Desde a década de 70, há um crescente interesse em simulações em computador de fenômenos físicos visto sua diversidade de aplicações. Dentre esses fenômenos, podem ser destacados a interação entre corpos rígidos, elásticos, plásticos, quebráveis e também fluidos. Neste trabalho realizamos a simulação de um desses fenômenos, o escoamento de fluidos, por um método conhecido como Smoothed Particles Hydrodynamics, uma abordagem lagrangeana baseada em partículas para resolução das equações que modelam o movimento do fluido. Várias são as vantagens de métodos lagrangeanos usando partículas sobre os que usam malhas, por exemplo, as propriedades do material transladam com as partículas como função do tempo, além da capacidade de lidar com grandes deformações. Dentre as desvantagem, destacamos uma deficiência relacionada ao ganho de energia total do sistema e estabilidade das partículas. Para lidar com isso, utilizamos uma abordagem baseada na lei da conservação da energia: em um sistema isolado a energia total se mantém constante e ela não pode ser criada ou destruida. Dessa forma, alterando o integrador temporal nós restringimos o aumento arbitrário de energia, tornando a simulação mais tolerante às condições iniciais / Since the late 70s, there is a growing interest in physically-based simulations due to its increasing range of application. Among these simulations, we may highlight interaction between rigid, elastic, plastic and breakable bodies and also fluids. In this work, one of these phenomena, fluid flow, is simulated using a technique known as Smoothed Particle Hydrodynamics, a meshless lagrangean method that solves the equations of the flow behavior of fluids. There are several advantages of meshless methods over mesh-based methods, for instance, the material properties are translated along with particles as a function of time and the ability to handle arbitrary deformations. Among the disadvantages, we may highlight a problem related to the gain of energy by the system and stability issues. In order to handle this, we used an approach based on the law of conservation of energy: in an isolated system the total energy remains constant and cannot be created or destroyed. Based on this, we used a technique that bounds the total energy and the simulation becomes less sensitive to initial conditions

Nos últimos anos, o estudo de métodos de animação de escoamento de fluidos tem sido uma área de intensa pesquisa em Computação Gráfica. O principal objetivo desse projeto é desenvolver novas técnicas em GPGPU baseadas na arquitetura CUDA para simular o escoamento de fluidos não-newtonianos, tais como fluidos viscoplásticos e viscoelásticos. Ao invés dos tradicionais métodos com malha diferenças finitas e elementos finitos, essas técnicas são baseadas em uma discretização lagrangeana das equações de governo desses fluidos através do método sem malha conhecido como SPH (Smoothed Particle Hydrodynamics) / I n recent years, the study of methods of animating fluid flow has been an area of intense research in Computer Graphics. The main objective of this project is to develop new techniques based on the CUDA GPGPU architecture to simulate the flow of non-Newtonian fluids, such as viscoelastic and viscoplastic fluids. Instead of traditional methods with mesh - finite differences and finite elements, these techniques are based on a Lagrangian discretization of the governing equations of these fluids through the mesh free method known as SPH (Smoothed Particle Hydrodynamics)

Ionising radiation is present in a variety of astrophysical problems, and it is particularly important for shaping the process of star formation in molecular clouds, containing hot, high-mass stars. In order to account for the effects of ionising radiation within numerical models of star formation, we need to combine a hydrodynamics method with a radiative transfer method and obtain a radiation hydrodynamics scheme (RHD). In this thesis I achieve live radiation hydrodynamics by coupling the Smoothed Particle Hydrodynamics (SPH) code Phantom with the Monte Carlo Radiative Transfer (MCRT) code CMacIonize. Since SPH is particle-based and MCRT is grid-based, I construct an unstructured, Voronoi grid in order to establish a link between the two codes. In areas with large density gradients, a Voronoi grid based purely on the SPH particle positions achieves insufficient resolution, and therefore I propose a novel algorithm for inserting a small number of additional grid cells to improve the local resolution. Furthermore, the MCRT calculations require the knowledge of an average density for each Voronoi cell. To address this, I develop an analytic density mapping from SPH to a Voronoi grid, by deriving an expression for the integrals of a series of kernel functions over the volume of a random polyhedron. Finally, I demonstrate the validity of the live RHD through the benchmark test of D-type expansion of an H II region, where good agreement is shown with the existing literature. The RHD implementation is then used to perform a proof-of-concept simulation of a collapsing cloud, which produces high-mass stars and is subsequently partially ionised by them. The presented code is a valuable tool for future star formation studies, and it can be used for modelling a broad range of additional astronomical problems involving ionising radiation and hydrodynamics.

