Este é o primeiro trabalho que apresenta simulações tridimensionais completamente adaptativas de um modelo de campo de fase para um fluido incompressível com densidade de massa constante e viscosidade variável, conhecido como Modelo H. Solucionando numericamente as equações desse modelo em malhas refinadas localmente com a técnica AMR, simulamos computacionalmente escoamentos bifásicos tridimensionais. Os modelos de campo de fase oferecem uma aproximação física sistemática para investigar fenômenos que envolvem sistemas multifásicos complexos, tais como fluidos com camadas de mistura, a separação de fases sob forças de cisalhamento e a evolução de micro-estruturas durante processos de solidificação. Como as interfaces são substituídas por delgadas regiões de transição (interfaces difusivas), as simulações de campo de fase requerem muita resolução nessas regiões para capturar corretamente a física do problema em estudo. Porém essa não é uma tarefa fácil de ser executada numericamente. As equações que caracterizam o modelo de campo de fase contêm derivadas de ordem elevada e intrincados termos não lineares, o que exige uma estratégia numérica eficiente capaz de fornecer precisão tanto no tempo quanto no espaço, especialmente em três dimensões. Para obter a resolução exigida no tempo, usamos uma discretização semi-implícita de segunda ordem para solucionar as equações acopladas de Cahn-Hilliard e Navier-Stokes (Modelo H). Para resolver adequadamente as escalas físicas relevantes no espaço, utilizamos malhas refinadas localmente que se adaptam dinamicamente para recobrir as regiões de interesse do escoamento, como por exemplo, as vizinhanças das interfaces do fluido. Demonstramos a eficiência e a robustez de nossa metodologia com simulações que incluem a separação dos componentes de uma mistura bifásica, a deformação de gotas sob cisalhamento e as instabilidades de Kelvin-Helmholtz. / This is the first work that introduces 3D fully adaptive simulations for a phase field model of an incompressible fluid with matched densities and variable viscosity, known as Model H. Solving numerically the equations of this model in meshes locally refined with AMR technique, we simulate computationally tridimensional two-phase flows. Phase field models offer a systematic physical approach to investigate complex multiphase systems phenomena such as fluid mixing layers, phase separation under shear and microstructure evolution during solidification processes. As interfaces are replaced by thin transition regions (diffuse interfaces), phase field simulations need great resolution in these regions to capture correctly the physics of the studied problem. However, this is not an easy task to do numerically. Phase field model equations have high order derivatives and intricate nonlinear terms, which require an efficient numerical strategy that can achieve accuracy both in time and in space, especially in three dimensions. To obtain the required resolution in time, we employ a semi-implicit second order discretization scheme to solve the coupled Cahn-Hilliard/Navier-Stokes equations (Model H). To resolve adequatly the relevant physical scales in space, we use locally refined meshes which adapt dynamically to cover special flow regions, e.g., the vicinity of the fluid interfaces. We demonstrate the efficiency and robustness of our methodology with simulations that include spinodal decomposition, the deformation of drops under shear and Kelvin-Helmholtz instabilities.

