In scientific computing, it is very common to visualize the approximate solution obtained by a numerical PDE solver by drawing surface or contour plots of all or some components of the associated approximate solutions. These plots are used to investigate the behavior of the solution and to display important properties or characteristics of the approximate solutions. In this thesis, we consider techniques for drawing such contour plots for the solution of two and three dimensional PDEs. We first present three fast contouring algorithms in two dimensions over an underlying unstructured mesh. Unlike standard contouring algorithms, our algorithms do not require a fine structured approximation. We assume that the underlying PDE solver generates approximations at some scattered data points in the domain of interest. We then generate a piecewise cubic polynomial interpolant (PCI) which approximates the solution of a PDE at off-mesh points based on the DEI (Differential Equation Interpolant) approach. The DEI approach assumes that accurate approximations to the solution and first-order derivatives exist at a set of discrete mesh points. The extra information required to uniquely define the associated piecewise polynomial is determined based on almost satisfying the PDE at a set of collocation points. In the process of generating contour plots, the PCI is used whenever we need an accurate approximation at a point inside the domain. The direct extension of the both DEI-based interpolant and the contouring algorithm to three dimensions is also investigated. The use of the DEI-based interpolant we introduce for visualization can also be used to develop effective Adaptive Mesh Refinement (AMR) techniques and global error estimates. In particular, we introduce and investigate four AMR techniques along with a hybrid mesh refinement technique. Our interest is in investigating how well such a `generic' mesh selection strategy, based on properties of the problem alone, can perform compared with a special-purpose strategy that is designed for a specific PDE method. We also introduce an \`{a} posteriori global error estimator by introducing the solution of a companion PDE defined in terms of the associated PCI.

In scientific computing, it is very common to visualize the approximate solution obtained by a numerical PDE solver by drawing surface or contour plots of all or some components of the associated approximate solutions. These plots are used to investigate the behavior of the solution and to display important properties or characteristics of the approximate solutions. In this thesis, we consider techniques for drawing such contour plots for the solution of two and three dimensional PDEs. We first present three fast contouring algorithms in two dimensions over an underlying unstructured mesh. Unlike standard contouring algorithms, our algorithms do not require a fine structured approximation. We assume that the underlying PDE solver generates approximations at some scattered data points in the domain of interest. We then generate a piecewise cubic polynomial interpolant (PCI) which approximates the solution of a PDE at off-mesh points based on the DEI (Differential Equation Interpolant) approach. The DEI approach assumes that accurate approximations to the solution and first-order derivatives exist at a set of discrete mesh points. The extra information required to uniquely define the associated piecewise polynomial is determined based on almost satisfying the PDE at a set of collocation points. In the process of generating contour plots, the PCI is used whenever we need an accurate approximation at a point inside the domain. The direct extension of the both DEI-based interpolant and the contouring algorithm to three dimensions is also investigated. The use of the DEI-based interpolant we introduce for visualization can also be used to develop effective Adaptive Mesh Refinement (AMR) techniques and global error estimates. In particular, we introduce and investigate four AMR techniques along with a hybrid mesh refinement technique. Our interest is in investigating how well such a `generic' mesh selection strategy, based on properties of the problem alone, can perform compared with a special-purpose strategy that is designed for a specific PDE method. We also introduce an \`{a} posteriori global error estimator by introducing the solution of a companion PDE defined in terms of the associated PCI.

