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Nous étudions dans ce travail une méthode de type éléments finis dans le but de simuler le mouvement de particules rigides immergées. La méthode développée ici est une méthode de type domaine fictif. L'idée est de chercher un prolongement régulier de la solution exacte à tout le domaine fictif afin d'obtenir une solution régulière sur tout le domaine et retrouver l'ordre optimal de l'erreur avec des éléments d'ordre 1. Le prolongement régulier est cherché en minimisant une fonctionnelle dont le gradient est donné par la solution d'un nouveau problème fluide faisant intervenir une distribution simple couche dans le second membre. Nous faisons une analyse numérique, dans le cas scalaire, de l'approximation de cette distribution par une combinaison de masse de Dirac. Un des avantages de cette méthode est de pouvoir utiliser des solveurs rapides sur maillages cartésiens tout en conservant l'ordre optimal de l'erreur. Un autre avantage de la méthode vient du fait que les opérateurs ne sont pas modifiés, seul les seconds membres dépendent de la géométrie du domaine initial. Nous avons de plus écrit un code C++ parallèle en deux et trois dimensions, permettant de simuler des écoulements fluide/particules rigides avec cette méthode. Nous présentons ainsi une description des principales composantes de ce code. / In this work, we study a finite element method in order to simulate the motion of immersed rigid bodies. This method is of the fictitious domain type. The idea is to look for a smooth extension in the whole domain of the exact solution and to recover the optimal order obtain with a conformal mesh. This smooth extension is sought by minimizing a functional whose gradient is the solution of another fluid problem with a single layer distribution as a right hand side. We make the numerical analysis, in the scalar case, of the approximation of this distribution by a sum of Dirac masses. One of the advantage of this method is to be able to use fast solvers on cartesian mesh while recovering the optimal order of the error. Another advantage of this method is that the operators are not modified at all. Only the right hand side depends on the geometry of the original problem. We write a parallel C++ code in two and three dimensions that simulate fluid/rigid bodies flows with this method. We present the core blocks of this code to show how it works.

Dans cette thèse, nous avons développé un code de simulation numérique directe pour l’étude des écoulements particulaires. Le schéma numérique est basé sur une technique de domaines fictifs. Le code est validé sur de nombreux cas test puis nous l’avons utilisé pour étudier la sédimentation de particules bidimensionnelles en milieu confiné. Trois cas ont été analysés : sédimentation d’une particule unique, d’un doublet de particules et d’un grand nombre de particules. Dans le premier cas nous retrouvons le phénomène de survitesse qui apparaît pour une particule excentrée à bas nombre de Reynolds. Nous montrons que cette survitesse est très sensible à l’inertie du fluide : elle diminue lorsqu’on augmente le nombre de Reynolds. Cet effet est retardé par le confinement. Dans le cas d’un doublet de particules, nous retrouvons les comportements complexes observés dans la littérature (hystérésis, cascade sous-harmonique et chaos). Nous montrons qu’une nouvelle série de bifurcations et un nouvel attracteur apparaissent pour des particules plus pesantes. Il s’agit là d’une transition vers le chaos par la voie de la quasi-périodicité. Nous donnons le diagramme de bifurcation étendu. La nouvelle branche correspond à une structure horizontale qui conduit à une sédimentation lente. Dans le cas d’un grand nombre de particules, nous montrons que la vitesse de chute de l’interface fluide-particules suit une loi de type Richardson-Zaki, mais avec un exposant d’environ 4. Comme pour des sphères, la valeur de cet exposant dépend du confinement. Enfin, nous observons un phénomène de blocage, inattendu pour des particules non-cohésives, dû au caractère bidimensionnel de la suspension / In the present thesis, a fully-resolved numerical code has been developed for the analysis of particle-laden flows. A fictitious domain method is used. First, this numerical tool has been validated by using classical benchmarks. It has then been used to simulate the complex sedimentation of particles in three generic two-dimensional configurations: a single particle, a particle pair and a large number of particles in a confined domain. In the first case, the peak-velocity of an off-centred inclusion is recovered at low-Reynolds number. It is shown that this peak-velocity is very sensitive to fluid inertia: the peak-velocity decreases when the Reynolds number increases. This effect is delayed by the confinement. The very complex dynamics of a pair of particles sedimenting in a confined domain, observed in the litterature, is recovered (hysteresis, period-doubling cascade and chaos). It is shown that a new series of bifurcations, leading to a new attractor, emerges when the non-dimensional particle weight is increased. This new transition corresponds to a quasi-periodic route. The extended bifurcation diagram is given. The new branch discovered in this work corresponds to a nearly horizontal particle doublet, with a slow settling velocity. In the case of the settling of large number of particles, a RZ-like law is recovered for the sedimentation velocity of the fluid-particle interface. The exponent is close to 4, in contrast with the case of spheres. Finally, the sedimentation velocity at the end of the settling process is observed to be significantly reduced, like for cohesive sediments. This unexpected behaviour is related to the two-dimensionality of the suspension

