A nested hybridizable discontinuous Galerkin method for computing second-harmonic generation in three-dimensional metallic nanostructures

In this paper, we develop a nested hybridizable discontinuous Galerkin (HDG) method to numerically solve the Maxwell's equations coupled with the hydrodynamic model for the conduction-band electrons in metals. By means of a static condensation to eliminate the degrees of freedom of the approximate solution defined in the elements, the HDG method yields a linear system in terms of the degrees of freedom of the approximate trace defined on the element boundaries. Furthermore, we propose to reorder these degrees of freedom so that the linear system accommodates a second static condensation to eliminate a large portion of the degrees of freedom of the approximate trace, thereby yielding a much smaller linear system. For the particular metallic structures considered in this paper, the resulting linear system obtained by means of nested static condensations is a block tridiagonal system, which can be solved efficiently. We apply the nested HDG method to compute the second harmonic generation (SHG) on a triangular coaxial periodic nanogap structure. This nonlinear optics phenomenon features rapid field variations and extreme boundary-layer structures that span multiple length scales. Numerical results show that the ability to identify structures which exhibit resonances at $\omega$ and $2\omega$ is paramount to excite the second harmonic response.Comment: 31 pages, 7 figure

Monolithic multiphysics simulation of hypersonic aerothermoelasticity using a hybridized discontinuous Galerkin method

This work presents implementation of a hybridized discontinuous Galerkin (DG) method for robust simulation of the hypersonic aerothermoelastic multiphysics system. Simulation of hypersonic vehicles requires accurate resolution of complex multiphysics interactions including the effects of high-speed turbulent flow, extreme heating, and vehicle deformation due to considerable pressure loads and thermal stresses. However, the state-of-the-art procedures for hypersonic aerothermoelasticity are comprised of low-fidelity approaches and partitioned coupling schemes. These approaches preclude robust design and analysis of hypersonic vehicles for a number of reasons. First, low-fidelity approaches limit their application to simple geometries and lack the ability to capture small scale flow features (e.g. turbulence, shocks, and boundary layers) which greatly degrades modeling robustness and solution accuracy. Second, partitioned coupling approaches can introduce considerable temporal and spatial inaccuracies which are not trivially remedied. In light of these barriers, we propose development of a monolithically-coupled hybridized DG approach to enable robust design and analysis of hypersonic vehicles with arbitrary geometries. Monolithic coupling methods implement a coupled multiphysics system as a single, or monolithic, equation system to be resolved by a single simulation approach. Further, monolithic approaches are free from the physical inaccuracies and instabilities imposed by partitioned approaches and enable time-accurate evolution of the coupled physics system. In this work, a DG method is considered due to its ability to accurately resolve second-order partial differential equations (PDEs) of all classes. We note that the hypersonic aerothermoelastic system is composed of PDEs of all three classes. Hybridized DG methods are specifically considered due to their exceptional computational efficiency compared to traditional DG methods. It is expected that our monolithic hybridized DG implementation of the hypersonic aerothermoelastic system will 1) provide the physical accuracy necessary to capture complex physical features, 2) be free from any spatial and temporal inaccuracies or instabilities inherent to partitioned coupling procedures, 3) represent a transition to high-fidelity simulation methods for hypersonic aerothermoelasticity, and 4) enable efficient analysis of hypersonic aerothermoelastic effects on arbitrary geometries

A weakly compressible hybridizable discontinuous Galerkin formulation for fluid-structure interaction problems

