Neste trabalho apresenta-se a formulação do problema de múltiplas fissuras baseada numa abordagem de superposição utilizada pelo Método da Partição (Splitting Method). Um dos objetivos principais deste trabalho refere-se à aferição da capacidade deste método na obtenção de fatores de intensidade de tensão, tendo em vista o seu desenvolvimento recente e a ausência de outras fontes de pesquisa além daquelas oriundas dos seus próprios autores. Segundo a abordagem do Método da Partição, os fatores de intensidade de tensão finais de uma estrutura podem ser encontrados a partir da sobreposição de três subproblemas. Deste modo, o problema é resolvido mediante imposição de que nas faces das fissuras as tensões que resultam da sobreposição sejam nulas. Sendo assim, apresenta-se a formulação do Método da Partição para uma ou mais fissuras e diversas análises numéricas que contemplam interação entre fissuras submetidas aos modos I e II de abertura. Outra etapa do trabalho refere-se à aplicação do Método dos Elementos Finitos Generalizados (MEFG) num dos subproblemas, dito local, ao invés do emprego do Método dos Elementos Finitos (MEF), que em sua forma convencional pode requerer um refinamento excessivo da malha, particularmente junto à ponta da fissura, aumentando o custo computacional da análise. Exemplos de simulação numérica são apresentados no sentido de comprovar que a utilização do MEFG viabiliza a obtenção de resultados com boa aproximação mesmo com malhas pouco refinadas, reduzindo significativamente o custo computacional de toda a análise. Além disto, é apresentada a formulação do Método da Partição para casos que contemplam também fissuras internas, uma vez que a formulação atual admite somente fissuras de borda. / This work presents the formulation of the problem of multiple cracks based on an superposition approach used by the Splitting Method. The main goal of this work concerns the verification of the ability of this method of obtaining stress intensity factors, in view of its recent development and the absence of other research sources beyond those derived from their own authors. According to the approach of Splitting Method, the final stress intensity factors of a structure can be found from the superposition of three subproblems. Thus, the problem is solved by superposition and then imposing the nullity of the stresses on the faces of cracks. Thus, the formulation of the Splitting Method is presented to one or more cracks and also several numerical simulations that consider the interaction between cracks subjected to opening mode I and II. Another part of this work concerns the application of the Generalized Finite Element Method (GFEM) in the local subproblem instead of the use of Finite Element Method (FEM), which in its conventional form may require an excessive mesh refinement, particularly near the tip the crack, increasing the computational cost of analysis. Examples of numerical simulation are presented in order to show that the use of GFEM enables to obtain results with good approximation even with little refined meshes, thus significantly reducing the computational cost of the entire analysis. Moreover, the formulation of the Splitting Method is presented for cases which also have internal cracks due to the current formulation admits only boundary cracks.

Este trabalho insere-se no âmbito das formulações não convencionais em elementos finitos. Particularmente, introduzem-se alguns aspectos do método dos elementos finitos generalizados (MEFG) e do clássico refino-p na consagrada formulação híbrida-Trefftz de tensão para a elasticidade bidimensional. A formulação apresentada aproxima diretamente dois campos independentes: o de tensões no domínio dos elementos e o de deslocamentos nas fronteiras dos elementos. Baseado na estrutura de enriquecimento centrada em nuvens, proposta pelo MEFG, podem ser selecionadas oportunamente regiões, formadas por um conjunto de elementos e fronteiras de elementos, onde o espaço da aproximação é adequadamente enriquecido mediante o refino-p. Neste contexto campos auto-equilibrados de tensões, derivados da solução da equação de Navier, são utilizados para compor a aproximação no domínio dos elementos, enquanto nas fronteiras dos elementos o campo de deslocamentos é construído a partir de bases específicas de aproximação; seja a base inicial, formada por funções de forma lineares, ou bases enriquecidas com polinômios hierárquicos, não hierárquicos e funções trigonométricas. Aborda-se também, ainda que preliminarmente, um estudo de painéis com múltiplas fissuras pelo método da partição em formulação híbrida-Trefftz com enriquecimento seletivo. As análises numéricas realizadas revelaram, em geral, uma formulação de ótimo desempenho, caracterizada por uma notável capacidade de aproximação dos campos de tensões e deslocamentos, elevada robustez numérica e reduzido dispêndio computacional. / This work is inserted in the context of unconventional formulations in the finite elements method. Particularly, some aspects of the generalized finite elements method (GFEM) and the classic p-refinement are introduced in the well known hybrid-Trefftz stress formulation for the two dimensional elasticity. The presented formulation approximates two independent fields: the one of stresses in the elements domain and the one of displacements in the boundaries of the elements. Based on the enrichment structure centered in clouds, proposed by the GFEM, some regions, formed by a group of elements and boundaries of elements where the approximation space is adequately enriched by the p-refinement, can be opportunely selected. In this context, self-equilibrated stress fields, derived from the solution of the Navier equation, are used to compose the approximation in the elements domain, whereas the displacements field in the borders of the elements is built from specific approximation bases, that is, the initial base formed by linear shape functions, or, bases enriched with hierarchical polynomials, nonhierarchical ones and trigonometric functions. Also, although preliminarily, a study of the multiple-cracked panels is done using the Splitting Method with a hybrid-Trefftz formulation and a selective enrichment. The numeric analyses done revealed, in general, a high performance formulation characterized by a great capacity of approximation the stress fields and displacements, high numeric robustness and reduced computer expenditure.

