Дисертації з теми "Physical boundary conditions"

To increase our understanding of how ice sheets and glaciers interact with the climate system, numerical models have become an indispensable tool. However, the complexity of these systems and the natural limitation in computational power is reflected in the simplifications of the represented processes and the spatial and temporal resolution of the models. Whether the effect of these limitations is acceptable or not, can be assessed by theoretical considerations and by validating the output of the models against real world data. Equally important is to verify if the numerical implementation and computational method accurately represent the mathematical description of the processes intended to be simulated. This thesis concerns a set of numerical models used in the field of glaciology, how these are applied and how they relate to other study areas in the same field. The dynamical flow of glaciers, which can be described by a set of non-linear partial differential equations called the Full Stokes equations, is simulated using the finite element method. To reduce the computational cost of the method significantly, it is common to lower the order of the used elements. This results in a loss of stability of the method, but can be remedied by the use of stabilization methods. By numerically studying different stabilization methods and evaluating their suitability, this work contributes to constraining the values of stabilization parameters to be used in ice sheet simulations. Erroneous choices of parameters can lead to oscillations of surface velocities, which affects the long term behavior of the free-surface ice and as a result can have a negative impact on the accuracy of the simulated mass balance of ice sheets. The amount of basal sliding is an important component that affects the overall dynamics of the ice. A part of this thesis considers different implementations of the basal impenetrability condition that accompanies basal sliding, and shows that methods used in literature can lead to a difference in velocity of 1% to 5% between the considered methods. The subglacial hydrological system directly influences the glacier's ability to slide and therefore affects the velocity distribution of the ice. The topology and dominant mode of the hydrological system on the ice sheet scale is, however, ill constrained. A third contribution of this thesis is, using the theory of R-channels to implement a simple numerical model of subglacial water flow, to show the sensitivity of subglacial channels to transient processes and that this limits their possible extent. This insight adds to a cross-disciplinary discussion between the different sub-fields of theoretical, field and paleo-glaciology regarding the characteristics of ice sheet subglacial hydrological systems. In the study, we conclude by emphasizing areas of importance where the sub-fields have yet to unify: the spatial extent of channelized subglacial drainage, to what degree specific processes are connected to geomorphic activity and the differences in spatial and temporal scales. As a whole, the thesis emphasizes the importance of verification of numerical models but also acknowledges the natural limitations of these to represent complex systems. Focusing on keeping numerical ice sheet and glacier models as transparent as possible will benefit end users and facilitate accurate interpretations of the numerical output so it confidently can be used for scientific purposes.

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 3: Manuscript. Paper 4: Manuscript.

Greenland Analogue Project

The accurate prediction of wave overtopping is one of the most important aspects in the design of coastal defence structures. This can be achieved by using three different approaches: by physical modelling using laboratory tests, by empirical formulae available in literature derived from physical modelling and field tests, or by numerical simulation of the hydraulic response of the structure. All of these prediction methods are subject to a certain level of uncertainty. One source of this is the requirement of a defined free surface elevation and velocity time series seaward boundary condition in any model. Often, these are not available but the modeller is instead provided with an incident energy density spectrum. A time series will then be reconstructed from this spectrum to be used as boundary conditions. Since the energy density spectrum provides only information on the amplitude of the components, it is usually assumed that the phases of these components are randomly distributed. To create the randomly generated phases, an initial seed value is required to generate a population of uniformly distributed random phases. By varying this value for each simulation a different time series will be produced. The overall objective of this research is to quantify the uncertainty in the prediction of overtopping due to this process. This research involved carrying out two sets of laboratory experiments. Firstly, those carried out in the 2D wave flume at HR Wallingford, which provided a reference case for the validation of a numerical model, as well as a measured incident wave spectra for the generation of the population of reconstructed offshore boundary time series. The second set of experiments was carried out in the smaller 2D flume at the University of Nottingham to investigate the effect of random seeding to generate the time series at the wave paddle on the resulting overtopping parameters. This was also carried out to allow a comparison in the variability between the physical and numerical results. It was found in the work, that when a measured free surface elevation is used as the input, good agreement between the numerical solver prediction and the overtopping measurements was observed. Subsequently, when a Monte Carlo approach was used to generate the population of reconstructed offshore boundary time series from the measured incident spectra the statistical analysis of the results showed that the variability was higher for the small numbers of overtopping waves and decreases as overtopping becomes more frequent. To allow for more generalised conclusions on the uncertainty, further numerical tests were then carried out with synthetic spectra allowing different hydraulic and structural parameters to be considered. These showed good agreement with the findings of the initial statistical analysis. Finally, the results from the physical model tests carried out at the University of Nottingham were analysed. The influence of laboratory effects were studied and analysis was carried out to establish the magnitude and sources of variability in these results. As with the numerical results, the characteristics of the distribution of the predicted overtopping parameters were also studied.

This work deals with the one-dimensional Stefan problem with a general time- dependent boundary condition at the fixed boundary. The solution will be obtained using a discrete random walk method and the results will be compared qualitatively with analytical- and finite difference method solutions. A critical part has been to model the moving boundary with the random walk method. The results show that the random walk method is competitive in relation to the finite difference method and has its advantages in generality and low effort to implement. The finite difference method has, on the other hand, higher accuracy for the same computational time with the here chosen step lengths. For the random walk method to increase the accuracy, longer execution times are required, but since the method is generally easily adapted for parallel computing, it is possible to speed up. Regarding applications for the Stefan problem, there are a large range of examples such as climate models, the diffusion of lithium-ions in lithium-ion batteries and modelling steam chambers for oil extraction using steam assisted gravity drainage.

This thesis describes the results of experimental work on polyelectrolyte gels and their interaction with oppositely charged surfactants, and presents two new algorithms applicable to the simulation of colloid and polymer systems. The model systems investigated were crosslinked poly(acrylate) (PA) and poly(styrene sulphonate) (PSS), and the surfactant was dodecyl trimethylammonium bromide (DoTAB). Pure gel materials were studied using dynamic light scattering. It was shown that the diffusion coefficient (D) increases with increasing degree of swelling and the concentration dependence is larger than predicted by scaling arguments. For gels at swelling equilibrium D increases with increasing degree of crosslinking. In subsequent studies on gel particles in DoTAB solution, Raman spectra were recorded at different positions in the gel. For both types of gels two distinct regions could be observed. For PA the surfactant is localised in the outer phase without any surfactant in the core, while for PSS the surfactant was distributed such that it had the same concentration relative to the polymer throughout the gel. In a second experiment, the kinetics for the deswelling of microscopic PSS particles in DoTAB solution was studied. It was found that the final volume varied linearly with the DoTAB concentration, and the rate of volume decrease could be fitted to a single exponential indicating stagnant layer diffusion to be the rate limiting process for the deswelling of the PSS particles. In the second part, I first describe an algorithm showing an efficient way to detect percolation in simulations, with periodic boundary conditions, using recursion. Spherical boundary conditions is an alternative to periodic boundary conditions for systems with long-range interactions. In the last part, the possibility to use the surface of a hypersphere in four dimensions for simulations of polymer systems is investigated, and algorithms for Monte Carlo and Brownian dynamics simulations are described.

