Dissertations / Theses on the topic 'Implicit incompressible smoothed particle hydrodynamics'

Context. Simulating realistic fluid behavior in incompressible fluids for computer graphics has been pioneered with the implicit incompressible smoothed particle hydrodynamics (IISPH) solver. The algorithm converges faster than other incompressible SPH-solvers, but real-time performance (in the perspective of video games, 30 frames per second) is still an issue when the particle count increases. Objectives. This thesis aims at improving the performance of the IISPH-solver by proposing a parallel solution that runs on the GPU using CUDA. The solution should not compromise the physical accuracy of the original solution. Investigated aspects are execution time, memory usage and physical accuracy. Methods. The proposed implementation uses a fine-grained approach where each particle is calculated on a separate thread. It is compared to a sequential and a parallel OpenMP implementation running on the CPU. Results and Conclusions. It is shown that the parallel CUDA solution allow for real-time performance for approximately 19 times the amount of particles than that of the sequential implementation. For approximately 175 000 particles the simulation runs at the constraint of real-time performance, more particles are still considered interactive. The visual result of the proposed implementation deviated slightly from the ones on the CPU.

Cette thèse porte sur les techniques de simulations des interactions dynamiques entre un véhicule sous-marin et l'eau qui l'entoure. L'objectif principal est de proposer une solution satisfaisante pour pouvoir, en amont du processus de conception, tester des algorithmes de contrôle et des formes de coques pour véhicules sous-marins. Il serait alors intéressant de pouvoir simuler en même temps la dynamique du solide et celle du fluide. L'idée développée dans cette thèse est d'utiliser la technique Smoothed Particles Hydrodynamics (SPH), qui est très récente et qui modélise le fluide comme un ensemble de particules sans maillage. Afin de valider les résultats de simulations une première étude a été réalisée avec un balancier hydrodynamique. Cette étude a permis la mise au point d'une méthode innovante d'estimation de paramètre hydrodynamique (forces de frottement et masse ajoutée) qui est plus robuste que les méthodes existantes lorsqu'il est nécessaire d'utiliser des dérivées numériques du signal mesuré. Ensuite, l'utilisation de deux types de solveur SPH : Weakly Compressible SPH et Incompressible SPH, est validée en suivant la démarche de validation proposée dans cette thèse. Sont étudiés, premièrement, le comportement du fluide seul, deuxièmement, un cas hydrostatique, et enfin un cas dynamique. L'utilisation de deux méthodes de modélisation de l'interaction fluide-solide : la méthode de réflexion de la pression et la méthode d'extrapolation est étudiée. La capacité d'atteindre une vitesse limite due aux forces de frottement est démontrée. Les résultats d'estimation des paramètres hydrodynamiques à partir des essais de simulation est finalement discutée. La masse ajoutée simulée du solide s'approche de la réalité, mais les forces de frottement semblent actuellement ne pas correspondre à la réalité. Des pistes d'améliorations pour pallier à ce problème sont proposées
This thesis focuses on techniques that allows the simulation of dynamic interactions between an underwater vehicle and the surrounding water. The main objective is to propose a satisfactory solution to be able to test control algorithms and hull shapes for underwater vehicles upstream of the design process. In those cases, it would be interesting to be able to simulate solid and fluid dynamics at the same time. The idea developed in this thesis is to use the Smoothed Particles Hydrodynamics (SPH) technique, which is very recent, and which models the fluid as a set of particles without mesh. In order to validate the simulation results a first study has been performed with a hydrodynamic pendulum. This study allowed the development of an innovative method for estimating the hydrodynamic parameters (friction forces and added mass) which is more robust than previous existing methods when it is necessary to use numerical derivatives of the measured signal. Then, the use of two types of SPH solver: Weakly Compressible SPH and Incompressible SPH, is validated following the validation approach proposed in this thesis. Firstly, the behaviour of the fluid alone is studied, secondly, a hydrostatic case, and finally a dynamic case. The use of two methods for modelling the fluid-solid interaction: the pressure mirroring method and the extrapolation method is studied. The ability to reach a limit velocity due to friction forces is demonstrated. The results of the hydrodynamic parameters estimation from simulation tests are finally discussed. The simulated added mass of the solid approaches reality, but the friction forces currently seem not to correspond to reality. Possible improvements to overcome this problem are proposed

