This thesis is concerned with the further development and analysis of a class of Computational Fluid Dynamics (CFD) methods for the numerical simulation of compressible flows on unstructured grids, known as Residual Distribution (RD).<p>The RD method constitutes a class of discretization schemes for hyperbolic systems <p>of conservation laws, which forms an attractive alternative to the more classical Finite Volume methods, particularly since it allows better representation of the flow physics by genuinely multi-dimensional upwinding and offers second-order accuracy on a compact stencil.<p><p>Despite clear advantages of RD schemes, they also have some unexpected anomalies in common with Finite Volume methods and an attempt to resolve them is presented. The most notable anomaly is the violation of the entropy condition, which as a consequence allows unphysical expansion shocks to exist in the numerical solution. In the thesis the genuinely multi-dimensional character of this anomaly is analyzed and a multi-dimensional entropy fix is presented and shown to avoid expansion shocks. Another infamous anomaly is the carbuncle phenomenon, an instability observed in many numerical solutions with strong shocks, such as the bow shock on a blunt body in hypersonic flow. The occurence of the carbuncle phenomenon with RD methods is analyzed and a novel formulation for a shock fix, based on an anisotropic diffusion term added in the shock layer, is presented and shown to cure the anomaly in 2D and 3D hypersonic flow problems.<p><p>In the present work an effort has been made also to an objective and quantitative assessment of the merits of the RD method for typical aerodynamical engineering applications, such as the transonic flow over airfoils and wings.<p>Validation examples including inviscid, laminar as well as high Reynolds number turbulent flows <p>and comparisons against results from state-of-the-art Finite Volume methods are presented.<p>It is shown that the second-order multi-dimensional upwind RD schemes have an accuracy which is at least as good as second-order FV methods using dimension-by-dimension upwinding and that their main advantage lies in providing excellent monotone shock capturing. / Doctorat en Sciences de l'ingénieur / info:eu-repo/semantics/nonPublished

Ce travail a pour objectif de générer des modèles vasculaires et de simuler des écoulements sanguins réalistes à l'intérieur de ces modèles. La première étape consiste à segmenter/reconstruire le volume 3D du réseau vasculaire. Une fois de tels volumes vasculaires segmentés et maillés, il est alors possible de simuler des écoulements sanguins à l'intérieur de ceux-ci. Pour la segmentation, nous utilisons une approche variationnelle. Nous proposons un premier modèle qui inclut un a priori de tubularité dans les modèles de débruitage ROF et TV-L1. Néanmoins, bien que ces modèles permettent de réhausser les vaisseaux, ils ne permettent pas de les segmenter. C'est pourquoi nous proposons un deuxième modèle amélioré qui inclut à la fois un a priori de tubularité et de direction dans le modèle de segmentation de Chan-Vese. Les résultats sont présentés sur des images synthétiques 2D, ainsi que sur des images rétiniennes. En ce qui concerne la simulation, nous nous intéressons d'abord au réseau veineux cérébral, encore peu étudié. Les équations de la dynamique des fluides qui régissent les écoulements sanguins dans notre géométrie sont alors les équations de Navier-Stokes. Pour résoudre ces équations, la méthode classique des caractéristiques est comparée avec un schéma d'ordre plus élevé. Ces deux schémas sont validés sur des solutions analytiques avant d'être appliqués aux cas réalistes du réseau veineux cérébral premièrement, puis du polygone artériel de Willis. / The aim of this work is to generate vascular models and simulate blood flows inside these models. A first step consists of segmenting/reconstructing the 3D volume of the vascular network. Once such volumes are segmented and meshed, it is then possible to simulate blood flows. For segmentation purposes, we use a variational approach. We first propose a model that embeds a vesselness prior in the denoising models ROF and TV-L1. Although these models can enhance vessels, they are not designed for segmentation. Then, we propose a second, improved model that includes both vesselness and direction priors in the Chan-Vese segmentation model. The results are presented on 2D synthetic images, as well as retinal images. In the second part, devoted to simulation, we first focus on the cerebral venous network, that has not been intensively studied. The equations governing blood flows inside our geometry are the Navier-Stokes equations. For their resolution, the classical method of characteristics is compared with a high-order scheme. Both schemes are validated on analytical solutions before their application on the realistic cases of the cerebral venous network, and the arterial polygon of Willis.