Ce travail de thèse concerne les méthodes numériques à base d'ondelettes pour la simulation de la turbulence incompressible. L'objectif principal est la prise en compte de conditions aux limites physiques dans la résolution des équations de Navier-Stokes. Contrairement aux travaux précédents où la vorticité était décomposée sur base d'ondelettes classiques, le point de vue qui est adopté ici vise à calculer le champ de vitesse de l'écoulement sous la forme d'une série d'ondelettes à divergence nulle. On est alors dans le cadre des équations de Navier-Stokes incompressibles en formulation vitesse-pression, pour lesquelles les conditions aux limites sur la vitesse s'écrivent explicitement, ce qui diffère de la formulation vitesse-tourbillon. Le principe de la méthode développée dans cette thèse consiste à injecter directement les conditions aux limites sur la base d'ondelettes. Ce travail prolonge la thèse de E. Deriaz réalisée dans le cas périodique. La première partie de ce travail a donc été la définition et la mise en œuvre de nouvelles bases d'ondelettes à divergence nulle ou à rotationnel nul sur [0,1]n, permettant la prise en compte de conditions aux limites, à partir des travaux originaux de P. G. Lemarié-Rieusset, K. Urban, E. Deriaz et V. Perrier. Dans une deuxième partie, des méthodes numériques efficaces utilisant ces nouvelles ondelettes sont proposées pour résoudre différents problèmes classiques : équation de la chaleur, problème de Stokes et calcul de la décomposition de Helmholtz-Hodge en non périodique. L'existence d'algorithmes rapides associés rend les méthodes compétitives. La dernière partie est consacrée à la définition de deux nouveaux schémas de résolution des équations de Navier-Stokes incompressibles par ondelettes, qui utilisent les ingrédients précédents. Des expériences numériques menées pour la simulation d'écoulement en cavité entraînée en dimension deux ou le problème de la reconnection de tubes de vortex en dimension trois montrent le fort potentiel des algorithmes développés
This work concerns wavelet numerical methods for the simulation of incompressible turbulent flow. The main objective of this work is to take into account physical boundary conditions in the resolution of Navier-Stokes equations on wavelet basis. Unlike previous work where the vorticity field was decomposed in term of classical wavelet bases, the point of view adopted here is to compute the velocity field of the flow in its divergence-free wavelet series. We are then in the context of velocity-pressure formulation of the incompressible Navier-Stokes equations, for which the boundary conditions are written explicitly on the velocity field, which differs from the velocity-vorticity formulation. The principle of the method implemented is to incorporate directly the boundary conditions on the wavelet basis. This work extends the work of the thesis of E. Deriaz realized in the periodic case. The first part of this work highlights the definition and the construction of new divergence-free and curl-free wavelet bases on [0,1]n, which can take into account boundary conditions, from original works of P. G. Lemarie-Rieusset, K. Urban, E. Deriaz and V. Perrier. In the second part, efficient numerical methods using these new wavelets are proposed to solve various classical problem: heat equation, Stokes problem and Helmholtz-Hodge decomposition in the non-periodic case. The existence of fast algorithms makes the associated methods more competitive. The last part is devoted to the definition of two new numerical schemes for the resolution of the incompressible Navier-Stokes equations into wavelets, using the above ingredients. Numerical experiments conducted for the simulation of driven cavity flow in two dimensions or the issue of reconnection of vortex tubes in three dimensions show the strong potential of the developed algorithms

This thesis presents the results of computer simulation studies of the structure, dynamics, and deformation of cross-linked polymer gels. Obtaining a fundamental understanding of the interrelation between the detailed structure and the properties of polymer gels is a challenge and a key issue towards designing materials for specific purposes. A new off-lattice method for constructing a closed network is presented that is free from defects, such as looping chains and dangling ends. Using these model networks in Brownian dynamics simulations, I show results for the structure and dynamics of bulk gels and describe a novel approach using spherical boundary conditions as an alternative to the periodic boundary conditions commonly used in simulations. This algorithm was also applied for simulating the diffusion of tracer particles within a static and dynamic network, to illustrate the quantitative difference and importance of including network mobility for large particles, as dynamic chains facilitate the escape of particles that become entrapped. I further investigate two technologically relevant properties of polymer gels: their stimuli-responsive behaviour and their mechanical properties. The collapse of core-shell nanogels was studied for a range of parameters, including the cross-linking degree and shell thickness. Two distinct regimes of gel collapse could be observed, with a rapid formation of small clusters followed by a coarsening stage. It is shown that in some cases, a collapsing shell may lead to an inversion of the core-shell particle which exposes the core polymer chains to the environment. This thesis also explores the deformation of bimodal gels consisting of both short and long chains, subject to uniaxial elongation, with the aim to understand the role of both network composition as well as structural heterogeneity on the mechanical response and the reinforcement mechanism of these materials. It is shown that a bimodal molecular weight distribution alone is sufficient to strongly alter the mechanical properties of networks compared to the corresponding unimodal networks with the same number-average chain length. Furthermore, it is shown that heterogeneities in the form of high-density short-chain clusters affect the mechanical properties relative to a homogeneous network, primarily by providing extensibility.

Thesis (M.Sc.Eng.)
To determine satellite trajectories in low earth orbit, engineers need to adequately estimate aerodynamic forces. But to this day, such a task su↵ers from inexact values of drag forces acting on complicated shapes that form modern spacecraft. While some of the complications arise from the uncertainty in the upper atmosphere, this work focuses on the problems in modeling the flow interaction with the satellite geometry. The only numerical approach that accurately captures e↵ects in this flow regime—like self-shadowing and multiple molecular reflections—is known as Test Particle Monte Carlo. This method executes a ray-tracing algorithm to follow particles that pass through a control volume containing the spacecraft and accumulates the momentum transfer to the body surfaces. Statistical fluctuations inherent in the approach demand particle numbers on the order of millions, often making this scheme too costly to be practical. This work presents a parallel Test Particle Monte Carlo method that takes advantage of both graphics processing units and multi-core central processing units. The speed at which this model can run with millions of particles enabled the exploration of regimes where a flaw was revealed in the model’s initial particle seeding. A new model introduces an analytical fix to this flaw—consisting of initial position distributions at the boundary of a spherical control volume and an integral for the correct number flux—which is used to seed the calculation. This thesis includes validation of the proposed model using analytical solutions for several simple geometries and demonstrates uses of the method for the aero-stabilization of the Phobos-Grunt Martian probe and pose-estimation for the ICESat mission.
2031-01-01

Thesis (S.B.)--Massachusetts Institute of Technology, Dept. of Physics, 2010.
Cataloged from PDF version of thesis.
Includes bibliographical references (p. 45-47).
We construct an effective action of General Relativity for small excitations from asymptotic transformations and use it to study conformal symmetry in the boundary of AdS3. By requiring finiteness of the boundary effective action(s) for certain asymptotic transformations, we derive the well known Virasoro algebra and central charge associated with the boundary of AdS3. Our Virasoro generating transformations are asymptotic symmetries of appropriately defined new asymptotically AdS3 spaces which are relaxed compared to the standard Brown-Henneaux ones but which yield the same asymptotic symmetry group and central charge. Thus one may view the effective action approach proposed in this thesis as a method for deriving boundary conditions for an asymptotic symmetry group. However, most importantly, we believe that the effective action approach is by itself an alternative independent way of obtaining and studying asymptotic conformal symmetry in the boundary of certain space-times based on well-grounded requirements of finite action.
by Achilleas P. Porfyriadis.
S.B.

The recent discovery of the Higgs boson by the ATLAS and CMS exper- iments and the subsequent measurements of it properties are the latest vindications of the Standard Model of particle physics. The SM has a number of well known flaws, and the continuing dearth of Beyond the Standard Model signatures from experiment has led to investigations into whether the SM is valid up to very high scales. The motivation for much of this work comes from the quartic Higgs coupling λ and its β function, which run to an extremely small values at high scales. These may be hints of new UV dynamics, in particular the Multiple Point Principle which posits the existence of a second degenerate minimum in the effective potential at the Planck scale, and Asymptotic Safety, where the dimensionless couplings of the potential run towards an interacting UV fixed point. In this work we will investi- gate the possibility for similar high scale boundary conditions in extensions of the Standard Model. Specifically, we look at the Real Singlet model, the Complex Sin- glet model, the Type-II Two Higgs Doublet Model, and the Inert Doublet Model. We will apply the relevant theoretical constraints to the parameter space of theses models, as well as experimental constraints such as those from ATLAS, CMS, LEP, the Tevatron, WMAP, Planck and LUX. Points that pass these constraints will also be investigated for their validity under a number of high scale boundary conditions on its scalar sector, and the valid parameter space will be checked for signatures in the mass spectrum that can be probed by current and future collider experiments.

Conformal field theories (CFT) constitute an interesting class of twodimensionalquantum field theories, with applications in string theoryas well as condensed matter physics. The symmetries of a CFT can beencoded in the mathematical structure of a conformal vertex algebra.The rational CFT’s are distinguished by the property that the categoryof representations of the vertex algebra is a modular tensor category.The solution of a rational CFT can be split off into two separate tasks, apurely complex analytic and a purely algebraic part.

The TFT-construction gives a solution to the second part of the problem.This construction gets its name from one of the crucial ingredients,a three-dimensional topological field theory (TFT). The correlators obtainedby the TFT-construction satisfy all consistency conditions of thetheory. Among them are the factorization constraints, whose implicationsfor boundary conditions are the main topic of this thesis.

