No description available.

This dissertation describes a new version of the Generalized Finite Element Method (GFEM), which is well suited for problems set in domains with a large number of internal features (e.g. voids, inclusions, etc.), which are practically impossible to solve using the standard FEM. The main idea is to employ the mesh-based handbook functions which are solutions of boundary value problems in domains extracted from vertex patches of the employed mesh and are pasted into the global approximation by the Partition of Unity Method (PUM). It is shown that the p-version of the Generalized FEM using mesh-based handbook functions is capable of achieving very high accuracy. It is also analyzed that the effect of the main factors affecting the accuracy of the method namely: (a) The data and the buffer included in the handbook domains, and (b) The accuracy of the numerical construction of the handbook functions. The robustness of the method is illustrated by several model problems defined in domains with a large number of closely spaced voids and/or inclusions with various shapes, including the heat conduction problem defined on domains with porous media and/or a real composite material.

Ces travaux portent sur le développement d’un matériau virtuel pour la simulation et l’optimisation des matériaux à microstructure hétérogène, en particulier des composites à matrice céramique de nouvelle génération. Pour ce faire une modélisation du fil est mise en place, au travers d’une démarche intégrée qui prend en compte la complexité de la microstructure et de sa variabilité issues du procédé de fabrication. La démarche proposée repose sur deux étapes : i) la construction d’une morphologie synthétique du fil, basée sur l’étude de micrographies et ii) une méthode de simulation multiéchelle inspirée de la méthode des éléments finis généralisée. L’originalité de cette dernière provient de l’utilisation de motifs, sorte de situations physiques ou topologiques élémentaires, pour décrire à la fois la microstructure et la cinématique locale. La démarche est validée et appliquée à diverses sections de fil synthétiques 2D, pour lesquelles le choix des motifs est discuté. L’extension au traitement de tronçon 3D du fil, ainsi qu’à la simulation de la fissuration à l’aide d’une méthode discrète est discutée et des premiers éléments de réponse sont apportés. / The thesis work focus on the development of a virtual material for heterogeneous materials simulation and optimization, especially in the case of now generation of ceramic-matrix composites. To do that, a model at the scale of the yarn is built up, by using an integrated approach that account for the complexity of the microstructure and its variability arising from the manufacturing process. This approach is made of two steps: i) the construction of a synthetic yarn, using micrographics studies and ii) a multiscale approach based on the generalized finites elements method. The originality of that method come from the use of pattern, sort of typical physical or topological situation, that describe both the local structure and kinematic. The approach is validated and applied to various 2D cross-sections of synthetic yarns, for which the choice of patterns is discussed. Extension to 3D section of the yarn, together with the simulation of the fracture in a discrete manner, is discussed and first elements of answer are proposed.

O Método dos Elementos Finitos Generalizados (MEFG) propõe, basicamente, uma ampliação no espaço de aproximação do Método dos Elementos Finitos (MEF) convencional por meio de funções de enriquecimento que representem bem comportamentos locais da solução do problema. Ele tem se apresentado como uma alternativa eficaz para a obtenção de soluções numéricas com boa precisão para problemas nos quais o MEF convencional requer custo computacional bastante elevado. Em relação ao controle sobre a precisão da resposta numérica obtida, o estudo e análise de erros de discretização, assim como a implementação de estratégias adaptativas, são temas que já foram amplamente abordados para o MEF e recentemente vêm sendo explorados no contexto do MEFG e suas versões estáveis. Neste trabalho, trata-se do tema de adaptatividade para o MEFG, objetivando melhor avaliar a precisão das soluções encontradas assim como garantir que elas atendam a limitações pré-especificadas para medidas dos erros. Em primeiro lugar, avalia-se a utilização de um estimador de erro a posteriori, recentemente proposto, como indicador de regiões onde a adaptatividade h ou p possa ser aplicada. Com o indicador adotado, estende-se para o MEFG estratégias h-adaptativas comumente utilizadas para o MEF, realizadas a partir de sucessivas gerações da malha. Além disso, explora-se neste trabalho uma técnica de agrupamento de partições da unidade, específica do MEFG, para tratar problemas de malhas irregulares e possibilitar análises h-adaptativas realizadas sobre sub-regiões do domínio do problema. Já no que se refere às análises p-adaptativas, a estratégia consiste em definir regiões de interesse para ativar o enriquecimento polinomial da solução aproximada. Exemplos numéricos ilustram a efetividade de todas as análises adaptativas implementadas, propostas para o MEFG e suas versões estáveis, as quais proporcionam respostas que atendem a limites de tolerância previamente estabelecidos. / The Generalized Finite Element Method (GFEM) proposes the generation of numerical approximations that belong to an space obtained by augmenting low-order standard finite element approximation spaces by enrichment functions that well represent local behaviours of the problem solution. The method has become an efficient alternative to obtain solutions with good accuracy for problems in which the standard Finite Element Method (FEM) would require excessively high computational cost. Regarding the control over the numerical solutions\' accuracy, discretization error analysis and study, as well as the implementation of adaptive strategies, are subjects largely studied for the FEM and they are recently being exploited in the GFEM and its stable versions context. In this work, adaptivity for the GFEM is addressed, looking for better evaluate the solutions\' accuracy and ensure that they meet users\' pre-specified limits for error measures. Firstly, the use of a recently proposed a posteriori error estimator as an indicator of the regions where h- or p-adaptivity can be performed is evaluated. With this chosen indicator, h-adaptive strategies commonly used for the FEM are extended to the GFEM by performing successive remeshings. Moreover, a partition of unity clustering technique is also exploited in order to treat nonmatching meshes and to enable h-adaptive analysis to be performed over some pre-defined domain subregions. Regarding the p-adaptive analysis, the basic strategy consists of defining some regions over which it will be set polynomial enrichments for the approximate solution using a particular GFEM stable version. Numerical examples show the effectiveness of all performed adaptive analysis, proposed for conventional and stable GFEMs. All implementations provide responses that can meet the users\' pre-specified tolerance.

