The Boltzmann Transport Equation (BTE) has been the keystone of the kinetic theory, which is at the center of Statistical Mechanics bridging the gap between the atomic structures and the continuum-like behaviors. The existence of solutions has been a great mathematical challenge and still remains elusive. As a grazing limit of the Boltzmann operator, the Fokker-Planck-Landau (FPL) operator is of primary importance for collisional plasmas. We have worked on the following three different projects regarding the most important kinetic models, the BTE and the FPL Equations. (1). A Discontinuous Galerkin Solver for Nonlinear BTE. We propose a deterministic numerical solver based on Discontinuous Galerkin (DG) methods, which has been rarely studied. As the key part, the weak form of the collision operator is approximated within subspaces of piecewise polynomials. To save the tremendous computational cost with increasing order of polynomials and number of mesh nodes, as well as to resolve loss of conservations due to domain truncations, the following combined procedures are applied. First, the collision operator is projected onto a subspace of basis polynomials up to first order. Then, at every time step, a conservation routine is employed to enforce the preservation of desired moments (mass, momentum and/or energy), with only linear complexity. The asymptotic error analysis shows the validity and guarantees the accuracy of these two procedures. We applied the property of ``shifting symmetries" in the weight matrix, which consists in finding a minimal set of basis matrices that can exactly reconstruct the complete family of collision weight matrix. This procedure, together with showing the sparsity of the weight matrix, reduces the computation and storage of the collision matrix from O(N3) down to O(N^2). (2). Spectral Gap for Linearized Boltzmann Operator. Spectral gaps provide information on the relaxation to equilibrium. This is a pioneer field currently unexplored form the computational viewpoint. This work, for the first time, provides numerical evidence on the existence of spectral gaps and corresponding approximate values. The linearized Boltzmann operator is projected onto a Discontinuous Galerkin mesh, resulting in a ``collision matrix". The original spectral gap problem is then approximated by a constrained minimization problem, with objective function the Rayleigh quotient of the "collision matrix" and with constraints the conservation laws. A conservation correction then applies. We also study the convergence of the approximate Rayleigh quotient to the real spectral gap. (3). A Conservative Scheme for Approximating Collisional Plasmas. We have developed a deterministic conservative solver for the inhomogeneous Fokker-Planck-Landau equations coupled with Poisson equations. The original problem is splitted into two subproblems: collisonless Vlasov problem and collisonal homogeneous Fokker-Planck-Landau problem. They are handled with different numerical schemes. The former is approximated using Runge-Kutta Discontinuous Galerkin (RKDG) scheme with a piecewise polynomial basis subspace covering all collision invariants; while the latter is solved by a conservative spectral method. To link the two different computing grids, a special conservation routine is also developed. All the projects are implemented with hybrid MPI and OpenMP. Numerical results and applications are provided. / text

La première partie du travail de thèse a consisté en l’établissement d’un état de l’art sur les méthodes de dimensionnement en fatigue sous environnement vibratoire. Nous avons ainsi étudié la représentation mathématique des vibrations aléatoires, les méthodes classiques de dimensionnement en fatigue pour des chargements uni-axiaux et multiaxiaux ainsi que les méthodes de dimensionnement en fatigue formulées dans le domaine spectral. Ces dernières ont été définies pour la problématique des vibrations aléatoires, dans le cadre d’un certain nombre d’hypothèses de travail, et montrent des temps de calculs nettement plus faibles que des méthodes de dimensionnement classiques. La deuxième partie de travail de la thèse a consisté à développer une méthode de dimensionnement en fatigue formulée dans le domaine spectral valide sur tout le domaine de fatigue, aussi bien en oligo-cyclique qu’en endurance. Les travaux ont portés sur l’intégration du comportement élasto-plastique confiné des matériaux dans les méthodes spectrales. L’approche proposée a fait l’objet d’une étude numérique comparative avec les méthodes existantes. Le dernier volet de la thèse est la comparaison de la méthode proposée avec des essais. Pour cela, le développement d’une éprouvette spécifique à la problématique a été réalisé. Les résultats obtenus montrent une bonne corrélation entre les approches numériques et expérimentales. / Mechanical structures are often subjected to random vibrations due to external forces (forces, acceleration...). Improving the safety of structures and reducing manufacturing and design costs is achieved thanks to accurate fatigue design methods. The first part of the thesis consisted in the establishment of a state of the art on design methods for fatigue under vibrational environment. The mathematical representation of random vibration, conventional fatigue design methods for uniaxial and multiaxial loads as well as fatigue design methods formulated in the spectral domain have been thoroughly studied. The latter have been applied to random vibrations cases, through a number of assumptions. They show significantly lower computation time than conventional design methods. The second part of the thesis is the development of a fatigue design method formulated in the spectral domain, which is valid over the whole domain of fatigue, from low cycle fatigue (LCH) to high cycle fatigue (HCF). The work focused on the integration of confined elasto-plastic behaviour of the materials in spectral methods. The proposed approach has been the subject of a comparative numerical study with existing methods. The last part of the thesis presents the comparison of proposed method to physical tests. A test specimen specific to this study has been developed. The results show a good correlation between the numerical and experimental approaches.

