73 research outputs found
Discretisations and Preconditioners for Magnetohydrodynamics Models
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
- 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 by choosing the Rayleigh number in the range
between 0 and 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
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Robust stabilised finite element solvers for generalised Newtonian fluid flows
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
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Smooth and Robust Solutions for Dirichlet Boundary Control of Fluid-Solid Conjugate Heat Transfer Problems
This work offers new computational methods for the optimal control of the conjugate heat transfer (CHT) problem in thermal science. Conjugate heat transfer has many important industrial applications, such as heat exchange processes in power plants and cooling in electronic packaging industry, and has been a staple of computational methods in thermal science for many years. This work considers the Dirichlet boundary control of fluid-solid CHT problems. The CHT system falls into the category of multi-physics problems. Its domain typically consists of two parts, namely, a solid region subject to thermal heating or cooling and a conjugate fluid region responsible for thermal convection transport. These two different physical systems are strongly coupled through the thermal boundary condition at the fluid-solid interface. The objective in the CHT boundary control problem is to select optimally the fluid inflow profile that minimizes an objective function that involves
the sum of the mismatch between the temperature distribution in the system and a prescribed temperature profile and the cost of the control. This objective is realized by minimizing a nonlinear objective function of the boundary control and the fluid temperature variables, subject to partial differential equations (PDE) constraints governed by the coupled heat diffusion equation in the solid region and mass, momentum and energy conservation equations in the fluid region.
Although CHT has received extensive attention as a forward problem, the optimal Dirichlet
velocity boundary control for the coupled CHT process to our knowledge is only very sparsely studied analytically or computationally in the literature [131]. Therefore, in Part I, we describe the formulation of the optimal control problem and introduce the building blocks for the finite element modeling of the CHT problem, namely, the diffusion equation for the solid temperature, the convection-diffusion equation for the fluid temperature, the incompressible viscous Navier-Stokes equations for the fluid velocity and pressure, and the model verification of CHT simulations.
In Part II, we provide theoretical analysis to explain the nonsmoothness issue which has been observed in this study and in Dirichlet boundary control of Navier-Stokes flows by other scientists. Based on these findings, we use either explicit or implicit numerical smoothing to resolve the nonsmoothness issue. Moreover, we use the numerical continuation on regularization parameters to alleviate the difficulty of locating the global minimum in one shot for highly nonlinear optimization problems even when the initial guess is far from optimal. Two suites of numerical experiments have been provided to demonstrate the feasibility, effectiveness and robustness of the optimization scheme.
In Part III, we demonstrate the strategy of achieving parallel scalable algorithms for CHT models in Simulations of Reactor Thermal Hydraulics. Our motivation originates from the observation that parallel processing is necessary for optimal control problems of very large scale, when the simulation of the underlying physics (or PDE constraints) involves millions or billions of degrees of freedom. To achieve the overall scalability of optimal control problems governed by PDE constraints, scalable components that resolve the PDE constraints and their adjoints are the key. In this Part, first we provide the strategy of designing parallel scalable solvers for each building blocks of the CHT modeling, namely, for the discrete diffusive operator, the discrete convection-diffusion operator, and the discrete Navier-Stokes operator. Second, we demonstrate a pair of effective, robust, parallel, and scalable solvers built with collaborators for simulations of reactor thermal hydraulics. Finally, in the the section of future work, we outline the roadmap of parallel and scalable solutions for Dirichlet boundary control of fluid-solid conjugate heat transfer processes
Preconditioning for linear systems arising from discretization of the Navier-Stokes equations using isogeometric analysis
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
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
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
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
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