Let us have a boundary value problem for the fourth order homogeneous linear ordinary differential equation with constant coefficients, four zero boundary conditions and one additional unilateral condition in the interior of the domain. We prove the existence of a sequence of non-trivial solutions for three types of boundary conditions.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Um código computacional para escoamentos bifásicos incorporando metodologia híbrida entre oMétodo dos Elementos Finitos e a descrição Lagrangeana-Euleriana Arbitrária do movimento é usado para simular a dinâmica de um jato transversal de gotas na zona primária de quebra. Os corpos dispersos são descritos por meio de um método do tipo front-tracking que produz interfaces de espessura zero através de malhas formadas pela união de elementos adjacentes em ambas as fases e de técnicas de refinamento adaptativo. Condições de contorno periódicas são implementadas de modo variacionalmente consistente para todos os campos envolvidos nas simulações apresentadas e uma versão modificada do campo de pressão é adicionada à formulação do tipo um-fluido usada na equação da quantidade de movimento linear. Simulações numéricas diretas em três dimensões são executadas para diferentes configurações de líquidos imiscí veis compatíveis com resultados experimentais encontrados na literatura. Análises da hidrodinâmica do jato transversal de gotas nessas configurações considerando trajetórias, variação de formato de gota, espectro de pequenas perturbações, além de aspectos complementares relativos à qualidade de malha são apresentados e discutidos. / A two-phase flow computational code taking a hybrid Arbitrary Lagrangian-Eulerian desciption of movement along with the Finite Element Method is used to simulate the dynamics of an incompressible drop jet in crossflow in the primary breakup zone. Dispersed entities are described by means of a front-tracking method which produces zero-thickness interfaces through contiguous element meshing and adaptive refinement techniques. Periodic boundary conditions are implemented in a variationally consistent way for all the scalar fields involved in the presented simulations and amodified version of the pressure field is added to the one-fluid formulation employed in the momentum equation. Three-dimensional direct numerical simulations for different flow configurations of immiscible liquids pertinent to experimental results found in literature. Analyses of the hydrodynamics of the drop jet in crossflow in these configurations considering trajectories, drop shape variations, spectrum of small disturbances, besides additional aspects relating to mesh quality are presented and discussed.

Fundação Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro / Como eventos de fissão induzida por nêutrons não ocorrem nas regiões nãomultiplicativas de reatores nucleares, e.g., moderador, refletor, e meios estruturais, essas regiões não geram potência e a eficiência computacional dos cálculos globais de reatores nucleares pode portanto ser aumentada eliminando os cálculos numéricos explícitos no interior das regiões não-multiplicativas em torno do núcleo ativo. É discutida nesta dissertação a eficiência computacional de condições de contorno aproximadas tipo albedo na formulação de ordenadas discretas (SN) para problemas de autovalor a dois grupos de energia em geometria bidimensional cartesiana. Albedo, palavra de origem latina para alvura, foi originalmente definido como a fração da luz incidente que é refletida difusamente por uma superfície. Esta palavra latina permaneceu como o termo científico usual em astronomia e nesta dissertação este conceito é estendido para reflexão de nêutrons. Este albedo SN nãoconvencional substitui aproximadamente a região refletora em torno do núcleo ativo do reator, pois os termos de fuga transversal são desprezados no interior do refletor. Se o problema, em particular, não possui termos de fuga transversal, i.e., trata-se de um problema unidimensional, então as condições de contorno albedo, como propostas nesta dissertação, são exatas. Por eficiência computacional entende-se analisar a precisão dos resultados numéricos em comparação com o tempo de execução computacional de cada simulação de um dado problema-modelo. Resultados numéricos para dois problemas-modelo com de simetria são considerados para ilustrar esta análise de eficiência. / As neutron fission events do not take place in the non-multiplying regions of nuclear reactors, e.g., moderator, reflector, and