Ces travaux portent sur le développement d'un matériau virtuel pour la simulation et l'optimisation des matériaux à microstructure hétérogène, en particulier des composites à matrice céramique de nouvelle génération. Pour ce faire une modélisation du fil est mise en place, au travers d'une démarche intégrée qui prend en compte la complexité de la microstructure et de sa variabilité issues du procédé de fabrication. La démarche proposée repose sur deux étapes : i) la construction d'une morphologie synthétique du fil, basée sur l'étude de micrographies et ii) une méthode de simulation multiéchelle inspirée de la méthode des éléments finis généralisée. L'originalité de cette dernière provient de l'utilisation de motifs, sorte de situations physiques ou topologiques élémentaires, pour décrire à la fois la microstructure et la cinématique locale. La démarche est validée et appliquée à diverses sections de fil synthétiques 2D, pour lesquelles le choix des motifs est discuté. L'extension au traitement de tronçon 3D du fil, ainsi qu'à la simulation de la fissuration à l'aide d'une méthode discrète est discutée et des premiers éléments de réponse sont apportés.

Among the plurality of numerical methods to simulate fracture and fatigue processes, the phase-field method gained a lot of attention during the past years. The strength of the method is located in its implicit ability to capture crack phenomena like crack initiation, branching, and merging without consulting further criteria. However, it lacks efficiency due to the fine finite element meshes required in order to accurately capture the high gradients of the phase-field as well as of the displacements across a crack by standard LAGRANGian shape functions. Adding enrichment functions (either gained from analytical considerations or obtained numerically) to the ansatz space, as in the XFEM/GFEM (extended/generalized finite element method), can improve the approximation quality significantly. Particularly functions like discontinuous functions, which otherwise can be approximated by LAGRANGian shape functions only on really fine meshes, are reproduced independently of the mesh orientation and size in the XFEM/GFEM. In the scope of this thesis, a novel framework called extended phase-field method (XPFM), based on a standard phase-field formulation for fracture processes in linear elastic fracture mechanics, is presented. The main idea is to introduce a transformed ansatz formulation for the approximation of the phase-field, which depends on second-order LAGRANGian shape functions. It can exactly reproduce, independently of the mesh, the exponential phase-field profile known from analytical considerations. From this transformed ansatz, enrichment functions are derived to improve the approximation quality of the displacements. In contrast to the XFEM, no additional tracking of the crack geometry is necessary because it is given directly by the phase-field. As a result, the XPFM can handle crack propagation simulations on meshes five to ten times coarser in each spatial direction than the original phase-field formulation without loss of accuracy. Thus, a remarkable reduction of the size of the equation systems to be solved is reached, leading to a considerable reduction of the numerical effort, especially for fatigue simulations. / In den vergangenen Jahren hat die Phasenfeldmethode zur Simulation von Riss- und Ermüdungsvorgängen viel Aufmerksamkeit erlangt. Der Vorteil der Methode besteht darin, dass Rissphänomene wie Rissinitiierung, -verzweigung und -vereinigung ohne die Berücksichtigung weiterer Kriterien implizit behandelt werden. Jedoch sind sehr feine Finite-Elemente-Netze notwendig, um die hohen Gradienten des Phasenfeldes und der Verschiebungen akkurat mit LAGRANGEschen Formfunktionen zu approximieren. Anreicherungsfunktionen (entweder basierend auf analytischen Lösungen des Problems oder auf numerischen Betrachtungen), wie sie in der XFEM/GFEM (erweiterte/generalisierte Finite-Elemente-Methode) eingesetzt werden, können die Approximationsqualität erheblich verbessern. Insbesondere Funktionen mit Diskontinuitäten, welche sonst nur auf äußerst feinen Netzen mit LAGRANGEschen Formfunktionen angenähert werden können, können mithilfe der XFEM/GFEM unabhängig von Elementgröße und -orientierung exakt reproduziert werden. Im Rahmen dieser Arbeit wird eine neuartige Methode, die erweitere Phasenfeldmethode (XPFM), vorgestellt. Sie basiert auf einem Standardphasenfeldmodell für Rissvorgänge in linear-elastischen Materialien. Die Grundidee der Methode besteht darin, einen transformierten Ansatz für das Phasenfeld zu konstruieren, welcher auf quadratischen LAGRANGE-Formfunktionen basiert, jedoch das aus der analytischen Lösung folgende exponentielle Phasenfeldprofil exakt reproduzieren kann. Darauf aufbauend werden Anreicherungsfunktionen für eine verbesserte Approximation der Verschiebungen abgeleitet. Im Gegensatz zur XFEM/GFEM wird kein gesonderter Algorithmus zur Verfolgung eines Risses benötigt, da die Rissgeometrie durch das Phasenfeld gegeben ist. Mit der XPFM können Rissprozesse auf fünf- bis zehnmal so groben Netzen (in jeder Raumrichtung) im Vergleich zum ursprünglichen Phasenfeldmodell ohne Verlust der Genauigkeit erfasst werden. Dadurch wird eine bemerkenswerte Reduktion der Größe der zu lösenden Gleichungssysteme und eine spürbare Reduktion des numerischen Aufwands, insbesondere bei der Simulation von Ermüdungsvorgängen, erreicht.