A scheme for the solution of fluid-structure interaction (FSI) problems with weakly compressible flows is proposed in this work. A novel hybridizable discontinuous Galerkin (HDG) method is derived for the discretization of the fluid equations, while the standard continuous Galerkin (CG) approach is adopted for the structural problem. The chosen HDG solver combines robustness of discontinuous Galerkin (DG) approaches in advection-dominated flows with higher order accuracy and efficient implementations. Two coupling strategies are examined in this contribution, namely a partitioned Dirichlet-Neumann scheme in the context of hybrid HDG-CG discretizations and a monolithic approach based on Nitsche's method, exploiting the definition of the numerical flux and the trace of the solution to impose the coupling conditions. Numerical experiments show optimal convergence of the HDG and CG primal and mixed variables and superconvergence of the postprocessed fluid velocity. The robustness and the efficiency of the proposed weakly compressible formulation, in comparison to a fully incompressible one, are also highlighted on a selection of two and three dimensional FSI benchmark problems.Comment: 49 pages, 20 figures, 2 table

Modélisation numérique de la propagation d'ondes en milieux complexes : application aux milieux granulaires non consolidés

Le sous-sol, qui contient de nombreuses ressources naturelles (eau, gaz, pétrole, etc.), peut également constituer un risque naturel en raison de ses caractéristiques lithologiques et topographiques. Par ailleurs, dans le contexte du changement climatique, il devient de plus en plus important d'estimer le taux de saturation des fluides dans ces milieux pour prévenir les catastrophes naturelles comme les glissements de terrain ou des inondations. Toutes ces raisons suscitent l'intérêt des géophysiciens qui cherchent à mieux comprendre la proche surface et donc à la caractériser. En Géophysique, différentes techniques sont utilisées pour caractériser le sous-sol parmi lesquelles des techniques sismiques non destructives. Lorsque les ondes sismiques traversent un matériau donné, elles sont diffractées, réfléchies ou converties et contiennent ainsi des informations sur les phases fluide et solide. Pour mieux comprendre les mesures acoustiques et sismiques dans les sédiments et les sols, de nombreuses études sur les milieux granulaires non consolidés ont été menées in situ et aussi à l'échelle du laboratoire où des modèles théoriques ont été développés. Dans cette thèse, nous souhaitons modéliser des milieux granulaires qui sont un type de milieu complexe difficile à caractériser. Pour atteindre cet objectif, nous avons suivi trois étapes. Premièrement, nous avons développé un outil numérique qui calcule l'ensemble du champ d'ondes d'un modèle élastique bidimensionnel avec des structures complexes. Nous proposons une méthode de volumes finis basée sur un solveur de Riemann (RFV-FSP/Riemann Finite Volume-Fluxes Frequency Shift PML method) pour calculer les champs d'ondes sismiques sur des grilles colocalisées ainsi qu'une formulation des conditions absorbantes de type PML spécifiquement conçue pour la méthode des volumes finis. Ces dernières sont optimisées à incidence rasante en utilisant des formulations convolutives avec décalage en fréquence (C-PML) ou non convolutives (ADE-PML). Ici, elles sont appliquées aux dérivées spatiales des flux, ce qui diffère des PML classiques qui sont généralement appliquées aux dérivées spatiales des variables primitives (vitesses et contraintes des particules). La méthode des volumes finis et les différents types de conditions aux limites sont test´es et validés sur différents cas synthétiques hétérogènes. Les volumes finis sont comparés à d'autres techniques comme les différences finies et les éléments finis d'ordre élevé. Nous appliquons aussi notre méthode à une configuration de couplage fluide-solide et à quelques modèles sismiques d'intérêt dans le contexte de milieux granulaires non consolidés présentant de fortes variations de propriétés avec la profondeur. En particulier nous concentrons notre attention sur la résolution numérique des ondes de surface comme les ondes de Rayleigh. Pour obtenir plus de précision, nous avons implémenté un schéma spatial décentré du quatrième ordre proche de la surface libre. Deuxièmement, nous avons mis en place des outils de traitement du signal qui détectent les temps des premières arrivées sismiques, et calculent les courbes de vitesse de phase et les modes de propagation des ondes. Ces derniers outils sont utilisés pour l'analyse de dispersion. Pour finir, nous revisitons une étude réalisée sur des milieux granulaires non consolidés à l'échelle du laboratoire en utilisant les différents outils développés. Nous comparons différents modèles (2D ou 3D) avec différentes rhéologies (élastique ou poro-élastique), différentes conditions aux limites (PML ou Dirichlet) et différentes modélisations numériques de la source (point source ou pot vibrant) afin de reproduire les données expérimentales. L'étude de la sensibilité des données sismiques à l'emplacement de la source