Le calcul d'écoulements dans les générateurs de vapeur des réacteurs à eau pressurisée est un problème complexe, faisant intervenir différents régimes d'écoulement et plusieurs échelles de temps et d'espace. Un scénario accidentel peut être caractérisé par des variations très rapides pour un nombre de Mach de l'ordre de l'unité. A l'inverse en régime nominal l'écoulement peut être stationnaire, à bas nombre de Mach. De plus quelque soit le régime considéré, la complexité de la géométrie d'un générateur de vapeur conduit à modéliser le faisceau de tubes par un milieu poreux, d'où le problème de couplage à l'interface avec le milieu fluide.Un schéma de correction de pression tout-Mach en volumes finis colocalisés a été introduit pour les équations d'Euler et de Navier-Stokes. L'existence d'une solution discrète, la consistance du schéma au sens de Lax et la positivité de l'énergie interne ont été démontrées. Le schéma a été ensuite étendu aux modèles diphasiques homogènes du code GENEPI développé au CEA. Enfin un algorithme Multigrille-AMR a été adaptée pour permettre de mettre en oeuvre notre schéma sur des maillages adaptatifs.Concernant la seconde problématique, une extension de la loi de Beavers-Joseph a été proposée pour le régime convectif. En introduisant un saut d'énergie cinétique à l'interface, on retrouve une loi de type Beavers-Joseph mais avec un coefficient de glissement non-linéaire, qui dépend de la vitesse fluide à l'interface et de la vitesse Darcy. La validité de cette nouvelle condition d'interface a été évaluée en réalisant des calculs de simulation numérique directe à différents nombres de Reynolds. / The numerical simulation of steam generators of pressurized water reactors is a complex problem, involving different flow regimes and a wide range of length and time scales. An accidental scenario may be associated with very fast variations of the flow with an important Mach number. In contrast in the nominal regime the flow may be stationary, at low Mach number. Moreover whatever the regime under consideration, the array of U-tubes is modelled by a porous medium in order to avoid taking into account the complex geometry of the steam generator, which entails the issue of the coupling conditions at the interface with the free-fluid.We propose a new pressure-correction scheme for cell-centered finite volumes for solving the compressible Navier-Stokes and Euler equations at all Mach number. The existence of a discrete solution, the consistency of the scheme in the Lax sense and the positivity of the internal energy were proved. Then the scheme was extended to the homogeneous two-phase flow models of the GENEPI code developed at CEA. Lastly a multigrid-AMR algorithm was adapted for using our pressure-correction scheme on adaptive grids.Regarding the second issue addressed in this work, an extension to the Beavers-Joseph law was proposed for the convective regime. By introducing a jump in the kinetic energy at the interface, we recover an interface condition close to the Beavers-Joseph law but with a non-linear slip coefficient, which depends on the free-fluid velocity at the interface and on the Darcy velocity. The validity of this new transmission condition was assessed with direct numerical simulations at different Reynolds numbers.

With the rapid increase in available compute power and memory, and bolstered by the advent of efficient formulations and algorithms, the role of 3D full-wave computational methods for accurate modelling of complex electromagnetic (EM) structures has gained in significance. The range of problems includes Radar Cross Section (RCS) computation, analysis and design of antennas and passive microwave circuits, bio-medical non-invasive detection and therapeutics, energy harvesting etc. Further, with the rapid advances in technology trends like System-in-Package (SiP) and System-on-Chip (SoC), the fidelity of chip-to-chip communication and package-board electrical performance parameters like signal integrity (SI), power integrity (PI), electromagnetic interference (EMI) are becoming increasingly critical. Rising pin-counts to satisfy functionality requirements and decreasing layer-counts to maintain cost-effectiveness necessitates 3D full wave electromagnetic solution for accurate system modelling. Method of Moments (MoM) is one such widely used computational technique to solve a 3D electromagnetic problem with full-wave accuracy. Due to lesser number of mesh elements or discretization on the geometry, MoM has an advantage of a smaller matrix size. However, due to Green's Function interactions, the MoM matrix is dense and its solution presents a time and memory challenge. The thesis focuses on formulation and development of novel techniques that aid in fast MoM based electromagnetic solutions. With the recent paradigm shift in computer hardware architectures transitioning from single-core microprocessors to multi-core systems, it is of prime importance to parallelize the serial electromagnetic formulations in order to leverage maximum computational benefits. Therefore, the thesis explores the possibilities to expedite an electromagnetic simulation by scalable parallelization of near-linear complexity algorithms like Fast Multipole Method (FMM) on a multi-core platform. Secondly, with the best of parallelization strategies in place and near-linear complexity algorithms in use, the solution time of a complex EM problem can