O trabalho apresenta uma análise do desempenho de vários métodos de extração de fatores de intensidade de tensão a partir de soluções numéricas obtidas com o método dos elementos finitos generalizados (MEFG). A convergência dos fatores de intensidade de tensão é comparada com a da energia de deformação a fim de investigar a superconvergência dos métodos. Para extração dos fatores de intensidade de tensão e o cálculo da taxa de energia disponibilizada para propagação da fissura, implementam-se os métodos da integral de contorno (MIC), da função cutoff (MFC) e da integral-J no contexto do MEFG. Desenvolve-se a formulação dos métodos de extração de forma a obter uma implementação independente da malha utilizada na modelagem do problema. Aplica-se a extração dos fatores de intensidade de tensão, para modos puros e mistos, em problemas clássicos da mecânica da fratura. Verifica-se a convergência dos fatores de intensidade de tensão e da taxa de energia disponibilizada para a propagação da fissura, obtidos com cada método de extração, com o enriquecimento da ordem polinomial da solução do MEFG. Investiga-se a robustez dos métodos com relação ao tamanho dos domínios de extração / The performance of several techniques to extract stress intensity factors (SIF) from numerical solutions computed with the generalized finite element method (GFEM) is investigated. The convergence of the stress intensity factors is compared with the convergence of strain energy with the aim of investigate the superconvergence of the methods. The contour integral (CIM), the cutoff function (CFM) and the J-integral methods are considered to compute stress intensity factors and energy release rate. The proposed implementation of the extraction techniques is completely independent of the discretization used. Several numerical examples demonstrating the convergence of the computed stress intensity factors and the energy release rate, with the increasing of p order of the GFEM solution, are presented

O trabalho apresenta uma análise do desempenho de vários métodos de extração de fatores de intensidade de tensão a partir de soluções numéricas obtidas com o método dos elementos finitos generalizados (MEFG). A convergência dos fatores de intensidade de tensão é comparada com a da energia de deformação a fim de investigar a superconvergência dos métodos. Para extração dos fatores de intensidade de tensão e o cálculo da taxa de energia disponibilizada para propagação da fissura, implementam-se os métodos da integral de contorno (MIC), da função cutoff (MFC) e da integral-J no contexto do MEFG. Desenvolve-se a formulação dos métodos de extração de forma a obter uma implementação independente da malha utilizada na modelagem do problema. Aplica-se a extração dos fatores de intensidade de tensão, para modos puros e mistos, em problemas clássicos da mecânica da fratura. Verifica-se a convergência dos fatores de intensidade de tensão e da taxa de energia disponibilizada para a propagação da fissura, obtidos com cada método de extração, com o enriquecimento da ordem polinomial da solução do MEFG. Investiga-se a robustez dos métodos com relação ao tamanho dos domínios de extração / The performance of several techniques to extract stress intensity factors (SIF) from numerical solutions computed with the generalized finite element method (GFEM) is investigated. The convergence of the stress intensity factors is compared with the convergence of strain energy with the aim of investigate the superconvergence of the methods. The contour integral (CIM), the cutoff function (CFM) and the J-integral methods are considered to compute stress intensity factors and energy release rate. The proposed implementation of the extraction techniques is completely independent of the discretization used. Several numerical examples demonstrating the convergence of the computed stress intensity factors and the energy release rate, with the increasing of p order of the GFEM solution, are presented

No description available.

