In order to prepare modern finite element analysis a program for the efficient parallel generation of tetrahedral meshes in a wide class of three dimensional domains having a generalized cylindric shape is presented. The applied mesh generation strategy is based on the decomposition of some 2D-reference domain into single con- nected subdomains by means of its triangulations the tetrahedral layers are built up in parallel. Adaptive grid controlling as well as nodal renumbering algorithms are involved. In the paper several examples are incorporated to demonstrate both program's capabilities and the handling with.

A parallel realisation of the Conjugate Gradient Method with Schur-Complement preconditioning, based on a domain decomposition approach, is described in detail. Special kinds of solvers for the resulting interiour and coupling systems are presented. A large range of numerical results is used to demonstrate the properties and behaviour of this solvers in practical situations.

This paper concerns the solution of plate bending problems in domains composed of rectangles. Domain decomposition (DD) is the basic tool used for both the parallelization of the conjugate gradient method and the construction of efficient parallel preconditioners. A so-called Dirich- let DD preconditioner for systems of linear equations arising from the fi- nite element approximation by non-conforming Adini elements is derived. It is based on the non-overlapping DD, a multilevel preconditioner for the Schur-complement and a fast, almost direct solution method for the Dirichlet problem in rectangular domains based on fast Fourier transform. Making use of Xu's theory of the auxiliary space method we construct an optimal preconditioner for plate problems discretized by conforming Bogner-Fox-Schmidt rectangles. Results of numerical experiments carried out on a multiprocessor sys- tem are given. For the test problems considered the number of iterations is bounded independent of the mesh sizes and independent of the number of subdomains. The resulting parallel preconditioned conjugate gradient method requiresO(h^-2 ln h^-1 ln epsilon^-11) arithmetical operations per processor in order to solve the finite element equations with the relative accuracy epsilon.

In dieser Arbeit entwickeln wir einen Löser für elliptische Randwertprobleme. Dazu diskretisieren wir ein Randwertproblem mittels der Methode der finiten Elemente und erhalten ein Gleichungssystem. Mittels Gebietszerlegungstechniken unterteilen wir das Gebiet der Differentialgleichung und können Teilprobleme des Randwertproblems definieren. Durch die Gebietszerlegung wird eine Hierarchie von Zerlegungen definiert, die wir mittels eines Gebietszerlegungsbaumes festhalten. Anhand dieses Baumes definieren wir nun einen Löser für das Randwertproblem. Dabei berechnen wir die verschiedenen Matrizen des Lösers durch den sogenannten HDD-Algorithmus (engl. hierarchical domain decomposition). Die meisten der zu erstellenden Matrizen sind dabei vollbesetzt, weshalb wir sie mittels Hierarchischer Matrizen approximieren. Mit Hilfe der Hierarchischen Matrizen können wir die Matrizen mit einem fast linearen Aufwand erstellen und speichern. Der Aufwand der Matrixoperationen ist ebenfalls fast linear. Damit wir die Hierarchischen Matrizen für den HDD-Algorithmus verwenden können, müssen wir die Technik der Hierarchischen Matrizen erweitern. Unter anderem führen wir eingeschränkte Clusterbäume, eingeschränkte Blockclusterbäume und die verallgemeinerte Addition für Hierarchische Matrizen ein. Zusätzlich führen wir eine neue Clusterbaum-Konstruktion ein, die auf den HDD-Algorithmus zugeschnitten ist. Die Kombination des HDD-Algorithmus mit Hierarchischen Matrizen liefert einen Löser, den wir mit einem fast linearen Aufwand berechnen können. Der Aufwand zur Berechnung einer Lösung sowie der Speicheraufwand ist ebenfalls fast linear. Des Weiteren geben wir noch einige Modifizierungen des HDD-Algorithmus für weitere Anwendungsmöglichkeiten an. Zusätzlich diskutieren wir die Möglichkeiten der Parallelisierung, denn durch die Verwendung der Gebietszerlegung wird das Randwertproblem in unabhängige Teilprobleme aufgeteilt, die sich sehr gut parallelisieren lassen. Wir schließen die Arbeit mit numerischen Tests ab, die die theoretischen Aussagen bestätigen.

