Este trabalho apresenta parte de um sistema de simulação integrado para escoamento de fluido incompressível bidimensional em malhas não estruturadas denominado UmFlow-2D. O sistema consiste de três módulos: um módulo modelador, um módulo simulador e um módulo visualizador. A parte do sistema apresentado neste trabalho é o módulo simulador. Este módulo, implementa as equações de Navier-Stokes. As equações governantes são discretizadas pelo método de diferenças finitas generalizadas e os termos convectivos pelo método semi-lagrangeano. Um método de projeção é empregado para desacoplar as componentes da velocidade e pressão. O gerenciamento da malha, não estruturada é feito pela estrutura de dados SHE. Os resultados numéricos obtidos pelo UmFlow-2D são comparados com soluções analíticas e soluções numéricas de outros trabalhos. / This work presents an integratc simulation system, called UmFlow-2D, wich aims a,t simulating two-dimensional íncompressible fluid flow using unstructed mesh. The system is divided three modules: modeling module, simulation module and visualization module. In this work we present the simulation module. The simulation module implements the Navier-Stokes equation. The governing equations are discretized by a generalized flnite dillerence method and the convective terms by semi-lagrangean method. A projection method is employed to uncouple the velocity componentes and pressure. The management at the unstructed mesh is ready using a data structure called SHE. The numérica! results are compared with analytical solutions and numerical simulations of other works.

Diploma thesis mainly deals with temperature management and calculation of temperature profile in oven by using SMD packages (PLCC, 1206) of different thermal capacitance on testing PCB. Above all shows theoretical consecution of temperature profile calculation in oven by using known mathematical method like the lumped capacitance method or finite difference method. Theoretical solution and measured values are compared. Diploma thesis also deals with fixation methods of thermocouples K type on assembly, comparison methods based on known and subexperiment, determines the deficiencies of methods. This thesis can perform as theoretical as well as experimental resource to prediction of temperature profiles of PCB´s with different assembly density.

The field of mathematical modelling for pedestrian dynamics has attracted significant scientific attention, with various models proposed from perspectives such as kinetic theory, statistical mechanics, game theory and partial differential equations. Often such investigations are seen as being a part of a new branch of study in the domain of applied physics, called sociophysics. Our study proposes three models that are tailored to specific scenarios of crowd dynamics. Our research focuses on two primary issues. The first issue is centred around pedestrians navigating through a partially dark corridor that impedes visibility, requiring the calculation of the time taken for evacuation using a Markov chain model. The second issue is posed to analyse how pedestrians move through a T-shaped junction. Such a scenario is motivated by the 2022 crowd-crush disaster took place in the Itaewon district of Seoul, Korea. We propose a lattice-gas-type model that simulates pedestrians’ movement through the grid by obeying a set of rules as well as a parabolic equation with special boundary conditions. By the means of numerical simulations, we investigate a couple of evacuation scenarios by evaluating the mean velocity of pedestrians through the dark corridor, varying both the length of the obscure region and the amount of uncertainty induced by the darkness. Additionally, we propose an agent-based-modelling and cellular automata inspired model that simulates the movement of pedestrians through a T-shaped grid, varying the initial number of pedestrians. We measure the final density and time taken to reach a steady pedestrian traffic state. Finally, we propose a parabolic equation with special boundary conditions that mimic the dynamic of the pedestrian populations in a T-junction. We solve the parabolic equation using a random walk numerical scheme and compare it with a finite difference approximation. Furthermore, we prove rigorously the convergence of the random walk scheme to a corresponding finite difference scheme approximation of the solution.