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
Em uma grande gama de problemas físicos, governados por equações diferenciais, muitas vezes é de interesse obter-se soluções para o regime transiente e, portanto, deve-se empregar técnicas de integração temporal. Uma primeira possibilidade seria a de aplicar-se métodos explícitos, devido à sua simplicidade e eficiência computacional. Entretanto, esses métodos frequentemente são somente condicionalmente estáveis e estão sujeitos a severas restrições na escolha do passo no tempo. Para problemas advectivos, governados por equações hiperbólicas, esta restrição é conhecida como a condição de Courant-Friedrichs-Lewy (CFL). Quando temse a necessidade de obter soluções numéricas para grandes períodos de tempo, ou quando o custo computacional a cada passo é elevado, esta condição torna-se um empecilho. A fim de contornar esta restrição, métodos implícitos, que são geralmente incondicionalmente estáveis, são utilizados. Neste trabalho, foram aplicadas algumas formulações implícitas para a integração temporal no método Smoothed Particle Hydrodynamics (SPH) de modo a possibilitar o uso de maiores incrementos de tempo e uma forte estabilidade no processo de marcha temporal. Devido ao alto custo computacional exigido pela busca das partículas a cada passo no tempo, esta implementação só será viável se forem aplicados algoritmos eficientes para o tipo de estrutura matricial considerada, tais como os métodos do subespaço de Krylov. Portanto, fez-se um estudo para a escolha apropriada dos métodos que mais se adequavam a este problema, sendo os escolhidos os métodos Bi-Conjugate Gradient (BiCG), o Bi-Conjugate Gradient Stabilized (BiCGSTAB) e o Quasi-Minimal Residual (QMR). Alguns problemas testes foram utilizados a fim de validar as soluções numéricas obtidas com a versão implícita do método SPH.
In a wide range of physical problems governed by differential equations, it is often of interest to obtain solutions for the unsteady state and therefore it must be employed temporal integration techniques. One possibility could be the use of an explicit methods due to its simplicity and computational efficiency. However, these methods are often only conditionally stable and are subject to severe restrictions for the time step choice. For advective problems governed by hyperbolic equations, this restriction is known as the Courant-Friedrichs-Lewy (CFL) condition. When there is the need to obtain numerical solutions for long periods of time, or when the computational cost for each time step is high, this condition becomes a handicap. In order to overcome this restriction implicit methods can be used, which are generally unconditionally stable. In this study, some implicit formulations for time integration are used in the Smoothed Particle Hydrodynamics (SPH) method to enable the use of larger time increments and obtain a strong stability in the time evolution process. Due to the high computational cost required by the particles tracking at each time step, the implementation will be feasible only if efficient algorithms were applied for this type of matrix structure such as Krylov subspace methods. Therefore, we carried out a study for the appropriate choice of methods best suited to this problem, and the methods chosen were the Bi-Conjugate Gradient (BiCG), the Bi-Conjugate Gradient Stabilized (BiCGSTAB) and the Quasi-Minimal Residual(QMR). Some test problems were used to validate the numerical solutions obtained with the implicit version of the SPH method.

Nesse trabalho, foi desenvolvido um simulador numérico (C/C++) para a resolução de escoamentos de fluidos newtonianos incompressíveis, baseado no método de partículas Lagrangiano, livre de malhas, Smoothed Particle Hydrodynamics (SPH). Tradicionalmente, duas estratégias são utilizadas na determinação do campo de pressões de forma a garantir-se a condição de incompressibilidade do fluido. A primeira delas é a formulação chamada Weak Compressible Smoothed Particle Hydrodynamics (WCSPH), onde uma equação de estado para um fluido quase-incompressível é utilizada na determinação do campo de pressões. A segunda, emprega o Método da Projeção e o campo de pressões é obtido mediante a resolução de uma equação de Poisson. No estudo aqui desenvolvido, propõe-se três métodos iterativos, baseados noMétodo da Projeção, para o cálculo do campo de pressões, Incompressible Smoothed Particle Hydrodynamics (ISPH). A fim de validar os métodos iterativos e o código computacional, foram simulados dois problemas unidimensionais: os escoamentos de Couette entre duas placas planas paralelas infinitas e de Poiseuille em um duto infinito e foram usadas condições de contorno do tipo periódicas e partículas fantasmas. Um problema bidimensional, o escoamento no interior de uma cavidade com a parede superior posta em movimento, também foi considerado. Na resolução deste problema foi utilizado o reposicionamento periódico de partículas e partículas fantasmas.
In this work, we have developed a numerical simulator (C/C++) to solve incompressible Newtonian fluid flows, based on the meshfree Lagrangian Smoothed Particle Hydrodynamics (SPH) Method. Traditionally, two methods have been used to determine the pressure field to ensure the incompressibility of the fluid flow. The first is calledWeak Compressible Smoothed Particle Hydrodynamics (WCSPH) Method, in which an equation of state for a quasi-incompressible fluid is used to determine the pressure field. The second employs the Projection Method and the pressure field is obtained by solving a Poissons equation. In the study developed here, we have proposed three iterative methods based on the Projection Method to calculate the pressure field, Incompressible Smoothed Particle Hydrodynamics (ISPH) Method. In order to validate the iterative methods and the computational code we have simulated two one-dimensional problems: the Couette flow between two infinite parallel flat plates and the Poiseuille flow in a infinite duct, and periodic boundary conditions and ghost particles have been used. A two-dimensional problem, the lid-driven cavity flow, has also been considered. In solving this problem we have used a periodic repositioning technique and ghost particles.

