The numerical solution of the convection-diffusion-reaction problem is considered in two and three dimensions. A stabilized finite element method of Galerkin/Least squares type accomodates diffusion-dominated as well as convection- and/or reaction- dominated situations. The resolution of boundary layers occuring in the singularly perturbed case is accomplished using anisotropic mesh refinement in boundary layer regions. In this paper, the standard analysis of the stabilized Galerkin method on isotropic meshes is extended to more general meshes with boundary layer refinement. Simplicial Lagrangian elements of arbitrary order are used.

The paper is concerned with the finite element resolution of layers appearing in singularly perturbed problems. A special anisotropic grid of Shishkin type is constructed for reaction diffusion problems. Estimates of the finite element error in the energy norm are derived for two methods, namely the standard Galerkin method and a stabilized Galerkin method. The estimates are uniformly valid with respect to the (small) diffusion parameter. One ingredient is a pointwise description of derivatives of the continuous solution. A numerical example supports the result. Another key ingredient for the error analysis is a refined estimate for (higher) derivatives of the interpolation error. The assumptions on admissible anisotropic finite elements are formulated in terms of geometrical conditions for triangles and tetrahedra. The application of these estimates is not restricted to the special problem considered in this paper.

The paper presents a method for generating curved surfaces of 3D finite element meshes by mesh refinement starting with a very coarse grid. This is useful for parallel implementations where the finest meshes should be computed and not read from large files. The paper deals with simple geometries as sphere, cylinder, cone. But the method may be extended to more complicated geometries. (with 45 figures)

Ce travail de thèse présente l'implémentation de la simulation d'écoulements diphasiques dans des conditions de réacteurs nucléaires à caloporteur eau, à l'échelle de bulles individuelles. Pour ce faire, nous étudions plusieurs modèles d'écoulements thermohydrauliques et nous focalisons sur une technique de capture d'interface mince entre phases liquide et vapeur. Nous passons ainsi en revue quelques techniques possibles de maillage adaptatif (AMR) et nous fournissons des outils algorithmiques et informatiques adaptés à l'AMR par patchs dont l'objectif localement la précision dans des régions d'intérêt. Plus précisément, nous introduisons un algorithme de génération de patchs conçu dans l'optique du calcul parallèle équilibré. Cette approche nous permet de capturer finement des changements situés à l'interface, comme nous le montrons pour des cas tests d'advection ainsi que pour des modèles avec couplage hyperbolique-elliptique. Les calculs que nous présentons incluent également la simulation du système de Navier-Stokes incompressible qui modélise la déformation de l'interface entre deux fluides non-miscibles. / This PhD work presents the implementation of the simulation of two-phase flows in conditions of water-cooled nuclear reactors, at the scale of individual bubbles. To achieve that, we study several models for Thermal-Hydraulic flows and we focus on a technique for the capture of the thin interface between liquid and vapour phases. We thus review some possible techniques for Adaptive Mesh Refinement (AMR) and provide algorithmic and computational tools adapted to patch-based AMR, which aim is to locally improve the precision in regions of interest. More precisely, we introduce a patch-covering algorithm designed with balanced parallel computing in mind. This approach lets us finely capture changes located at the interface, as we show for advection test cases as well as for models with hyperbolic-elliptic coupling. The computations we present also include the simulation of the incompressible Navier-Stokes system, which models the shape changes of the interface between two non-miscible fluids.

