Recent development in Graphics Processing Units (GPUs) has enabled inexpensive high-performance computing for general-purpose applications. The K-Nearest Neighbors problem is widely used in applications ranging from classification to gathering of photons in the Photon Mapping algorithm. Using the euclidean distance measure when gathering photons can cause false bleeding of colors between surfaces. Ellipsoidical search boundaries for photon gathering are shown to reduce artifacts due to this false bleeding. Shifted Sorting has been found to yield high performance on GPUs while simultaneously retaining a high approximation rate. This study presents an algorithm for approximatively solving the K-Nearest Neighbors problem modified to use a distance measure creating an ellipsoidical search boundary. The ellipsoidical search boundary is used to alleviate the issue of false bleeding of colors between surfaces in Photon Mapping. The Approximative K-Nearest Neighbors algorithm presented is a modification of the Shifted Sorting algorithm. The algorithm is found to be highly parallelizable and performs to a factor of 86% queries processed per millisecond compared to a reference implementation using spherical search boundaries implied by the euclidean distance. The rate of compression from spherical to ellipsoidical search boundary is appropriately chosen in the range 3.0 to 7.0. The algorithm is found to scale well in respect to increases in both number of data points and number of query points. / Grafikprocessorer (GPU-er) har på senare tid möjliggjort högprestandaberäkningar till låga kostnader för generella applikationer. K-Nearest Neighbors problemet har vida applikationsområden, från klassifikation inom maskininlärning till insamlande av fotoner i Photon Mapping för rendering av tredimensionella scener. Användning av euklidiska avstånd vid insamling av fotoner kan leda till en felaktig bladning av färger mellan ytor. Ellipsoidiska sökområden vid fotoninsamling har visats reducera artefakter oraskade av denna typ av felaktiga färgutblandning. Shifted Sorting har visats ge hög prestanda på GPU-er utan att förlora kvalitet av approximationsgrad. Denna rapport undersöker hur den approximativa varianten av K-Nearest Neighborsalgoritmen med Shifted Sorting presterar på GPU-er med avståndsmåttet modifierat sådant att ett ellipsoidiskt sökområde bildas. Algoritmen används för att reduceras problemet av felaktig blanding av färg i Photon Mapping. Algoritmen visas vara mycket parallelliserbar och presterar till en grad av 86% behandlade sökpunkter per millisekund i jämförelse med en referensimplementation som använder sfäriska sökområden. Kompressionsgraden längs sökpunktens ytnormal väljs fördelaktligen till ett värde i intervallet 3,0 till 7,0. Algoritmen visas skala väl med avseende på både ökningar i antal data punkter och antal sökpunkter.

The ability of Reynolds Averaged Navier Stokes (RANS) models to predict the characteristics of a non-Body of Revolution (non-BOR) Ellipsoidal model is studied to establish the feasibility of utilizing RANS as a non-BOR concept design tool. Data unable to be obtained experimentally, such as streamwise and spanwise pressure gradients and yaw turn boundary layer characteristics, are also established. A range of conditions are studied including ahead, pitched up, steady 10 and 15 degree yaw turns, and unsteady 10 and 15 degree yaw turns. Simulation results show good agreement for ahead and pitched forces and moments. Straight ahead skin friction values also showed good agreement, providing even improved agreement over an LES model which utilized wall functions. Yaw turn conditions also showed good agreement for roll angles up to 10 degrees. Steady maneuvering forces and moments showed good agreement up to 10 degrees roll and separation calculations also showed good agreement up to 10 degrees roll. Unsteady maneuvering characteristics showed mixed results, with the normal force and pitching moment trends generally agreeing with experimental data, whereas the unsteady rolling moment did not tend to follow experimental trends. Two primary conditions, the change in curvature between the mid-body and elliptical ends and the accuracy of modeling of 3D flows with RANS, are discussed as sources of discrepancies between the experimental data and steady simulations greater than 10 degrees roll and unsteady rolling simulations. / Master of Science