Ce travail se consacre à la simulation numérique tridimensionnelle de suspensions denses, monodispersés, non-inertielles et non-colloïdales. Nous avons pour ce faire développé une méthode numérique basée sur une approche de type domaine fictif. Le modèle inclut également une modélisation détaillée des forces de lubrification ainsi que des forces de contact avec prise en compte des rugosités et du frottement. Un résultat majeur est le rôle important du frottement entre particules sur la rhéologie de la suspension – en particulier sur la viscosité de cisaillement et les contraintes normales – mais aussi sur la viscosité normale ou la diffusion des particules. Le frottement contribue à augmenter fortement la contrainte de contact alors que la contrainte hydrodynamique n’est quasiment pas affectée. Cette contrainte de contact s’avère être la contrainte majoritaire dans les suspensions denses. La prise en compte du frottement dans les simulations permet de se rapprocher notablement des résultats expérimentaux. Le rôle du confinement est également étudié et les parois s’avèrent conduire à une organisation locale marquée de type hexagonal ainsi qu’à un glissement. Cette organisation entraîne des effets sensibles sur les propriétés rhéologiques surtout pour la première différence de contraintes normales N1 qui peut localement devenir positive. Enfin, nous abordons l’impact d’amas percolants de particules dans la suspension. La fraction volumique de percolation se situe entre 0,3 et 0,4 avec un effet marqué de la rugosité, du frottement et de la taille du domaine. / This work is devoted to three-dimensional numerical simulations of monodisperse non-inertial non-colloidal concentrated suspensions. To this end, a numerical method based on a fictitious domain technique is developed. It includes a detailed lubrication model as well as a contact model allowing for particle roughness and friction. One major result is the strong effect of friction on rheology, especially on shear viscosity and normal stresses. It also alters markedly normal viscosity or particle diffusion. Friction acts mostly through an increase in the contact stress since the hydrodynamic stress remains unaffected. This contact stress occurs to be the prevailing stress in dense suspensions. Overall, frictional results are in much better agreement with available experiments. The role of confinement is investigated as well and walls are shown to induce a strong local hexagonal ordering with a significant wall slip. This wall-induced ordering has a notable effect on rheology, especially on the first normal stress difference N1 that can be locally positive. Finally, we have studied the percolation of particle clusters across the suspension. The critical volume fraction is found to be in the range 0.3~0.4, with a significant dependence on roughness, friction, and domain size. Percolating clusters characteristics can globally be described by an isotropic percolation theory, with discrepancies regarding some critical exponents however. The role of percolating clusters on rheology is found to be very limited.

Chemotaxis, the phenomenon through which cells respond to external chemical signals, is one of the most important and universally observable in nature. It has been the object of considerable modelling effort in the last decades. The models for chemotaxis available in the literature cannot reconcile the dynamics of external chemical signals and the intracellular signalling pathways leading to the response of the cells. The reason is that models used for cells do not contain the distinction between the extracellular and intracellular domains. The work presented in this dissertation intends to resolve this issue. We set up a numerical hybrid simulation framework containing such description and enabling the coupling of models for phenomena occurring at extracellular and intracellular levels. Mathematically, this is achieved by the use of the fictitious domain method for finite elements, allowing the simulation of partial differential equations on evolving domains. In order to make the modelling of the membrane binding of chemical signals possible, we derive a suitable fictitious domain method for Robin boundary elliptic problems. We also display ways to minimise the computational cost of such simulation by deriving a suitable preconditioner for the linear systems resulting from the Robin fictitious domain method, as well as an efficient algorithm to compute fictitious domain specific linear operators. Lastly, we discuss the use of a simpler cell model from the literature and match it with our own model. Our numerical experiments show the relevance of the matching, as well as the stability and accuracy of the numerical scheme presented in the thesis.

