Cette thèse porte sur l'étude des ballotements dans les réservoirs de satellites à poste, lors des phases de manoeuvre à faible accélération. En effet la bulle de gaz d'hélium servant à pressuriser le réservoir se met en mouvement, générant ainsi des perturbations sur la stabilité globale du satellite. Afin de mener à bien cette étude, des méthodes numériques ont été développées, avec une méthode de frontières immergées pour prendre en compte les parois du réservoir.Le code est utilise la méthode Level-Set pour capturer l'interface, et gère les sauts à l'aide de la méthode Ghost-Fluid. Un solveur BlackBox Multigrid est également développé pour améliorer lesperformances de calcul. Une étude est présentée dans le dernier chapitre pour définir quelques lois de comportements en fonction des vitesses et accélérations générées lors des manoeuvres. / The core study of this PhD thesis is the sloshing in satellite tanks, during low acceleration maneuvers. Indeed the helium bubble used to pressurize the tank moves, thus generating perturbations on the global stability of the satellite. In order to understand this problem, numerical schemes have been developed, such as an immersed boundary method to model the tank wall. The numerical tool uses a Level-Set function coupled to a Ghost Fluid Method to track the interface and to account for the jump conditions.A BlackBox Multigrid Solver have been developed to improve computational cost. Finally a study is presented in the last chapter to predict the behaviour of the fluids with a varying rotational speed generated during some classical maneuvers.

O foco principal do presente trabalho é estender uma metodologia numérica embasada no uso de uma técnica de refinamento adaptativo de malha (AMR - Adaptive Mesh Refinement) e no uso de esquemas temporais multipasso implícitos-explícitos (IMEX) a aplicações envolvendo escoamentos reativos com baixo número de Mach. Originalmente desenvolvida para escoamentos incompressíveis, a formulação euleriana daquela metodologia emprega as equações de Navier-Stokes como modelo matemático para descrever a dinâmica do escoamento e o Método da Projeção, baseado no divergente nulo da velocidade do escoamento, para tratar o acoplamento pressão-velocidade presente na formulação com variáveis primitivas. Tal formulação euleriana original é estendida para acomodar novas equações agregadas ao modelo matemático da fase contínua: conservação de massa, fração de mistura (para representar as concentrações de combustível e oxidante), e energia. Além disso, uma equação termodinâmica de estado é integrada ao modelo matemático estendido e é empregada juntamente com a equação de conservação de massa para produzir uma nova restrição (não nula desta vez) ao divergente do campo de velocidade. Assume-se que o escoamento ocorre a baixo número de Mach (hipótese principal). O Método de Diferença Finita é empregado na discretização espacial das variáveis eulerianas de estado, empregando-se uma malha AMR. As vantagens e dificuldades desta extensão são cuidadosamente investigadas e reportadas. Pela importância, do ponto de vista de aplicações práticas, alguns estudos numéricos preliminares envolvendo escoamentos incompressíveis turbulentos com sprays são realizados (as gotículas compõem a fase dispersa). Num primeiro momento, apenas sprays com gotículas inertes são considerados. Embora ainda apenas iniciais, tais estudos já se mostram importantes pois identificam com clareza, em primeira instância, algumas das dificuldades inerentes a serem enfrentadas ao se tratar dentro desta nova metodologia um conjunto relativamente grande de gotículas lagrangianas. No caso de escoamentos incompressíveis turbulentos com sprays, a integração temporal se dá com métodos IMEX para a fase contínua e com o Método de Euler Modificado para a fase dispersa. A turbulência, em todos os casos que a envolvem, é tratada pelo modelo de Simulação das Grandes Escalas (LES - Large Eddy Simulation). As simulações computacionais se dão em um domínio tridimensional, um parelelepípedo, e empregam uma extensão (resultante do presente trabalho) do código AMR3D, um programa de computador sequencial implementado em Fortran90, oriundo de uma colaboração de longa data entre o IME-USP e o MFLab/FEMEC-UFU (Laboratório de Dinâmica de Fluidos da Universidade Federal de Uberlândia). O processamento foi efetuado no LabMAP (Laboratório da Matemática Aplicada do IME-USP). / It is the main goal of the present work to extend a numerical methodology based on both the use of an adaptive mesh refinement technique (AMR) and the use of a multistep, implicit-explicit time-step strategy (IMEX) to applications involving low Mach number reactive flows. Originally developed for incompressible flows, the Eulerian formulation of that methodology employs the Navier-Stokes equations to model the flow dynamics and the Projection Method, based on the vanishing divergence of the velocity field, to tackle the pressure-velocity coupling present when using primitive variables. That Eulerian formulation is extended by adding a new set of equations to the original mathematical model, describing the various properties of the continuous phase: mass conservation, mixture fraction (to represent concentrations of fuel and oxidizer) and energy. Also, a thermodynamic equation of state is included into the extended mathematical model which is employed, along with the equation for the conservation of mass, to derive a new restriction (this time, different from zero) to the divergence of the velocity field. It is assumed that one is dealing with a low Mach number flow (the main hipothesis). The discretization in space employs the Finite Difference Method for the Eulerian variables on a AMR mesh. Advantages and difficulties of such an extension of the previous methodology are carefully investigated and reported. For its importance in the real-world applications, few preliminary numerical studies involving incompressible turbulent flows with sprays are performed (the droplets form what it is called the dispersed phase). Only sprays formed by inert droplets are considered. Even though initial yet, such studies are most important because they clearly identify, first hand, certain difficulties in handling relatively large sets of Lagrangian droplets in the context of this new AMR methodology. In the context of turbulent incompressible flows with sprays, the overall time-step scheme is given by IMEX methods for the continuous phase and by the Improved Euler Method for the dispersed phase. In all the cases in which it is considered, turbulence is modeled by the Large Eddy Simulation (LES) model. The computational simulations are held in a tridimensional domain given by a paralellepiped and all of them employ the extention (resulting of the present work) of the AMR3D code, a sequencial computer program implemented in Fortran90, whose origin is the collaborative work between IMEUSP and MFLab/FEMEC-UFU (Fluid Dynamics Laboratory, Federal University of Uberlândia). Computations were performed at LabMAP (Applied Mathematics Laboratory at IME-USP).