La 3D s'impose comme un nouveau média dont l'adoption généralisée passe par la conception d'outils, accessibles au grand public, de création et de manipulation de formes tridimensionnelles quelconques. Les outils actuels reposent fortement sur la modélisation sous-jacente des formes, généralement surfacique, et sont alors peu intuitifs ou limitatifs dans l'expressivité offerte à l'utilisateur.Nous souhaitons, dans ces travaux, définir une approche ne présentant pas ces défauts et permettant à l'utilisateur de se concentrer sur le processus créatif. En nous inspirant de l'utilisation séculaire de l'argile, nous proposons une approche modélisant la matière sous forme lagrangienne.Une forme est ainsi décrite par un système de particules, où chaque particule représente un petit volume du volume global.Dans ce cadre lagrangien, la méthode Smoothed Particle Hydrodynamics (SPH) permet l'approximation de grandeurs physiques en tout point de l'espace. Nous proposons alors une modélisation de matériaux à deux couches, l'une décrivant la topologie et l'autre décrivant la géométrie du système global.La méthode SPH permet, entre autres, d'évaluer la densité de matière. Ceci nous permet de définir une surface implicite basée sur les propriétés physiques du système de particules pour redonner un aspect continu à la matière.Ces matériaux peuvent alors être manipulés au moyen d'interactions locales reproduisant le maniement de la pâte à modeler, et de déformations globales. L'intérêt de notre approche est démontrée par plusieurs prototypes fonctionnant sur des stations de travail standard ou dans des environnements immersifs
3D is emerging as a new media. Its widespread adoption requires the implementation of userfriendly tools to create and manipulate three-dimensional shapes. Current softwares heavily rely on underlying shape modeling, usually a surfacic one, and are then often counter-intuitive orlimiting. Our objective is the design of an approach alleviating those limitations and allowing the user to only focus on the process of creating forms. Drawing inspiration from the ancient use of clay,we propose to model a material in a lagrangian description. A shape is described by a particles system, where each particle represents a small fraction of the total volume of the shape. In this framework, the Smoothed Particle Hydrodynamics method enables to approximate physical values anywhere in space. Relying on this method, we propose a modeling of material with two levels, one level representing the topology and the other one describing local geometry of the shape.The SPH method especially enables to evaluate a density of matter. We use this property todefine an implicit surface based on the physical properties of the particles system to reproduce the continuous aspect of matter. Those virtual materials can then be manipulated locally through interactions reproducing the handling of dough in the real world or through global shape deformation. Our approach is demonstrated by several prototypes running either on typical desktop workstation or in immersive environment system