Calcium treatment is an established operation in the production of steels. Most importantly, it serves to modify detrimental inclusions in the melt for improved castability and superior product properties. Due to calcium’s low melting point and high vapour pressure, its addition to liquid steel is challenging and yields are generally low. The current standard method for addition is cored wire feeding, in which a calcium-bearing compound is injected into the melt inside a sacrificial metal tube. The depth at which the compound is released has a pronounced impact on its yield and depends on the rate at which the wire is injected. A mathematical model is constructed to predict the time after which release occurs and thus estimate injection depth. It is based on one-dimensional heat transfer and respects phase change effects, i.e., solidification of melt on the cold wire surface and subsequent melting of the wire and compound. Equations are derived using the finite difference method and the model is solved numerically using an implicit method. The model is applied to two different steel grades – one austenitic (AISI 316L), one super-duplex (SAF 2507) – and predictions are tested for the former grade in a 75-ton industrial ladle furnace. Results indicate that a decreased feeding rate (<100 m/min) leads to better calcium yield. Positive effects were also observed with downward melt stirring and a good slag coverage. Improved yield can lead to savings in process and material cost. The results also provide insights into the cored wire feeding process which may be useful for future process development, both for calcium and other additions. / Kalciumbehandling är en etablerad process vid tillverkning av stål. Framförallt används den för modifieringen av skadliga inneslutningar i smältan för att förbättra gjutbarheten och produktegenskaper. På grund av kalciums låga smältpunkt och höga ångtryck är det svårt att tillsätta det till flytande stål och utbytet är i allmänhet låg. Den nuvarande standardmetoden för tillsats är inmatning av tråd, där ett kalciumhaltig ämne matas in i smältan inuti en metallhölje som smälter bort. Djupet på vilket ämnet frigörs påverkar utbytet och beror på hur snabbt tråden matas. En matematisk modell konstrueras för att beräkna den tid efter vilken frisättning sker. Den bygger på endimensionell värmeöverföring och tar hänsyn till fasförändringseffekter, dvs. stelning av smältan på den kalla trådytan och smältning av tråd och kalciumämne efteråt. Ekvationer härleds med hjälp av finita differensmetoden och modellen löses numeriskt med hjälp av en implicit metod. Modellen tillämpas på två olika stålsorter – en austenitisk (AISI 316L) och en superduplex (SAF 2507) – och förutsägelser testas för den förstnämnda sorten i en 75-tons industriell skänkugn. Resultaten visar att en minskad matningshastighet (< 100 m/min) leder till ett ökat kalciumutbyte. Positiva effekter observerades med nedåtgående smältomrörning och god slaggtäckning. Ett förbättrat utbyte kan leda till besparingar i fråga om process- och materialkostnader. Resultaten ger också insikter i processen för inmatning av tråd, vilket kan vara användbart för framtida processutveckling, både för kalcium och andra tillsatser.

Spread options are derivative securities, which are written on the difference between the values of two underlying market variables. They are very important tools to hedge the correlation risk. American style spread options allow the holder to exercise the option at any time up to and including maturity. Although they are widely used to hedge and speculate in financial market, the valuation of the American spread option is very challenging. Because even under the classic assumptions that the underlying assets follow the log-normal distribution, the resulting spread doesn't have a distribution with a simple closed formula. In this dissertation, we investigate the American spread option pricing problem. Several approaches for the geometric Brownian motion model and the stochastic volatility model are developed. We also implement the above models and the numerical results are compared among different approaches.

Deformation of materials due to heat exposure can be a challenge in vehicle design. Sun exposure and heat build up in the vehicle cabin contribute to accelerated aging of the materials used. To understand how this process affects the durability, numerical simulation is an important tool. This thesis work investigates whether it is possible to, with weather data and knowledge of how a car is used, simulate the deformation process of the trim parts due to heat exposure from the sun. To answer this question, two simulation methods have been compared: The first method is to only look at the aging of the material over time. This method is used today at Volvo Cars, but has been found to be insufficient from a number of aspects. The second method includes the effects of the reversible deformations. This method has not previously been tested in a similar context but was considered interesting as it models more true to life. In addition, the effects of modeling the car according to a virtual customer behavior, rather than stationary, were investigated. The main simulation tools used are TAITherm and Abaqus. The results show that modeling the vehicle with an imagined user routine yields a lower mean temperature as the car is cooled down when driven. The result of the comparison between the two methods shows that inclusion of the effects of the reversible deformations contribute to a better model of the aging of the trim parts due to heat exposure from the sun. / Deformation av material till följd av hög värmelast är ofta en utmaning inom fordonsdesign. Materialet i kupén exponeras både för direkt solljus och den värme som ackumuleras i det stängda utrymmet. För att få förståelse för hur denna process påverkar hållbarheten är numerisk analys ett viktigt verktyg. Föreliggande examensarbete undersöker om det är möjligt att, med väderdata och kunskap om hur en bil används, simulera deformationsprocessen i trim- delarna som uppkommer av värmexponering från solen. För att besvara denna fråga har två simuleringsmetoder jämförts: Den första metoden är att endast titta på materialets åldrande över tiden. Den används idag på Volvo Cars, men har befunnits vara otillräcklig utifrån ett antal aspekter. Den andra metoden inkluderar effekterna av de elastiska deformationerna av materialet. Denna metod har inte tidigare prövats i ett liknande sammanhang men bedömdes intressant då den modellerar mer verklighetstroget. Dessutom undersöktes effekterna av att modellera bilen enligt ett virtuellt kundbeteen- de snarare än som stillastående. De huvudsakliga simuleringsverktygen som använts här är TAITherm och Abaqus. Resultaten visar att modellering av fordonet med en tänkt användarrutin ger en lägre medeltemperatur eftersom bilen kyls ner när den körs. Resultatet av jämförelsen mellan de två metoderna visar att inkludering av effekterna av de elastiska deformationerna bidrar till en bättre modellering av åldrandet av trimdelarna till följd av värmexponering från solen.