[pt] A simulação numérica é uma ferramenta essencial para a engenharia de reservatórios moderna, em particular no desenvolvimento de campos de óleo marítimos. A maioria das simulações de reservatórios utilizam malhas estruturadas em três dimensões, com tamanho variando de alguns milhares a dezenas de milhões de células. Algumas simulações apresentam um alto custo computacional que pode dificultar os estudos de desenvolvimento de um campo, mesmo com a alta capacidade computacional disponível hoje. Malhas de simulação não-estruturadas são uma alternativa para reduzir o tamanho dos modelos de reservatórios (e, consequentemente, o tempo de execução das simulações), sem sacrificar a qualidade dos resultados. Este trabalho utiliza malhas de Voronoi, também conhecidas como malhas de bissetores perpendiculares, uma vez que suas propriedades permitem simplificar as equações discretizadas do escoamento em comparação com outros tipos de malhas não-estruturadas. Dois passos são críticos para a criação de um modelo não-estruturado de reservatórios a partir de um modelo geológico refinado: geração da malha e transferência de escala das propriedades. A maioria dos métodos propostos para ambas as tarefas utilizam informações de simulações na malha refinada. Embora essa abordagem apresente bons resultados, pode ser muito custosa e precisa ser refeita caso haja alterações significativas nas condições de escoamento. Este trabalho discute técnicas para geração de malha e transferência de escala que não dependam de simulações na escala fina. As técnicas utilizam apenas a distribuição de propriedades de reservatórios e o posicionamento de poços, falhas e outras feições discretas. A abordagem adotada para geração da malha parte de uma disposição regular de pontos que são redistribuídos de acordo com um mapa de espaçamento previamente definido. Dois algoritmos iterativos para redistribuição desses pontos baseados em modelos físicos são propostos. Diversos critérios de espaçamento também são investigados. Dois algoritmos de transferência de escala em malhas não-estruturadas são propostos. Estes métodos se baseiam nas técnicas de Cardwell and Parsons e de renormalização para transferência de escala em malhas estruturadas. Por fim, exemplos representativos são utilizados para demonstrar as potencialidades e eficácia das estratégias propostas. / [en] Numerical simulation represents an essential tool for modern reservoir engineering, especially for the development of offshore oil fields. Most reservoir simulations are performed on three-dimensional structured grids, with a size ranging from a few thousands to tens of millions of cells. Some simulations can have a high computational cost that hinders the field development studies, even using the processing power available nowadays. Unstructured meshes are an effective alternative to reduce the size of reservoir models (and, consequently, the overall simulation time) without sacrificing the quality of the results. In this work, we adopt Voronoi meshes, also known as perpendicular bisector grids, since their properties simplify the discretized flow equations in reservoir simulations when compared to other types of unstructured meshes. Two main steps are critical to creating an unstructured reservoir model from a refined geological model: grid generation and upscaling of the reservoir properties. Most methods employed for both steps rely on information obtained from simulations using fine-scale meshes. Although this approach yields good results, it can be time-consuming and may be optimal only for the specified set of flow conditions. This work discusses the generation of unstructured grids and upscaling techniques that do not require any previous simulations. Instead, they are based only on reservoir property distributions and the location of discrete features such as wells and faults. The proposed grid generation strategy starts from a regular set of points and then redistributes them according to a previously defined spacing map. Two iterative redistribution algorithms based on physical models are presented, and several criteria for spacing maps are also investigated. Two upscaling algorithms for unstructured grids are proposed, based on the Cardwell and Parsons and renormalization techniques for structured meshes. Finally, representative examples are presented to demonstrate the capabilities and effectiveness of the proposed strategies.

Mathematical models are important tools in systems biology, since the regulatory networks in biological cells are too complicated to understand by biological experiments alone. Analytical solutions can be derived only for the simplest models and numerical simulations are necessary in most cases to evaluate the models and their properties and to compare them with measured data. This thesis focuses on the mesoscopic simulation level, which captures both, space dependent behavior by diffusion and the inherent stochasticity of cellular systems. Space is partitioned into compartments by a mesh and the number of molecules of each species in each compartment gives the state of the system. We first examine how to compute the jump coefficients for a discrete stochastic jump process on unstructured meshes from a first exit time approach guaranteeing the correct speed of diffusion. Furthermore, we analyze different methods leading to non-negative coefficients by backward analysis and derive a new method, minimizing both the error in the diffusion coefficient and in the particle distribution. The second part of this thesis investigates macromolecular crowding effects. A high percentage of the cytosol and membranes of cells are occupied by molecules. This impedes the diffusive motion and also affects the reaction rates. Most algorithms for cell simulations are either derived for a dilute medium or become computationally very expensive when applied to a crowded environment. Therefore, we develop a multiscale approach, which takes the microscopic positions of the molecules into account, while still allowing for efficient stochastic simulations on the mesoscopic level. Finally, we compare on- and off-lattice models on the microscopic level when applied to a crowded environment.

