We present an $hp$-adaptive strategy that is based on estimating the decay of the expansion coefficients when a function is expanded in $L^2$-orthogonal polynomails on a triangle or a tetrahedron. This method is justified by showing that the decay of the coefficients is exponential if and only if the function is analytic. Numerical examples illustrate the performance of this approach, and we compare it with two other $hp$-adaptive strategies.

Mit 3D-Netzmodellen werden Objekte der materiellen Welt oder unserer Vorstellung computergestützt abgebildet. In digitalen Produktentwicklungsprozessen werden mit ihnen sowohl die Objektgestalt als auch anwendungsspezifische Informationen von Objekten und von Prozessen definiert. Mit flächenhaften Netzen (z. B. Dreiecksnetze) wird die Oberfläche von Objekten in diskreter Form repräsentiert, mit volumenhaften Netzen (z. B. Tetraedernetze) zusätzlich das Objektinnere. 3D-Netzmodelle werden bei der Erzeugung und der Manipulation, der Analyse und der Validierung, in fertigungsvorbereitenden Prozessen sowie zur Präsentation digitaler 3D-Objekte angewandt.

Computational Neuroscience deals with spatio-temporal scales which vary considerably.For example interactions at synaptic contact regions occur on the scale of nanometers and nanoseconds to milliseconds (micro-scale) whereas networks of neurons can measure up to millimeters and signals are processed on the scale of seconds (macro-scale). Whole-cell calcium dynamics models (meso-scale) mediate between the multiple spatio-temporal scales. Of crucial importance is the calcium propagation mediated by the highly complex endoplasmic reticulum network. Most models do not account for the intricate intracellular architecture of neurons and consequently cannot resolve the interplay between structure and calcium-mediated function. To incorporate the detailed cellular architecture in intracellular Calcium models, a novel mesh generation methodology has been developed to allow for the efficient generation of computational meshes of neurons with a three-dimensionally resolved endoplasmic reticulum. Mesh generation routines are compiled into a versatile and fully automated reconstruct-and-simulation toolbox for multi-scale physics to be utilized on high-performance or regular computing infrastructures. First-principle numerical simulations on the neuronal reconstructions reveal that intracellular Calcium dynamics are effected by morphological features of the neurons, for instance a change of endoplasmic reticulum diameter leads to a significant spatio-temporal variability of the calcium signal at the soma. / Math & Science Education

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This thesis addresses the problem of relaxing a finite element mesh to more closely match a size function. The main contributions include new methods for performing size function based mesh relaxation, as well as an algorithm for measuring the performance of size function based mesh relaxation methods.

A new methodology for the determination of target element sizes for the construction of finite element meshes applicable to the simulation of tidal flow in coastal and oceanic domains is developed and tested. The methodology is consistent with the discrete physics of tidal flow, and includes the effects of bottom stress. The method enables the estimation of the localized truncation error of the nonconservative momentum equations throughout a triangulated data set of water surface elevation and flow velocity. The method's domain-wide applicability is due in part to the formulation of a new localized truncation error estimator in terms of complex derivatives. More conventional criteria that are often used to determine target element sizes are limited to certain bathymetric conditions. The methodology developed herein is applicable over a broad range of bathymetric conditions, and can be implemented efficiently. Since the methodology permits the determination of target element size at points up to and including the coastal boundary, it is amenable to coastal domain applications including estuaries, embayments, and riverine systems. These applications require consideration of spatially varying bottom stress and advective terms, addressed herein. The new method, called LTEA-CD (localized truncation error analysis with complex derivatives), is applied to model solutions over the Western North Atlantic Tidal model domain (the bodies of water lying west of the 60° W meridian). The convergence properties of LTEACD are also analyzed. It is found that LTEA-CD may be used to build a series of meshes that produce converging solutions of the shallow water equations. An enhanced version of the new methodology, LTEA+CD (which accounts for locally variable bottom stress and Coriolis terms) is used to generate a mesh of the WNAT model domain having 25% fewer nodes and elements than an existing mesh upon which it is based; performance of the two meshes, in an average sense, is indistinguishable when considering elevation tidal signals. Finally, LTEA+CD is applied to the development of a mesh for the Loxahatchee River estuary; it is found that application of LTEA+CD provides a target element size distribution that, when implemented, outperforms a high-resolution semi-uniform mesh as well as a manually constructed, existing, documented mesh.

Técnicas para gerar malhas triangulares ou tetraedrais a partir de imagens, ou assumem como entrada uma imagem pré-processada, ou geram uma malha sem distinguir as diferentes estruturas contidas na imagem. O pré-processamento e a ausência de estruturas bem definidas podem apresentar dificuldades na utilização das malhas geradas em algumas aplicações, tais como simulações numéricas. Neste trabalho, apresentamos uma nova técnica que elimina a necessidade do pré-processamento embutindo a segmentação dentro do processo de geração de malha. Além disto, a técnica proposta considera critérios de qualidade nas malhas geradas, mostrando-se apropriada para aplicações de simulação numérica assim como modelagem de imagens com malhas / Techniques devoted to generate triangular or tetrahedral meshes from images either take as starting point a pre-processed image or generate a mesh without distinguishing different structures contained in the image. The pre-process and the absence of well defined structures may impose difficulties in using the resulting mesh in some applications as, for example, numerical simulations. In this work, we present a new technique that aims at eliminating the need for pre-processing by building the segmentation into the mesh generation process. Furthermore, the proposed technique consider quality criteria in the generated meshes, being appropriated for applications in numerical simulation as well as in image modeling