A drag study was conducted on an oblate ellipsoid body in the Virginia Tech Stability Wind Tunnel. Two-dimensional wake surveys were taken with a seven-hole probe and an integral momentum method was applied to the results to calculate the drag on the body. Several different model configurations were tested; these included the model oriented at a 0° and 10° angle of attack with respect to the oncoming flow. For both angles, the model was tested with and without flow trip strips. At the 0° angle of attack orientation, data were taken at a speed of 44 m/s. Data with the model at a 10° angle of attack were taken at 44 m/s and 16 m/s. The high speed flow corresponded to a length-based Reynolds number of about 4.3 million; the low speed flow gave a Reynolds number of about 1.6 million. The results indicated that the length-squared drag coefficients ranged from around 0.0026 for the 0° angle of attack test cases and 0.0035 for the 10° angle of attack test cases. The 10° angle of attack cases had higher drag due to the increase in the frontal profile area of the model and the addition of induced drag. The flow trip strips appeared to have a tiny effect on the drag; a slight increase in drag coefficient was seen by their application but it was not outside of the uncertainty in the calculation. At the lower speed, uncertainties in the calculation were so high that the drag results could not be considered with much confidence, but the drag coefficient did decrease from the higher Reynolds number cases. Uncertainty in the drag calculations derived primarily from spatial fluctuations of the mean velocity and total pressure in the wake profile; uncertainty was estimated to be about 16% or less for the 44 m/s test cases. / Master of Science

The most common equipment used for attitude determination in small satellites are magnetometers. However, using magnetometers gives rise to many challenges. One of these challenges is the calibration of the magnetometer. Magnetometer calibration takes many factors into account. There are external and internal factors. External factors can be the satellite itself. Satellites are built of many complex subsystems. These subsystems can produce magnetic disturbances and affect the measurements taken by the magnetometer ,which also affects the attitude determination of the satellite. Internal factors are nonorthogonalityand scale factors. In this project, we aim to test different calibration methods and compare the results. Another objective is to provide a complete procedure for a calibration of the magnetometer using the Helmholtz coils. The comparison of the results with other methods can help with the decision of which should be used to calibrate the magnetometer onboard the satellite for future calibrations for MIST satellites. / En av de mest vanliga verktyg för attitydbestämning i små satteliter är magnetometer, men att använda magnetometer kan leda till många utmaningar, en av de kalibrering av magnetometern. Magnetometer kalibrering är beroende av många faktorer. Det finns inre och yttre faktorer. Yttre faktorer kan vara själva satelliten. Del system som bildar satelliten kan påverka mätningar i magnetometern och då påverkar attitydbestämning av hela satelliten. Inre faktorer är icke ortogonalitet och skalära faktorer. I det här projektet vi ska testa olika kalibrerings metoder och jämföra resultaten. Ett annat mål är att bygga en komplett procedur för att kalibrera magnetometer med hjälp av Helmholtz spolar. Jämförelsen och resultaten från kalibreringen visar hur det är möjligt att kalibrnetometer som är integrerad i satelliten för kommande kalibreringar i MIST. / Kandidatexjobb i elektroteknik 2022, KTH, Stockholm

Investigation of momentum and heat transfer between the fluid and solid phase is critical to the study of fluid-particle systems. Dense suspensions are characterized by the solid fraction (ratio of solid volume to total volume), the particle Reynolds number, and the shape of the particle. The behavior of non-spherical particles deviates considerably from spherical particle shapes which have been studied extensively in the literature. Momentum transfer, to first-order, is driven by drag forces experienced by the particles in suspension, followed by lift and lateral forces, and also through the transmission of fluid torque to the particles. The subject of this thesis is a family of prolate ellipsoidal particle geometries of aspect ratios (AR) 2.5, 5.0 and 10.0 at nominal solid fractions (φ) between 0.1 and 0.3, and suspensions of cylinders of AR=0.25. The nominal particle Reynolds number (Re) is varied between 10 to 200, representative of fluidized beds. Fluid forces and heat transfer coefficients are obtained numerically by Particle Resolved Simulations (PRS) using the Immersed Boundary Method (IBM). The method enables the calculation of the interstitial