O escoamento em torno de corpos tem sido objeto de estudo de muitos pesquisadores e é muito explorado experimental e computacionalmente, devido a sua grande aplicabilidade na engenharia. No entanto, simular computacionalmente este tipo de escoamento requer uma atenção especial ao escolher o tipo malha a ser utilizado. Em muitos casos faz-se necessário o uso de uma malha que se adapte ao contorno do obstáculo, o que pode ocasionar um aumento no esforço computacional. Um maneira de contornar este problema é a utilização do Método das Fronteiras Imersas, que possibilita o uso de malha cartesiana na simulação computacional do escoamento em torno de obstáculos. Isso é possível através da adição de um termo forçante nas equações que modelam o escoamento, e assim as forças que agem sobre o contorno do corpo são transferidas diretamente para a malha. O objetivo deste trabalho de mestrado foi implementar o método das Fronteiras Imersas e simular o escoamento em torno de um cilindro circular em repouso, movimentando-se na mesma direção do escoamento, na direção perpendicular ao escoamento, ou rotacionando em torno do próprio eixo. As simulações computacionais possibilitaram a captura do fenômeno de Atrelagem Síncrona, caracterizado pela sincronia entre a frequência de desprendimento natural de vórtices e a frequência de oscilação do mesmo. O Método das Fronteiras Imersas mostrou um ótimo desempenho quando comparado a resultados experimentais e numéricos encontrados na literatura / The flow around bodies have been studied by many researchers. Both experimental and computational approaches have been extensively explored in researches on flow around bodies and have been applied in many engeneering problems. However, to choose an appropriate type of mesh to perform computational simulations of this type of problem requires special attention. In many cases, it is necessary to use a mesh that is able to conform to the boundary if a given obstacle. The need to perform this adaptation may increase the computational effort. The Immersed Boundary Method enables the use of cartesian meshes to perform computational simulations of flows around obstacles. The idea of this method is to add a forcing term in the equations that model the flow. Thus, the forces applied on the body boundaries are directly transfered to the mesh. The aim of this work was to perform a computational implementation of the Immersed Boundary Method to simulate the flow over a oscilating circular cylinder. This oscilation may be inline with the flow, cross-flow, or rotating. The computational simulations enabled the capture of the lock-in phenomena, which consists of the syncronization between the vortex shedding frequency and the cylinder oscilation frequency. The results obtained from the computational simulations using the Immersed Boundary Method were in good agreement with the numerical and experimental results found in the literature

No desenvolvimento de códigos paralelos, a biblioteca PETSc se destaca como uma ferramenta prática e útil. Com o uso desta ferramenta, este trabalho apresenta um estudo sobre resolvedores de sistemas lineares aplicados a escoamentos incompressíveis de fluidos em microescala, além de uma análise de seu comportamento em paralelo. Após um estudo dos diversos aspectos dos métodos de fronteira imersa, é apresentado um método de fronteira imersa paralelo de primeira ordem. Na sequência, é apresentada uma proposta de melhoria na precisão do método, baseada na minimização da distância entre a condição de contorno exata e aproximada, no sentido de mínimos quadrados. O desenvolvimento de uma ferramenta paralela eficiente é demonstrado na solução numérica de problemas envolvendo escoamentos incompressíveis de fluidos viscosos com fronteiras imersas / In the development of parallel codes, PETSc library has an important position as a practical and useful tool. With this tool, this work presents a study about linear system solvers applied to incompressible flow in microscale problems, furthermore an analysis of the parallel behavior of these methods is presented. After a study of several aspects of immersed boundary methods, and taking advantage of the flexibility of PETSc, a parallel first order immersed boundary method is presented. Thereafter, an improvement in the accuracy of the method is presented, based on the minimization of the distance between exact and approximated boundary conditions, in the least square sense. The development of a parallel and efficient tool is demonstrated in the numerical solution of incompressible viscous flow problems with immersed boundary

Neste trabalho, desenvolve-se um método de fronteira imersa para o estudo de escoamento compressível modelado pelas equações de Euler bidimensionais. O método de discretização de diferenças finitas é empregado, usando o método de Steger-Warming de ordem dois para discretizar as variáveis espaciais e o esquema de Runge-Kutta de ordem quatro para discretizar as variáveis temporais. O método da fronteira imersa foi empregado para o estudo de aeroelasticidade computacional em uma seção típica de aerofólio bidimensional com dois movimentos prescritos: torsional e vertical, com o objetivo de se verifcar a eficiência do método e sua aplicabilidade para problemas em aeroelasticidade computacional. Neste estudo desenvolveu-se também um programa de computador para simular escoamentos compressíveis de fluido invíscido utilizando a metodologia proposta. A verificação do código gerado foi feita utilizando o método das soluções manufaturadas e o problema de reflexão de choque oblíquo. A validação foi realizada comparando-se os resultados obtidos para o escoamento ao redor de uma seção circular e de uma seção de aerofólio NACA 0012 com os resultados experimentais, para cada caso. / In this work, an immersed boundary method is developed to study compressible flow modeled by the two-dimensional Euler equations. The finite difference method is employed, using the second order Steger-Warming method to discretizate the space variables and the fourth order Runge-Kutta method to discretizate the time variables. The immersed boundary method was employed to study computational aeroelasticity on a typical two-dimensional airfoil section with two prescribed motion: pitching and plunging, in order to verify the efficiency of the numerical method and its applicability in computational aeroelasticity problems. In this work, a computer program was developed to simulate compressible flows for inviscid fluids using the methodology proposed. The verification of the computational code was performed using the method of manufactured solutions and the oblique shock wave reflection problem. The validation was performed comparing the obtained results for flows around a circular section and a NACA 0012 airfoil section with the experimental results, for each case.

