Le Contrôle Non Destructif (CND) permet de sonder l’intérieur d’un milieu dans le but desurveiller son intégrité et son vieillissement. Assisté d’outils de simulation il permet de détecter, caractériseret localiser des défauts de structure du milieu inspecté mais sa fiabilité dépend de la précision des méthodesde simulation. Dans le cadre du CND ultrasonore, les méthodes usuelles (numériques et asymptotiques) sontbien souvent inadéquates pour simuler la diffraction par les défauts. On leur préfère des techniques hybrides.On propose dans cette thèse une nouvelle approche hybride pour la simulation numérique de la diffractionhaute fréquence en milieu étendu (configuration critique pour le CND). Combinant la méthode des équationsintégrales et la méthode des rayons, cette approche exploite le caractère multi-échelle du problème hautefréquence en proposant un modèle d’obstacle à deux échelles. Elle permet le calcul précis de la diffraction etla propagation rayon des champs. D’abord mise au point dans le cadre de la diffraction d’ondes acoustiquespar un obstacle de taille inférieure à la longueur d’onde (méthode barycentrique), l’approche est ensuiteétendue à des configurations de diffraction par des obstacles de l’ordre de la longueur d’onde grâce àl’introduction d’un partitionnement de l’unité de sa surface (méthode multi-centres). Pour accélérerl’approche hybride, on propose une procédure de résolution Online-Offline, basée sur un pré-calcul de lamatrice de diffraction associée à un ensemble réduit de directions d’incidence et d’observation et sur uneinterpolation polynomiale de ses vecteurs singuliers pour son évaluation dans des directions quelconquesd’émetteurs et de récepteurs. On étudie ensuite la stratégie dans le cadre de l’acoustique 3D puis on en faitune extension de principe à l’élastodynamique. On donne enfin un ensemble de pistes pour étendre l’approchehybride dans des cas de diffraction par un ou plusieurs obstacles pouvant être proches des bords du milieu. / Non Destructive Testing (NDT) aims at probing a medium to check its integrity and aging. Withthe help of simulation tools, it allows to detect, caracterize and locate flaws inside a material with a precisiondepending on that of the simulation methods. Usual numerical and asymptotic methods nevertheless often failat precisely computing diffraction for ultrasonic NDT. Hybrid approaches are thus prefered in this framework.In this thesis, we propose a new hybrid strategy combining the boundary integral equation method and raytracing to compute high frequency diffraction of an obstacle in a large medium (critical NDT configuration).This strategy allows to compute precisely the diffraction effects and to convert and propagate the diffractedfield as rays. The proposed strategy relies on a two-scale model of the diffracting obstacle. First developpedto simulate acoustic waves diffraction on an obstacle of size less than the wave length (barycentric method),the hybrid strategy is then extended to compute the diffraction by an obstacle of size some wave lengths(polycentric method) thanks to the introduction of a partition of unity of the obstacle surface. Besides, inorder to accelerate the hybrid approach, we propose an Online-Offline resolution procedure based on theOffline computation of the scattering matrix for a reduced set of incidence and observation directions and onthe use of a polynomial interpolation of its singular vectors for the Online evaluation of the scattering matrixfor any incidence and observation directions. We then study the possibility of extension of the hybrid strategyto 3D acoutics and elastodynamics. We finally give several perspectives for the adaptation of the approach todeal with diffraction by one or several obstacles potentially close to the propagating medium boundaries.

<p>This investigation is motivated by the problem of optimal design of cooling elements in modern battery systems used in hybrid/electric vehicles. We consider a simple model of two-dimensional steady-state heat conduction generated by a prescribed distribution of heat sources and involving a one-dimensional cooling element represented by a closed contour. The problem consists in finding an optimal shape of the cooling element which will ensure that the temperature in a given region is close (in the least squares sense) to some prescribed distribution. We formulate this problem as PDE-constrained optimization and use methods of the shape-differential calculus to obtain the first-order optimality conditions characterizing the locally optimal shapes of the contour. These optimal shapes are then found numerically using the conjugate gradient method where the shape gradients are conveniently computed based on adjoint equations. A number of computational aspects of the proposed approach is discussed and optimization results obtained in several test problems are presented.</p> / Master of Science (MSc)