Uma grande limitação dos métodos de diferenças finitas é que eles estão restritos a malhas e domínios retangulares. Para descrever escoamentos em domínios complexos, como, por exemplo, problemas com superfícies livres, faz-se necessário o uso de técnicas acessórias. O método de interfaces imersas é uma dessas técnicas. Nesse trabalho, primeiramente foi desenvolvido um método de projeção, totalmente livre de pressão, para as equações de Navier-Stokes com variáveis primitivas em malha deslocada. Esse método é baseado em diferenças finitas compactas, possuindo segunda ordem temporal e quarta ordem espacial. Esse método foi combinado com o método de interface imersa de Linnick e Fasel [2] para resolver numericamente as equações de Stokes com quarta ordem de precisão. A verificação do código foi feita por meio do método das soluções manufaturadas e da comparação com resultados de outros autores em problemas clássicos da literatura. / A great limitation of finite differences methods is that they are restricted to retangular meshes and domains. In order to describe flows in complex domains, e.g. free surface problems, it is necessary to use accessory techniques. The immersed interface method is one of such techniques. In the present work, firstly, a projection method was developed, which is completely pressure-free, for the Navier-Stokes equations with primitive variables in a staggered mesh. This method is based on compact finite differences, with temporal second-order precision and spatial foruth-order precision. This method was combined with the immersed interface method from Linnick e Fasel [2] in order to numerically solve the Stokes equations with fourth-order precision. The verification of the code was performed with the manufactured solutions method and by comparing results with other authors for some classical problems in the literature.

Le cœur d'un réacteur nucléaire est un milieu très hétérogène encombré de nombreux obstacles solides et les phénomènes thermohydrauliques à l'échelle macroscopique sont directement impactés par les phénomènes locaux. Toutefois les ressources informatiques actuelles ne suffisent pas à effectuer des simulations numériques directes d'un cœur complet avec la précision souhaitée. Cette thèse est consacré au développement de méthodes d'éléments finis multi-échelles (MsFEMs) pour simuler les écoulements incompressibles dans un milieu hétérogène avec un coût de calcul raisonnable. Les équations de Navier-Stokes sont approchées sur un maillage grossier par une méthode de Galerkin stabilisé, dans laquelle les fonctions de base sont solutions de problèmes locaux sur des maillages fins prenant précisément en compte la géométrie locale. Ces problèmes locaux sont définis par les équations de Stokes ou d'Oseen avec des conditions aux limites ou des termes sources appropriés. On propose plusieurs méthodes pour améliorer la précision des MsFEMs, en enrichissant l'espace des fonctions de base locales. Notamment, on propose des MsFEMs d'ordre élevée dans lesquelles ces conditions aux limites et termes sources sont choisis dans des espaces de polynômes dont on peut faire varier le degré. Les simulations numériques montrent que les MsFEMs d'ordre élevés améliorent significativement la précision de la solution. Une chaîne de simulation multi-échelle est construite pour simuler des écoulements dans des milieux hétérogènes de dimension deux et trois. / The nuclear reactor core is a highly heterogeneous medium crowded with numerous solid obstacles and macroscopic thermohydraulic phenomena are directly affected by localized phenomena. However, modern computing resources are not powerful enough to carry out direct numerical simulations of the full core with the desired accuracy. This thesis is devoted to the development of Multiscale Finite Element Methods (MsFEMs) to simulate incompressible flows in heterogeneous media with reasonable computational costs. Navier-Stokes equations are approximated on the coarse mesh by a stabilized Galerkin method, where basis functions are solutions of local problems on fine meshes by taking precisely local geometries into account. Local problems are defined by Stokes or Oseen equations with appropriate boundary conditions and source terms. We propose several methods to improve the accuracy of MsFEMs, by enriching the approximation space of basis functions. In particular, we propose high-order MsFEMs where boundary conditions and source terms are chosen in spaces of polynomials whose degrees can vary. Numerical simulations show that high-order MsFEMs improve significantly the accuracy of the solution. A multiscale simulation chain is constructed to simulate successfully flows in two- and three-dimensional heterogeneous media.