Moldagem por injeção é um dos mais importantes processos industriais para produção de produtos plásticos finos. Esse processo é dividido essencialmente em quatro estágios: plastificação, preenchimento, empacotamento e resfriamento. O escoamento de um fluido caracterizado por alta viscosidade em uma cavidade estreita é um problema tipicamente encontrado em processos de moldagem por injeção.Neste caso, o escoamento pode ser descrito por uma formulação conhecida como aproximação de Hele-Shaw. Tal formulação pode ser derivada das equações de conservação tridimensionais usando um número de suposições a respeito do polímero injetado e da geometria da cavidade do molde, juntamente com a integração e o acoplamento das equações da conservação da quantidade de movimento e da continuidade. Essa formulação, referindo às limitações da geometria do molde como sendo canais estreitos e quase sem curvatura, é comumente denominada formulação 2 1/2D. Neste trabalho, é apresentada uma técnica para a simulação da fase de preenchimento de um processo de moldagem por injeção, usando essa formulação 2 1/2D, com um método de volumes finitos e malhas não estruturadas. O modelo de Cross modificado com dependência da temperatura de Arrhenius é empregado para descrever a viscosidade do polímero fundido. O campo de distribuição de temperatura é tridimensional e é resolvido usando um esquema semi-Lagrangeano baseado em volumes finitos. As malhas não estruturadas utilizadas são geradas por triangulação de Delaunay e o método numérico implementado usa a estrutura de dados topológica SHE - Singular Handle Edge, que é capaz de lidar com condições de contorno e singularidades, aspectos comumente encontrados em simulações numéricas de escoamento de fluidos. / Injection molding is one of the most important industrial processes for the manufacturing of thin plastic products. This process can be divided into four stages: plastic melting, filling, packing and cooling phases. The flow of a fluid characterized by high viscosity in a narrow gap is a problem typically found in injection molding processes. In this case, the flow can be described by a formulation known as Hele-Shaw approach. Such formulation can be btained from the three-dimensional conservation equation using a number of assumptions regarding the injected polymer and the geometry of the mold, together with the integration and the coupling of the momentum and continuity equations. This approach, referring to limitations of the mould geometry to narrow, weakly curved channels, is usually called 2 1/2D approach. In this work a technique for the simulation of the filling stage of the injection molding process, using this 2 1/2D approach, with a finite volume method and unstructured meshes, is presented. The modified-Cross model with Arrhenius temperature dependence is employed to describe the viscosity of the melt. The temperature field is 3D and it is solved using a semi-Lagrangian scheme based on the finite volume method. The employed unstructured meshes are generated by Delaunay triangulation and the implemented numerical method uses the topological data structure SHE - Singular Handle Edge, capable to deal with boundary conditions and singularities, aspects commonly found in numerical simulation of fluid flow.

