O desenvolvimento de uma ferramenta de Dinâmica de Fluidos Computacional que utiliza Método de Elementos Finitos baseada na discretização de Galerkin descontínuo é apresentado neste trabalho com objetivo de resolver a equação de Euler linearizada para escoamento compressível em duas dimensões usando malhas estruturadas e não estruturadas. Procuramos utilizar esta ferramenta como um propagador de ondas sonoras para estudar fenômenos aeroacústicos. O problema de Riemann presente no fluxo convectivo da equação de Euler é tratado com um método upwind HLL e para o avanço da solução no tempo é usado o método de Runge-Kutta explícito de 4 estágios com segunda ordem de precisão. A eficiência computacional, a convergência do método e a precisão são testadas através de simulações de escoamentos já apresentadas na literatura. A taxa de convergência para altas ordens de aproximação é assintótica que é um resultado compatível com a formulação Galerkin descontínuo. / The development of a Computation Fluid Dynamic Tool based on the Finite Element Method with discontinuous Galerkin discretization is presented in this work. The aim of this study is to solve the compressible linearized Euler\'s equation in two dimensions on structured and non structured meshes. This tool has been used as a means to study aeroacoustics phenomena. The Riemann\'s problem presented on a convective flow in Euler\'s equation is tackled by a HLL\'s method and the time integration being used is the four-stage Runge-Kutta explicit method with second order of accuracy. The computational efficiency, the convergence of the method and the accuracy are tested by comparing our results for flow simulations with those that are available in the literature. The convergence rate to high approximation order is asymptotic and it shows a result which is compatible with a discontinuous Galerkin formulation.

We consider high order discontinuous-Galerkin finite element methods for partial differential equations, with a focus on the neutron transport equation. We begin by examining a method for preprocessing block-sparse matrices, of the type that arise from discontinuous-Galerkin methods, prior to factorisation by a multifrontal solver. Numerical experiments on large two and three dimensional matrices show that this pre-processing method achieves a significant reduction in fill-in, when compared to methods that fail to exploit block structures. A discontinuous-Galerkin finite element method for the neutron transport equation is derived that employs high order finite elements in both space and angle. Parallel Krylov subspace based solvers are considered for both source problems and $k_{eff}$-eigenvalue problems. An a-posteriori error estimator is derived and implemented as part of an h-adaptive mesh refinement algorithm for neutron transport $k_{eff}$-eigenvalue problems. This algorithm employs a projection-based error splitting in order to balance the computational requirements between the spatial and angular parts of the computational domain. An hp-adaptive algorithm is presented and results are collected that demonstrate greatly improved efficiency compared to the h-adaptive algorithm, both in terms of reduced computational expense and enhanced accuracy. Computed eigenvalues and effectivities are presented for a variety of challenging industrial benchmarks. Accurate error estimation (with effectivities of 1) is demonstrated for a collection of problems with inhomogeneous, irregularly shaped spatial domains as well as multiple energy groups. Numerical results are presented showing that the hp-refinement algorithm can achieve exponential convergence with respect to the number of degrees of freedom in the finite element space

Cette thèse s'inscrit dans un projet d'innovation duale RAPID financé par DGA/DS/MRIS et appelé GREAT faisant intervenir la société Axessim, l'ONERA, INRIA, l'IRMA et le CEA. Ce projet a pour but la mise en place d'une solution industrielle de simulation électromagnétique basée sur une méthode Galerkin Discontinue (GD) parallèle sur maillage hexaédrique. Dans un premier temps, nous établissons un schéma numérique adapté à un système de loi de conservation. Nous pouvons ainsi appliquer cette approche aux équations de Maxwell, mais également à tout système hyperbolique. Dans un second temps, nous mettons en place une parallélisation à deux niveaux de ce schéma. D'une part, les calculs sont parallélisés sur carte graphique au moyen de la bibliothèque OpenCL. D'autre part, plusieurs cartes graphiques peuvent être utilisées, chacune étant pilotée par un processus MPI. De plus, les communications MPI et les calculs OpenCL sont asynchronisés permettant d'obtenir une forte accélération. / This thesis is part of a dual innovation project funded by RAPID DGA/DS/MRIS and called GREAT involving Axessim company, ONERA, INRIA, IRMA and the CEA. This project aims at the establishment of an industrial solution of electromagnetic simulation based on a method Discontinuous Galerkin (DG) on parallel hexahedral mesh. First, we establish a numerical scheme adapted to a conservation law system. We can apply this approach to the Maxwell equations but also to any hyperbolic system. In a second step, we set up a two-level parallelization of this scheme. On the one hand, the calculations are parallelized on graphics card using the OpenCL library. On the other hand, multiple graphics cards can be used, each driven by a MPI process. In addition, MPI communications and OpenCL computations are launched asynchronously in order to obtain a strong acceleration.

