Arbitrary high order numerical methods for time-harmonic acoustic scattering problems originally defined on unbounded domains are constructed. This is done by coupling recently developed high order local absorbing boundary conditions (ABCs) with finite difference methods for the Helmholtz equation. These ABCs are based on exact representations of the outgoing waves by means of farfield expansions. The finite difference methods, which are constructed from a deferred-correction (DC) technique, approximate the Helmholtz equation and the ABCs to any desired order. As a result, high order numerical methods with an overall order of convergence equal to the order of the DC schemes are obtained. A detailed construction of these DC finite difference schemes is presented. Details and results from an extension to heterogeneous media are also included. Additionally, a rigorous proof of the consistency of the DC schemes with the Helmholtz equation and the ABCs in polar coordinates is also given. The results of several numerical experiments corroborate the high order convergence of the proposed method. A novel local high order ABC for elastic waves based on farfield expansions is constructed and preliminary results applying it to elastic scattering problems are presented.

A numerical high order difference method is developed for solution of the incompressible Navier-Stokes equations. The solution is determined on a staggered curvilinear grid in two dimensions and by a Fourier expansion in the third dimension. The description in curvilinear body-fitted coordinates is obtained by an orthogonal mapping of the equations to a rectangular grid where space derivatives are determined by compact fourth order approximations. The time derivative is discretized with a second order backward difference method in a semi-implicit scheme, where the nonlinear terms are linearly extrapolated with second order accuracy. An approximate block factorization technique is used in an iterative scheme to solve the large linear system resulting from the discretization in each time step. The solver algorithm consists of a combination of outer and inner iterations. An outer iteration step involves the solution of two sub-systems, one for prediction of the velocities and one for solution of the pressure. No boundary conditions for the intermediate variables in the splitting are needed and second order time accurate pressure solutions can be obtained. The method has experimentally been validated in earlier studies. Here it is validated for flow past a circular cylinder as an example of a physical test case and the fourth order method is shown to be efficient in terms of grid resolution. The method is applied to external flow past a parabolic body and internal flow in an asymmetric diffuser in order to investigate the performance in two different curvilinear geometries and to give directions for future development of the method. It is concluded that the novel formulation of boundary conditions need further investigation. A new iterative solution method for prediction of velocities allows for larger time steps due to less restrictive convergence constraints. / QC 20101221

The derivation of centered compact schemes at interior and boundary grid points is performed and an analysis of stability and computational efficiency is given. Compact schemes are high order implicit methods for numerical solutions of initial and/or boundary value problems modeled by differential equations. These schemes generally require smaller stencils than the traditional explicit finite difference counterparts. To avoid numerical instabilities at and near boundaries and in regions of mesh non-uniformity, a numerical filtering technique is employed. Experiments for non-stationary linear problems (convection, heat conduction) and also for nonlinear problems (Burgers' and KdV equations) were performed. The compact solvers were combined with Euler and fourth-order Runge-Kutta time differencing. In most cases, the order of convergence of the numerical solution to the exact solution was the same as the formal order of accuracy of the compact schemes employed.