For the purpose of numerically simulating continuum mechanical structures, different types of material may be represented by the extreme values {<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" />,1}, where 0&lt;<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" /><img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cll" />1, of a varying coefficient <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha" /> in the governing equations. The paramter <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" /> is not allowed to vanish in order for the equations to be solvable, which means that the exact conditions are approximated. For example, for linear elasticity problems, presence of material is represented by the value <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha" /> = 1, while <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha" /> = <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" /> provides an approximation of void, meaning that material-free regions are approximated with a weak material. For acoustics applications, the value <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha" /> = 1 corresponds to air and <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha" /> = <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" /> to an approximation of sound-hard material using a dense fluid. Here we analyze the convergence properties of such material approximations as <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" />!0, and we employ this type of approximations to perform design optimization. In Paper I, we carry out boundary shape optimization of an acoustic horn. We suggest a shape parameterization based on a local, discrete curvature combined with a fixed mesh that does not conform to the generated shapes. The values of the coefficient <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha" />, which enters in the governing equation, are obtained by projecting the generated shapes onto the underlying computational mesh. The optimized horns are smooth and exhibit good transmission properties. Due to the choice of parameterization, the smoothness of the designs is achieved without imposing severe restrictions on the design variables. In Paper II, we analyze the convergence properties of a linear elasticity problem in which void is approximated by a weak material. We show that the error introduced by the weak material approximation, after a finite element discretization, is bounded by terms that scale as <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" /> and <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" />1/2hs, where h is the mesh size and s depends on the order of the finite element basis functions. In addition, we show that the condition number of the system matrix scales inversely proportional to <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" />, and we also construct a left preconditioner that yields a system matrix with a condition number independent of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" />. In Paper III, we observe that the standard sound-hard material approximation with <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha" /> = <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" /> gives rise to ill-conditioned system matrices at certain wavenumbers due to resonances within the approximated sound-hard material. To cure this defect, we propose a stabilization scheme that makes the condition number of the system matrix independent of the wavenumber. In addition, we demonstrate that the stabilized formulation performs well in the context of design optimization of an acoustic waveguide transmission device. In Paper IV, we analyze the convergence properties of a wave propagation problem in which sound-hard material is approximated by a dense fluid. To avoid the occurrence of internal resonances, we generalize the stabilization scheme presented in Paper III. We show that the error between the solution obtained using the stabilized soundhard material approximation and the solution to the problem with exactly modeled sound-hard material is bounded proportionally to <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" />.

In this work, flow through synthetic arrangements of contacting spheres is studied as a model problem for porous media and packed bed type flows. Direct numerical simulations are performed for moderate pore Reynolds numbers in the range, 10 ≤ Re ≤ 600, where non-linear porescale flow features are known to contribute significantly to macroscale properties of engineering interest. To first choose and validate appropriate computational models for this problem, the relative performance of two numerical approaches involving body conforming and non-conforming grids for simulating porescale flows is examined. In the first approach, an unstructured solver is used with tetrahedral meshes, which conform to the boundaries of the porespace. In the second approach, a fictitious domain formulation (Apte et al., 2009. J Comput. Phys. 228 (8), 2712-2738) is used, which employs non-body conforming Cartesian grids and enforces the no-slip conditions on the pore boundaries implicitly through a rigidity constraint force. Detailed grid convergence studies of both steady and unsteady flow through prototypical arrangements of spheres indicate that for a fixed level of uncertainty, significantly lower grid densities may be used with the fictitious domain approach, which also does not require complex grid generation techniques. Next, flows through both random and structured arrangements of spheres are simulated at pore Reynolds numbers in the steady inertial ( 10 ≲ Re ≲ 200) and unsteady inertial (Re ≈ 600) regimes, and used to analyze the characteristics of porescale vortical structures. Even at similar Reynolds numbers, the vortical structures observed in structured and random packings are remarkably different. The interior of the structured packings are dominated by multi-lobed vortex rings structures that align with the principal axes of the packing, but perpendicular to the mean flow. The random packing is dominated by helical vortices, elongated parallel to the mean flow direction. The unsteady dynamics observed in random and structured arrangements are also distinct, and are linked to the behavior of the porescale vortices. Finally, to investigate the existence and behavior of transport barriers in packed beds, a numerical tool is developed to compute high resolution finite-time Lyapunov exponent (FTLE) fields on-the-fly during DNS of unsteady flows. Ridges in this field are known to correspond to Lagrangian Coherent Structures (LCS), which are invariant barriers to transport and form the skeleton of time dependent Lagrangian fluid motion. The algorithm and its implementation into a parallel DNS solver are described in detail and used to explore several flows, including unsteady inertial flow in a random sphere packing. The resulting FTLE fields unambiguously define the boundaries of dynamically distinct porescale features such as counter rotating helical vortices and jets, and capture time dependent phenomena including vortex shedding at the pore level. / Graduation date: 2013

