In this dissertation, a novel method to extrude near-body meshes from surface meshes of arbitrary topology that exploits topologically adaptive generalized elements to improve mesh quality is presented. Specifically, an advancing layer algorithm to generate near-body meshes which are appropriate for viscous fluid flows is discussed. First, an orthogonal two-layer algebraic reference mesh is generated. The reference mesh is then smoothed using a locally three-dimensional Poisson-type mesh generation equation that is generalized to smooth extruded meshes of arbitrary surface topology. Local quality improvement operations such as edge collapse, face refinement, and local reconnection are performed in each layer to drive the mesh toward isotropy. An automatic marching thickness reduction algorithm is used to extrude from multiple geometries in close proximity. A global face refinement algorithm is used to improve the transition from the extruded mesh to the voidilling tetrahedral mesh. A few example meshes along with quality plots are presented to demonstrate the efficacy of the algorithms developed.

A block-based adaptive mesh refinement (AMR) finite-volume scheme is developed for solution of hyperbolic conservation laws on two-dimensional hybrid multi-block meshes. A Godunov-type upwind finite-volume spatial-discretization scheme, with piecewise limited linear reconstruction and Riemann-solver based flux functions, is applied to the quadrilateral cells of a hybrid multi-block mesh and these computational cells are embedded in either body-fitted structured or general unstructured grid partitions of the hybrid grid. A hierarchical quadtree data structure is used to allow local refinement of the individual subdomains based on heuristic physics-based refinement criteria. An efficient and scalable parallel implementation of the proposed algorithm is achieved via domain decomposition. The performance of the proposed scheme is demonstrated through application to solution of the compressible Euler equations for a number of flow configurations and regimes in two space dimensions. The efficiency of the AMR procedure and accuracy, robustness, and scalability of the hybrid mesh scheme are assessed.

A block-based adaptive mesh refinement (AMR) finite-volume scheme is developed for solution of hyperbolic conservation laws on two-dimensional hybrid multi-block meshes. A Godunov-type upwind finite-volume spatial-discretization scheme, with piecewise limited linear reconstruction and Riemann-solver based flux functions, is applied to the quadrilateral cells of a hybrid multi-block mesh and these computational cells are embedded in either body-fitted structured or general unstructured grid partitions of the hybrid grid. A hierarchical quadtree data structure is used to allow local refinement of the individual subdomains based on heuristic physics-based refinement criteria. An efficient and scalable parallel implementation of the proposed algorithm is achieved via domain decomposition. The performance of the proposed scheme is demonstrated through application to solution of the compressible Euler equations for a number of flow configurations and regimes in two space dimensions. The efficiency of the AMR procedure and accuracy, robustness, and scalability of the hybrid mesh scheme are assessed.

La compréhension des écoulements souterrains est importante pour de nombreuses applications comme l’énergie ou le stockage des déchets nucléaires. Cette thèse, effectuée en collaboration avec le Bureau de Recherches Géologiques et Minières (BRGM), est dédiée à la simulation des écoulements diphasiques compositionnels thermiques en milieux poreux et ses applications à la géothermie haute énergie et plus particulièrement au champ géothermique de Bouillante (Guadeloupe). Tout d’abord, deux formulations à variables persistantes sont comparées en termes d’implémentation et de convergence numérique. Dans ces deux formulations, les fractions molaires d’une phase absente sont étendues par celles à l’équilibre thermodynamique avec la phase présente. Il en résulte que l’ensemble des variables principales et des équations ne dépend pas de l’ensemble de phases présentes. De plus, l’équilibre thermodynamique est exprimé par une contrainte de complémentarité pour chacune des phases, ce qui permet l’utilisation de méthodes de type semi-smooth Newton pour résoudre les systèmes non-linéaires. D’autre part, cette thèse présente une nouvelle méthodologie combinant des discrétisations centrées aux noeuds (le schéma Vertex Approximate Gradient - VAG) et aux faces (le schéma Hybrid Finite Volume - HFV) sur une partition arbitraire des ensembles de mailles ou de faces, dans le but d’adapter le choix du schéma aux différentes parties du maillage. En effet, les maillages hybrides composés de différents types de mailles sont plus adaptés à la discrétisation de la géologie et de la géométrie des différents domaines d’un système géothermique. Ainsi le schéma peut être choisi localement en fonction de la géométrie de la maille et des propriétés pétrophysiques. L’analyse de convergence est effectuée dans le cadre des discrétisations Gradient pour des problèmes de diffusion du second ordre et la