Smoothed Particle Hydrodynamics (SPH) is analysed as the weighted residual method. In particular the analysis focuses on the collocation aspect of the method. Using Monte Carlo experiments we demonstrate that SPH is highly sensitive to node disorder, especially in its symmetrised energy and momentum conserving form. This aspect of the method is related to low [Beta] MHD instabilities observed by other authors. A remedy in the form of the Weighted Differences Method is suggested, which addresses this problem to some extent, but at a cost of losing automatic conservation of energy and momentum. ¶ The Weighted Differences Method is used to simulate propagation of Alfven and magnetosonic wave fronts in [Beta] = 0 plasma, and the results are compared with data obtained with the NCSA Zeus3D code with the Method of Characteristics (MOC) module. ¶ SPH is then applied to two interesting astrophysical situations: accretion on to a white dwarf in a compact binary system, which results in a formation of an accretion disk, and gravitational collapse of a magnetised vortex. Both models are 3 dimensional. ¶ The accretion disk which forms in the binary star model is characterised by turbulent flow: the Karman vortex street is observed behind the stream-disk interaction region. The shock that forms at the point of stream-disk interaction is controlled by the means of particle merges, whereas Monaghan-Lattanzio artificial viscosity is used to simulate Smagorinsky closure. ¶ The evolution of the collapsing magnetised vortex ends up in the formation of an expanding ring in the symmetry plane of the system. We observe the presence of spiralling inward motion towards the centre of attraction. That final state compares favourably with the observed qualitative and quantitative characteristics of the circumnuclear disk in the Galactic Centre. That simulation has also been verified with the NCSA Zeus3D run. ¶ In conclusions we contrast the result of our Monte Carlo experiments with the results delivered by our production runs. We also compare SPH and Weighted Differences against the new generation of conservative finite differences methods, such as the Godunov method and the Piecewise Parabolic Method. We conclude that although SPH cannot match the accuracy and performance of those methods, it appears to have some advantage in simulation of rotating flows, which are of special interest to astrophysics.

Cette thèse porte sur le développement, la validation et l'application d'un code de calcul numérique à surface libre. Smoothed Particle Hydrodynamics (SPH) est une méthode particulaire,Lagrangienne, imaginée à l'origine pour la simulation de corps gravitaires en Astrophysique. Depuis les années 90, elle a été étendue à la modélisation d'écoulements à surface libre. En effet, lors de grandes déformations des interfaces (déferlement par exemple), l'absence de maillage est particulièrement intéressante. Cette propriété est également intéressante pour la simulation de plusieurs phases liquides aux propriétés différentes (eau/fond sédimentaire par exemple). Les applications visées sont la propagation de la houle et son interaction avec des obstacles immergés. Le modèle SPH a donc été établi en se basant sur la bibliographie et validé sur le cas de l'étirement d'une goutte en l'absence de gravité.Ensuite, la prise en compte de conditions aux limites sur des parois solides est étudiée puis appliquée à l'effondrement d'une colonne d'eau suivi d'un impact sur un mur vertical. Enfin, le modèle est étendu aux écoulements visqueux. Une nouvelle méthode de prise en compte des conditions limites de type adhérence, inspirée des frontières immergées en différences finies, a ainsi été développée. Elle autorise la simulation d'écoulements autour d'objets de formes complexes immergés dans le fluide. Le modèle est finalement appliqué à la propagation de la houle en canal et à son amortissement par une plaque horizontale immergée.

