Dissertations / Theses on the topic 'Numerical Limit Analysis'

Accurate prediction of the future state of the atmosphere is important throughout society, ranging from the weather forecast in a few days time to modelling the effects of a changing climate over decades and generations. The equations which govern how the atmosphere evolves have long been known; these are the Navier-Stokes equations, the laws of thermodynamics and the equation of state. Unfortunately the nonlinearity of the equations prohibits analytic solutions, so simplified models of particular flow phenomena have historically been, and continue to be, used alongside numerical models of the full equations. In this thesis, the two-dimensional Eady model of shear-driven frontogenesis (the creation of atmospheric fronts) was used to investigate how errors made in a localised region can affect the global solution. Atmospheric fronts are the boundary of two different air masses, typically characterised by a sharp change in air temperature and wind direction. This occurs across a small length of O(10 km), whereas the extent of the front itself can be O(1000 km). Fronts are a prominent feature of mid-latitude weather systems and, despite their narrow width, are part of the large-scale, global solution. Any errors made locally in the treatment of fronts will therefore affect the global solution. This thesis uses the convergence of the Euler equations to the semigeostrophic equations, a simplified model which is representative of the large-scale flow, including fronts. The Euler equations were solved numerically using current operational techniques. It was shown that highly predictable solutions could be obtained, and the theoretical convergence rate maintained, even with the presence of near-discontinuous solutions given by intense fronts. Numerical solutions with successively increased resolution showed that the potential vorticity, which is a fundamental quantity in determining the large-scale, balanced flow, approached the semigeostrophic limit solution. Regions of negative potential vorticity, indicative of local areas of instability, were reduced at high resolution. In all cases, the width of the front reduced to the grid-scale. While qualitative features of the limit solution were reproduced, a stark contrast in amplitude was found. The results of this thesis were approximately half in amplitude of the limit solution. Some attempts were made at increasing the intensity of the front through spatial- and temporal-averaging. A scheme was proposed that conserves the potential vorticity within the Eady model.

Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, 2004.
Includes bibliographical references (leaves62-63).
This thesis reviews limit theorems and their applications for stability problems in geotechnical engineering. Rigorous numerical solutions of limit analyses can be obtained through finite element discretization of the soil mass and formulation of the limit theorems within a linear programming framework. The current research uses a formulation proposed by Sloan et al. (1988) and extended in a recent Ph.D. thesis by Ukritchon (1998) to include soil-structure interactions. The thesis details the input and output required for numerical limit analysis and presents an example application for the stability of a broad excavation for the MUNI Metro Turnback project in San Francisco. This well documented case study involves a 13 M deep excavation within a deep deposit of May Mud that was supported by an SPTC wall with three levels of cross-lot bracing. The numerical limit analyses calculate factors of safety, FS = 1.03 - 1.36, against basal instability. The factor of safety used in the original design (FS = 1.2) is contained in this range. The results illustrate that numerical limit analysis offers a practical alternative to limit equilibrium methods in evaluating the stability of braced excavations.
by Fernando G. Degwitz.
M.Eng.

COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR
O presente trabalho apresenta um estudo de problemas de estabilidade, freqüentemente,encontrados, em Engenharia Geotécnica, através da técnica da Análise Limite associada ao Método dos Elementos Finitos (MEF).Inicialmente, faz-se uma revisão das formulações da Análise Limite, via MEF,encontradas, com maior freqüência, na literatura técnica especializada.Uma formulação mista que é descrita em detalhe na tese foi escolhida para implementação. Extensões das formulações da Análise Limite de meios contínuos são propostas para contemplar características de maciços rochosos fraturados. É proposto,também, um procedimento numérico para tratar de problemas de estabilidade de meios que exibem fluxo plástico não associado.As implementações realizadas foram validadas através de problemas cujas soluções podem ser obtidas por via analítica.Finalmente, um número considerável de problemas de interesse em Engenharia Geotécnica é estudado utilizando a implementação realizada. Os resultados destes estudos sugerem a viabilidade da utilização da técnica estudada na solução de problemas práticos de Engenharia Civil.
This work presents a study of stability problems often encountered in Geotechnical Engineering, through the use of Limit Analysis in conjunction with the Finite Element Method (FEM).Initially, a literature survey of the most often found formulations in Limit Analysis through the FEM is carried out.A mixed formulation of Limit Analysis was chosen for implementation and its details are fully described. Extensions of the formulation to deal with stability problems in fractured rock media are also proposed and described. A numerical procedure to take into account the effect of non associative plastic flow is proposed.The implementations carried out were validated through problems to which analytical solutions could be found.Finally, a considerable number of problems of interest to Geotechnical Engineering is studied with the implemented formulation. The results of these studies suggest that Limit Analysis can be considered as a viable tool in the solution of practical problems in Geotechnical Engineering.

Current research in the field of limit analysis is focussing on the development of numerical tools which are sufficiently efficient and robust to be used in engineering practice. This places demands on the numerical discretisation strategy adopted as well as on the mathematical programming tools applied, which are the key ingredients of a typical computational limit analysis procedure. In this research, the Element-Free Galerkin (EFG) discretisation strategy is used to approximate the displacement and moment fields in plate and slab problems, and second-order cone programming (SOCP) is used to solve the resulting discretised formulations. A numerical procedure using the EFG method and second-order cone programming for the kinematic limit analysis problem was developed first. The moving least squares technique was used in combination with a stabilised conforming nodal integration scheme, both to keep the size of the optimisation problem small and to provide stable and accurate solutions. The formulation was expressed as a problem of minimizing a sum of Euclidean norms, which was then transformed into a form suitable for solution using SOCP. To improve the accuracy of solutions and to speed-up the computational process, an efficient h-adaptive EFG scheme was also developed. The naturally conforming property of meshfree approximations (with no nodal connectivity required) facilitates the implementation of h-adaptivity. The error in the computed displacement field was estimated accurately using the Taylor expansion technique. A stabilised conforming nodal integration scheme was also extended to error estimators, leading to an efficient and truly meshfree adaptive method. To obtain an indication of bounds on the solutions obtained, an equilibrium formulation was also developed. Pure moment fields were approximated using a moving least squares technique. The collocation method was used to enforce the strong form of the equilibrium equations and a stabilised conforming nodal integration scheme was introduced to eliminate numerical instability problems. The von Mises and Nielsen yield criteria were then enforced by introducing second-order cone constraints.

The great engineers, architects and master builders of 17th and 19th centuries constructed the most amazing vaults in the entire world. Considering that the advanced technologies were massively developed only in the 20th century, the traditional approach for constructing masonry vaults were based in the geometry. As Heyman said, the correct shape of a masonry structure ensures its safety. Nowadays, it can be consider that their practice was a success because many of these vaults still stand after all these years. This thesis describes the considerable existing approaches about funicular solution for masonry vaults with the principal features of each method and their extensions. All methods derives compression-only structural lower-bound solutions, which guarantees the safety of the vaulted structures with at least one admissible equilibrium state given its geometry and boundary conditions. The review comprehends the old methods, Graphic Statics and Limit Analysis that are a strong base to the recent ones, also presented in this paper, Thrust Network Method, Force Density Method, Thrust Network Analysis and Thrust Surface Method. In 2007, the Thrust Network Analysis was introduced by Block and Ochsendorf, since then some extensions of their work were brought up to the scientific community and they are summarized in this thesis due to the importance for upcoming improved methods. Lastly, it is presented an application of the software RhinoVAULT 2 for funicular form-finding using the Thrust Network Analysis approach. The project designed is the "Tettoia Nervi", placed in Bologna, Italy.

PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO
COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR
PROGRAMA DE EXCELENCIA ACADEMICA
Nesse trabalho é avaliada a solução numérica do colapso na frente de escavação em túneis rasos, através da teoria de análise limite numérico, usando o teorema do limite inferior, a partir da condição de equilíbrio para as condições plásticas, além de considerar o comportamento do material rígido perfeitamente plástico. O teorema de limite inferior implica em maximizar o fator multiplicador na carga atuante, por isso a análise limite se torna um problema de otimização, nele tem que se usar a programação matemática para ser resolvido. É avaliada a solução numérica tridimensional da análise limite através do método dos elementos finitos, usando malha de elementos hexaédricos de oito nós, a análise dos elementos finitos é feita com o próprio código gerado na linguagem de programação do MATLAB 2017.As metodologias de programação matemática empregadas são: programação cônica de segunda ordem e programação semidefinida. Antes deve-se adaptar os critérios de ruptura de Drucker Prager à programação cônica de segunda ordem e Mohr-Coulomb tridimensional à programação semidefinida. Para a otimização se usa o algoritmo comercial MOSEK Aps 7.1 baseado no método do ponto interior em grande escala, na linguagem do MATLAB 2017. Além disso, obteve-se o mecanismo de colapso através da propriedade da dualidade do problema de otimização, dualidade que é cumprida pelos teoremas de limite superior e inferior.
In this work the numerical solution of the collapse in the front of excavation in shallow tunnels is evaluated through the theory of numerical limit analysis, using the lower limit theorem, from the equilibrium condition for the plastic conditions, considering the behavior of the perfectly plastic rigid material. The lower limit theorem implies maximizing the multiplier factor in the acting load, so that the limit analysis becomes an optimization problem. The three-dimensional numerical solution of the limit analysis using the finite element method is evaluated using a mesh of eight-node hexahedral elements. The finite element analysis is done using the code generated in the MATLAB 2017 programming language. The mathematical programming methodologies used are: second order conic programming and semidefinite programming. The Drucker-Prager three-dimensional criteria should be adapted to the conic programming of the second order and Mohr-Coulomb three-dimensional to the semidefinite programming. For the optimization, the MOSEK Aps 7.1 commercial algorithm based on the large-scale interior point method is used in the MATLAB 2017 language. In addition, the collapse mechanism was obtained through the duality property of the optimization problem, duality that is fulfilled by the upper and lower limit theorems.

Análises de estabilidade de taludes, em mineração, são uma das atividades mais importantes do processo de obtenção de um bem mineral. Assegurar a estabilidade das escavações e a segurança do empreendimento são vitais para o processo de extração e, para tanto, a escolha da forma de análise (2D, quasi-tridimensional ou 3D) ou do método (equilíbrio limite ou análise numérica) mais propício para esse tipo de estudo é de suma importância. Muitas vezes o engenheiro responsável acaba baseando sua escolha na familiaridade com determinada metodologia ou na praticidade de determinado programa, em detrimento a fatores técnicos e de relevância para o estudo. Dessa forma, visando preencher tal lacuna existente atualmente e facilitar a escolha da melhor metodologia a ser utilizada, o presente estudo reúne os principais métodos conhecidos, suas vantagens e desvantagens e, assim, propõem uma metodologia para que essa escolha seja baseada em parâmetros técnicos relevantes. A metodologia proposta é, de certa forma, restritiva na escolha dos procedimentos que serão considerados, uma vez que, buscou-se utilizar métodos que tenham sua aplicabilidade comprovada, que possuam vantagens consideráveis sobre outros métodos e que sejam rigorosos, ou seja, satisfaçam todas as condições de equilíbrio (forças e momentos). Para corroborar com a metodologia proposta, quatro situações práticas diferentes foram apresentadas, demonstrando a aplicabilidade da metodologia.
Analysis of slope stability in mining is one of the most important activities of the process of obtaining a mineral resources. Ensuring the excavation stability and the safety of the project are vital to the extraction process. The choice of the form of analysis (2D, quasi-threedimensional or 3D) or of the method (limit equilibrium or numerical analysis) more suitable for this type of study is of paramount importance since, the responsible engineer often end up basing the analysis on familiarity with a particular methodology or on the availability of a particular program, to the detriment of technical factors and relevance to the study. Thus, in order to fill this existing gap and facilitate the selection of the best methodology to be used, the present study brings together the main known methods, their advantages and disadvantages and, therefore, presents a methodology for such a choice to be based on relevant technical parameters. The proposed methodology is somewhat restrictive in the choice of procedures that will be considered, since it has been tried to use methods that have proven applicability, that have considerable advantages over other methods and that are rigorous, satisfying all equilibrium conditions (forces and moments). In order to corroborate the proposed methodology, four different practical situations were presented, demonstrating the applicability of the methodology.

