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Since the 1970s, the Rubik’s Cube puzzle has intrigued millions of people. It did not take long until people tried to solve the cube as fast as possible, and started to compete. However, in recent years, a new trend of robots solving the cube has flourished. The purpose of this project was to investigate time and precision aspects of a Rubik’s cube solving robot. A four armed robot was designed and a system was built. The construction made it possible to first scan the cube using a webcam and solve the cube by letting the robot perform the movements necessary for solving. Differences in time for different scrambles was analyzed as well as the robot’s precision. Since the robot only had four arms, the results showed differences in turning the up- and downside of the cube. To get access to these sides, the robot had to perform a flip motion. Since the software used was not optimized to minimize these moves there were differences in solving time. Moreover, the robot’s precision differed depending on its setup and how fast it was programmed to go. Two different setups was compared, one being slow and having big margins between each move and the other one being more rapid. Both setups worked, although calibration was more crucial for the faster one. / Sedan 70-talet har Rubiks kub fängslat miljontals människor. Det tog inte lång tid innan kampen om att lösa kuben så snabbt som möjligt uppstod och med det även tävlingar. Under de senaste åren har däremot en ny trend kommit med robotar som löser kuben. Projektets syfte var att undersöka tids- och precisionsaspekter för en robot som löser en Rubiks kub. I det här projeketet byggdes en fyrarmad robot med uppgift att lösa kuben. Systemet skapades så att kuben först skannas med en webkamera och sedan sätts på plats i roboten som löser kuben. Skillnader i tid för olika blandningar samt robotens precision analyserades. Eftersom roboten konstruerades med fyra armar var det tydliga skillnader i tid för upp- och nersidans drag. För att få tillgång till dessa sidor var hela kuben tvungen att roteras. Eftersom mjukvaran som användes inte var optimerad för att minimera nyttjandet av dessa rörelser varierade lösningstiderna. Dessutom var det skillnader i robotens precision beroende på systemets kalibrerade hastighet. Två olika inställningar jämfördes där en var långsammare med större marginaler mellan varje drag och den andra var snabbare. Båda fungerade för att lösa kuben men kalibreringen blev mycket avgörande för den snabbare inställningen.

In gas stirred steelmaking ladles, entrainment of slag into metal and vice versa takes place. The slag entrainment has been shown to abruptly increase the mass transfer rates of refining reactions through high temperature and water modeling studies of the past. However such an effect has not been correlated with the degree of entrainment, since the latter has not been quantified in terms of operating parameters like gas injection rate and fluid properties. Much of the past works are limited to finding the critical conditions for onset of entrainment. The difficulty lies in measuring the degree of entrainment in industrial ladles or even in a water model. Mathematical modeling is also challenging due to the complexity of the multiphase phenomena. So in this thesis, a modular mathematical modeling approach is presented wherein the phenomena of slag entrainment into metal is resolved into four aspects, models developed for each and finally integrated to study its role. The individual models are (1) multiphase large eddy simulations to simulate slag entrainment in a narrow domain that receives its boundary conditions from (2) single phase RANS simulation of a full ladle, (3) a Lagrangian particle tracking method to compute the residence times of slag droplets in metal phase and (4) a kinetic model that integrates the above three models to compute mass transfer rate as a function of degree of entrainment. Mass transfer rate predictions comparable to a literature correlation were obtained. This supports the modeling approach and also the assessment of role of various system parameters on entrainment characteristics. In essence, the present work shows a systematic approach to model and study the complex multiphase phenomena. / Thesis / Doctor of Philosophy (PhD) / The entrainment of liquid slag into liquid steel in gas stirred-steelmaking ladles is known to increase the rate of refining drastically. However, there is lack of correlation between degree of entrainment and ladle operating conditions, which this thesis addresses through mathematical modeling.

A preconditioner to improve the convergence properties of Krylov subspace solvers is derived and analyzed in this work. This method is adapted to linear systems arising from a finite-element discretization of a phase-field crystal equation.

