Dans cette thèse on s'intéresse aux jets en écoulement transverse dans une configuration générique de celle du refroidissement par effusion de chambres de combustion aéronautiques. L'amélioration des modèles de paroi avec transfert de masse passe par une meilleure connaissance de l'interaction entre les jets et l’écoulement principal. Nous avons donc réalisé la simulation numérique directe d'un jet issu d'un perçage incliné avec ou sans giration, pour des écoulements isothermes, turbulents et à bas nombre de Mach, dans un contexte compressible. Pour cela nous avons travaillé avec la bibliothèque AeroSol d'éléments finis continus et discontinus sur maillage hybride. En particulier nous nous sommes intéressés à la stabilité des flux numériques pour le compressible instationnaire associés à la méthode de Galerkin discontinue lorsque le nombre de Mach tend vers zéro. Nous avons pu mettre en évidence des comportements instables lors de l'utilisation de discrétisation temporelle explicite que nous avons corrigés en proposant un nouveau flux. Dans un deuxième temps, nous avons effectué les développements nécessaires à la réalisation des calculs. Nous nous sommes en particulier intéressés à la génération d'un champ de vitesse turbulent synthétique par la méthode SEM (Synthetic Eddy Method) que nous avons implantée dans AeroSol et validée. Grâce aux outils de post-traitement développés, nous avons conduit l'analyse de nos résultats. Dans le cas sans giration, les comparaisons avec les résultats expérimentaux et les résultats de simulations RANS que nous avons obtenus en parallèle sur la configuration du banc d'essai MAVERIC sont encourageants. La structure moyenne d'ensemble du jet est notamment correctement reproduite. En ce qui concerne la cas avec giration, le comportement attendu de déflexion successive du jet dans les deux plans caractéristiques (plan d'injection et plan de l'écoulement transverse) est bien reproduit et illustre tout le potentiel prévisionnel de la librairie de calcul que nous avons contribué à développer. / In this work we are interested in jet in crossflow in a generic configuration to the one used in effusion cooling for combustion chambers. Improved wall models with mass transfer requires a better knowledge of the interaction between the jets and the main flow. We therefore carried out the direct numerical simulation of a jet issuing from an inclined hole with or without gyration, for isothermal turbulent flow at low Mach number, in a compressible context. To achieved this, we worked with the continuous and discontinuous finite element library : AeroSol on hybrid grid. In particular we studied the stability of numerical flux for the unsteady compressible flow associated with discontinuous Galerkin method when the Mach number tends to zero. We were able to demonstrate unstable behavior when using explicit time discretization and we corrected them by providing a new flux. In a second time, we have performed the necessary development to achieve the calculations. We have been especially interested in the generation of a synthetic turbulent velocity field using the SEM method (Synthetic Eddy Method) that we have implemented in aerosol and validate. Thanks to the developed post-processing tools, we have conducted an analysis of our results. In the case without gyration, comparisons with experimental results and the results of RANS simulations we obtained on the Maveric test-bench configuration are encouraging. The mean flow of the jet is correctly reproduced. In the case with gyration, the expected behavior of successive deflection of the jet in both planes (injection plane and transverse plane of the flow) is reproduced and shows all the potential of the AeroSol library we helped to develop.

Title: Numerical analysis of approximation of nonpolygonal domains for discon- tinuous Galerkin method Author: Filip Klouda Department: Department of Numerical Mathematics Supervisor: prof. RNDr. Vít Dolejší, Ph.D., DSc., KNM MFF UK Abstract: In this work we use the discontinuous Galerkin finite element method for the semidiscretization of a nonlinear nonstationary convection-diffusion pro- blem defined on a nonpolygonal two-dimensional domain. Using so called appro- ximating curved elements we define a piecewise polynomial approximation of the boundary of the domain and a space on which we search for a solution. We study the convergence of the method considering a symmetric as well as nonsymmetric discretization of diffusion terms and with the interior and boundary penalty. The obtained results allow us to derive an error estimate for the Discontinuous Galer- kin method employing the approximating curved elements. This estimate depends on the order of the approximation of the solution and also on the order of the approximation of the boundary. We describe one possibility of the construction of the approximating curved elements with the aid of a polynomial mapping given by an interpolation of points on the boundary. We present numerical experiments. Keywords: nonlinear convection-diffusion equation, discontinuous...