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No desenvolvimento de códigos paralelos, a biblioteca PETSc se destaca como uma ferramenta prática e útil. Com o uso desta ferramenta, este trabalho apresenta um estudo sobre resolvedores de sistemas lineares aplicados a escoamentos incompressíveis de fluidos em microescala, além de uma análise de seu comportamento em paralelo. Após um estudo dos diversos aspectos dos métodos de fronteira imersa, é apresentado um método de fronteira imersa paralelo de primeira ordem. Na sequência, é apresentada uma proposta de melhoria na precisão do método, baseada na minimização da distância entre a condição de contorno exata e aproximada, no sentido de mínimos quadrados. O desenvolvimento de uma ferramenta paralela eficiente é demonstrado na solução numérica de problemas envolvendo escoamentos incompressíveis de fluidos viscosos com fronteiras imersas / In the development of parallel codes, PETSc library has an important position as a practical and useful tool. With this tool, this work presents a study about linear system solvers applied to incompressible flow in microscale problems, furthermore an analysis of the parallel behavior of these methods is presented. After a study of several aspects of immersed boundary methods, and taking advantage of the flexibility of PETSc, a parallel first order immersed boundary method is presented. Thereafter, an improvement in the accuracy of the method is presented, based on the minimization of the distance between exact and approximated boundary conditions, in the least square sense. The development of a parallel and efficient tool is demonstrated in the numerical solution of incompressible viscous flow problems with immersed boundary

Ce travail est dédié à la simulation d’écoulements multidimensionnels de gaz raréfiés dans un domaine où l’interface avec le solide est mobile. Le comportement du gaz est modélisé par un modèle de type BGK de l’équation de Boltzmann et une méthode déterministe de vitesses discrètes est utilisée pour discrétiser l’espace des vitesses microscopiques.Dans ce document, nous proposons tout d’abord trois discrétisations spatiales du modèle qui permettent la prise en compte du mouvement des parois solides, grâce à un traitement spécifique des conditions aux limites. Ces approches sont implémentées et validées pour plusieurs cas unidimensionnels et à la suite de cette étude, la méthode maille coupée est choisie pour une extension à des écoulements de dimensions plus élevées.La suite du travail présente l’algorithme utilisé pour la simulation d’écoulements 2D et 3D. La précision et la robustesse de l’implémentation sont mises en avant grâce à la simulation de nombreux cas tests, dont les résultats sont comparés à ceux issus de la littérature. La méthode maille coupée a notamment été optimisée par une technique de raffinement de maillage adaptatif. La simulation instationnaire 3D de la rotation des pâles du radiomètre de Crookes illustre pleinement le potentiel de la méthode. / This work is devoted to the multidimentional simulation of rarefied gases in a domain with moving boundary. The governing equation is given by BGKtype model of Boltzmann equation and velocity space is discretized with a standard discrete velocity method.We first propose three space discretizations that take boundary motion into account by specific treatment of the boundary conditions. These approaches are implemented and validated for several 1D flows. Based on this study, the cut cell method is chosen to be extend to multidimentional flows.Then we detail the cut cell algorithm for 2D and 3D flow simulations. Robustness and accuracy of the implementation are investigated through the simulation of numerous test cases. Our results are rigorously compared to the ones coming from the literature and good agreement is shown. The cut cell method has been optimized with an adaptive refinement mesh technique. The 3D unstationary simulation of the Crookes radiometer rotating vanes is a perfect illustration of the method potential.