We construct a mathematical model of an order driven market where traders can submit limit orders and market orders to buy and sell securities. We adapt the notion of no free lunch of Harrison and Kreps and Jouini and Kallal to our setting and we prove a no-arbitrage theorem for the model of the order driven market. Furthermore, we compute signatures of order books of different financial markets. Signatures, i.e. the full sequence of definite iterated integrals of a path, are one of the fundamental elements of the theory of rough paths. The theory of rough paths provides a framework to describe the evolution of dynamical systems that are driven by rough signals, including rough paths based on Brownian motion and fractional Brownian motion (see the work of Lyons). We show how we can obtain the solution of a polynomial differential equation and its (truncated) signature from the signature of the driving signal and the initial value. We also present and analyse an ODE method for the numerical solution of rough differential equations. We derive error estimates and we prove that it achieves the same rate of convergence as the corresponding higher order Euler schemes studied by Davie and Friz and Victoir. At the same time, it enhances stability. The method has been implemented for the case of polynomial vector fields as part of the CoRoPa software package which is available at http://coropa.sourceforge.net. We describe both the algorithm and the implementation and we show by giving examples how it can be used to compute the pathwise solution of stochastic rough differential equations driven by Brownian rough paths and fractional Brownian rough paths.

The main work for Qiujing PAN’s PhD thesis is to develop the stability analysis for underground structures, which contains two parts, deterministic model and probabilistic analysis. During his 1st year of PhD research, he has mainly finished the deterministic model study. In the 2nd year, I developed a probabilistic model for high dimensional problems
In the contemporary society, the utilization and exploitation of underground space has become an inevitable and necessary measure to solve the current urban congestion. One of the most important requirements for successful design and construction in tunnels and underground engineering is to maintain the stability of the surrounding soils of the engineering. But the stability analysis requires engineers to have a clear ideal of the earth pressure, the pore water pressure, the seismic effects and the soil variability. Therefore, the research aimed at employing an available theory to design tunnels and underground structures which would be a hot issue with high engineering significance. Among these approaches employed to address the above problem, limit analysis is a powerful tool to perform the stability analysis and has been widely used for real geotechnical works. This research subject will undertake further research on the application of upper bound theorem to the stability analysis of tunnels and underground engineering. Then this approach will be compared with three dimensional analysis and experimental available data. The final goal is to validate new simplified mechanisms using limit analysis to design the collapse and blow-out pressure at the tunnel face. These deterministic models will then be used in a probabilistic framework. The Collocation-based Stochastic Response Surface Methodology will be used, and generalized in order to make possible at a limited computational cost a complete parametric study on the probabilistic properties of the input variables. The uncertainty propagation through the models of stability and ground movements will be evaluated, and some methods of reliability-based design will be proposed. The spatial variability of the soil will be taken into account using the random field theory, and applied to the tunnel face collapse. This model will be developed in order to take into account this variability for much smaller computation times than numerical models, will be validated numerically and submitted to extensive random samplings. The effect of the spatial variability will be evaluated

Incremental sheet forming (ISF) is a novel metal shaping technology that is economically viable for low-volume manufacturing, customisation and rapid-prototyping. It uses a small tool that is controlled by a computer-numerically controlled sequence and the path taken by this tool over the sheet defines the product geometry. Little is currently known about how to design the tool-path to minimise geometric errors in the formed part. The work here addresses this problem by developing a model based tool-path optimisation scheme for ISF. The key issue is how to generate an efficient model for ISF to use within a path optimisation routine, since current simulation methods are too slow. A proportion of this thesis is dedicated to evaluating the applicability of the rigid plastic assumption for this purpose. Three numerical models have been produced: one based on small strain deformation, one based on limit analysis theory and another that approximates the sheet to a network of rods. All three models are formulated and solved as second-order cone programs (SOCP) and the limit analysis based model is the first demonstration of an upper-bound shell finite element (FE) problem solved as an SOCP. The models are significantly faster than commercially available FE software and simulations are compared with experimental and numerical data, from which it is shown the rigid plastic assumption is suitable for modelling deformation in ISF. The numerical models are still too slow for the path optimisation scheme, so a novel linearised model based on the concept of spatial impulse responses is also formulated and used in an optimal control based tool-path optimisation scheme for producing axisymmetric products with ISF. Off-line and on-line versions of the scheme are implemented on an ISF machine and it is shown that geometric errors are significantly reduced when using the proposed method. This work provides a new structured framework for tool-path design in ISF and it is also a novel use of feedback to compensate for geometrical errors in ISF.

Les chaînes plissées d'avant-pays sont des objets géologiques qui se forment dans un contexte compressif et représentent la partie externe des orogènes.Elles sont composées de nombreuses structures plissées associées à des chevauchements généralement enracinés au sein d'un niveau de décollement peu profond situé dans la partie basale de la couverture sédimentaire. Ces objets géologiques ont été beaucoup étudiés au XXème siècle.Ils ont été modélisés cinématiquement, analogiquement et mécaniquement.S’il est indispensable de tenir compte de l’aspect mécanique en géologie structurale, les modèles mécaniques restent cependant trop peu utilisés par le géologue structuraliste.L'objectif de cette thèse est de montrer comment apporter des contraintes mécaniques à l'étude des structures géologiques, grâce à une théorie mécanique facilement utilisable en géologie structurale.Cette théorie, le calcul à la rupture (Limit Analysis), représente en effet un bon intermédiaire entre les modèles cinématiques et les modèles mécaniques en éléments finis, très complets mais relativement complexes d'utilisation.Nous étudions ici des exemples appartenant à la chaîne plissée d’avant-pays du Jura et utilisons les logiciels Optum G2 et SLAMTec.Nous procédons alors selon deux approches. La première approche, présentée en première partie de ce manuscrit consiste à étudier la déformation passée.Nous nous focalisons sur la région de Saint-Ursanne dans le Nord-Est du Jura, en Suisse et réalisons dans un premier temps une coupe géologique d'une structure que nous étendons ensuite en 3D via une série de coupes balayant l'ensemble de la zone d'étude.Pour réaliser ces coupes nous utilisons à la fois des principes d’équilibrage cinématique, des modélisations mécaniques par calcul à la rupture et des modélisations analogiques en boîte à sable.Nous montrons ainsi l’importance de l’héritage tectonique sur la mise en place des structures de cette région et modélisons mécaniquement l’influence d’un décalage du niveau de décollement par une faille normale héritée, lors d’un épisode compressif ultérieur.Nous montrons alors que ce décalage représente un générateur de failles ainsi qu’un point d’accroche qui ralenti la propagation du front de déformation, contrôlant par ailleurs la direction de la structure qui se développera.Nous montrons également que la topographie joue un rôle majeur dans la mise en place des structures géologiques.Dans la deuxième partie du manuscrit nous abordons une seconde approche qui consiste à étudier la déformation actuelle.Nous changeons cette fois-ci d’échelle pour nous intéresser à la tectonique actuelle de l’ensemble du Jura.Nos modélisations prédisent que cette chaîne plissée, formée par une tectonique de couverture, est aujourd’hui partiellement affectée par une tectonique de socle.Alors que la partie ouest du Jura serait toujours affectée par une tectonique de couverture, le socle devrait aujourd’hui être impliqué dans la déformation au niveau de la partie nord-est.Nous montrons cependant qu’une tectonique de socle n’empêche pas l’activation simultanée des niveaux de décollements situés dans la couverture (les évaporites triasiques, dans le cas du Jura).L’intérêt pratique de ces méthodes est illustré par des questions industrielles : le dernier chapitre présente une étude des coupes géologiques d’un terrain d’étude de la Nagra, situé dans l’est du Jura.Nous regardons la réponse mécanique actuelle en compression de l’interprétation structurale proposée.Les outils numériques utilisés dans l’ensemble de ce travail ont, par ailleurs, fait l’objet d’un tutoriel réalisé dans le cadre d’une mission au sein de l’entreprise Total
Fold-and-thrust-belts form in a compressive geological context and represent the external part of orogens.They are composed of numerous folds and thrusts generally rooted in a shallow décollement level located in the basal part of the sedimentary cover.These belts have been studied extensively during the XX century. They have been modeled kinematically, analogically and mechanically.The mechanical aspect of deformation must be taken into account in structural geology, but yet, mechanical models remain underused by the structural geologist.The objective of this thesis is to show how to bring mechanical constraints to the study of geological structures with a mechanical theory easily usable in structural geology, the Limit Analysis.This theory represents a good compromise between the kinematic models and the finite element mechanical models, very complete but relatively complicated to use.We study here examples from the Jura fold-and-thrust belt and use the softwares Optum G2 and SLAMTec.We proceed with two different approaches.The first one is presented in the first part of this manuscript and consists in studying the past deformation.We focus on Saint-Ursanne area, in the North-East Switzerland Jura.We construct first a 2D geological cross section of the Mont Terri structure and then we extend this work in 3D by a series of cross sections that scan the whole Saint Ursanne area.These cross-sections are drawn using kinematical rules, mechanical modeling with Limit Analysis and analog sandbox modeling.We show the importance of the tectonic inheritance on the onset of structures in the studied region and we mechanically model the influence of a décollement offset due to an inherited normal fault during a subsequent compressive episode.We show that this offset represents a “fault generator” and a catching point that slows the propagation of the deformation front and furthermore controls the orientation of the structure that develops.We also show that the topography plays a major role in the establishment of geological structures.In the second part of the manuscript we follow a second approach that consists in studying the current deformation.We change of scale and study the current tectonics of the entire Jura.Our models predict that this belt, formed in thin-skin tectonics, is currently partially affected by thick-skin tectonics.The western Jura would be still affected by a thin-skin tectonics, but the basement should be involved in the deformation in the north-east Jura.However, we show that a thick-skin tectonics does not preclude the simultaneous activation of the shallower décollements contained by the sedimentary cover (Triassic evaporites, in the Jura).Practical interest of these methods is illustrated by industrial questions: the last chapter presents a study of geological cross-sections from a studied area of Nagra in the eastern Jura.We look at the current mechanical answer under compression of the proposed structural interpretations.The numerical tools used in this work have been the subject of a tutorial made during a mission at Total company