abstract: The dynamics of a fluid flow inside 2D square and 3D cubic cavities under various configurations were simulated and analyzed using a spectral code I developed. This code was validated against known studies in the 3D lid-driven cavity. It was then used to explore the various dynamical behaviors close to the onset of instability of the steady-state flow, and explain in the process the mechanism underlying an intermittent bursting previously observed. A fairly complete bifurcation picture emerged, using a combination of computational tools such as selective frequency damping, edge-state tracking and subspace restriction. The code was then used to investigate the flow in a 2D square cavity under stable temperature stratification, an idealized version of a lake with warmer water at the surface compared to the bottom. The governing equations are the Navier-Stokes equations under the Boussinesq approximation. Simulations were done over a wide range of parameters of the problem quantifying the driving velocity at the top (e.g. wind) and the strength of the stratification. Particular attention was paid to the mechanisms associated with the onset of instability of the base steady state, and the complex nontrivial dynamics occurring beyond onset, where the presence of multiple states leads to a rich spectrum of states, including homoclinic and heteroclinic chaos. A third configuration investigates the flow dynamics of a fluid in a rapidly rotating cube subjected to small amplitude modulations. The responses were quantified by the global helicity and energy measures, and various peak responses associated to resonances with intrinsic eigenmodes of the cavity and/or internal retracing beams were clearly identified for the first time. A novel approach to compute the eigenmodes is also described, making accessible a whole catalog of these with various properties and dynamics. When the small amplitude modulation does not align with the rotation axis (precession) we show that a new set of eigenmodes are primarily excited as the angular velocity increases, while triadic resonances may occur once the nonlinear regime kicks in. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2019

Spectral methods for partial differential equations with boundary conditions in complex domains are developed with the help of a fictitious domain approach. For rectangular embedding, spectral-Galerkin formulations with special trial and test functions are presented and discussed, as well as the well-posedness and the error analysis. For circular and annular embedding, dimension reduction is applied and a sequence of 1-D problems with artificial boundary values are solved. Applications of our methods include the fractional Laplace problem and the Helmholtz equations. In numerical examples, our methods show good performance on the boundary value problems in both smooth and polygonal complex domains, and the L2 errors decay exponentially for smooth solutions. For singular problems, high-order convergence rates can also be obtained.

Grouping is a vital precursor to object recognition. The complexity of the object recognition process can be reduced to a large extent by using a frontend grouping process. In this dissertation, a grouping framework based on spectral methods for graphs is used. The objects are segmented from the background by means of an associated learning process that decides on the relative importance of the basic salient relationships such as proximity, parallelism, continuity, junctions and common region. While much of the previous research has been focussed on using simple relationships like similarity, proximity, continuity and junctions, this work differenciates itself by using all the relationships listed above. The parameters of the grouping process is cast as probabilistic specifications of Bayesian networks that need to be learned: the learning is accomplished by a team of stochastic learning automata. One of the stages in the grouping process is graph partitioning. There are a variety of cut measures based on which partitioning can be obtained and different measures give different partitioning results. This work looks at three popular cut measures, namely the minimum, average and normalized. Theoretical and empirical insight into the nature of these partitioning measures in terms of the underlying image statistics are provided. In particular, the questions addressed are as follows: For what kinds of image statistics would optimizing a measure, irrespective of the particular algorithm used, result in correct partitioning? Are the quality of the groups significantly different for each cut measure? Are there classes of images for which grouping by partitioning is not suitable? Does recursive bi-partitioning strategy separate out groups corresponding to K objects from each other? The major conclusion is that optimization of none of the above three measures is guaranteed to result in the correct partitioning of K objects, in the strict stochastic order sense, for all image statistics. Qualitatively speaking, under very restrictive conditions when the average inter-object feature affinity is very weak when compared to the average intra-object feature affinity, the minimum cut measure is optimal. The average cut measure is optimal for graphs whose partition width is less than the mode of distribution of all possible partition widths. The normalized cut measure is optimal for a more restrictive subclass of graphs whose partition width is less than the mode of the partition width distributions and the strength of inter-object links is six times less than the intra-object links. The learning framework described in the first part of the work is used to empirically evaluate the cut measures. Rigorous empirical evaluation on 100 real images indicates that in practice, the quality of the groups generated using minimum or average or normalized cuts are statistically equivalent for object recognition, i.e. the best, the mean, and the variation of the qualities are statistically equivalent. Another conclusion is that for certain image classes, such as aerial and scenes with man-made objects in man-made surroundings, the performance of grouping by partitioning is the worst, irrespective of the cut measure.