Numerical solutions of the Navier-Stokes Equations became more popular in recent decades with increasingly accessible and powerful computational resources. Simulations in reconstructed or artificial pore geometries are often performed to gain insight into microscopic fluid flow structures or are used for upscaling quantities of interest, like hydraulic conductivity. A physically adequate representation of pore-scale flow fields requires analysis of large domains. We solve the incompressible NSE in artificial ordered and random pore-space structures. A simple cubic and face-centred packings of spheres placed in a square duct are analysed. For the fluid flow simulations of random media, packings of spheres, icosahedra, and cubes forming unconsolidated sediments are generated using a rigid body simulation software. The Direct Numerical Simulation method is used for the solution of the NSE implemented in the open-source computational fluid dynamics software OpenFOAM. The influence of the number of spheres in ordered packings, the mesh type, and the mesh resolution is investigated for fluid flow up to Reynolds numbers of 100 based on the spheres' diameter. The random media mesh generation method relies on approximate surface reconstruction. The resulting tetrahedral meshes are then used for steady-state simulations and refined based on an a-posteriori error estimator. The fluid flow simulation results can further be used twofold: 1) They provide homogenized hydro-mechanical properties of the analysed medium for the larger meso and macro groundwater flow simulations. A concept of one-way binding for large-scale simulations is presented. 2) Visualisation: A post-processing image rendering technique was employed in interactive and still image visualisation environments allowing better overview over local fluid flow structures. The ogs FEM code for the solution of large-scale groundwater processes was inspected for computational efficiency. The conclusions drawn from this analysis formed the~basis for the implementation of the~new version of the code---ogs6. The improvements include comparison of linear algebra software realisations and an implementation of optimized memory access patterns in FEM-local assembler part.

Ice sheets, such as the Greenland Ice Sheet or Antarctic Ice Sheet, have a fundamental impact on landscape formation, the global climate system, and on sea level rise. The slow, creeping flow of ice can be represented by a non-linear version of the Stokes equations, which treat ice as a non-Newtonian, viscous fluid. Large spatial domains combined with long time spans and complexities such as a non-linear rheology, make ice sheet simulations computationally challenging. The topic of this thesis is the efficiency and error control of large simulations, both in the sense of mathematical modelling and numerical algorithms. In the first part of the thesis, approximative models based on perturbation expansions are studied. Due to a thick boundary layer near the ice surface, some classical assumptions are inaccurate and the higher order model called the Second Order Shallow Ice Approximation (SOSIA) yields large errors. In the second part of the thesis, the Ice Sheet Coupled Approximation Level (ISCAL) method is developed and implemented into the finite element ice sheet model Elmer/Ice. The ISCAL method combines the Shallow Ice Approximation (SIA) and Shelfy Stream Approximation (SSA) with the full Stokes model, such that the Stokes equations are only solved in areas where both the SIA and SSA is inaccurate. Where and when the SIA and SSA is applicable is decided automatically and dynamically based on estimates of the modeling error. The ISCAL method provides a significant speed-up compared to the Stokes model. The third contribution of this thesis is the introduction of Radial Basis Function (RBF) methods in glaciology. Advantages of RBF methods in comparison to finite element methods or finite difference methods are demonstrated. / eSSENCE