Les fuites non détectées dans les réseaux d'eau potable sont responsables en moyenne de la perte de 30% de l'eau transportée. Il s'avère donc primordial de pouvoir contrôler ces fuites. Pour atteindre cet objectif, la modélisation de l'écoulement de l'eau dans les conduites en tenant compte des fuites a été formulée de différente manière. La première formulation est un système d'équations différentielles ordinaires représentant des fuites constantes, réparties uniformément le long des conduites. Le système peut s'avérer être numériquement raide lorsque des organes hydrauliques sont rajoutés. Deux méthodes implicites ont été proposées pour sa résolution : la méthode de Rosenbrock et la méthode de Gear. Les résultats obtenus montrent que le débit varie linéairement le long des conduites et que les pertes en eau par unité de longueur sont identiques sur chaque conduite. La seconde formulation prend en compte la relation entre les fuites et la pression. Un système de deux équations aux dérivées partielles a été proposé. L'EDP de transport-diffusion-réaction, contenant l'opérateur du p-Laplacien, est résolue par une méthode à pas fractionnaires. Deux méthodes ont été testées. Dans la première, la réaction est couplée avec la diffusion et dans la seconde, elle est couplée avec le transport. Les résultats indiquent que les pertes en eau ne sont pas réparties de façon homogène sur le réseau. Cette formulation décrit de manière plus réaliste les réseaux d'eau potable. Enfin, le problème du contrôle du volume des fuites par action sur la pression a été étudié. Pour cela, un problème d'optimisation est résolu sous la contrainte que la pression doit être minimale pour réduire les fuites et être suffisante pour garantir un bon service aux consommateurs. Les résultats trouvés confirment que la réduction de la pression permet de réduire le volume des fuites de façon significative et que le choix de l'emplacement du ou des points de contrôle est primordial pour optimiser cette réduction. / Leakage represents a large part, in average more than 30%, of the water supplied. Consequently, it is important to control leakage in Water Distribution System (WDS). For this purpose different methods, which take leakage into account, are proposed to model the hydraulics of WDS. The first formulation considers constant leakage in a network and leads to an ordinary differential equation. It turns out to be a hydraulic stiff problem due to valve and pump operations. This equation is solved using two methods: the first one is a generalised Runge-Kutta method and the second one the Gear method. The results show that the flow rate varies linearly along a pipe and that the water loss per unit of length is identical for each pipe. Magnitude of inertia terms has also been studied. The second formulation takes pressure-dependent leakage into account. We propose to introduce partial differential equations in order to predict more accurately hydraulic flows in WDS. Thus, the physical advection-diffusion-reaction model is presented. A nonlinear operator, called p-Laplacian, related to the diffusion is included into the model. Two resolutions of this model based on a splitting method are detailed. The results confirm that losses vary nonlinearly with pressure. Finally, the leakage-control problem is studied. For this purpose, we solve an optimisation problem with the objective to minimize the distributed volume in order to reduce leakage. The condition of sufficient pressure to satisfy consumers is imposed in this optimisation. The results prove that pressure control significantly reduces leakage and that the emplacement of the valve is important to optimise this reduction.