flow and pressure field surrounding each particle in suspension leading to the direct integration of fluid forces acting on each particle in the suspension. A substantial outcome of the research is the development of a new drag force correlation for random suspensions of prolate ellipsoids over the full range of geometries and conditioned studied. In many practical applications, especially as the deviation from the spherical shape increases, particles are not oriented randomly to the flow direction, resulting in suspensions which have a mean preferential orientation. It is shown that the mean suspension drag varies linearly with the orientation parameter, which varies from -2.0 for particles oriented parallel to the flow direction to 1.0 for particles normal to the flow direction. This result is significant as it allows easy calculation of drag force for suspension with any preferential orientation. The heat transfer coefficient or Nusselt number is investigated for prolate ellipsoid suspensions. Significantly, two methods of calculating the heat transfer coefficient in the literature are reconciled and it is established that one asymptotes to the other. It is also established that unlike the drag force, at low Reynolds number the suspension mean heat transfer coefficient is very sensitive to the spatial distribution of particles or local-to-particle solid fractions. For the same mean solid fraction, suspensions dominated by particle clusters or high local solid fractions can exhibit Nusselt numbers which are lower than the minimum Nusselt number imposed by pure conduction on a single particle in isolation. This results from the dominant effect of thermal wakes at low Reynolds numbers. As the Reynolds number increases, the effect of particle clusters on heat transfer becomes less consequential. For the 0.25 aspect ratio cylinder, it was found that while existing correlations under predicted the drag forces, a sinusoidal function F_(d,θ)=F_(d,θ=0°)+(F_(d,θ=90°)-F_(d,θ=0°) )sin⁡(θ) captured the variation of normalized drag with respect to inclination angle over the range 10≤Re≤300 and 0≤φ≤0.3. Further the mean ensemble drag followed F_d=F_(d,θ=0°)+1/2(F_(d,θ=90°)-F_(d,θ=0°)). It was shown that lift forces were between 20% to 80% of drag forces and could not be neglected in models of fluid-particle interaction forces. Comparing the pitching fluid torque to collision torque during an elastic collision showed that as the particle equivalent diameter, density, and collision velocities decreased, fluid torque could be of the same order of magnitude as collisional torque and it too could not be neglected from models of particle transport in suspensions. / Doctor of Philosophy / Momentum and heat exchange between the fluids (air, water…) and suspensions of solid particles plays a critical role in power generation, chemical processing plants, pharmaceuticals, in the environment, and many other applications. One of the key components in momentum exchange are the forces felt by the particles in the suspension due to the flow of the fluid around them and the amount of heat the fluid can transfer to or from the particles. The fluid forces and heat transfer depend on many factors, chief among them being the properties of the fluid (density, viscosity, thermal properties) and the properties of the particles in the suspension (size, shape, density, thermal properties, concentration). This introduces a wide range of parameters that have the potential to affect the way the fluid and particles behave and move. Experimental measurements are very difficult and expensive to conduct in these systems and computational modeling can play a key role in characterization. For accuracy, computational models have to have the correct physical laws encoded in the software. The objective of this thesis is to use very high-fidelity computer models to characterize the forces and heat transfer under different conditions to develop general formulas or correlations which can then be used in less expensive computer models. Three basic particle shapes are considered in this study, a sphere, a disk like cylindrical particles, and particles of ellipsoidal shapes. More specifically, Particle Resolved Simulations of flow through suspensions of ellipsoids with aspect ratio of 2.5, 5, 10 and cylinders with aspect ratio of 0.25 are performed. The Reynolds number range covered is [10, 200] for ellipsoids and [10, 300] for cylinders with solid fraction range of [0.1, 0.3]. New fluid drag force correlations are proposed for the ellipsoid and cylinder suspensions, respectively, and heat transfer behavior is also investigated.