still be exceedingly large due to over-meshing of the geometry to achieve a desired level of accuracy. Hence, the thesis focuses on judicious placement of mesh elements on the geometry to capture the physics of the problem without compromising on accuracy- a technique called Adaptive Mesh Refinement. This facilitates a reduction in the number of solution variables or degrees of freedom in the system and hence the solution time. For multi-scale structures as encountered in chip-package-board systems, the MoM formulation breaks down for parts of the geometry having dimensions much smaller as compared to the operating wavelength. This phenomenon is popularly known as low-frequency breakdown or low-frequency instability. It results in an ill-conditioned MoM system matrix, and hence higher iteration count to converge when solved using an iterative solver framework. This consequently increases the solution time of simulation. The thesis thus proposes novel formulations to improve the spectral properties of the system matrix for real-world complex conductor and dielectric structures and hence form well-conditioned systems. This reduces the iteration count considerably for convergence and thus results in faster solution. Finally, minor changes in the geometrical design layouts can adversely affect the time-to-market of a commodity or a product. This is because the intermediate design variants, in spite of having similarities between them are treated as separate entities and therefore have to follow the conventional model-mesh-solve workflow for their analysis. This is a missed opportunity especially for design variant problems involving near-identical characteristics when the information from the previous design variant could have been used to expedite the simulation of the present design iteration. A similar problem occurs in the broadband simulation of an electromagnetic structure. The solution at a particular frequency can be expedited manifold if the matrix information from a frequency in its neighbourhood is used, provided the electrical characteristics remain nearly similar. The thesis introduces methods to re-use the subspace or Eigen-space information of a matrix from a previous design or frequency to solve the next incremental problem faster.

Ce travail de thèse s’inscrit dans le cadre de l’étude de l’Interaction mécanique Pastille-Gaine (IPG) se produisant dans les crayons combustibles des réacteurs à eau pressurisée. Ce mémoire porte sur le développement de méthodes de raffinement de maillage permettant de simuler plus précisément le phénomène d’IPG tout en conservant des temps de calcul et un espace mémoire acceptables pour des études industrielles. Une stratégie de raffinement automatique basée sur la combinaison de la méthode multi-grilles Local Defect Correction (LDC) et l’estimateur d’erreur a posteriori de type Zienkiewicz et Zhu est proposée. Cette stratégie s’appuie sur l’erreur fournie par l’estimateur pour détecter les zones à raffiner constituant alors les sous-grilles locales de la méthode LDC. Plusieurs critères d’arrêt sont étudiés afin de permettre de stopper le raffinement quand la solution est suffisamment précise ou lorsque le raffinement n’apporte plus d’amélioration à la solution globale.Les résultats numériques obtenus sur des cas tests 2D élastiques avec discontinuité de chargement permettent d’apprécier l’efficacité de la stratégie proposée.Le raffinement automatique de maillage dans le cas de problèmes de contact unilatéral est ensuite abordé. La stratégie proposée dans ce travail s’étend aisément au raffinement multi-corps à condition d’appliquer l’estimateur d’erreur sur chacun des corps séparément. Un post-traitement est cependant souvent nécessaire pour garantir la conformité des zones de raffinement vis-à-vis des frontières de contact. Une variété de tests numériques de contact entre solides élastiques confirme l’efficacité et la généricité de la stratégie proposée. / This Ph.D. work takes place within the framework of studies on Pellet-Cladding mechanical Interaction (PCI) which occurs in the fuel rods of pressurized water reactor. This manuscript focuses on automatic mesh refinement to simulate more accurately this phenomena while maintaining acceptable computational time and memory space for industrial calculations. An automatic mesh refinement strategy based on the combination of the Local Defect Correction multigrid method (LDC) with the Zienkiewicz and Zhu a posteriori error estimator is proposed. The estimated error is used to detect the zones to be refined, where the local subgrids of the LDC method are generated. Several stopping criteria are studied to end the refinement process when the solution is accurate enough or when the refinement does not improve the global solution accuracy anymore.Numerical results for elastic 2D test cases with pressure discontinuity shows the efficiency of the proposed strategy.The automatic mesh refinement in case of unilateral contact problems is then considered. The strategy previously introduced can be easily adapted to the multibody refinement by estimating solution error on each body separately. Post-processing is often necessary to ensure the conformity of the refined areas regarding the contact boundaries. A variety of numerical experiments with elastic contact (with or without friction, with or without an initial gap) confirms the efficiency and adaptability of the proposed strategy.