Le but de ce travail était la modélisation mécanique et thermique du laminage asymétrique. Celle-ci a été effectuée à l'aide de diverses méthodes. Deux approches, l'une analytique et l'autre semi-analytique, ont été utilisées pour la modélisation mécanique et deux autres, celles des différences finies et des éléments finis, pour la modélisation thermique. La première approche est celle des champs uniformes ; on a développé un modèle simple qui peut donner des ordres de grandeurs des différents paramètres du procédé. Une deuxième méthode plus complète mais plus complexe se base sur un champ de vitesse déduit d'une estimation parabolique de la forme des trajectoires. L’avantage de ce modèle est qu’il est capable de prévoir des grandeurs locales. Enfin la méthode des éléments finis a été aussi utilisée pour la modélisation mécanique. Dans un deuxième temps, l’aspect thermique du laminage asymétrique a été étudié. Pour la modélisation du procédé, nous avons alors adopté deux méthodes numériques, les différences finies et les éléments finis, du fait de la complexité de celui-ci. Le modèle thermique développé par différences finies a été ensuite intégré dans notre modèle mécanique (méthode des trajectoires). Celui-ci a donc pu être étendu pour pouvoir utiliser des lois d’écoulement qui dépendent de la température et donc pouvoir simuler le laminage asymétrique à chaud. Les prédictions du champ de température ont été confrontées aux résultats de calculs par éléments finis obtenus avec le code Abaqus. Finalement, une étude simple du développement de la texture cristallographique au cours du laminage a été réalisée. / The aim of this work was to propose a mechanical and thermal modelling of asymmetric rolling. This was achieved using various methods. Two approaches, the first one analytical and the second semi-analytical, were employed for mechanical modeling. Two others methods, the finite difference and finite element methods, were used for thermal modeling. The first analytical model is an upper bound method based on a uniform strain field; a simple model was developed that can give orders of magnitude of the various process parameters. The second model uses a more refined analytical velocity field based on the classical parabolic estimation of the material flow lines in rolling. The advantage of this model is its ability to provide local quantities. Finally, the finite element method was also used for mechanical modeling.The thermal aspects of asymmetric rolling were investigated in turn by two methods. For modeling the process, we adopted two numerical approaches, the finite difference and finite element methods, because of the complexity of the latter. The thermal model developed by finite differences was integrated into our mechanical model (flow line model). It was then extended to investigate asymmetrical hot rolling. The predictions of the temperature field were compared with results from finite element calculations obtained with the Abaqus code. Finally, a simple study of crystallographic texture development was achieved.

