Este trabalho trata da combinação entre a formulação híbrida-mista de tensão (FHMT) (Freitas et al. (1996)), para a elasticidade plana, com o método dos elementos finitos generalizados (MEFG), Duarte et al. (2000). O MEFG se caracteriza como uma forma não-convencional do método dos elementos finitos (MEF) que resulta da incorporação a este de conceitos e técnicas dos métodos sem malha, como o enriquecimento nodal proposto do método das nuvens “hp". Como na FHMT são aproximados dois campos no domínio (tensão e deslocamento) e um no contorno (deslocamento), diferentes possibilidades de enriquecimento nodal são exploradas. Para a discretização do modelo híbrido-misto empregam-se elementos finitos quadrilaterais com funções de forma bilineares para o domínio e elementos lineares para o contorno. Essas funções são enriquecidas por funções polinomiais, trigonométricas, polinômios que proporcionam distribuição de tensões auto-equilibradas ou mesmo funções especiais relacionadas às soluções dos problemas de fratura. Uma extensão do teste numérico abordado em Zienkiewicz et al. (1986) é proposta como investigação inicial das condições necessárias para garantia de estabilidade da resposta numérica. O estudo da estabilidade é completado com a análise da condição de Babuška-Brezzi (inf-sup). Esta condição é aplicada nos elementos finitos quadrilaterais híbridos-mistos enriquecidos por meio de um teste numérico, denominado de inf-sup teste, desenvolvido com base no trabalho de Chapelle e Bathe (1993). Exemplos numéricos revelam que a FHMT é uma interessante alternativa para obtenção de boas estimativas para os campos de tensões e deslocamentos, usando-se enriquecimento sobre alguns nós de malhas pouco refinadas / This work presents a combination of hybrid-mixed stress model formulation (HMSMF) (Freitas et al. (1996)), to treat plane elasticity problems, with generalized finite element method (GFEM), (Duarte et al. (2000)). GFEM is characterized as a nonconventional formulation of the finite element method (FEM). GFEM is the result of the incorporation of concepts and techniques from meshless methods. One example of these techniques is the nodal enrichment that was formulated in the “hp" clouds method. Since two fields in domain (stress and displacement) and one in boundary (displacement) are approximated in the HMSMF, different possibilities of nodal enrichment are tested. For the discretization of the hybrid-mixed model quadrilateral finite elements with bilinear shape functions for the domain and linear elements for the boundary were employed. These functions are enriched with polynomial functions, trigonometric functions, polynomials that generate self-equilibrated stress distribution, or, even special functions connected with solutions of fracture problems. An extension of the numerical test cited in Zienkiewicz et al. (1986) is proposed as initial investigation of necessary conditions to assure the stability of the numerical answer. The stability study is completed with the analysis of the Babuška-Brezzi (inf-sup) condition. This last condition is applied to hybrid-mixed enrichment quadrilaterals finite elements by means of a numerical test, denominated inf-sup test, which was developed based on paper of Chapelle and Bathe (1993). Numerical examples reveal that HMSMF is an interesting alternative to obtain good estimates of the stress and displacement fields, using enrichment over some nodes of poor meshes

Este trabalho trata da combinação entre a formulação híbrida-mista de tensão (FHMT) (Freitas et al. (1996)), para a elasticidade plana, com o método dos elementos finitos generalizados (MEFG), Duarte et al. (2000). O MEFG se caracteriza como uma forma não-convencional do método dos elementos finitos (MEF) que resulta da incorporação a este de conceitos e técnicas dos métodos sem malha, como o enriquecimento nodal proposto do método das nuvens “hp”. Como na FHMT são aproximados dois campos no domínio (tensão e deslocamento) e um no contorno (deslocamento), diferentes possibilidades de enriquecimento nodal são exploradas. Para a discretização do modelo híbrido-misto empregam-se elementos finitos quadrilaterais com funções de forma bilineares para o domínio e elementos lineares para o contorno. Essas funções são enriquecidas por funções polinomiais, trigonométricas, polinômios que proporcionam distribuição de tensões auto-equilibradas ou mesmo funções especiais relacionadas às soluções dos problemas de fratura. Uma extensão do teste numérico abordado em Zienkiewicz et al. (1986) é proposta como investigação inicial das condições necessárias para garantia de estabilidade da resposta numérica. O estudo da estabilidade é completado com a análise da condição de Babuška-Brezzi (inf-sup). Esta condição é aplicada nos elementos finitos quadrilaterais híbridos-mistos enriquecidos por meio de um teste numérico, denominado de inf-sup