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In this paper, we consider the capture dynamics of a particle undergoing a random walk above a sheet of absorbing traps. In particular, we seek to characterize the distribution in time from when the particle is released to when it is absorbed. This problem is motivated by the study of lymphocytes in the human blood stream; for a particle near the surface of a lymphocyte, how long will it take for the particle to be captured? We model this problem as a diffusive process with a mixture of reflecting and absorbing boundary conditions. The model is analyzed from two approaches. The first is a numerical simulation using a Kinetic Monte Carlo (KMC) method that exploits exact solutions to accelerate a particle-based simulation of the capture time. A notable advantage of KMC is that run time is independent of how far from the traps one begins. We compare our results to the second approach, which is asymptotic approximations of the FPT distribution for particles that start far from the traps. Our goal is to validate the efficacy of homogenizing the surface boundary conditions, replacing the reflecting (Neumann) and absorbing (Dirichlet) boundary conditions with a mixed (Robin) boundary condition.

This manuscript consists of two papers that formulate novel methodologies pertaining to time series analysis in the state-space framework. In Chapter 1, we introduce an innovative time series forecasting procedure that relies on model-based clustering and model averaging. The clustering algorithm employs a state-space model comprised of three latent structures: a long-term trend component; a seasonal component, to capture recurring global patterns; and an anomaly component, to reflect local perturbations. A two-step clustering algorithm is applied to identify series that are both globally and locally correlated, based on the corresponding smoothed latent structures. For each series in a particular cluster, a set of forecasting models is fit, using covariate series from the same cluster. To fully utilize the cluster information and to improve forecasting for a series of interest, multi-model averaging is employed. We illustrate the proposed technique in an application that involves a collection of monthly disease incidence series. In Chapter 2, to effectively characterize a count time series that arises from a zero-inflated binomial (ZIB) distribution, we propose two classes of statistical models: a class of observation-driven ZIB (ODZIB) models, and a class of parameter-driven ZIB (PDZIB) models. The ODZIB model is formulated in the partial likelihood framework. Common iterative algorithms (Newton-Raphson, Fisher Scoring, and Expectation Maximization) can be used to obtain the maximum partial likelihood estimators (MPLEs). The PDZIB model is formulated in the state-space framework. For parameter estimation, we devise a Monte Carlo Expectation Maximization (MCEM) algorithm, using particle methods to approximate the intractable conditional expectations in the E-step of the algorithm. We investigate the efficacy of the proposed methodology in a simulation study, and illustrate its utility in a practical application pertaining to disease coding.

This thesis is concerned with the numerical simulation of high Reynolds number, three-dimensional, incompressible flows in open domains. Many problems treated in Computational Fluid Dynamics (CFD) occur in free space: e.g., external aerodynamics past vehicles, bluff bodies or aircraft; shear flows such as shear layers or jets. In observing all these flows, we can remark that they are often unsteady, appear chaotic with the presence of a large range of eddies, and are mainly dominated by convection. For years, it was shown that Lagrangian Vortex Element Methods (VEM) are particularly well appropriate for simulating such flows. In VEM, two approaches are classically used for solving the Poisson equation. The first one is the Biot-Savart approach where the Poisson equation is solved using the Green's function approach. The unbounded domain is thus implicitly taken into account. In that case, Parallel Fast Multipole (PFM) solvers are usually used. The second approach is the Vortex-In-Cell (VIC) method where the Poisson equation is solved on a grid using fast grid solvers. This requires to impose boundary conditions or to assume periodicity. An important difference is that fast grid solvers are much faster than fast multipole solvers. We here combine these two approaches by taking the advantages of each one and, eventually, we obtain an efficient VIC-PFM method to solve incompressible flows in open domain. The major interest of this combination is its computational efficiency: compared to the PFM solver used alone, the VIC-PFM combination is 15 to 20 times faster. The second major advantage is the possibility to run Large Eddy Simulations (LES) at high Reynolds number. Indeed, as a part of the operations are done in an Eulerian way (i.e. on the VIC grid), all the existing subgrid scale (SGS) models used in classical Eulerian codes, including the recent "multiscale" models, can be easily implemented.

