In this thesis, we study a numerical tool named multi-mesh method within the framework of the adaptive finite element method. The aim of this method is to minimize the size of the linear system to get the optimal performance of simulations. Multi-mesh methods are typically used in multi-physics problems, where more than one component is involved in the system. During the discretization of the weak formulation of partial differential equations, a finite-dimensional space associated with an independently refined mesh is assigned to each component respectively. The usage of independently refined meshes leads less degrees of freedom from a global point of view. To our best knowledge, the first multi-mesh method was presented at the beginning of the 21st Century. Similar techniques were announced by different mathematics researchers afterwards. But, due to some common restrictions, this method is not widely used in the field of numerical simulations. On one hand, only the case of two-mesh is taken into scientists\' consideration. But more than two components are common in multi-physics problems. Each is, in principle, allowed to be defined on an independent mesh. Besides that, the multi-mesh methods presented so far omit the possibility that coefficient function spaces live on the different meshes from the trial and test function spaces. As a ubiquitous numerical tool, the multi-mesh method should comprise the above circumstances. On the other hand, users are accustomed to improving the performance by taking the advantage of parallel resources rather than running simulations with the multi-mesh approach on one single processor, so it would be a pity if such an efficient method was only available in sequential. The multi-mesh method is actually used within local assembling process, which should not be conflict with parallelization. In this thesis, we present a general multi-mesh method without the limitation of the number of meshes used in the system, and it can be applied to parallel environments as well. Chapter 1 introduces the background knowledge of the adaptive finite element method and the pioneering work, on which this thesis is based. Then, the main idea of the multi-mesh method is formally derived and the detailed implementation is discussed in Chapter 2 and 3. In Chapter 4, applications, e.g. the multi-phase flow problem and the dendritic growth, are shown to prove that our method is superior in contrast to the standard single-mesh finite element method in terms of performance, while accuracy is not reduced.

Este trabalho apresenta um estudo sobre segmentação de volumes médicos e uma solução para se obter malhas poligonais de pacientes específicos para uso em simulações de cirurgia. Malhas de pacientes específicos são importantes para planejamento de intervenções cirúrgicas e permitem uma melhor visualização de condições patológicas em um paciente, coisa não obtível em malhas geradas artisticamente. Nós analisamos quais são os fatores complicantes para se obter estas malhas de um paciente específico usando apenas imagens médicas obtidas em exames padrões. Para isso, nós revisamos diversos métodos existentes para segmentação de volumes médicos. Isso nos levou a definir os problemas com as técnicas existentes, e a desenvolver um método que não sofra destes problemas, utilizando pouca interação humana e não tendo dependências de mais dados que não o exame do paciente. Nosso alvo para obter malhas especificas foram órgãos de tecido mole, que são um caso especialmente complicado da área, graças a várias questões relacionadas às imagens médicas e à anatomia humana. Atacamos esse problema aplicando modificações geométricas em malhas especiais, que deformam até atingir a forma dos órgãos que se deseja segmentar. Os resultados mostram que nossa técnica conseguiu obter malhas específicas de pacientes a partir de volumes médicos com qualidade superior a de outros algoritmos de mesma classe. Graças a simplicidade do método desenvolvido, nossos resultados são facilmente implementáveis e reproduzidos. / This work presents a study about medical-volume segmentation and a solution to generate patient-specific meshes to use in patient-specific surgery simulations. Patientspecific meshes are useful assets for surgery planning and to allow better visualization of certain pathological conditions of a given patient, which are not obtainable by artistically designed meshes. We analyzed what are the complications to obtain a patient-specific mesh using only standard medical imagery exams. For that, we reviewed several medical volume segmentation techniques. It led us to define the problems within the existing techniques and to develop a method that does not suffer from these problems, with the least possible user interaction or relying on any other data other then the patient exam. Our target for obtaining specific meshes were soft tissue organs, which are a specially complicated case due to various issues related to the medical images and human anatomy. This is accomplished by geometrical operations over special meshes that deform until achieving the shape of the desired organ. Results show that our technique was able to obtain patient-specific meshes from medical images with superior quality than algorithms of the same class. Thanks to the simplicity of the developed approach, its also easy to implement and to reproduce our obtained results.