Electromagnetic flow meters are simple, rugged, non-invasive flow measuring instruments, which are extensively employed in many applications. In particular, they are ideally suited for the flow rate measurement of liquid metals, which serve as coolants in fast breeder reactors. In such applications, theoretical evaluation of the sensitivity turns out to be the best possible choice. Invariably, an evaluation of the associated electromagnetic fields forms the first step. However, due to the complexity of the problem, only numerical field computational approach would be feasible. In the pertinent literature, couple of e orts could be found which employ the well-known Galerkin Finite Element Method (GFEM) for the required task. However, GFEM is known to suffer from the numerical stability problem even at moderate flow rates. This problem is quite common in fluid dynamics area and several stabilization schemes have been suggested as a remedial measure. Among such schemes, the Streamline Upwinding Petrov Galerkin (SU/PG) method is a simple and widely employed approach. The same has been adopted in some of the moving conductor literatures for obtaining a stable solution. Nevertheless, in fluid dynamics literature, it has been shown that the SU/PG solution can suffer from distortion/peaking at the boundary. The remedial measures proposed are nonlinear in nature and hence are computationally demanding. Also, even the SU/PG scheme by itself requires significant additional computation for quadratic and higher order elements. Further, the value of stabilization parameter is not accurately known for 2D and 3D problems. The present work is basically an attempt to address the above problem for flow meter and other rectilinearly moving conductor problems. More specifically, but for the requirement of (graded) structured mesh along the flow direction, it basically aims to address a more general class of problems not just limited to the flow meter. Following the classical approach employed in fluid dynamics literature, first the problem is studied in its 1D form. It was observed that a relatively better performance of GFEM over FDM scheme is basically due to the difference in their Right Hand Side (RHS) terms, which represents the applied magnetic field. Taking clue from this, it was envisaged that a better insight to the numerical problem can be obtained by using the control system theory's transfer function approach. An application of FDM or GFEM to the 1D form of the governing equation, leads to flalge-braic equations with space variable in discrete form. Hence, a Z-transform based approach is employed to relate the applied magnetic field to the vector potential of the resulting reaction magnetic field. It is then shown that the presence of a pole at Z = -1 is basically responsible for the oscillations in the numerical solution. It is then proposed that by using the control systems pole-zero cancellation principle, stability can be brought into the numerical solution. This requires suitable modification of RHS terms in the discretised equations and accordingly, two novel schemes have been proposed which works within the framework of GFEM. In author's considered opinion, the use of Z-transform for analysing the stability of the numerical schemes and the idea of employing pole-zero cancellation to bring in stability, are first of its kind. In the first of the proposed schemes, the pole-zero cancellation is achieved by simply restating the input magnetic field in terms its vector potential. Solving the difference equations given by the application of FDM or GFEM to 1D version of the governing equation, it is analytically shown that the proposed scheme is absolutely stable at high flow rates. However, at midrange of flow rates there is a small error, which is analytically quantified. Then the scheme is applied to the original flow meter problem which has only axially varying applied field and the stability is demonstrated for an extensive range of flow rates. Note that the discretisation along the flow direction was restricted in the above exercise to graded regular mesh, which can readily be realised for problems involving rectilinearly moving conductors. In order to cater for more general cases in which the applied field varies in both axial and transverse directions, a second scheme is developed. Here the RHS term representing the input magnetic field is considered in a generic weighted average form. The required weights are evaluated by imposing apart from the need for an essential zero yielding term, the flux preservation and other symmetry conditions. The stability of this scheme is proven analytically for both 1D and 2D version of the problem using respectively, the 1D and 2D Z-transform based approaches. The analytical inferences are adequately validated with numerical exercises. Also, the small error present for the midrange of flow rates is analytically quantified. Then the second scheme is applied to the actual flow meter with a general magnetic field pro le. The proposed scheme is shown to be very stable and accurate even at very high flow rates. As before, the discretisation was restricted to graded regular mesh along the flow direction. By solving for the standard TEAM No. 9 benchmark problem, applicability of the second scheme for other rectilinearly moving conductor problem has been adequately demonstrated. Even though the problems considered in this work readily permits the use of a graded regular mesh along the flow direction, for the sake of completeness, discretisation with arbitrary quadrilateral and triangular mesh is also considered. The performance of the proposed schemes for such cases even though found to deteriorate, is still shown to be considerably better than the GFEM. In summary, this work has successfully proposed two novel, computationally effcient and stable GFEM schemes for the simulation of electromagnetic flow meters and other rectilin early moving conductor problems.