Cette thèse a pour objectif de développer une approche mathématique pour la modélisation des systèmes vivants en mettant l’accent sur les équations hyperboliques et cinétiques décrivant les systèmes multicellulaires en biologie, la dynamique de foule, et les comportements collectifs des individus en sciences sociales et économiques considérées comme des sciences comportementales, appelées parfois “sciences douces”. Plus précisément, les points traités dans cette thèse ont été les suivants : 1) Le développement de ce qu’on appelle la théorie cinétique des particules actives pour la dérivation d’une structure mathématique pour la modélisation des systèmes vivants, qui tient compte des caractéristiques et complexités de ces systèmes complexes, où la dynamique des entités est développée aussi sur la variable d’espace. Cette structure mathématique générale offre un cadre conceptuel pour la dérivation des modèles spécifiques correspondant à des classes de systèmes bien définies et remplace les approches classiques utilisées pour modéliser les systèmes inertes qui s’avèrent inappropriés pour la modélisation des systèmes vivants. 2) Le développement de méthodes mathématiques pour la dérivation de modèles à l’échelle macroscopique de type Keller-Segel et de type Cattaneo à partir d’une description cinétique basée sur la théorie des particules actives, ainsi que le développement et l’implémentation des schémas numériques préservant la limite asymptotique, en particulier des méthodes de volumes finis pour les systèmes de lois de conservations sont utilisées pour l’approximation des modèles macroscopiques. 3) L’application à la modélisation, l’analyse qualitative et les simulations des systèmes sociaux. Plus précisément les applications ont été adressées aux systèmes sociaux-économiques et à la dynamique comportementale de la foule en mettant en œuvre l’évacuation d’un espace dangereux où la géométrie est complexe et en tenant compte de la propagation du stress. Des simulations numériques ont été obtenues par un développement approprié des méthodes de Monte Carlo. 4) L’étude de la convergence de développement de Hilbert pour la dérivation d’équations macroscopiques à partir de la description mésoscopique basée sur la théorie cinétique des particules actives, et l’analyse qualitative liée à l’existence et l’unicité des solutions des systèmes cinétiques
This thesis tackles the challenging aim of developing a mathematical theory of living systems with focus on hyperbolic and kinetic equations, to multicellular systems in biology, crowd dynamics, and social sciences and economy viewed as behavioral sciences, occasionally called soft sciences. In more details, the following topics have been tackled: 1) Development of the theory and application of the kinetic theory of the scalled active particles, with the main objective of deriving a general mathematical structure, consistent with the complexity features of living systems, where the dynamics are developed over the space variable. This structure offers the conceptual background for the derivation of specific models corresponding to well-defined classes of systems and substitutes the field theories, which classically offers the natural support in the sciences of the inert matter that cannot be applied in the case of living systems. Applications have also motivated development of simulation tools. 2) Mathematical methods to derive macroscopic tissue equations, of Keller– Segel and Cattaneo type, from the underlying description at the microscopic scale delivered by kinetic type models and development of computational schemes towards simulations both of kinetic transport models and hyperbolic macroscopic models. In more details, finite volume methods for hyperbolic conservative laws equations have been developed for the simulations of macroscopic models. 3) Applications to modeling, qualitative analysis, and simulations of social systems. Applications have been addressed to social systems and behavioral crowd dynamics with a special focus on evacuation dynamics from venues with complex geometry with special focus to a dy- namics, where panic propagates. Simulations have been obtained by a suitable developments of the socalled Monte Carlo particle methods. 4) Analytical problems generated by the convergence of the Hilbert approach to the derivation of macroscopic equations from the kinetic theory approach, and a qualitative analysis related to existence and uniqueness of the solutions of the initial value problems of the kinetic systems

The celebrated Schrödinger equation is the key to understanding the dynamics of quantum mechanical particles and comes in a variety of forms. Its numerical solution poses numerous challenges, some of which are addressed in this work. Arguably the most important problem in quantum mechanics is the so-called harmonic oscillator due to its good approximation properties for trapping potentials. In Chapter 2, an algebraic correspondence-technique is introduced and applied to construct efficient splitting algorithms, based solely on fast Fourier transforms, which solve quadratic potentials in any number of dimensions exactly - including the important case of rotating particles and non-autonomous trappings after averaging by Magnus expansions. The results are shown to transfer smoothly to the Gross-Pitaevskii equation in Chapter 3. Additionally, the notion of modified nonlinear potentials is introduced and it is shown how to efficiently compute them using Fourier transforms. It is shown how to apply complex coefficient splittings to this nonlinear equation and numerical results corroborate the findings. In the semiclassical limit, the evolution operator becomes highly oscillatory and standard splitting methods suffer from exponentially increasing complexity when raising the order of the method. Algorithms with only quadratic order-dependence of the computational cost are found using the Zassenhaus algorithm. In contrast to classical splittings, special commutators are allowed to appear in the exponents. By construction, they are rapidly decreasing in size with the semiclassical parameter and can be exponentiated using only a few Lanczos iterations. For completeness, an alternative technique based on Hagedorn wavepackets is revisited and interpreted in the light of Magnus expansions and minor improvements are suggested. In the presence of explicit time-dependencies in the semiclassical Hamiltonian, the Zassenhaus algorithm requires a special initiation step. Distinguishing the case of smooth and fast frequencies, it is shown how to adapt the mechanism to obtain an efficiently computable decomposition of an effective Hamiltonian that has been obtained after Magnus expansion, without having to resolve the oscillations by taking a prohibitively small time-step. Chapter 5 considers the Schrödinger eigenvalue problem which can be formulated as an initial value problem after a Wick-rotating the Schrödinger equation to imaginary time. The elliptic nature of the evolution operator restricts standard splittings to low order, ¿ < 3, because of the unavoidable appearance of negative fractional timesteps that correspond to the ill-posed integration backwards in time. The inclusion of modified potentials lifts the order barrier up to ¿ < 5. Both restrictions can be circumvented using complex fractional time-steps with positive real part and sixthorder methods optimized for near-integrable Hamiltonians are presented. Conclusions and pointers to further research are detailed in Chapter 6, with a special focus on optimal quantum control.
Bader, PK. (2014). Geometric Integrators for Schrödinger Equations [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/38716
TESIS
Premiado

Thesis (Sc.D.)--Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, 1998.
Includes bibliographical references.
This thesis describes the development and application of numerical limit analyses for planar, undrained stability problems in geotechnical engineering. The proposed analyses formulate upper and lower bound theorems for perfectly plastic, cohesive­frictional soils as linear programming problems, with spatial variations of the unknown velocities and stresses approximated by standard finite element interpolation functions. The thesis includes three substantial modifications of the pre-existing software: l) inclusion of structural beam and joint elements, that enable the program to model failure in combined bending, shear and axial loading; 2) implementation of functions to represent the undrained strength anisotropy of soft clays; and 3) representation of pore water pressures and effective stress strength parameters for stability analyses that include free draining soil layers. In all of the examples presented in this thesis, the proposed numerical limit analyses are able to bracket the true collapse load within ±5%. The stability of surface footings under combined effects of vertical, horizontal and moment (V, H, M) loading is solved in the form of three-dimensional failure envelopes, that include effects of underbase suction and non-homogeneous undrained strength profiles. Existing empirical bearing capacity factors for inclined, eccentric loading are shown to be conservative, often underestimating the exact collapse solutions for footings on homogeneous clay by more than 25%. However, the same correction factors can become completely unreliable when there is a significant undrained strength gradient. The numerically-derived failure envelope for footings on homogeneous clay is described approximately by curve fitting techniques using relatively simple functions that can be used to update the existing bearing capacity factors. The undrained (short-term) pullout capacity of a single caisson cell in clay is initially investigated as a planar problem. Limit analyses show that the capacity can be presented as a normalized function of the undrained strength gradient in the clay, and interior surface area of the caisson. The results show that there are significant errors in limit equilibrium calculations which assume mechanisms of reverse end-bearing. Pullout predictions are compared with measured data from 1) small scale laboratory element tests on miniature caissons, 2) centrifuge models and 3) field tests on a four-cell caisson unit. These comparisons introduce approximations in the scaling of the planar solutions to axisymmetric cell geometries, and inclined loading cases. Good agreement is obtained with all three sets of experiments, and confirms that the proposed limit analyses are at least comparable in predictive capability to the limit equilibrium calculations currently used in practice. The limit analyses represent the undrained strength anisotropy of soft clays using a yield function originally proposed by Davis and Christian ( 1970). Undrained strength anisotropy is found to have a major influence on the stability and failure mechanisms for unbraced vertical cuts in clay. In contrast, there is minimal effect on the failure mechanisms for kinematically constrained problems such as the lateral load capacity of deeply embedded pile or pipe sections. Applications of limit analyses for braced excavations show important errors in existing calculations of basal stability. The analyses also provide the first theoretical solutions that quantify effects of wall embedment and bending strength on the stability of well-braced cuts. The results have also been compared with previous displacement-based finite element calculations based on the anisotropic MIT-E3 soil model. Good matching is obtained using the limit analyses together with anisotropic strength parameters mobilized at shear strains, y = 0.6 - 1.0%, corresponding to failure conditions in the finite element solutions. Limit analyses have been used to re-assess the stability of tied-back walls at Bird Island Flats in East Boston. Detailed analyses were carried out for two cases l) estimated excavation and support geometry at an unstable section; and 2) at final excavated grade after placement of a remedial buttress constructed by deep soil-mixing and jet grouting. Reliability analyses show that uncertainties in the undrained strength profile represent the major factor affecting predictions of stability. Successful performance of the improved cross-section is not consistently explained by strength parameters derived from the unstable cross-section. Additional calculations compare the stabilizing effects of different designs for the wall, tiebacks and remedial buttress. The failure of an offshore breakwater in Brazil was analyzed extensively using conventional slope stability methods in a recent Ph.D thesis by Lee ( 1995). Limit analyses, using the same soil parameters, shows similar mechanisms of failure controlled by the undrained shear strength of the underlying Sergipe clay. However, the factors of safety computed by Spencer's method are shown to be unrealistically low due to the assumed directions of interslice forces in the overlying sand and rockfill layers.
by Boonchai Ukritchon.
Sc.D.

La maintenance, la réparation et le renforcement de murs en maçonnerie par matériaux composites collés nécessitent le développement de méthodes et techniques d'évaluation de l'aptitude au service et de requalification tant pour leur restauration fonctionnelle que pour leur adaptation à de nouvelles contraintes (évolution du zonage sismique par exemple). Le présent travail de thèse a pour objectif de proposer un outil d'évaluation par calcul de l'état limite ultime de murs en maçonnerie, renforcés par composites collés, chargés dans leur plan.Les travaux effectués concernent la modélisation par homogénéisation de murs en maçonnerie et une campagne expérimentale de caractérisation du transfert de charge entre un renfort en tissu de fibres de carbone et un support en blocs creux de béton par le biais d'un joint de colle. Pour modéliser la résistance des murs maçonnés, nous proposons une loi élastique parfaitement plastique pour le comportement dans le plan d'une maçonnerie constituée de blocs liés par des joints de mortier. Le convexe de plasticité est obtenu par une technique d'homogénéisation périodique qui prend en compte la nature tridimensionnelle de la cellule de base. On obtient alors un convexe limité par plusieurs surfaces de charge. Un algorithme numérique original est ensuite proposé et programmé dans le logiciel aux Eléments Finis ABAQUS. Des simulations numériques utilisant le module développé sont présentées

This PhD thesis treats some problems concerning nonlinear differential equations. In the first two papers computer-assisted proofs are used. The differential equations there are rewritten as fixed point problems, and the existence of solutions are proved. The problem in the first paper is one-dimensional; with one boundary condition given by an integral. The problem in the second paper is three-dimensional, and Dirichlet boundary conditions are used. Both problems have their origins in fluid dynamics. Paper III describes an inverse problem for the heat equation. Given the solution, a solution dependent diffusion coefficient is estimated by intervals at a finite set of points. The method includes the construction of set-valued level curves and two-dimensional splines. In paper IV we prove that there exists a unique, globally attracting fixed point for a differential equation system. The differential equation system arises as the number of peers in a peer-to-peer network, which is described by a suitably scaled Markov chain, goes to infinity. In the proof linearization and Dulac's criterion are used.

