to deal with the convective term, as much in the Navier-Stokes equations as in the energy equation. Even without the discretization of the convective term, an efficient iterative solver, which constitutes the only viable alternative for three dimensional problems, must be designed for the class of Generalized Stokes Problems (GSP), which could be able to behave well independently of the mesh Reynolds number, as it can vary greatly for coupled fluid-thermal analysis. Moreover, it offers a natural framework to treat free-surface problems like wave breaking and rough fluid-structure contact. On one hand, the convection of the mesh during one time step after the resolution of the non-linear system provides explicitly the locus of the domain to be considered. On the other hand, fluid-to-fluid and fluid-to-wall contact, as well as the update of the domain due to the remeshing, must be accurately and efficiently performed. Finally, the solidification of the fluid coupled with its motion through a variable viscosity is considered An efficient overall algorithm must be designed to bring the method effective, particularly in a three dimensional context, which is the ambition of this monograph. Various numerical examples are included to validate and highlight the potential of the method

    Incompressible Lagrangian fluid flow with thermal coupling

    Get PDF
    In this monograph is presented a method for the solution of an incompressible viscous fluid flow with heat transfer and solidification usin a fully Lagrangian description on the motion. The originality of this method consists in assembling various concepts and techniques which appear naturally due to the Lagrangian formulation.Postprint (published version
    corecore