L’adaptation de maillage est un outil puissant pour l’obtention de simulations aérodynamiques précises à coût limité. Dans le cas particulier des simulations visant au calcul de fonctions aérodynamiques (efforts, moments, rendements...), plusieurs méthodes dites de raffinement ciblé (ou, en anglais, « goal-oriented ») basées sur le vecteur adjoint de la fonction d’intérêt ont été proposées. L’objectif de la thèse est l’extension d’une méthode de ce type basée sur la dérivée totale dJ/dX de la grandeur aérodynamique d’intérêt, J, par rapport aux coordonnées du maillage volumique, X. Les trois méthodes usuelles de calcul de gradient discret – la méthode de différentiation directe, la méthode adjointe-"paramètres" et la méthode adjointe-"maillage" évaluant dJ/dX – ont tout d’abord été étudiées et codées dans le logiciel elsA de l’ONERA pour des maillages non-structurés, pour des écoulements compressibles de fluide parfait et des écoulements laminaires. La seconde étape du travail a consisté à créer un senseur local θ basé sur dJ/dX qui identifie les zones du maillage volumique où la position des nœuds a une forte incidence sur l’évaluation de la fonction J. Ce senseur sert d’indicateur pour l’adaptation de différents maillages, pour différents régimes d’écoulement (subsonique, transsonique, supersonique), pour des configurations d’aérodynamique interne (aube et tuyère) et externe (profil d’aile). La méthode proposée est comparée à une méthode de raffinement ciblée très populaire (Venditti et Darmofal, 2001) et à une méthode de raffinement basée sur les caractéristiques de l’écoulement (ou, en anglais, « feature-based ») ; elle conduit à des résultats très satisfaisants. / Mesh adaptation is a powerful tool to obtain accurate aerodynamic simulations with limited cost. In the specific case of computation of aerodynamic functions (forces, moments, efficiency ...), goal-oriented methods based on the adjoint vector have been proposed. The aim of the thesis is the extension of a method of this type based on the total derivative dJ/dX of the aerodynamic output of interest, J, with respect to the volume mesh coordinates, X. The three common methods for calculating discrete gradient – the direct differentiation method, the parameter-adjoint method and mesh-adjoint method evaluating dJ/dX – have been studied first and coded in the elsA ONERA software for unstructured grids, for compressible inviscid and laminar flows. The second part of this work was has been to define a local sensor θ based on dJ/dX in order to identify zones where the volume mesh nodes position has a strong impact on the evaluation of the function J. This sensor is the selected indicator for different mesh adaptations for different flow regimes (subsonic, transonic, supersonic) for internal (blade and nozzle) and external (wing profile) aerodynamic configurations. The proposed method is compared to a well-known goal-oriented method (Darmofal and Venditti, 2001) and to a feature-based method ; it leads to very consistent results. very consistent results.

Heat transfer and solidification models were developed for use in a numerical model of a continuous caster to provide a means of predicting how the developing shell would react under variable operating conditions. Measurement data of the operating conditions leading up to a breakout occurrence were provided by an industrial collaborator and were used to define the model boundary conditions. Steady-state and transient simulations were conducted, using boundary conditions defined from time-averaged measurement data. The predicted shell profiles demonstrated good agreement with thickness measurements of a breakout shell segment – recovered from the quarter-width location. Further examination of the results with measurement data suggests pseudo-steady assumption may be inadequate for modeling shell and flow field transition period following sudden changes in casting speed. An adaptive mesh refinement procedure was established to increase refinement in areas of predicted shell growth and to remove excess refinement from regions containing only liquid. A control algorithm was developed and employed to automate the refinement procedure in a proof-of-concept simulation. The use of adaptive mesh refinement was found to decrease the total simulation time by approximately 11% from the control simulation – using a static mesh.