This dissertation presents innovative unified approaches to understand and predict the motion between potential wells. The theoretical-computational framework, based on the tube dynamics, will reveal how the dissipative and gyroscopic forces change the phase space structure that governs the escape (or transition) from potential wells. In higher degree of freedom systems, the motion between potential wells is complicated due to the existence of multiple escape routes usually through an index-1 saddle. Thus, this dissertation firstly studies the local behavior around the index-1 saddle to establish the criteria of escape taking into account the dissipative and gyroscopic forces. In the analysis, an idealized ball rolling on a surface is selected as an example to show the linearized dynamics due to its special interests that the gyroscopic force can be easily introduced by rotating the surface. Based on the linearized dynamics, we find that the boundary of the initial conditions of a given energy for the trajectories that transit from one side of a saddle to the other is a cylinder and ellipsoid in the conservative and dissipative systems, respectively. Compared to the linear systems, it is much more challenging or sometimes impossible to get analytical solutions in the nonlinear systems. Based on the analysis of linearized dynamics, the second goal of this study is developing a bisection method to compute the transition boundary in the nonlinear system using the dynamic snap-through buckling of a buckled beam as an example. Based on the Euler-Bernoulli beam theory, a two degree of freedom Hamiltonian system can be generated via a two mode-shape truncation. The transition boundary on the Poincar'e section at the well can be obtained by the bisection method. The numerical results prove the efficiency of the bisection method and show that the amount of trajectories that escape from the potential well will be smaller if the damping of the system is increasing. Finally, we present an alternative idea to compute the transition boundary of the nonlinear system from the perspective of the invariant manifold. For the conservative systems, the transition boundary of a given energy is the invariant manifold of a periodic orbit. The process of obtaining such invariant manifold compromises two parts, including the computation of the periodic orbit by solving a proper boundary-value problem (BVP) and the globalization of the manifold. For the dissipative systems, however, the transition boundary of a given energy becomes the invariant manifold of an index-1 saddle. We present a BVP approach using the small initial sphere in the stable subspace of the linearized system at one end and the energy at the other end as the boundary conditions. By using these algorithms, we obtain the nonlinear transition tube and transition ellipsoid for the conservative and dissipative systems, respectively, which are topologically the same as the linearized dynamics. / Doctor of Philosophy / Transition or escape events are very common in daily life, such as the snap-through of plant leaves and the flipping over of umbrellas on a windy day, the capsize of ships and boats on a rough sea. Some other engineering problems related to escape, such as the collapse of arch bridges subjected to seismic load and moving trucks, and the escape and recapture of the spacecraft, are also widely known. At first glance, these problems seem to be irrelated. However, from the perspective of mechanics, they have the same physical principle which essentially can be considered as the escape from the potential wells. A more specific exemplary representative is a rolling ball on a multi-well surface where the potential energy is from gravity. The purpose of this dissertation is to develop a theoretical-computational framework to understand how a transition event can occur if a certain energy is applied to the system. For a multi-well system, the potential wells are usually connected by saddle points so that the motion between the wells generally occurs around the saddle. Thus, knowing the local behavior around the saddle plays a vital role in understanding the global motion of the nonlinear system. The first topic aims to study the linearized dynamics around the saddle. In this study, an idealized ball rolling on both stationary and rotating surfaces will be used to reveal the dynamics. The effect of the gyroscopic force induced by the rotation of the surface and the energy dissipation will be considered. In the second work, the escape dynamics will be extended to the nonlinear system applied to the snap-through of a buckled beam. Due to the nonlinear behavior existing in the system, it is hard to get the analytical solutions so that numerical algorithms are needed. In this study, a bisection method is developed to search the transition boundary. By using such method, the transition boundary on a specific Poincar'e section is obtained for both the conservative and dissipative systems. Finally, we revisit the escape dynamics in the snap-through buckling from the perspective of the invariant manifold. The treatment for the conservative and dissipative systems is different. In the conservative system, we compute the invariant manifold of a periodic orbit, while in the dissipative system we compute the invariant manifold of a saddle point. The computational process for the conservative system consists of the computation of the periodic orbit and the globalization of the corresponding manifold. In the dissipative system, the invariant manifold can be found by solving a proper boundary-value problem. Based on these algorithms, the nonlinear transition tube and transition ellipsoid in the phase space can be obtained for the conservative and dissipative systems, respectively, which are qualitatively the same as the linearized dynamics.