O trabalho trata da utilização de formulações não-convencionais de elementos finitos na obtenção de fatores de intensidade de tensão associados a múltiplas fissuras distribuídas num domínio bidimensional. A formulação do problema de múltiplas fissuras baseia-se numa abordagem de sobreposição proposta pelo Método da Partição (\"Splitting Method\"). Segundo essa abordagem a solução do problema pode ser encontrada a partir da sobreposição de três subproblemas combinados de tal forma que o fluxo de tensão resultante nas faces das fissuras seja nulo. O uso do Método dos Elementos Finitos (MEF) em sua forma convencional pode requerer um refinamento excessivo da rede nesse tipo de problema, aumentando o custo computacional da análise. Objetivando reduzir este custo, empregam-se duas formulações não-convencionais, de forma independente, num dos subproblemas, dito local: a formulação híbrido-Trefftz de tensão e o Método dos Elementos Finitos Generalizados (MEFG). Na formulação híbrido-Trefftz é adotado o recurso do enriquecimento seletivo mediante o refrno- p na aproximação dos campos de deslocamento no contorno do elemento. Já com relação ao MEFG, empregam-se funções polinomiais e a solução analítica da mecânica da fratura como funções enriquecedoras. Exemplos de simulação numérica são apresentados no sentido de comprovar que a utilização dessas formulações não-convencionais juntamente com o Método da Partição viabiliza a obtenção de resultados com boa aproximação com recurso a redes pouco refinadas, reduzindo significativamente o custo computacional de toda a análise. / This paper treats with the use of non-conventional finite element formulations to obtain the stress intensity factor of multiple cracks located in a two-dimensional domain. The formulation of the multiple cracks problem is based on an overlapping approach suggested by the Splitting Method. Accordingly, the solution of the problem can be achieved by dividing the problem in three steps, combined so that the resulting stress flux is zero on the cracks face. The use of the Finite Element Method (FEM) in its conventional formulation requires a mesh refinement in this kind of problem, then increasing the computational cost. Aiming to reduce this cost, two non-conventional formulations are used independently to solve the local problem: the Hybrid-Trefftz stress formulation and the Generalized Finite Elements Method (GFEM). The Hybrid-Trefftz formulation is applied with selective enrichment using p-refinement in the displacements field on the element boundaries. The GFEM employs polynomial functions and analytical solutions of the fracture mechanics as enrichment functions. Examples of numerical simulations are presented in order to show that non- conventional formulations and the Splitting Method can provide accurate results with coarse mesh, thus reducing the computational cost.

The design of complex structures demands the prediction of possible fracture-dominant failure processes, due to the existence of unavoidable preexistent flaws and other defects, as well as sharps and cracks. On one hand, the complexity of the structure and the presence of many defects to be accounted for in the modeling can become the computational effort impracticable. On the other hand, it is important to seek the development of a computational framework based on some numerical method to study these problems. A way to overcome the difficulties mentioned, therefore making feasible the analysis of complex structures with many cracks, flaws and other defects, consists of combining a representative mechanical modeling with an efficient numerical method. This is precisely the fundamental aim of this work. Firstly, the Splitting Method is used aiming to build a representative modeling. Secondly, the Generalized Finite Element Method (GFEM) is chosen as an efficient numerical method, in which enrichment strategies of the approximated solution using stress functions in particular can be explored. The GFEM framework also allows avoiding the excessive refinement of the mesh, which increases the computational effort in conventional finite element analysis. In the Splitting Method, a kind of decomposition method, the original problem is subdivided in local and global problems which are then combined by imposing null traction at the crack surfaces. In this work, the Splitting Method was completely programmed in Python language and its use extended to analyze crack propagation including