As análises de erros são conduzidas antes de qualquer projeto a ser desenvolvido. A necessidade do conhecimento do comportamento do erro numérico em malhas estruturadas e não-estruturadas surge com o aumento do uso destas malhas nos métodos de discretização. Desta forma, o objetivo deste trabalho foi criar uma metodologia para analisar os erros de discretização gerados através do truncamento na Série de Taylor, aplicados às equações de Poisson e de Advecção-Difusão estacionárias uni e bidimensionais, utilizando-se o Método de Volumes Finitos em malhas do tipo Voronoi. A escolha dessas equações se dá devido a sua grande utilização em testes de novos modelos matemáticos e função de interpolação. Foram usados os esquemas Central Difference Scheme (CDS) e Upwind Difference Scheme(UDS) nos termos advectivos. Verificou-se a influência do tipo de condição de contorno e a posição do ponto gerador do volume na solução numérica. Os resultados analíticos foram confrontados com resultados experimentais para dois tipos de malhas de Voronoi, uma malha cartesiana e outra triangular comprovando a influência da forma do volume finito na solução numérica obtida. Foi percebido no estudo que a discretização usando o esquema CDS tem erros menores do que a discretização usando o esquema UDS conforme literatura. Também se percebe a diferença nos erros em volumes vizinhos nas malhas triangulares o que faz com que não se tenha uma uniformidade nos gráficos dos erros estudados. Percebeu-se que as malhas cartesianas com nó no centróide do volume tem menor erro de discretização do que malhas triangulares. Mas o uso deste tipo de malha depende da geometria do problema estudado / The analyses of errors are conducted before any project to be developed. The necessity of studying the behavior of the numerical error on structured and unstructured grids comes up with the increasing use of these methods of discretization meshes. Thus, the objective was to create a methodology to analyze the errors generated by discretization of the truncation in the Taylor series, applied to the equations of Poisson and Advection-Diffusion stationary and uni and bi-dimensional, using the Finite Volume Method on Voronoi mesh. The choice of these equations is due to its wide use in testing new mathematical models and interpolation function. The schemes used were the Central Difference Scheme (CDS) and the Upwind Difference Scheme (UDS) in the advective terms. There was the influence of boundary condition and position of the generator in the numerical solution of the volume. The analytical results were compared with experimental results for two types of Voronoi meshes, a Cartesian mesh and a triangular shape showing the influence of finite volume in the numerical solution obtained. It was perceived that the discretization in the study using the CDS scheme has smaller errors than the discretization scheme using the UDS as literature. Also notice the difference in the errors in neighboring volumes in triangular meshes which means that there has been no uniformity in the graphs of errors studied. It was noticed that the Cartesian meshes with node at the centroid of the volume is smaller than discretization error triangular meshes. But using this type of meshes depends on the geometry of the problem studied

In this thesis we address several analytical and numerical problems related with the general relativistic study of black hole space-times and boson stars. We have developed a new centered finite volume method based on the flux splitting approach. The techniques for dealing with the singularity, steep gradients and apparent horizon location, are studied in the context of a single Schwarzschild black hole, in both spherically symmetric and full 3D simulations. We present an extended study of gauge instabilities related with a class of singularity avoiding slicing conditions and show that, contrary to previous claims, these instabilities are not generic for evolved gauge conditions. We developed an alternative to the current space coordinate conditions, based on a generalized Almost Killing Equation. We performed a general relativistic study regarding the long term stability of Mixed-State Boson Stars configurations and showed that they are suitable candidates for dark matter models. / En esta tesis abordamos varios problemas analíticos y numéricos relacionados con el estudio de agujeros negros relativistas y modelos de materia oscura. Hemos desarrollado un nuevo método de volúmenes finitos centrados basado en el enfoque de la división de flujo. Discutimos las técnicas para tratar con la singularidad, los gradientes abruptos y la localización del horizonte aparente en el contexto de un solo agujero negro de Schwarzschild, en simulaciones tanto con simetría esférica como completamente tridimensionales. Hemos extendido el estudio de una familia de condiciones de foliaciones evitadoras de singularidad y mostrado que ciertas inestabilidades no son genéricas para condiciones de gauge dinámicas. Desarrollamos una alternativa a las prescripciones actuales basada en una Almost Killing Equation generalizada. Hemos realizado también un estudio con respecto a la estabilidad a largo plazo de configuraciones de Mixed-State Boson Stars, el cual sugiere que estas podrían ser candidatas apropiadas para modelos de materia oscura.