We will consider the scenario of the co-rotation of a fluid star (in specific, a neutron star) and a black hole. The neutron star (or primary)is assumed to have constant angular velocity. The tidal effects on the primary are investigated. First, the centrally condensed approximation is applied, where both bodies are considered as point sources. In the second treatment, the primary is treated as an incompressible and homogeneous fluid mass, which in addition to its own gravity is subject to centrifugal and Coriolis forces, derived from fluid motions. The black hole (or secondary) is treated as a rigid sphere and can be regarded as a point mass. The equilibrium figure is derived. The problem is then adapted to include vorticity and a pseudo-Newtonian potential. The coalescence of neutron star - black hole binaries and their importance to gravitational wave detection is also discussed. / Mathematical Sciences / M. Sc. (Applied Mathematics)

The problem of packing ellipsoids consists in arranging a given collection of ellipsoids within a particular set. The ellipsoids can be freely rotated and translated, and must not overlap each other. A particular case of this problem arises when the ellipsoids are balls. The problem of packing balls has been the subject of intense theoretical and empirical research. In particular, many works have tackled the problem with optimization tools. On the other hand, the problem of packing ellipsoids has received more attention only in the past few years. This problem appears in a large number of practical applications, such as the design of high-density ceramic materials, the formation and growth of crystals, the structure of liquids, crystals and glasses, the flow and compression of granular materials, the thermodynamics of liquid to crystal transition, and, in biological sciences, in the chromosome organization in human cell nuclei. In this work, we deal with the problem of packing ellipsoids within compact sets from an optimization perspective. We introduce continuous and differentiable nonlinear programming models and algorithms for packing ellipsoids in the n-dimensional space. We present two different models for the non-overlapping of ellipsoids. As these models have quadratic numbers of variables and constraints, we also propose an implicit variables models that has a linear number of variables and constraints. We also present models for the inclusion of ellipsoids within half-spaces and ellipsoids. By applying a simple multi-start strategy combined with a clever choice of starting guesses and a nonlinear programming local solver, we present illustrative numerical experiments that show the capabilities of the proposed models. / O problema de empacotamento de elipsoides consiste em arranjar uma dada coleção de elipsoides dentro de um determinado conjunto. Os elipsoides podem ser rotacionados e transladados e não podem se sobrepor. Um caso particular desse problema surge quando os elipsoides são bolas. O problema de empacotamento de bolas tem sido alvo de intensa pesquisa teórica e experimental. Em particular, muitos trabalhos têm abordado esse problema com ferramentas de otimização. O problema de empacotamento de elipsoides, por outro lado, começou a receber mais atenção apenas recentemente. Esse problema aparece em um grande número de aplicações práticas, como o projeto de materiais cerâmicos de alta densidade, na formação e crescimento de cristais, na estrutura de líquidos, cristais e vidros, no fluxo e compressão de materiais granulares e vidros, na termodinâmica e cinética da transição de líquido para cristal e em ciências biológicas, na organização de cromossomos no núcleo de células humanas. Neste trabalho, tratamos do problema de empacotamento de elipsoides dentro de conjuntos compactos do ponto de vista de otimização. Introduzimos modelos de programação não-linear contínuos e diferenciáveis e algoritmos para o empacotamento de elipsoides no espaço n-dimensional. Apresentamos dois modelos diferentes para a não-sobreposição de elipsoides. Como esses modelos têm números quadráticos de variáveis e restrições em função do número de elipsoides a serem empacotados, também propomos um modelo com variáveis implícitas que possui uma quantidade linear de variáveis e restrições. Também apresentamos modelos para a inclusão de elipsoides em semi-espaços e dentro de elipsoides. Através da aplicação de uma estratégia multi-start simples combinada com uma escolha inteligente de pontos iniciais e um resolvedor para otimização local de programas não-lineares, apresentamos experimentos numéricos que mostram as capacidades dos modelos propostos.

