Nous construisons une méthode numérique fiable pour simuler un écoulement dans un milieu poreux modélisé par une équation elliptique. La simulation est rendue difficile par les hétérogénéités du milieu, la taille et la géométrie complexe du domaine de calcul. Un maillage d'hexaèdres réguliers ne permet pas de représenter fidèlement les couches géologiques du domaine. Par conséquent, nous sommes amenés à travailler avec un maillage de cubes déformés. Il existe différentes méthodes de volumes finis ou d'éléments finis qui résolvent ce problème avec plus ou moins de succès. Pour la méthode que nous proposons, nous nous imposons d'avoir seulement un degré de liberté par maille pour la pression et un degré de liberté par face pour la vitesse de Darcy, pour rester au plus près des habitudes des codes industriels. Comme les méthodes d'éléments finis mixtes standards ne convergent pas, notre méthode est basée sur un élément fini mixte composite. En deux dimensions, une maille polygonale est découpée en triangles en ajoutant un point au barycentre des sommets, et une expression explicite des fonctions de base a pu être obtenue. En dimension 3, la méthode s'étend naturellement au cas d'une maille pyramidale. Dans le cas d'un hexaèdre ou d'un cube déformé quelconque, la maille est divisée en 24 tétraèdres en ajoutant un point au barycentre des sommets et en divisant les faces en 4 triangles. Les fonctions de base de l'élément sont alors construites en résolvant un problème discret. Les méthodes proposées ont été analysées théoriquement et complétées par des estimateurs a posteriori. Elles ont été expérimentées sur des exemples académiques et réalistes en utilisant le calcul parallèle. / We develop a reliable numerical method to approximate a flow in a porous media, modeled by an elliptic equation. The simulation is made difficult because of the strong heterogeneities of the medium, the size together with complex geometry of the domain. A regular hexahedral mesh does not allow to describe accurately the geological layers of the domain. Consequently, this leads us to work with a mesh made of deformed cubes. There exists several methods of type finite volumes or finite elements which solve this issue. For our method, we wish to have only one degree of freedom per element for the pressure and one degree of freedom per face for the Darcy velocity, to stay as close to the habits of industrial software. Since standard mixed finite element methods does not converge, our method is based on composite mixed finite element. In two dimensions, a polygonal mesh is split into triangles by adding a node to the vertices's barycenter, and explicit formulation of the basis functions was obtained. In dimension 3, the method extend naturally to the case of pyramidal mesh. In the case of a hexahedron or a deformed cube, the element is divided into 24 tetrahedra by adding a node to the vertices's barycenter and splitting the faces into 4 triangles. The basis functions are then built by solving a discrete problem. The proposed methods have been theoretically analyzed and completed by a posteriori estimators. They have been tested on academical and realistic examples by using parallel computation.

Interpolation error estimates in terms of geometric quality measures are established for harmonic coordinates on polytopes in two and three dimensions. First we derive interpolation error estimates over convex polygons that depend on the geometric quality of the triangles in the constrained Delaunay triangulation of the polygon. This characterization is sharp in the sense that families of polygons with poor quality triangles in their constrained Delaunay triangulations are shown to produce large error when interpolating a basic quadratic function. Non-convex polygons exhibit a similar limitation: large constrained Delaunay triangles caused by vertices approaching a non-adjacent edge also lead to large interpolation error. While this relationship is generalized to convex polyhedra in three dimensions, the possibility of sliver tetrahedra in the constrained Delaunay triangulation prevent the analogous estimate from sharply reflecting the actual interpolation error. Non-convex polyhedra are shown to be fundamentally different through an example of a family of polyhedra containing vertices which are arbitrarily close to non-adjacent faces yet the interpolation error remains bounded.