The linear elastodynamic two-dimensional anti-plane stress problem, where deformations occur in only one direction is considered for one sided non-planar faults. Fault dynamics are modeled using purely velocity dependent friction laws, and applied on boundaries with complex geometry. Summation-by-parts operators and energy estimates are used to couple a high-order finite difference method with an unstructured finite volume method. The unstructured finite volume method is used near the fault and the high-order finite difference method further away from the fault where no complex geometry is present. Boundary conditions are imposed weakly on characteristic form using the simultaneous approximation term technique, allowing explicit time integration to be used. Numerical computations are performed to verify the accuracy and time stability, of the method.

As análises de erros são conduzidas antes de qualquer projeto a ser desenvolvido. A necessidade do conhecimento do comportamento do erro numérico em malhas estruturadas e não-estruturadas surge com o aumento do uso destas malhas nos métodos de discretização. Desta forma, o objetivo deste trabalho foi criar uma metodologia para analisar os erros de discretização gerados através do truncamento na Série de Taylor, aplicados às equações de Poisson e de Advecção-Difusão estacionárias uni e bidimensionais, utilizando-se o Método de Volumes Finitos em malhas do tipo Voronoi. A escolha dessas equações se dá devido a sua grande utilização em testes de novos modelos matemáticos e função de interpolação. Foram usados os esquemas Central Difference Scheme (CDS) e Upwind Difference Scheme(UDS) nos termos advectivos. Verificou-se a influência do tipo de condição de contorno e a posição do ponto gerador do volume na solução numérica. Os resultados analíticos foram confrontados com resultados experimentais para dois tipos de malhas de Voronoi, uma malha cartesiana e outra triangular comprovando a influência da forma do volume finito na solução numérica obtida. Foi percebido no estudo que a discretização usando o esquema CDS tem erros menores do que a discretização usando o esquema UDS conforme literatura. Também se percebe a diferença nos erros em volumes vizinhos nas malhas triangulares o que faz com que não se tenha uma uniformidade nos gráficos dos erros estudados. Percebeu-se que as malhas cartesianas com nó no centróide do volume tem menor erro de discretização do que malhas triangulares. Mas o uso deste tipo de malha depende da geometria do problema estudado / The analyses of errors are conducted before any project to be developed. The necessity of studying the behavior of the numerical error on structured and unstructured grids comes up with the increasing use of these methods of discretization meshes. Thus, the objective was to create a methodology to analyze the errors generated by discretization of the truncation in the Taylor series, applied to the equations of Poisson and Advection-Diffusion stationary and uni and bi-dimensional, using the Finite Volume Method on Voronoi mesh. The choice of these equations is due to its wide use in testing new mathematical models and interpolation function. The schemes used were the Central Difference Scheme (CDS) and the Upwind Difference Scheme (UDS) in the advective terms. There was the influence of boundary condition and position of the generator in the numerical solution of the volume. The analytical results were compared with experimental results for two types of Voronoi meshes, a Cartesian mesh and a triangular shape showing the influence of finite volume in the numerical solution obtained. It was perceived that the discretization in the study using the CDS scheme has smaller errors than the discretization scheme using the UDS as literature. Also notice the difference in the errors in neighboring volumes in triangular meshes which means that there has been no uniformity in the graphs of errors studied. It was noticed that the Cartesian meshes with node at the centroid of the volume is smaller than discretization error triangular meshes. But using this type of meshes depends on the geometry of the problem studied