Two numerical models have been developed to describe the volumetric changes during solidification in pure metals and alloys and to predict shrinkage defects in the castings of general three-dimensional configuration. The first model is based on the full system of the Continuity, Navier-Stokes and Enthalpy Equations. Volumetric changes are described by introducing a source term in the Continuity Equation which is a function of the rate of local phase transformation. The model is capable of simulating both volumetric shrinkage and expansion. The second simplified shrinkage model involves the solution of only the Enthalpy Equation. Simplifying assumptions that the feeding flow is governed only by gravity and solidification rate and that phase transformation proceeds only from liquid to solid allowed the fluid flow equations to be excluded from consideration. The numerical implementation of both models is based on an existing proprietary general purpose CFD code, FLOW-3D, which already contains a numerical algorithm for incompressible fluid flow with heat transfer and phase transformation. An important part of the code is. the Volume Of Fluid (VOF) algorithm for tracking multiple free surfaces. The VOF function is employed in both shrinkage models to describe shrinkage cavity formation. Several modifications to FLOW-3D have been made to improve the accuracy and efficiency of the metal/mould heat transfer and solidification algorithms. As part of the development of the upwind differencing advection algorithm used in the simulations, the Leith's method is incorporated into the public domain twodimensional SOLA code. It is shown that the resulting scheme is unconditionally stable despite being explicit.

In the present work we implemented parallel version of a computational fluid dynamics code. This code is based on Discontinuous Galerkin Method (DGM), which is due to its favourable properties suitable for parallelization. In the work we describe the Navier-Stokes equations and their discretization using DGM. We explain the advantages of usage of the DGM and formulate the serial algorithm. Next we focus on the parallel implementation of the algorithm and several particular issues connected to the parallelization. We present the numerical experiments showing the efficiency of the parallel code in the last chapter.

This thesis is concerned on the simulation of wet steam flow using discontinuous Galerkin method. Wet steam flow equations consist of Naviere-Stokes equations for compressible flow and Hill's equations for condensation of water vapor. The first part of this thesis describes the mathematical formulation of wet steam model and the derivation of Hill's equations. The model equations are discretized with the aid of discontinuous Galerkin method and backward difference formula which leads to implicit scheme represented by nonlinear algebraic system. This system is solved using Newton-like method. The derived scheme was implemented in program ADGFEM which is used for solving non-stationary convective-diffusive problems. The numerical results are presented in the last part of this thesis. 1

O presente trabalho de pesquisa tem por objetivo estudar a paralelização de algoritmos voltados à solução de equações diferenciais parciais. Esses algoritmos são utilizados para gerar a solução numérica das equações de Navier-Stokes em um escoamento bidimensional incompressível de um fluido newtoniano. As derivadas espaciais são calculadas através de um método de diferenças finitas compactas com a utilização de aproximações de altas ordens de precisão. Uma vez que o cálculo de derivadas espaciais com alta ordem de precisão da forma compacta adotado no presente estudo requer a solução de sistemas lineares tridiagonais, é importante realizar estudos voltados a resolução desses sistemas, para se obter uma boa performance. Ressalta-se ainda que a solução de sistemas lineares também faz-se presente na solução numérica da equação de Poisson. Os resultados obtidos decorrentes da solução das equações diferenciais parciais são comparados com os resultados onde se conhece a solução analítica, de forma a verificar a precisão dos métodos implementados. Os resultados do código voltado à resolução das equações de Navier-Stokes paralelizado para simulação de escoamentos incompressíveis são comparados com resultados da teoria de estabilidade linear, para validação do código final. Verifica-se a performance e o speedup do código em questão, comparando-se o tempo total gasto em função do número de elementos de processamento utilizados / The objective of the present work is to study the parallelization of partial differential equations. The aim is to achieve an effective parallelization to generate numerical solution of Navier-Stokes equations in a two-dimensional incompressible and isothermal flow of a Newtonian fluid. The spatial derivatives are calculated using compact finite differences approximations of higher order accuracy. Since the calculation of spatial derivatives with high order adopted in the present work requires the solution of tridiagonal systems, it is important to conduct studies to solve these systems and achieve good performance. In addiction, linear systems solution is also present in the numerical solution of a Poisson equation. The results generated by the solution of partial differential equations are compared to analytical solution, in order to verify the accuracy of the implemented methods. The numerical parallel solution of a Navier-Stokes equations is compared with linear stability theory to validate the final code. The performance and the speedup of the code in question is also checked, comparing the execution time in function of the number of processing elements