Lors de la rentrée atmosphérique, l’écoulement raréfié de l’air autour de l’objet rentrant est régi par un modèle cinétique dérivé de l’équation de Boltzmann ; celui-ci décrit l’évolution d’une fonction de distribution des particules de gaz dans l’espace des phases, de dimension 6 dans le cas général. La simulation numérique déterministe de cet écoulement requiert donc le traitement d’une quantité considérable de données, soit un espace mémoire et un temps de calcul importants. Nous étudions dans ce travail différents moyens de réduire le coût de ces calculs. La première approche est une méthode permettant d’optimiser la taille de la grille de vitesses discrètes employée dans le calcul par une prédiction de l’allure des fonctions de distribution dans l’espace des vitesses, en supposant un faible déséquilibre thermodynamique du gaz. La seconde approche consiste à essayer d’exploiter les propriétés de préservation asymptotique des schémas Galerkin Discontinu, déjà établies dans le cadre du transport linéaire des neutrons, qui permettent de tenir compte des effets de la couche limite cinétique sans que celle-ci soit résolue par le maillage, alors que les méthodes classiques (comme les Volumes Finis) imposent l’utilisation de maillages très raffinés en zone de proche paroi. Dans une dernière partie, nous comparons les performances respectives de ces schémas Galerkin Discontinu et de quelques schémas Volumes Finis, appliqués au modèle BGK sur un cas simple, en étudiant en particulier leur comportement près des parois et les conditions aux limites numériques. / During the atmospheric re-entry of a space engine, the rarefied air flow around the body is determined by a kinetic model derived from the Boltzmann equation, which describes the evolution of a distribution function of gas molecules in the phase space, this means a 6-dimensional space in the general case. Consequently, a deterministic numerical simulation of this flow requires large computational ressources, both in memory storage and CPU time. The aim of this work is to reduce those ressources, using two different approaches. The first one is a method allowing to optimize the size of the discrete velocity grid used for the computation by a prediction of the shape of the distributions in the velocity space, assuming that the gas is close to thermodynamic equilibrium. The second approach is an attempt to use the asymptotic preservation properties of Discontinuous Galerkin schemes, already established for neutron transport, which allow to take into account the effects of kinetic boundary layers even if they are not resolved by the mesh, while classical methods (such as Finite Volumes) require very refined meshes along the direction normal to the walls. In a last part, we compare the performances of these Discontinuous Galerkin schemes with some classical Finite Volumes schemes, applied to the BGK equation in a simple case, and pay particular attention to their near-wall behavior and numerical boundary conditions.

L’Analyse isogéométrique (AIG) est une méthode innovante de résolution numérique des équations différentielles, proposée à l’origine par Thomas Hughes, Austin Cottrell et Yuri Bazilevs en 2005. Cette technique de discrétisation est une généralisation de l’analyse par éléments finis classiques (AEF), conçue pour intégrer la conception assistée par ordinateur (CAO), afin de combler l’écart entre la description géométrique et l’analyse des problèmes d’ingénierie. Ceci est réalisé en utilisant des B-splines ou des B-splines rationnelles non uniformes (NURBS), pour la description des géométries ainsi que pour la représentation de champs de solutions inconnus.L’objet de cette thèse est d’étudier la méthode isogéométrique dans le contexte des problèmes hyperboliques en utilisant les fonctions B-splines comme fonctions de base. Nous proposons également une méthode combinant l’AIG avec la méthode de Galerkin discontinue (GD) pour résoudre les problèmes hyperboliques. Plus précisément, la méthodologie de GD est adoptée à travers les interfaces de patches, tandis que l’AIG traditionnelle est utilisée dans chaque patch. Notre méthode tire parti de la méthode de l’AIG et la méthode de GD.Les résultats numériques sont présentés jusqu’à l’ordre polynomial p= 4 à la fois pour une méthode deGalerkin continue et discontinue. Ces résultats numériques sont comparés pour un ensemble de problèmes de complexité croissante en 1D et 2D. / Isogeometric Analysis (IGA) is a modern strategy for numerical solution of partial differential equations, originally proposed by Thomas Hughes, Austin Cottrell and Yuri Bazilevs in 2005. This discretization technique is a generalization of classical finite element analysis (FEA), designed to integrate Computer Aided Design (CAD) and FEA, to close the gap between the geometrical description and the analysis of engineering problems. This is achieved by using B-splines or non-uniform rational B-splines (NURBS), for the description of geometries as well as for the representation of unknown solution fields.The purpose of this thesis is to study isogeometric methods in the context of hyperbolic problems usingB-splines as basis functions. We also propose a method that combines IGA with the discontinuous Galerkin(DG)method for solving hyperbolic problems. More precisely, DG methodology is adopted across the patchinterfaces, while the traditional IGA is employed within each patch. The proposed method takes advantageof both IGA and the DG method.Numerical results are presented up to polynomial order p= 4 both for a continuous and discontinuousGalerkin method. These numerical results are compared for a range of problems of increasing complexity,in 1D and 2D.