O carste pode ser entendido como um sistema aberto, uma forma de relevo que resulta de processos que operam em conjunto nos subsistemas hidrológico e geoquímico. Os mecanismos de gênese e desenvolvimento de aquíferos cársticos levam à formação de meios anisotrópicos altamente descontínuos e heterogêneos, na sua estrutura espacial e no seu funcionamento temporal, podendo apresentar estrutura de permeabilidade hierárquica com fluxo turbulento. Apesar das limitações impostas pela anisotropia do meio, a descarga de aquíferos cársticos tende a concentrar-se em uma única ressurgência, permitindo, para entender suas propriedades, o uso da análise de sistemas baseada em estatística de séries temporais. O arranjo espacial complexo da drenagem subterrânea tem implicações no sinal hidrológico do exutório do sistema, de forma que a correta interpretação dos espectros obtidos prescindem de uma adequada caracterização da estrutura da drenagem, por meio testes com traçadores. Neste contexto, este trabalho teve como objetivo investigar o sinal hidrológico e sua relação com os padrões de recarga e a geometria e distribuição espacial das drenagens subterrâneas dos aquíferos do planalto cárstico Alambari-Ouro Grosso, localizados na região cárstica do Vale do Ribeira, município de Iporanga, São Paulo. Os métodos e técnicas utilizados envolveram análises por meio de fotointerpretação, morfometria de depressões poligonais, identificação e tipificação in situ e em gabinete das feições de recarga e descarga do sistema, testes qualitativos e quantitativos com traçadores corantes, obtenção de séries históricas de parâmetros hidrológicos e estatística descritivas exploratórias em séries temporais. Os resultados obtidos a partir de testes com traçadores corantes e análises morfométricas demonstraram a existência, no mesmo sinclinal, de dois sistemas independentes e funcionalmente distintos: i) Sistema Alambari, responsável por drenar quase 80% da área das depressões poligonais do planalto e a maior área da contribuição alóctone aos sistemas (65,9%), apresentando assinaturas hidrológicas típicas de um carste bem desenvolvido; e ii) Ouro Grosso, composta por depressões poligonais menos expressivas, recargas difusas, armazenamento epicárstico/pedológico e comportamento mais inercial. O sistema Alambari apresenta ainda depressões poligonais com as maiores declividades e a maior amplitude altimétrica, o que representa recargas potencialmente mais energéticas que no sistema Ouro Grosso. As séries temporais demonstraram que, em qualquer estação climática, o sistema Ouro Grosso é mais inercial que o sistema Alambari e o efeito memória da vazão menos variável em Alambari, característica provavelmente associada a maior capacidade de drenagem do sistema. Comportamento semelhante ao da vazão foi observado na condutividade elétrica, com exceção no inverno de 2017 (o mais seco), onde observou-se uma subida gradativa da mesma, demonstrando que a memória da condutividade elétrica pode refletir a resposta rápida do sistema a impulsos de chuva ou o comportamento de longo prazo associado à lenta e constante mineralização das águas do sistema. O comportamento da condutividade elétrica no sistema Ouro Grosso é anômalo, uma vez que as injeções de água meteórica no sistema resultam em picos de saturação e não diluição. Este fenômeno foi explicado a partir da noção de \"efeito pistão\", ou seja, um impulsionamento de águas saturadas na rede fraturada e nos reservatórios epicársticos/pedológicos, resultando na disponibilização de água mineralizada ao sistema. A análise das correlações cruzadas mostrou que na relação chuva versus condutividade elétrica, nos períodos relativamente mais secos, a desmineralização da água é menor. As análises também evidenciaram o comportamento dos reservatórios dinâmicos cujo fluxo de água mineralizada aumenta quando a recarga é mais intensa. / Karsts can be understood as an open system, a form of relief resulting from processes that operate together in the hydrological and geochemical subsystems. The mechanisms of genesis and development of karst aquifers lead to the formation of highly discontinuous and heterogeneous anisotropic media, in their spatial structure and in their temporal functioning, being able to present a hierarchical permeability structure with turbulent flow. Despite the limitations imposed by the environmental anisotropy, the discharge of karst aquifers tends to concentrate in a single resurgence, allowing, to understand its properties, the use of systems analysis based on time series statistics. The complex spatial arrangement of the underground drainage has implications in the hydrological signal of the exudate of the system, so that the correct interpretation of the obtained spectra doesn\'t have an adequate characterization of the structure of the drainage, through tests with tracers. In this context, the objective of this work was to investigate the hydrological sign and its relation with the recharge patterns and the geometry and spatial distribution of the subterranean drainage of the aquifer of the Alambari-Ouro Grosso karst plateau, located in the karst region of the Ribeira Valley, southeastern Brazil. The methods and techniques used involved analysis through photointerpretation, morphometry of polygonal depressions, in situ identification and typing, and in the recharge and discharge characteristics of the system, qualitative and quantitative tests with dye tracers, obtaining historical series of hydrological parameters and descriptive statistics in time series. The results obtained from tests with dye tracers and morphometric analysis showed the existence, in the same syncline, of two independent and functionally distinct systems: i) Alambari System, responsible for draining almost 80% of the area of the polygonal depressions of the plateau and the largest area of the allochthonous contribution to the systems (65.9%), presenting hydrological signatures typical of a well developed karst; and ii) Ouro Grosso, composed of less expressive polygonal depressions, diffuse refills, epikarst/pedological storage and more inertial behavior. The Alambari system also has polygonal depressions with the highest slopes and the highest altimetric amplitude, which represents potentially more energetic recharges than in the Ouro Grosso system. The time series showed that, in any weather season, the Ouro Grosso system is more inertial than Alambari system and the flow memory effect is less variable in Alambari, a characteristic probably associated to the greater drainage capacity of the system. Similar behavior to the flow rate was observed in the electrical conductivity, except in the winter of 2017 (the driest), where it was observed a gradual increase, demonstrating that the electrical conductivity memory can reflect the rapid response of the system to impulses of rainfall or the long-term behavior associated with the slow and steady mineralization of the system\'s waters. The behavior of the electrical conductivity in the Ouro Grosso system is anomalous, since the injections of meteoric water into the system result in saturation and non-dilution peaks. This phenomenon was explained by the notion of \"piston effect\", a saturated water propulsion in the fractured network and in the epikarst/pedological reservoirs, resulting in the availability of mineralized water to the system. The analysis of the cross correlations showed that in the relationship between rainfall versus electrical conductivity, in the relatively drier periods, water demineralization is lower. The analyzes also showed the behavior of the dynamic reservoirs whose mineralized water flow increases when the recharge is more intense.