Ce travail est motivé par une demande croissante d'informations quantitatives sur l'évolution du littoral. Nous avons étudié deux approches pour l'analyse de la dynamique sédimentaire. Les deux techniques aboutissent à la résolution de modèles non-locaux pour le fond. L'étude mathématique a porté sur l'analyse de l'existence et l'unicité de perturbations autour des ondes progressives solutions du modèle de Fowler. Nous avons montré que les solutions constantes de l'équation de Fowler sont instables. Pour la simulation numérique de ce modèle, nous avons dans un premier temps considéré des schémas aux différences finies explicites pour lesquels nous avons obtenu des critères de stabilité numérique. Dans un second temps, nous avons utilisé une approche par splitting de sorte à pouvoir résoudre la convection, puis la diffusion et l'anti-diffusion fractionnaire de façon exacte. Ensuite, il est apparu que nous pouvions utiliser les principes de minimisation pour décrire l'évolution d'un lit érodable sous l'action de l'eau où le fond est considéré comme une structure déformable de faible rigidité s'adaptant en minimisant une certaine fonctionnelle d'énergie. Il est intéressant de constater que cette seconde approche peut être liée à la première car elle débouche aussi sur une équation de type Exner avec un terme non-local. En nous inspirant du modèle morphodynamique non-local de Fowler, nous concluons cette thèse par une application exotique au traitement de signal où nous proposons une nouvelle méthode de filtrage. / This work is motivated by a growing demand for quantitative information on the evolution of the coastline.We have studied two approaches for the analysis of sand morphodynamics.Both techniques lead to the resolution of nonlocal models for the seabottom.The mathematical study focused on the analysis of the existence and uniqueness of perturbations around the travelling-waves solutions of the Fowler model. We have shown that constant solutions of Fowler's equation are unstable.For the numerical simulation of this model, we have first considered explicit finite difference schemes for which we got numerical stability criteria. We have next used an approach by splitting method in order to solve first the convection, then the diffusion/fractional anti-diffusion exactly. We have also used minimization principles to describe the evolution of an erodible bed sheared by a fluid flow where the seabed is considered as a deformable structure with low stiffness whichadapts itself by minimizing a certain energy functional. It is interesting to note that this secondapproach can be linked to the first one because it also leads to a new Exner equation with a nonlocal term for the flux. Inspired by Fowler's morphodynamical model, we conclude this dissertation with an unexpected application to signal processing.

Cette thèse vise à étudier le couplage entre méthodes de Galerkine discontinue (DG) et méthodes de différences finies (DF) en maillages hybrides non structuré / cartésien, en vue d'applications en aéroacoustique numérique. L'idée d'une telle approche consiste à pouvoir tirer profit localement des avantages respectifs de ces méthodes, soit, en d'autres termes, à pouvoir prendre en compte la présence de géométries complexes par une méthode DG en maillage non structuré, et les zones qui en sont suffisamment éloignées par une méthode DF en maillage cartésien, moins coûteuse. Plus précisément, il s'agit de concevoir un algorithme d'hybridation de ces deux types de schémas pour l'approximation des équations d'Euler linéarisées, puis d'évaluer avec attention le comportement numérique des solutions qui en sont issues. De par le fait qu'aucun résultat théorique ne semble actuellement atteignable dans un cas général, cette étude est principalement fondée sur une démarche d'expérimentation numérique. Par ailleurs, l'intérêt d'une telle hybridation est illustré par son application à un calcul de propagation acoustique dans un cas réaliste / This thesis aims at studying coupling techniques between Discontinuous Galerkin (DG) and finite difference (FD) schemes in a non-structured / Cartesian hybrid-mesh context,in the framework of Aeroacoustics computations. The idea behind such an approach is the possibility to locally take advantage of the qualities of each method. In other words, the goal is to be able to deal with complex geometries using a DG scheme on a non-structured mesh in their neighborhood, while solving the rest of the domain using a FD scheme on a cartesian grid, in order to alleviate the needs in computational resources. More precisely, this work aims at designing an hybridization algorithm between these two types of numerical schemes, in the framework of the approximation of the solutions of the Linearized Euler Equations. Then, the numerical behaviour of hybrid solutions is cautiously evaluated. Due to the fact that no theoretical result seems achievable at the present time, this study is mainly based on numerical experiments. What's more, the interest of such an hybridization is illustrated by its application to an acoustic propagation computation in a realistic case