The main result reviewed in this thesis is that the factorization constraintsgive rise to a semisimple commutative associative complex algebrawhose irreducible representations are the so-called reflection coefficients.The reflection coefficients capture essential information aboutboundary conditions, such as ground-state degeneracies and Ramond-Ramond charges of string compactifications. We also show that the annuluspartition function can be derived fromthis classifying algebra andits representation theory.

Techniques for numerically constructing initial data in the 3+1 formalism of general relativity (GR) are studied, using the theoretical framework described in Bowen and York (1980), Physical Review D 21(8), 2047-2056. The two main assumptions made are maximal slicing and 3-conformal flatness of the generated spaces. For ease of numerical solution, axisymmetry is also assumed, but all the results should extend without difficulty to the non-axisymmetric case. The numerical code described in this thesis may be used to construct vacuum spaces containing arbitrary numbers of black holes, each with freely specifyable (subject to the axisymmetry assumption) position, mass, linear momentum, and angular momentum. It should be emphasised that the time evolution of these spaces has not yet been attempted. There are two significant innovations in this work: the use of a new boundary condition for the surfaces of the black holes, and the use of multiple coordinate patches in the numerical solution. The new boundary condition studied herein requires the inner boundary of the numerical grid to be a marginally trapped surface. This is in contrast to the approach used in much previous work on this problem area, which requires the constructed spaces to be conformally isometric under a "reflection mapping" which interchanges the interior of a specified black hole with the remainder of the space. The new boundary condition is found to be easy to implement, even for multiple black holes. It may also prove useful in time evolution problems. The coordinate choice scheme introduced in this thesis uses multiple coordinate patches in the numerical solution, each with a coordinate system suited to the local physical symmetries of the region of space it covers. Because each patch need only cover part of the space, the metrics on the individual patches can be kept simple, while the overall patch system still covers a complicated topology. The patches are linked together by interpolation across the interpatch boundaries. Bilinear interpolation suffices to give accuracy comparable with that of common second order difference schemes used in numerical GR. This use of multiple coordinate patches is found to work very well in both one and two black hole models, and should generalise to a wide variety of other numerical GR problems. Patches are also found to be a useful (if somewhat over-general) way of introducing spatially varying grid sizes into the numerical code. However, problems may arise when trying to use multiple patches in time evolution problems, in that the interpatch boundaries must not become spurious generators or reflectors of gravitational radiation, due to the interpolation errors. These problems have not yet been studied. The code described in this thesis is tested against Schwarzschild models and against previously published work using the Bowen and York formalism, reproducing the latter within the limits of error of the codes involved. A number of new spaces containing one and two black holes with linear or angular momentum are also constructed to demonstrate the code, although little analysis of these spaces has yet been done.
Science, Faculty of
Physics and Astronomy, Department of
Graduate

This thesis presents results of lattice QCD computations of the K → π semi-leptonic (Kl3) and pion electromagnetic form factors using partially twisted boundary conditions. These form factors parameterize low-energynon-perturbative strong interaction effects and cannot therefore be calculated in perturbative QCD. The pion electromagnetic form factor provides information on its charge distribution. The Kl3 form factor at zero momentum transfer (q2 = 0) can be used in the determination of the |Vus| element of the CKM matrix. An accurate determination of these form factors is therefore important. Using partially twisted boundary conditions we calculate the Kl3 form factor directly at q2 = 0, removing the need for the q2 interpolation required in previous lattice QCD simulations, thus eliminating one source of systematic error in this calculation. We also use partially twisted boundary conditions to calculate the pion form factor at values of q2 close to q2 = 0 allowing for a direct evaluation of the charge radius of the pion. The simulations are performed on an ensemble of the RBC/UKQCD collaboration’s gauge configurations with Domain Wall Fermions and the Iwasaki gauge action with an inverse lattice spacing of 1.73(3) GeV at light quark masses corresponding to a pion mass of 330 MeV. We calculate the form factors at these simulated quark masses and then use chiral perturbation theory to extrapolate our results to physical light quark masses. We find for the charge radius of the physical pion fir2π fi = 0.418(31) fm2, in agreement with the experimentally determined result. For the value of the Kl3 form factor, fKπ + (q2), at q2 = 0 and physical quark masses we find fKπ + (0) = 0.960(+5 −6). This result is then used to determine a value for |Vus|. Together with a recent determination of |Vud| we find that the current results are consistent with unitarity of the CKM matrix

In aircraft with jet propulsion engine intakes at supersonic speed, strong pressure waves referred to as shockwaves occur, which may interact with any present boundary layers along the intake surface. The adverse pressure gradients associated with Shock Wave-Boundary Layer Interaction (SWBLI) may cause boundary layer flow separation, which can result in disturbances of the flow that can be harmful to the device or decrease engine performance. A common way in dealing with the adverse effects of SWBLI is through removal of low-momentum flow in the boundary layer, a process referred to as boundary layer bleed. In the process of bleed, the boundary layer is subjected to a pressure difference promoting flow out of the system, through a porous surface, and into a plenum. The porous surfaces used in the mass flow removal process contain orifices in small scales. Thus, in Computational Fluid Dynamics (CFD), creating a mesh resolving both the orifice scales and the bulk flow is a cumbersome task, and the computational cost becomes substantially increased. To this end, several boundary conditions which effectively model the large-scale effects of bleed have been developed. The aim of this study is to implement the Boundary Condition (BC) developed by John W. Slater into M-EDGE, the in-house compressible CFD-solver of SAAB Aeronautics. The bleed boundary condition model is based on a dimensionless surface sonic flow coefficient, which is derived from empirical wind-tunnel measurements of the bleed mass flow. In previous work, the Slater bleed BC has been shown to correlate well with wind-tunnel data. Furthermore, a simple transpiration law formulated by Reynald Bur was implemented in order get familiarized with the M-EDGE Fortran source code. However, this model is expected to yield unsatisfactory results, as reported in previous work in the field. The implemented Slater BC is tested on two different two-dimensional flow cases; flow over a flat plate without SWBLI, and flow including a shock wave generator creating SWBLI. In the flat plate case, simulations were run at Mach numbers 1.27, 1.58, 1.98 and 2.46 over a 6.85cm plate of 19% porosity. In the SWBLI-case, only flow at Mach 2.46 was considered, with a 9.53cm plate of 21% porosity. The Reynolds number range used throughout was 1.39−1.76·10^7/m. Simulations were run at different bleed rates over a structured grid using steady state RANS with the Spalart-Allmaras one-equation turbulence model. The boundary condition performance was assessed by its ability to recreate the sonic flow coefficients on which it is based. Further, the shape of downstream pitot pressure profiles are compared with experimental data. Results from the studies indicate that the implementation manages to recreate the data for the sonic flow coefficient with small error margins. The implementation can be used to simulate porous plates of different dimensions and porosities, even though the bleed model is based on empirical mass flow measurements of a 6.85cmplate of 19% porosity. The implementation is able to predict global bleed effects in the flow field, as indicated by comparisons of pitot pressure profiles at various downstream reference planes, despite differences in reference boundary layer intake profiles. Further, the overall flow field was compared visually with other simulation-studies, indicating that the global Mach distributions of the geometries were in accordance with the reference data. However, pitot profiles should be further studied with better matched intake boundary layer profiles. The main limitation of the boundary condition is that it relies on the wind-tunnel data of the surface sonic flow coefficients for specific bleed plate configurations. Furthermore, the implementation has only been verified to work within specific Mach number range of the underlying empirical measurements. In future work, the generality of the model could be increased by extending the data to other configurations and Mach numbers by conducting new experiments or using other published empirical data.

The finite difference time domain (FDTD) method has become a main stream analysis tool for engineers solving complex electromagnetic wave interaction problems. Its first principles approach affords it a wide range of applications from radar cross section (RCS) predictions of electrically large structures to molecular scale analysis of complex materials. This wide area of application may be attributed to the coupling of auxiliary differential equations with Maxwell's equations to describe the physical properties of a given problem. Previous extensions have included sub-cell models for describing lumped circuit elements within a single Yee cell, transformation of near-field information to the far-field for the analysis of antenna problems, dispersive material models and mesh truncation techniques. A review of these extensions is presented. What has not been previously developed is the ability to truncate lossy dielectric materials at the boundary of the simulation domain. Such outer boundary conditions (OBCs) are required in simulations dealing with ground penetrating radar, integrated circuits and many microwave devices such as stripline and microstrip structures. We have developed such an OBC by surrounding the exterior of the simulation domain with a lossy dispersive material based on a two time-derivative Lorentz model (L2TDLM). We present the development of the material as an absorber and ultimately as a full 3D OBC. Examples of microstrip, structures are presented to re-enforce the importance of modeling losses in dielectric structures. Finally, validation of the FDTD simulator and demonstration of the L2TDLM OBC's effectiveness is achieved by comparison with measured results from these microwave devices.