L'objectif de cette thèse est d'étudier les comportements typiques d'une famille de transformations du cylindre discret Z×T (où T=R/Z est le cercle de longueur un). Appliquez une rotation à chaque cercle du cylindre puis coupez tous les cercles en deux et déplacez une moitié de chaque cercle d'un niveau vers le bas et une moitié d'un niveau vers le haut. Nous utilisons pour cela des résultats existants en théorie des échanges d'intervalles et en théorie des surfaces de translation compactes. Tout d'abord, nous avons prouvé que pour presque toute suite bi-infinie de rotations, le système obtenu est conservatif (c'est à dire il n'y a pas d'ensemble errant de mesure strictement positive). Ensuite, nous avons prouvé que pour un ensemble Gδ-dense de paramètres, le système est en même temps conservatif, minimal et ergodique. Ce système a un rapport heuristique avec une famille de billards planaires, ainsi qu'une traduction dans des flots sur des surfaces de translation de genre infini. Cela est expliqué dans la thèse. / The main objective of this thesis is the study of the typical dynamical behaviour of a family of transformations on the discrete cylinder Z×T (where T=R/Z is the length one circle). Apply a rotation to every single circle of the cylinder then cut every circle in two and move half of each circle one level down and the other half one level up. To achieve this goal, we use existing results about interval exchange transformations and about compact translation surfaces. First, we proved that for almost every bi-infinite sequence of rotations, the obtained system is conservative (i.e. there is not wandering set of positive measure). Next, we proved that for a Gδ-dense set of parameters, the described system is ergodic, minimal and conservative. This system is heuristically related to a family of planar billiards, it has also a translation into flows on infinite genus translation surfaces.

We present a method for extracting contours from digital images, using techniques from computational geometry. Our approach is different from traditional pixel-based methods in image processing. Instead of working directly with pixels, we extract a set of oriented feature points from the input digital images, then apply classical geometric techniques, such as clustering, linking, and simplification, to find contours among these points. Experiments on synthetic and natural images show that our method can effectively extract contours, even from images with considerable noise; moreover, the extracted contours have a very compact representation.

La formule de trace est un outil privilégié pour l'étude du problème spectral inverse puisqu'elle établit, sur une variété riemannienne compacte, une relation entre le spectre du laplacien et les longueurs des géodésiques périodiques. Cette thèse étend ce type de formule dans deux situations présentant des singularités ponctuelles. Dans ces deux cas, on commence par étudier l'équation des ondes et par établir la propagation des singularités associée. Sur une variété $M$ de dimension $3$, on place un potentiel Dirac en un point $p$. Cela revient à considérer une extension autoadjointe du laplacien, défini sur ${\cal C}^\infty( M\backslash \{p\} )$, différente du laplacien riemannien de $M.$ Le propagateur de l'équation des ondes associée est construit en faisant apparaître des diffractions successives au point $p$, ce qui donne alors la propagation des singularités. La formule de trace en découle~; on montre notamment que les courbes obtenues en suivant successivement un ou plusieurs lacets géodésiques joignant $p$ à $p$ donnent une contribution dont on calcule la partie principale. Sur une surface euclidienne à singularités coniques, il faut commencer par étendre la notion de géodésique en admettant le passage par les points coniques. Au voisinage d'une géodésique $g$, la géométrie locale de l'ensemble des géodésiques (éventuellement) diffractives dépend d'un nombre (appelé {\it complexité classique\/}) que l'on relie à la suite des angles de diffractions le long de $g.$ On montre alors que la propagation des singularités se fait en suivant ces géodésiques généralisées. La trace fait alors apparaître la contribution de géodésiques périodiques diffractives dont on calcule la partie principale sous certaines hypothèses.

Polygonal meshes are a common way of representing 3D surface models in many different areas of computer graphics and geometry processing. However, these models are becoming more and more complex which increases the cost of processing these models. In order to reduce this cost, mesh simplification algorithms are developed. Another important property of a polygonal mesh model is that whether it is regular or not. Regular meshes have many advantages over the irregular ones in terms of memory requirements, efficient processing, rendering etc. In this thesis work, both mesh simplification and regular remeshing algorithms are studied. Moreover, some of the popular mesh libraries are compared with respect to their approaches and performance to the mesh simplification. In addition, mesh models with disk topology are remeshed and converted to regular ones.