fatigue crack growth. The generated code presents in addition to several features related to Fracture Mechanics concepts, as the computation of the stress intensity factor (mode I and II) trough J Integral. Some examples are presented to depict the propagation of the cracks in multisite damage structures. It is shown that for this kind of problems the enrichment strategy provided by GFEM is essential. Moreover, the final example demonstrates that the computational tool allows for investigation of different possible crack scenarios with a low cost analysis. One concludes about the representativeness and efficiency of the methodology hereby proposed. / O projeto de estruturas complexas demanda a previsão de possíveis processos de ruptura governados por fraturamento, devido à existência de inevitáveis defeitos pré-existentes, como entalhes e fissuras. Por um lado, a complexidade da estrutura e a presença de muitos defeitos a serem considerados no modelo podem tornar a análise inviável devido ao esforço computacional necessário. Por outro lado, é importante procurar desenvolver uma estrutura computacional baseada em métodos numéricos para estudar estes problemas. Um modo de superar as dificuldades mencionadas, portanto tornando possível a análise de estruturas complexas com muitas fissuras e outros defeitos, consiste em combinar um modelo mecânico que seja representativo com um método numérico eficiente. Este é precisamente o objetivo fundamental deste trabalho. Primeiramente, o Método da Partição é utilizado para a construção de um modelo representativo. Em segundo lugar, o Método dos Elementos Finitos Generalizados (GFEM) é empregado por ser um método numérico eficiente, no qual as estratégias de enriquecimento da solução aproximada usando funções de tensão, em particular, podem ser exploradas. A estrutura do GFEM também permite evitar o excessivo refinamento da malha, que aumenta o esforço computacional em análises convencionais nas quais se utiliza o método dos elementos finitos. No Método da Partição, um tipo de método de decomposição, o problema original é subdividido em problemas locais e globais que são então combinados impondo-se a nulidade do vetor de tensões na superfície da fissura. Neste trabalho, o Método da Partição foi completamente programado em linguagem Python® e sua utilização estendida para analisar a propagação de fissuras, incluindo-se a associação do crescimento com a resposta em fadiga. Além disso, o código gerado apresenta diversas características relacionadas aos conceitos da Mecânica da Fratura, como o cálculo do fator de intensidade de tensão (modos I e II) mediante a Integral J. Alguns exemplos são apresentados para ilustrar a propagação de fissuras em estruturas multi-fraturadas. Mostra-se que para este tipo de problemas a estratégia de enriquecimento fornecida pelo GFEM é essencial. Além disso, o exemplo final comprova que a ferramenta computacional permite a investigação de diferentes possíveis cenários de fissuras com uma análise de baixo custo. Conclui-se sobre a representatividade e eficiência da metodologia proposta.

In the quest for a high fidelity numerical scheme for CFD it is necessary to satisfy demands on accuracy, conservation, positivity and upwinding. Recently the requirement of rotational invariance has been added to this list. In the present work we are mainly interested in upwinding and rotational invariance of Least Squares Kinetic Upwind Method (LSKUM). The standard LSKUM achieves upwinding by stencil division along co-ordinate axes which is referred to as co-ordinate splitting method. This leads to symmetry breaking and rotational invariance is lost. Thus the numerical solution becomes co-ordinate frame dependent. To overcome this undesirable feature of existing numerical schemes, a new algorithm called KUMARI (Kinetic Upwind Method Avec Rotational Invariance, 'Avec' in French means 'with') has been developed. The interesting mathematical relation between directional derivative, Fourier series and divergence operator has been used effectively to achieve upwinding as well as rotational invariance and hence making the scheme truly or genuinely multidimensional upwind scheme. The KUMARI has been applied to the test case of standard 2D shock reflection problem, flow past airfoils, then to 2D blast wave problem and lastly to 2D Riemann problem (Lax's 3rd test case). The results show that either KUMARI is comparable to or in some cases better than the usual LSKUM.