L'objectif principal de cette thèse est l'étude des problèmes et des applications qu'ils se développent dans le domaine de la nanophotonique. Plus précisément, nous considérons les structures de métaux nobles où les modèles de dispersion locaux sont insuffisants et la non-localité doit être incluse dans le modèle. Ici, le système physique sous-jacent est typiquement modélisé comme des équations de Maxwell couplées à des lois de dispersion spatio-temporelles dans le régime des longueurs d'onde optiques. Bien que les solutions analytiques puissent être dérivées pour un petit nombre de problèmes, cela n'est généralement pas possible pour les dispositifs du monde réel, qui présentent souvent des géométries complexes et des compositions de matériaux. Suite à une analyse rigoureuse des propriétés physiques et mathématiques du modèle continu original, nous proposons une méthode de type à éléments finis d'ordre élevé pour discrétiser le modèle continu dans l'espace et le temps. Les méthodes discontinues Galerkin (DG) sont bien établies pour la discrétisation spatiale des équations de Maxwell. Cette thèse prolonge les travaux antérieurs sur les systèmes couplés des équations de Maxwell et les lois de dispersion spatiale. Nous utilisons des méthodes explicites de Runge-Kutta (RK) d'ordre élevé pour la discrétisation temporelle. L'intégration temporelle RK garantit un ordre de convergence espace-temps élevé du schéma entièrement discret, qui repose sur un schéma de preuve de convergence. Parallélisme MPI (Message Passing Interface), éléments curvilignes et PML (Perfectly Matched Layers) autour des aspects d'implémentation et d'évaluation des performances dans le cadre du logiciel développé à Inria Sophia Antipolis-Méditerannée (DIOGENES). La méthode développée est appliquée à de nombreuses simulations nanophotoniques réelles de dispositifs où des observables tels que la réflexion, la section transversale (CS) et la spectroscopie de perte d'énergie électronique (EELS) sont étudiés. Entre autres, nous élaborons une feuille de route pour un étalonnage expérimental robuste du modèle de dispersion non local linéarisé basé sur la solution de problèmes inverses et la quantification d'incertitude (UQ) des paramètres géométriques stochastiques. Nous avons également amélioré les accords de simulations numériques non locales et les résultats expérimentaux pour la résonance des plasmons d'espacement des nano-cubes d'argent. Cela démontre la pertinence de simulations non locales précises. / The main objective of this thesis is the study of problems and applications as they arise in the field of nanophotonics. More speci cally, we consider noble metal structures where local dispersion models are insu cient and nonlocality has to be included in the model. Here, the underlying physical system is typically modeled as Maxwell’s equations coupled to spatio- temporal dispersion laws in the regime of optical wavelengths. While analytical solutions can be derived for a small number of problems, this is typically not possible for real-world devices, which often feature complicated geometries and material compositions. Following a rigorous analysis of the physical and mathematical properties of the original continuous model, we propose a high order finite element type method for discretizing the continuous model in space and time. Discontinuous Galerkin (DG) methods are well established for the spatial discretization of Maxwell’s equations. This thesis extends previous work on the coupled systems of Maxwell’s equations and spatial dispersion laws. We use explicit high-order Runge-Kutta (RK) methods for the subsequent time discretiz- ation. RK time integration guarantees a high space-time convergence order of the fully-discrete scheme, which is underpinned by a sketch of a convergence proof. Message Passing Interface (MPI) parallelization, curvilinear elements and Perfectly Matched Layers (PMLs) round of implementation aspects and performance assessments in the scope of the Software developed at Inria Sophia Antipolis-Méditerannée (DIOGENeS). The developed method is applied to numerous real-world nanophotonics simulations of devices where observables like re ectance, Cross Section (CS) and Electron Energy Loss Spectroscopy (EELS) are studied. Inter alia, we elaborate a roadmap for a robust experimental calibration of the linearized nonlocal disper- sion model based on the solution of inverse problems and Uncertainty Quanti cation (UQ) of stochastic geometric parameters. We also find improved agreements of nonlocal numerical simulations and exper- imental results for the gap-plasmon resonance of silver nano-cubes. This demonstrates the relevance of accurate nonlocal simulations.

Moment methods derived from the kinetic theory of gases can be used for the prediction of continuum and non-equilibrium flows and offer numerical advantages over other methods, such as the Navier-Stokes model. Models developed in this fashion are described by first-order hyperbolic partial differential equations (PDEs) with stiff local relaxation source terms. The application of discontinuous-Galerkin (DG) methods for the solution of such models has many benefits. Of particular interest is the third-order accurate, coupled space-time discontinuous-Galerkin Hancock (DGH) method. This scheme is accurate, as well as highly efficient on large-scale distributed-memory computers. The current study outlines a general implementation of the DGH method used for the parallel solution of moment methods in one, two, and three dimensions on modern distributed clusters. An algorithm for adaptive mesh refinement (AMR) was developed alongside the implementation of the scheme, and is used to achieve even higher accuracy and efficiency. Many different first-order hyperbolic and hyperbolic-relaxation PDEs are solved to demonstrate the robustness of the scheme. First, a linear convection-relaxation equation is solved to verify the order of accuracy of the scheme in three dimensions. Next, some classical compressible Euler problems are solved in one, two, and three dimensions to demonstrate the scheme's ability to capture discontinuities and strong shocks, as well as the efficacy of the implemented AMR. A special case, Ringleb's flow, is also solved in two-dimensions to verify the order of accuracy of the scheme for non-linear PDEs on curved meshes. Following this, the shallow water equations are solved in two dimensions. Afterwards, the ten-moment (Gaussian) closure is applied to two-dimensional Stokes flow past a cylinder, showing the abilities of both the closure and scheme to accurately compute classical viscous solutions. Finally, the one-dimensional fourteen-moment closure is solved.