Aplicações industriais envolvendo escoamentos multifásicos são inúmeras, sendo que, o aprimoramento de alguns desses processos pode resultar em um grande salto tecnológico com significativo impacto econômico. O estudo numérico dessas aplicações é imprescindível, pois fornece informações precisas e mais detalhadas do que a realização de testes experimentais. Um grande desafio é o estudo numérico de escoamentos viscoelásticos multifásicos envolvendo altas taxa de elasticidade, devido às instabilidades causadas por altas tensões elásticas, grandes deformações, e até mudanças topológicas na interface. Assim, a investigação numérica desse tipo de problema exige uma formulação precisa e robusta. No presente trabalho, um novo resolvedor de escoamentos bifásicos envolvendo fluidos complexos é apresentado, com particular interesse em escoamentos com altas taxas de elasticidade. A formulação proposta é baseada no método Volume-of-fluid (VOF) para representação da interface e no algoritmo Continuum Surface Force (CSF) para o balanço de forças na interface. A curvatura e advecção da interface são calculados via métodos geométricos para garantir a precisão dos resultados. Métodos de estabilização são utilizados quando números críticos de Weissenberg (Wi) são encontrados, devido ao famoso problema do alto número de Weissenberg (HWNP). O método da projeção, combinado com um método implícito para solução da equação da quantidade de movimento, são discretizados por um esquema de diferenças finitas em uma malha deslocada. Problemas de benchmarks foram resolvidos para acessar a precisão numérica da formulação em diferentes níveis de complexidade física, tal como representação e advecção da interface, influência das forças interfaciais, e características reológicas do fluido. A fim de demonstrar a capacidade do novo resolvedor, dois problemas bifásicos transientes, envolvendo fluidos viscoelásticos, foram resolvidos: o efeito de Weissenberg e o reômetro extensional (CaBER). O efeito de Weissenberg ou rod-climbing effect consiste em um bastão que gira dentro de um recipiente com fluido viscoelástico e, devido às forças elásticas, o fluido escala o bastão. Os resultados foram comparados com dados teóricos, numéricos e experimentais, encontrados na literatura para pequenas velocidades angulares. Além disso, resultados obtidos com altas velocidades angulares (alta elasticidade) são apresentados com o modelo Oldroyd-B, em que escaladas muito elevadas foram observadas. Valores críticos da velocidade angular foram identificados, e para valores acima foi observada a ocorrência de instabilidades elásticas, originadas pela combinação de tensões elásticas, curvatura interfacial, e escoamentos secundários. Até onde sabemos, numericamente, essas instabilidades nunca foram capturadas antes. O CaBER consiste no comportamento e colapso de um filamento de fluido viscoelástico, formado entre duas placas paralelas devido às forças capilares. Esse experimento envolve consideráveis dificuldades, dentre as quais podemos destacar a grande influência das forças capilares e a diferença de escalas de comprimento no escoamento. Em grande parte dos resultados encontrados na literatura, o CaBER é resolvido por modelos simplificados em uma dimensão. Resultados obtidos foram comparados com tais resultados da literatura e com soluções teóricas, apresentando admirável precisão. / Industrial applications involving multiphase flow are numerous. The improvement of some of these processes can result in a major technological leap with significant economic impact. The numerical study of these applications is essential because it provides accurate and more detailed information than conducting experiments. A challenge is the numerical study of high viscoelastic multiphase flows due to instabilities caused by the high elastic tension, large deformations and even topological changes in the interface. Thus the numerical investigation of this problem requires a robust formulation. In this study a new two-phase solver involving complex fluids is presented, with particular interest in the solution of highly elastic flows of viscoelastic fluids. The proposed formulation is based on the volume-of-fluid method (VOF) to interface representation and continuum surface force algorithm (CSF) for the balance of forces in the interface. The curvature and interface advection are calculated via geometric methods to ensure the accuracy of the results. Stabilization methods are used when critical Weissenberg numbers are found due to the famous high Weissenberg number problem (HWNP). The projection method combined with an implicit method for the solution of the momentum equation are discretized by a finite difference scheme in a staggered grid. Benchmark test problems are solved in order to access the numerical accuracy of different levels of physical complexities, such as the dynamic of the interface and the role of fluid rheology. In order to demonstrate the ability of the new resolver, two-phase transient problems involving viscoelastic fluids have been solved, theWeissenberg effect problem and the extensional rheometer (CaBER). The Weissenberg effect problem or rod-climbing effect consists of a rod that spins inside of a container with viscoelastic fluid and due to the elastic forces the fluid climbs the rod. The results were compared with numerical and experimental data from the literature for small angular velocities. Moreover results obtained for high angular velocities are presented using the Oldroyd-B model, which showed high climbing heights. Critical values of the angular speed have been identified. For values above a critical level were observed the occurrence of elastic instabilities caused by the combination of elastic tension, interfacial curvature and secondary flows. To our knowledge, numerically these instabilities were never captured before. The CaBER consists of the behavior and collapse of a viscoelastic fluid filament formed between two parallel plates due to capillary forces. This experiment involves considerable difficulties, among which we can highlight the great influence of the capillary forces and the difference of the length scales in the flow. In much of the results found in the literature, the CaBER is solved by simplified models. The results were compared with results reported in the literature and theoretical solutions, which showed remarkable accuracy.