teste, desenvolvido com base no trabalho de Chapelle e Bathe (1993). Exemplos numéricos revelam que a FHMT é uma interessante alternativa para obtenção de boas estimativas para os campos de tensões e deslocamentos, usando-se enriquecimento sobre alguns nós de malhas pouco refinadas / This work presents a combination of hybrid-mixed stress model formulation (HMSMF) (Freitas et al. (1996)), to treat plane elasticity problems, with generalized finite element method (GFEM), (Duarte et al. (2000)). GFEM is characterized as a nonconventional formulation of the finite element method (FEM). GFEM is the result of the incorporation of concepts and techniques from meshless methods. One example of these techniques is the nodal enrichment that was formulated in the “hp” clouds method. Since two fields in domain (stress and displacement) and one in boundary (displacement) are approximated in the HMSMF, different possibilities of nodal enrichment are tested. For the discretization of the hybrid-mixed model quadrilateral finite elements with bilinear shape functions for the domain and linear elements for the boundary were employed. These functions are enriched with polynomial functions, trigonometric functions, polynomials that generate self-equilibrated stress distribution, or, even special functions connected with solutions of fracture problems. An extension of the numerical test cited in Zienkiewicz et al. (1986) is proposed as initial investigation of necessary conditions to assure the stability of the numerical answer. The stability study is completed with the analysis of the Babuška-Brezzi (inf-sup) condition. This last condition is applied to hybrid-mixed enrichment quadrilaterals finite elements by means of a numerical test, denominated inf-sup test, which was developed based on paper of Chapelle and Bathe (1993). Numerical examples reveal that HMSMF is an interesting alternative to obtain good estimates of the stress and displacement fields, using enrichment over some nodes of poor meshes

Dans ces travaux de thèse on s’intéresse au développement d’un outil numérique pour résoudre le problème de conduction instationnaire avec changement de phase dans un milieu composite constitué d’une mousse de graphite infiltrée par un matériau à changement de phase tel que le sel, dans le contexte du stockage de l’énergie thermique solaire.Au chapitre 1, on commence par présenter le modèle sur lequel on va travailler. Il estséparé en trois sous-parties : un problème de conduction de chaleur dans la mousse, un problème de changement de phase dans les pores remplis de sel et une condition de résistance thermique de contact entre les deux matériaux qui est traduite par une discontinuité du champ de température.Au chapitre 2, on étudie le problème stationnaire de conduction thermique dans un milieu composite avec résistance de contact. Ceci permet de se focaliser sur la plus grande difficulté présente dans le problème qui est le traitement de la condition de saut à l’interface.Deux méthodes d’éléments finis sont proposées pour résoudre ce problème : une méthode basée sur les éléments finis Lagrange P1 et une méthode hybride-duale utilisant les éléments finis Raviart-Thomas d’ordre 0 et P0. L’analyse numérique des deux méthodes est effectuée et les résultats de tests numériques attestent des efficacités des deux méthodes [10]. Les matériaux à changement de phase qu’on étudie dans le cadre de cette thèse sont des matériaux pures, par conséquent le changement de phase s’effectue en une valeur de température fixe qui est la température de fusion. Ceci est modélisé par un saut dans la fonction fraction liquide et par conséquent dans la fonction enthalpie du matériau. Cette discontinuité représente une difficulté numérique supplémentaire qu’on propose de surmonter en introduisant un intervalle de régularisation autour de la température de fusion.Cette procédure est présentée dans le chapitre 3 où une étude analytique et numérique montre que l’erreur sur la température se comporte comme " en dehors de la zone de mélange, où " est la largeur de l’intervalle de régularisation. Cependant, à l’intérieur l’erreur se comporte comme p " et on montre que cette estimation est optimale. Cette diminution de vitesse de convergence est due à l’énergie qui reste bloquée dans la zone de mélange [58].Dans le chapitre 4 on présente quatre des schémas les plus utilisés pour le traitement de la non-linearité due au changement de phase: mise à jour du terme source, linéarisation de l’enthalpie, la capacité thermique apparente et le schéma de Chernoff. Différents tests numériques sont réalisés afin de tester et comparer ces quatre méthodes pour différents types de problèmes. Les résultats montrent que le schéma de linéarisation de l’enthalpie est le plus précis à chaque pas de temps tans dis que le schéma de la capacité thermique apparente donne de meilleurs résultats au bout d’un