convergence est confirmée numériquement sur différents types de maillages hybrides 3D. Ensuite la discrétisation VAG-HFV est étendue au cas des écoulements de Darcy diphasiques non-isothermes compositionnels et est appliquée au cas test 2D représentant le plan de faille vertical du réservoir géothermique de Bouillante. Un autre aspect important de la modélisation des flux géothermiques consiste à prendre en compte les interactions entre le flux dans le milieu poreux et l’atmosphère. Puisque le couplage entre le modèle poreux et un modèle 2D surfacique ou 3D atmosphérique n’est pas réaliste en terme de coût de calcul aux échelles spatiale et temporelle géologiques, l’interaction sol-atmosphère est modélisée grâce à une condition limite prenant en compte l’équilibre de matière et d’énergie à l’interface. Ce modèle considère une couche limite atmosphérique avec transfert convectif molaire et thermique (en supposant l’évaporation de la phase liquide), une condition de débordement liquide aux surfaces d’infiltration, ainsi que le rayonnement thermique et la recharge en eau douce due aux précipitations. Cette condition limite est évaluée à l’aide d’une solution de référence couplant les écoulements non-isothermes liquide-gaz en milieu poreux et le gaz dans le milieu libre. Elle est ensuite étudiée numériquement en terme de convergence et de solution sur des cas tests géothermiques, dont le plan de faille vertical du réservoir géothermique de Bouillante. En complément est présenté le travail issu d’une collaboration lors de l’école d’été du CEMRACS 2016. Le projet consistait à ajouter un modèle de puits multi-branche thermique au code ComPASS, un nouveau simulateur géothermique parallèle basé sur des maillages non-structurés avec la possibilité de représenter des fractures. / The study of the subsurface flows is important for various applications such as energy or nuclear waste storage. This thesis, performed in collaboration with the French Geological Survey (BRGM), is dedicated to the simulation of non-isothermal compositional two-phase flows in porous media and its applications to high-energy geothermal fields and more precisely to the Bouillante field (Guadeloupe, French West Indies). First of all, two persistent variable formulations are compared in terms of implementation and numerical convergence. In these two formulations, the choice of the principal variables is based on with the extension of the phase molar fractions by the one at thermodynamic equilibrium with the present phase. It results that the set of principal variables and equations does not depend on the set of present phases. It also has the advantage to express the thermodynamic equilibrium as complementarity constraints, which allows the use of semi-smooth Newton methods to solve the non-linear systems. Moreover, this thesis presents a new methodology to combine a node-centered discretization (the Vertex Approximate Gradient scheme - VAG) and a face-centered discretization (the Hybrid Finite Volume scheme - HFV) on arbitrary subsets of cells or faces in order to choose the best-suited scheme in different parts of the mesh. Indeed, hybrid meshes composed of different types of cells are best suited to discretize the geology and geometry of the different parts of the geothermal system. Then, the scheme is adapted locally to the type of mesh/ cells and to petrophysical properties. The convergence analysis is performed in the gradient discretization framework over second order diffusion problems and the convergence is checked numerically on various types of hybrid three-dimensional meshes. Then, the VAG-HFV discretization is extended to non-isothermal compositional liquid-gas Darcy flows and is applied on the two dimensional cross-section of the Bouillante high temperature geothermal reservoir. Another important aspect of the geothermal flows modelling consists in considering the interactions between the porous medium and the atmosphere. Since the coupling between the porous medium and the 2D surface of 3D atmospheric flows is not computationally realistic at the space and time scales of a geothermal flow, the soil-atmosphere interaction is modelled using an advanced boundary condition accounting for the matter (mole) and energy balance at the interface. The model considers an atmospheric boundary layer with convective molar and energy transfers (assuming the vaporization of the liquid phase in the atmosphere), a liquid outflow condition at seepage surfaces, as well as the heat radiation and the precipitation influx. This boundary condition is assessed using a reference solution coupling the Darcy flow to a full-dimensional gas free flow. Then, it is studied numerically in terms of solution and convergence of the Newton-min non-linear solvers on several geothermal test cases including two-dimensional simulations of the Bouillante geothermal field. In addition is presented the collaborative project which took place during the CEMRACS summer school 2016. The project consisted in adding a multibranch thermal well model into the ComPASS code, a new geothermal simulator based on unstructured meshes and adapted to parallel distributed architectures with the ability to represent fractures.