The kinetic schemes, also known as Boltzmann schemes are based on the moment-method-strategy, where an upwind scheme is first developed at the Boltzmann level and after taking suitable moments we arrive at an upwind scheme for the governing Euler or Navier-Stokes equations. The Kinetic Flux Vector Splitting (KFVS)scheme, which belongs to the family of kinetic schemes is being extensively used to compute inviscid as well as viscous flows around many complex configurations of practical interest over the past two decades. To resolve many flow features accurately, like suction peak, minimising the loss in stagnation pressure, shocks, slipstreams, triple points, vortex sheets, shock-shock interaction, mixing layers, flow separation in viscous flows require an accurate and low dissipative numerical scheme. The first order KFVS method even though is very robust suffers from the problem of having much more numerical diffusion than required, resulting in very badly smearing of the above features. However, numerical dissipation can be reduced considerably by using higher order kinetic schemes. But they require more points in the stencil and hence consume more computational time and memory. The second order schemes require flux or slope limiters in the neighbourhood of discontinuities to avoid spurious and physically meaningless wiggles or oscillations in pressure, temperature or density. The limiters generally restrict the residue fall in second order schemes while in first order schemes residue falls up to machine zero. Further, pressure and density contours or streamlines are much smoother for first order accurate schemes than second order accurate schemes. A question naturally arises about the possibility of constructing first order upwind schemes which retain almost all advantages mentioned above while at the same time crisply capture the flow features. In the present work, an attempt has been made to address the above issues by developing yet another kinetic scheme, known as the low dissipative modified KFVS (m-KFVS) method based on modified CIR (MCIR) splitting with molecular velocity dependent dissipation control function. Different choices for the dissipation control function are presented. A detailed mathematical analysis and the underlying physical arguments behind these choices are presented. The expressions for the m-KFVS fluxes are derived. For one of the choices, the expressions for the split fluxes are similar to the usual first order KFVS method. The mathematical properties of 1D m-KFVS fluxes and the eigenvalues of the corresponding flux Jacobians are studied numerically. The analysis of numerical dissipation is carried out both at Boltzmann and Euler levels. The expression for stability criterion is derived. In order to be consistent with the interior scheme, modified solid wall and outer boundary conditions are derived by extending the MCIR idea to boundaries. The cell-centred finite volume method based on m-KFVS is applied to several standard test cases for 1D, 2D and 3D inviscid flows. In the case of subsonic flows, the m-KFVS method produces much less numerical entropy compared to first order KFVS method and the results are comparable to second order accurate q-KFVS method. In transonic and supersonic flows, m-KFVS generates much less numerical dissipation compared to first order KFVS and even less compared to q-KFVS method. Further, the m-KFVS method captures the discontinuities more sharply with contours being smooth and near second order accuracy has been achieved in smooth regions, by still using first order stencil. Therefore, the numerical dissipation generated by m-KFVS is considerably reduced by suitably choosing the dissipation control variables. The Euler code based on m-KFVS method almost takes the same amount of computational time as that of KFVS method. Although, the formal accuracy is of order one, the m-KFVS method resolves the flow features much more accurately compared to first order KFVS method but the numerical dissipation generated by m-KFVS method may not be minimal. Hence, the dissipation control vector is in general not optimal. If we can find the optimal dissipation control vector then we will be able to achieve the minimal dissipation. In the present work, the above objective is attained by posing the minimisation of numerical dissipation in m-KFVS method as an optimal control problem. Here, the control variables are the dissipation control vector. The discrete form of the cost function, which is to be minimised is considered as the sum of the squares of change in entropy at all cells in the computational domain. The number of control variables is equal to the total number of cells or finite volumes in the computational domain, as each cell has only one dissipation control variable. In the present work, the minimum value of cost function is obtained by using gradient based optimisation method. The sensitivity gradients of the cost function with respect to the control variables are obtained using discrete adjoint approach. The discrete adjoint equations for the optimisation problem of minimising the numerical dissipation in m-KFVS method applied to 2D and 3D Euler equations are derived. The method of steepest descent is used to update the control variables. The automatic differentiation tool Tapenade has been used to ease the development of adjoint codes. The m-KFVS code combined with discrete adjoint code is applied to several standard test cases for inviscid flows. The test cases considered are, low Mach number flows past NACA 0012 airfoil and two element Williams airfoil, transonic and supersonic flows past NACA 0012 airfoil and finally, transonic flow past Onera M6 wing. Numerical results have shown that the m-KFVS-adjoint method produces even less numerical dissipation compared to m-KFVS method and hence results in more accurate solution. The m-KFVS-adjoint code takes more computational time compared to m-KFVS code. The present work demonstrates that it is possible to achieve near second order accuracy by formally first order accurate m-KFVS scheme while retaining advantages of first order accurate methods.