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<p>The present dissertation focuses on the numerical solution of compressible flows with an emphasis on simulations of transitional hypersonic boundary layers. Initially, general concepts such as the governing equations, numerical approximations and theoretical modeling strategies are addressed. These are used as a basis to introduce two innovative techniques, the Quasi-Spectral Viscosity (QSV) method, applied to high-order finite difference settings and the Legendre Spectral Viscosity (LSV) approach, used in high-order flux reconstruction schemes. Such techniques are derived based on the mathematical formalism of the filtered compressible Navier-Stokes equations. While the latter perspective is only typically used for turbulence modeling in the context of Large-Eddy Simulations (LES), both the QSV and LSV subfilter scale (SFS) closure models are capable of performing simulations in the presence of shock-discontinuities. On top of that, the QSV approach is also shown to support dynamic subfilter turbulence modeling capabilities.</p> <p>QSV’s innovation lies in the introduction of a physical-space implementation of a spectral-like subfilter scale (SFS) dissipation term by leveraging residuals of filter operations, achiev- ing two goals: (1) estimating the energy of the resolved solution near the grid cutoff; (2) imposing a plateau-cusp shape to the spectral distribution of the added dissipation. The QSV approach was tested in a variety of flows to showcase its capability to act interchangeably as a shock capturing method or as a SFS turbulence closure. QSV performs well compared to previous eddy-viscosity closures and shock capturing methods. In a supersonic TGV flow, a case which exhibits shock/turbulence interactions, QSV alone outperforms the simple super- position of separate numerical treatments for SFS turbulence and shocks. QSV’s combined capability of simulating shocks and turbulence independently, as well as simultaneously, effectively achieves the unification of shock capturing and Large-Eddy Simulation.</p> <p>The LSV method extends the QSV idea to discontinuous numerical schemes making it suitable for unstructured solvers. LSV exploits the set of hierarchical basis functions formed by the Legendre polynomials to extract the information on the energy content near the resolution limit and estimate the overall magnitude of the required SFS dissipative terms, resulting in a scheme that dynamically activates only in cells where nonlinear behavior is important. Additionally, the modulation of such terms in the Legendre spectral space allows for the concentration of the dissipative action at small scales. The proposed method is tested in canonical shock-dominated flow setups in both one and two dimensions. These include the 1D Burgers’ problem, a 1D shock tube, a 1D shock-entropy wave interaction, a 2D inviscid shock-vortex interaction and a 2D double Mach reflection. Results showcase a high-degree of resolution power, achieving accurate results with a small number of degrees of freedom, and robustness, being able to capture shocks associated with the Burgers’ equation and the 1D shock tube within a single cell with discretization orders 120 and higher.</p> <p>After the introduction of these methods, the QSV-LES approach is leveraged to perform numerical simulations of hypersonic boundary layer transition delay on a 7^◦-half-angle cone for both sharp and 2.5 mm-nose tip radii due to porosity representative of carbon-fibre-reinforced carbon-matrix ceramics (C/C) in the Reynolds number range Re_m = 2.43 · 106 – 6.40 · 10⁶ m⁻¹ at the freestream Mach number of M_∞ = 7.4. A low-order impedance model was fitted through experimental measurements of acoustic absorption taken at discrete frequencies yielding a continuous representation in the frequency domain that was imposed in the simulations via a broadband time domain impedance boundary condition (TDIBC). The stability of the base flow is studied over impermeable and porous walls via pulse-perturbed axisymmetric simulations with second-mode spatial growth rates matching linear predictions. This shows that the QSV-LES approach is able to dynamically deactivate its dissipative action in laminar portions of the flow making it possible to accurately capture the boundary layer’s instability dynamics. Three-dimensional transitional LES were then performed with the introduction of grid independent pseudorandom pressure perturbations. Comparison against previous experiments were made regarding the frequency content of the disturbances in the transitional region with fairly good agreement capturing the shift to lower frequencies. Such shift is caused by the formation of near-wall low-temperature streaks that concentrate the pressure disturbances at locations with locally thicker boundary layers forming trapped wavetrains that can persist into the turbulent region. Additionally, it is shown that the presence of a porous wall representative of a C/C material does not affect turbulence significantly and simply shifts its onset downstream.</p>

Optimization problems are ubiquitous in all quantitative scientific disciplines, from computer science and engineering to operations research and economics. Developing algorithms for solving various optimization problems has been the focus of mathematical research for years. In the last decade, optimization research has become even more popular due to its applications in the rapidly developing field of machine learning. In this thesis, we discuss a few fundamental and well-studied optimization problem classes: decentralized distributed optimization (Chapters 2 to 4), distributed optimization under similarity (Chapter 5), affinely constrained optimization (Chapter 6), minimax optimization (Chapter 7), and high-order optimization (Chapter 8). For each problem class, we develop the first provably optimal algorithm: the complexity of such an algorithm cannot be improved for the problem class given. The proposed algorithms show state-of-the-art performance in practical applications, which makes them highly attractive for potential generalizations and extensions in the future.

Projection methods for the incompressible Navier-Stokes equations (NSE) are efficient, but introduce numerical boundary layers and have limited temporal accuracy due to their fractional step nature. The Pressure Poisson Equation (PPE) reformulations represent a class of methods that replace the incompressibility constraint by a Poisson equation for the pressure, with a suitable choice of the boundary condition so that the incompressibility is maintained. PPE reformulations of the NSE have important advantages: the pressure is no longer implicitly coupled to the velocity, thus can be directly recovered by solving a Poisson equation, and no numerical boundary layers are generated; arbitrary order time-stepping schemes can be used to achieve high order accuracy in time. In this thesis, we focus on numerical approaches of the PPE reformulations, in particular, the Shirokoff-Rosales (SR) PPE reformulation. Interestingly, the electric boundary conditions, i.e., the tangential and divergence boundary conditions, provided for the velocity in the SR PPE reformulation render classical nodal finite elements non-convergent. We propose two alternative methodologies, mixed finite element methods and meshfree finite differences, and demonstrate that these approaches allow for arbitrary order of accuracy both in space and in time. / Mathematics