Dans cette thèse, nous nous intéressons à la résolution des équations SPN du transport de neutrons au sein des cœurs de réacteurs nucléaires à eau pressurisée. Ces équations forment un problème aux valeurs propres généralisé. Dans notre étude nous commençons par le problème source associé et ensuite nous étudions le problème aux valeurs propres. Un cœur de réacteur est composé de différents milieux: le combustible, le fluide caloporteur, le modérateur... à cause de ces hétérogénéités de la géométrie, le flux solution du problème source peut être peu régulier. Nous proposons l’analyse numérique de l’approximation de la solution par la méthode des éléments finis du problème source dans le cas où la solution est peu régulière. Pour le problème aux valeurs propres, dans le cas mixte, les théories déjà développées ne s’appliquent pas. Nous proposons ici une nouvelle méthode pour étudier la convergence de la méthode des éléments finis mixtes pour les problèmes aux valeurs propres. Pour les solutions peu régulières, la montée en ordre de la méthode des éléments finis n’améliore pas l’approximation du problème, il faut raffiner le maillage aux alentours des singularités de la solution. La géométrie des cœurs de réacteur se prête bien aux maillages cartésiens, mais leur raffinement augmente vite leur nombre de degrés de liberté. Pour palier à cette augmentation, nous proposons ici une méthode de décomposition de domaine qui permet d’utiliser des maillages globalement non-conformes. / In this thesis, we investigate the resolution of the SPN neutron transport equations in pressurized water nuclear reactor. These equations are a generalized eigenvalue problem. In our study, we first considerate the associated source problem and after we concentrate on the eigenvalue problem. A nuclear reactor core is composed of different media: the fuel, the coolant, the neutron moderator... Due to these heterogeneities of the geometry, the solution flux can have a low-regularity. We propose the numerical analysis of its approximation with finite element method for the low regular case. For the eigenvalue problem under its mixed form, we can not rely on the theories already developed. We propose here a new method for studying the convergence of the SPN neutron transport eigenvalue problem approximation with mixed finite element. When the solution has low-regularity, increasing the order of the method does not improve the approximation, the triangulation need to be refined near the singularities of the solution. Nuclear reactor cores are well-suited for Cartesian grids, but the refinement of these sort of triangulations increases rapidly their number of degrees of freedom. To avoid this drawback, we propose domain decomposition method which can handle globally non-conforming triangulations.

L’objectif de ce travail est d’étudier les capacités de la méthode de Boltzmann sur réseau (LBM) dans un cadre numériquement contraignant : celui de la simulation aéroacoustique en maillage non-uniforme, à très haut nombre de Reynolds et à nombre de Mach non négligeable (Ma > 0.1), appliquée à l’automobile. La problématique industrielle est celle du calcul du bruit intérieur d’origine aérodynamique, dont le calcul du champ de pression pariétal instationnaire sur le vitrage conducteur est la première étape décisive. Il a été constaté qu’un manque de précision sur la faible part acoustique du champ de pression total sur le vitrage, provenant très probablement d’erreurs au niveau des transitions de résolution du maillage, était la cause d’une surestimation du bruit intérieur. Nous présentons d’abord une construction cohérente et unifiée de la méthode de Boltzmann sur réseau à partir de l’équation de Boltzmann, dans un cadre athermal faiblement compressible. Nous étudions ensuite en détail les propriétés aéroacoustiques de la LBM, en parcourant toutes les grandes familles d’opérateurs de collision de la littérature. Une variante de modèle à temps de relaxation multiples, utilisable pour l’aéroacoustique, est présentée et testée. Un modèle alternatif simplifié de filtrage sélectif, rapide et compact, est développé et validé. La problématique des maillages non-uniformes est abordée. Un recensement exhaustif des études LBM menées dans ce cadre dans la littérature montre qu’aucune ne correspond à nos contraintes. Des algorithmes alternatifs aux transitions sont développés. Enfin, des applications industrielles sont réalisées à l’aide des modèles développés dans le mémoire. / The main goal of this work is to study the capacities of the Lattice Boltzmann Method in a constrained numerical framework : that of numerical simulation in automotive aeroacoustics with non-uniform meshes, at high Reynolds number and non egligible Mach number (Ma > 0.1). The industrial problem is the computation of the interior aerodynamic noise, which includes as its first decisive step the computation of the unsteady wall pressure field on the car windows. It was observed that a lack of precision on the weak acoustic part of the total pressure field on the driver-side window, which is most probably due to errors at mesh refinement interfaces, caused an overestimation of the interior noise. We first present a coherent and unified construction of the Lattice BoltzmannMethod from the Boltzmann equation, in an athermal weakly compressible framework. Then, we study in details the aeroacoustic properties of the LBM by reviewingall the main families of collisional operators that exist in the literature. A variant of multiple relaxation time operator that can be used for aeroacoustics is presented and tested. A simplified alternative selective filter, fast and compact, is developped and numerically validated. The problem of non-uniform meshes is discussed. An exhaustive review of the LBM studies that have been carried out within that framework shows that none of them corresponds to our constraints. Alternative transition nodes algorithms are developed. Finally, all the developed models of this work are applied to industrial cases.