This thesis deals with numerical simulation of turbulentflows in geometrically simple cases. Both plane and cylindricalgeometries are used. The simplicity of the geometry allows theuse of spectral methods which yield a very high accuracy usingrelatively few grid points. A spectral method for planegeometries is implemented on a parallel computer. Thetransitional Reynolds number for plane Couette flow is verifiedto be about 360, in accordance with earlier findings. TurbulentCouette flow at twice the transitional Reynolds number isstudied and the findings of large scale structures in earlierstudies of Couette flow are substantiated. These largestructures are shown to be of limited extent and give anintegral length scale of six half channel heights, or abouteight times larger than in pressure-driven channel flow.Despite this, they contain only about 10 \% of the turbulentenergy. This is demonstrated by applying a very smallstabilising rotation, which almost eliminates the largestructures. A comparison of the Reynolds stress budget is madewith a boundary layer flow, and it is shown that the near-wallvalues in Couette flow are comparable with high-Reynolds numberboundary layer flow. A new spectrally accurate algorithm isdeveloped and implemented for cylindrical geometries andverified by studying the evolution of eigenmodes for both pipeflow and annular pipe flow. This algorithm is a generalisationof the algorithm used in the plane channel geometry. It usesFourier transforms in two homogeneous directions and Chebyshevpolynomials in the third, wall-normal, direction. TheNavier--Stokes equations are solved with a velocity-vorticityformulation, thereby avoiding the difficulty of solving for thepressure. The time advancement scheme used is a mixedimplicit/explicit second order scheme. The coupling between twovelocity components, arising from the cylindrical coordinates,is treated by introducing two new components and solving forthem, instead of the original velocity components. TheChebyshev integration method and the Chebyshev tau method isboth implemented and compared for the pipe flow case.

This work develops a nonlinear, three-dimensional spectral collocation method for the simulation of the incompressible Navier-Stokes equations for geophysical and environmental flows. These flows are often driven by the interaction of stratified fluid with topography, which is accurately accounted for in this model using a mapped coordinate system. The spectral collocation method used here evaluates derivatives with a Fourier trigonometric or Chebyshev polynomial expansion as appropriate, and it evaluates the nonlinear terms directly on a collocated grid. The coordinate mapping renders ineffective fast solution methods that rely on separation of variables, so to avoid prohibitively expensive matrix solves this work develops a low-order finite-difference preconditioner for the implicit solution steps. This finite-difference preconditioner is itself too expensive to apply directly, so it is solved pproximately with a geometric multigrid method, using semicoarsening and line relaxation to ensure convergence with locally anisotropic grids. The model is discretized in time with a third-order method developed to allow variable timesteps. This multi-step method explicitly evaluates advective terms and implicitly evaluates pressure and viscous terms. The model’s accuracy is demonstrated with several test cases: growth rates of Kelvin-Helmholtz billows, the interaction of a translating dipole with no-slip boundaries, and the generation of internal waves via topographic interaction. These test cases also illustrate the model’s use from a high-level programming perspective. Additionally, the results of several large-scale simulations are discussed: the three-dimensional dipole/wall interaction, the evolution of internal waves with shear instabilities, and the stability of the bottom boundary layer beneath internal waves. Finally, possible future developments are discussed to extend the model’s capabilities and optimize its performance within the limits of the underlying numerical algorithms.