Mathematical models describing the Q-switched laser generation, which is a widely used laser technique for producing short intense pulses of light, belong to the class of semi-nonlinear models where only source terms nonlinearly depend on the solution. Numerical methods for solution of systems of semi-nonlinear partial differential equations have been extensively studied in many papers. Schrödinger-type equations, parabolic-type equations or general diffusion-reaction models arise in nonlinear optics. Such differential problems are solved mainly by finite-difference and Galerkin methods. The convergence analysis is based on the stability analysis of the linearized problems. The construction and theoretical analysis of discrete schemes for one-dimensional problem give a basis for a numerical solution of more general two-dimensional and three-dimensional problems where a diffraction process is taken into account. The two-dimensional problem simulates the dynamics of high-power semiconductor lasers. To solve the problems simulating propagation of photon fluxes in the nonlinear disperse medium, the finite-difference time-domain method is used. However, the major drawback of this method is that the computational domain must be sufficiently large. In order to restrict the computational domain and to solve the problem only in the region of interest, special artificial boundary conditions are investigated. The three-dimensional problem simulates an interaction of counter propagating... [to full text]
Disertacijoje nagrinėjami kai kurių lazerių fizikos ir netiesinės optikos uždavinių skaitinės analizės metodai. Tiriami trys pagrindiniai atvejai: bėgančias plokščias bangas aprašantis vienmatis, bėgančias difraguojančias bangas nagrinėjantis dvimatis ir lazerio pluoštų sąveiką netiesinėje Kero terpėje modeliuojantis trimatis modeliai. Šiuos uždavinius sieja pernešimo diferencialinės lygtys dalinėmis išvestinėmis, aprašančios į priešingas puses sklindančias lazerio bangas. Dvimačiame ir trimačiame uždaviniuose sprendžiamos dalinių išvestinių Šrėdingerio (ang. Schrödinger) tipo diferencialinės lygtys. Šiems matematiniams modeliams sudarytos baigtinių skirtumų schemos, atlikta jų analizė ir pagrindimas. Skaitinių eksperimentų realizacijai sukurti lygiagretieji algoritmai, jie yra būtini atliekant didelių resursų reikalaujančius skaičiavimus. Disertaciją sudaro įvadas, keturi skyriai, rezultatų apibendrinimas, naudotos literatūros ir autoriaus publikacijų disertacijos tema sarašai. Įvadiniame skyriuje aptariama tiriamoji problema, darbo aktualumas, aprašomas tyrimų objektas, formuluojamas darbo tikslas bei uždaviniai, aprašoma tyrimų metodika, darbo mokslinis naujumas, darbo rezultatų praktinė reikšmė, ginamieji teiginiai. Įvado pabaigoje pristatomos disertacijos tema autoriaus paskelbtos publikacijos ir pranešimai konferencijose bei disertacijos struktūra. Pirmasis skyrius skirtas mokslinės literatūros apžvalgai ir supažindinimui su netiesinės optikos sąvokomis bei... [toliau žr. visą tekstą]

Cette thèse traite de la diffusion par des surfaces rugueuses monodimensionnelles. Les surfaces présentant des petites échelles de variations nécessitent une discrétisation fine pour représenter les effets de diffusion sur le champ diffracté, ce qui augmente les coûts numériques. Deux aspects sont considérés : la réduction de la taille du problème en construisant une condition aux limiteséquivalente traduisant les effets des variations rapides et la réduction du nombre d’itérations nécessaires pour résoudre le système linéaire issu de la méthode des moments par une méthode basée sur les sous-espaces de Krylov. En ce qui concerne la réduction de la taille du problème, une technique d’homogénéisation est utilisée pour transformer la condition aux limites posée sur lasurface rugueuse par des paramètres effectifs. Ces paramètres sont déterminés par des problèmes auxiliaires qui tiennent compte des échelles fines de la surface. Dans le cas de surfaces parfaitement métalliques, la procédure est appliquée en polarisation Transverse Magnétique (TM) et Transverse Électrique (TE). Une impédance équivalente de Léontovich d’ordre 1 est déduite.Le procédure est automatique et les ordres supérieurs sont dérivés pour la polarisation TM. La procédure d’homogénéisation est aussi appliquée pour des interfaces rugueuses séparant deux milieux diélectriques. En ce qui concerne la réduction du nombre d’itérations, un préconditionneur, basé sur des considérations physiques, est construit à partir des modes de Floquet. Bien que le préconditionneur soit initialement élaboré pour des surfaces périodiques, nous montrons qu’il est aussi efficace pour des surfaces tronquées éclairées par une onde plane. L’efficacité des deux aspects présentés dans cette thèse est numériquement illustrée pour des configurations d’intérêt
This work is about the scattering by monodimensional rough surfaces. Surfaces presenting small scales of variations need a very refined mesh to finally capture the scattering field behaviour what increases the computational cost. Two aspects are considered : the reduction of the problemsize through an effective boundary condition incorporating the effect of rapid variations and the reduction of the number of iterations to solve the linear system arising from method of moments by a method based on Krylov subspace. Firstly, an homogenization process is used to convert the boundary condition on the rough interface into effective parameters. These parameters are determined by the solutions of auxiliary problems which involve the detailed profile of the interface. In the case of perfectly metallic surfaces, the process is applied to the E- and H-polarization and an Leontovich impedance of order 1 is deduced. The process is automatic and higher orders are derived for E-polarization. The homogenization process is also applied to dielectric rough interfaces. Secondly, a physically-based preconditioner is built with Floquet’s modes. Although the preconditioner has been designed for periodical surfaces, it was shown to be efficient in the case of truncated surfaces illuminated by a plane wave. The efficiency of both aspects is numerically illustrated for some configurations of interest

La dynamique des flammes de prémélange est étudiée par deux approches numériques différentes. La première résout les équations compressibles de Navier-Stokes avec une chimie simplifiée (DNS). Afin de réduire les coûts de calcul, nous analysons et développons un schéma numérique à grille décalée. Le traitement des ondes acoustiques aux sorties est connu pour rendre les flammes cylindriques légèrement carrées. Ces déformations non-physiques sont expliquées en mettant en évidence la modélisation insuffisamment précise de l'accélération du fluide lorsque l'écoulement est oblique à la sortie. Une étude paramétrique et statistique de flammes turbulentes est menée en 2D et une simulation parallèle 3D est réalisée dans un domaine de (3cm)3. En considérant la flamme infiniment mince, l'approche EEM diminue considérablement les coûts de calcul. Les mêmes simulations sont réalisées et comparées aux résultats de DNS pour tester la capacité du modèle EEM à fournir des résultats quantitatifs.

Diploma thesis is focused on the description of the parameters of nonlinear material models of concrete, which are implemented in a computational system LS-DYNA, interacting with performance of nonlinear test calculations in system LS-DYNA on selected problems, which are formed mainly by simulations of tests of mechanical and physical properties of concrete in uniaxial compressive and tensile on cylinders with applying different boundary conditions and by simulation of bending slab, with subsequent comparison of some results of test calculations with results of the experiment. The thesis includes creation of appropriate geometric models of selected problems, meshing of these geometric models, description of parameters and application of nonlinear material models of concrete on selected problems, application of loads and boundary conditions on selected problems and performance of nonlinear calculations in a computational system LS-DYNA. Evaluation of results is made on the basis of stress-strain diagrams and load-displacement diagrams based on nonlinear calculations taking into account strain rate effects and on the basis of hysteresis curves based on nonlinear calculations in case of application of cyclic loading on selected problems. Verification of nonlinear material models of concrete is made on the basis of comparison of some results of test calculations with results obtained from the experiment.