We have extended the HOLA molecular dynamics (MD) code to run slider-on-block friction experiments for Al and Cu. Both objects are allowed to evolve freely and show marked deformation despite the hardness difference. We recover realistic coefficients of friction and verify the importance of cold-welding and plastic deformations in dry sliding friction. Our first data also show a mechanism for decoupling between load and friction at high velocities. Such a mechanism can explain an increase in the coefficient of friction of metals with velocity. The study of the effects of currents on our system required the development of a suitable electrodynamic (ED) solver, as the disparity of MD and ED time scales threatened the efficiency of our code. Our first simulations combining ED and MD are presented. / text

With the advances in the computing world, computational fluid dynamics (CFD) is becoming more and more critical tool in the field of fluid dynamics. In the past few decades, a huge number of CFD models have been developed with ever improved performance. In this research a robust CFD model, called Riemann2D, is extended to model flow over a mobile bed and applied to a full scale dam break problem. Riemann2D, an object oriented hyperbolic solver that solves shallow water equations with an unstructured triangular mesh and using high resolution shock capturing methods, provides a generic framework for the solution of hyperbolic problems. The object-oriented design of Riemann2D has the flexibility to apply the model to any type of hyperbolic problem with the addition of new information and inheriting the common components from the generic part of the model. In a part of this work, this feature of Riemann2D is exploited to enhance the model capabilities to compute flow over mobile beds. This is achieved by incorporating the two dimensional version of the one dimensional non-capacity model for erodible bed hydraulics by Cao et al. (2004). A few novel and simple algorithms are included, to track the wet/dry and dry/wet fronts over abruptly varying topography and stabilize the solution while using high resolution shock capturing methods. The negative depths computed from the surface gradient by the limiters are algebraically adjusted to ensure depth positivity. The friction term contribution in the source term, that creates unphysical values near the wet/dry fronts, are resolved by the introduction of a limiting value for the friction term. The model is validated using an extensive variety of tests both on fixed and mobile beds. The results are compared with the analytical, numerical and experimental results available in the literature. The model is also tested against the actual field data of 1957 Malpasset dam break. Finally, the model is applied to simulate dam break flow of Warsak Dam in Pakistan. Remotely sensed topographic data of Warsak dam is used to improve the accuracy of the solution. The study reveals from the thorough testing and application of the model that the simulated results are in close agreement with the available analytical, numerical and experimental results. The high resolution shock capturing methods give far better results than the traditional numerical schemes. It is also concluded that the object oriented CFD model is very easy to adapt and extend without changing the generic part of the model.

On considère des méthodes de Galerkin stochastiques pour des systèmes hyperboliques faisant intervenir des données en entrée incertaines de lois de distribution connues paramétrées par des variables aléatoires. On s'intéresse à des problèmes où un choc apparaît presque sûrement en temps fini. Dans ce cas, la solution peut développer des discontinuités dans les domaines spatial et stochastique. On utilise un schéma de Volumes Finis pour la discrétisation spatiale et une projection de Galerkin basée sur une approximation polynomiale par morceaux pour la discrétisation stochastique. On propose un solveur de type Roe avec correcteur entropique pour le système de Galerkin, utilisant une technique originale pour approcher la valeur absolue de la matrice de Roe et une adaptation du correcteur entropique de Dubois et Mehlmann. La méthode proposée reste coûteuse car une discrétisation stochastique très fine est nécessaire pour représenter la solution au voisinage des discontinuités. Il est donc nécessaire de faire appel à des stratégies adaptatives. Comme les discontinuités sont localisées en espace et évoluent en temps, on propose des représentations stochastiques dépendant de l'espace et du temps. On formule cette méthodologie dans un contexte multi-résolution basé sur le concept d'arbres binaires pour décrire la discrétisation stochastique. Les étapes d'enrichissement et d'élagage adaptatifs sont réalisées en utilisant des critères d'analyse multi-résolution. Dans le cas multidimensionnel, une anisotropie de la procédure adaptative est proposée. La méthodologie est évaluée sur le système des équations d'Euler dans un tube à choc et sur l'équation de Burgers en une et deux dimensions stochastiques / This work is concerned with stochastic Galerkin methods for hyperbolic systems involving uncertain data with known distribution functions parametrized by random variables. We are interested in problems where a shock appears almost surely in finite time. In this case, the solution exhibits discontinuities in the spatial and in the stochastic domains. A Finite Volume scheme is used for the spatial discretization and a Galerkin projection based on piecewise poynomial approximation is used for the stochastic discretization. A Roe-type solver with an entropy correction is proposed for the Galerkin system, using an original technique to approximate the absolute value of the Roe matrix and an adaptation of the Dubois and Mehlman entropy corrector. Although this method deals with complex situations, it remains costly because a very fine stochastic discretization is needed to represent the solution in the vicinity of discontinuities. This fact calls for adaptive strategies. As discontinuities are localized in space and time, stochastic representations depending on space and time are proposed. This methodology is formulated in a multiresolution context based on the concept of binary trees for the stochastic discretization. The adaptive enrichment and coarsening steps are based on multiresolution analysis criteria. In the multidimensional case, an anisotropy of the adaptive procedure is proposed. The method is tested on the Euler equations in a shock tube and on the Burgers equation in one and two stochastic dimensions