structural core, these regions do not generate power and the computational efficiency of nuclear reactor global calculations can hence be improved by eliminating the explicit numerical calculations within the non-multiplying regions around the active domain. Discussed here is the computational efficiency of approximate discrete ordinates (SN) albedo boundary conditions for two-energy group eigenvalue problems in X,Y geometry. Albedo, the Latin word for whiteness, was originally defined as the fraction of incident light reflected diffusely by a surface. This Latin word has remained the usual scientific term in astronomy and in this dissertation this concept is extended for the reflection of neutrons. The non-standard SN albedo substitutes approximately the reflector region around the active domain, as we neglect the transverse leakage terms within the nonmultiplying reflector. Should the problem have no transverse leakage terms, i.e., onedimensional slab geometry, then the offered albedo boundary conditions are exact. By computational efficiency we mean analyzing the accuracy of the numerical results versus the CPU execution time of each run for a given model problem. Numerical results to two symmetric test problems are shown to illustrate this efficiency analysis.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Um código computacional para escoamentos bifásicos incorporando metodologia híbrida entre oMétodo dos Elementos Finitos e a descrição Lagrangeana-Euleriana Arbitrária do movimento é usado para simular a dinâmica de um jato transversal de gotas na zona primária de quebra. Os corpos dispersos são descritos por meio de um método do tipo front-tracking que produz interfaces de espessura zero através de malhas formadas pela união de elementos adjacentes em ambas as fases e de técnicas de refinamento adaptativo. Condições de contorno periódicas são implementadas de modo variacionalmente consistente para todos os campos envolvidos nas simulações apresentadas e uma versão modificada do campo de pressão é adicionada à formulação do tipo um-fluido usada na equação da quantidade de movimento linear. Simulações numéricas diretas em três dimensões são executadas para diferentes configurações de líquidos imiscí veis compatíveis com resultados experimentais encontrados na literatura. Análises da hidrodinâmica do jato transversal de gotas nessas configurações considerando trajetórias, variação de formato de gota, espectro de pequenas perturbações, além de aspectos complementares relativos à qualidade de malha são apresentados e discutidos. / A two-phase flow computational code taking a hybrid Arbitrary Lagrangian-Eulerian desciption of movement along with the Finite Element Method is used to simulate the dynamics of an incompressible drop jet in crossflow in the primary breakup zone. Dispersed entities are described by means of a front-tracking method which produces zero-thickness interfaces through contiguous element meshing and adaptive refinement techniques. Periodic boundary conditions are implemented in a variationally consistent way for all the scalar fields involved in the presented simulations and amodified version of the pressure field is added to the one-fluid formulation employed in the momentum equation. Three-dimensional direct numerical simulations for different flow configurations of immiscible liquids pertinent to experimental results found in literature. Analyses of the hydrodynamics of the drop jet in crossflow in these configurations considering trajectories, drop shape variations, spectrum of small disturbances, besides additional aspects relating to mesh quality are presented and discussed.

Neste trabalho definimos o operador não local, raiz quadrada do Laplaciano com condição de fronteira de Neumann, através do método de extensão harmônica. O estudo foi feito com o auxílio das séries de Fourier em domínios limitados, como sendo o intervalo, o quadrado e a bola. Posteriormente, aplicamos nosso estudo, à problemas elípticos não lineares envolvendo o operador não local raiz quadrada do Laplaciano com condição de fronteira de Neumann. / In this work we define the non-local operator, square root of the Laplacian with Neumann boundary condition, using the method of harmonic extension. The study was done with the aid of Fourier series in bounded domains, as the interval, the square and the ball. Subsequently, we apply our study, the nonlinear elliptic problems involving non-local operator square root of the Laplacian with Neumann boundary condition.