On s'intéresse dans ce travail au problème de propagation acoustique dans des guides à parois traitées avec des matériaux absorbants à réaction localisée ou non localisée en présence d'écoulement. En effet, dans les systèmes industriels comme les turboréacteurs d'avions, les silencieux d'échappement et les systèmes de ventilation, le bruit est le plus souvent canalisé vers l'extérieur par des guides de géométries plus ou moins complexes. Une étude des guides d'ondes permet donc de prédire et de comprendre les phénomènes physiques tels que la réfraction, la convection, l'absorption et l'atténuation des ondes. Dans l'étude des guides d'ondes, on considère souvent qu'ils sont infiniment longs afin de s'affranchir de certains phénomènes (réflexion par exemple) à leurs extrémités. Résoudre le problème de propagation dans les guides infinis par la méthode des éléments finis nécessite de tronquer le domaine infini par des frontières artificielles sur lesquelles des conditions limites transparentes doivent être écrites. Dans ce travail, les conditions limites transparentes sont écrites sous forme d'un opérateur Dirichlet-to-Neumann (DtN) basé sur une décomposition de la pression acoustique sur la base des modes propres du guide étudié tout en prenant en compte l'influence des paramètres comme l'écoulement et le traitement acoustique avec des matériaux absorbants. La propagation acoustique dans le guide est régie par un modèle scalaire basé sur l'équation de Helmholtz et les matériaux absorbants utilisés sont des matériaux absorbants d'impédance locale Z et des matériaux poreux. Nous nous sommes intéressés en particulier aux matériaux poreux ? squelette rigide que l'on modélise par un fluide équivalent car la propagation acoustique dans ces matériaux est aussi gouvernée par l'équation de Helmholtz comme dans un milieu fluide. Des résultats d'étude de la propagation acoustique dans des guides rectilignes uniformes traités en présence d'un écoulement uniforme ont permis de valider la méthode développée pour tronquer les domaines infinis. L'étude a aussi été menée avec succés pour des guides non uniformes traités en présence d'un écoulement potentiel.

Nous proposons une étude numérique bidimensionnelle des écoulements laminaires de convection mixte en conduite rectangulaire horizontale chauffée par le bas (encore appelés "écoulements de Poiseuille-Rayleigh-Bénard"). La méthode de résolution est de type volumes finis; elle est basée sur la méthode du Lagrangien augmenté. Nous présentons une revue bibliographique, historique et complète, des travaux effectués sur ce sujet. Nous rapportons de manière détaillée les résultats de trois thèses, analysant la stabilité linéaire de l'écoulement de Poiseuille et la stabilité faiblement non-linéaire des rouleaux thermoconvectifs transverses à l'axe de la conduite, et fournissant des mesures expérimentales très fines des champs de vitesse dans les domaines linéaires et non-linéaires. L'objectif de ce travail est d'étudier dans quelle mesure les simulations numériques s'accordent avec théories et expériences en effectuant des comparaisons quantitatives systématiques avec ces trois thèses. Nous étudions l'influence des conditions aux limites ouvertes et périodiques sur la stabilité et sur le développement spatial et temporel des rouleaux transversaux, en prenant en compte le caractère convectif ou absolu de l'instabilité. Cinq sortes de conditions aux limites ouvertes sont testées. On montre que l'amplitude de la perturbation en amont de la frontière de sortie peut considérablement varier d'une condition à l'autre, alors que la longueur de la zone perturbée varie très peu, et que la structure de l'écoulement en dehors de la zone perturbée se conserve. Avec les conditions aux limites périodiques, les rouleaux transversaux transitent vers l'écoulement de Poiseuille au passage du seuil entre instabilité convective et stabilité linéaire. Avec les conditions aux limites ouvertes, cette transition a lieu près du seuil entre instabilité absolue et instabilité convective. Tous les résultats de la stabilité faiblement non-linéaire sont reproduits par la simulation 2D, et certains résultats expérimentaux 3D sont retrouvés avec une grande précision. On montre que les points de désaccord qui sont mis en évidence résultent d'effets purement tridimensionnels et qu'ils ne pourront donc être traités que par la simulation numérique 3D.

Le bruit est devenu une nuisance importante en zone urbaine au point que selon l'Organisation Mondiale de la Santé, 40% de la population européenne est exposée à des niveaux de bruit excessifs, principalement dû aux transports terrestres. Il devient donc nécessaire de trouver de nouveaux moyens de lutter contre le bruit en zone urbaine. Dans ce travail, on étudie une solution possible à ce problème: un écran bas antibruit. Il s'agit d'un écran de hauteur inférieure à un mètre placé près d'une source, conçu pour réduire le niveau de bruit pour les piétons et les cyclistes à proximité. Ce type de protection est étudié numériquement et expérimentalement. Nous nous intéressons particulièrement aux écrans adaptés au bruit du tramway puisque dans ce cas les sources sont proches du sol et peuvent être atténuées efficacement. La forme ainsi que le traitement de surface de l'écran sont optimisés par une méthode de gradient couplée à une méthode 2D d'éléments finis de frontière. Les variables à optimiser sont les coordonnées de nœuds de contrôle et les paramètres servant à décrire l'impédance de surface. Les sensibilités sont calculées efficacement par la méthode de l'état adjoint. Les formes générées par l'algorithme d'optimisation sont assez irrégulières mais induisent une nette amélioration par rapport à des formes simples, d'au moins 5 dB (A). Il est également montré que l'utilisation de traitement absorbant du côté source de l'écran peut améliorer la performance sensiblement. Ce dernier point est confirmé par des mesures effectuées sur modèle réduit. De plus, un prototype à l'échelle 1 d'écran bas antibruit a été construit et testé en conditions réelles, le long d'une voie de tramway à Grenoble. Les mesures montrent que la protection réduit le niveau de 10 dB (A) pour un récepteur proche situé à hauteur d'oreilles. Ces résultats semblent confirmer l'applicabilité de ces protections pour réduire efficacement le bruit en zone urbaine
Noise has become a main nuisance in urban areas to the point that according to the World Health Organization 40% of the European population is exposed to excessive noise levels, mainly due to ground transportation. There is therefore a need to find new ways to mitigate noise in urban areas. In this work, a possible device to achieve this goal is studied: a low-height noise barrier. It consists of a barrier typically less than one meter high placed close to a source, designed to decrease the noise level for nearby pedestrians and cyclists. This type of device is studied both numerically and experimentally. Tramway noise barriers are especially studied since the noise sources are in this case very close to the ground and can therefore be attenuated efficiently. The shape and the surface treatment of the barrier are optimized using a gradient-based method coupled to a 2D boundary element method (BEM). The optimization variables are the node coordinates of a control mesh and the parameters describing the surface impedance. Sensitivities are calculated efficiently using the adjoint state approach. Numerical results show that the shapes generated by the optimization algorithm tend to be quite irregular but provide a significant improvement of more than 5 dB (A) compared to simpler shapes. Utilizing an absorbing treatment on the source side of the barrier is shown to be efficient as well. This second point has been confirmed by scale model measurements. In addition, a full scale low height noise barrier prototype has been built and tested in situ close to a tramway track in Grenoble. Measurements show that the device provides more than 10 dB (A) of attenuation for a close receiver located at the typical height of human ears. These results therefore seem to confirm the applicability of such protections to efficiently decrease noise exposure in urban areas