Single ventricle heart defects, where systemic and pulmonary venous returns mix in the single functional ventricle, represent the most complex form of congenital heart defect, affecting 2 babies per 1000 live births. Surgical repairs, termed "Fontan Repairs," reroute the systemic venous return directly to the pulmonary arteries, thus preventing venous return mixing and restoring normal oxygenation saturation levels. Unfortunately, these repairs are only palliative and Fontan patients are subjected to a multitude of chronic complications. It has long been suspected that hemodynamics play a role in determining patient outcome. However, the number of anatomical and functional variables that come into play and the inability to conduct large scale clinical evaluations, due to too small a patient population, has hindered decisive progress and there is still not a good understanding of the optimal care strategies on a patient-by-patient basis. Over the past decades, image-guided computational fluid dynamics (CFD) has arisen as an attractive option to accurately model such complex biomedical phenomena, providing a high degree of freedom regarding the geometry and flow conditions to be simulated, and carrying the potential to be automated for large sample size studies. Despite these theoretical advantages, few CFD studies have been able to account for the complexity of patient-specific anatomies and in vivo pulsatile flows. In this thesis, we develop an unstructured Cartesian immersed-boundary flow solver allowing for high resolution, time-accurate simulations in arbitrarily complex geometries, at low computational costs. Combining the proposed and validated CFD solver with an interactive virtual-surgery environment, we present an image-based surgical planning framework that: a) allows for in depth analysis of the pre-operative in vivo hemodynamics; b) enables surgeons to determine the optimum surgical scenario prior to the operation. This framework is first applied to retrospectively investigate the in vivo pulsatile hemodynamics of different Fontan repair techniques, and quantitatively compare their efficiency. We then report the prospective surgical planning investigations conducted for six failing Fontan patients with an interrupted inferior vena cava and azygous continuation. In addition to a direct benefit to the patients under consideration, the knowledge derived from these surgical planning studies will also have a larger impact for the clinical management of Fontan patients as they shed light onto the impact of caval offset, vessel flaring and other design parameters upon the Fontan hemodynamics depending on the underlying patient anatomy. These results provide useful surgical guidelines for each anatomical template, which could benefit the global surgical community, including centers that do not have access to patient-specific surgical planning interfaces.

Fluid-structure interaction (FSI), occurrent in many areas of engineering and in the natural world, has been the subject of much research using a wide range of modelling strategies. However, problems with high levels of structural deformation are difficult to resolve and this is particularly the case for biomedical flows. A Lagrangian flow model coupled with a robust model for nonlinear structural mechanics seems a natural candidate since large distortion of the computational geometry is expected. Smoothed particle Hydrodynamics (SPH) has been widely applied for nonlinear interface modelling and this approach is investigated here. Biomedical applications often involve thin flexible structures and a consistent approach for modelling the interaction of fluids with such structures is also required. The Lagrangian weakly compressible SPH method is investigated in its recent delta-SPH form utilising inter-particle density fluxes to improve stability. Particle shifting is also used to maintain particle distributions sufficiently close to uniform to enable stable computation. The use of artificial viscosity is avoided since it introduces unphysical dissipation. First, solid boundary conditions are studied using a channel flow test. Results show that when the particle distribution is allowed to evolve naturally instabilities are observed and deviations are noted from the expected order of accuracy. A parallel development in the SPH group at Manchester has considered SPH in Eulerian form (for different applications). The Eulerian form is applied to the channel flow test resulting in improved accuracy and stability due to the maintenance of a uniform particle distribution. A higher-order accurate boundary model is developed and applied for the Eulerian SPH tests and third-order convergence is achieved. The well documented case of flow past a thin plate is then considered. The immersed boundary method (IBM) is now a natural candidate for the solid boundary. Again, it quickly becomes apparent that the Lagrangian SPH form has limitations in terms of numerical noise arising from anisotropic particle distributions. This corrupts the predicted flow structures for moderate Reynolds numbers (O(102)). Eulerian weakly compressible SPH is applied to the problem with the IBM and is found to give accurate and convergent results without any numerical stability problems (given the time step limitation defined by the Courant condition). Modelling highly flexible structures using the discrete element model is investigated where granular structures are represented as bonded particles. A novel vector-based form (the V-Model) is identified as an attractive approach and developed further for application to solid structures. This is shown to give accurate results for quasi-static and dynamic structural deformation tests. The V-model is applied to the decay of structural vibration in a still fluid modelled using Eulerian SPH with no artificial stabilising techniques. Again, results are in good agreement with predictions of other numerical models. A more demanding case representative of pulsatile flow through a deep leg vein valve is also modelled using the same form of Eulerian SPH. The results are free of numerical noise and complex FSI features are captured such as vortex shedding and non-linear structural deflection. Reasonable agreement is achieved with direct in-vivo observations despite the simplified two-dimensional numerical geometry. A robust, accurate and convergent method has thus been developed, at present for laminar two-dimensional low Reynolds number flows but this may be generalised. In summary a novel robust and convergent FSI model has been established based on Eulerian SPH coupled to the V-Model for large structural deformation. While these developments are in two dimensions the method is readily extendible to three-dimensional, laminar and turbulent flows for a wide range of applications in engineering and the natural world.