A reconstrução de superfícies a partir de nuvens de pontos faz parte de um novo paradigma de modelagem em que modelos computacionais para objetos reais são reconstruídos a partir de dados amostrados sobre a superfície dos mesmos. O principal problema que surge nesse contexto é o fato de que não são conhecidas relações de conectividade entre os pontos que compõe a amostra. Os objetivos do presente trabalho são estudar métodos implícitos para a reconstrução de superfícies e propor algumas melhorias pouco exploradas por métodos já existentes. O uso de funções implícitas no contexto da reconstrução conduz a métodos mais robustos em relação a ruídos, no entanto, uma das principais desvantagens de tais métodos está na dificuldade de capturar detalhes finos e sharp features. Nesse sentido, o presente trabalho propõe o uso de abordagens adaptativas, tanto na poligonalização de superfícies quanto na aproximação de superfícies. Além disso, questões relativas à robustez das soluções locais e à qualidade da malha também são abordadas. Por fim, o método desenvolvido é acoplado aumsoftware traçador de raios afimde se obterumamaneira de modelar cenas tridimensionais utilizando nuvens de pontos, além dos objetos gráficos tradicionais. Os resultados apresentados mostram que muitas das soluções propostas oferecem um incremento à qualidade dos métodos de reconstrução anteriormente propostos / Surface reconstruction from point clouds is part of a new modeling paradigm in which computational models for real objects are reconstructed from data sampled from their surface. The main problem that arises in this context is the fact that there are no known connectivity relationships amongst the points that compose the sample. The objectives of the present work are to study implicit methods for surface reconstruction and to propose some improvements scarcely explored by previous work. The use of implicit functions in the context of surface reconstruction leads to less noise sensitive methods; however, one major drawback of such methods is the difficulty in capturing fine details and sharp features. Towards this, the present work proposes the use of adaptive approaches, not only in the polygonization but also in the surface approximation. Besides, robustness issues in local solutions and mesh quality are also tackled. Finally, the developed method is embedded in a ray tracer software in order to set a basis for modeling tridimensional scenes using point sets, in addition to traditional graphic objects. The presented results show that a great deal of the proposed solutions offer a quality increase to the reconstruction method previously proposed

"This thesis presents an implementation of the Finite Difference Time Domain (FDTD) method on a massively parallel computer system, for the analysis of electromagnetic phenomenon. In addition, the implementation of an efficient mesh generator is also presented. For this research we selected the MasPar system, as it is a relatively low cost, reliable, high performance computer system. In this thesis we are primarily concerned with the selection of an efficient algorithm for each of the programs written for our selected application, and devising clever ways to make the best use of the MasPar system. This thesis has a large emphasis on examining the application performance."

Com a popularização de equipamentos tradicionais de captura de imagens, como câmeras digitais, e o avanço tecnológico dos dispositivos não invasivos, como tomografia e ressonância, cresce também a necessidade e consequente uso de métodos numéricos para simulação de fenômenos físicos em domínios definidos por imagens. Um dos pré-requisitos para a aplicação de tais métodos numéricos consiste na discretização do domínio em questão, num processo denominado geração de malhas. Embora diversos métodos de geração de malha tenham sido propostos para discretizar domínios definidos por primitivas geométricas, pouco tem sido feito no sentido de gerar uma decomposição diretamente a partir de imagens. Neste trabalho, apresentamos uma abordagem de geração de malhas de qualidade a partir de domínios definidos por imagens com textura. Mais especificamente, a pesquisa descrita nesta tese contribui com a melhoria do algoritmo Imesh, ao sanar três de suas principais limitações: tratamento de imagens com texturas; controle do nível de refinamento da malha e suporte a outros tipos de elementos. Estas contribuições flexibilizam o processo de geração da malha, e ainda ampliam o domínio de aplicações do algoritmo Imesh, à medida que são considerados domínios definidos por imagens com textura e o uso de métodos numéricos para elementos não simpliciais torna-se possível. O algoritmo de melhoria da malha gerada utiliza uma abordagem inovadora de remalhamento baseada em templates e guiada por retalhos de Bézier / With the spreading of traditional image capturing devices, such as digital cameras, and the technological advancement of more specific imaging devices such as CT and MRI, also increased the need and the following use of numerical methods for simulation of physical phenomena in domains defined by images. One of the prerequisites for the application of such numerical methods is the discretization of the corresponding domain, in a process called mesh generation. Although several methods of mesh generation have been proposed to discretize domains defined by geometric primitives, little has been done to generate a decomposition directly from images. We present an approach to generate quality meshes from domains defined by images with texture. More specifically, the research described in this thesis contributes to the improvement of the Imesh algorithm, removing three of its main limitations: treatment textured images, control of the level of the mesh refinement and support for other types of non-simplicial elements. These contributions provide flexibility to the mesh generation process, and extend the range of applications of Imesh by both handling textured images and considering the use of numerical methods for non-simplicial elements. The mesh quality improvement algorithm uses a new approach based on mesh templates and it is guided by Bezier patches