était également cru- ciale pour améliorer l'amplitude des signaux et la détection des différents modes sismiques. Cela nous permettra à l'avenir de mieux imager et comprendre ces milieux complexes.The subsurface, which contains many natural resources (water, gas, oil, etc.), can also constitute a natural risk because of its lithological and topographical characteristics. In the context of climate change, it becomes more and more important to estimate the rate of saturation of fluids in these media to prevent natural disasters like landslides or flash floods. All these reasons arouse the in- terest of geophysicists who seek to better understand the near surface and therefore to characterize it. In geophysics, different techniques are used to characterize the subsurface among them seismic techniques which are non-destructive. When seismic waves are crossing a given material, they are diffracted, reflected or converted and thus contain information on fluid and solid phases. To better understand acoustic and seismic measurements in sediments and soils, many studies on unconsol- idated granular media have been conducted in situ, and at the laboratory scale where theoretical models have been developed. In this thesis, we want to model granular media which are a type of complex medium difficult to characterize. To achieve this objective, we followed three steps. First, we developed a numerical tool which calculates the entire wave field of a two dimensional geometric elastic model with complex structures. And we compare its accuracy to other techniques like the classical staggered-fine difference or the high-order spectral element methods. We propose a finite volume method based on a Riemann solver (RFV-FSP/Riemann Finite Volume-Fluxes frequency Shift PML method) to compute seismic wave fields on collocated grids as well as a formulation of perfectly matched layer (PML) absorbing boundary conditions that are more specifically designed to the finite volume method. The PML boundary conditions are optimized at grazing incidence by using frequency shift convolutional (C-PML) or non convolutional formulations (ADE-PML). Here, they are applied to the spatial fluxes derivatives, which is a different formulation than clas- sical PMLs that are generally applied to the spatial derivatives of the primitive variables (particle velocities and stresses). The finite volume method and the different kinds of boundary conditions are tested and validated on different heterogeneous synthetic cases. The finite volume method is compared to other techniques like finite differences and high order finite elements. Finally, we apply our method to a fluid-solid coupling configuration and to some seismic models of interest in the context of unconsolidated granular media presenting sharp property variations with depth. In particular we focus our attention on the implementation of the numerical resolution of surface waves like the Rayleigh waves, which is not trivial with classical staggered finite differences. We thus implemented a non-centered fourth-order spatial scheme at the free surface to achieve more accuracy. Second, we implemented signal processing tools that calculate phase velocity curves and detect first arrival travel times and wave propagation modes of seismic data. These tools are used for dispersion analysis. Third, we revisit a study carried out on unconsolidated granular media at the laboratory scale using the different tools (finite differences or finite volumes). We compare different models with different rheologies (elastic or poro-elastic), different dimensions (3D or 2D), different boundary conditions (PML or Dirichlet) and different numerical modeling of the source (stick or point) in order to reproduce the experimental data. The study of the sensitivity of the seismic data to the source location was also crucial to improve the amplitude of the signals and the detection of the different seismic modes. This will allow us in the future to better image and understand these complex media

HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB

This paper presents HDGlab, an open source MATLAB implementation of the hybridisable discontinuous Galerkin (HDG) method. The main goal is to provide a detailed description of both the HDG method for elliptic problems and its implementation available in HDGlab. Ultimately, this is expected to make this relatively new advanced discretisation method more accessible to the computational engineering community. HDGlab presents some features not available in other implementations of the HDG method that can be found in the free domain. First, it implements high-order polynomial shape functions up to degree nine, with both equally-spaced and Fekete nodal distributions. Second, it supports curved isoparametric simplicial elements in two and three dimensions. Third, it supports non-uniform degree polynomial approximations and it provides a flexible structure to devise degree adaptivity strategies. Finally, an interface with the open-source high-order mesh generator Gmsh is provided to facilitate its application to practical engineering problems