This dissertation develops adaptive hp-Finite Element (FE) technology and a parallel sparse direct solver enabling the accurate modeling of the absorption of Electro-Magnetic (EM) energy in the human head. With a large and growing number of cell phone users, the adverse health effects of EM fields have raised public concerns. Most research that attempts to explain the relationship between exposure to EM fields and its harmful effects on the human body identifies temperature changes due to the EM energy as the dominant source of possible harm. The research presented here focuses on determining the temperature distribution within the human body exposed to EM fields with an emphasis on the human head. Major challenges in accurately determining the temperature changes lie in the dependence of EM material properties on the temperature. This leads to a formulation that couples the BioHeat Transfer (BHT) and Maxwell equations. The mathematical model is formed by the time-harmonic Maxwell equations weakly coupled with the transient BHT equation. This choice of equations reflects the relevant time scales. With a mobile device operating at a single frequency, EM fields arrive at a steady-state in the micro-second range. The heat sources induced by EM fields produce a transient temperature field converging to a steady-state distribution on a time scale ranging from seconds to minutes; this necessitates the transient formulation. Since the EM material properties depend upon the temperature, the equations are fully coupled; however, the coupling is realized weakly due to the different time scales for Maxwell and BHT equations. The BHT equation is discretized in time with a time step reflecting the thermal scales. After multiple time steps, the temperature field is used to determine the EM material properties and the time-harmonic Maxwell equations are solved. The resulting heat sources are recalculated and the process continued. Due to the weak coupling of the problems, the corresponding numerical models are established separately. The BHT equation is discretized with H¹ conforming elements, and Maxwell equations are discretized with H(curl) conforming elements. The complexity of the human head geometry naturally leads to the use of tetrahedral elements, which are commonly employed by unstructured mesh generators. The EM domain, including the head and a radiating source, is terminated by a Perfectly Matched Layer (PML), which is discretized with prismatic elements. The use of high order elements of different shapes and discretization types has motivated the development of a general 3D hp-FE code. In this work, we present new generic data structures and algorithms to perform adaptive local refinements on a hybrid mesh composed of different shaped elements. A variety of isotropic and anisotropic refinements that preserve conformity of discretization are designed. The refinement algorithms support one- irregular meshes with the constrained approximation technique. The algorithms are experimentally proven to be deadlock free. A second contribution of this dissertation lies with a new parallel sparse direct solver that targets linear systems arising from hp-FE methods. The new solver interfaces to the hierarchy of a locally refined mesh to build an elimination ordering for the factorization that reflects the h-refinements. By following mesh refinements, not only the computation of element matrices but also their factorization is restricted to new elements and their ancestors. The solver is parallelized by exploiting two-level task parallelism: tasks are first generated from a parallel post-order tree traversal on the assembly tree; next, those tasks are further refined by using algorithms-by-blocks to gain fine-grained parallelism. The resulting fine-grained tasks are asynchronously executed after their dependencies are analyzed. This approach effectively reduces scheduling overhead and increases flexibility to handle irregular tasks. The solver outperforms the conventional general sparse direct solver for a class of problems formulated by high order FEs. Finally, numerical results for a 3D coupled BHT with Maxwell equations are presented. The solutions of this Maxwell code have been verified using the analytic Mie series solutions. Starting with simple spherical geometry, parametric studies are conducted on realistic head models for a typical frequency band (900 MHz) of mobile phones. / text