Foi aplicada a técnica de multigrid na implementação de dois programas computacionais visando a solução numérica de problemas em regime permanente de escoamentos laminares com geometrias simples e de um problema simples de transferência de calor em que o campo de velocidade é conhecido e constante, todos bidimensionais em coordenadas cartesianas. Os programas empregaram malhas computacionais estruturadas e ortogonais, estando os mesmos generalizados ao uso de malhas não-uniformes. As equações algébricas foram obtidas segundo uma formulação em volumes finitos, com as variáveis armazenadas no centro dos volumes elementares segundo um arranjo localizado e foram utilizados esquemas de interpolação distintos para cada classe de problema. O sistema de equações resultante foi relaxado através dos algoritmos Gauss-Seidel e TDMA - TriDiagonal Matrix Algorithm. O acoplamento pressão-velocidade foi feito segundo o método SIMPLE - Semi-Implicit Method for Pressure-Linked Equations. O algoritmo multigrid foi implementado na formulação correction storage em ambos os programas. A técnica foi demonstrada para alguns problemas bench-mark, com os resultados apresentando uma aceleração significativa do processo de convergência da solução numérica multigrid em relação às soluções em malha única, especialmente nas situações em que foram empregadas malhas bastante refinadas e foi exigida elevada precisão.

Este trabalho investiga a eficiência do método multigrid quando usado na obtenção de campos de velocidade e temperatura bidimensionais laminares em domínios retangulares. A análise numérica é baseada no esquema de discretização em volumes finitos aplicado a malhas estruturadas ortogonais regulares. O desempenho do algoritmo multigrid correction storage (CS) é analisado para diferentes números de Reynolds na entrada (Rein) e para um número distinto de malhas. Até quatro malhas computacionais foram usadas para ambos os ciclos V e W. Soluções para os campos de temperatura e velocidade obtidos simultaneamente e não-simultaneamente foram investigados. As vantagens do uso de mais de uma malha computacional são discutidas. Para as soluções simultâneas os resultados indicam um aumento do esforço computacional com o incremento do número de Reynold Rein. O número ótimo de relaxações intermadiárias para os ciclos V e W é discutido.