在這篇論文中，我們考察亞音速流體流進流出一給定的有限長管道的問題，目的在於找到管道入口和出口處內蘊的（物理上可接受的）邊界條件。我們首先刻劃一組物理邊界條件，借此可以確定長方形管道中亞音速無旋流的存在性和唯一性。在給定管道入口處的流量及水平來流方向和出口處的適當壓力條件下，存在兩個正的常數m₀和m₁(m₀ < m₁)，使得當流量m ∈ [m₀,m₁)，在長方形管道中存在唯一的亞音速無旋流。流體的水平速度是恒正的，並且當流量m趨於m₁時，流體的最大速度也將趨於音速。問題的困難來自于由流量蘊含的非局部項和出口處的壓力條件。我們首先引入伯努利常數作參數的附屬問題来處理非局部项，然後建立流量和伯努利常數之間的單調關係。為處理壓力條件引起的斜導數缺失問題，我們利用角速度和壓力來重新描述這一問題。我們利用Moser迭代來得到角速度的最大模估計，以確保流體的水平速度恒為正。
我們接下來考察彎曲管道中一般來流方向和管道壁的幾何結構對亞音速無旋流的影響。我們發現來流方向和管道壁的傾斜角度和出口壓力起著相似的作用。管道壁的曲率也起著很重要的作用。我們的結果也可以推廣到二維亞音速歐拉流和三維軸對稱亞音速歐拉流的情形。
接下來我們考慮三維有限長管道中的亞音速歐拉流的情形，這是最有趣和最困難的情形，也是論文的核心部份。我們在二維管道中給定的邊界條件在三維有一個自然的推廣。這些重要的提示對我們尋求歐拉方程組新的分解及借此理解其中的雙曲與橢圓耦合的結構是至關重要的。我們的新的分解的關鍵想法在於利用伯努利定律來約化速度場。具體的做法是通過定義新的變量 [附圖] 及通過 [附圖] ，利用伯努利函數B 來代替u₁。這樣我們可以更深入地挖掘伯努利定律的作用，期望借此可以稍微簡化一下複雜的歐拉方程組。對伯努利函數為常數的流體，我們找到了一個新的守恆量，這跟二維的約化的旋度的情形相似。讓人驚奇的是，我們還可以找到一組新的守恆律，這一情況即使在二維也從未被人注意到。我們利用這一分解來證明長方體管道中靠近某些特殊亞音速流并滿足給定的入口處的來流方向及伯努利函數和出口處的壓力條件的亞音速歐拉流的存在性和唯一性。同樣的想法可以用於不可壓歐拉方程組、自相似歐拉方程組、帶阻尼項的歐拉方程組、定常歐拉泊松方程組和定常的歐拉麥克斯韋方程組。
最後我們考慮歐拉泊松方程組某些定常亞音速解的結構穩定性。如果帶亞音速背景電荷的背景解的馬赫數和電場都比較小的話，那麼背景解關於背景電荷、來流方向、伯努利函數、出口壓力的小撓動是結構穩定的。在我們的數學分析中新的元素在於求解帶斜導數邊界條件和Dirichlet邊界條件的混合型的二階強耦合的橢圓型方程組。
In this thesis, we investigate an inflow-outflow problem for subsonic gas flows in a nozzle with finite length, aiming at finding intrinsic (physically acceptable) boundary conditions on upstream and downstream. We first characterize a set of physical boundary conditions that ensure the existence and uniqueness of a subsonic irrotational flow in a rectangle. Our results show that suppose we prescribe the horizontal incoming flow angle at the inlet and an appropriate pressure at the exit, there exists two positive constants m₀ and m₁ with m₀ < m₁, such that a global subsonic irrotational flow exists uniquely in the nozzle, provided that the incoming mass flux m ∈ [m₀,m₁). The maximum speed will approach the sonic speed as the mass flux m tends to m₁. The new difficulties arise from the nonlocal term involved in the mass flux and the pressure condition at the exit. We first introduce an auxiliary problem with the Bernoulli’s constant as a parameter to localize the nonlocal term and then establish a monotonic relation between the mass flux and the Bernoulli’s constant to recover the original problem. To deal with the loss of obliqueness induced by the pressure condition at the exit, we employ the formulation in terms of the angular velocity and the density. A Moser iteration is applied to obtain the L∞ estimate of the angular velocity, which guarantees that the flow possesses a positive horizontal velocity in the whole nozzle.
As a continuation, we investigate the influence of the incoming flow angle and the geometry structure of the nozzle walls on subsonic flows in a finitely long curved nozzle. It turns out to be interesting that the incoming flow angle and the angles of inclination of nozzle walls play the same role as the end pressure. The curvatures of the nozzle walls play an important role. We also extend our results to subsonic Euler flows in the 2-D and 3-D asymmetric cases.
Then it comes to the most interesting and difficult casethe 3-D subsonic Euler flow in a bounded nozzle, which is also the essential part of this thesis. The boundary conditions we have imposed in the 2-D case have a natural extension in the 3-D case. These important clues help us a lot to develop a new formulation to get some insights on the coupling structure between hyperbolic and elliptic modes in the Euler equations. The key idea in our new formulation is to use the Bernoulli’s law to reduce the dimension of the velocity field by defining new variables [with formula] and replacing u₁ by the Bernoulli’s function B through [with formula] In this way, we can explore the role of the Bernoulli’s law in greater depth and hope that may simplify the Euler equations a little bit. We find a new conserved quantity for flows with a constant Bernoulli’s function, which behaves like the scaled vorticity in the 2-D case. More surprisingly, a system of new conservation laws can be derived, which is never been observed before, even in the two dimensional case. We employ this formulation to construct a smooth subsonic Euler flow in a rectangular cylinder by assigning the incoming flow angles and the Bernoulli’s function at the inlet and the end pressure at the exit, which is also required to be adjacent to some special subsonic states. The same idea can be applied to obtain similar information for the incompressible Euler equations, the self-similar Euler equations, the steady Euler equations with damping, the steady Euler-Poisson equations and the steady Euler-Maxwell equations.
Last, we are concerned with the structural stability of some steady subsonic solutions for the Euler-Poisson system. A steady subsonic solution with subsonic background charge is proven to be structurally stable with respect to small perturbations of the background charge, the incoming flow angles and the end pressure, provided the background solution has a low Mach number and a small electric field. The new ingredient in our mathematical analysis is the solvability of a new second order elliptic system supplemented with oblique derivative conditions at the inlet and Dirichlet boundary conditions at the exit of the nozzle.
Detailed summary in vernacular field only.
Detailed summary in vernacular field only.
Detailed summary in vernacular field only.
Detailed summary in vernacular field only.
Weng, Shangkun.
Thesis (Ph.D.)--Chinese University of Hong Kong, 2012.
Includes bibliographical references (leaves 176-187).
Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web.
Abstract also in Chinese.
Chapter 1 --- Introduction --- p.11
Chapter 2 --- Subsonic irrotational flows in a rectangular nozzle --- p.26
Chapter 2.1 --- Introduction --- p.27
Chapter 2.2 --- Reduction of the problem and main results --- p.33
Chapter 2.2.1 --- An auxiliary problem --- p.34
Chapter 2.2.2 --- Restrictions on the end pressure --- p.34
Chapter 2.2.3 --- Main results --- p.36
Chapter 2.3 --- Unique solvability of the auxiliary problem --- p.38
Chapter 2.3.1 --- Reformulation of the auxiliary problem --- p.38
Chapter 2.3.2 --- Proof of Theorem 2.2.2 --- p.40
Chapter 2.4 --- The relationship between the mass flux m and the Bernoulli’s constant B --- p.56
Chapter 3 --- Subsonic irrotational flows in a 2-D finitely long curved nozzle --- p.61
Chapter 3.1 --- Introduction --- p.62
Chapter 3.2 --- Reduction of the problem and main results --- p.65
Chapter 3.2.1 --- An auxiliary problem --- p.65
Chapter 3.2.2 --- Main results --- p.66
Chapter 3.3 --- Unique solvability of the auxiliary problem --- p.68
Chapter 3.3.1 --- Reformulation of the auxiliary problem --- p.68
Chapter 3.3.2 --- Proof of Theorem 3.2.1 --- p.69
Chapter 4 --- Subsonic Euler flows in a divergent nozzle --- p.85
Chapter 4.1 --- Introduction --- p.86
Chapter 4.2 --- Subsonic Euler flows in a 2-D divergent nozzle --- p.87
Chapter 4.2.1 --- Formulation of the problem and main results . --- p.87
Chapter 4.2.2 --- Proof of Theorem 4.2.2 --- p.92
Chapter 4.3 --- Subsonic Euler flows in a three-dimensional divergent conic nozzle with an asymmetric end pressure --- p.97
Chapter 4.3.1 --- Formulation of the problem and main results . --- p.97
Chapter 4.3.2 --- Proof of Theorem 4.3.2 --- p.102
Chapter 5 --- A new formulation for the 3-D compressible Euler equations --- p.108
Chapter 5.1 --- Introduction --- p.109
Chapter 5.2 --- A new formulation for the 3-D compressible Euler equations --- p.114
Chapter 5.3 --- The 3-D compressible Euler equations with a constant Bernoulli’s function --- p.117
Chapter 5.3.1 --- A new conserved quantity --- p.117
Chapter 5.3.2 --- A system of new conservation laws --- p.122
Chapter 5.4 --- Subsonic Euler flows in a rectangular cylinder --- p.128
Chapter 5.4.1 --- Extension to the domain Ωe = [0, 1] × T² --- p.130
Chapter 5.4.2 --- Main results --- p.130
Chapter 5.4.3 --- Proof of Theorem 5.4.1 --- p.132
Chapter 5.5 --- A new formulation for the 3-D incompressible Euler equations --- p.139
Chapter 5.5.1 --- A new formulation for the 3-D incompressible Euler equations --- p.139
Chapter 5.5.2 --- The 3-D incompressible Euler equations with a constant Bernoulli’s function --- p.142
Chapter 5.6 --- Appendix: The verification of (5.3.4)-(5.3.6) --- p.145
Chapter 6 --- On steady subsonic flows for the Euler-Poisson models --- p.148
Chapter 6.1 --- Introduction --- p.149
Chapter 6.2 --- Preliminary --- p.152
Chapter 6.2.1 --- A new formulation for the Euler-Poisson equations --- p.152
Chapter 6.2.2 --- Background solutions --- p.153
Chapter 6.3 --- Structural stability of background solutions --- p.156
Chapter 6.3.1 --- Extension to the domain Ωe = [0, 1] × T² --- p.158
Chapter 6.3.2 --- Main results --- p.159
Chapter 6.3.3 --- Proof of Theorem 6.3.1 --- p.162
Chapter 7 --- Discussions and Future works --- p.170
Chapter 7.1 --- Subsonic flows in a finitely long nozzle --- p.170
Chapter 7.2 --- The transonic shock problem in a 3-D divergent nozzle --- p.172
Chapter 7.3 --- Dynamical stability of a transonic shock in nozzles --- p.174
Bibliography --- p.175