La simulation d'écoulements dans des géométries complexes nécessite la création de maillages parfois difficile à réaliser. La méthode de pénalisation proposée dans ce travail permet de simplifier cette étape. En effet, la résolution des équations qui gouvernent l'écoulement se fait sur un maillage plus simple mais non-adapté à la géométrie du problème. Les conditions aux limites sur les parties du domaine physique immergées dans le maillage sont prises en compte à travers l'ajout d'un terme de pénalisation dans les équations. Nous nous sommes intéressés à l'approximation du terme de pénalisation pour une discrétisation par volumes finis sur maillages décalés et colocatifs. Les cas tests de vérification réalisés attestent d'un ordre de convergence spatial égal à 2 pour la méthode de pénalisation appliquée à la résolution d'une équation de type Poisson ou des équations de Navier-Stokes. Enfin, on présente les résultats obtenus pour la simulation d'écoulements turbulents autour d'un cylindre à Re=3900 et à l'intérieur d'une partie d'un assemblage combustible à Re=9500. / The simulations of fluid flows in complex geometries require the generation of body-fitted meshes which are difficult to create.The penalty method developed in this work is useful to simplify the mesh generation task.The governing equations of fluid flow are discretized using a finite volume method on an unfitted mesh.The immersed boundary conditions are taken into account through a penalty term added to the governing equations.We are interested in the approximation of the penalty term using a finite volume discretization with collocated and staggered grid.The penalty method is second-order spatial accurate for Poisson and Navier-Stokes equations.Finally, simulations of turbulent flows around a cylinder at Re=3900 and turbulent motions in a rod bundle at Re=9500 are performed.

[pt] Este trabalho apresenta uma nova formulação implícita e totalmente acoplada para o problema de escoamentos tridimensionais com corpos rígidos suspensos. Esta é a principal contribuição deste trabalho. A formulação foi implementada em C mais mais e testada para o problema de sedimentação de uma partícula esférica. Os resultados indicam comportamento físico plausível apesar de serem limitados por inacurácia de malha. O programa resolve numericamente as Equações de Navier-Stokes acopladas com as Equações da Dinâmica de Corpo Rígido usando o Método de Elementos Finitos. O acoplamento entre os domínios fluido e sólido é feito pela Técnica do Domínio Fictício, que evita a geração de malha a cada passo de tempo. O escoamento tridimensional sem partículas também é estudado neste trabalho e é a base para a formulação do escoamento com partículas. / [en] This work presents a novel implicit and fully coupled formulation for the problem of 3D flows with suspended rigid bodies. This is the main contribution of the work. The formulation was implemented in C plus plus and tested for the sedimentation problem of one spherical particle. The results indicate plausible physical behavior in spite of being limited by mesh accuracy. The software solves numerically the Navier-Stokes Equations coupled with Rigid Body Dynamics Equations using the Finite Elements Method. The coupling between fluid and solid domains is done by means of the Fictitious Domain Technique, which avoids mesh generation for every time step. The 3D flow of non particulate flow is also studied in this work and is the basis for the particulate flow formulation.

This thesis presents a finite element methodology for simulating fluid–solid interactions in the left ventricle (LV) under LVAD support. The developed model was utilised to study the passive and active characteristics of ventricular function in anatomically accurate LV geometries constructed from normal and patient image data. A non–conforming ALE Navier–Stokes/finite–elasticity fluid–solid coupling system formed the core of the numerical scheme, onto which several novel numerical additions were made. These included a fictitious domain (FD) Lagrange multiplier method to capture the interactions between immersed rigid bodies and encasing elastic solids (required for the LVAD cannula), as well as modifications to the Newton–Raphson/line search algorithm (which provided a 2 to 10 fold reduction in simulation time). Additional developments involved methods for extending the model to ventricular simulations. This required the creation of coupling methods, for both fluid and solid problems, to enable the integration of a lumped parameter representation of the systemic and pulmonary circulatory networks; the implementation and tuning of models of passive and active myocardial behaviour; as well as the testing of appropriate element types for coupling non–conforming fluid– solid finite element models under high interface tractions (finding that curvilinear spatial interpolations of the fluid geometry perform best). The behaviour of the resulting numerical scheme was investigated in a series of canonical test problems and found to be convergent and stable. The FD convergence studies also found that discontinuous pressure elements were better at capturing pressure gradients across FD boundaries. The ventricular simulations focused firstly on studying the passive diastolic behaviour of the LV both with and without LVAD support. Substantially different vortical flow features were observed when LVAD outflow was included. Additionally, a study of LVAD cannula lengths, using a particle tracking algorithm to determine recirculation rates of blood within the LV, found that shorter cannulas improved the recirculation of blood from the LV apex. Incorporating myocardial contraction, the model was extended to simulate the full cardiac cycle, converging on a repeating pressure–volume loop over 2 heart beats. Studies on the normal LV geometry found that LVAD implementation restricts the recirculation of early diastolic inflow, and that fluid–solid coupled models introduce greater heterogeneity of myocardial work than was observed in equivalent solid only models. A patient study was undertaken using a myocardial geometry constructed using image data from an LVAD implant recipient. A series of different LVAD flow regimes were tested. It was found that the opening of the aortic valve had a homogenising effect on the spatial variation of work, indicating that the synchronisation of LVAD outflow with the cardiac cycle is more important if the valve remains shut. Additionally, increasing LVAD outflow during systole and decreasing it during diastole led to improved mixing of blood in the ventricular cavity – compared with either the inverse, or holding outflow constant. Validation of these findings has the potential to impact the treatment protocols of LVAD patients.