Flooding has always been and is still today a disastrous event with agricultural, infrastructural, economical and not least humanitarian ramifications. Understanding the behaviour of floods is crucial to be able to prevent or mitigate future catastrophes, a task which can be accomplished by modelling the water flow. In this thesis the finite element method is employed to solve the shallow water equations, which govern water flow in shallow environments such as rivers, lakes and dams, a methodology that has been widely used for flooding simulations. Alternative approaches to model floods are however also briefly discussed. Since the finite element method suffers from numerical instabilities when solving nonlinear conservation laws, the shallow water equations are stabilised by introducing a high-order nonlinear artificial viscosity, constructed using a multi-mesh strategy. The accuracy, robustness and well-balancedness of the solution are examined through a variety of benchmark tests. Finally, the equations are extended to include a friction term, after which the effectiveness of the method in a real-life scenario is verified by a prolonged simulation of the Malpasset dam break.

Le travail présenté ici avait pour but le développement d'un schéma de type Galerkin discontinu (GD) d'ordre élevé pour la résolution des équations de la dynamique des gaz écrites dans un formalisme Lagrangien total, sur des maillages bi-dimensionnels totalement déstructurés. À cette fin, une méthode progressive a été utilisée afin d'étudier étape par étape les difficultés numériques inhérentes à la discrétisation Galerkin discontinue ainsi qu'aux équations de la dynamique des gaz Lagrangienne. Par conséquent, nous avons développé dans un premier temps des schémas de type Galerkin discontinu jusqu'à l'ordre trois pour la résolution des lois de conservation scalaires mono-dimensionnelles et bi-dimensionnelles sur des maillages déstructurés. La particularité principale de la discrétisation GD présentée est l'utilisation des bases polynomiales de Taylor. Ces dernières permettent, dans le cadre de maillages bi-dimensionnels déstructurés, une prise en compte globale et unifiée des différentes géométries. Une procédure de limitation hiérarchique, basée aux noeuds et préservant les extrema réguliers a été mise en place, ainsi qu'une forme générale des flux numériques assurant une stabilité globale L_2 de la solution. Ensuite, nous avons tâché d'appliquer la discrétisation Galerkin discontinue développée aux systèmes mono-dimensionnels de lois de conservation comme celui de l'acoustique, de Saint-Venant et de la dynamique des gaz Lagrangienne. Nous avons noté au cours de cette étude que l'application directe de la limitation mise en place dans le cadre des lois de conservation scalaires, aux variables physiques des systèmes mono-dimensionnels étudiés provoquait l'apparition d'oscillations parasites. En conséquence, une procédure de limitation basée sur les variables caractéristiques a été développée. Dans le cas de la dynamique des gaz, les flux numériques ont été construits afin que le système satisfasse une inégalité entropique globale. Fort de l'expérience acquise, nous avons appliqué la discrétisation GD mise en place aux équations bi-dimensionnelles de la dynamique des gaz, écrites dans un formalisme Lagrangien total. Dans ce cadre, le domaine de référence est fixe. Cependant, il est nécessaire de suivre l'évolution temporelle de la matrice jacobienne associée à la transformation Lagrange-Euler de l'écoulement, à savoir le tenseur gradient de déformation. Dans le travail présent, la transformation résultant de l'écoulement est discrétisée de manière continue à l'aide d'une base Éléments Finis. Cela permet une approximation du tenseur gradient de déformation vérifiant l'identité essentielle de Piola. La discrétisation des lois de conservation physiques sur le volume spécifique, le moment et l'énergie totale repose sur une méthode Galerkin discontinu. Le schéma est construit de sorte à satisfaire de manière exacte la loi de conservation géométrique (GCL). Dans le cas du schéma d'ordre trois, le champ de vitesse étant quadratique, la géométrie doit pouvoir se courber. Pour ce faire, des courbes de Bézier sont utilisées pour la paramétrisation des bords des cellules du maillage. Nous illustrons la robustesse et la précision des schémas mis en place à l'aide d'un grand nombre de cas tests pertinents, ainsi que par une étude de taux de convergence. / The intent of the present work was the development of a high-order discontinuous Galerkin scheme for solving the gas dynamics equations written under total Lagrangian form on two-dimensional unstructured grids. To achieve this goal, a progressive approach has been used to study the inherent numerical difficulties step by step. Thus, discontinuous Galerkin schemes up to the third order of accuracy have firstly been implemented for the one-dimensional and two-dimensional scalar conservation laws on unstructured grids. The main feature of the presented DG scheme lies on the use of a polynomial Taylor basis. This particular choice allows in the two-dimensional case to take into general unstructured grids account in a unified framework. In this frame, a vertex-based hierarchical limitation which preserves smooth extrema has been implemented. A generic form of numerical fluxes ensuring the global stability of our semi-discrete discretization in the $L_2$ norm has also been designed. Then, this DG discretization has been applied to the one-dimensional system ofconservation laws such as the acoustic system, the shallow-water one and the gas dynamics equations system written in the Lagrangian form. Noticing that the application of the limiting procedure, developed for scalar equations, to the physical variables leads to spurious oscillations, we have described a limiting procedure based on the characteristic variables. In the case of the one-dimensional gas dynamics case, numerical fluxes have been designed so that our semi-discrete DG scheme satisfies a global entropy inequality. Finally, we have applied all the knowledge gathered to the case of the two-dimensional gas dynamics equation written under total Lagrangian form. In this framework, the computational grid is fixed, however one has to follow the time evolution of the Jacobian matrix associated to the Lagrange-Euler flow map, namely the gradient deformation tensor. In the present work, the flow map is discretized by means of continuous mapping, using a finite element basis. This provides an approximation of the deformation gradient tensor which satisfies the important Piola identity. The discretization of the physical conservation laws for specific volume, momentum and total energy relies on a discontinuous Galerkin method. The scheme is built to satisfying exactly the Geometric Conservation Law (GCL). In the case of the third-order scheme, the velocity field being quadratic we allow the geometry to curve. To do so, a Bezier representation is employed to define the mesh edges. We illustrate the robustness and the accuracy of the implemented schemes using several relevant test cases and performing rate convergences analysis.