In computational fluid dynamics, we often need to solve complex problems with high precision and efficiency. We propose a three-pronged approach to attain this goal. First, we use the discontinuous Galerkin spectral element method (DG-SEM) for its high accuracy. Second, we use graphics processing units (GPUs) to perform our computations to exploit available parallel computing power. Third, we implement a parallel adaptive mesh refinement (AMR) algorithm to efficiently use our computing power where it is most needed. We present a GPU DG-SEM solver with AMR and dynamic load balancing for the 2D wave equation. The DG-SEM is a higher-order method that splits a domain into elements and represents the solution within these elements as a truncated series of orthogonal polynomials. This approach combines the geometric flexibility of finite-element methods with the exponential convergence of spectral methods. GPUs provide a massively parallel architecture, achieving a higher throughput than traditional CPUs. They are relatively new as a platform in the scientific community, therefore most algorithms need to be adapted to that new architecture. We perform most of our computations in parallel on multiple GPUs. AMR selectively refines elements in the domain where the error is estimated to be higher than a prescribed tolerance, via two mechanisms: p-refinement increases the polynomial order within elements, and h-refinement splits elements into several smaller ones. This provides a higher accuracy in important flow regions and increases capabilities of modeling complex flows, while saving computing power in other parts of the domain. We use the mortar element method to retain the exponential convergence of high-order methods at the non-conforming interfaces created by AMR. We implement a parallel dynamic load balancing algorithm to even out the load imbalance caused by solving problems in parallel over multiple GPUs with AMR. We implement a space-filling curve-based repartitioning algorithm which ensures good locality and small interfaces. While the intense calculations of the high order approach suit the GPU architecture, programming of the highly dynamic adaptive algorithm on GPUs is the most challenging aspect of this work. The resulting solver is tested on up to 64 GPUs on HPC platforms, where it shows good strong and weak scaling characteristics. Several example problems of increasing complexity are performed, showing a reduction in computation time of up to 3× on GPUs vs CPUs, depending on the loading of the GPUs and other user-defined choices of parameters. AMR is shown to improve computation times by an order of magnitude or more.

No description available.

[pt] A otimização topológica tem como objetivo encontrar a distribuição mais eficiente de material (ótima topologia) em uma determinada região, satisfazendo as restrições de projeto estabelecidas pelo usuário. Na abordagem tradicional atribui-se uma variável de projeto, constante, denominada densidade, para cada elemento finito da malha. Dessa forma, a qualidade da representação dos novos contornos da estrutura depende do nível de discretização da malha: quanto maior a quantidade de elementos, mais bem definida será a topologia da estrutura otimizada. No entanto, a utilização de malhas super-refinadas implica em um elevado custo computacional, principalmente na etapa de solução numérica das equações de equilíbrio pelo método dos elementos finitos. Este trabalho propõe uma nova estratégia computacional para o refinamento adaptativo local de malhas utilizando elementos finitos poligonais em domínios bidimensionais arbitrários. A ideia consiste em realizar um refinamento da malha nas regiões de concentração de material, sobretudo nos contornos internos e externos, e um desrefinamento nas regiões de baixa concentração de material, como por exemplo, nos furos internos. Desta forma, é possível obter topologias ótimas, com alta resolução e relativamente baixo custo computacional. Exemplos representativos são apresentados para demonstrar a robustez e a eficiência da metodologia proposta por meio de comparações com resultados obtidos com malhas super-refinadas e mantidas constantes durante todo o processo de otimização topológica. / [en] Topology optimization aims to find the most efficient distribution of material (optimal topology) in a given domain, subjected to design constraints defined by the user. The quality of the new boundary representation depends on the level of mesh refinement: the greater the number of elements in the mesh, the better will be the representation of the optimized structure. However, the use of super refined meshes implies in a high computational cost, especially regarding the numerical solution of the linear systems of equations that arise from the finite element method. This work proposes a new computational strategy for adaptive local mesh refinement using polygonal finite elements in arbitrary two-dimensional domains. The idea is to perform a mesh refinement in regions of material concentration, mostly in inner and outer boundaries, and a mesh derefinement in regions of low material concentration such as the internal holes. Thus, it is possible to obtain optimal topologies with high resolution and relatively low computational cost. Representative examples are presented to demonstrate the robustness and efficiency of the proposed methodology by comparing the results obtained herein with the ones from the literature where super refined meshes are held constant throughout all topology optimization process.