We report methodological and computational details of our Kohn-Sham density functional method with Gaussian orbitals for systems with periodic boundary conditions (PBC). When solving iterative self-consistent field (SCF) equations of density functional theory (DFT), the most computationally demanding tasks are Kohn-Sham (or Fock) matrix formation and the density matrix update step. The former requires evaluation of the Coulomb interactions and the exchange-correlation quadrature, and in our code both of them are computed via O (N) techniques. An O (N) approach for the Coulomb problem in electronic structure calculations with PBC is developed here and is based on the direct space fast multipole method (FMM). The FMM achieves not only linear scaling of computational time with system size but also high accuracy, which is pivotal for avoiding numerical instabilities that have previously plagued calculations with large bases, especially those containing diffuse functions. The density matrix update step is carried out via the conventional O (N3) diagonalization of the Fock matrix, which for systems with less than ≈3000 basis functions is cheaper than the recently developed O (N) algorithms. In addition to evaluating energy, our code also computes analytic energy gradients with respect to atomic positions and cell dimensions (forces). Combining the latter with the developed in this work redundant internal coordinate algorithm for optimization of periodic systems, it becomes possible to optimize geometries of periodic structures with great efficiency and accuracy. We demonstrate the capabilities of our method with benchmark calculations on polyacetylene, poly(p-phenylenevinylene) (PPV), and a series of carbon and boron-nitride single wall nanotubes employing basis sets of double zeta plus polarization quality, in conjunction with generalized gradient approximation and kinetic energy density dependent functionals. We also present vibrational frequencies for PPV obtained from finite differences of forces. The largest calculation reported in this work contains 244 atoms and 1344 contracted Gaussians in the unit cell.

Non-equilibrium boundary conditions based upon kinetic theory and linear irreversible thermodynamics are applied to the interface kinetics in vapor crystal growth of unitary and binary materials. These are compared to equilibrium boundary conditions in a simple, 1D closed ampoule physical vapor transport model. It is found that in cases where the diffusive impedance is negligible and when system pressure is low, surface kinetics play an important role in limiting the mass transport. In cases where diffusion is the dominant transport impedance, and/or when the pressure in the system is high, the kinetic impedances at the interfaces are negligible, as impedances due to diffusion and latent heat transport at the interfaces become more significant. The non-equilibrium boundary conditions are dependent upon the sticking coefficient of the surface. An experiment to estimate the sticking coefficient on solid surfaces is proposed. The non-equilibrium theory also predicts significant temperature jumps at the interfaces.
Graduate

We analyze a number of systems of evolution equations that arise in the study of physical chemistry. First we discuss the well-posedness of a system of mixing compressible barotropic multicomponent flows. We discuss the regularity of these variational solutions, their existence and uniqueness, and we analyze the emergence of a novel type of entropy that is derived for the system of equations. Next we present a numerical scheme, in the form of a discontinuous Galerkin (DG) finite element method, to model this compressible barotropic multifluid. We find that the DG method provides stable and accurate solutions to our system, and that further, these solutions are energy consistent; which is to say that they satisfy the classical entropy of the system in addition to an additional integral inequality. We discuss the initial-boundary problem and the existence of weak entropy at the boundaries. Next we extend these results to include more complicated transport properties (i.e. mass diffusion), where exotic acoustic and chemical inlets are explicitly shown. We continue by developing a mixed method discontinuous Galerkin finite element method to model quantum hydrodynamic fluids, which emerge in the study of chemical and molecular dynamics. These solutions are solved in the conservation form, or Eulerian frame, and show a notable scale invariance which makes them particularly attractive for high dimensional calculations. Finally we implement a wide class of chemical reactors using an adapted discontinuous Galerkin finite element scheme, where reaction terms are analytically integrated locally in time. We show that these solutions, both in stationary and in flow reactors, show remarkable stability, accuracy and consistency.
text

Sung, Siu Chung = 鎂矽酸鹽(MgSiO₃)在核幔邊界條件下的第一性原理研究 / 宋紹聰.
Thesis (M.Phil.)--Chinese University of Hong Kong, 2008.
Includes bibliographical references (p. 110-115).
Abstracts in English and Chinese.
Sung, Siu Chung = Mei xi suan yan (MgSiO₃) zai he man bian jie tiao jian xia de di yi xing yuan li yan jiu / Song Shaocong.
Chapter 1 --- Introduction --- p.1
Chapter 2 --- Review on MgSiO3 --- p.5
Chapter 2.1 --- Interior of the Earth --- p.5
Chapter 2.1.1 --- The importance of MgSiO3 in geosciences --- p.6
Chapter 2.1.2 --- "Anomalies in lower mantle, D"" layer and the core-mantle boundary" --- p.7
Chapter 2.2 --- Review on experimental and theoretical studies on MgSiO3 --- p.9
Chapter 2.2.1 --- The perovskite structure --- p.9
Chapter 2.2.2 --- MgSiO3 pv --- p.12
Chapter 2.3 --- ppv structure --- p.14
Chapter 2.3.1 --- MgSiO3 ppv --- p.15
Chapter 2.3.2 --- MgSiO3 liquid --- p.18
Chapter 3 --- Physical quantities in geoscience and molecular dynamics sim- ulations --- p.20
Chapter 3.1 --- Equation of state --- p.21
Chapter 3.2 --- Gruneisen parameter --- p.22
Chapter 3.3 --- Thermoelasticity --- p.22
Chapter 3.4 --- Phase transition --- p.24
Chapter 3.5 --- Correlation function --- p.25
Chapter 3.5.1 --- Pair Distribution function --- p.25
Chapter 3.5.2 --- Coordination number --- p.27
Chapter 3.5.3 --- Time correlation function and mean square displacement --- p.27
Chapter 3.6 --- Seismic velocities --- p.28
Chapter 4 --- Theoretical Methods --- p.30
Chapter 4.1 --- Density Functional Theory --- p.30
Chapter 4.2 --- Approximating exchange-correlation energy functional --- p.33
Chapter 4.3 --- Car-Parrinello Molecular Dynamics --- p.34
Chapter 4.4 --- Variable cell dynamics --- p.36
Chapter 4.5 --- Nose-Hoover Thermostat --- p.37
Chapter 5 --- Simulation method and details --- p.39
Chapter 5.1 --- Structure at 0 K --- p.40
Chapter 5.1.1 --- Initialization of simulation cells --- p.40
Chapter 5.1.2 --- Convergence test --- p.41
Chapter 5.1.3 --- "Electronic minimization, fictitious electronic mass and time step" --- p.42
Chapter 5.2 --- Electronic and ionic minimization --- p.43
Chapter 5.3 --- Cell optimization and structure at 0 K --- p.44
Chapter 5.3.1 --- Optimized simulation cell of pv and ppv --- p.44
Chapter 5.4 --- Equation of state and stability of solid --- p.46
Chapter 5.5 --- Melting --- p.48
Chapter 5.6 --- Statistical average --- p.50
Chapter 6 --- MgSiO3 perovskite and post-perovskite at CMB conditions --- p.51
Chapter 6.1 --- Equations of state of pv and ppv at 0 K --- p.51
Chapter 6.2 --- Enthalpy of pv and ppv at 0 K --- p.54
Chapter 6.3 --- Equations of state of pv and ppv at different temperatures --- p.55
Chapter 6.4 --- Fluctuation of stress components of pv and ppv --- p.59
Chapter 6.5 --- Pair distribution function of pv and ppv --- p.61
Chapter 6.5.1 --- Pair distribution function at different temperatures with similar cell volume --- p.61
Chapter 6.5.2 --- Pair distribution function at 4000 K and different volumes --- p.66
Chapter 6.5.3 --- Pair distribution function at 6000 K and different volumes --- p.70
Chapter 6.5.4 --- Coordination numbers --- p.74
Chapter 7 --- Liquid structure at CMB conditions --- p.78
Chapter 7.1 --- Equations of state of liquid --- p.78
Chapter 7.2 --- Stress components of liquid --- p.80
Chapter 7.3 --- Pair distribution function of liquid --- p.83
Chapter 7.4 --- Coordination numbers of liquid --- p.88
Chapter 7.4.1 --- Mean square displacement --- p.88
Chapter 8 --- Phase diagram of MgSiO3 --- p.92
Chapter 8.1 --- Pressure-temperature relations --- p.92
Chapter 8.1.1 --- Enthalpy --- p.94
Chapter 8.2 --- Internal energy --- p.96
Chapter 8.3 --- Phase boundaries and phase diagram --- p.99
Chapter 9 --- Discussions --- p.105
Chapter 9.1 --- Phase diagram --- p.105
Chapter 9.2 --- LDA vs GGA --- p.107
Chapter 9.3 --- Pv and ppv at low pressure --- p.107
Chapter 9.4 --- Two-phase method --- p.108
Bibliography --- p.110
Chapter A --- Rotation and shape optimization --- p.116