Ce memoire presente une etude numerique et cinematique de la stabilite des barrages et des ouvrages en terre. Nous exposons tout d'abord une synthese de formulations des cfficients de securite etablis par les principales methodes anterieures. Nous elaborons ensuite une nouvelle approche de l'analyse de la stabilite des ouvrages en terre pour une forme de rupture circulaire. L'evaluation du cfficient de securite par cette approche ne tient pas compte, comme dans la theorie classique, de scenarios de rupture fictifs (reduction generale de la resistance mecanique du massif du sol), mais de la proximite du champ de vitesse de deplacement calcule par elements finis par rapport au champ de vitesse de deplacement de solide rigide. Cette methode peut se greffer sur un calcul par elements finis et evite la dichotomie entre les calculs en deplacement et les calculs a la rupture. La validation de cette approche se fait tout d'abord en examinant l'influence de la variation de plusieurs parametres (pente, caracteristiques mecaniques et chargement) sur le cfficient de securite cinematique. Les resultats obtenus sont ensuite compares a ceux fournis par les methodes anterieures (classiques et recentes). Nous developpons aussi une extension de l'approche cinematique de la stabilite des ouvrages en terre aux cas de lignes de glissement plus complexes. Les resultats obtenus sont confrontes d'une part a ceux fournis par la procedure cinematique dans le cas d'un glissement circulaire et d'autre part a ceux obtenus par une methode utilisant un scenario d'equilibre limite.

This PhD thesis consists of a summary and seven papers, where various applications of auto-validated computations are studied. In the first paper we describe a rigorous method to determine unknown parameters in a system of ordinary differential equations from measured data with known bounds on the noise of the measurements. Papers II, III, IV, and V are concerned with Abelian integrals. In Paper II, we construct an auto-validated algorithm to compute Abelian integrals. In Paper III we investigate, via an example, how one can use this algorithm to determine the possible configurations of limit cycles that can bifurcate from a given Hamiltonian vector field. In Paper IV we construct an example of a perturbation of degree five of a Hamiltonian vector field of degree five, with 27 limit cycles, and in Paper V we construct an example of a perturbation of degree seven of a Hamiltonian vector field of degree seven, with 53 limit cycles. These are new lower bounds for the maximum number of limit cycles that can bifurcate from a Hamiltonian vector field for those degrees. In Papers VI, and VII, we study a certain kind of normal form for real hyperbolic saddles, which is numerically robust. In Paper VI we describe an algorithm how to automatically compute these normal forms in the planar case. In Paper VII we use the properties of the normal form to compute local invariant manifolds in a neighbourhood of the saddle.

The steel bandage is an economical and effective widely used method for strengthening RC columns. Diploma thesis reviews most common design proposals for the types of the bandage with directly loaded and not directly loaded steel angles. This work also includes a comparison of the different constitutive laws for confined concrete. In order to simulate real behaviour of strengthened structure the numerical analysis has been carried out using non-linear software ATENA. According to the European standards and adopted hypotheses the simplified analytical model has been derived to be able to calculate load-bearing capacity of a strengthened column under normal force and bending moment. The proposed model considers the contribution in confinement pressure caused by steel bandage and the contribution of directly loaded angles. In addition, the direct application on real frame structure has been performed.

La striction diffuse et localisée sont des phénomènes précurseurs à la rupture ductile et représentent l'une des principales causes de rebut des pièces métalliques au cours de leur mise en forme. La mise en œuvre d'outils théoriques et numériques capables de prédire l'apparition de ces défauts s'avère nécessaire pour des raisons économiques et environnementales. Ces outils nécessitent en partie l'intégration d'un modèle de comportement adéquat permettant de reproduire les phénomènes physiques mis en jeu. Un tel modèle de comportement est ensuite couplé à un indicateur d'instabilité plastique offrant la possibilité de prédire de manière fiable les phénomènes de striction diffuse et localisée. Dans ce travail de thèse, nous avons considéré certains modèles d'endommagement micromécaniques basés sur l'approche de Gurson, et qui ont été couplés à différents critères d'instabilités plastiques reposant sur l'Analyse de Bifurcation. L'implantation numérique des modèles retenus a été réalisée dans le code de calcul par éléments finis Abaqus/Standard. En ce qui concerne les modèles d'endommagement, plusieurs schémas d'intégrations ont été testés afin d'analyser leur performance et leur robustesse lorsque le comportement présente un effet adoucissant. L'approche combinant le modèle de comportement et les critères de striction a été utilisée pour prédire les limites de formabilité en striction diffuse et localisée de plusieurs matériaux métalliques. Les résultats obtenus ont permis de dresser une hiérarchisation théorique et numérique des critères de striction utilisés
Diffuse and localized necking are precursors to ductile fracture, and represent one of the main causes of metal parts rejection during forming operations. The implementation of theoretical and computational tools to predict the occurrence of these defects turns out to be necessary for economic and environmental reasons. These tools require in part the introduction of an appropriate behavior model in order to reproduce the physical phenomena involved during forming operations. Such a behavior model is then coupled to a plastic instability indicator providing the ability to reliably predict diffuse and localized necking. In the present work, we considered a micromechanical damage models based on Gurson's approach, which were coupled to different plastic instabilities criteria, based on Bifurcation Analysis. The numerical implementation of these models was carried within the implicit finite element code Abaqus/Standard. With regard to the damage models, several integration schemes were tested to analyze their performance and robustness when the behavior exhibits softening effect. The approach combining the Gurson's damage model and necking criteria has been applied for the prediction of formability limits of several metallic materials. The obtained results allowed establishing a theoretical and numerical classification between the necking criteria used in this work

L’opérateur du p-Laplacien non local, l’équation d’évolution et la régularisation variationnelle associées régies par un noyau donné ont des applications dans divers domaines de la science et de l’ingénierie. En particulier, ils sont devenus des outils modernes pour le traitement massif des données (y compris les signaux, les images, la géométrie) et dans les tâches d’apprentissage automatique telles que la classification. En pratique, cependant, ces modèles sont implémentés sous forme discrète (en espace et en temps, ou en espace pour la régularisation variationnelle) comme approximation numérique d’un problème continu, où le noyau est remplacé par la matrice d’adjacence d’un graphe. Pourtant, peu de résultats sur la consistence de ces discrétisations sont disponibles. En particulier, il est largement ouvert de déterminer quand les solutions de l’équation d’évolution ou du problème variationnel des tâches basées sur des graphes convergent (dans un sens approprié) à mesure que le nombre de sommets augmente, vers un objet bien défini dans le domaine continu, et si oui, à quelle vitesse. Dans ce manuscrit, nous posons les bases pour aborder ces questions.En combinant des outils de la théorie des graphes, de l’analyse convexe, de la théorie des semi- groupes non linéaires et des équations d’évolution, nous interprétons rigoureusement la limite continue du problème d’évolution et du problème variationnel du p-Laplacien discrets sur graphes. Plus précisé- ment, nous considérons une suite de graphes (déterministes) convergeant vers un objet connu sous le nom de graphon. Si les problèmes d’évolution et variationnel associés au p-Laplacien continu non local sont discrétisés de manière appropriée sur cette suite de graphes, nous montrons que la suite des solutions des problèmes discrets converge vers la solution du problème continu régi par le graphon, lorsque le nombre de sommets tend vers l’infini. Ce faisant, nous fournissons des bornes d’erreur/consistance.Cela permet à son tour d’établir les taux de convergence pour différents modèles de graphes. En parti- culier, nous mettons en exergue le rôle de la géométrie/régularité des graphons. Pour les séquences de graphes aléatoires, en utilisant des inégalités de déviation (concentration), nous fournissons des taux de convergence nonasymptotiques en probabilité et présentons les différents régimes en fonction de p, de la régularité du graphon et des données initiales
The non-local p-Laplacian operator, the associated evolution equation and variational regularization, governed by a given kernel, have applications in various areas of science and engineering. In particular, they are modern tools for massive data processing (including signals, images, geometry), and machine learning tasks such as classification. In practice, however, these models are implemented in discrete form (in space and time, or in space for variational regularization) as a numerical approximation to a continuous problem, where the kernel is replaced by an adjacency matrix of a graph. Yet, few results on the consistency of these discretization are available. In particular it is largely open to determine when do the solutions of either the evolution equation or the variational problem of graph-based tasks converge (in an appropriate sense), as the number of vertices increases, to a well-defined object in the continuum setting, and if yes, at which rate. In this manuscript, we lay the foundations to address these questions.Combining tools from graph theory, convex analysis, nonlinear semigroup theory and evolution equa- tions, we give a rigorous interpretation to the continuous limit of the discrete nonlocal p-Laplacian evolution and variational problems on graphs. More specifically, we consider a sequence of (determin- istic) graphs converging to a so-called limit object known as the graphon. If the continuous p-Laplacian evolution and variational problems are properly discretized on this graph sequence, we prove that the solutions of the sequence of discrete problems converge to the solution of the continuous problem governed by the graphon, as the number of graph vertices grows to infinity. Along the way, we provide a consistency/error bounds. In turn, this allows to establish the convergence rates for different graph models. In particular, we highlight the role of the graphon geometry/regularity. For random graph se- quences, using sharp deviation inequalities, we deliver nonasymptotic convergence rates in probability and exhibit the different regimes depending on p, the regularity of the graphon and the initial data