A space-time hybridizable discontinuous Galerkin method for linear free-surface waves

We present and analyze a novel space-time hybridizable discontinuous Galerkin (HDG) method for the linear free-surface problem on prismatic space-time meshes. We consider a mixed formulation which immediately allows us to compute the velocity of the fluid. In order to show well-posedness, our space-time HDG formulation makes use of weighted inner products. We perform an a priori error analysis in which the dependence on the time step and spatial mesh size is explicit. We provide two numerical examples: one that verifies our analysis and a wave maker simulation

Simulation de la propagation d'ondes élastiques en domaine fréquentiel par des méthodes Galerkine discontinues

The scientific context of this thesis is seismic imaging which aims at recovering the structure of the earth. As the drilling is expensive, the petroleum industry is interested by methods able to reconstruct images of the internal structures of the earth before the drilling. The most used seismic imaging method in petroleum industry is the seismic-reflection technique which uses a wave equation model. Seismic imaging is an inverse problem which requires to solve a large number of forward problems. In this context, we are interested in this thesis in the modeling part, i.e. the resolution of the forward problem, assuming a time-harmonic regime, leading to the so-called Helmholtz equations. The main objective is to propose and develop a new finite element (FE) type solver characterized by a reduced-size discrete operator (as compared to existing such solvers) without hampering the accuracy of the numerical solution. We consider the family of discontinuous Galerkin (DG) methods. However, as classical DG methods are much more expensive than continuous FE methods when considering steady-like problems, because of an increased number of coupled degrees of freedom as a result of the discontinuity of the approximation, we develop a new form of DG method that specifically address this issue: the hybridizable DG (HDG) method. To validate the efficiency of the proposed HDG method, we compare the results that we obtain with those of a classical upwind flux-based DG method in a 2D framework. Then, as petroleum industry is interested in the treatment of real data, we develop the HDG method for the 3D elastic Helmholtz equations.Le contexte scientifique de cette thèse est l'imagerie sismique dont le but est de reconstituer la structure du sous-sol de la Terre. Comme le forage a un coût assez élevé, l'industrie pétrolière s'intéresse à des méthodes capables de reconstituer les images de la structure terrestre interne avant de le faire. La technique d'imagerie sismique la plus utilisée est la technique de sismique-réflexion qui est basée sur le modèle de l'équation d'ondes. L'imagerie sismique est un problème inverse qui requiert de résoudre un grand nombre de problèmes directs. Dans ce contexte, nous nous intéressons dans cette thèse à la résolution du problème direct en régime harmonique, soit à la résolution des équations d'Helmholtz. L'objectif principal est de proposer et de développer un nouveau type de solveur élément fini (EF) caractérisé par un opérateur discret de taille réduite (comparée à la taille des solveurs déjà existants) sans pour autant altérer la précision de la solution numérique. Nous considérons les méthodes de Galerkine discontinues (DG). Comme les méthodes DG classiques sont plus coûteuses que les méthodes EF continues si l'on considère un même problème à cause d'un grand nombre de degrés de liberté couplés, résultat des approximations discontinues, nous développons une nouvelle classe de méthode DG réduisant ce problème : la méthode DG hybride (HDG). Pour valider l'efficacité de la méthode HDG proposée, nous comparons les résultats obtenus avec ceux obtenus avec une méthode DG basée sur des flux décentrés en 2D. Comme l'industrie pétrolière s'intéresse au traitement de données réelles, nous développons ensuite la méthode HDG pour les équations élastiques d'Helmholtz 3D