This thesis is concerned with the numerical simulation of high Reynolds number, three-dimensional, incompressible flows in open domains. Many problems treated in Computational Fluid Dynamics (CFD) occur in free space: e.g., external aerodynamics past vehicles, bluff bodies or aircraft; shear flows such as shear layers or jets. In observing all these flows, we can remark that they are often unsteady, appear chaotic with the presence of a large range of eddies, and are mainly dominated by convection. For years, it was shown that Lagrangian Vortex Element Methods (VEM) are particularly well appropriate for simulating such flows. In VEM, two approaches are classically used for solving the Poisson equation. The first one is the Biot-Savart approach where the Poisson equation is solved using the Green's function approach. The unbounded domain is thus implicitly taken into account. In that case, Parallel Fast Multipole (PFM) solvers are usually used. The second approach is the Vortex-In-Cell (VIC) method where the Poisson equation is solved on a grid using fast grid solvers. This requires to impose boundary conditions or to assume periodicity. An important difference is that fast grid solvers are much faster than fast multipole solvers. We here combine these two approaches by taking the advantages of each one and, eventually, we obtain an efficient VIC-PFM method to solve incompressible flows in open domain. The major interest of this combination is its computational efficiency: compared to the PFM solver used alone, the VIC-PFM combination is 15 to 20 times faster. The second major advantage is the possibility to run Large Eddy Simulations (LES) at high Reynolds number. Indeed, as a part of the operations are done in an Eulerian way (i.e. on the VIC grid), all the existing subgrid scale (SGS) models used in classical Eulerian codes, including the recent "multiscale" models, can be easily implemented.

This thesis examines the mixing times for one-dimensional interacting particle systems. We use the coupling method to study the mixing rates for particle systems on the circle which move according to specific permutations e.g., transpositions and 3-cycles. / Graduation date: 2008

Dans cette thèse nous explorons les possibilités offertes par l'implémentation de méthodes hybrides sur des machines de calcul hétérogènes dans le but de réaliser des simulations numériques de problèmes multiéchelles. La méthode hybride consiste à coupler des méthodes de diverses natures pour résoudre les différents aspects physiques et numériques des problèmes considérés. Elle repose sur une méthode particulaire avec remaillage qui combine les avantages des méthodes Lagrangiennes et Eulériennes. Les particules sont déplacées selon le champ de vitesse puis remaillées à chaque itération sur une grille en utilisant des formules de remaillage d'ordre élevés. Cette méthode semi-Lagrangienne bénéficie des avantages du maillage régulier mais n'est pas contrainte par une condition de CFL.Nous construisons une classe de méthodes d'ordre élevé pour lesquelles les preuves de convergence sont obtenues sous la seule contrainte de stabilité telle que les trajectoires des particules ne se croisent pas.Dans un contexte de calcul à haute performances, le développement du code de calcul a été axé sur la portabilité afin de supporter l'évolution rapide des architectures et leur nature hétérogène. Une étude des performances numériques de l'implémentation GPU de la méthode pour la résolution d'équations de transport est réalisée puis étendue au cas multi-GPU. La méthode hybride est appliquée à la simulation du transport d'un scalaire passif dans un écoulement turbulent 3D. Les deux sous-problèmes que sont l'écoulement turbulent et le transport du scalaire sont résolus simultanément sur des architectures multi-CPU et multi-GPU. / In this work, we investigate the implementation of hybrid methods on heterogeneous computers in order to achieve numerical simulations of multi-scale problems. The hybrid numerical method consists of coupling methods of different natures to solve the physical and numerical characteristics of the problem. It is based on a remeshed particle method that combines the advantages of Lagrangian and Eulerian methods. Particles are pushed by local velocities and remeshed at every time-step on a grid using high order interpolation formulas. This forward semi-lagrangian method takes advantage of the regular mesh on which particles are reinitialized but is not limited by CFL conditions.We derive a class of high order methods for which we are able to prove convergence results under the sole stability constraint that particle trajectories do not intersect.In the context of high performance computing, a strong portability constraint is applied to the code development in order to handle the rapid evolution of architectures and their heterogeneous nature. An analysis of the numerical efficiency of the GPU implementation of the method is performed and extended to multi-GPU platforms. The hybrid method is applied to the simulation of the transport of a passive scalar in a 3D turbulent flow. The two sub-problems of the flow and the scalar calculations are solved simultaneously on multi-CPU and multi-GPU architectures.