In this dissertation, we develop Adaptive Mesh Refinement (AMR) techniques for the steady-state multigroup SN neutron transport equation using a higher-order Discontinuous Galerkin Finite Element Method (DGFEM). We propose two error estimations, a projection-based estimator and a jump-based indicator, both of which are shown to reliably drive the spatial discretization error down using h-type AMR. Algorithms to treat the mesh irregularity resulting from the local refinement are implemented in a matrix-free fashion. The DGFEM spatial discretization scheme employed in this research allows the easy use of adapted meshes and can, therefore, follow the physics tightly by generating group-dependent adapted meshes. Indeed, the spatial discretization error is controlled with AMR for the entire multigroup SNtransport simulation, resulting in group-dependent AMR meshes. The computing efforts, both in memory and CPU-time, are significantly reduced. While the convergence rates obtained using uniform mesh refinement are limited by the singularity index of transport solution (3/2 when the solution is continuous, 1/2 when it is discontinuous), the convergence rates achieved with mesh adaptivity are superior. The accuracy in the AMR solution reaches a level where the solution angular error (or ray effects) are highlighted by the mesh adaptivity process. The superiority of higherorder calculations based on a matrix-free scheme is verified on modern computing architectures. A stable symmetric positive definite Diffusion Synthetic Acceleration (DSA) scheme is devised for the DGFEM-discretized transport equation using a variational argument. The Modified Interior Penalty (MIP) diffusion form used to accelerate the SN transport solves has been obtained directly from the DGFEM variational form of the SN equations. This MIP form is stable and compatible with AMR meshes. Because this MIP form is based on a DGFEM formulation as well, it avoids the costly continuity requirements of continuous finite elements. It has been used as a preconditioner for both the standard source iteration and the GMRes solution technique employed when solving the transport equation. The variational argument used in devising transport acceleration schemes is a powerful tool for obtaining transportconforming diffusion schemes. xuthus, a 2-D AMR transport code implementing these findings, has been developed for unstructured triangular meshes.

Efficient, accurate numerical simulation of coupled heat transfer and fluid dynamics systems continues to be a challenge. Direct numerical simulation (DNS) packages like FLU- ENT exist and are sufficient for design and predicting flow in a static system, but in larger systems where input parameters can change rapidly, the cost of DNS increases prohibitively. Major obstacles include handling the scales of the system accurately - some applications span multiple orders of magnitude in both the spatial and temporal dimensions, making an accurate simulation very costly. There is a need for a simulation method that returns accurate results of multi-scale systems in real time. To address these challenges, the Multi- Resolution Discontinuous Galerkin (MRDG) method has been shown to have advantages over other reduced order methods. Using multi-wavelets as the local approximation space provides an inherently efficient method of data compression, while the unique features of the Discontinuous Galerkin method make it well suited to composition with wavelet theory. This research further exhibits the viability of the MRDG as a new approach to efficient, accurate thermal system simulations. The development and execution of the algorithm will be detailed, and several examples of the utility of the MRDG will be included. Comparison between the MRDG and the "vanilla" DG method will also be featured as justification of the advantages of the MRDG method.