Comprendre, prédire et finalement retarder la transition vers la turbulence dans les écoulements sont d'importants problèmes posés aux scientifiques depuis les travaux pionniers d'Osborne Reynolds en 1883. Ces questions ont été principalement adressées à l'aide de la théorie des instabilités hydrodynamiques. A cause des ressources informatiques limitées, les analyses de stabilité linéaire reposent essentiellement sur d'importantes hypothèses simplificatrices telles que celle d'un écoulement parallèle. Dans ce cadre, connu sous le nom de stabilité locale, seule la stabilité d'écoulement ayant un fort intérêt académique mais relativement peu d'applications pratiques a pu être étudiée. Néanmoins, au cours de la décennie passée, l'hypothèse d'écoulement parallèle a été relaxée au profit de celle d'un écoulement bidimensionnel conduisant alors à ce que l'on appelle la stabilité globale. Ce nouveau cadre permet alors d'étudier les mécanismes d'instabilité et de transition ayant lieu au sein d'écoulements plus réalistes. Plus particulièrement, la stabilité d'écoulements fortement non-parallèles pouvant présenter des décollements massifs, une caractéristique fréquente dans les écoulements d'intérêt industriel, peut maintenant être étudiée. De plus, avec l'accroissement constant des moyens de calcul et le développement de nouveaux algorithmes de recherche de valeurs propres itératifs, il est aujourd'hui possible d'étudier la stabilité d'écoulements pleinement tridimensionnels pour lesquels aucune hypothèse simplificatrice n'est alors nécessaire. Dans la continuité des travaux présentés par Bagheri et al. en 2008, le but de la présente thèse est de développer les outils nécessaires à l'analyse de la stabilité d'écoulements 3D. Trois écoulements ont été choisis afin d'illustrer les nouvelles capacités de compréhension apportées par l'analyse de la stabilité globale appliquée à des écoulements tridimensionnels réels : i) l'écoulement au sein d'une cavité entraînée 3D, ii) l'écoulement se développant dans un tuyau sténosé, et enfin iii) l'écoulement de couche limite se développant au passage d'une rugosité cylindrique montée sur une plaque plane. Chacun de ces écoulements a différentes applications pratiques allant d'un intérêt purement académique à une application biomédicale et aérodynamique. Ce choix d'écoulements nous permet également d'illustrer les différents aspects des outils développés au cours de cette thèse ainsi que les limitations qui leur sont inhérentes
Understanding, predicting and eventually delaying transition to turbulence in fluid flows have been challenging issues for scientists ever since the pioneering work of Osborne Reynolds in 1883. These problems have mostly been addressed using the hydrodynamic linear stability theory. Yet, due to limited computational resources, linear stability analyses have essentially relied until recently on strong simplification hypotheses such as the “parallel flow” assumption. In this framework, known as “local stability theory”, only the stability of flows with strong academic interest but limited practical applications can be investigated. However, over the course of the past decade, simplification hypotheses have been relaxed from the “parallel flow” assumption to a two-dimensionality assumption of the flow resulting in what is now known as the “global stability theory”. This new framework allows one to investigate the instability and transition mechanisms taking place in more realistic flows. More particularly, the stability of strongly non-parallel flows exhibiting separation, a common feature of numerous flows of practical interest, can now be studied. Moreover, with the continuous increase of computational power available and the development of new iterative eigenvalue algorithms, investigating the global stability of fully three-dimensional flows, for which no simplification hypothesis is necessary, is now feasible. Following the work presented in 2008 by Bagheri et al., the aim of the present thesis is thus to develop the tools mandatory to investigate the stability of 3D flows. Three flow configurations have been chosen to illustrate the new investigation capabilities brought by global stability theory when it is applied to realistic three-dimensional flows: i) the flow within a cuboid lid-driven cavity, ii) the flow within an asymmetric stenotic pipe and iii) the boundary layer flow developing over a cylindrical roughness element mounted on a flat plate. Each of these flows have different practical applications ranging from purely academic interests to biomedical and aerodynamical applications. They also allow us to put in the limelight different aspects and possible limitations of the various tools developed during this PhD thesis

L'équation de transport, dans le régime fortement collisionnel admet une limite asymptotique de diffusion. Les discrétisations angulaires comme la méthode des ordonnées discrètes Sn où le développement tronqué en harmonique sphérique Pn préservent aussi cette limite de diffusion. Par conséquent, il est intéressant de construire pour de tels systèmes des méthodes de volumes finis sur maillages non structurés qui préservent cette limite de diffusion pour des grilles grossières. En effet, ces modèles peuvent être couplés avec des codes hydrodynamiques Lagrangiens qui génèrent des maillages très tordus. Pour commencer, on considère la discrétisation angulaire la plus simple de l'équation de transport appelée le modèle P1. Après une rapide introduction sur les méthodes 1D, on commence par modifier le schéma acoustique en dimension deux avec la méthode de Jin-Levermore. Le schéma ainsi obtenu n'est pas convergent dans le régime de diffusion car le schéma de diffusion valide n'est pas consistant sur maillages non structurés. Pour résoudre ce problème, on a proposé de nouvelles méthodes valides sur maillages non structurés. Ces méthodes sont basées sur un autre formalisme des méthodes de volumes finis ou les flux sont localisés aux interfaces, couplé avec la méthode de Jin-Levermore. On obtient deux schémas convergents qui dérivent sur les schémas asymptotic preserving 1D. Le schéma limite de diffusion obtenu est un nouveau schéma pour lequel on a donné une preuve de convergence. Dans un second temps, on a proposé une extension du travail réalisé pour le modèle P1 dans le cadre des discrétisations angulaires d'ordres élevés. Pour obtenir une discrétisation asymptotic preserving pour ces modèles on a utilisé une décomposition entre la discrétisation angulaire de premier ordre et les discrétisations angulaires d'ordres supérieurs. Enfin on a étudié la discrétisation du problème d'absorption/émission présent en transfert radiatif ainsi que la discrétisation du modèle non linéaire M1. L'approximation du modèle M1 est basé sur un couplage entre un schéma Lagrange+projection pour une reformulation du modèle M1 et la méthode de Jin-Levermore. La méthode numérique obtenue préserve la limite asymptotique, l'inégalité d'entropie et le principe du maximum associé au système sur maillages non structurés.

Orientador: Petronio Pulino
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica
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Resumo: O Método dos Elementos Finitos representou nos últimos anos uma ferramenta fundamental no estudo de problemas de contorno. A evolução desde sua formulação fundamental a partir do Método de Galerkin clássico até sua versão com refinamento hp, se tornou na base dos métodos numéricos mais avançados como é o Método de Galerkin Descontinuo. O Método dos Elementos Finitos de Alta Ordem juntamente com os Métodos Espectrais usados na obtenção de soluções numéricas para problemas de contorno com fronteira descontinua, serão nosso objeto de estudo nesta Dissertação. Desde sua formulação matemática fundamental, por intermédio da escolha apropriada das funções hierárquicas que compõem os espaços de aproximação, assim como a montagem dos sistemas lineares locais e sua respectiva utilização no sistema linear global esparso, cuja solução é obtida pelo método iterativo de Gradiente Conjugado usando diversos Precondicionadores, será o caminho a seguir
Abstract: The Finite Element Method developed in the last decades has been the most important tool in the study of Boundary Value Problems. Your evolution from its fundamental formulation using the Galerkin Method to the hp-adaptive finite element methods (hp-FEM), provided the necessary foundation for more advanced Numerical Methods like the Galerkin Discontinuous Method. The Finite Element Method of Higher Order, together with the Spectral formulation as a numerical method to solve Boundary Problems with Discontinuous Boundary, is the objective of study to this dissertation. The fundamental mathematical formulation of the finite element methods, passing through of to choose of hierarchical basis functions, also the assembly of local linear systems and it posteriorly use to construct a Sparse Linear System, whose solution is obtained for an iterative Preconditioner Gradient Method
Mestrado
Matematica Aplicada
Mestre em Matemática Aplicada

Cette thèse est essentiellement composée de deux parties. Dans la première partie, on étudie le système d'Euler-Maxwell. En utilisant la méthode d'intégration de l'énergie classique, on montre l'existence et l'unicité de solutions régulières du système avec données initiales petites. Ensuite, on étudie la limite de relaxation en montrant que, le sytème d'Euler-Maxwell converge vers les équations de dérive-diffusion quand le temps de relaxation tend vers zéro. Dans la deuxième partie, on cherche la contrôlabilité et l'observabilité exactes frontières de systèmes hyperboliques quasi-linéaires dans un réseau du type d'arbre. On établit des résultats d'existences de la contrôlabilité et l'observabilité par des méthodes constructives qui sont basées sur la théorie de la solution C1 semi-globale du système hyperbolique quasi-linéaire du premier ordre avec conditions initiales et frontières. Ensuite, on trouve des dualités de la contrôlabilité et l'observabilité.

Mieux comprendre les spécificités de la dynamique des couches limites à grand nombre de Reynolds malgré les contraintes métrologiques et son coût de simulation numérique est crucial. A titre d'exemple, cette dynamique peut déterminer plus de la moitié de la traînée d'un avion en croisière. Décrire la turbulence pariétale peut guider la résolution numérique d'une partie des fluctuations à un coût maîtrisé par des stratégies WMLES (simulation des grandes échelles avec modèle de paroi). Les présentes analyses physiques de couches limites turbulentes incompressibles à gradient de pression nul et à grand nombre de Reynolds s'appuient sur des simulations numériques avancées. Après validation d'une base de données, le frottement moyen pariétal est décomposé selon l'identité FIK (Fukagata et al. (2002)), dont l'application malgré le développement spatial est discutée. Une analyse spectrale montre que les grandes échelles (\lambda_x > \delta) contribuent à environ la moitié du frottement vers Re_\theta = 10^4. Les limitations de l'identité FIK motivent la dérivation d'une décomposition physique de la génération du frottement dont le comportement asymptotique est alors relié à la production d'énergie cinétique turbulente dans la zone logarithmique. Pour mieux reconstruire les spectres spatiaux, une nouvelle méthode d'estimation de la vitesse de convection turbulente en fonction de la longueur d'onde des fluctuations, adaptée au développement spatial et à des signaux temporels de durée finie, est dérivée, interprétée et évaluée à Re_\theta = 13000. Certaines des conclusions éclairent des modifications d'une stratégie WMLES, le mode III de la méthode ZDES
Better understanding the specificities of the dynamics of high-Reynolds number boundary layers despite metrological constraints and its numerical simulation cost is crucial. For instance, this dynamics can determine more than half of the drag of a cruising aircraft. Describing wall turbulence can guide the numerical resolution of some of the fluctuations at a limited cost by WMLES strategies (wall-modelled large eddy simulation). The present physical analyses of zero-pressure gradient incompressible turbulent boundary layers at high Reynolds number rely on advanced numerical simulations. After validating a database, mean skin friction is decomposed by means of the FIK identity (Fukagata et al. (2002)), whose application despite the spatial growth is discussed. A spectral analysis shows that the large scales (\lambda_x > \delta) contribute approximately half of the friction near Re_\theta = 10^4. The limitations of the FIK identity motivate the derivation of a physical decomposition of the generation of friction whose asymptotic behaviour is then related to turbulent kinetic energy production in the logarithmic layer. In order to better reconstruct spatial spectra, a new method to estimate the turbulent convection velocity as a function of the wavelength of the fluctuations, adapted to spatial growth and to temporal signals of finite duration, is derived, interpreted, and assessed at Re_\theta = 13000. Some of the conclusions enlighten modifications to a WMLES strategy, mode III of the ZDES method

Cette thèse étudie les aspects mathématiques et numériques de phénomènes d'ondes dans les plasmas magnétiques. La réflectométrie, une technique de sonde des plasmas de fusion, est modélisée par les équations de Maxwell. Le tenseur de permittivité présente dans ce modèle des valeurs propres ainsi que des termes diagonaux qui s'annulent. La relation de dispersion met en évidence deux phénomènes cruciaux : coupures et résonances, lorsque le nombre d'onde s'annule ou tend vers l'infini. La partie I rassemble les résultats numériques. La grande nouveauté réside dans la définition d'une solution résonante. En effet, à cause des coefficients s'annulant continument en changeant de signe, la solution peut être singulière, i.e. avoir une composante non intégrable. Cependant, grâce au principe d'absorption limite, une solution résonante est explicitement définie comme la limite de solutions intégrables du problème régularisé. L'expression théorique de la singularité est validée par des tests numériques du passage à la limite. La partie II concerne l'approximation numérique. Elle comprend la mise en place d'une nouvelle méthode numérique adaptée aux coefficients réguliers. Celle-ci est basée sur la formulation variationnelle Ultra Faible mais nécessite des fonctions de base spécifiques, construites comme approximations locales du problème adjoint. L'analyse de convergence est effectuée en dimension un, en dimension deux la construction des fonctions de base et leur propriété d'interpolation sont détaillées. La méthode d'ordre élevé obtenue permet de simuler le phénomène de coupure tandis que simuler le phénomène de résonance en dimension deux reste un défi.