Chan Kai Yam. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves 80-82). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Parameter Identification in Heat Conduction Systems --- p.1 / Chapter 1.2 --- Inverse Problems --- p.3 / Chapter 1.3 --- Challenges in Inverse Problems --- p.6 / Chapter 2 --- Tools in Parameter Identification --- p.9 / Chapter 2.1 --- Output Least Squares Method --- p.10 / Chapter 2.2 --- Tikhonov Regularization --- p.11 / Chapter 2.3 --- Our Approach --- p.14 / Chapter 3 --- Numerical Implementations --- p.20 / Chapter 3.1 --- Finite Element Discretization and Its Convergence --- p.20 / Chapter 3.2 --- Steepest Descent Method --- p.22 / Chapter 3.3 --- Multigrid Techniques --- p.26 / Chapter 4 --- Numerical Experiments --- p.29 / Chapter 4.1 --- One Dimensional Examples --- p.30 / Chapter 4.1.1 --- Selection of mk --- p.31 / Chapter 4.1.2 --- Selection of nk --- p.34 / Chapter 4.1.3 --- Selection of Number of Levels in the Coarse Grid Correction Step --- p.37 / Chapter 4.1.4 --- Convergence with Different Regularization Pa- rameters γ --- p.39 / Chapter 4.1.5 --- Convergence with Different Initial Guesses --- p.42 / Chapter 4.1.6 --- Comparisons between MG and SG Methods --- p.44 / Chapter 4.1.7 --- Comparisons between MG and RMG Methods --- p.46 / Chapter 4.1.8 --- More Examples --- p.49 / Chapter 4.1.9 --- Coarse Grid Correction in Another Approach --- p.60 / Chapter 4.2 --- Two Dimensional Examples --- p.71 / Chapter 4.3 --- Conclusions --- p.78 / Bibliography --- p.80

Very fine discretizations of differential operators often lead to large, sparse matrices A, where the condition number of A is large. Such ill-conditioning has well known effects on both solving linear systems and eigenvalue computations, and, in general, computing solutions with relative accuracy independent of the condition number is highly desirable. This dissertation is divided into two parts. In the first part, we discuss a method of preconditioning, developed by Ye, which allows solutions of Ax=b to be computed accurately. This, in turn, allows for accurate eigenvalue computations. We then use this method to develop discretizations that yield accurate computations of the smallest eigenvalue of the biharmonic operator across several domains. Numerical results from the various schemes are provided to demonstrate the performance of the methods. In the second part we address the role of the condition number of A in the context of multigrid algorithms. Under various assumptions, we use rigorous Fourier analysis on 2- and 3-grid iteration operators to analyze round off errors in floating point arithmetic. For better understanding of general results, we provide detailed bounds for a particular algorithm applied to the 1-dimensional Poisson equation. Numerical results are provided and compared with those obtained by the schemes discussed in part 1.

The effectiveness of the Additive Correction Multigrid (ACM) algorithm, a line-byline Tri-diagonal Matrix Algorithm (TDMA), and simple Gauss-Seidel (GS) iteration in numerical heat transfer analysis is investigated on a conventional single processor computer and on a distributed memory parallel computer. The performance of these methods is studied by solving a two-dimensional, steady heat conduction problem. The execution time of ACM on a single processor is proportional to the number of unknowns to the 1.5 power. This is in contrast to the execution time of the TDMA for which the execution time is proportional to the number of unknowns to the 2.0 power. The GS , TDMA and ACM algorithms are adapted to a model IPSC2 Intel hypercube which has a 32 processing nodes each with 8 MBytes oflocal memory. Because GS is a local method, it has almost perfect speed up, but it also converges more slowly than TDMA, The TDMA, on the other hand, is affected by domain decomposition to a greater extent than GS. As the number of processors used to solve the problem is increased, the execution times for GS and TDMA are essentially equal. Solving the model problem with 32 processors on a 192x192 grid resulted in parallel efficiencies of 95%, 80% and 78% for the GS, TDMA, and ACM algorithms, respectively. Though the parallel efficiency of ACM was the lowest of the three, the parallel ACM algorithm required an order of magnitude less time to solve the model than either parallel GS or parallel TDMA without multigrid.