This paper is concerned with mesh refinement techniques for treating elliptic boundary value problems in domains with re- entrant edges and corners, and focuses on numerical experiments. After a section about the model problem and discretization strategies, their realization in the experimental code FEMPS3D is described. For two representative examples the numerically determined error norms are recorded, and various mesh refinement strategies are compared.

Dans un contexte industriel, l’utilisation de modèles diphasiques d’ordre réduit est nécessaire pour pouvoir effectuer des simulations numériques prédictives d’injection de combustible liquide dans les chambres de combustion automobiles et aéronautiques, afin de concevoir des équipements plus performants et moins polluants. Le processus d’atomisation du combustible, depuis sa sortie de l’injecteur sous un régime de phases séparées, jusqu’au brouillard de gouttelettes dispersées, est l’un des facteurs clés d’une combustion de bonne qualité. Aujourd’hui cependant, la prise en compte de toutes les échelles physiques impliquées dans ce processus nécessite une avancée majeure en termes de modélisation, de méthodes numériques et de calcul haute performance (HPC). Ces trois aspects sont abordés dans cette thèse. Premièrement, des modèles de mélange, dérivés par le principe variationnel de Hamilton et le second principe de la thermodynamique sont étudiés. Ils sont alors enrichis afin de pouvoir décrire des pulsations des interfaces au niveau de la sous-échelle. Des comparaisons avec des données expérimentales dans un contexte de milieux à bulles permettent de vérifier la cohérence physique des modèles et de valider la méthodologie. Deuxièmement, une stratégie de discrétisation est développée, basée sur une séparation d’opérateur, permettant la résolution indépendante de la partie convective des systèmes à l’aide de solveurs de Riemann approchés standards et les termes sources à l’aide d’intégrateurs d’équations différentielles ordinaires. Ces différentes méthodes répondent aux particularités des systèmes diphasiques compressibles, ainsi qu’au choix de l’utilisation de maillages adaptatifs (AMR). Pour ces derniers, une stratégie spécifique est développée : il s’agit du choix de critères de raffinement et de la projection de la solution d’une grille à une autre (plus fine ou plus grossière). Enfin, l’utilisation de l’AMR dans un cadre HPC est rendue possible grâce à la bibliothèque AMR p4est, laquelle a montré une excellente scalabilité jusqu’à plusieurs milliers de coeurs de calcul. Un code applicatif, CanoP, a été développé et permet de simuler des écoulements fluides avec des méthodes de volumes finis sur des maillages AMR. CanoP pourra être utilisé pour des futures simulations d’atomisation liquide. / In an industrial context, reduced-order two-phase models are used in predictive simulations of the liquid fuel injection in combustion chambers and help designing more efficient and less polluting devices. The combustion quality strongly depends on the atomization process, starting from the separated phase flow at the exit of the nozzle down to the cloud of fuel droplets characterized by a disperse-phase flow. Today, simulating all the physical scales involved in this process requires a major breakthrough in terms of modeling, numerical methods and high performance computing (HPC). These three aspects are addressed in this thesis. First, we are interested in mixture models, derived through Hamilton’s variational principle and the second principle of thermodynamics. We enrich these models, so that they can describe sub-scale pulsations mechanisms. Comparisons with experimental data in a context of bubbly flows enables to assess the models and the methodology. Based on a geometrical study of the interface evolution, new tracks are then proposed for further enriching the mixture models using the same methodology. Second, we propose a numerical strategy based on finite volume methods composed of an operator splitting strategy, approximate Riemann solvers for the resolution of the convective part and specific ODE solvers for the source terms. These methods have been adapted so as to handle several difficulties related to two-phase flows, like the large acoustic impedance ratio, the stiffness of the source terms and low-Mach issues. Moreover, a cell-based Adaptive Mesh Refinement (AMR) strategy is considered. This involves to develop refinement criteria, the setting of the solution values on the new grids and to adapt the standard methods for regular structured grids to non-conforming grids. Finally, the scalability of this AMR tool relies on the p4est AMR library, that shows excellent scalability on several thousands cores. A code named CanoP has been developed and enables to solve fluid dynamics equations on AMR grids. We show that CanoP can be used for future simulations of the liquid atomization.