<p>Captured and generated 3D data can be so large that it creates a problem for today's computers since they do not fit into the main or graphics card memory. Therefore methods for handling and rendering the data must be developed. This thesis presents a way to pre-process and render out-of-core height map data for real time use. The pre-processing uses a mesh decimation API called Simplygon developed by Donya Labs to optimize the geometry. From the height map a normal map can also be created and used at render time to increase the visual quality. In addition to the 3D data textures are also supported. To decrease the time to load an object the normal and texture maps can be compressed on the graphics card prior to rendering. Three different methods for covering gaps are explored of which one turns out to be insufficient for rendering cylindrical equidistant projected data.At render time two threads work in parallel. One thread is used to page the data from the hard drive to the main and graphics card memory. The other thread is responsible for rendering all data. To handle precision errors caused by spatial difference in the data each object receives a local origin and is then rendered relative to the camera. An atmosphere which handles views from both space and ground is computed on the graphics card.The result is an application adapted to current graphics card technology which can page out-of-core data and render a dataset covering the entire earth at 500 meters spatial resolution with a realistic atmosphere.</p>

Cette thèse pose les fondations du modèle « the Second-generation Louvain-la-Neuve Ice-ocean Model » (SLIM) qui est basé sur la méthode des éléments finis et les maillages non-structurés. Ce modèle fait partie d'une seconde génération de modèles numériques de circulation marine ou océanique. Notre travail a principalement porté sur les aspects géométriques liés à l'utilisation des maillages non-structurés. Nous avons implémenté un algorithme de triangulation qui génère automatiquement des maillages anisotropes non-structurés sur le plan et la sphère et nous avons défini des stratégies de raffinement de maillage adaptées aux applications marines. Ces stratégies orchestrent la distribution de la taille et de la forme des éléments du maillage afin d'optimiser la précision et le coût en temps de calcul du nouveau modèle. Nous avons aussi abordé l'interpolation contrainte de champs scalaires et vectoriels d'un premier maillage vers un second. L'utilisation conjointe de ces trois outils combinée avec un estimateur d'erreur a posteriori permettra l'adaptation dynamique de maillages au cours de simulations transitoires. Finalement, nous avons bâti les outils géométriques nécessaires à l'écriture d'une formulation discrète des équations de la dynamique des fluides géophysiques sur la sphère. Basée sur un système de coordonnées curvilignes propre à chaque élément du maillage, cette approche originale ne possède aucune des difficultés mathématiques et numériques liées aux singularités des pôles et auxquelles les modèles de la première génération n'ont pu apporter de solution entièrement satisfaisante.

Cette thèse pose les fondations du modèle « the Second-generation Louvain-la-Neuve Ice-ocean Model » (SLIM) qui est basé sur la méthode des éléments finis et les maillages non-structurés. Ce modèle fait partie d'une seconde génération de modèles numériques de circulation marine ou océanique. Notre travail a principalement porté sur les aspects géométriques liés à l'utilisation des maillages non-structurés. Nous avons implémenté un algorithme de triangulation qui génère automatiquement des maillages anisotropes non-structurés sur le plan et la sphère et nous avons défini des stratégies de raffinement de maillage adaptées aux applications marines. Ces stratégies orchestrent la distribution de la taille et de la forme des éléments du maillage afin d'optimiser la précision et le coût en temps de calcul du nouveau modèle. Nous avons aussi abordé l'interpolation contrainte de champs scalaires et vectoriels d'un premier maillage vers un second. L'utilisation conjointe de ces trois outils combinée avec un estimateur d'erreur a posteriori permettra l'adaptation dynamique de maillages au cours de simulations transitoires. Finalement, nous avons bâti les outils géométriques nécessaires à l'écriture d'une formulation discrète des équations de la dynamique des fluides géophysiques sur la sphère. Basée sur un système de coordonnées curvilignes propre à chaque élément du maillage, cette approche originale ne possède aucune des difficultés mathématiques et numériques liées aux singularités des pôles et auxquelles les modèles de la première génération n'ont pu apporter de solution entièrement satisfaisante.