Moldagem por injeção é um dos mais importantes processos industriais para produção de produtos plásticos finos. Esse processo é dividido essencialmente em quatro estágios: plastificação, preenchimento, empacotamento e resfriamento. O escoamento de um fluido caracterizado por alta viscosidade em uma cavidade estreita é um problema tipicamente encontrado em processos de moldagem por injeção.Neste caso, o escoamento pode ser descrito por uma formulação conhecida como aproximação de Hele-Shaw. Tal formulação pode ser derivada das equações de conservação tridimensionais usando um número de suposições a respeito do polímero injetado e da geometria da cavidade do molde, juntamente com a integração e o acoplamento das equações da conservação da quantidade de movimento e da continuidade. Essa formulação, referindo às limitações da geometria do molde como sendo canais estreitos e quase sem curvatura, é comumente denominada formulação 2 1/2D. Neste trabalho, é apresentada uma técnica para a simulação da fase de preenchimento de um processo de moldagem por injeção, usando essa formulação 2 1/2D, com um método de volumes finitos e malhas não estruturadas. O modelo de Cross modificado com dependência da temperatura de Arrhenius é empregado para descrever a viscosidade do polímero fundido. O campo de distribuição de temperatura é tridimensional e é resolvido usando um esquema semi-Lagrangeano baseado em volumes finitos. As malhas não estruturadas utilizadas são geradas por triangulação de Delaunay e o método numérico implementado usa a estrutura de dados topológica SHE - Singular Handle Edge, que é capaz de lidar com condições de contorno e singularidades, aspectos comumente encontrados em simulações numéricas de escoamento de fluidos. / Injection molding is one of the most important industrial processes for the manufacturing of thin plastic products. This process can be divided into four stages: plastic melting, filling, packing and cooling phases. The flow of a fluid characterized by high viscosity in a narrow gap is a problem typically found in injection molding processes. In this case, the flow can be described by a formulation known as Hele-Shaw approach. Such formulation can be btained from the three-dimensional conservation equation using a number of assumptions regarding the injected polymer and the geometry of the mold, together with the integration and the coupling of the momentum and continuity equations. This approach, referring to limitations of the mould geometry to narrow, weakly curved channels, is usually called 2 1/2D approach. In this work a technique for the simulation of the filling stage of the injection molding process, using this 2 1/2D approach, with a finite volume method and unstructured meshes, is presented. The modified-Cross model with Arrhenius temperature dependence is employed to describe the viscosity of the melt. The temperature field is 3D and it is solved using a semi-Lagrangian scheme based on the finite volume method. The employed unstructured meshes are generated by Delaunay triangulation and the implemented numerical method uses the topological data structure SHE - Singular Handle Edge, capable to deal with boundary conditions and singularities, aspects commonly found in numerical simulation of fluid flow.

[pt] A visualização volumétrica é uma importante técnica para a exploração de dados tridimensionais complexos, como, por exemplo, o resultado de análises numéricas usando o método dos elementos finitos. A aplicação eficiente dessa técnica a malhas não-estruturadas tem sido uma importante área de pesquisa nos últimos anos. Há dois métodos básicos para a visualização dos dados volumétricos: extração de superfícies e renderização direta de volumes. Na primeira, iso-superfícies de um campo escalar são extraídas explicitamente. Na segunda, que é a utilizada neste trabalho, dados escalares são classificados a partir de uma função de transferência, que mapeia valores do campo escalar em cor e opacidade, para serem visualizados. Com a evolução das placas gráficas (GPU) dos computadores pessoais, foram desenvolvidas novas técnicas para visualização volumétrica interativa de malhas não-estruturadas. Os novos algoritmos tiram proveito da aceleração e da possibilidade de programação dessas placas, cujo poder de processamento cresce a um ritmo superior ao dos processadores convencionais (CPU). Este trabalho avalia e compara dois algoritmos para visualização volumétrica de malhas não-estruturadas, baseados em GPU: projeção de células independente do observador e traçado de raios. Adicionalmente, são propostas duas adaptações dos algoritmos estudados. Para o algoritmo de projeção de células, propõe-se uma estruturação dos dados na GPU para eliminar o alto custo de transferência de dados para a placa gráfica. Para o algoritmo de traçado de raios, propõe-se fazer a integração da função de transferência na GPU, melhorando a qualidade da imagem final obtida e permitindo a alteração da função de transferência de maneira interativa. / [en] Volume visualization is an important technique for the exploration of threedimensional complex data sets, such as the results of numerical analysis using the finite elements method. The efficient application of this technique to unstructured meshes has been an important area of research in the past few years. There are two basic methods to visualize volumetric data: surface extraction and direct volume rendering. In the first, the iso-surfaces of the scalar field are explicitly extracted. In the second, which is the one used in this work, scalar data are classified by a transfer function, which maps the scalar values to color and opacity, to be visualized. With the evolution of personal computer graphics cards (GPU), new techniques for volume visualization have been developed. The new algorithms take advantage of modern programmable graphics cards, whose processing power increases at a faster rate than the one observed in conventional processors (CPU). This work evaluates and compares two GPU- based algorithms for volume visualization of unstructured meshes: view- independent cell projection (VICP) and ray-tracing. In addition, two adaptations of the studied algorithms are proposed. For the cell projection algorithm, we propose a GPU data structure in order to eliminate the high costs of the CPU to GPU data transfer. For the raytracing algorithm, we propose to integrate the transfer function in the GPU, which increases the quality of the generated image and allows to interactively change the transfer function.