Modern turbomachines are designed towards thinner, lighter and highly loaded blades. This gives rise to increased sensitivity to flow induced vibrations such as flutter, which leads to structure failure in a short period of time if not sufficiently damped. Although numerical tools are more and more reliable, flutter prediction still depends on a large degree on simplified models. In addition, the critical nature of flutter, resulting in poor welldocumented real cases in the open literature, and the lack of experimental database typical of engine flows make its apprehension even more challenging. In that context, the present thesis is dedicated to study flutter in recent turbines through aerodynamic analysis of subsonic or supersonic flows in response to a prescribed vibratory mode of the structure. The objective is to highlight some mechanisms potentially responsible for flutter in order to be in better position when designing blades. The strategy consists in leading both experimental and numerical investigations. The experimental part is based on a worldwide unique annular turbine sector cascade employed for measuring the aeroelastic response by means of the aerodynamic influence coefficient technique. The cascade comprises seven low pressure gas turbine blades one of which can oscillate in a controlled way as a rigid body. Aeroelastic responses are measured at various mechanical and aerodynamic parameters: pure and combined modeshapes, reduced frequency, Mach number, incidence angle. In addition to turbulence level measurements, the database aims at assessing the influence of these parameters on the aerodynamic damping, at validating the linear combination principle and at providing input for numerical tools. The numerical part is based on unsteady computations linearized in the frequency domain and performed in the traveling wave mode. The focus is put on two industrial space turbines: 2D computations are performed on an integrally bladed disk, also called blisk; its very low viscous material damping results in complex motions with combined modes and extremely high reduced frequency. The blisk operates at low subsonic conditions without strong non-linearities. Although the blades have been predicted aeroelastically stable, an original methodology based on elementary decompositions of the blade motion is presented to identify the destabilizing movements. The results suggest that the so-called classical flutter is surprisingly prone to occur. Moreover, the aerodynamic damping has been found extremely sensitive to the interblade phase angle and cut-on/cut-off conditions.* 3D computations are then performed on a supersonic turbine, which features shockwaves and boundary layer separation. In contrast, the blade motion is of elementary nature, i.e. purely axial. The blades have been predicted aeroelastically unstable for backward traveling waves and stable for forward traveling waves. The low reduced frequencies allow quasi-steady analysis, which still account for flutter mechanisms: the shock wave motion establishes the boundary between stable and unstable configurations.

In der Arbeit wird das asymptotische Verhalten von geeignet normierten und zentrierten Summen von Zufallsgrößen untersucht, die entweder unabhängig sind oder im Falle der Abhängigkeit als Martingaldifferenzfolge oder stark multiplikatives System auftreten. Neben der klassischen Summationstheorie werden die Limitierungsverfahren mit einer unendlichen Summationsmatrix oder einer angepaßten Folge von Gewichtsfunktionen betrachtet. Es werden die Methode der charakteristischen Funktionen und besonders die direkte Methode der konjugierten Verteilungsfunktionen weiterentwickelt, um quantitative Aussagen über gleichmäßige und ungleichmäßige Restgliedabschätzungen in zentralen Grenzwertsatz zu beweisen. Die Untersuchungen werden dabei in der Lp-Metrik, 1
In the work the asymptotic behavior of suitably centered and normalized sums of random variables is investigated, which are either independent or occur in the case of dependence as a sequence of martingale differences or a strongly multiplicative system. In addition to the classical theory of summation limiting processes are considered with an infinite summation matrix or an adapted sequence of weighting functions. It will be further developed the method of characteristic functions, and especially the direct method of the conjugate distribution functions to prove quantitative statements about uniform and non-uniform error estimates of the remainder term in central limit theorem. The investigations are realized in the Lp metric, 1

This thesis aims at analysing the predictive capabilities of statistical URANS and hybrid RANS-LES methods to model complex flows at high Reynolds numbers and carrying out a physical analysis of the near-region turbulence and coherent structures. This study handles configurations included in the European research programmes ATAAC (Advanced Turbulent Simulation for Aerodynamics Application Challenges) and TFAST (Transition Location Effect on Shock Wave Boundary Layer Interaction). First, the detached flow in a configuration of a tandem of cylinders, positionned behind one another, is investigated at Reynolds number 166000. A static case, corresponding to the layout of the support of a landing gear, is initially considered. The fluid-structure interaction is then studied in a dynamic case where the downstream cylinder, situated in the wake of the upstream one, is given one degree of freedom in translation in the crosswise direction. A parametric study of the structural parameters is carried out to identify the various regimes of interaction. Secondly, the physics of the transonic buffet is studied by means of time-frequency analysis and proper orthogonal decomposition (POD), in the Mach number range 0.70–0.75. The interactions between the main shock wave, the alternately detached boundary layer and the vortices developing in the wake are analysed. A stochastic forcing, based on reinjection of synthetic turbulence in the transport equations of kinetic energy and dissipation rate by using POD reconstruction, has been introduced in the so-called organised-eddy simulation (OES) approach. This method introduces an upscale turbulence modelling, acting as an eddy-blocking mechanism able to capture thin shear-layer and turbulent/non-turbulent interfaces around the body. This method highly improves the aerodynamic forces prediction and opens new ensemble-averaged approaches able to model the coherent and random processes at high Reynolds number. Finally, the shock-wave/boundary-layer interaction (SWBLI) is investigated in the case of an oblique shock wave at Mach number 1.7 in order to contribute to the so-called "laminar wing design" studies at European level. The performance of statistical URANS and hybrid RANS-LES models is analysed with comparison, with experimental results, of integral boundary-layer values (displacement and momentum thicknesses) and wall quantities (friction coefficient). The influence of a transitional boundary layer on the SWBLI is featured.

Des simulations numériques directes (DNS) sont calculées afin d’étudier le processus complet de transition laminaire-turbulent d’une couche limite qui se développe sur une surface concave. On constate que l’écoulement traversant une telle géométrie est susceptible de développer des instabilités centrifuges sous forme de tourbillons de Görtler. La transition est déclenchée au moyen d’éléments de rugosité de paroi qui sont également utilisés pour prédéfinir la longueur d’onde transversale des tourbillons de Görtler. Les différentes régions rencontrées dans le processus de transition, c’est-à-dire linéaire, non-linéaire, transition et complètement turbulente, sont identifiées et caractérisées. Les instabilités primaires et secondaires (sinueux et variqueux) sont également identifiées et analysées. Des études paramétriques montrant l’effet de plusieurs paramètres physiques (le rayon de courbure, la longueur d’onde des tourbillons, l’amplitude et la localisation de la perturbation, et la géométrie des éléments de rugosité de paroi) sur le point de départ de la transition sont présentées. De plus, des analyses thermiques sont menées afin d’étudier la modification de la couche limite thermique produite par le mouvement rotatif des tourbillons de Görtler. L’évolution du transfert thermique de la surface est étudiée et on constate qu’il est considérablement augmenté dans la région non-linéaire dépassant les valeurs de la région turbulente. On constate également que l’analogie de Reynolds entre la quantité de mouvement dans le sens de l’écoulement et le transfert de chaleur est suivie tout au long du processus de transition
Direct numerical simulations (DNS) are computed in order to study the complete laminar-to-turbulent transition process of a boundary layer developing over a concave surface. It is found that the flow passing through such geometry is prone to develop centrifugal instabilities in the form of Görtler vortices. Transition is triggered by means of wall-roughness elements that are also utilized to preset the spanwise wavelength of the Görtler vortices. The different regions encountered in the transition process, i.e. linear, nonlinear, transition, and fully turbulent, are identified and characterized. Primary and secondary (varicose and sinuous) instabilities are identified and analyzed as well. Parametric studies showing the effect of several physical parameters (radius of curvature, the vortices wave-length, the perturbation amplitude and streamwise location, and the wall-roughness perturbation geometry) on the transition starting point are presented. Furthermore, thermal analyses are conducted in order to study the modification of the thermal boundary layer due to the Görtler vortices swirl motion. The streamwise evolution of the surface heat transfer is investigated finding that it is considerably enhanced in the non-linear region surpassing the turbulence-region values. It is also found that the Reynolds analogy between streamwise-momentum and heat transfer is followed throughout the whole transition process

Ette thèse traite d’écoulements de fluides à seuil (ou viscoplastiques) en milieu confiné. Les difficultés analytiques et numériques sont dues à la multivaluation du tenseur des contraintes dans les zones plastiques ainsi qu’à la non-différentiabilité du problème de minimisation associé. Cette thèse s’articule en deux parties.Dans un premier temps, des simulations numériques parallèles très précises à l’aide d’algorithmes de dualité ont été effectuées. Elles ont permis de retrouver des résultats observés expérimentalement dont l’existence d’une ligne de glissement pour l’écoulement au dessus d’un obstacle et le caractère quasi-Poiseuille de la vitesse au-delà de cette ligne. Par ailleurs, la théorie de couche limite viscoplastique définie par Oldroyd (1947, à nombre de Bingham asymptotiquement grand) a été revisitée à nombre de Bingham modéré en milieu confiné. L’étude a mis en œuvre des allers-retours entre ces simulations et les expériences physiques de Luu et al. d’IRSTEA ainsi qu’une dérivation théorique. L’approximation de couche limite est vérifiée dans une certaine mesure à l’intérieur de la cavité. Une adaptation de la notion de couche limite viscoplastique est alors exhibée et permet d’étendre les scalings dérivés par Oldroyd (1947) et Balmforth et al. (J. of Fluid Mech, 2017). Ces scalings sont aussi généralisés au cas de la loi d’Herschel-Bulkley. Dans un second temps, on présente une analyse asymptotique des champs de vitesses et de contraintes pour des écoulements en faible épaisseur (ε). Un développement à l’ordre ε2 de la vitesse permet de trouver une équation de Reynolds à la même précision. Cette équation de Reynolds prolonge les résultats déjà existants dans le cadre newtonien, d’une part et dans le cadre fluide à seuil avec une surface libre, d’autre part
This thesis is devoted to the flow of yield stress (or viscoplastic) fluids in pipes.Analytical and numerical difficulties lie in the multivaluation of the stress tensor in the plastic regions and in the non-differentiability of the associated minimization problem. This manuscript is organized following two main axes.First, very accurate numerical simulations were carried out using duality methods and parallel multifrontal solvers. Thus, experimental observations were recovered, namely the existence of a slip line for the flow over an obstacle and the Poiseuille-like behaviour of the velocity above this line. Moreover, the viscoplastic boundary layer theory defined by Oldroyd (1947 at high Bingham numbers) was revisited at moderate Bingham numbers in confined areas. This study provided an opportunity to go back and forth between these simulations and the physical measures of Luu et al. from IRSTEA and to perform a theoretical derivation. The boundary layer approximation is valid up to a certain extent in the cavity. An adaptation of the viscoplastic boundary layer definition is then given and allows to generalize the scalings shown by Oldroyd (1947) and Balmforth et al. (JFM 2017). These scalings are also generalized to the Herschel-Bulkley case. Then, an asymptotic analysis of the velocity and stress fields for thin layer (ε) flows is presented. A velocity development up to ε2 lets us find a Reynolds equation of same accuracy. This Reynolds equation extends the already existing results, on the one hand in the newtonian case and on the second hand for free surface flows