Cette thèse porte sur le développement d’une méthode multigrilles parallèle visant à réduire les temps de calculs des simulations éléments finis dans le domaine de la mise en forme de pièces forgées en 3D. Ces applications utilisent une méthode implicite, caractérisées par une formulation mixte en vitesse/pression et une gestion du contact par pénalisation. Elles impliquent de grandes déformations qui rendent nécessaires des remaillages fréquents sur les maillages tétraédriques non structurés utilisés. La méthode multigrilles développée suit une approche hybride, se basant sur une construction géométrique des niveaux grossiers par déraffinement de maillage non emboîtés et sur une construction algébrique des systèmes linéaires intermédiaires et grossiers. Un comportement asymptotique quasi-linéaire et une bonne efficacité parallèle sont attendus afin de permettre la réalisation de simulations à grand nombre de degrés de liberté dans des temps plus raisonnables qu’aujourd’hui. Pour cela, l’algorithme de déraffinement de maillages est compatible avec le calcul parallèle, ainsi que les opérateurs permettant les transferts de champs entre les différents niveaux de maillages partitionnés. Les spécificités des problèmes à traiter ont mené à la sélection d'un lisseur plus complexe que ceux utilisés plus fréquemment dans la littérature. Sur la grille la plus grossière, une méthode de résolution directe est utilisée, en séquentiel comme en calcul parallèle. La méthode multigrilles est utilisée en tant que préconditionneur d’une méthode de résidu conjugué et a été intégrée au logiciel FORGE NxT et montre un comportement asymptotique et une efficacité parallèle proches de l’optimal. Le déraffinement automatique de maillages permet une compatibilité avec les remaillages fréquents et permet à la méthode multigrilles de simuler un procédé du début à la fin. Les temps de calculs sont significativement réduits, même sur des simulations avec des écoulements particuliers, sur lesquelles la méthode multigrilles ne peut être utilisée de manière optimale. Cette robustesse permet, par exemple, de réduire de 4,5 à 2,5 jours le temps de simulation d’un procédé. / A parallel multigrid method is developed to reduce large computational costs involved by the finite element simulation of 3D metal forming applications. These applications are characterized by a mixed velocity/pressure implicit formulation with a penalty formulation to enforce contact and lead to large deformations, handled by frequent remeshings of unstructured meshes of tetrahedral. The developed multigrid method follows a hybrid approach where the different levels of non-nested meshes are geometrically constructed by mesh coarsening, while the linear systems of the intermediate and coarse levels result from the algebraic approach. A close to linear asymptotical behavior is expected along with parallel efficiency in order to allow simulations with large number of degrees of freedom under reasonable computation times. These objectives lead to a parallel mesh coarsening algorithm and parallel transfer operators allowing fields transfer between the different levels of partitioned meshes. Physical specificities of metal forming applications lead to select a more complex multigrid smoother than those classically used in literature. A direct resolution method is used on the coarsest mesh, in sequential and in parallel computing. The developed multigrid method is used as a preconditioner for a Conjugate Residual algorithm within FORGE NxT software and shows an asymptotical behavior and a parallel efficiency close to optimal. The automatic mesh coarsening algorithm enables compatibility with frequent remeshings and allows the simulation of a forging process from beginning to end with the multigrid method. Computation times are significantly reduced, even on simulations with particular material flows on which the multigrid method is not optimal. This robustness allows, for instance, reducing from 4.5 to 2.5 days the computation of a forging process.