We deal with the numerical solution of the Navier-Stokes equations describing a motion of viscous compressible flows. The governing equations are discretized with the aid of discontinuous Galerkin finite element method which is based on a discontinuous piecewise polynomial approximation. The discretizations leads to a large nonlinear algebraic system. In order to solve this system efficiently, we develop the so-called p-multigrid solution strategy which employ as a projec- tion and a restriction operators the L2 -projection in the spaces of polynomial functions on each element separately. The p-multigrid technique is studied, deve- loped and implemented in the code ADGFEM. The computational performance of the method is presented.

The development of computational fluid dynamics has given us a very powerful tool for investigation of fluid dynamics. However, in order to maintain the progress, it is necessary to improve the numerical algorithms. Nowadays, the high-order methods based on the discontinuous projection seem to have the largest potential for the future. In the work, we used open-source framework Nektar++, which provides the high-order discretization method. We tested the abilities of the framework for computing the compressible sonic and transonic flow. We successfully obtained simulations of the viscous and inviscid flow. We computed the lift and the drag coefficients and showed that for a higher polynomial order we can obtain the same accuracy with less degrees of freedom and lower computational time. Also, we tested the shock capturing method for the computation of the inviscid transonic flow and confirmed the potential of the high order methods. 1

A posteriori error estimation is an inseparable component of any reliable numerical method for solving partial differential equations. The aim of the goal-oriented a posteriori error estimates is to control the computational error directly with respect to some quantity of interest, which makes the method very convenient for many engineering applications. The resulting error estimates may be employed for mesh adaptation which enables to find a numerical approximation of the quantity of interest under some given tolerance in a very efficient manner. In this thesis, the goal-oriented error estimates are derived for discontinuous Galerkin discretizations of the linear scalar model problems, as well as of the Euler equations describing inviscid compressible flows. It focuses on several aspects of the goal-oriented error estimation method, in particular, higher order reconstructions, adjoint consistency of the discretizations, control of the algebraic errors arising from iterative solutions of both algebraic systems, and linking the estimates with the hp-anisotropic mesh adaptation. The computational performance is demonstrated by numerical experiments.

A two-dimensional discontinuous Galerkin finite element method algorithm in the time domain was developed for calculation of the radar cross-section of an arbitrary object. The algorithm was formed using local nodal basis functions in each element and coupling them via numerical upwind flux. Both transverse electric and transverse magnetic polarization, as well as three different dispersive material models, were handled. The computational domain was effectively truncated with low reflections using the uniaxial perfectly matched layer method. Two different time stepping methods were used, low-storage explicit Runge-Kutta and Leap-Frog, to allow for flexibility in the time step and application of a stabilization method. The algorithm was verified with geometries, which have analytical expressions, and an existing validated code. The algorithm was also compared to an existing algorithm, which utilized the continuous finite element method with implicit time stepping, and showed outstanding performance regarding computation time and memory allocation. Since the developed algorithm had explicit time stepping could no general conclusions favoring any of the methods beyond these specific algorithms be made. The results still encouraged continued development of the DGFEM algorithm, where the expansion into three dimensions and optimizations could be explored further.

Indiana University-Purdue University Indianapolis (IUPUI) / The interplay between biomechanics and blood perfusion in the optic nerve head (ONH) has a critical role in ocular pathologies, especially glaucomatous optic neuropathy. Elucidating the complex interactions of ONH perfusion and tissue structure in health and disease using current imaging methodologies is difficult, and mathematical modeling provides an approach to address these limitations. The biophysical phenomena governing the ONH physiology occur at different scales in time and space and porous media theory provides an ideal framework to model them. We critically review fundamentals of porous media theory, paying particular attention to the assumptions leading to a continuum biphasic model for the phenomenological description of fluid flow through biological tissues exhibiting viscoelastic behavior. The resulting system of equations is solved via a numerical method based on a novel hybridizable discontinuous Galerkin finite element discretization that allows accurate approximations of stresses and discharge velocities, in addition to solid displacement and fluid pressure. The model is used to theoretically investigate the influence of tissue viscoelasticity on the blood perfusion of the lamina cribrosa in the ONH. Our results suggest that changes in viscoelastic properties of the lamina may compromise tissue perfusion in response to sudden variations of intraocular pressure, possibly leading to optic disc hemorrhages.