certain temps de calcul. Cela indique que si l’on s’intéresse aux états transitoires du matériaux le premier schéma est lemeilleur choix. Cependant, si l’on s’intéresse au comportement thermique asymptotique du matériau le second schéma est plus adapté. Les résultats montrent également que le schéma de Chernoff est le plus rapide parmi les quatre schémas en terme de temps de calcul et donne des résultats comparables à ceux des deux plus précis.Enfin, dans le chapitre 5 on utilise le schéma de Chernoff avec la méthode d’éléments finis hybride-duale Raviart-Thomas d’ordre 0 et P0 pour résoudre le problème non-linéaire de conduction thermique dans un milieu composite réel avec matériau à changement de phase. Le but étant de déterminer si un matériau composite avec une distribution uniforme de pores est assimilable à un matériau à changement de phase homogènes avec des propriétés thermo-physiques équivalentes. Pour toutes les expériences numériques exposées dans ce manuscrit on a utilisé le logiciel libre d’éléments finis FreeFem++ [41]. / In this thesis we aim to develop a numerical tool that allow to solve the unsteady heatconduction problem in a composite media with a graphite foam matrix infiltrated witha phase change material such as salt, in the framework of latent heat thermal energystorage.In chapter 1, we start by explaining the model that we are studying which is separated in three sub-parts : a heat conduction problem in the foam, a phase change problem in the pores of the foam which are filled with salt and a contact resistance condition at the interface between both materials which results in a jump in the temperature field.In chapter 2, we study the steady heat conduction problem in a composite media withcontact resistance. This allow to focus on the main difficulty here which is the treatment of the thermal contact resistance at the interface between the carbon foam and the salt. Two Finite element methods are proposed in order to solve this problem : a finite element method based on Lagrange P1 and a hybrid dual finite element method using the lowest order Raviart-Thomas elements for the heat flux and P0 for the temperature. The numerical analysis of both methods is conducted and numerical examples are given to assert the analytic results. The work presented in this chapter has been published in the Journal of Scientific Computing [10].The phase change materials that we study here are mainly pure materials and as a consequence the change in phase occurs at a single point, the melting temperature. This introduces a jump in the liquid fraction and consequently in the enthalpy. This discontinuity represents an additional numerical difficulty that we propose to overcome by introducing a smoothing interval around the melting temperature. This is explained in chapter 3 where an analytical and numerical study shows that the error on the temperature behaves like " outside of the mushy zone, where _ is the width of the smoothing interval. However, inside the error behaves like p " and we prove that this estimation is optimal due to the energy trapped in the mushy zone. This chapter has been published in Communications in Mathematical Sciences [58].The next step is to determine a suitable time discretization scheme that allow to handle the non-linearity introduced by the phase change. For this purpose we present in chapter 4 four of the most used numerical schemes to solve the non-linear phase change problem : the update source method, the enthalpy linearization method, the apparent heat capacity method and the Chernoff method. Various numerical tests are conducted in order to test and compare these methods for various types of problems. Results show that the enthalpy linearization is the most accurate at each time step while the apparent heat capacity gives better results after a given time. This indicates that if we are interestedin the transitory states the first scheme is the best choice. However, if we are interested in the asymptotic thermal behavior of the material the second scheme is better. Results also show that the Chernoff scheme is the fastest in term of calculation time and gives comparable results to the one given by the first two methods.Finally, in chapter 5 we use the Chernoff method combined with the hybrid-dual finiteelement method with P0 and the lowest order Raviart-Thomas elements to solve thenon-linear heat conduction problem in a realistic composite media with a phase change material. Numerical simulations are realised using 2D-cuts of X-ray images of two real graphite matrix foams infiltrated with a salt. The aim of these simulations is to determine if the studied composite materials could be assimilated to an equivalent homogeneous phase change material with equivalent thermo-physical properties. For all simulationsconducted in this work we used the free finite element software FreeFem++ [41].