Mon travail pendant cette thèse a d’abord été le développement d’une source de génération d’harmoniques d’ordre élevé basée sur une chaîne laser femtoseconde amplifiée (Saphir-Titane) fonctionnant à une cadence de 1 kHz (AURORE). Une ligne de lumière construite au CELIA permet de fournir un faisceau focalisé VUV-XUV femtoseconde, monochromatique dans une région spectrale comprise entre 10 nm et 73 nm environ (17 eV à 120 eV). Cette installation expérimentale est en fonctionnement et est parmi les toutes premières lignes à être mise en service pour la communauté scientifique française et étrangère. J’ai également mis en place une installation d'étude des cinétiques de luminescence avec résolution temporelle sub-picoseconde (450 fs) par mélange de fréquences. Le thème général de ce travail est l’étude des processus de relaxation et d'interaction entre les excitations électroniques créées par des impulsions ultra brèves femtosecondes de photons IR, UV et VUV-XUV dans les solides diélectriques massifs et des nano particules. L’observable principale utilisée est la luminescence émise par ces systèmes, résolue spectralement et en temps sur des échelles allant de la µs à des temps sub picosecondes. Ce travail a abouti à une avancée sensible de la description des processus principaux de formation et d’évolution des excitations électroniques. La comparaison et l’interprétation des données expérimentales obtenues pour des nano particules et des cristaux ont permis d’élucider certaines propriétés spécifiques de ces systèmes. / The work during this Ph.D. was a development of a source of high order harmonics generation based on amplified Ti:Sapphire femtosecond laser with repetition rate 1kHz (AURORE). The beam line constructed in CELIA has on its exit a VUV-XUV focalized beam; it may has wide spectrum or monochromatic in spectral range from 10nm up to 73nm (17-120eV). This beam line is in operation and is using for experiments for solid state VUV spectroscopy, photoelectron spectroscopy etc. Also it was installed a system for detection of luminescence with sub-picosecond time resolution (450fs) based on the nonlinear effect – generation of sum of two light frequencies. The main subject of this work was the study of processes of relaxation and interaction of electronic excitations, created by ultra-short pulse of IR, UV or XUV in dielectric crystals and nanoparticles. Out method is based on observation of luminescence with spectral and time resolution up to sub-picosecond temporal resolution. This study has given new experimental results for description of fundamental processes of creation and evolution of electronic excitations. Comparison and interpretation of experimental data of semiconductor nano-particles and monocrystals gave some interpretations of extra-fast luminescence of these systems.