A block-based adaptive mesh refinement (AMR) finite-volume scheme is developed for solution of hyperbolic conservation laws on two-dimensional hybrid multi-block meshes. A Godunov-type upwind finite-volume spatial-discretization scheme, with piecewise limited linear reconstruction and Riemann-solver based flux functions, is applied to the quadrilateral cells of a hybrid multi-block mesh and these computational cells are embedded in either body-fitted structured or general unstructured grid partitions of the hybrid grid. A hierarchical quadtree data structure is used to allow local refinement of the individual subdomains based on heuristic physics-based refinement criteria. An efficient and scalable parallel implementation of the proposed algorithm is achieved via domain decomposition. The performance of the proposed scheme is demonstrated through application to solution of the compressible Euler equations for a number of flow configurations and regimes in two space dimensions. The efficiency of the AMR procedure and accuracy, robustness, and scalability of the hybrid mesh scheme are assessed.

A block-based adaptive mesh refinement (AMR) finite-volume scheme is developed for solution of hyperbolic conservation laws on two-dimensional hybrid multi-block meshes. A Godunov-type upwind finite-volume spatial-discretization scheme, with piecewise limited linear reconstruction and Riemann-solver based flux functions, is applied to the quadrilateral cells of a hybrid multi-block mesh and these computational cells are embedded in either body-fitted structured or general unstructured grid partitions of the hybrid grid. A hierarchical quadtree data structure is used to allow local refinement of the individual subdomains based on heuristic physics-based refinement criteria. An efficient and scalable parallel implementation of the proposed algorithm is achieved via domain decomposition. The performance of the proposed scheme is demonstrated through application to solution of the compressible Euler equations for a number of flow configurations and regimes in two space dimensions. The efficiency of the AMR procedure and accuracy, robustness, and scalability of the hybrid mesh scheme are assessed.

Captured and generated 3D data can be so large that it creates a problem for today's computers since they do not fit into the main or graphics card memory. Therefore methods for handling and rendering the data must be developed. This thesis presents a way to pre-process and render out-of-core height map data for real time use. The pre-processing uses a mesh decimation API called Simplygon developed by Donya Labs to optimize the geometry. From the height map a normal map can also be created and used at render time to increase the visual quality. In addition to the 3D data textures are also supported. To decrease the time to load an object the normal and texture maps can be compressed on the graphics card prior to rendering. Three different methods for covering gaps are explored of which one turns out to be insufficient for rendering cylindrical equidistant projected data.At render time two threads work in parallel. One thread is used to page the data from the hard drive to the main and graphics card memory. The other thread is responsible for rendering all data. To handle precision errors caused by spatial difference in the data each object receives a local origin and is then rendered relative to the camera. An atmosphere which handles views from both space and ground is computed on the graphics card.The result is an application adapted to current graphics card technology which can page out-of-core data and render a dataset covering the entire earth at 500 meters spatial resolution with a realistic atmosphere.

Polygonal meshes are a common way of representing 3D surface models in many different areas of computer graphics and geometry processing. However, these models are becoming more and more complex which increases the cost of processing these models. In order to reduce this cost, mesh simplification algorithms are developed. Another important property of a polygonal mesh model is that whether it is regular or not. Regular meshes have many advantages over the irregular ones in terms of memory requirements, efficient processing, rendering etc. In this thesis work, both mesh simplification and regular remeshing algorithms are studied. Moreover, some of the popular mesh libraries are compared with respect to their approaches and performance to the mesh simplification. In addition, mesh models with disk topology are remeshed and converted to regular ones.

In this work, we propose, implement and test two fully automated mesh adaptation methods for 1-D multigroup eigenproblems. The first method is the standard hp-adaptive refinement strategy and the second technique is a goal-oriented hp-adaptive refinement strategy. The hp-strategies deliver optimal guaranteed solutions obtained with exponential convergence rates with respect to the number of unknowns. The goal-oriented method combines the standard hp-adaptation technique with a goal-oriented adaptivity based on the simultaneous solution of an adjoint problem in order to compute quantities of interest, such as reaction rates in a sub-domain or point-wise fluxes or currents. These algorithms are tested for various multigroup 1-D diffusion problems and the numerical results confirm the optimal, exponential convergence rates predicted theoretically.