Many important engineering problems, ranging from antenna design to seismic imaging, require the numerical solution of problems of time-domain propagation and scattering of acoustic, electromagnetic, elastic waves, etc. These problems present several key difficulties, including numerical dispersion, the need for computational boundary conditions, and the extensive computational cost that arises from the extremely large number of unknowns that are often required for adequate spatial resolution of the underlying three-dimensional space. In this thesis a new class of numerical methods is developed. Based on the recently introduced Fourier continuation (FC) methodology (which eliminates the Gibbs phenomenon and thus facilitates accurate Fourier expansion of nonperiodic functions), these new methods enable fast spectral solution of wave propagation problems in the time domain. In particular, unlike finite difference or finite element approaches, these methods are very nearly dispersionless---a highly desirable property indeed, which guarantees that fixed numbers of points per wavelength suffice to solve problems of arbitrarily large extent. This thesis further puts forth the mathematical and algorithmic elements necessary to produce highly scalable implementations of these algorithms in challenging parallel computing environments---such as those arising in GPU architectures---while preserving their useful properties regarding convergence and dispersion.

Additionally, this thesis develops a fast method for evaluation of computational boundary conditions which is based on Kirchhoff's integral formula in conjunction with the FC methodology and an accelerated equivalent source integration method introduced recently for solution of integral equation problems. The combination of these ideas gives rise to a physically exact radiating boundary condition that is nonlocal but fast. The only known alternatives that provide all three of these features are only applicable to a highly restrictive class of domains such as spheres or cylinders, whereas the Kirchhoff-based approach considered here only requires a bounded domain with nonvanishing thickness. As is the case with the FC scattering solvers mentioned above, the boundary-conditions algorithm is modified into a formulation that admits efficient implementation in GPU and other parallel infrastructures.

Finally, this thesis illustrates the character of the newly developed algorithms, in both GPU and parallel CPU infrastructures, with a variety of numerical examples. In particular, it is shown that the GPU implementations result in thirty- to fiftyfold speedups over the corresponding single CPU implementations. An extension of the boundary-condition algorithm, further, is demonstrated, which enables for propagation of time-domain solutions over arbitrarily large spans of empty space at essentially null computational cost. Finally, a hybridization of the FC and boundary condition algorithm is presented, which is also part of this thesis work, and which provides an interface of the newly developed algorithms with legacy finite-element representations of geometries and engineering structures. Thus, combining spectral and classical PDE solvers and propagation methods with novel GPU and parallel CPU implementations, this thesis demonstrates a computational capability that enables solution, in novel computational architectures, of some of the most challenging problems in the broad field of computational wave propagation and scattering.

This paper matches four different classes of axisymmetric solutions to Einstein's Field equations (Weyl and Levi-Civita static solutions, Kerr solution, Papapetrou solutions or Lewis and Van Stockum solutions) across a thin axisymmetric rotating shell to a flat interior metric. No assumptions about the behavior of the gravitational field at infinity are imposed, including the usual assumptions of asymptotic flatness or compactness. The purpose is to see how changing the boundary conditions at infinity affects frame dragging inside the shell The first case discussed is that of a spherical rigidly rotating shell of uniform density. Excluding the limit of the shell radius equal to the gravitational radius, the only matching exterior solutions finite on the rotation axis are trivial flat and Schwarzschild metrics. The freedom gained by not imposing boundary conditions at infinity is not sufficient to allow other matchings across such a shell to a flat interior metric Allowing the shell to be any of a class of axisymmetric prolate spheroidal shells with rotation and density dependent on latitude, I find matchings in cases other than the gravitational limit for both the Weyl and Levi-Civita static case and the Papapetrou case. Both contain asymptotically flat solutions as well as solutions with other behavior at infinity. The component of the stress energy tensor for time and angle of rotation as seen in an inertial frame is zero in the Weyl and Levi-Civita static case for all boundary conditions. It may however be nonzero in the Papapetrou case depending upon the conditions at infinity imposed. The interpretation is that changing the boundary conditions in this case changes the frame dragging of the interior inertial frames
acase@tulane.edu

The fast-paced growth in microelectromechanical systems (MEMS), microfluidic fabrication, porous media applications, biomedical assemblies, space propulsion, and vacuum technology demands accurate and practical transport equations for rarefied gas flows. It is well-known that in rarefied situations, due to strong deviations from the continuum regime, traditional fluid models such as Navier-Stokes-Fourier (NSF) fail. The shortcoming of continuum models is rooted in nonequilibrium behavior of gas particles in miniaturized and/or low-pressure devices, where the Knudsen number (Kn) is sufficiently large. Since kinetic solutions are computationally very expensive, there has been a great desire to develop macroscopic transport equations for dilute gas flows, and as a result, several sets of extended equations are proposed for gas flow in nonequilibrium states. However, applications of many of these extended equations are limited due to their instabilities and/or the absence of suitable boundary conditions. In this work, we concentrate on regularized 13-moment (R13) equations, which are a set of macroscopic transport equations for flows in the transition regime, i.e., Kn≤1. The R13 system provides a stable set of equations in Super-Burnett order, with a great potential to be a powerful CFD tool for rarefied flow simulations at moderate Knudsen numbers. The goal of this research is to implement the R13 equations for problems of practical interest in arbitrary geometries. This is done by transformation of the R13 equations and boundary conditions into general curvilinear coordinate systems. Next steps include adaptation of the transformed equations in order to solve some of the popular test cases, i.e., shear-driven, force-driven, and temperature-driven flows in both planar and curved flow passages. It is shown that inexpensive analytical solutions of the R13 equations for the considered problems are comparable to expensive numerical solutions of the Boltzmann equation. The new results present a wide range of linear and nonlinear rarefaction effects which alter the classical flow patterns both in the bulk and near boundary regions. Among these, multiple Knudsen boundary layers (mechanocaloric heat flows) and their influence on mass and energy transfer must be highlighted. Furthermore, the phenomenon of temperature dip and Knudsen paradox in Poiseuille flow; Onsager's reciprocity relation, two-way flow pattern, and thermomolecular pressure difference in simultaneous Poiseuille and transpiration flows are described theoretically. Through comparisons it is shown that for Knudsen numbers up to 0.5 the compact R13 solutions exhibit a good agreement with expensive solutions of the Boltzmann equation.