Cette thèse est consacrée à la simulation d'écoulements turbulents, incompressibles ou à faible nombre de Mach pour des applications touchant à la sûreté nucléaire. En particulier, nous nous concentrons sur le développement et l'analyse mathématique de schémas numériques performants pour la méthode dite de Simulation des Grandes Echelles. Ces schémas sont basés sur des méthodes à pas fractionnaires de type correction de pression et des éléments finis non conformes de bas degré. Deux arguments semblent essentiels à la construction de tels schémas: le contrôle de l'énergie cinétique et la précision pour des écoulements à convection dominante. Concernant la discrétisation en temps, nous proposons un schéma de type Crank-Nicolson et nous montrons qu'il satisfait un contrôle de l'énergie cinétique. Ce schéma présente de plus l'avantage d'être peu dissipatif numériquement (résidu d'ordre deux en temps). Concernant le défaut de précision de la discrétisation par l'élément fini de Rannacher-Turek, nous envisageons deux approches. La première consiste à construire un schéma pénalisé contraignant les degrés de liberté tangents aux faces des cellules à s'écrire comme combinaison linéaire des degrés de liberté normaux alentour. La deuxième approche repose sur l'enrichissement de l'espace discret d'approximation pour la pression. Enfin, différents tests numériques sont présentés en dimensions deux et trois et pour des maillages généraux, afin d'illustrer les capacités des schémas étudiés et de confronter les résultats théoriques et expérimentaux. / This thesis is devoted to the simulation of incompressible or low Mach turbulent flows, for nuclear safety applications. In particular, we focus on the development and analysis of performing numerical schemes for the Large Eddy Simulation technique. These schemes are based on fractional step methods of pressure correction type and on nonconforming low degree finite elements. Two requirements seems essential to build such schemes, namely a control of kinetic energy and the accuracy for convection dominated flows. Concerning the time marching algorithm, we propose a Crank-Nicolson like scheme for which we prove a kinetic energy control. This scheme has the advantage to be numerically low dissipative (numerical dissipation residual is second order in time). Concerning the low accracy of the Rannacher-Turek discretization, two approaches are investigated in this work. The first one consists in building a penalized scheme constraining the velocity degrees of freedom tangent to the faces to be written as a linear combination of the normal ones. The second approach relies on the enrichment of the pressure approximation discrete space. Finally, various numerical tests are presented in both two and three dimensions and for general meshes, to illustrate the capacity of the schemes and compare theoretical and experimental results.

This work deals with discretizing viscous fluxes in the context of unstructured data based finite volume and meshless solvers, two competing methodologies for simulating viscous flows past complex industrial geometries. The two important requirements of a viscous discretization procedure are consistency and positivity. While consistency is a fundamental requirement, positivity is linked to the robustness of the solution methodology. The following advancements are made through this work within the finite volume and meshless frameworks. Finite Volume Method: Several viscous discretization procedures available in the literature are reviewed for: 1. ability to handle general grid elements 2. efficiency, particularly for 3D computations 3. consistency 4. positivity as applied to a model equation 5. global error behavior as applied to a model equation. While some of the popular procedures result in inconsistent formulation, the consistent procedures are observed to be computationally expensive and also have problems associated with robustness. From a systematic global error study, we have observed that even a formally inconsistent scheme exhibits consistency in terms of global error i.e., the global error decreases with grid refinement. This observation is important and also encouraging from the view point of devising a suitable discretization scheme for viscous fluxes. This study suggests that, one can relax the consistency requirement in order to gain in terms of robustness and computational cost, two key ingredients for any industrial flow solver. Some of the procedures are analysed for positivity as applied to a Laplacian and it is found that the two requirements of a viscous discretization procedure, consistency(accuracy) and positivity are essentially conflicting. Based on the review, four representative schemes are selected and used in HIFUN-2D(High resolution Flow Solver on UNstructured Meshes), an unstructured data based cell center finite volume flow solver, to simulate standard laminar and turbulent flow test cases. From the analysis, we can advocate the use of Green Gauss theorem based diamond path procedure which can render high level of robustness