L'objectif de cette thèse est d'améliorer la compréhension des phénomènes jouant un rôle dans la mesure effectuée dans un analyseur sanguin, en particulier le comptage et la mesure de volumétrie d'une population de globules rouges reposant sur l'effet Coulter. Des simulations numériques sont effectuées dans le but de prédire la dynamique des globules rouges dans les zones de mesure et pour reproduire la mesure électrique associée, servant au comptage et à la volumétrie des cellules. Ces simulations sont effectuées à l'intérieur de configurations industrielles d'analyseur sanguin, en utilisant un outil numérique développé à l'IMAG, le solveur YALES2BIO. En utilisant la méthode des frontières immergées avec suivi de front, un modèle de particule déformable est introduit, celui-ci prend en compte le contraste de viscosité ainsi que les effets mécaniques de la courbure et de l'élasticité sur la membrane. Le solveur est validé grâce à de nombreux cas tests parcourant différents régimes et effets physiques. L'écoulement fluide dans cette géométrie d'analyseur sanguin est caractérisée par un fort gradient de vitesse axial dans la direction de l'écoulement, impliquant la présence d'un écoulement extensionnel au niveau du micro-orifice, là où a lieu la mesure. La dynamique des globules rouges est étudiée par des simulations numériques pour différentes conditions initiales, telles que sa position ou son orientation. Il est observé que les globules rouges vont se réorienter selon l'axe principal de l'analyseur sanguin dans tous les cas. Pour comprendre le phénomène, des modèles analytiques sont adaptés au cas des écoulements extensionnels et reproduisent correctement les tendances de réorientation.Cette thèse présente également la reproduction de la mesure électrique utilisée pour le comptage et la mesure de la distribution des volumes de globules rouges. De nombreuses simulations de la dynamique des globules rouges sont effectuées et utilisées pour générer l'impulsion électrique correspondant au passage du globule rouge dans le micro-orifice. Les amplitudes d'impulsions électriques résultantes permettent la caractérisation de la réponse électrique en fonction des paramètres initiaux de la simulation par une approche statistique. Un algorithme de Monte-Carlo est utilisé pour la quantification des erreurs de mesure liées à l'orientation et la position des globules rouges dans le micro-orifice. Ceci permet la génération d'une distribution de volume mesurée pour une population de globules rouges bien définie et la caractérisation des erreurs de mesure associées. / The aim of this thesis is to improve the understanding of the phenomena involved in the measurement performed in a blood analyzer, namely the counting and sizing of red blood cells based on the Coulter effect. Numerical simulations are performed to predict the dynamics of red blood cells in the measurement regions, and to reproduce the associated electrical measurement used to count and size the cells. These numerical simulations are performed in industrial configurations using a numerical tool developed at IMAG, the YALES2BIO solver. Using the Front-Tracking Immersed Boundary Method, a deformable particle model for the red blood cell is introduced which takes the viscosity contrast as well as the mechanical effects of the curvature and elasticity on the membrane into account. The solver is validated against several test cases spreading over a large range of regimes and physical effects.The velocity field in the blood analyzer geometry is found to consist of an intense axial velocity gradient in the direction of the flow, resulting in a extensional flow at the micro-orifice, where the measurement is performed. The dynamics of the red blood cells is studied with numerical simulations with different initial conditions, such as its position or orientation. They are found to reorient along the main axis of the blood analyzer in all cases. In order to understand the phenomenon, analytical models are adapted to the case of extensional flows and are found to reproduce the observed trends.This thesis also presents the reproduction of the electrical measurement used to count red blood cells and measure their volume distribution. Numerous dynamics simulations are performed and used to generate the electrical pulse corresponding to the passage of a red blood cell inside the micro-orifice. The resulting electrical pulse amplitudes are used to characterize the electrical response depending on the initial parameters of the simulation by means of a statistical approach. A Monte-Carlo algorithm helps quantifying the errors on the measurement of cell depending on its orientation and position inside the micro-orifice. This allows the generation of a measured volume distribution of a well defined red blood cell population and the characterization of the associated measurement errors.