Cette thèse présente quelques résultats dans le domaine des équations aux dérivées partielles stochastiques. Une majeure partie d'entre eux concerne l'étude de limites diffusives de modèles cinétiques perturbés par un terme aléatoire. On présente également un résultat de régularité pour une classe d'équations aux dérivées partielles stochastiques ainsi qu'un résultat d'existence et d'unicité de mesures invariantes pour une équation de Fokker-Planck stochastique. Dans un premier temps, on présente trois travaux d'approximation-diffusion dans le contexte stochastique. Le premier s'intéresse au cas d'une équation cinétique avec opérateur de relaxation linéaire dont l'équilibre des vitesses a un comportement de type puissance à l'infini. L'équation est perturbée par un processus Markovien. Cela donne lieu à une limite fluide stochastique fractionnaire. Les deux autres résultats concernent l'étude de l'équation de transfert radiatif qui est un problème cinétique non linéaire. L'équation est bruitée dans un premier temps avec un processus de Wiener cylindrique et dans un second temps par un processus Markovien. Dans les deux cas, on obtient à la limite une équation de Rosseland stochastique. Dans la suite, on présente un résultat de régularité pour les équations aux dérivées partielles quasi-linéaires de type parabolique dont la partie aléatoire est gouvernée par un processus de Wiener cylindrique. Enfin, on étudie une équation de Fokker-Planck qui présente un terme de forçage aléatoire régi par un processus de Wiener cylindrique. On prouve d'une part l'existence et l'unicité des solutions de ce problème et d'autre part l'existence et l'unicité de mesures invariantes pour la dynamique de cette équation
This thesis presents several results about stochastic partial differential equations. The main subject is the study of diffusive limits of kinetic models perturbed with a random term. We also present a result about the regularity of a class of stochastic partial differential equations and a result of existence and uniqueness of invariant measures for a stochastic Fokker-Planck equation.First, we give three results of approximation-diffusion in a stochastic context. The first one deals with the case of a kinetic equation with a linear operator of relaxation whose velocity equilibrium has a power tail distribution at ininity. The equation is perturbed with a Markovian process. This gives rise to a stochastic fluid fractional limit. The two remaining results consider the case of the radiative transfer equation which is a non-linear kinetic equation. The equation is perturbed successively with a cylindrical Wiener process and with a Markovian process. In both cases, we are led to a stochastic Rosseland fluid limit.Then, we introduce a result of regularity for a class of quasilinear stochastic partial differential equations of parabolic type whose random term is driven by a cylindrical Wiener process.Finally, we study a Fokker-Planck equation with a noisy force governed by a cylindrical Wiener process. We prove existence and uniqueness of solutions to the problem and then existence and uniqueness of invariant measures to the equation

Research Doctorate - Doctor of Philosophy (PhD)
Failure of pavements associated with moving repeated traffic loads are typically caused by gradual deterioration rather than sudden collapse. This type of failure behaviour is mostly due to the accumulation of plastic strains under repeated loadings. There are different from the geotechnical engineering solution of the stability problems of soil masses under simple loading programs. It can be shown theoretically that there is a load magnitude below which a protective residual stress will develop in the road, and above which the pavement will undergo an incremental failure. This load is known as the 'shakedown limit load' and the protective residual stresses associated with this shakedown limit load are the optimal residual stresses for the life of the pavement. This thesis thus aims at describing a general numerical method for the shakedown solutions of pavements subjected to repeated moving traffic loads. To achieve this goal, a novel numerical formulation based on Bleich-Melan shakedown theorem is developed by making use of finite element techniques and mathematical programming. The proposed numerical procedure can be used to solve for the shakedown limit load and the associated developed residual stresses for both two- and three- dimensional continua under complex load distribution and boundary geometry, layered material, and inhomogeneous soil deposit.

Risk analysis is an important step in the design of rock slopes in open pit mining. Risk is defined as the product of the probability of slope failure and the consequences of the failure, and is generally evaluated in terms of safety and economic risk. Most of the risk analysis carried out at present is based on the use of Limit equilibrium (LE) techniques in evaluating the probability of failure (POF) of the slopes. The approach typically makes use of full Monte Carlo simulations of the Limit Equilibrium models, with all uncertain variables randomly varied. The number of required simulations is generally over a thousand, at times as high as 20000, in order to produce statistically valid results of the POF. Such an approach is clearly not amenable to the use of numerical modelling programs due to the high computational effort required, hence the major use of LE methods. This project explores the impact of using numerical modelling (with the program PHASE 2) instead of the traditional LE techniques (using the program SLIDE) in evaluating the probability of slope failure. The difference in the overall assessed risk, in terms of economic impact, for the mining operation was then evaluated. Instead of full Monte Carlo simulations within the numerical analysis stability model, an alternative method called the Response Surface Methodology was used for the probabilistic analysis. The use of numerical modelling in the assessment of risk results in a significant difference in the assessed risk. LE models tend to be conservative in terms of stability and POF results. However, they give lower estimates of failure volumes than the numerical models. As a result, the assessed risk from LE models can be lower or higher than that from numerical analyses. A case study of an open pit mine was used to validate these findings. The effect of using 2D plane strain models instead of 3D analysis was also investigated. Numerical models built with FLAC were compared with FLAC3D models. Circular and elliptical pits were considered, with varying radii of curvature. The results showed that for circular pits 2D assumptions are adequate if the radius of curvature is sufficiently large or, for an elliptical pit, if the distance from the small radius ends is sufficiently large. In situations other than these the 2D models give conservative estimates of stability and probability of failure. A detailed comparison of failure volumes was difficult due to the fact that the 2D models do not give the out of plane extent of the failure. However, using simplistic assumptions, the overall assessed risk from FLAC and FLAC3D is shown to be different.

碩士
國立臺中教育大學
數學教育學系
96
Due to Gerretsen - inequality always can solve many tough question in proving effectively. So, this research use method of primary substitution operation and arithmetic average greater than geometry average to conclude Gerretsen -inequality as below: 16Rr-5r^2le s^2 le 4R^2+4Rr+3r^2. And then, using the numerical analysis to conclude the deviation for upper and under limit of Gerretsen - inequality. The research found : 1.Gerretsen- inequality only use the character of arithmetic average greater than geometry average to prove the inequality . The rest is just routine primary substitution operation. 2.Using Excel to calculate upper limit of Gerretsen - inequality, the difference of inequality inclines to 0.33333. 3.Using Excel to calculate under limit of Gerretsen - inequality, the difference of inequality inclines to 0.33333.

Conservation of cultural heritage against seismic risk constitutes one of the greatest challenges of the scientific community, which is engaged in refining efficient solutions for practitioners as well as theoretical mechanical models. The response of distinctive architectural elements, like arches, domes and vaults, has attracted the interest of historical scientists, but still today comprehensive and general formulations lack a full dynamic perspective. Among architectural elements, the arch is certainly an iconography of mechanics applied to architecture. Indeed, extensive investigations are available in the literature on the response of circular arches to vertical loads, and a few full dynamic models for horizontal acceleration load can be found as well. Pointed arches, even though spread in seismic prone areas, received much less of interest. An amazing case study on which working on, consisting in a giant pointed arched system located in Afghanistan, motivated a seminal interest on the issue, toughened by the lacking literature. For these reasons, this dissertation reports on a parametric analysis of the vulnerability of pointed arches made of two circular arcs, which is the simplest thinkable pointed arch. The analysis considers variations of arch slenderness and sharpness that result from different positions of centres of circular arcs. Firstly, the arch is addressed as a rigid macro-block system, and limit analysis with the kinematic approach is exploited to determine the collapse acceleration through Non-Linear Programming optimisation. The pattern of hinges at collapse differs considerably from the one that occurs for circular arches. Moreover, the effect of arch slenderness on collapse accelerations turns out to be significantly conditioned by sharpness. Acceleration necessary to initiate motion grows with the rise, as opposed to what occurs for circular shapes. Dynamic behaviour of pointed arches for rectangular shaped and harmonic inputs are investigated as well transforming arch mechanisms into four-bar linkages. Systematic integration of the non-linear form of the distinctive ODE of the problem revealed that failure during the second half cycle of motion occurs for most of the profiles. Such a trend would have never been tackled in the framework of linearized motion, which however provides more conservative estimations. Moreover, failure during the second half cycle of motion for harmonic inputs, especially for low and medium frequencies, is found to be deeply influenced by the adopted impact model and, more importantly, by the position of hinges. A dedicated sensitivity analysis validates the procedure predicting failure of circular arches as a particular case of pointed. Considering also a micro-block approach, a wide experimental campaign addressed the equivalent static and full dynamic response of a set of 11 reduced scale model of pointed arches made of prismatic Autoclaved Aerated Concrete blocks. Global geometric characteristics of models are the same considered in the macro-block approach. Tilt tests and shake table tests uncovered the inherent sliding vulnerability of these profiles. Thus, a kinematic model capable of considering sliding, independently from the adopted friction coefficient can be represented by a two-bar model hinged at the ground and connected by a slider; outcomes tackle global sliding mechanism of thick and sharp profiles. A similar aim justified the use of the Distinct Element Method through the commercial code 3DEC. As for equivalent static tests, the range of friction coefficients necessary to initiate a hinging mechanism vary with variation in geometry of the profile, and most important, for thick and sharp profiles, the rocking mechanism can hardly be activated unless a perfect hinging interface is not assumed. Regarding dynamic tests, harmonic pulses with frequency ranging between 2Hz and 10Hz have been considered and results of analytical, numerical and experimental models have been compared. Given the stated vulnerability of pointed arches to crown sliding, the four bar linkage model will always be lacking a fundamental aspect, especially for sharp and stocky profiles subjected to high-frequency inputs. Future investigations should address vulnerability to complete time histories, and in a probabilistic framework, sliding phenomena when overloading is considered and 3D structures.