L’objectif de ce mémoire est d’introduire une nouvelle méthode smoothed particle hydrodynamics (SPH) implicite purement lagrangienne, pour la résolution des équations de Navier- Stokes incompressibles bidimensionnelles en présence d’une surface libre. Notre schéma de discrétisation est basé sur celui de Kéou Noutcheuwa et Owens [19]. Nous avons traité la surface libre en combinant la méthode multiple boundary tangent (MBT) de Yildiz et al. [43] et les conditions aux limites sur les champs auxiliaires de Yang et Prosperetti [42]. Ce faisant, nous obtenons un schéma de discrétisation d’ordre $\mathcal{O}(\Delta t ^2)$ et $\mathcal{O}(\Delta x ^2)$, selon certaines contraintes sur la longueur de lissage $h$. Dans un premier temps, nous avons testé notre schéma avec un écoulement de Poiseuille bidimensionnel à l’aide duquel nous analysons l’erreur de discrétisation de la méthode SPH. Ensuite, nous avons tenté de simuler un problème d’extrusion newtonien bidimensionnel. Malheureusement, bien que le comportement de la surface libre soit satisfaisant, nous avons rencontré des problèmes numériques sur la singularité à la sortie du moule.
The objective of this thesis is to introduce a new implicit purely lagrangian smoothed particle hydrodynamics (SPH) method, for the resolution of the two-dimensional incompressible Navier-Stokes equations in the presence of a free surface. Our discretization scheme is based on that of Kéou Noutcheuwa et Owens [19]. We have treated the free surface by combining Yildiz et al. [43] multiple boundary tangent (MBT) method and boundary conditions on the auxiliary fields of Yang et Prosperetti [42]. In this way, we obtain a discretization scheme of order $\mathcal{O}(\Delta t ^2)$ and $\mathcal{O}(\Delta x ^2)$, according to certain constraints on the smoothing length $h$. First, we tested our scheme with a two-dimensional Poiseuille flow by means of which we analyze the discretization error of the SPH method. Then, we tried to simulate a two-dimensional Newtonian extrusion problem. Unfortunately, although the behavior of the free surface is satisfactory, we have encountered numerical problems on the singularity at the output of the die.

碩士
國立臺灣科技大學
機械工程系
100
An incompressible smoothed particle hydrodynamics (ISPH) method based on the incremental pressure projection method is developed in this study. The Rayleigh-B\'enard convection in a square enclosure is used as a validation case and the results obtained by the proposed ISPH model are compared to the benchmark solutions. The comparison shows that the estabished ISPH method has a good performance in terms of accuracy. Subsequently, the proposed ISPH method is employed to simulate natural convection from a heated cylinder in a square enclosure. It shows that the predictions obtained by the ISPH method are in good agreements with the results obtained by previous studies using alternative numerical methods. A rotating and heated cylinder is also considered to study the effect of the rotation on the heat transfer process in the enclosure space. The numerical results show that for a square enclosure at Ra/Re_D^2 ≥ 1.0, the addition of kinetic energy in the form of rotation does not enhance the heat transfer process. In terms of results, it turns out that the proposed ISPH model is capable to simulate heat transfer problems with the complex and moving boundaries without adapting a mesh.

Fluid structure interactions in the presence of a free surface includes complex phenomena, such as slamming, air entrainment, transient loads, complex free surface profiles and turbulence. Hence, an appropriate and efficient numerical method is required to deal with these type of problems (efficient both in problem setup and numerical solution). Eulerian mesh-based methods can be used to solve different types of problems, however they have difficulties in problems involving moving boundaries and discontinuities (e.g. fluid structure interactions in the presence of a free surface). Smoothed Particle Hydrodynamics (SPH) is a mesh-less Lagrangian particle method, ideal for solving problems with large deformation and fragmentation such as complex free surface flows. The SPH method was originally invented to study astrophysical applications and requires modifications in order to be applied for hydrodynamic applications. Applying solid boundary conditions for hydrodynamic applications in SPH is a key difference to the original SPH developed for astrophysics. There are several methods available in literature to apply solid boundaries in SPH. In this research, an accurate solid boundary condition is used to calculate the pressure at the boundary particles based on the surrounding fluid particles. The two main methods to calculate the pressure in the SPH method are the weakly compressible SPH (WCSPH) and the incompressible SPH (ISPH) approaches. The WCSPH uses the equation of state while ISPH solves Poisson's equation to determine the pressure. In this dissertation, an explicit incompressible SPH (ISPH) method is used to study nonlinear free surface applications. In the explicit ISPH method, Poisson's equation is explicitly solved to calculate the pressure within a projection based algorithm. This method does not require solving a set of algebraic equations for pressure at each time step unlike the implicit method. Here, an accurate boundary condition along with an accurate source term for Poisson's equation is used within the explicit method. Also, the sub-particle turbulent calculation is applied to the explicit ISPH method (which handles large-scale turbulent structures implicitly) in order to calculate the flow field quantities and consequently forces on the device more accurately. The SPH method is typically computationally more expensive than Eulerian-based CFD methods. Therefore, parallelization methods are required to improve the performance of the method, especially for 3D simulations. In this dissertation, two novel parallel schemes are developed based on Open Multi Processing (OpenMP) and Message Passing Interface (MPI) standards. The explicit ISPH approach is an advantage for parallel computing but our proposed method could also be applied to the WCSPH or implicit ISPH. The proposed SPH model is used to simulate and analyze several nonlinear free surface problems. First, the proposed explicit ISPH method is used to simulate a transient wave overtopping on a horizontal deck. Second, a wave impacting on a scaled oscillating wave surge converter (OWSC) is simulated and studied. Third, the performance and accuracy of the code is tested for a dam-break impacting on tall and short structures. Forth, the hydrodynamic loads from the spar of a scaled self-reacting point absorber wave energy converter (WEC) design is studied. Finally, a comprehensive set of landslide generated waves are modeled and analyzed and a new technique is proposed to calculate the motion of a slide on an inclined ramp implicitly without using a prescribed motion.
Graduate