Adaptive anisotropic refinement of finite element meshes allows to reduce the computational effort required to achieve a specified accuracy of the solution of a PDE problem. We present a new approach to adaptive refinement and demonstrate that this allows to construct algorithms which generate very flexible and efficient anisotropically refined meshes, even improving the convergence order compared to adaptive isotropic refinement if the problem permits.:1 Introduction 2 Extension of FEM ansatz spaces 3 Optimality of the extension 4 Application 1: graded refinement 5 Application 2: anisotropic refinement in 2D 6 Numerical experiments 7 Conclusions and outlook

The effect of building geometry on the wind environment of cities is such that it can cause problems like wind danger, discomfort and poor ventilation of airborne pollutants. Computational fluid dynamics (CFD) can play a role in assessing changes in wind environment caused by building projects before realisation at little cost. However, the current state-of-the-art methods, RANS and LES, force a steep trade-off between accuracy and computational cost, and neither method is truly predictive. Time-resolved adaptive direct finite element simulation (DFS) is a method for CFD that is predictive and automatically optimises the mesh for a goal quantity, making it both efficient and accurate. In this thesis, DFS was implemented in FEniCS and used on basic validation cases to provide a proof of concept for the use of this method in the building aerodynamics, on resources freely available to anyone. The results show that the method is accurate to within 10% of the validation data with respect to the goal quantity. Visually, the expected flow features are clearly identifiable. DFS was successfully applied to a relatively complicated building geometry, with a total computation time of about 120 core-hours. We conclude that DFS has significant potential as a method for evaluating urban wind environments. Furthermore, because of its ease of use and lack of parameters, DFS can play an important role in helping architects, designers and students understand the effect of urban geometries on the wind environment. This report provides a basis for further research on DFS for building aerodynamics, as validation on more diverse urban geometries is still necessary. / Effekten av byggnaders form och geometri är så viktig att den kan ge problem för ventilation av t.ex. föroreningar, för energieffektivitet, och för vindfaror med t.ex. hög vindhastihet som kan vara farligt eller skapa obehag. Beräkningsströmningsdynamik (CFD) kan ha en roll i bedömningen av byggnadsprojekt i ett tidigt skede till liten kostnad. Dock är de etablerade och ledande metodikerna, RANS och LES, inte prediktiva och tvingar fram en kompromiss mellan beräkningskosnad och noggrannhet. Vår metodik “Time-resolved adaptive direct finite element simulation” (DFS) är en metod för CFD som är prediktiv och automatiskt optimerar beräkningsnätet (och därmed beräkningskostnaden) för en given målkvantitet, som ger både effektivitet och noggrannhet. I denna avhandling implementerades DFS i FEniCS och användes i grundläggande valideringsfall för att ge ett proof of conceptför användning av denna metod i byggnadsaerodynamik, på resurser som är fritt tillgängliga för alla. Resultaten visar att metoden är korrekt inom 10% av valideringsdata med avseende på målkvantiteten. Visuellt är de förväntade flödesfunktionerna tydligt identifierbara. DFS applicerades framgångsrikt på en relativt komplicerad byggnadsgeometri med en total beräkningstid på cirka 120 kärntimmar, vilket är en försumbar kostnad. Vi drar slutsatsen att DFS har en betydande potential som metod för utvärdering av stadsvindmiljöer. Dessutom, på grund av dess användarvänlighet och frihet från parametrar, kan DFS spela en viktig roll för att hjälpa arkitekter, designers och studenter att förstå effekterna av stadsgeometrier på vindmiljön. Denna rapport ger en grund för vidare forskning om DFS för aerodynamik, eftersom validering av mer olika stadsgeometrier fortfarande är nödvändig.