A research report submitted to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, Johannesburg, in partial fulfilment of the requirements for the degree of Master of Science in Engineering
The planning of open pit mines, and road and rail cuttings constitutes one of the activities usually undertaken by geotechnical engineers. However, this endeavour faces major challenges such as the correct design of rock slopes, the evaluation of their stability and the risk associated with them. Two main analytical methods are used in this process: the Limit Equilibrium (LE) analysis and the Numerical Modelling (NM) method; Slide and Phase2 programmes will be used respectively in this regard. Previous studies have shown some discrepancies between their results in assessing the probabilities of slope failure and the consequent economic risks. This research project aims to understand the reasons behind these divergences and possibly to find some ways of reducing them. To attain these objectives, a homogeneous slope model was adopted. It required a detailed validation study depending on the analytical method, such that the settings would carefully be calibrated to avoid any further misinterpretation of the results. For Phase2, attention was given with regard to the number of mesh elements and their type, and for Slide, the number of slices. In addition, for both methods, attention was given to the adequate distribution of rock mass variables, the adequate failure criterion, etc. Deterministic and probabilistic assessments were performed to better interpret the differences to be found from these methods and results. The response surface methodology (RSM) facilitated the probabilistic studies, to avoid the constraints of long computer run times and to ease the study of the influence of the rock mass parameters on the slope stability. For the considered model, 25 slices and 1500 mesh elements were found adequate to better assess the probability of failure (POF), while 1000 slices and 50 000 mesh elements provided results of estimated failure volumes. Well defined distributions of rock mass variables have proven indispensable to better assess the POF as well as the risk associated with the slope failure. Application of the Hoek Brown criterion resulted in the LE analysis predicting higher failure volumes than when Mohr Coulomb criterion was used. With the NM method, cares were taken not to under or overestimate the resultant failure volume when extracting the path of the failure surface. Deterministic assessments showed that the risk determined from NM analyses is not always greater than that from LE analyses. The addition of rock mass parameters not taken into account in LE analyses results does not sensibly influence the POF and the failure volume outcomes, but can in some conditions influence the behaviour of the outcome risk of slope failure. In essence, LE and NM methods can be relied on for probabilistic studies, or even for risk assessments conditioned by carefully setting the models, and in case of LE being adopted to assess the risk, it is recommended to introduce a multiplying factor for cases similar to those that have already been analysed.

The aim of this thesis is to investigate and develop different numerical methodologies for modeling the Dielectric Barrier discharge (DBD) plasma actuators for flow control purposes. Two different modeling approaches were considered; one based on Plasma-fluid model and the other based on a phenomenological model. A three component Plasma fluid model based on the transport equations of charged particles was implemented in this thesis in OpenFOAM, using several techniques to reduce the numerical issues. The coupled plasma-fluid problem involves wide range of length and time scales which make the numerical simulation difficult. Therefore, to obtain stable and accurate results in a reasonable computational run time, several numerical procedures were implemented including: semi-implicit treatment of coupling of Poisson equation and charge density equation, super-time-stepping and operator splitting algorithm. We examined our code for a constant positive voltage, testing for the dependency of the behavior of the current density to the selected numerical scheme. In addition, although there is no clear numerical or experimental benchmark case for DBD plasma actuator problem, the developed plasma solver was compared quantitively and qualitively with several numerical works in the literature. Afterward, the developed numerical methodology was used to explore the possibility of influencing the flow, with higher speed, using nano-second (NS) pulsed DBD plasma actuator. Therefore, the interaction of the transonic flow and actuation effects of DBD plasma actuator with nano second pulsed voltage was simulated. The effect of gas heating and body force was calculated by the plasma solver and was supplied into the gas dynamic solver for simulating the flow field. Moreover, the results of the plasma fluid model were used to develop an energy deposition model. It was shown that the energy deposition model is able to capture the main features of the effect of NS DBD plasma actuators correctly, with less computational time. It was also shown that fast energy transfer, from plasma to fluid, leads to the formation of micro-shock waves that modify locally the features of the transonic flow. Although the numerical efficiency of the plasma fluid model was improved, the computational cost of simulating the effect of DBD plasma actuator on a real scale flow situation was still high. Therefore, a simple model for plasma discharge and its effect on the flow was developed based on scaling of the thrust generated by DBD plasma actuators. The scaled thrust model correctly predicts the nonlinear dependency of the thrust produced and the applied voltage. These scales were then introduced into a simple phenomenological model to estimate and simulate the body force distribution generated by the plasma actuator. Although the model includes some experimental correlations, it does not need any fitting parameter. The model was validated with experimental results and showed better accuracy compared to previous plasma models. Using a simple phenomenological model that was developed here, a numerical study was conducted to investigate and compare the effect of steady and unsteady actuation for controlling the flow at relatively high Reynolds number. Firstly it was shown that the size of the time-averaged separation bubble is greatly reduced and the flow structure is sensitive to the frequency of burst modulation of DBD plasma actuators. The results also confirmed that in the case of unsteady actuation, the burst frequency and burst ratio are crucial parameters for influencing the capability of the actuators to control the flow. It was found that burst frequencies near the natural frequencies of the system were able to excite the flow structure in a resonance mode. This observation also confirmed that with proper frequencies of excitation, the flow structure can be well rearranged and the flow losses can be reduced. In the end, Plasma actuators were used for controlling the flow over the Coanda surface of the ACHEON nozzle. When the plasma actuator was used, it was possible to postpone separation of the flow and increase the deflection angle of the exit jet of the nozzle. To find the optimum position of the actuators, seven DBD actuators in forward forcing mode were placed over the Coanda surface considering the numerically obtained separation points. Results show that when the actuator is placed slightly before the separation point, enhanced thrust vectorizing with the use of DBD actuator is achievable. Preliminary results of the experiments agree with planned/foreseen deflection angle obtained from numerical computation.
O objetivo deste trabalho visa a investigação e desenvolvimento de diferentes métodos numéricos para modelação de actuadores a plasma de Descarga em Barreira Dieléctrica, (DBD), tendo em vista o controlo do escoamento na camada limite. Esta modelação numérica foi abordada de duas formas diferentes, uma baseada num modelo de “plasma-fluid” e outra fundamentada num modelo fenomenológico. Neste trabalho é usado um modelo “plasma-fluid” de três componentes que é baseado numa equação de transporte para as partículas electricamente carregadas. Este foi implementado no software OpenFOAM fazendo uso de diversas técnicas para minimização de problemas numéricos que ocorriam na resolução das equações. O cálculo de um problema com acoplamento entre plasma e fluido envolve uma gama diversa de escalas, tanto temporais como dimensionais, trata-se então de uma simulação numérica delicada. Como tal, e por forma a obter resultados estáveis e precisos num tempo de cálculo considerado razoável, foram implementados diversos procedimentos numéricos, tais como o tratamento semiimplícito do acoplamento da equação de Poisson com a equação da densidade de carga, o super-passo-tempo e ainda um algoritmo do tipo divisão de operador. Foi considerado o caso de uma diferença de potencial positiva, constante, e testada a dependência da densidade de corrente com os diferentes esquemas numéricos. Apesar de não existir atualmente uma base de dados, de tipo numérica ou experimental, com casos de teste para actuadores a plasma tipo DBD, o modelo computacional desenvolvido para calcular o plasma foi validado qualitativamente, bem como quantitativamente, usando os vários trabalhos numéricos disponíveis na literatura. Após esta validação inicial, a metodologia numérica desenvolvida foi utilizada para explorar a possibilidade de influenciar um escoamento de maior velocidade, através de actuadores a plasma tipo DBD com impulsos de tensão da ordem de nano-segundos (NS). Desta forma foi simulada a interacção entre um escoamento transónico e o efeito dos actuadores a plasma tipo DBD sobre o escoamento, usando pulsos de nano-segundos. O efeito térmico do gás, assim como a força resultante, foram calculados usando o modelo numérico para cálculo de plasmas desenvolvido neste trabalho. O resultado obtido é acoplado ao modelo de cálculo para a dinâmica de gases, o que torna possível simular as condições do escoamento resultante. Adicionalmente, os resultados do modelo de “plasma-fluid” foram reaproveitados para desenvolver um modelo de deposição de energia. Este demonstrou ter a capacidade de capturar correctamente as características principais do efeito de actuadores de plasma, de tipo NS-DBD, com um tempo de computação menor. Foi demonstrada que uma rápida transferência de energia, do plasma para o fluido, leva à formação de micro-ondas de choque que alteram localmente as características do escoamento transónico. Apesar da eficiência numérica do modelo de “plasma-fluid” ter sido melhorada, o seu custo computacional para a simulação de actuadores a plasma tipo DBD à escala real continua bastante elevado. Neste sentido, a partir de uma escala de propulsão gerada pelo actuador plasma DBD, foi desenvolvido um modelo mais simples para a descarga do plasma e para determinar os seus efeitos sobre o escoamento. O modelo inicial previa correctamente uma dependência não-linear entre a força propulsiva gerada e a diferença de potencial aplicada. Estas escalas foram então introduzidas num modelo fenomenológico mais simples para estimar, e simular, a distribuição de forças geradas pelo actuador a plasma. Apesar de o modelo incluir algumas correlações experimentais, este não requer qualquer parâmetro de afinação. O modelo foi validado com resultados experimentais, demonstrando melhores resultados quando comparado com outros modelos de plasma . Utilizando um modelo fenomenológico simplificado, que foi desenvolvido no presente trabalho, foi feito um estudo numérico com o objetivo de investigar, e comparar, os efeitos que uma actuação estacionária e não-estacionária exibe sobre o controlo do escoamento a números de Reynolds relativamente elevados. Foi demostrado que a dimensão da bolha de separação é reduzida em muito e que a estrutura do escoamento é sensível à frequência da modulação “burst” do actuador a plasma tipo DBD. Os resultados também confirmaram que, para o caso de actuação não-estacionária, a frequência de “burst” e o “burst ratio”, são parâmetros cruciais para influenciar a capacidade de controlo do escoamento por parte dos actuadores a plasma. Determinou-se que as frequências “burst”, semelhantes às frequências naturais do sistema, são capazes de excitar as estruturas do escoamento num modo de ressonância. Esta observação confirma igualmente que, com frequências de excitação apropriadas, a estrutura de um escoamento de camada limite consegue ser correctamente modificada, e que as perdas no escoamento são reduzidas. Por fim, os actuadores a plasma foram utilizados para o controlo do escoamento sobre uma superfície Coanda de uma tubeira. Quando nesta foi aplicado um plasma, tornou-se possível retardar a separação do escoamento e aumentar o ângulo de deflexão do jacto gerado pelo propulsor. Por forma a encontrar a posição óptima para os actuadores, sete actuadores de tipo DBD foram distribuídos ao longo da superfície Coanda, tendo em consideração os pontos de separação do escoamento na camada limite obtidos numericamente. Os resultados mostram que quando o actuador DBD é colocado ligeiramente antes do ponto de separação do escoamento, há um aumento da capacidade de controlo e vectorização do jacto gerado. Os resultados preliminares das experiências efectuadas estão de acordo com o ângulo de deflexão do jacto previsto pelo modelo computacional.

In der Arbeit wird das asymptotische Verhalten von geeignet normierten und zentrierten Summen von Zufallsgrößen untersucht, die entweder unabhängig sind oder im Falle der Abhängigkeit als Martingaldifferenzfolge oder stark multiplikatives System auftreten. Neben der klassischen Summationstheorie werden die Limitierungsverfahren mit einer unendlichen Summationsmatrix oder einer angepaßten Folge von Gewichtsfunktionen betrachtet. Es werden die Methode der charakteristischen Funktionen und besonders die direkte Methode der konjugierten Verteilungsfunktionen weiterentwickelt, um quantitative Aussagen über gleichmäßige und ungleichmäßige Restgliedabschätzungen in zentralen Grenzwertsatz zu beweisen. Die Untersuchungen werden dabei in der Lp-Metrik, 1In the work the asymptotic behavior of suitably centered and normalized sums of random variables is investigated, which are either independent or occur in the case of dependence as a sequence of martingale differences or a strongly multiplicative system. In addition to the classical theory of summation limiting processes are considered with an infinite summation matrix or an adapted sequence of weighting functions. It will be further developed the method of characteristic functions, and especially the direct method of the conjugate distribution functions to prove quantitative statements about uniform and non-uniform error estimates of the remainder term in central limit theorem. The investigations are realized in the Lp metric, 1