The Scotian margin of the east coast of Canada has a large theoretical gas hydrate stability zone (GHSZ) yet review of extensive industry seismic data reveals a prominent BSR at only one location. 3D seismic reflection and long offset (9 km) pre-stack 2D multichannel seismic data were used to study the velocity structure and geophysical characteristics of the hydrate zone and surrounding regions. The Mohican Channel study area shows a unique double BSR at 300 to 450 m below the seafloor in the western section of the study area immediately adjacent to the Mohican Channel in a water depth range of 1500- 1930m. The topmost BSR (BSR 1) is the more extensive of the two covering an area of 150 km2 in the 3D volume and a calculated area of 280 km2 using 2D industry and single-channel seismic profiles outside of the study area. BSR 2 covers an area of ~50 km2 and occurs approximately 80m below BSR 1. A system of polygonal faults is prominent in the area and some faults appear as conduits for gas leakage into the GHSZ. Fluid escape features are common on the surface of BSR 1 but rare on the seafloor suggesting that fluid flux is at lower levels than in the past.

The Eshelby-type problems of an arbitrary-shape polygonal or polyhedral inclusion embedded in an infinite homogeneous isotropic elastic material are analytically solved using a simplified strain gradient elasticity theory (SSGET) that contains a material length scale parameter. The Eshelby tensors for a plane strain inclusion with an arbitrary polygonal cross section and for an arbitrary-shape polyhedral inclusion are analytically derived in general forms in terms of three potential functions. These potential functions, as area integrals over the polygonal cross section and volume integrals over the polyhedral inclusion, are evaluated. For the polygonal inclusion problem, the three area integrals are first transformed to three line integrals using the Green's theorem, which are then evaluated analytically by direct integration. In the polyhedral inclusion case, each of the three volume integrals is first transformed to a surface integral by applying the divergence theorem, which is then transformed to a contour (line) integral based on Stokes' theorem and using an inverse approach. In addition, the Eshelby tensor for an anti-plane strain inclusion with an arbitrary polygonal cross section embedded in an infinite homogeneous isotropic elastic material is analytically solved. Each of the newly derived Eshelby tensors is separated into a classical part and a gradient part. The latter includes the material length scale parameter additionally, thereby enabling the interpretation of the inclusion size effect. For homogenization applications, the area or volume average of each newly derived Eshelby tensor over the polygonal cross section or the polyhedral inclusion domain is also provided in a general form. To illustrate the newly obtained Eshelby tensors and their area or volume averages, different types of polygonal and polyhedral inclusions are quantitatively studied by directly using the general formulas derived. The numerical results show that the components of the each SSGET-based Eshelby tensor for all inclusion shapes considered vary with both the position and the inclusion size. It is also observed that the components of each averaged Eshelby tensor based on the SSGET change with the inclusion size.

Discontinuous planes often develop in soils; examples include shear bands, desiccation cracks, polygonal faults, and hydraulic fractures. These discontinuities affect the mechanical behavior (stiffness and strength) and transport properties of sediments (fluid migration and diffusion). Contrary to discontinuities in solid materials, granular materials such as soils are already separated at the particle scale. Therefore, the fundamental understanding of the development of discontinuities in soils must recognize their inherent granular nature and effective-stress dependent behavior. This research focuses on particle-scale mechanisms involved in contraction-driven shear failure due to mineral dissolution, desiccation cracks, and hydraulic fractures. Complementary experimental, analytical and numerical methods are used to study three cases. Contraction-driven polygonal fault formation under the seabed. Shear failure planes are often found in sediments that formed under near horizontal burial conditions. Particle-scale volume contraction due to mineral dissolution causes a decrease in the state of stress from the insitu K0-condition to the active failure Ka stress field. Shear strain localization follows in sediment with post-peak strain softening response. Desiccation cracks in saturated fine soils. The formation of desiccation cracks in soils is often interpreted in terms of tensile strength, which contradicts the cohesionless, effective stress dependent frictional behavior of fine grained soils. Experimental results monitored using high resolution time lapse photography point to a proper effective stress-dependent mechanism centered on the invasion of the air-water interface membrane. Miscible and immiscible fluid-driven fracture formation. Hydraulic fracture in granular materials cause grain separation and the development of conduits for preferential fluid flow leading to fracture formation due to the forced invasion of either immiscible or miscible fluids. Capillary, seepage, and skeletal interparticle forces define particle scale mechanisms at the fracture tip. Numerical simulations confirm that the effective stress remains in compression everywhere throughout the granular medium in the three localization mechanisms.