Este trabalho apresenta a aplicação de técnicas de processamento paralelo na resolução de equações algébricas oriundas do Método de Elementos Finitos aplicado ao Eletromagnetismo, nos regimes estático e harmônico. As técnicas de programação paralelas utilizadas foram OpenMP, CUDA e GPUDirect, sendo esta última para as plataformas do tipo Multi-GPU. Os métodos iterativos abordados incluem aqueles do subespaço Krylov: Gradientes Conjugados, Gradientes Biconjugados, Conjugado Residual, Gradientes Biconjugados Estabilizados, Gradientes Conjugados para equações normais (CGNE e CGNR) e Gradientes Conjugados ao Quadrado. Todas as implementações fizeram uso das bibliotecas CUSP, CUSPARSE e CUBLAS. Para problemas estáticos, os seguintes pré-condicionadores foram adotados, todos eles com implementações paralelizadas e executadas na GPU: Decomposições Incompletas LU e de Cholesky, Multigrid Algébrico, Diagonal e Inversa Aproximada. Para os problemas harmônicos, apenas os dois primeiros pré-condicionadores foram utilizados, porém na sua versão sequencial, com execução na CPU, resultando em uma implementação híbrida CPU-GPU. As ferramentas computacionais desenvolvidas foram testadas na simulação de problemas de aterramento elétrico. No caso do regime harmônico, em que o fenômeno é regido pela Equação de Onda completa com perdas e não homogênea, a formulação adotada foi aquela em dois potenciais, A-V aresta-nodal. Em todas as situações, os aplicativos desenvolvidos para GPU apresentaram speedups apreciáveis, demonstrando a potencialidade dessa tecnologia para a simulação de problemas de larga escala na Engenharia Elétrica, com excelente relação custo-benefício. / This work presents the use of parallel processing techniques in Graphics Processing Units (GPU) for the solution of algebraic equations arising from the Finite Element modeling of electromagnetic phenomena, both in steadystate and time-harmonic regime. The techniques used were parallel programming OpenMP, CUDA and GPUDirect, the latter for those platforms of type Multi-GPU. The iterative methods discussed include those of the Krylov subspace: Conjugate Gradients, Bi-conjugate Gradients, Conjugate Residual, Bi-conjugate Gradients Stabilized, Conjugate Gradients for Normal Equations (CGNE and CGNR) and Conjugate Gradients Squared. All implementations have made use of CUSP, CUSPARSE and CUBLAS libraries. For the static problems, the following pre-conditioners were adopted, all with parallelized implementations and executed on the GPU: Incomplete decompositions, both LU and Cholesky, Algebraic Multigrid, Diagonal and Approximate Inverse. For the time-harmonic varying problems, only the first two pre-conditioners were used, but in their sequential version and running in the CPU, which yielded a hybrid CPU-GPU implementation. The developed computational tools were tested in the simulation of electrical grounding systems. In the case of the harmonic regime, in which the phenomenon is governed by the driven, lossy wave equation, the formulation adopted was that in two potential, the ungauged edge A-V formulation. In all cases, the developed GPU-based tools showed considerable speedups, showing that this is a promising technology for the simulation of large-scale Electrical Engineering problems, with excellent cost-benefit.

No desenvolvimento de códigos paralelos, a biblioteca PETSc se destaca como uma ferramenta prática e útil. Com o uso desta ferramenta, este trabalho apresenta um estudo sobre resolvedores de sistemas lineares aplicados a escoamentos incompressíveis de fluidos em microescala, além de uma análise de seu comportamento em paralelo. Após um estudo dos diversos aspectos dos métodos de fronteira imersa, é apresentado um método de fronteira imersa paralelo de primeira ordem. Na sequência, é apresentada uma proposta de melhoria na precisão do método, baseada na minimização da distância entre a condição de contorno exata e aproximada, no sentido de mínimos quadrados. O desenvolvimento de uma ferramenta paralela eficiente é demonstrado na solução numérica de problemas envolvendo escoamentos incompressíveis de fluidos viscosos com fronteiras imersas / In the development of parallel codes, PETSc library has an important position as a practical and useful tool. With this tool, this work presents a study about linear system solvers applied to incompressible flow in microscale problems, furthermore an analysis of the parallel behavior of these methods is presented. After a study of several aspects of immersed boundary methods, and taking advantage of the flexibility of PETSc, a parallel first order immersed boundary method is presented. Thereafter, an improvement in the accuracy of the method is presented, based on the minimization of the distance between exact and approximated boundary conditions, in the least square sense. The development of a parallel and efficient tool is demonstrated in the numerical solution of incompressible viscous flow problems with immersed boundary