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.

Background. Geometry-Independent Path Planning (GIPP) can be done by generating a navigation mesh and computing paths on that mesh in real-time for parallel and dynamic path planning. However, many of the existing algorithms are not suitable for the Graphics card, therefore a new path planning algorithm is created. Hardware Accelerated Line Of Sight (HALOS) performs parallel path planning on grid maps in real-time using the GPU.   Objectives. This thesis aims to implement an automatic navigation mesh generation algorithm using the GPU rendering pipeline and a GPU-bound path planning algorithm for a grid-based map. The proposed method should generate accurate paths and run in real-time. To gather results, the methods are measured in run-time on different types of hardware and scenarios. Methods. Multiple experiments are conducted. A navigation mesh is generated in real-time using the rendering pipeline of the GPU. In addition, a novel path planning algorithm generates paths in real-time using the GPU by utilizing line of sight on the navigation mesh. GIPP is a multi-source, single-destination algorithm. The path planning is done parallel and dynamically to navigate around moving obstacles. Results. The experiments show that GIPP can generate a dynamic navigation mesh in real-time. However, the coverage of GIPP is poor, and some resolutions of GIPP result in agents being unable to reach the goal node. The performance effect of dynamic worlds on path planning is not noticeable compared to static worlds. Conclusions. The proposed method can perform real-time navigation mesh generation and path planning. Different resolutions of GINT show inconsistencies in the length of the path generated. This method, GIPP, is well suited for complex, dynamic, single-floor meshes that more traditional navigation mesh generators are not guaranteed to handle in real-time. The main performance bottleneck for GIPP is the number of layers created during path planning. / Bakgrund. Geometrioberoende vägplanering (GIPP) kan utföras genom att generera ett navigationsnät och beräkna vägar på detta nät i realtid för parallell och dynamisk vägplanering. Många vägplaneringsalgoritmer kan inte köras i realtid på grafikkortet. Därför har Hardware Accelerated Line Of Sight (HALOS) skapats, vilket utför parallell vägplanering i realtid med hjälp av GPU:n. Mål. Denna avhandling syftar till att implementera en automatisk algoritm för generering av navigationsnät med hjälp av GPU:ns renderingspipeline och implementera en GPU-bunden vägplaneringsalgoritm för en rutbaserad karta. Den föreslagna metoden genererar vägar och körs i realtid. För att samla in resultat mäts metoderna i körtid på olika typer av hårdvara och scenarier. \newline\textbf{Metoder.} Flertalet experiment utförst på GIPP. Ett navigationsnät genereras i realtid med hjälp av GPU:ns renderingspipeline och en ny vägplaneringsalgoritm genererar vägar i realtid med hjälp av sikten längs navigationsnätet. Denna algoritm har flera källor med en destination (MSSD) för att hantera ett stort antal agenter. Vägplaneringen görs parallellt och dynamiskt för att navigera runt rörliga hinder. Resultat. Experimenten visar att GIPP kan generera ett navigationsnät i realtid. GIPP har dock dålig precision när det gäller att generera effektiva vägar mot målet.  Vissa upplösningar leder till att agenter inte når slutmålet. Dynamiska världar har omärkbar påverkan på prestandan i jämförelse med statiska världar när det gäller vägplanering. Slutsatser. Den föreslagna metoden kan utföra navigationsnätsgenerering och vägplanering i realtid. Olika upplösningar av navigationsnätet visar att vägplanering har olikeheter i avstånd. Denna metod, GIPP, lämpar sig väl för komplexa, dynamiska, enplansvärldar. GIPPs flaskhals i prestandan är mängden lager som skapas under vägplaneringen.

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."