Orientador: Jose Ricardo Figueiredo / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecanica / Made available in DSpace on 2018-08-14T21:37:27Z (GMT). No. of bitstreams: 1 Oliveira_KeteriPolianeMoraesde_D.pdf: 4537142 bytes, checksum: 86df7a67611ff0f668587669a2e1ed3d (MD5) Previous issue date: 2009 / Resumo: Este trabalho apresenta uma comparação das precisões das soluções numéricas entre as malhas fundamentais (deslocada tipo MAC, semi-deslocada, co-localizada nos vértices e co-localizada nos centros) para equações de Navier-Stokes em escoamentos incompressíveis com variáveis primitivas em regime permanente. Emprega-se os esquemas central, exponencial e UNIFAES (Unified Finite Approaches Exponential-type Scheme) para discretização dos termos advectivos e difusivos das equações de Navier-Stokes. As equações de quantidade de movimento são integradas explicitamente após a solução de uma equação de Poisson para o campo de pressão. Foram resolvidos os problemas bidimensionais da cavidade com a velocidade da tampa uniforme, da cavidade hidrodinâmica quadrada na forma regularizada sem descontinuidade na velocidade da tampa e do degrau. É empregada a metodologia da extrapolação de Richardson para estimar a solução correta nos casos que não possuem solução de referência precisa; para baixos números de Reynolds, os resultados extrapolados no caso do problema da cavidade com velocidade da tampa uniforme coincidem satisfatoriamente com os valores de referência encontrados na literatura. Para o problema da cavidade com a velocidade da tampa uniforme, a malha deslocada (MAC) e a malha co-localizada nos centros apresentam os melhores resultados, seguidas da malha co-localizada nos vértices e por ultimo a malha semi-deslocada, cuja acuidade é afetada pelas descontinuidades nos cantos. De fato, para o problema da cavidade hidrodinâmica quadrada na forma regularizada a malha semi-deslocada apresenta freqüentemente os melhores resultados; em seguida a malha deslocada (MAC) e a malha co-localizada nos centros apresentam resultados comparáveis, e a malha co-localizada nos vértices mostra os piores resultados. Para o degrau foi empregada apenas a malha semi-deslocada. Em geral, o esquema UNIFAES provou-se estável mesmo para os valores mais altos do número de Reynolds e mais acurado que os esquemas central e exponencial. / Abstract: This work presents a comparison of the accuracy of the numerical solutions of the fundamental meshes (MAC staggered mesh, semi-staggered mesh, vertex-centered mesh, cell-centered mesh) for the incompressible Navier-Stokes equations in primitive variables. It employs the central differencing, the exponential scheme and UNIFAES (Unified Finite Approaches Exponentialtype Scheme) for discretization of the advective and diffusive terms of Navier-Stokes equations. The momentum equations are explicitly integrated after the solution of a Poisson pressure equation. The 2D uniform velocity driven lid cavity, the 2D lid-driven cavity in the regularized form without corner discontinuities, and the backward facing step test problems are employed. Richardson extrapolation is employed to estimate the correct solution in cases which have no precise reference solution, for low Reynolds numbers, the extrapolated results of the uniform lid velocity cavity problem coincide well with the reference values found in literature. For the 2D uniform lid velocity driven cavity test problem, the MAC staggered and the cell-centered collocated meshes show the best results, followed by the vertex-centered mesh and at last the semi-staggered mesh, whose accuracy is affected by the corner discontinuities. Indeed, for the 2D lid-driven cavity in the regularized form test problem, the semi-staggered mesh often presents the best results, and then the MAC staggered mesh and the cell-centered collocated mesh presents comparable results, and the vertex-centered mesh shows the worst results. For the step test problem only the semi-staggered mesh was employed. In general, the UNIFAES proved to be stable even at higher values of Reynolds number; and more accurate than the central differencing and then exponential. / Doutorado / Termica e Fluidos / Doutor em Engenharia Mecânica

Submitted by Geandra Rodrigues (geandrar@gmail.com) on 2018-10-24T12:40:05Z No. of bitstreams: 1 filipefernandesdepaula.pdf: 19865574 bytes, checksum: dfebb62a2a39cd7c70ab5c775d5441ce (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2018-11-23T12:10:30Z (GMT) No. of bitstreams: 1 filipefernandesdepaula.pdf: 19865574 bytes, checksum: dfebb62a2a39cd7c70ab5c775d5441ce (MD5) / Made available in DSpace on 2018-11-23T12:10:30Z (GMT). No. of bitstreams: 1 filipefernandesdepaula.pdf: 19865574 bytes, checksum: dfebb62a2a39cd7c70ab5c775d5441ce (MD5) Previous issue date: 2018-09-12 / O desenvolvimento de técnicas adequadas para extração eficiente de óleo de reservatórios de petróleo passa pela simulação precisa de tais fenômenos, que é alcançada através do estudo de modelos matemáticos e métodos computacionais robustos, eficientes e precisos. Neste contexto, este trabalho visa o estudo de métodos numéricos para a simulação de escoamentos bifásicos em meios porosos heterogêneos. Para tanto, propomos uma abordagem numérica do tipo staggered para estes modelos, que se baseia na aproximação de forma desacoplada dos sistemas de equações diferenciais parciais referentes aos problemas de Darcy e da saturação das fases. Dessa forma, podem ser empregados métodos numéricos específicos para cada sistema, que melhor se adequem às suas caractrerísticas. Assim, propomos o estudo de métodos de elementos finitos mistos, estáveis e estabilizados, clássicos e híbridos e localmente conservativos para o cômputo da velocidade da mistura e de um método de volumes finitos não-oscilatório de alta ordem, baseado em esquemas centrais, para a equação hiperbólica não-linear que governa o transporte da saturação das fases. Resultados numéricos comprovam a flexibilidade, a taxa de convergência e o custo computacional dos métodos adotados, além de demonstrar a eficácia dos métodos quando aplicados a simulação de problemas associados a extração de petróleo em cenários fortemente heterogêneos. / The development of techniques for efficient oil extraction from reservoirs passes through the simulation of such phenomena, which is achieved by the study of mathematical models and robust, precise and efficient computational methods. This dissertation studies methods for the simulation of two-phase flows in heterogeneous porous media. To do so, we propose a “staggered” numerical approach for the numerical methods, that is based on the approximation of uncoupled systems of differential equations related to Darcy’s problems and saturation of the phases. Then, appropriate methods for each system, that best suit its characteristics can be applied. Therefore, we propose studying locally conservative finite element methods, stable and stabilized, classical and hybrid to approximate the velocity field and a non-oscillatory high order finite volume method, based on central schemes, to approximate the non-linear hyperbolic equation that governs the transport of phases. Numerical results attest to the flexibility, convergence rate and computational cost of the adopted methods, and demonstrate the effectiveness of such methods when applied to oil extraction in various heterogeneous porous media scenarios.