Le but de cette étude est d'améliorer les performances, en termes d'espace mémoire et de temps de calcul, des simulations actuelles de l'Interaction mécanique Pastille-Gaine (IPG), phénomène complexe pouvant avoir lieu lors de fortes montées en puissance dans les réacteurs à eau sous pression. Parmi les méthodes de raffinement de maillage, méthodes permettant de simuler efficacement des singularités locales, une approche multi-grille locale a été choisie car elle présente l'intérêt de pouvoir utiliser le solveur en boîte noire tout en ayant un faible nombre de degrés de liberté à traiter par niveau. La méthode Local Defect Correction (LDC), adaptée à une discrétisation de type éléments finis, a tout d'abord été analysée et vérifiée en élasticité linéaire, sur des configurations issues de l'IPG, car son utilisation en mécanique des solides est peu répandue. Différentes stratégies concernant la mise en oeuvre pratique de l'algorithme multi-niveaux ont également été comparées. La combinaison de la méthode LDC et de l'estimateur d'erreur a posteriori de Zienkiewicz-Zhu, permettant d'automatiser la détection des zones à raffiner, a ensuite été testée. Les performances obtenues sur des cas bidimensionnels et tridimensionnels sont très satisfaisantes, l'algorithme proposé se montrant plus performant que des méthodes de raffinement h-adaptatives. Enfin, l'algorithme a été étendu à des problèmes mécaniques non linéaires. Les questions d'un raffinement espace/temps mais aussi de la transmission des conditions initiales lors du remaillage ont entre autres été abordées. Les premiers résultats obtenus sont encourageants et démontrent l'intérêt de la méthode LDC pour des calculs d'IPG. / The aim of this study is to improve the performances, in terms of memory space and computational time, of the current modelling of the Pellet-Cladding mechanical Interaction (PCI),complex phenomenon which may occurs during high power rises in pressurised water reactors. Among the mesh refinement methods - methods dedicated to efficiently treat local singularities - a local multi-grid approach was selected because it enables the use of a black-box solver while dealing few degrees of freedom at each level. The Local Defect Correction (LDC) method, well suited to a finite element discretisation, was first analysed and checked in linear elasticity, on configurations resulting from the PCI, since its use in solid mechanics is little widespread. Various strategies concerning the implementation of the multilevel algorithm were also compared. Coupling the LDC method with the Zienkiewicz-Zhu a posteriori error estimator in orderto automatically detect the zones to be refined, was then tested. Performances obtained on two-dimensional and three-dimensional cases are very satisfactory, since the algorithm proposed is more efficient than h-adaptive refinement methods. Lastly, the LDC algorithm was extended to nonlinear mechanics. Space/time refinement as well as transmission of the initial conditions during the remeshing step were looked at. The first results obtained are encouraging and show the interest of using the LDC method for PCI modelling.