Yes
This research paper presents an incompressible smoothed particle hydrodynamics (ISPH) technique to investigate a regular wave overtopping on the coastal structure of different types. The SPH method is a mesh-free particle modeling approach that can efficiently treat the large deformation of free surface. The incompressible SPH approach employs a true hydrodynamic formulation to solve the fluid pressure that has less pressure fluctuations. The generation of flow turbulence during the wave breaking and overtopping is modeled by a subparticle scale (SPS) turbulence model. Here the ISPH model is used to investigate the wave overtopping over a coastal structure with and without the porous material. The computations disclosed the features of flow velocity, turbulence, and pressure distributions for different structure types and indicated that the existence of a layer of porous material can effectively reduce the wave impact pressure and overtopping rate. The proposed numerical model is expected to provide a promising practical tool to investigate the complicated wave-structure interactions.
Nazarbayev University Seed Grant, entitled “Environmental assessment of sediment pollution impact on hydropower plants”. S. Shao also acknowledges the Royal Society Research Grant (2008/R2 RG080561)

Recent technological advances are based on effectively using complex multiphysics concepts. Therefore, there is an ever increasing need for accurate numerical al-gorithms of reduced complexity for solving multiphysics problems. Traditional mesh-based simulation methods depend on a neighbor connectivity information for formulation of operators like derivatives. In large deformation problems, de-pendence on a mesh could prove a limitation in terms of accuracy and cost of preprocessing. Meshless methods obviate the need to construct meshes thus al-lowing simulations involving severe geometric deformations such as breakup of a contiguous domain into multiple fragments. Smoothed Particle Hydrodynamics (SPH) is a meshless particle based Lagrangian numerical method that has the longest continuous history of development ever since it was introduced in 1977. Commensurate with the significant growth in computational power, SPH has been increasingly applied to solve problems of greater complexity in fluid mechanics, solid mechanics, interfacial flows and astrophysics to name a few. The SPH approximation of the continuity and momentum equations govern-ing fluid flow traditionally involves a stiff equation of state relating pressure and density, when applied to incompressible flow problems. Incompressible Smoothed Particle Hydrodynamics (ISPH) is a variant of SPH that replaces this weak com-pressibility approach with a pressure equation that gives a hydrostatic pressure field which ensures a divergence-free (or density invariant) velocity field. The present study explains the development of an ISPH algorithm and its implementa-tion with focus on application to free surface flows, interaction of fluid with rigid bodies and coupling of incompressible fluids with a compressible second phase. Several improvements to the exiting ISPH algorithm are proposed in this study. A semi-analytic free surface model which is more accurate and robust compared to existing algorithms used in ISPH methods is introduced, validated against experi-ments and grid based CFD results. A surface tension model with specific applica-bility to free surfaces is presented and tested using 2D and 3D simulations. Using theoretical arguments, a volume conservation error in existing particle methods in general is demonstrated. A deformation gradient based approach is used to derive a new pressure equation which reduces these errors. The method is ap-plied to both free surface and internal flow problems and is shown to have better volume conservation and therefore reduced density fluctuations. Also, comments on instabilities arising from particle distributions are made and the role of the smoothing functions in such instabilities is discussed. The challenges in imple-menting the ISPH algorithm in a computer code are discussed and the experience of developing an in-house ISPH code is described. A parametric study on water entry of cylinders of different shapes, angular velocity and density is performed and aspects such as surface profiles, impact pressures and penetration velocities are compared. An analysis on the energy transfer between the solid and the fluid is also performed. Low Froude number water entry of a sphere is studied and the impact pressure is compared with the theoretical estimates. The Incompressible SPH formulation, employing the proposed improvements from the study is then coupled with a compressible SPH formulation to perform two phase flow simulations interacting compressible and incompressible fluids. To gain confidence in its applicability, the simulations are compared against the theoretical predication given by the Rayleigh-Plesset equation for the problem of compressible drop in an incompressible fluid.