Nous considérons deux modèles EDP-EDO couplés: un pour modéliser des goulots d’étranglementmobiles et l’autre pour décrire la distribution du trafic sur une bretelle d’accès. Le premier modèle a étéintroduit pour décrire le mouvement d’un bus, qui roule à une vitesse inférieure à celle des autresvoitures, en réduisant la capacité de la route et générant ainsi un goulot d’étranglement. Une loi deconservation scalaire avec une contrainte mobile sur le flux décrit le trafic et une EDO décrit latrajectoire du bus. Nous présentons un résultat d’existence des solutions du modèle et nous proposonsune méthode numérique “front/capturing" et une méthode basée sur une technique de reconstructiondes ondes de chocs. Dans la deuxième partie, nous introduisons un nouveau modèle macroscopique dejonction pour les bretelles d’autoroute. Nous considérons le modèle de trafic de Lighthill-Whitham-Richards sur une jonction composée d’une voie principale, une bretelle d’accès et une bretelle de sortie,toutes reliées par un nœud. Une loi de conservation scalaire décrit l’évolution de la densité des véhiculessur la voie principale et une EDO décrit l’évolution de la longueur de la file d’attente sur la bretelled’accès. La définition de la solution du problème de Riemann à la jonction est basée sur la résolutiond’un problème d’optimisation linéaire et sur l’utilisation d’un paramètre de priorité. Ensuite, ce modèleest étendu aux réseaux et discrétisé en utilisant un schéma de Godunov qui prend en compte les effetsde la bretelle d’accès. Enfin, nous présentons un modèle d’optimisation de la circulation sur les ronds points. / In this thesis we consider two coupled PDE-ODE models. One to model moving bottlenecks and theother one to describe traffic flow at junctions. First, we consider a strongly coupled PDE-ODE systemthat describes the influence of a slow and large vehicle on road traffic. The model consists of a scalarconservation law accounting for the main traffic evolution, while the trajectory of the slower vehicle isgiven by an ODE depending on the downstream traffic density. The moving constraint is expressed byan inequality on the flux, which models the bottleneck created in the road by the presence of the slowerDépôt de thèse – Donnéescomplémentairesvehicle. We prove the existence of solutions to the Cauchy problem for initial data of bounded variation.Moreover, two numerical schemes are proposed. The first one is a finite volume algorithm that uses alocally nonuniform moving mesh. The second one uses a reconstruction technique to display thebehavior of the vehicle. Next, we consider the Lighthill-Whitham-Richards traffic flow model on ajunction composed by one mainline, an onramp and an offramp, which are connected by a node. Theonramp dynamics is modeled using an ordinary differential equation describing the evolution of thequeue length. The definition of the solution of the Riemann problem at the junction is based on anoptimization problem and the use of a right of way parameter. The numerical approximation is carriedout using a Godunov scheme, modified to take into account the effects of the onramp buffer. Aftersuitable modification, the model is used to solve an optimal control problem on roundabouts. Two costfunctionals are numerically optimized with respect to the right of way parameter.