The design of complex structures demands the prediction of possible fracture-dominant failure processes, due to the existence of unavoidable preexistent flaws and other defects, as well as sharps and cracks. On one hand, the complexity of the structure and the presence of many defects to be accounted for in the modeling can become the computational effort impracticable. On the other hand, it is important to seek the development of a computational framework based on some numerical method to study these problems. A way to overcome the difficulties mentioned, therefore making feasible the analysis of complex structures with many cracks, flaws and other defects, consists of combining a representative mechanical modeling with an efficient numerical method. This is precisely the fundamental aim of this work. Firstly, the Splitting Method is used aiming to build a representative modeling. Secondly, the Generalized Finite Element Method (GFEM) is chosen as an efficient numerical method, in which enrichment strategies of the approximated solution using stress functions in particular can be explored. The GFEM framework also allows avoiding the excessive refinement of the mesh, which increases the computational effort in conventional finite element analysis. In the Splitting Method, a kind of decomposition method, the original problem is subdivided in local and global problems which are then combined by imposing null traction at the crack surfaces. In this work, the Splitting Method was completely programmed in Python language and its use extended to analyze crack propagation including fatigue crack growth. The generated code presents in addition to several features related to Fracture Mechanics concepts, as the computation of the stress intensity factor (mode I and II) trough J Integral. Some examples are presented to depict the propagation of the cracks in multisite damage structures. It is shown that for this kind of problems the enrichment strategy provided by GFEM is essential. Moreover, the final example demonstrates that the computational tool allows for investigation of different possible crack scenarios with a low cost analysis. One concludes about the representativeness and efficiency of the methodology hereby proposed. / O projeto de estruturas complexas demanda a previsão de possíveis processos de ruptura governados por fraturamento, devido à existência de inevitáveis defeitos pré-existentes, como entalhes e fissuras. Por um lado, a complexidade da estrutura e a presença de muitos defeitos a serem considerados no modelo podem tornar a análise inviável devido ao esforço computacional necessário. Por outro lado, é importante procurar desenvolver uma estrutura computacional baseada em métodos numéricos para estudar estes problemas. Um modo de superar as dificuldades mencionadas, portanto tornando possível a análise de estruturas complexas com muitas fissuras e outros defeitos, consiste em combinar um modelo mecânico que seja representativo com um método numérico eficiente. Este é precisamente o objetivo fundamental deste trabalho. Primeiramente, o Método da Partição é utilizado para a construção de um modelo representativo. Em segundo lugar, o Método dos Elementos Finitos Generalizados (GFEM) é empregado por ser um método numérico eficiente, no qual as estratégias de enriquecimento da solução aproximada usando funções de tensão, em particular, podem ser exploradas. A estrutura do GFEM também permite evitar o excessivo refinamento da malha, que aumenta o esforço computacional em análises convencionais nas quais se utiliza o método dos elementos finitos. No Método da Partição, um tipo de método de decomposição, o problema original é subdividido em problemas locais e globais que são então combinados impondo-se a nulidade do vetor de tensões na superfície da fissura. Neste trabalho, o Método da Partição foi completamente programado em linguagem Python® e sua utilização estendida para analisar a propagação de fissuras, incluindo-se a associação do crescimento com a resposta em fadiga. Além disso, o código gerado apresenta diversas características relacionadas aos conceitos da Mecânica da Fratura, como o cálculo do fator de intensidade de tensão (modos I e II) mediante a Integral J. Alguns exemplos são apresentados para ilustrar a propagação de fissuras em estruturas multi-fraturadas. Mostra-se que para este tipo de problemas a estratégia de enriquecimento fornecida pelo GFEM é essencial. Além disso, o exemplo final comprova que a ferramenta computacional permite a investigação de diferentes possíveis cenários de fissuras com uma análise de baixo custo. Conclui-se sobre a representatividade e eficiência da metodologia proposta.