Le bon fonctionnement et les performances des turbines à vapeur sont liés à l’état de la vapeur et notamment au taux d’humidité qu’elle contient. EDF souhaite pouvoir maîtriser les phénomènes spécifiques à ces problématiques afin d’améliorer l’utilisation et l’évolution de ses turbines. Le sujet de recherche concerne la modélisation de la formation de l’humidité dans un corps de turbine et l’étude des couplages entre la phase liquide et les instationnarités. Dans ce contexte, la démarche adoptée est la suivante : la présence d’humidité est prise en compte à l’aide d’un modèle homogène, couplé à des modèles de condensation permettant de prendre en compte les phénomènes hors-équilibre thermodynamique : le grossissement et la nucléation des gouttes d’eau dans la vapeur. Pour mener à bien les calculs, des méthodes numériques adaptées aux gaz réels ont été utilisées et testées à l’aide d’un code monodimensionnel avant d’être intégrées dans le code 3D elsA. Deux types de modèles de condensation ont été mis en œuvre, considérant ou non la polydispersion des gouttes dans la vapeur. Les couplages instationnaires entre la condensation et l’écoulement principal ont été étudiés à différents niveaux d’observations (1D, 1D − 3D, 3D). Il a été montré que la méthode des moments apporte une richesse supplémentaire par rapport à un modèle mono-dispersé, et permet de mieux capter les couplages instationnaires entre l’humidité et le champ principal. / In addition to conventional turbomachinery problems, both the behavior and performances of steam turbines are highly dependent on the vapour thermodynamic state and the presence of a liquid phase. EDF, the main French electricity producer, is interested in further developing its’ modelling capabilities and expertise in this area to allow for operational studies and long-term planning. This PhD thesis explores the modelling of wetness formation and growth in a steam turbine and an analysis of the coupling between the liquid phase and the main flow unsteadiness. To this end, the work in this thesis took the following approach. Wetness was accounted for using a homogeneous model coupled with transport equations to take into account the effects of non-equilibrium phenomena, such as the growth of the liquid phase and nucleation. The real gas attributes of the problem demanded adapted numerical methods. Before their implementation in the 3D elsA solver, the accuracy of the chosen models was tested using a developed one-dimensional nozzle code. In this manner, various condensation models were considered, including both polydispersed and monodispersed behaviours of the steam. Finally, unsteady coupling effects were observed from several perspectives (1D, 1D − 3D, 3D), demonstrating the ability of the method of moments to sustain unsteady phenomena which were not apparent in a simple monodispersed model.

La problématique de la stabilisation des flammes non-prémélangées reste primordiale. Il faut pour la résoudre déterminer l’importance relative des phénomènes aérodynamiques, thermiques et chimiques intervenant dans les mécanismes de stabilisation. La démarche expérimentale utilisée pour cela au cours de cette thèse porte une attention particulière sur l’influence des conditions aux limites thermiques, et comporte deux volets à travers lesquels le rôle des transferts thermiques est mis en exergue. D’abord, la zone d’attachement d’une flamme stabilisée derrière la lèvre du brûleur est examinée, en considérant les couplages entre le positionnement du bout de flamme à proximité du brûleur et la température de la lèvre. Différentes régions ont ainsi pu être identifiées selon le comportement du bout de flamme, qui évolue depuis une nature diffusive vers une nature propagative à l’approche des limites aérodynamiques de stabilité. Par ailleurs, une modification des propriétés thermiques du brûleur a permis de mettre en évidence une évolution, avec la température de la lèvre, du rôle relatif des modes de coincement thermique et chimique de la flamme par la paroi du brûleur. Ensuite, l’étude concerne non plus un état stabilisé de la flamme, mais les transitions entre les différents régimes de combustion, et plus particulièrement le décrochage d’une flamme attachée. L’examen des conditions conduisant à la déstabilisation de la flamme est un moyen d’apporter des éléments-clefs de compréhension quant aux couplages et aux équilibres aérothermochimiques prévalant préalablement au décrochage. Une évolution du processus de décrochage a ainsi été mise en avant avec l’augmentation de la température initiale des réactants, en lien avec l’évolution de phénomènes transitoires d’extinction locale de la zone de réaction. / Non-premixed flame stabilization is still an important issue in combustion. Addressing this issue requires to evaluate the relative importance of aerodynamic, thermal and chemical phenomena involved in the stabilization mechanisms. This thesis develops to this end an experimental approach, with a particular focus on the influence of thermal boundary conditions, examining the role of heat transfer in a twofold analysis. At first, the attachment zone of a rim-stabilized jet-flame is investigated, by careful consideration of the coupling existing between the burner lip temperature and the flame attachment location relative to the burner. Several regions have been identified according to the flame leading edge behavior, which evolves from diffusive to propagative closer to the aerodynamic stability limits. Besides, by modifying the burner thermal properties, a change has been shown in the relative roles of thermal and chemical quenching of the flame by the burner wall, depending on the burner lip temperature. Secondly, the attention is directed to transitions between different combustion regimes, namely attached and lifted flames. In particular, beyond the stable state of an attached flame, its lifting process is investigated. Examining in which conditions destabilization of the flame occurs indeed appears to be an ideal opportunity to gain insight into the aerothermochemical coupling and equilibriums existing prior to lift-off. Thus, the lifting process has been shown to be modified by the reactant initial temperature, in close relationship with the change in occurrence of localized transitory extinction events of the reaction zone.