Optical amplifiers and lasers continue to play its crucial role and they have become an indispensable part of the every fiber optic communication systems being installed from optical network to ultra-long haul systems. It seems that they will keep on to be a promising future technology for high speed, long-distance fiber optic transmission systems. The numerical simulations of the model equations have been already commercialized by the photonic system designers to meet the future challenges. One of the challenging problems for designing Raman amplifiers or lasers is to develop a numerical method that meets all the requirements such as accuracy, robustness and speed. In the last few years, there have been much effort towards solving the coupled differential equations of Raman model with high accuracy and stability. The techniques applied in literature for solving propagation equations are mainly based on the finite differences, shooting or in some cases relaxation methods. We have described a new method to solve the nonlinear equations such as Newton-Krylov iteration and performed numerical simulations using spectral methods. A novel algorithm implementing spectral method (pseuodspectral) for solving the two-point boundary value problem of propagation equations is proposed, for the first time to the authors&#039 / knowledge in this thesis. Numerical results demonstrate that in a few iterations great accuracy is obtained using fewer grid points.

<p>This thesis deals with numerical simulation of turbulentflows in geometrically simple cases. Both plane and cylindricalgeometries are used. The simplicity of the geometry allows theuse of spectral methods which yield a very high accuracy usingrelatively few grid points. A spectral method for planegeometries is implemented on a parallel computer. Thetransitional Reynolds number for plane Couette flow is verifiedto be about 360, in accordance with earlier findings. TurbulentCouette flow at twice the transitional Reynolds number isstudied and the findings of large scale structures in earlierstudies of Couette flow are substantiated. These largestructures are shown to be of limited extent and give anintegral length scale of six half channel heights, or abouteight times larger than in pressure-driven channel flow.Despite this, they contain only about 10 \% of the turbulentenergy. This is demonstrated by applying a very smallstabilising rotation, which almost eliminates the largestructures. A comparison of the Reynolds stress budget is madewith a boundary layer flow, and it is shown that the near-wallvalues in Couette flow are comparable with high-Reynolds numberboundary layer flow. A new spectrally accurate algorithm isdeveloped and implemented for cylindrical geometries andverified by studying the evolution of eigenmodes for both pipeflow and annular pipe flow. This algorithm is a generalisationof the algorithm used in the plane channel geometry. It usesFourier transforms in two homogeneous directions and Chebyshevpolynomials in the third, wall-normal, direction. TheNavier--Stokes equations are solved with a velocity-vorticityformulation, thereby avoiding the difficulty of solving for thepressure. The time advancement scheme used is a mixedimplicit/explicit second order scheme. The coupling between twovelocity components, arising from the cylindrical coordinates,is treated by introducing two new components and solving forthem, instead of the original velocity components. TheChebyshev integration method and the Chebyshev tau method isboth implemented and compared for the pipe flow case.</p>

This work develops a nonlinear, three-dimensional spectral collocation method for the simulation of the incompressible Navier-Stokes equations for geophysical and environmental flows. These flows are often driven by the interaction of stratified fluid with topography, which is accurately accounted for in this model using a mapped coordinate system. The spectral collocation method used here evaluates derivatives with a Fourier trigonometric or Chebyshev polynomial expansion as appropriate, and it evaluates the nonlinear terms directly on a collocated grid. The coordinate mapping renders ineffective fast solution methods that rely on separation of variables, so to avoid prohibitively expensive matrix solves this work develops a low-order finite-difference preconditioner for the implicit solution steps. This finite-difference preconditioner is itself too expensive to apply directly, so it is solved pproximately with a geometric multigrid method, using semicoarsening and line relaxation to ensure convergence with locally anisotropic grids. The model is discretized in time with a third-order method developed to allow variable timesteps. This multi-step method explicitly evaluates advective terms and implicitly evaluates pressure and viscous terms. The model’s accuracy is demonstrated with several test cases: growth rates of Kelvin-Helmholtz billows, the interaction of a translating dipole with no-slip boundaries, and the generation of internal waves via topographic interaction. These test cases also illustrate the model’s use from a high-level programming perspective. Additionally, the results of several large-scale simulations are discussed: the three-dimensional dipole/wall interaction, the evolution of internal waves with shear instabilities, and the stability of the bottom boundary layer beneath internal waves. Finally, possible future developments are discussed to extend the model’s capabilities and optimize its performance within the limits of the underlying numerical algorithms.