The solution of the Burgers' equation computed by the standard finite element method is degraded by oscillations, which are the manifestation of the Gibbs phenomenon. In this work we study the following numerical me- thods: Discontinuous Galerkin method, stable low order schemes and the flux corrected technique method in order to prevent the undesired Gibbs phenomenon. The focus is on the reduction of severe overshoots and under- shoots and the preservation of the smoothness of the solution. We consider a simple 1D problem on the interval (0, 1) with different initial conditions to demonstrate the properties of the presented methods. The numerical results of individual methods are provided. 1

La thèse porte sur la simulation, en régime fréquentiel, du rayonnement acoustique en écoulement subsonique quelconque et dans un domaine infini. L'approche choisie s'appuie sur la résolution d'un système équivalent aux équations d'Euler linéarisées : le modèle de Galbrun. Ce modèle repose sur une représentation mixte Lagrange-Euler et aboutit à une équation dont l'unique inconnue est la perturbation du déplacement Lagrangien. Une des difficultés de l'approche de Galbrun est qu'une discrétisation directe de cette équation par une méthode d'éléments finis standard n'est pas stable. Un moyen de contourner cet obstacle est d'écrire une équation augmentée en ajoutant une nouvelle inconnue, le rotationnel du déplacement, appelée par abus vorticité. Cette approche conduit à un système qui couple une équation de type équation des ondes avec une équation de transport en régime fréquentiel. Et elle permet l'utilisation de couches parfaitement adaptées (PML) pour borner le domaine de calcul. La première partie du manuscrit est dédiée à l’étude de l’équation de transport harmonique et de sa résolution numérique, en particulier par un schéma de type Galerkin discontinu. Un des points délicats est lié au caractère oscillant des solutions de l'équation. Une fois cette étape franchie, la résolution du problème de propagation acoustique a été abordée. Une approximation basée sur l'utilisation d'éléments finis mixtes continus-discontinus avec couches parfaitement adaptées (PML) a été étudiée. En particulier, les caractères bien posés des problèmes continu et discret ainsi que la convergence du schéma numérique ont été démontrés sous certaines conditions sur l'écoulement porteur. Enfin, une mise en œuvre a été effectuée. Les résultats montrent la validité de cette approche mais aussi sa pertinence dans le cas d'écoulements complexes, voire d'écoulements dits instables / This thesis deals with the numerical simulation of time harmonic acoustic propagation in an arbitrary mean flow in an unbounded domain. Our approach is based on an equation equivalent to the linearized Euler equations called the Galbrun equation. It is derived from a mixed Eulerian-Lagrangian formulation and results in a single equation whose only unknown is the perturbation of the Lagrangian displacement. A direct solution using finite elements is unstable but this difficulty can be overcome by using an augmented equation which is constructed by adding a new unknown, the vorticity, defined as the curl of the displacement. This leads to a set of equations coupling a wave like equation with a time harmonic transport equation which allows the use of perfectly matched layers (PML) at artificial boundaries to bound the computational domain. The first part of the thesis is a study of the time harmonic transport equation and its approximation by means of a discontinuous Galerkin scheme, the difficulties coming from the oscillating behaviour of its solutions. Once these difficulties have been overcome, it is possible to deal with the resolution of the acoustic propagation problem. The approximation method is based on a mixed continuous-Galerkin and discontinuous-Galerkin finite element scheme. The well-posedness of both the continuous and discrete problems is established and the convergence of the approximation under some mean flow conditions is proved. Finally a numerical implementation is achieved and numerical results are given which confirm the validity of the method and also show that it is relevant in complex cases, even for unstable flows