Numerical simulations of the complex flows of complex (viscoelastic) fluids are investigated. The primary fluid investigated in this thesis is human blood, a complex fluid which can be modelled via viscoelastic constitutive models. The most commonly used constitutive models for viscoelastic fluids include the OldroydB, Giesekus, Johnson-Segalman, Finitely Extensible Non-Linear Elastic (FENE), Phan-Thein-Tanner (PTT) models etc. Our Numerical approach is based on the finite volume methods implemented on the OpenFOAM platform. We employ the Giesekus, Oldroyd-B, and Generalized Oldroyd-B viscoelastic constitutive models in this thesis, depending on the underlying context. Numerical validation of our results is conducted via the most used benchmark flow problems for viscoelastic fluid flow. The robust and efficient numerical methodologies are then deployed to investigate the flow characteristics, and hence illustrate various novel behavior, for blood flow in stenosed and bifurcated arteries. The present work took advantage of the availability of a reasonable set of viscoelastic constitutive model solvers within OpenFOAM, specifically the viscoelasticFluidFoam solver which we modified and developed to suit our focused needs for blood flow computations. The modified computational algorithms were successfully validated against well-known benchmark flow problems in the literature. Noting that the Giesekus viscoelastic constitutive model is a generalization of both the Oldroyd-B and Generalized Oldroyd-B models, the validation of results is carried out via the Giesekus model enabling us to develop a general-purpose code capable of simulating several viscoelastic constitutive models. The main results were otherwise presented for the Oldroyd-B and Generalized Oldroyd-B models as these are the most applicable to blood flow modelling. The results demonstrate that the velocity spurt through the stenosis is directly proportional to the constriction caused by the stenosis. The higher the blockage from the constriction, the higher the corresponding velocity spurt through the constriction. This velocity behavior, as the constriction blockage increases, correspondingly increase the wall shear stresses. High wall shear stresses significantly increase the possibility of rupture of the stenosis/blockage. This can lead to catastrophic consequences in the usual case where the stenosis is caused by tumor growth.

Les méthodes hybrides Moments/PDF sont bien adaptées pour la description des écoulements diphasiques turbulents, polydispersés, hors équilibre thermodynamique. Ces méthodes permettent d'avoir une description assez fine de la polydispersion, de la convection et des termes sources non-linéaires. Cependant, les approximations issues de telles simulations sont bruitées ce qui, dans certaines situations, occasionne un biais. L'approche alternative étudiée dans ce travail consiste à coupler une description Eulerienne des moments avec une description stochastique Lagrangienne à l'intérieur de la phase dispersée, permettant de réduire l'erreur statistique et d'éliminer le biais. La mise en oeuvre de cette méthode nécessite le développement de schémas numériques robustes. Les approches proposées sont basées sur l'utilisation simultanée des techniques de relaxation et de décentrement, et permettent d'obtenir des approximations stables des solutions instationnaires du système d'équations aux dérivées partielles, avec des données peu régulières provenant du couplage avec le modèle stochastique. Une comparaison des résultats de la méthode hybride Moments-Moments/PDF avec ceux issus de la méthode hybride "classique'' est présentée en termes d'analyse des erreurs numériques sur un cas de jet co-courant gaz-particules. / Hybrid Moments/PDF methods have shown to be well suitable for the description of polydispersed turbulent two-phase flows in non-equilibrium which are encountered in some industrial situations involving chemical reactions, combustion or sprays. hey allow to obtain a fine enough physical description of the polydispersity, non-linear source terms and convection phenomena. However, their approximations are noised with the statistical error, which in several situations may be a source of a bias. An alternative hybrid Moments-Moments/PDF approach examined in this work consists in coupling the Moments and the PDF descriptions, within the description of the dispersed phase itself. This hybrid method could reduce the statistical error and remove the bias. However, such a coupling is not straightforward in practice and requires the development of accurate and stable numerical schemes. The approaches introduced in this work rely on the combined use of the upwinding and relaxation-type techniques. They allow to obtain stable unsteady approximations for a system of partial differential equations containing non-smooth external data which are provided by the PDF part of the model. A comparison of the results obtained using the present method with those of the ``classical'' hybrid approach is presented in terms of the numerical errors for a case of a co-current gas-particle wall jet.

O objetivo desse trabalho consiste no estudo de métodos de identificação de parâmetros em problemas envolvendo a equação de advecção-difusão 2D. Essa equação é resolvida utilizando o método dos volumes finitos, sendo empregada métodos de reconstrução de alta ordem em malhas não-estruturadas de triângulos para calcular os fluxos nas faces dos volumes de controle. Como ferramenta de busca dos parâmetros é empregada a técnica baseadas em gradientes, sendo os mesmos calculados utilizando processos baseados em métodos adjuntos. / The aim of this work concern to study parameter identification methods on problems involving the advection-diffusion equation in two dimensions. This equation is solved employing the finite volume methods, and high-order reconstruction methods, on triangle unstructured meshes to solve the fluxes across the faces of control volumes. As parameter searching tool is employed technicals based on gradients. The gradients are solved using processes based on adjoint methods.