The objective of this thesis is to develop a more efficient solver for a large system of linear equations arising from finite element discretization on unstructured tetrahedral meshes for a scalar elliptic partial differential equation of second order for pressure in a commercial computational fluid dynamics (CFD) simulation. Segregated solution methods (or pressure correction type methods) are a widely used approach to obtain solutions of Navier-Stokes equations during numerical simulation by many commercial CFD codes. At each time step, these simulations usually require the approximate solution of a series of scalar equations for velocity, pressure and temperature. Even if the simulation does not require high-accuracy approximations, the large systems of linear equations for pressure may not be efficiently solved. The matrices of these systems of linear equations of real-life industry problems often strongly violate weak diagonal dominance and the numerical simulation often requires solutions of very large systems with over a few hundred thousands degrees of freedom. These conditions produce very ill-conditioned systems of linear equations. Therefore, it is very difficult to solve such systems of linear equations efficiently using most currently available common iterative solvers. A survey of solvers for systems of linear equations was undertaken to determine the preferred solution methodology. An algebraic multigrid preconditioned conjugate gradient (AMGPCG) method solver was chosen for these problems. This solver uses the algebraic multigrid (AMG) cycle as a preconditioner for the conjugate gradient (CG) method. The disadvantages of the conventional AMG method are an expensive setup time and large memory requirements, particularly for three dimensional problems. The disadvantage of an expensive setup time needs to be overcome because the simulation usually requires only low-accuracy approximations for pressure. Also it is important to overcome the disadvantage of the large memory requirements for use in commercial software. In this work, an efficient AMGPCG solver is developed by overcoming the disadvantages of the conventional AMG method. The robustness of AMGPCG is shown theoretically so that the solver is always convergent. Optimum or close to optimum rates of convergence behavior for the solver are shown numerically so that the number of necessary iterations to obtain the estimated solution is approximately independent of mesh resolution. Furthermore, numerical experiments of solving pressure for some industry problems were carried out and compared with other efficient solvers including a fast commercial AMGPCG solver (SAMG, release 20b1). It was found that the developed AMGPCG solver was the fastest among these solvers for solving these problems and its algorithm has been numerically proven to be efficient. In addition, the memory requirement is at an acceptable level for commercial CFD codes.

Image analysis problems, posed mathematically as variational principles or as partial differential equations, are amenable to numerical solution by relaxation algorithms that are local, iterative, and often parallel. Although they are well suited structurally for implementation on massively parallel, locally-interconnected computational architectures, such distributed algorithms are seriously handicapped by an inherent inefficiency at propagating constraints between widely separated processing elements. Hence, they converge extremely slowly when confronted by the large representations necessary for low-level vision. Application of multigrid methods can overcome this drawback, as we established in previous work on 3-D surface reconstruction. In this paper, we develop efficient multiresolution iterative algorithms for computing lightness, shape-from-shading, and optical flow, and we evaluate the performance of these algorithms on Synthetic images. The multigrid methodology that we describe is broadly applicable in low-level vision. Notably, it is an appealing strategy to use in conjunction with regularization analysis for the efficient solution of a wide range of ill-posed visual reconstruction problems.

In this dissertation, we study the mixed finite element method for the linear plane elasticity problem and iterative solvers for the resulting discrete system. We use the Arnold-Winther Element in the mixed finite element discretization. An overlapping Schwarz preconditioner and a multigrid preconditioner for the discrete system are developed and analyzed. We start by introducing the mixed formulation (stress-displacement formulation) for the linear plane elasticity problem and its discretization. A detailed analysis of the Arnold-Winther Element is given. The finite element discretization of the mixed formulation leads to a symmetric indefinite linear system. Next, we study efficient iterative solvers for the symmetric indefinite linear system which arises from the mixed finite element discretization of the linear plane elasticity problem. The preconditioned Minimum Residual Method is considered. It is shown that the problem of constructing a preconditioner for the indefinite linear system can be reduced to the problem of constructing a preconditioner for the H(div) problem in the Arnold-Winther finite element space. Our main work involves developing an overlapping Schwarz preconditioner and a multigrid preconditioner for the H(div) problem. We give condition number estimates for the preconditioned systems together with supporting numerical results.