The main emphasis of this thesis is to study a posteriori error analysis of discontinuous Galerkin (DG) methods for the elliptic variational inequalities. The DG methods have become very pop-ular in the last two decades due to its nature of handling complex geometries, allowing irregular meshes with hanging nodes and different degrees of polynomial approximation on different ele-ments. Moreover they are high order accurate and stable methods. Adaptive algorithms refine the mesh locally in the region where the solution exhibits irregular behaviour and a posteriori error estimates are the main ingredients to steer the adaptive mesh refinement. The solution of linear elliptic problem exhibits singularities due to change in boundary con-ditions, irregularity of coefficients and reentrant corners in the domain. Apart from this, the solu-tion of variational inequality exhibits additional irregular behaviour due to occurrence of the free boundary (the part of the domain which is a priori unknown and must be found as a component of the solution). In the lack of full elliptic regularity of the solution, uniform refinement is inefficient and it does not yield optimal convergence rate. But adaptive refinement, which is based on the residuals ( or a posteriori error estimator) of the problem, enhance the efficiency by refining the mesh locally and provides the optimal convergence. In this thesis, we derive a posteriori error estimates of the DG methods for the elliptic variational inequalities of the first kind and the second kind. This thesis contains seven chapters including an introductory chapter and a concluding chap-ter. In the introductory chapter, we review some fundamental preliminary results which will be used in the subsequent analysis. In Chapter 2, a posteriori error estimates for a class of DG meth-ods have been derived for the second order elliptic obstacle problem, which is a prototype for elliptic variational inequalities of the first kind. The analysis of Chapter 2 is carried out for the general obstacle function therefore the error estimator obtained therein involves the min/max func-tion and hence the computation of the error estimator becomes a bit complicated. With a mild assumption on the trace of the obstacle, we have derived a significantly simple and easily com-putable error estimator in Chapter 3. Numerical experiments illustrates that this error estimator indeed behaves better than the error estimator derived in Chapter 2. In Chapter 4, we have carried out a posteriori analysis of DG methods for the Signorini problem which arises from the study of the frictionless contact problems. A nonlinear smoothing map from the DG finite element space to conforming finite element space has been constructed and used extensively, in the analysis of Chapter 2, Chapter 3 and Chapter 4. Also, a common property shared by all DG methods allows us to carry out the analysis in unified setting. In Chapter 5, we study the C0 interior penalty method for the plate frictional contact problem, which is a fourth order variational inequality of the second kind. In this chapter, we have also established the medius analysis along with a posteriori analy-sis. Numerical results have been presented at the end of every chapter to illustrate the theoretical results derived in respective chapters. We discuss the possible extension and future proposal of the work presented in the Chapter 6. In the last chapter, we have documented the FEM codes used in the numerical experiments.

L’objectif de cette thèse est de développer une méthode Galerkine discontinue d’ordre élevé capable de prendre en considération des simulations réalistes liées à la nanophotonique. Au cours des dernières décennies, l’évolution des techniques de lithographie a permis la création de structure géométriques de tailles nanométriques, révélant ainsi une large gamme de phénomènes nouveaux nés de l’interaction lumière-matière à ces échelles. Ces effets apparaissent généralement pour des objets de taille égale ou (très) inférieure à la longueur d’onde du champ incident. Ce travail repose sur le développement et l’implémentation de modèles de dispersion appropriés (principalement pour les métaux), ainsi que sur un large éventail de méthodes computationnelles classiques. Deux développements méthodologiques majeurs sont présentés et étudiés en détails: (i) les éléments courbes, et (ii) l’ordre d’approximation local. Ces études sont accompagnées de plusieurs cas-tests réalistes tirés de la nanophotonique. / The goal of this thesis is to develop a discontinuous Galerkin time-domain method to be able to handle realistic nanophotonics computations. During the last decades, the evolution of lithography techniques allowed the creation of geometrical structures at the nanometer scale, thus unveiling a variety of new phenomena arising from light-matter interactions at such levels. These effects usually occur when the device is of comparable size or (much) smaller than the wavelength of the incident field. This work relies on the development and implementation of appropriate models for dispersive materials (mostly metals), as well as on a large panel of classical computational techniques. Two major methodological developments are presented and studied in details: (i) curvilinear elements, and (ii) local order of approximation. This work is complemented with several physical studies of real-life nanophotonics applications.