Une capsule est une goutte de liquide enveloppée par une membrane fine et déformable. Les propriétés mécaniques de la membrane sont essentielles pour le mouvement de la capsule. L’analyse de l’écoulement d’une suspension de capsules dans un canal microfluidique au moyen d’un modèle mécanique est une technique permettant de déterminer les propriétés élastiques de la membrane. Un modèle numérique tridimensionnel a été développé pour résoudre ce problème d’interaction fluide-structure en écoulement confiné. Il couple une méthode des intégrales de frontières pour les écoulements des fluides et une méthode éléments finis pour la déformation de la membrane. Le modèle est utilisé pour étudier l’écoulement d’une capsule initialement sphérique dans des canaux de différentes sections. Dans un canal cylindrique, on montre que l’effet de confinement du canal conduit à la compression de la capsule. Cela engendre la formation de plis sur la membrane autour de l’axe de l’écoulement, phénomène également observé expérimentalement. Dans un canal de section carrée, les effets de la loi constitutive de la membrane, du rapport de taille et du débit d’écoulement sur la déformation de la capsule sont systématiquement étudiés. La comparaison entre les résultats expérimentaux et numériques nous permet de déduire les propriétés mécaniques de la membrane d’une population de capsules artificielles. Ce travail démontre la faisabilité de la mesure de propriétés mécaniques d’une membrane en utilisant une technique microfluidique en canal carré. Il pourrait être étendu par l’étude d’écoulements instationnaires dans un canal de section variable ou avec bifurcations. / A capsule is a liquid droplet enclosed by a thin and deformable membrane. The membrane mechanical properties are critical for the deformation and motion of capsules. The flow of a capsule suspension through a microfluidic channel with dimensions comparable to those of the suspended particles can be used to infer the membrane elastic properties. However a mechanical model of the process is necessary. We present a three-dimensional numerical model to simulate such fluid-structure interaction problem. We use a novel numerical model that couples a boundary integral method for the internal and external fluid flows and a finite element method for the membrane deformation. The model is applied to study the flow of an initially spherical capsule in channels with different cross-sections. In a cylindrical channel with circular cross-section, we show that the confinement effect leads to the compression of the capsule in the hoop direction. The membrane tends to buckle and to fold as observed experimentally. In a microfluidic channel with a square cross-section, the effects of the membrane constitutive law, size ratio and flow strength on the capsule deformation are systematically studied. The comparison between experimental and numerical results allows us to deduce the membrane mechanical properties of a population of artificial capsules. The present work shows that it is possible to measure the membrane mechanical properties by using a microfluidic channel with a square cross-section. It can be extended to unsteady capsule flows in a channel with variable cross-sections or bifurcations.

[pt] Este trabalho trata do escoamento na escala das gotas e da Reologia de emulsões diluídas. Técnicas analíticas e numéricas são empregadas na solução do problema. Nas vizinhan»cas das gotas o escoamento pode ser considerado livre de efeitos de inércia e conseqüentemente as equações governantes são as equações de Stokes. Esse limite é conhecido na literatura como Microhidrodinâmica. O campo de velocidade e de tensão sobre a superfície das gotas é calculado. Um processo de média espacial é realizado em um volume representativo da suspensão tal que a mesma possa ser estudada como um ruido contínuo equivalente. Métodos assintóticos baseados em aproximações de pequenas deformações das gotas são empregados para produzir teorias de primeira e segunda ordens da razão de viscosidade. Uma extensão da teoria para emulsões diluídas polidispersas é desenvolvida. Uma teoria viscoelástica quasi-linear é construída para emulsões diluídas de alta razão de viscosidade em cisalhamento oscilatório. Em regimes de grandes deformações utiliza-se o Método Integral de Contorno para determinar-se a forma da gota e o campos de velocidade sobre a mesma. O método é descrito em detalhes, tanto do ponto de vista teórico como de sua implementação numérica. A validação da metodologia numérica é feita utilizando resultados teóricos e experimentais, disponíveis na literatura. A reologia da emulsão é estudada em escoamentos de cisalhamento simples, oscilatório, pura extensão e cisalhamento quadrático (escoamento de Poiseuille). Os resultados numéricos para cisalhamento simples são utilizados para determinar constantes materiais da teoria assintótica de segunda ordem para a tensão. Limites não-lineares de escoamento em regimes de razões de viscosidade moderadas para os cisalhamentos simples, oscilatório e quadrático são estudados / [en] This work deals with the flow in the scale of the drops and the Rheology of diluted emulsions. Analytic and numerical techniques are employed in order to solve the problem. In the drop neighborhoods the flow may be considered as free of inertia effects and consequently governed by Stokes equations. In the literature this limit is known as Microhydrodynamics. The flow field and the stress tensor on the drop surface are calculated. A spatial mean process was taken, in a representative suspension volume, in order to study the emulsion as an homogeneous and continuous fluid. Asymptotic methods based in small drop deformation approximation are used to produce first and second orders theories which the parameter is the viscosity ratio. An extension of these theories for polydisperse diluted emulsion is developed. A quasi-linear viscoelasticity theory is constructed for diluted emulsion of high viscosity ratios in oscillatory shear flows. In the regimes of large deformations, the velocity and the stress on the particles are evaluated by a numerical procedure based on the Boundary Integral Method for deformable drops. The theoretical and numerical aspects of the Boundary Integral Method are described in details. The code is validated by comparison the numerical results with the experimental data presented in the literature, and also by comparison with the theoretical results of small deformation. The emulsion rheology is studied in simple shear, oscillatory shear, extensional and also in pressure driven flows. The numerical results are used to determine material constants of the stress theory of the second order. Non linear flow regimes of moderate viscosity ratios in simple shear, oscillatory shear and pressure driven flows are also studied.