Cette thèse se place dans le contexte général de la modélisation de réseaux d’antennes de grande taille, avec pour objectif d’atteindre un niveau de précision suffisamment élevé pour permettre une optimisation complète des performances et en particulier une amélioration de l’efficacité énergétique. Partant du constat que l’optimisation électromagnétique de grands réseaux représente un verrou si les couplages doivent être modélisés efficacement, cette thèse propose la mise en œuvre d’une méthode permettant la modélisation fine de grands réseaux d'antennes tout en réduisant les temps de calcul et en conservant un haut degré de précision. L'objectif est de montrer qu'une approche dérivée des formalismes périodiques infinis connus de la littérature permet d'obtenir une matrice [S] complète d'un réseau depuis l'étude d'une cellule unitaire. Après avoir présenté un état de l'art sur les réseaux d'antennes et leurs méthodes d'analyse, l'approche de modélisation proposée est détaillée. Des véhicules de test numériques et expérimentaux, permettant de valider cette méthode de modélisation, sont ensuite réalisés. L'approche est enfin utilisée avec succès pour deux types d'application. / This thesis is organized in the general context of modeling a large antenna arrays with the aim of achieving a high level of precision. This modeling allows a complete optimization of the performances and an enhancement of the energy efficiency. Given that the electromagnetic optimization of large arrays still represents a challenge if the mutual coupling is not efficiently modeled. This work offers the implementation of a method allowing the precise modeling of large antenna arrays while reducing the computation time and maintaining a high degree of precision. The aim is to demonstrate that the derived approach from the infinite periodic formalisms makes it possible to obtain a full [S] matrix of an array based on the study of a unit cell. After presenting a state of the art of antenna arrays and their methods of analysis, the proposed modeling approach is explained. Numerical and Experimental demonstrators are then made for the validation. Finally, this method is successfully used for two types of applications.

This paper presents a methodology for determining eigenfunctions and frequencies of the Euler-Bernoulli model for elastic beams that can include damping and devices located at intermediate or end points of the beam. The eigenfunctions or vibration modes of the beam are obtained by using solution basis generated by the dynamic solution of a fourth-order differential equation, through a block matrix formulation of the boundary and compatibility conditions. The use of the dynamic basis has been often used to reduce the calculations in obtaining the modes and frequencies. Forced responses are obtained with the Galerkin method by modifying the classical modal analysis with the inclusion of new conditions of orthogonality between modes that are suitable for problems with viscous damping or non-classical boundary conditions. / Este trabalho apresenta uma metodologia para determinar as autofunções e as frequências de um modelo Euler-Bernoulli para vigas elásticas que podem incluir amortecimento e dispositivos localizados num ponto intermediário ou nos extremos da viga. As autofunções ou modos de vibração da viga são obtidos usando uma base de solução gerada pela solução dinâmica de uma equação diferencial de quarta ordem, através de uma formulação matricial em blocos para as condições de contorno e de compatibilidade. O uso da base dinâmica tem sido frequentemente utilizada para reduzir os cálculos na obtenção dos modos e das frequências. Respostas forçadas são obtidas usando o método de Galerkin, modificando a análise modal clássica com a inclusão de novas condições de ortogonalidade entre os modos que são adequadas para problemas com amortecimento viscoso ou com condições de contorno não-clássicas

Cette étude documente la succession de phases de rift qui se sont formées le long d'une marge continentale dont l'évolution géologique a été particulièrement longue. Ce type d'évolution géologique est bien admise dans la littérature mais reste encore peu comprise en terme de processus géologiques. Ici, il est spécifiquement étudié comment ce type de rifts peut évoluer au cours du temps et il est tout particulièrement mis en évidence les processus de réactivation de failles et la formation des éléments géologiques associés dans l'évolution des bassins en extension. Cette étude résume les résultats principaux de deux projets de recherche menés sur(1) le Bassin de Porcupine (offshore irlandais) qui s'est formé pendant la propagation du système de rift nord-atlantique et (2) la région de la Mer de Corail (Papouasie Nouvelle Guinée) dont l'évolution en contexte de supra-subduction est liée à l'affrontement des plaques Pacifique et Australie. Ces deux projets de recherche discutent spécifiquement de l'enregistrement sédimentaire et de l'évolution des failles à partir de données sismiques corrélées à des données de puits, de magnétisme ou de gravimétrie.Les deux bassins ont évolué en bordure d'orogènes depuis le Paléozoïque supérieur. Il est démontré que l'extension s'est initiée