Cette thèse vise à contribuer à l'adoption du virtual testing, pratique industrielle encore embryonnaire qui consistera à optimiser et certifier par la simulation numérique le dimensionnement de pièces industrielles critiques. Le virtual testing permettra des économies colossales dans la conception des pièces mécaniques et un plus grand respect de l'environnement grâce à des designs optimisés. Afin d'atteindre un tel objectif, de nouvelles méthodes de calcul doivent être mises en place, plus sûres, plus respectueuses des architectures matérielles, plus rapides, compatibles avec les contraintes temporelles de l'ingénierie. Nous nous intéressons à la résolution parallèle de problèmes non linéaires de grande taille par des méthodes de décomposition de domaine. Notre objectif est d'atteindre une approximation de la solution exacte en minimisant les communications entre les sous-domaines. Pour cela nous souhaitons maximiser les calculs réalisés indépendamment par sous-domaine à l'aide d'approches de relocalisation non linéaire, contrôler les critères de convergence des solveurs imbriqués de manière à éviter la sur-résolution et les divergences, améliorer la construction de conditions d'interface mixtes, et non linéariser l'étape de préconditionnement du solveur. L'objectif à terme étant de traiter des problèmes de complexité industrielle, la robustesse des méthodes sera un souci constant. De manière classique, les problèmes non linéaires sont résolus en construisant une suite de systèmes linéaires qui peuvent être résolus en parallèle à l'aide de méthodes itératives, telles que les solveurs de Krylov. Nous souhaitons remettre en question cette procédure usuelle en essayant de construire une suite de petits systèmes non linéaires indépendants à résoudre en parallèle. Une telle technique implique l'utilisation de solveurs itératifs imbriqués dont les critères de convergence doivent être syntonisés dynamiquement de manière à éviter à la fois la sur-résolution et la perte de convergence. La robustesse de la méthode pourra notamment être assurée par l'emploi de conditions d'interface mixtes bien construites et de préconditionneurs bien choisis. / This thesis is aimed to contribute to the adoption of virtual testing, an industrial practice still embryonic which consists in optimizing and certifying by numerical simulations the dimensioning of critical industrial structures. The virtual testing will allow colossal savings in the design of mechanical parts and a greater respect for the environment, thanks to optimized designs. In order to achieve this goal, new calculation methods must be implemented, satisfying more requirements concerning safety, respect for hardware architectures, fastness, and compatibility with the time constraints of engineering.We are interested in the parallel resolution of large nonlinear problems by domain decomposition methods. Our goal is to approximate the exact solution by minimizing communication between subdomains. In order to do this, we want to maximize the computations performed independently by subdomain, using nonlinear relocation approaches. We also try to control the convergence criteria of the nested solvers in order to avoid over-resolution and divergences, to improve the construction of conditions Of mixed interface, and non-linearizing the preconditioning step of the solver. The ultimate objective being to deal with problems of industrial complexity, the robustness of the methods we develop will be a constant concern.Conventionally, non-linear problems are solved by constructing a sequence of linear systems that can be solved in parallel using iterative methods, such as Krylov solvers. We wish to question this usual procedure by trying to construct a sequence of small independent nonlinear systems to be solved in parallel. Such a technique involves the use of interleaved iterative solvers, whose convergence criteria must be dynamically tuned in order to avoid both over-resolution and loss of convergence. The robustness of the method can be ensured in particular by the use of well-constructed mixed interface conditions and well-chosen preconditioners;

No description available.

Physical lab experiments have shown that the pressure caused by an impinging jet on a granular bed has the potential to form craters. This poses a danger to landing success and nearby spacecraft for future rocket missions. Current numerical simulations for this process do not accurately reproduce experimental results. Our goal is to produce improved simulations to more accurately and effi- ciently model the changes in pressure as gas flows through a porous medium. A two-dimensional model in space known as the nonlinear Porous Medium Equation as it is derived from Darcy’s law is used. An Alternating-Direction Implicit (ADI) temporal scheme is presented and implemented which reduces our multidimensional problem into a series of one-dimensional problems. We take advantage of explicit approximations for the nonlinear terms using extrapolation formulas derived from Taylor-series, which increases efficiency when compared to other common methods. We couple our ADI temporal scheme with different spatial discretizations including a second-order Finite Difference (FD) method, a fourth-order Orthogonal Spline Collocation (OSC) method, and an Nth-order Chebyshev Spectral method. Accuracy and runtime are compared among the three methods for comparison in a linear analogue of our problem. We see the best results for accuracy when using an ADI-Spectral method in the linear case, but discuss possibilities for increased effi- ciency using an ADI-OSC scheme. Nonlinear results are presented using the ADI-Spectral method and the ADI-FD method.

No description available.