No
Two numerical particle models, the Smoothed Particle Hydrodynamics (SPH) and Moving Particle Semi-implicit (MPS) methods, coupled with a sub-particle scale (SPS) turbulence model, are presented to simulate free surface flows. Both SPH and MPS methods have the advantages in that the governing Navier¿Stokes equations are solved by Lagrangian approach and no grid is needed in the computation. Thus the free surface can be easily and accurately tracked by particles without numerical diffusion. In this paper different particle interaction models for SPH and MPS methods are summarized and compared. The robustness of two models is validated through experimental data of a dam-break flow. In addition, a series of numerical runs are carried out to investigate the order of convergence of the models with regard to the time step and particle spacing. Finally the efficiency of the incorporated SPS model is further demonstrated by the computed turbulence patterns from a breaking wave. It is shown that both SPH and MPS models provide a useful tool for simulating free surface flows

Two numerical particle models, the Smoothed Particle Hydrodynamics (SPH) and Moving Particle Semi-implicit (MPS) methods, coupled with a sub-particle scale (SPS) turbulence model, are presented to simulate free surface flows. Both SPH and MPS methods have the advantages in that the governing Navier¿Stokes equations are solved by Lagrangian approach and no grid is needed in the computation. Thus the free surface can be easily and accurately tracked by particles without numerical diffusion. In this paper different particle interaction models for SPH and MPS methods are summarized and compared. The robustness of two models is validated through experimental data of a dam-break flow. In addition, a series of numerical runs are carried out to investigate the order of convergence of the models with regard to the time step and particle spacing. Finally the efficiency of the incorporated SPS model is further demonstrated by the computed turbulence patterns from a breaking wave. It is shown that both SPH and MPS models provide a useful tool for simulating free surface flows.