au long de structures anciennes qui ont été réactivées lors de l'étalement orogénique mis en évidence par la migration de dépocentres. L'extension s'est ensuite développée de façon discontinue au cours de divers épisodes extensifs pulsés. Les premiers épisodes montrent en général des bassins étroits, plus ou moins discrets et qui sont remplis de dépôts continentaux qui passent progressivement à des environnements marins. L'extension est d'abord diffuse puis localisée le long de failles bordières qui accommodent l'ensemble de l'extension et le boudinage de la croûte. Au fur et à mesure que l'extension progresse, l'influence structurale de la croûte continentale diminue et la déformation évolue vers l'océan avant le stade ultime de déchirure continentale de façon très localisée. Le rifting se termine en général dès que les conditions aux limites changent, résultant en la production de bassins océaniques en propagateur qui sont transversaux aux bassins néoformés. Dans les marges classiques de type atlantique (p.ex. Bassin de Porcupine), ceci est lié à la propagation de croûte océanique dans des bassins plus ou moins perpendiculaires tels que le Golfe deGascogne dont l'ouverture stoppe la déformation extensive. Dans les zones en supra-subduction (par ex. Mer de Corail), la fin de l'extension est plutôt liée à un changement dans la dynamique de subduction qui peut contrôler l'ensemble de la propagation de l'océanisation.Ainsi, il apparait que l'extension migre progressivement depuis une zone initiale en réactivant d'anciennes structures orogéniques quelles que soit les conditions géodynamiques initiales. Ici, il est proposé qu'une phase de déformation souligne un mégacycle extensif durant lequel des grabens individuels, des bassins riftés ou des systèmes de rifts se forment successivement et surimposent les uns aux autres. Chaque mégacycle peut se définir sédimentologiquement et tectoniquement. La surimposition montre en général une déformation qui évolue progressivement vers l'océan par la réactivation spécifique de structures pré-existantes.L'initiation de l'extension et le passage d'un mégacycle à un autre est en général lié par un changement des conditions aux limites. / This study documents the succession of rifting phases that evolve along continental margins of longstanding history. This type of evolution, well admitted in the literature, remains poorly understood in terms of geological processes. Herein, the way rifts evolve through is specifically studied to highlight the fault reactivation processes and the formation of associated geological elements during the evolution of extensional basins. This study summarises some principal results of two research projects focussing on (1) the Porcupine Basin (offshore Ireland) that formed from the North-Atlantic rift propagation and (2) the Coral Sea region (Papua New Guinea) which the evolution within a supra-subduction context links to the convergence of the Pacific and Australian lithospheric plates. These two research projects specifically discuss the sedimentary record and the faulting evolution from seismic data correlated to wells, magnetism and gravimetry.Both basins evolved from the Late Palaeozoic on sides of orogens since the Late Palaeozoic. It is demonstrated that extension initiated along old structures that were reactivated during a general orogenic collapse evidenced by migration of depocentres. Then, extension discontinuously developed during several rifting events that are usually internally pulsed. The first rift events are usually very narrow and discreet, and evolve in a continental domain. Deformation progressively passes to localized normal faulting implying that major bounding faults progressively accommodate the extension. As long as extension progresses, the influence of continental structures tend to decrease and the deformation evolves oceanward prior continental tear. Extension generally stops once boundary conditions change, resulting in oceanic crust that may propagate transversally (oceanic propagators).Along classical Atlantic-type margins (e.g. Porcupine Basin), this links to the propagation of the oceanic crust but internal crosscutting oceanic basins (e.g. Bay of Biscay) stop the deformation prior to seafloor spreading that form the overall oceanic crust.In supra-subduction zones (e.g. Coral Sea), it is rather related to a change in the dynamics of the subduction which may control the overall direction of the spreading propagation.Thus, extension migrates progressively away from the initial deformation core by reactivating pre-existing structures, whatever geodynamic conditions are but with a general decrease in the influence of old continental fabrics. Herein, it is proposed herein that a deformation phase expresses as an extensional megacycle during which graben, basins or rift settings develop successively and overprint the ones another. Each extensional megacycle may be defined sedimentologically and tectonically. Their overlap generally shows an oceanward migration which reactivates specific pre-existing structures. The initiation of extension and the evolution from a megacycle another is generally linked associated to a variation of the boundary conditions.