O trabalho trata da utilização de formulações não-convencionais de elementos finitos na obtenção de fatores de intensidade de tensão associados a múltiplas fissuras distribuídas num domínio bidimensional. A formulação do problema de múltiplas fissuras baseia-se numa abordagem de sobreposição proposta pelo Método da Partição (\"Splitting Method\"). Segundo essa abordagem a solução do problema pode ser encontrada a partir da sobreposição de três subproblemas combinados de tal forma que o fluxo de tensão resultante nas faces das fissuras seja nulo. O uso do Método dos Elementos Finitos (MEF) em sua forma convencional pode requerer um refinamento excessivo da rede nesse tipo de problema, aumentando o custo computacional da análise. Objetivando reduzir este custo, empregam-se duas formulações não-convencionais, de forma independente, num dos subproblemas, dito local: a formulação híbrido-Trefftz de tensão e o Método dos Elementos Finitos Generalizados (MEFG). Na formulação híbrido-Trefftz é adotado o recurso do enriquecimento seletivo mediante o refrno- p na aproximação dos campos de deslocamento no contorno do elemento. Já com relação ao MEFG, empregam-se funções polinomiais e a solução analítica da mecânica da fratura como funções enriquecedoras. Exemplos de simulação numérica são apresentados no sentido de comprovar que a utilização dessas formulações não-convencionais juntamente com o Método da Partição viabiliza a obtenção de resultados com boa aproximação com recurso a redes pouco refinadas, reduzindo significativamente o custo computacional de toda a análise. / This paper treats with the use of non-conventional finite element formulations to obtain the stress intensity factor of multiple cracks located in a two-dimensional domain. The formulation of the multiple cracks problem is based on an overlapping approach suggested by the Splitting Method. Accordingly, the solution of the problem can be achieved by dividing the problem in three steps, combined so that the resulting stress flux is zero on the cracks face. The use of the Finite Element Method (FEM) in its conventional formulation requires a mesh refinement in this kind of problem, then increasing the computational cost. Aiming to reduce this cost, two non-conventional formulations are used independently to solve the local problem: the Hybrid-Trefftz stress formulation and the Generalized Finite Elements Method (GFEM). The Hybrid-Trefftz formulation is applied with selective enrichment using p-refinement in the displacements field on the element boundaries. The GFEM employs polynomial functions and analytical solutions of the fracture mechanics as enrichment functions. Examples of numerical simulations are presented in order to show that non- conventional formulations and the Splitting Method can provide accurate results with coarse mesh, thus reducing the computational cost.

Une modélisation numérique des ondes poroélastiques, décrites par le modèle de Biot, est proposée dans le domaine temporel. La dissipation visqueuse à l'intérieur des pores est décrite par le modèle de perméabilité dynamique de Johnson-Koplik-Dashen (JKD). Certains coefficients du modèle de Biot-JKD sont proportionnels à la racine carrée de la fréquence, introduisant dans le domaine temporel des dérivées fractionnaires décalées d'ordre 1/2, revenant à un produit de convolution. Basé sur une représentation diffusive, le produit de convolution est remplacé par un nombre fini de variables de mémoire satisfaisant une équation différentielle ordinaire locale en temps, menant au modèle de Biot-DA (diffusive approximation). Les propriétés des deux modèles sont analysées : hyperbolicité, décroissance de l'énergie, dispersion. On montre que la meilleure méthode de détermination des coefficients de l'approximation diffusive - quadratures de Gauss, optimisation linéaire ou non-linéaire sur la plage de fréquence d'intérêt - est l'optimisation non-linéaire. Une méthode de splitting est utilisée numériquement : la partie propagative est discrétisée par un schéma aux différences finies ADER d'ordre 4, et la partie diffusive est intégrée exactement. Les conditions de saut aux interfaces sont discrétisées avec une méthode d'interface immergée. Des simulations numériques sont présentées pour des milieux isotropes et isotropes transverses. Des comparaisons avec des solutions analytiques montrent l'efficacité et la précision de cette approche. Des simulations numériques en milieux complexes sont réalisées : influence de la porosité d'os spongieux, diffusion multiple en milieu aléatoire. / A time-domain numerical modeling of Biot poroelastic waves is proposed. The viscous dissipation in the pores is described using the dynamic permeability model of Johnson-Koplik-Dashen (JKD). Some of the coefficients in the Biot-JKD model are proportional to the square root of the frequency: in the time-domain, these coefficients introduce shifted fractional derivatives of order 1/2, involving a convolution product. Based on a diffusive representation, the convolution product is replaced by a finite number of memory variables that satisfy local-in-time ordinary differential equations, resulting in the Biot-DA (diffusive approximation). The properties of the two models are analyzed: hyperbolicity, decrease of energy, dispersion. To determine the coefficients of the diffusive approximation, different methods of quadrature are analyzed: Gaussian quadratures, linear or nonlinear optimization procedures in the frequency range of interest. The nonlinear optimization is shown to be the best way of determination. A splitting strategy is applied numerically: the propagative part is discretized using a fourth-order ADER scheme on a Cartesian grid, and the diffusive part is solved exactly. An immersed interface method is implemented to discretize the jump conditions at interfaces. Numerical experiments are presented for isotropic and transversely isotropic media. Comparisons with analytical solutions show the efficiency and the accuracy of this approach. Some numerical experiments are performed in complex media: influence of the porosity of a cancellous bone, multiple scattering across a set of random scatterers.