The problem of packing ellipsoids consists in arranging a given collection of ellipsoids within a particular set. The ellipsoids can be freely rotated and translated, and must not overlap each other. A particular case of this problem arises when the ellipsoids are balls. The problem of packing balls has been the subject of intense theoretical and empirical research. In particular, many works have tackled the problem with optimization tools. On the other hand, the problem of packing ellipsoids has received more attention only in the past few years. This problem appears in a large number of practical applications, such as the design of high-density ceramic materials, the formation and growth of crystals, the structure of liquids, crystals and glasses, the flow and compression of granular materials, the thermodynamics of liquid to crystal transition, and, in biological sciences, in the chromosome organization in human cell nuclei. In this work, we deal with the problem of packing ellipsoids within compact sets from an optimization perspective. We introduce continuous and differentiable nonlinear programming models and algorithms for packing ellipsoids in the n-dimensional space. We present two different models for the non-overlapping of ellipsoids. As these models have quadratic numbers of variables and constraints, we also propose an implicit variables models that has a linear number of variables and constraints. We also present models for the inclusion of ellipsoids within half-spaces and ellipsoids. By applying a simple multi-start strategy combined with a clever choice of starting guesses and a nonlinear programming local solver, we present illustrative numerical experiments that show the capabilities of the proposed models. / O problema de empacotamento de elipsoides consiste em arranjar uma dada coleção de elipsoides dentro de um determinado conjunto. Os elipsoides podem ser rotacionados e transladados e não podem se sobrepor. Um caso particular desse problema surge quando os elipsoides são bolas. O problema de empacotamento de bolas tem sido alvo de intensa pesquisa teórica e experimental. Em particular, muitos trabalhos têm abordado esse problema com ferramentas de otimização. O problema de empacotamento de elipsoides, por outro lado, começou a receber mais atenção apenas recentemente. Esse problema aparece em um grande número de aplicações práticas, como o projeto de materiais cerâmicos de alta densidade, na formação e crescimento de cristais, na estrutura de líquidos, cristais e vidros, no fluxo e compressão de materiais granulares e vidros, na termodinâmica e cinética da transição de líquido para cristal e em ciências biológicas, na organização de cromossomos no núcleo de células humanas. Neste trabalho, tratamos do problema de empacotamento de elipsoides dentro de conjuntos compactos do ponto de vista de otimização. Introduzimos modelos de programação não-linear contínuos e diferenciáveis e algoritmos para o empacotamento de elipsoides no espaço n-dimensional. Apresentamos dois modelos diferentes para a não-sobreposição de elipsoides. Como esses modelos têm números quadráticos de variáveis e restrições em função do número de elipsoides a serem empacotados, também propomos um modelo com variáveis implícitas que possui uma quantidade linear de variáveis e restrições. Também apresentamos modelos para a inclusão de elipsoides em semi-espaços e dentro de elipsoides. Através da aplicação de uma estratégia multi-start simples combinada com uma escolha inteligente de pontos iniciais e um resolvedor para otimização local de programas não-lineares, apresentamos experimentos numéricos que mostram as capacidades dos modelos propostos.

We present a practical algorithm for computing the minimum volume n-dimensional ellipsoid that must contain m given points a₁,..., am ∈ Rn. This convex constrained problem arises in a variety of applied computational settings, particularly in data mining and robust statistics. Its structure makes it particularly amenable to solution by interior-point methods, and it has been the subject of much theoretical complexity analysis. Here we focus on computation. We present a combined interior-point and active-set method for solving this problem. Our computational results demonstrate that our method solves very large problem instances (m = 30,000 and n = 30) to a high degree of accuracy in under 30 seconds on a personal computer. / Singapore-MIT Alliance (SMA)