Dans cette thèse, nous présentons une nouvelle méthode smoothed particle hydrodynamics (SPH) pour la résolution des équations de Navier-Stokes incompressibles, même en présence des forces singulières. Les termes de sources singulières sont traités d'une manière similaire à celle que l'on retrouve dans la méthode Immersed Boundary (IB) de Peskin (2002) ou de la méthode régularisée de Stokeslets (Cortez, 2001). Dans notre schéma numérique, nous mettons en oeuvre une méthode de projection sans pression de second ordre inspirée de Kim et Moin (1985). Ce schéma évite complètement les difficultés qui peuvent être rencontrées avec la prescription des conditions aux frontières de Neumann sur la pression. Nous présentons deux variantes de cette approche: l'une, Lagrangienne, qui est communément utilisée et l'autre, Eulerienne, car nous considérons simplement que les particules SPH sont des points de quadrature où les propriétés du fluide sont calculées, donc, ces points peuvent être laissés fixes dans le temps. Notre méthode SPH est d'abord testée à la résolution du problème de Poiseuille bidimensionnel entre deux plaques infinies et nous effectuons une analyse détaillée de l'erreur des calculs. Pour ce problème, les résultats sont similaires autant lorsque les particules SPH sont libres de se déplacer que lorsqu'elles sont fixes. Nous traitons, par ailleurs, du problème de la dynamique d'une membrane immergée dans un fluide visqueux et incompressible avec notre méthode SPH. La membrane est représentée par une spline cubique le long de laquelle la tension présente dans la membrane est calculée et transmise au fluide environnant. Les équations de Navier-Stokes, avec une force singulière issue de la membrane sont ensuite résolues pour déterminer la vitesse du fluide dans lequel est immergée la membrane. La vitesse du fluide, ainsi obtenue, est interpolée sur l'interface, afin de déterminer son déplacement. Nous discutons des avantages à maintenir les particules SPH fixes au lieu de les laisser libres de se déplacer. Nous appliquons ensuite notre méthode SPH à la simulation des écoulements confinés des solutions de polymères non dilués avec une interaction hydrodynamique et des forces d'exclusion de volume. Le point de départ de l'algorithme est le système couplé des équations de Langevin pour les polymères et le solvant (CLEPS) (voir par exemple Oono et Freed (1981) et Öttinger et Rabin (1989)) décrivant, dans le cas présent, les dynamiques microscopiques d'une solution de polymère en écoulement avec une représentation bille-ressort des macromolécules. Des tests numériques de certains écoulements dans des canaux bidimensionnels révèlent que l'utilisation de la méthode de projection d'ordre deux couplée à des points de quadrature SPH fixes conduit à un ordre de convergence de la vitesse qui est de deux et à une convergence d'ordre sensiblement égale à deux pour la pression, pourvu que la solution soit suffisamment lisse. Dans le cas des calculs à grandes échelles pour les altères et pour les chaînes de bille-ressort, un choix approprié du nombre de particules SPH en fonction du nombre des billes N permet, en l'absence des forces d'exclusion de volume, de montrer que le coût de notre algorithme est d'ordre O(N). Enfin, nous amorçons des calculs tridimensionnels avec notre modèle SPH. Dans cette optique, nous résolvons le problème de l'écoulement de Poiseuille tridimensionnel entre deux plaques parallèles infinies et le problème de l'écoulement de Poiseuille dans une conduite rectangulaire infiniment longue. De plus, nous simulons en dimension trois des écoulements confinés entre deux plaques infinies des solutions de polymères non diluées avec une interaction hydrodynamique et des forces d'exclusion de volume.
In this thesis we develop a new smoothed particle hydrodynamics (SPH) method suitable for solving the incompressible Navier-Stokes equations, even with singular forces. Singular source terms are handled in a manner similar to that in the immersed boundary (IB) method of Peskin (2002) or in the method of regularized Stokeslets (Cortez, 2001). The numerical scheme implements a second-order pressure-free projection method due to Kim and Moin (1985) and completely obviates the difficulties that may be faced in prescribing Neumann pressure boundary conditions. We present two variants of this approach, one Langrangian which is commonly used and one Eulerian, simply because we consider that the SPH particles are quadrature points on which the fluid properties are calculated, therefore, these points can be kept fixed in time. The proposed SPH method is first tested on the planar start-up Poiseuille problem and a detailed error analysis is performed. For this problem, the results are similar whether the SPH particles are free to move or fixed on a regular grid. Our hybrid SPH-IB method is then used to calculate the dynamics of a stretched immersed elastic membrane. The membrane is represented by a cubic spline along which the tension in the membrane is computed and transmitted to the surrounding fluid. The Navier-Stokes equations with singular force due to the membrane are then solved to determine the velocity of the fluid in which the membrane is immersed. The fluid velocity thus obtained is interpolated on the interface, to determine its displacement. We discuss the advantages, in this problem, of fixing the SPH particles, rather than allowing them to move with the fluid. A new coupled Brownian dynamics-SPH method for the computation of confined flows of non-dilute polymer solutions with full hydrodynamic interaction and excluded volume forces is next presented. The starting point for the algorithm is the system of coupled Langevin equations for polymer and solvent (CLEPS) (see Oono and Freed (1981) and Öttinger and Rabin (1989), for example) describing, in the present case, the microscopic dynamics of a flowing polymer solution with a bead-spring representation of the macromolecules. Numerical tests of some two-dimensional channel flows reveal that use of a second-order projection scheme coupled with fixed SPH quadrature points leads to second-order velocity convergence and almost second-order pressure convergence, provided that the solution is sufficiently smooth. In the case of large-scale dumbbell and bead-spring chain calculations, an appropriate scaling of the number of grid points as a function of the number of beads N ensures, in the absence of excluded volume forces, that the cost of our algorithm is O(N) flops. Finally, we begin calculations in three dimensions with our SPH model. To this end, we solve in three dimensions the problem of Poiseuille flow between two infinite and parallel plates and the problem of Poiseuille flow in a rectangular infinitely long duct. In addition, we carry out three dimensional computations of confined flows of non-dilute polymer solutions with full hydrodynamic interaction and excluded volume forces.