Asynchronous iterative algorithms are parallel iterative algorithms in which communications and iterations are not synchronized among processors. Thus, as soon as a processing unit finishes its own calculations, it starts the next cycle with the latest data received during a previous cycle, without waiting for any other processing unit to complete its own calculation. These algorithms increase the number of updates in some processors (as compared to the synchronous case) but suppress most idle times. This usually results in a reduction of the (execution) time to achieve convergence. Optimized Schwarz methods (OSM) are domain decomposition methods in which the transmission conditions between subdomains contain operators of the form \linebreak $\partial/\partial \nu +\Lambda$, where $\partial/\partial \nu$ is the outward normal derivative and $\Lambda$ is an optimized local approximation of the global Steklov-Poincar\'e operator. There is more than one family of transmission conditions that can be used for a given partial differential equation (e.g., the $OO0$ and $OO2$ families), each of these families containing a particular approximation of the Steklov-Poincar\'e operator. These transmission conditions have some parameters that are tuned to obtain a fast convergence rate. Optimized Schwarz methods are fast in terms of iteration count and can be implemented asynchronously. In this thesis we analyze the convergence behavior of the synchronous and asynchronous implementation of OSM applied to solve partial differential equations with a shifted Laplacian operator in bounded rectangular domains. We analyze two cases. In the first case we have a shift that can be either positive, negative or zero, a one-way domain decomposition and transmission conditions of the $OO2$ family. In the second case we have Poisson's equation, a domain decomposition with cross-points and $OO0$ transmission conditions. In both cases we reformulate the equations defining the problem into a fixed point iteration that is suitable for our analysis, then derive convergence proofs and analyze how the convergence rate varies with the number of subdomains, the amount of overlap, and the values of the parameters introduced in the transmission conditions. Additionally, we find the optimal values of the parameters and present some numerical experiments for the second case illustrating our theoretical results. To our knowledge this is the first time that a convergence analysis of optimized Schwarz is presented for bounded subdomains with multiple subdomains and arbitrary overlap. The analysis presented in this thesis also applies to problems with more general domains which can be decomposed as a union of rectangles. / Mathematics

A l'échelle macroscopique, la mécanique des milieux continus (MMC) rencontre parfois des difficultés à représenter correctement le comportement d'un système physique, du fait d'une modélisation insuffisante des phénomènes. Ces faiblesses sont particulièrement marquées dans les systèmes où les interfaces, qui font apparaître des échelles d'espace très différentes, jouent un rôle prépondérant : microfluidique, écoulements polyphasiques etc.. Or, dans de nombreux domaines, et notamment dans le milieu pétrolier, les modèles macroscopiques existants semblent insuffisants pour pouvoir traiter correctement les cas proposés. Par ailleurs, la méconnaissance des paramètres d’entrée d'une simulation macroscopique tels que les propriétés de transport, introduit parfois une mauvaise représentation de l’ensemble des processus diffusifs. La simulation à l'échelle microscopique, en l'occurrence la dynamique moléculaire classique (DM), peut pallier certains problèmes rencontrés par les approches macroscopiques, en permettant de mieux appréhender les divers processus physiques, notamment aux interfaces. Elle permet également de suppléer l’expérimentation, en permettant de calculer pour un fluide modèle les propriétés physiques du mélange étudié. Ainsi, à partir des ces données générées, il est possible de construire des corrélations palliant aux différents manques. Néanmoins, de par son caractère microscopique, cette approche ne permet de simuler que des échelles sub-micrométriques qui sont bien éloignées de la taille indispensable à la plupart des cas réalistes, qu’ils soient académiques ou industriels. En couplant les deux démarches, macroscopique et microscopique, de manière directe ou indirecte, il est donc envisageable d’accéder à des informations que l’une ou l’autre des ces approches ne peut fournir seule. / Hybrid atomistic-continuum methods allow the simulation of complex flows, depending on the intimate connection of many spatiotemporal scales : from the nanoscale to the microscale and beyond. By limiting the molecular description within a small localized region, for example near fluid/fluid or fluid/solid interfaces (breakdown of the continuum), these methods are useful to study large systems for reasonable times. Besides, there is a wide variety of applications for such hybrid methods, ranging from the micro- or nano-scale devices, and other industrial processes such as wetting, droplet formation, and biomolecules near interfaces. In this work, we present one scheme for coupling the Navier-Stokes set of equations with Molecular Dynamics. Among the existing alternatives to couple these two approaches, we have chosen to implement a domain decomposition algorithm based on the alternating Schwarz method. In this method, the flow domain is decomposed into two overlapping regions : an atomistic region described by molecular dynamics and a continuum region described by a finite volume discretization of the incompressible Navier-Stokes equations. The fundamental assumption is that the atomistic and the continuum descriptions match in the overlapping region, where the exchange of information is performed. The information exchange, requires the imposition of velocity from one sub-domain in the form of boundary conditions (Dirichlet)/constraints on the solver of the other subdomain and vice versa. The spatial coupling as well as the temporal coupling of the two approaches has been investigated in this work. To show the feasibility of such a coupling, we have applied the multiscale method to a classical fluid mechanics problems.