Dans cette thèse, nous calculons la forme d’un objet immergé d’aire donnée qui minimise la résistance de vague. Le corps, considéré lisse, avance à vitesse constante sous la surface libre d’un fluide qui est supposé parfait et incompressible. La résistance de vague est la traînée, c’est-à-dire la composante horizontale de la force exercée par le fluide sur l’obstacle. Nous utilisons les équations de Neumann-Kelvin 2D, qui s’obtiennent en linéarisant les équations d’Euler irrotationnelles avec surface libre. Le problème de Neumann-Kelvin est formulé comme une équation intégrale de frontière basée sur une solution fondamentale qui intègre la condition linéarisée à la surface libre. Nous utilisons une méthode de descente de gradient pour trouver un minimiseur local du problème de résistance de vague. Un gradient par rapport à la forme est calculé par la méthode de variation de frontières. Nous utilisons une approche level-set pour calculer la résistance de vague et gérer les déplacements de la frontière de l’obstacle. Nous obtenons une grande variété de formes optimales selon la profondeur de l’objet et sa vitesse. / In this thesis, we compute the shape of a fully immersed object with a given area which minimises the wave resistance. The smooth body moves at a constant speed under the free surface of a fluid which is assumed to be inviscid and incompressible. The wave resistance is the drag, i.e. the horizontal component of the force exerted by the fluid on the obstacle. We work with the 2D Neumann-Kelvin equations, which are obtained by linearising the irrotational Euler equations with a free surface. The Neumann-Kelvin problem is formulated as a boundary integral equation based on a fundamental solution which handles the linearised free surface condition. We use a gradient descent method to find a local minimiser of the wave resistance problem. A gradient with respect to the shape is calculated by a boundary variation method. We use a level-set approach to calculate the wave-making resistance and to deal with the displacements of the boundary of the obstacle. We obtain a great variety of optimal shapes depending on the depth of the object and its velocity.

Le travail présenté dans cette thèse consiste à modéliser les transports de particules dans un écoulement de Stokes en tenant compte des caractéristiques des particules, du régime d’écoulement de fluide qui les transporte et de l’effet de confinement qui pourrait être lié par exemple à la prise en compte d’une structure d’un milieu poreux. Le phénomène de transport que nous étudions est basé sur le mécanisme d’effet cliquet qui apparaît quand on soumet le système à un pompage alternatif dans le temps. L’étude de la dynamique de la particule est effectuée en utilisant la méthode de continuation qui permet de suivre la solution périodique selon le paramètre désiré et d’identifier les types de bifurcation ainsi que les lieux de ces points critiques. Cette étude nécessite la connaissance de la force de traînée exercée sur la particule et que nous calculons en utilisant la méthode des équations intégrales de frontières. / The work presented in this Phd Thesis is the modelling of particle transport in a Stokes flow taking into account the characteristics of the particle, the regime flow, and the effect of the confinement which could be related for example to the consideration of a porous media structure. The phenomena of the transport is based on the mecansim of ratchet effect which appears when the system is undergoes an alternative pumping. The study of the particle dynamics is performed using the continuation method which allows to follow the periodic solutions according to the desired parameter, to identify the types of bifurcation and critical points. This study requires the knowledge of the drag force exerted on the particle and which we compute using the boundary integral equation.

Dans cette thèse, nous menons à l'aide de la méthode des équations intégrales des analyses de sensitivité de solutions ou de spectres de l'équation de conductivité par rapport aux variations géométriques ou de paramètres de l'équation. En particulier, nous considérons le problème de conductivité dans des milieux à forts contrastes, le problème de perturbation du bord d'une inclusion de conductivité, le problème de valeurs propres du Laplacien dans des domaines perturbés et le problème d'ouverture de gap dans le spectre des cristaux photoniques.