Foreign object ingestion (FOI) into jet engines is a recurring scenario during the operation life of aircraft. Objects can range from as small as a pebble on the tarmac to the size of a large bird. Among the potential ingestion scenarios, damage caused by smaller objects may be considered to be negligible. Alternatively, larger objects can initiate progressive damage, potentially leading to catastrophic failure, compromising the integrity of the structure, and endangering the safety of passengers. Considering the dramatic increase in air traffic, FOI represents a crucial safety hazard, and must be better understood to minimize possible damage and structural failure. The main purpose of this study is to develop a unique methodology to assess the response and dynamic damage progression of an advanced, high-bypass propulsion system in the event of an FOI during operation. Using a finite element framework, a unique modeling methodology has been proposed in order to characterize the FOI response of the system. In order to demonstrate versatility of the computational analysis, the impact characteristics of two most common foreign object materials, bird and ice, were investigated. These materials were then defined in finite element domain, verified computationally, and then validated against the existing physical experiments. In addition to the mechanics of the two FOI materials, other material definitions, used to characterize the structures of the high-bypass propulsion system, were also explored. Both composite materials and rate dependent definitions for metal alloys were investigated to represent the damage mechanics in the event of an FOI. Subsequently, damage sequence of high-bypass propulsion systems subject to FOI was developed and assessed, using a uniquely devised Fluid-Structure Interaction (FSI) technique. Using advanced finite element formulation, this approach enabled the accurate simulation of the comprehensive damage progression of the propulsion systems by including aerodynamic interaction. Through this strategy, fluid mechanics was combined with structural mechanics in order to simulate the mutual interaction between both continua, allowing the interpretation of both the additional damage caused by the fluid flow and disrupted aerodynamics induced by the dynamic deformation of the fan blade. Subsequently, this multidisciplinary-multiphysics computational approach, in the framework of the comprehensive analysis methodology introduced, enabled the effective determination of details on the overall progressive impact damage, not traditionally available to propulsion designers. / PHD

Les méthodes particulaires sont des méthodes numériques adaptées à la résolution d'équations de conservation. Leur principe consiste à introduire des particules ``numériques'' conservant localement l'inconnue sur un petit volume, puis à les transporter le long de leur trajectoire. Lorsqu'un terme source est présent dans les équations, l'évolution de la solution le long des caractéristiques est prise en compte par une intéraction entre les particules. Ces méthodes possèdent de bonnes propriétés de conservation et ne sont pas soumises aux conditions habituelles de CFL qui peuvent être contraignantes pour les méthodes Eulériennes. Cependant, une contrainte de recouvrement entre les particules doit être satisfaite pour vérifier des propriétés de convergence de la méthode. Pour satisfaire cette condition de recouvrement, un remaillage périodique des particules est souvent utilisé. Elle consiste à recréer régulièrement de nouvelles particules uniformément réparties, à partir de celles ayant été advectées à l'itération précédente. Quand cette étape de remaillage est effectuée à chaque pas de temps, l'analyse numérique de ces méthodes particulaires remaillées nécessite d'être reconsidérée, ce qui représente l'objectif de ces travaux de thèse. Pour mener à bien cette analyse, nous nous basons sur une analogie entre méthodes particulaires avec remaillage et schémas de grille. Nous montrons que pour des grands pas de temps les schémas numériques obtenus souffrent d'une perte de précision. Nous proposons des méthodes de correction, assurant la consistance des schémas en tout point de grille, le pas de temps étant contraint par une condition sur le gradient du champ de vitesse. Cette méthode est construite en dimension un. Des techniques de limitation sont aussi introduites de manière à remailler les particules sans créer d'oscillations en présence de fortes variations de la solution. Enfin, ces méthodes sont généralisées aux dimensions plus grandes que un en s'inspirant du principe de splitting d'opérateurs. Les applications numériques présentées dans cette thèse concernent la résolution de l'équation de transport sous forme conservative en dimension un à trois, dans des régimes linéaires ou non-linéaires. / Particle methods are numerical methods designed to solve advection dominated conservation equations. Their principle is to introduce ``numerical'' particles that concentrate the unknown locally on a small volume, and to transport them along their trajectories. These methods have good conservation properties and are not subject to the usual CFL conditions that can be binding for the Eulerian methods. However, an overlap condition must be satisfied between the particles to ensure convergence properties of the method. To satisfy this condition, a periodic remeshing of the particles is often used. New particles uniformly distributed are created on a regular mesh. When this remeshing step is performed at every time step, numerical analysis of particle methods needs to be revisited. This is the purpose of this thesis. To carry out this analysis, we rely on an analogy between remeshed particle methods and grid schemes. We show that for large time step the numerical schemes have a loss of accuracy. We propose correction methods wich ensure consistency at any grid point, provided the time step satisfies a condition based on the gradient of the velocity field. Limitation techniques are also introduced to remesh particles without creating any oscillations in the presence of strong variations of the solution. Finally, these methods are generalized to dimensions greater than one. Numerical example on various transport equations are given to illustrate the benefit of the proposed algorithms.