Cette thèse traite plusieurs aspects scientifiques inhérents à la simulation numérique de problèmes d'interaction fluide-structure impliquant de fines membranes déformables. Deux cas spécifiques relatifs à la biomécanique cardiovasculaire sont considérés : l'interaction de l'écoulement sanguin avec la valve aortique (qui se produit à l'échelle macroscopique), et l'interaction de la membrane des globules rouges avec ses fluides interne et externe (qui se produit à l'échelle microscopique). Dans les deux cas, le couplage fluide-structure est géré par l'intermédiaire d'un formalisme de frontières immergées, en représentant la membrane par un maillage Lagrangien se mouvant au travers d'un maillage fluide Eulérien. Lorsque l'on traite la dynamique des globules rouges, la membrane est considérée comme étant une structure sans masse et infiniment fine. La première question à laquelle on s'intéresse dans cette thèse est la manière de modéliser la microstructure complexe de la membrane des globules rouges. Un moyen possible pour caractériser un modèle de membrane adapté est de simuler l'expérience des pinces optiques, qui consiste en une configuration expérimentale bien contrôlée qui permet d'étudier la mécanique individuelle d'un globule rouge isolé dans une large gamme de déformations. Plusieurs modèles pertinents sont identifiés, mais les caractéristiques de déformation mesurées durant l'expérience des pinces optiques se révèlent n'être pas assez sélectives pour être utilisées dans un contexte de validation. Des mesures de déformation additionnelles sont proposées, qui pourraient permettre une meilleure caractérisation de la mécanique de la membrane des globules rouges. En ce qui concerne les configurations macroscopiques, une méthode numérique innovante est proposée afin de gérer des simulations numériques de membranes 3D continues, en conservant le formalisme de frontières immergées. Dans cette méthode, appelée méthode des frontières immergées épaisses, la membrane a une épaisseur finie. La précision et la robustesse de la méthode sont démontrées par l'intermédiaire d'une variété de cas tests bien choisis. La méthode proposée est ensuite appliquée à un problème d'interaction fluide-structure réaliste, à savoir l'interaction d'un écoulement (sanguin) pulsé avec une valve aortique biomimétique. Une étude combinée expérimentale et numérique est menée, montrant que la méthode est capable de capturer la dynamique globale de la valve, ainsi que les principales caractéristiques de l'écoulement en aval de la valve. Tous les développements ont été effectués dans le solveur YALES2BIO (http://www.math.univ-montp2.fr/~yales2bio/) développé à l'IMAG, qui est donc disponible pour toutes autres améliorations, validations et études applicatives. / This thesis deals with several scientific aspects inherent to the numerical simulation of fluid-structure interaction problems involving thin deformable membranes. Two specific cases relevant to cardiovascular biomechanics are considered: the interaction of the blood flow with the aortic valve (which occurs at the macroscopic scale), and the interaction of the red blood cells membrane with its inner and outer fluids (which occurs at the microscopic scale). In both cases, the fluid-structure interaction coupling is handled using an immersed boundary formalism, representing the membrane by a Lagrangian mesh moving through an Eulerian fluid mesh.When dealing with red blood cells dynamics, the membrane is considered to be an infinitely thin and massless structure. The first question which is addressed in the present thesis work is how to model the complex microstructure of the red blood cells membrane. A possible way to characterize a suitable membrane model is to simulate the optical tweezers experiment, which is a well-controlled experimental configuration enabling to study the individual mechanics of an isolated red blood cell in a large range of deformation. Some relevant membrane models are identified, but the deformation characteristics measured during the optical tweezers experiment reveal to be not selective enough to be used in a validation context. Additional deformation measurements are proposed, which could allow a better characterization of the red blood cell membrane mechanics.Regarding the macroscopic configurations, an innovative numerical method is proposed to handle numerical simulations of 3D continuum membranes, still within the immersed boundary formalism. In this method, called immersed thick boundary method, the membrane has a finite thickness. The accuracy and robustness of the method are demonstrated through a variety of well-chosen test cases. Then, the proposed method is applied to a realistic fluid-structure interaction problem, namely the interaction of a pulsatile (blood) flow with a biomimetic aortic valve. A combined experimental and numerical study is led, showing that the method is able to capture the global dynamics of the valve, as well as the main features of the flow downstream of the valve.All the developments were performed within the YALES2BIO solver (http://www.math.univ-montp2.fr/~yales2bio/) developed at IMAG, which is thus available for further improvements, validations and applicative studies.