Fluid mechanics of blood can be well approximated by a mixture model of a Newtonian fluid and deformable particles representing the red blood cells. Experimental and theoretical evidence suggests that the deformation and rheology of red blood cells is similar to that of phospholipid vesicles. Vesicles and red blood cells are both area preserving closed membranes that resist bending. Beyond red blood cells, vesicles can be used to investigate the behavior of cell membranes, intracellular organelles, and viral particles. Given the importance of vesicle flows, in this thesis we focus in efficient numerical methods for such problems: we present computationally scalable algorithms for the simulation of dilute suspension of deformable vesicles in two and three dimensions. Our method is based on the boundary integral formulation of Stokes flow. We present new schemes for simulating the three-dimensional hydrodynamic interactions of large number of vesicles with viscosity contrast. The algorithms incorporate a stable time-stepping scheme, high-order spatiotemporal discretizations, spectral preconditioners, and a reparametrization scheme capable of resolving extreme mesh distortions in dynamic simulations. The associated linear systems are solved in optimal time using spectral preconditioners. The highlights of our numerical scheme are that (i) the physics of vesicles is faithfully represented by using nonlinear solid mechanics to capture the deformations of each cell, (ii) the long-range, N-body, hydrodynamic interactions between vesicles are accurately resolved using the fast multipole method (FMM), and (iii) our time stepping scheme is unconditionally stable for the flow of single and multiple vesicles with viscosity contrast and its computational cost-per-simulation-unit-time is comparable to or less than that of an explicit scheme. We report scaling of our algorithms to simulations with millions of vesicles on thousands of computational cores.

Three-dimensional mathematical problems of the interaction between thermoelastic and scalar oscillation fields are considered in a general anisotropic case. An elastic structure is assumed to be a bounded homogeneous anisortopic body occupying domain $\Omega^+\sub\R^3$ , where the thermoelastic field is defined, while in the physically anisotropic unbounded exterior domain $\Omega^-=\R^3\\ \overline{\Omega^+}$ there is defined the scalar field. These two fields satisfy the differential equations of steady state oscillations in the corresponding domains along with the transmission conditions of special type on the interface $\delta\Omega^{+-}$. Uniqueness and existence theorems, for the non-resonance case, are proved by the reduction of the original interface problems to equivalent systems of boundary pseudodifferential equations ($\Psi DEs$) . The invertibility of the corresponding matrix pseudodifferential operators ($\Psi DO$) in appropriate functional spaces is shown on the basis of generalized Sommerfeld-Kupradze type thermoradiation conditions for anisotropic bodies. In the resonance case, the co-kernels of the $\Psi DOs$ are analysed and the efficent conditions of solvability of the transmission problems are established.

Cavitation is the spontaneous vaporisation of a liquid to its gaseous state due to the local absolute pressure falling to the liquid's vapour pressure (Douglas, Gasiorek et al. 1995). Cavitation is present in a wide range of mechanical systems ranging from ship screws to journal bearing. Generally, cavitation is unavoidable and may cause considerable damage and efficiency losses to these systems. This thesis considers hydraulic systems specifically, and uses a modified Greens equation to develop a boundary integral method to simulate the effect that suspended solid bodies have on a single cavitation bubble. Because of the limitations of accurately modelling cavitation bubbles beyond touchdown, results are only presented for cases up to touchdown. The aim of the model is to draw insight into the reasons there is a measurable change in cavitation erosion rate with increasing oil-in-water emulsion percentage. This principle was extended to include the effect that ingested particulates may have on cavitation in hydraulic machinery. Two particular situations are modelled; the first consists of stationary rigid particles in varying proximity to a cavitation bubble near a rigid boundary. The second case is similar; however the suspended particle is allowed to move under the influence of the pressure differential caused by the expanding/contracting cavitation bubble. Numerous characteristics of the domain are considered, including domain pressures and fluid field motion, and individual boundary surface characteristics. The conclusion of the thesis is that solid bodies, either stationary or moving, have little effect on the cavity from an energy perspective. Regardless of size or density, all energy transferred from the cavity to the solid body is returned indicating that there is no net change. As this energy is ultimately responsible for the peak pressure experienced by the domain (and hence responsible for eroding the rigid boundary) as the cavity rebounds, it then serves that a cavity with a solid body will rebound at the same pressure as a cavity without a suspended body present. If this is coupled with the observation that the cavity centroid at touchdown is largely unaffected by the presence of a suspension, then it would appear that the bubble near a solid would rebound at a very similar position as a cavity without a solid. Consequently, the damage potential of a cavity is unaffected by a suspension. However, there is one point of contention as the profile of the re-entrant jet of the cavity is altered by the presence of a suspension. As energy is radiated away from the cavity during penetration, it is possible that the shape of the jet may alter the rate that energy is radiated away during penetration. However, this requires further research to be definitive.