Problemas envolvendo interação entre fluido e estrutura são desafiadores para a engenharia e, ao mesmo tempo em que abrangem dois meios com características físicas distintas, demandam uma descrição matemática para cada um deles que seja compatível, de forma a permitir o acoplamento. Assim, este trabalho apresenta uma formulação em descrição Lagrangeana para análises dinâmicas de sólidos, fluidos incompressíveis e interação fluido-estrutura (IFE). Nos problemas de IFE é comum a estrutura apresentar grandes deslocamentos, o que torna imprescindível considerar o efeito da não-linearidade geométrica. Levando isso em consideração, é empregada uma formulação do método dos elementos finitos (MEF) baseada em posições, cuja aplicação em análises dinâmicas de estruturas em regime de grandes deslocamentos vem se mostrando bastante robusta. Já no âmbito da dinâmica dos fluidos, sabe-se que uma descrição Lagrangeana acaba por eliminar os termos convectivos das equações de Navier-Stokes, dispensando o uso de métodos estabilizantes nessas equações. Por outro lado, a dificuldade é então transferida para o uso de técnicas eficientes de remesh, preservação da qualidade da malha e de identificação do contorno, uma vez que os fluidos podem deformar-se indefinidamente quando submetidos a forças de cisalhamento. Assim, uma combinação do método dos elementos finitos e do método de partículas é utilizada, onde as forças de interação entre as partículas de fluido são calculadas por meio de uma malha de elementos finitos que é renovada para cada passo de tempo. Por meio de técnicas que reconstroem automaticamente o contorno, é possível simular problemas de superfície livre que sofram severas alterações e, até mesmo, uma eventual separação de partículas do domínio inicial, representando, por exemplo, a formação de gotas. Por fim, o sistema de acoplamento entre o fluido e o sólido é simplificado devido a ambos os domínios serem descritos através de um referencial Lagrangeano, não necessitando de métodos para a adaptação da malha do fluido de modo a acompanhar o movimento da estrutura. / Problems involving fluid-structure interaction are challenging for engineering and, while involving two different materials with distinct physical properties, they require a compatible mathematical description for both solid and fluid domain in order to allow the coupling. Thus, this work introduces a formulation, under Lagrangian description, for the solution of solid, incompressible fluid dynamics and fluid-structure interaction (FSI). In FSI problems, the structure usually presents large displacements thus making mandatory a geometric non-linear analysis. Considering it, we adopt a position based formulation of the finite element method (FEM) which has been shown to be very robust when applied to large displacement solid dynamics. For the fluid mechanics problem it is well known that a Lagrangian description eliminates the convective terms from the Navier-Stokes equations and thus, no stabilization technique is required. However, the difficulty is then transferred to the need of efficient re-meshing, mesh quality and external boundary identification techniques, since the fluid presents no resistance to shear stresses and may deform indefinitely. In this sense, we employ a combination of finite element and particle methods in which the particle interaction forces are computed by mean of a finite element mesh which is re-constructed at every time step. Free surface flows are simulated by a boundary recognition technique enabling large domain distortions or even the particles separation from the main domain, representing for instance a water drop. Finally, the fluid-structure coupling is simplified due to the Lagrangian description adopted for both materials, with no need for extra adaptive mesh-moving technique for the fluid computational domain to follow the structure motion.