to the flow solver for industrial computations. Meshless Method: An Upwind-Least Squares Finite Difference(LSFD-U) meshless solver is developed for simulating viscous flows. Different viscous discretization procedures are proposed and analysed for positivity and the procedure which is found to be more positive is employed. Obtaining suitable point distribution, particularly for viscous flow computations happens to be one of the important components for the success of the meshless solvers. In principle, the meshless solvers can operate on any point distribution obtained using structured, unstructured and Cartesian meshes. But, the Cartesian meshing happens to be the most natural candidate for obtaining the point distribution. Therefore, the performance of LSFD-U for simulating viscous flows using point distribution obtained from Cartesian like grids is evaluated. While we have successfully computed laminar viscous flows, there are difficulties in terms of solving turbulent flows. In this context, we have evolved a strategy to generate suitable point distribution for simulating turbulent flows using meshless solver. The strategy involves a hybrid Cartesian point distribution wherein the region of boundary layer is filled with high aspect ratio body-fitted structured mesh and the potential flow region with unit aspect ratio Cartesian mesh. The main advantage of our solver is in terms of handling the structured and Cartesian grid interface. The interface algorithm is considerably simplified compared to the hybrid Cartesian mesh based finite volume methodology by exploiting the advantage accrue out of the use of meshless solver. Cheap, simple and robust discretization procedures are evolved for both inviscid and viscous fluxes, exploiting the basic features exhibited by the hybrid point distribution. These procedures are also subjected to positivity analysis and a systematic global error study. It should be remarked that the viscous discretization procedure employed in structured grid block is positive and in fact, this feature imparts the required robustness to the solver for computing turbulent flows. We have demonstrated the capability of the meshless solver LSFDU to solve turbulent flow past complex aerodynamic configurations by solving flow past a multi element airfoil configuration. In our view, the success shown by this work in computing turbulent flows can be considered as a landmark development in the area of meshless solvers and has great potential in industrial applications.

Computing 3D high lift flows has been a challenge to the CFD community because of three important reasons: complex physics, complex geometries and large computational requirements. In the recent years, considerable progress has been made in understanding the suitability of various CFD solvers in computing 3D high lift flows, through the systematic studies carried out under High Lift Prediction workshops. The primary focus of these workshops is to assess the ability of the CFD solvers to predict CLmax and αmax associated with the high lift flows, apart from the predictability of lift and drag of such flows in the linear region. Now there is a reasonable consensus in the community about the ability of the CFD solvers to predict these quantities and fresh efforts to further understand the ability of the CFD solvers to predict more complex physics associated with these flows have already begun. The goal of this thesis is to assess the capability of the computational methods in predicting such complex flow phenomena associated with the 3D High-Lift systems. For evaluation NASA three element Trapezoidal wing configuration which poses a challenging task in numerical modeling was selected. Unstructured data based 3D RANS solver HiFUN (HiFUN stands for High Resolution Flow Solver for UNstructured Meshes) is used in investigating the high lift flow. The computations were run fully turbulent, using the one equation Spalart-Allmaras turbulence model. A summary of the results obtained using the flow solver HiFUN for the 3D High lift NASA Trapezoidal wing are presented. Hybrid unstructured grids have been used for the computations. Grid converged solution obtained for the clean wing and the wing with support brackets, are compared with experimental data. The ability of the solver to predict critical design parameters associated with the high lift flow, such as αmax and CLmax is demonstrated. The utility of the CFD tools, in predicting change in aerodynamic parameters in response to perturbational changes in the configuration is brought out. The solutions obtained for the high lift configuration from two variants of the Spalart-Allmaras turbulence model are compared. To check the unsteadiness in the flow, particularly near stall, unsteady simulations were performed on static grid. Lastly, hysteresis on lower leg of lift curve is discussed, the results obtained for quasi-steady and dynamic unsteady simulations are presented. Inferences from the study on useful design practices pertaining to the 3D high lift flow simulations are summarized.