A parallel GPU compatible Lagrangian mesh free particle solver for multiphase fluid flow based on SPH scheme is developed and used to capture the interface evolution during droplet impact. Surface tension is modeled employing the multiphase scheme of Hu et al. (2006). In order to precisely simulate the wetting phenomena, a method based on the work of Šikalo et al. (2005) is jointly used with the model proposed by Afkhami et al. (2009) to ensure accurate dynamic contact angle calculations. Accurate predictions were obtained for droplet contact angle during spreading. A two-dimensional analytical model is developed as an expansion to the work of Chandra et al. (1991). Results obtain from the solver agrees well to this analytical results. Effects of memory management techniques along with a variety of task assigning algorithms on GPU are studied. GPU speedups of up to 120 times faster than a single processor CPU were obtained.

A parallel GPU compatible Lagrangian mesh free particle solver for multiphase fluid flow based on SPH scheme is developed and used to capture the interface evolution during droplet impact. Surface tension is modeled employing the multiphase scheme of Hu et al. (2006). In order to precisely simulate the wetting phenomena, a method based on the work of Šikalo et al. (2005) is jointly used with the model proposed by Afkhami et al. (2009) to ensure accurate dynamic contact angle calculations. Accurate predictions were obtained for droplet contact angle during spreading. A two-dimensional analytical model is developed as an expansion to the work of Chandra et al. (1991). Results obtain from the solver agrees well to this analytical results. Effects of memory management techniques along with a variety of task assigning algorithms on GPU are studied. GPU speedups of up to 120 times faster than a single processor CPU were obtained.

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

In this thesis, I study numerical and semi-analytical models of self-gravitating protostellar discs, with the aim of furthering our understanding of the role of disc-self gravity in planet formation. At the time of writing, the ALMA era of observational astronomy is upon us. Therefore, I place my research into this context with synthetic images of both numerical and semi-analytical models. I begin with an examination into the apparent lack of convergence, with increasing resolution, of the fragmentation boundary in Smoothed Particle Hydrodynamics (SPH) simulations of a protostellar disc. I run a suite of SPH with different numerical implementations, and find that even very similar implementations can fundamentally change the final answer. I analyse a suite of SPH simulations that fragment to form gravitationally bound objects, with the motivation of informing future population synthesis model development. I find that fragment-fragment and fragment-disc interaction dominates the orbital evolution of the system even at very early times, and any attempt to produce a population of objects from the gravitational instability process must include these interactions. Before a disc fragments, it will go through a self-gravitating phase. If the disc cools globally on a timescale such that it is balanced by heating due to gravitational stresses, the disc will be in a state of quasi-equilibrium. So long as the disc mass is sufficiently low, and spirals are sufficiently tightly wound, then angular momentum transport can be described by the local approximation, for which there is an analytical description. Using this analytical description, I develop an existing 1D model into 3D, and examine a wide range of parameter space for which disc self-gravity produces significant non-axisymmetry. Using radiative transfer calculations coupled with synthetic observations, I determine that there is a very narrow range of parameter space in which a disc will have sufficiently large gravitational stresses so as to produce detectable spirals, but the stresses not be so large as to cause the disc to fragment. By developing a simple analytical prescription for dust, I show that this region of parameter space can be broadened considerably. However, it requires grains that are large enough to become trapped by pressure maxima in the disc, so I conclude that if self-gravitating spiral arms are detected in the continuum, it is likely that at least some grain growth has taken place.

Nos últimos anos, o estudo de métodos de animação de escoamento de fluidos tem sido uma área de intensa pesquisa em Computação Gráfica. O principal objetivo desse projeto é desenvolver novas técnicas em GPGPU baseadas na arquitetura CUDA para simular o escoamento de fluidos não-newtonianos, tais como fluidos viscoplásticos e viscoelásticos. Ao invés dos tradicionais métodos com malha diferenças finitas e elementos finitos, essas técnicas são baseadas em uma discretização lagrangeana das equações de governo desses fluidos através do método sem malha conhecido como SPH (Smoothed Particle Hydrodynamics) / I n recent years, the study of methods of animating fluid flow has been an area of intense research in Computer Graphics. The main objective of this project is to develop new techniques based on the CUDA GPGPU architecture to simulate the flow of non-Newtonian fluids, such as viscoelastic and viscoplastic fluids. Instead of traditional methods with mesh - finite differences and finite elements, these techniques are based on a Lagrangian discretization of the governing equations of these fluids through the mesh free method known as SPH (Smoothed Particle Hydrodynamics)