Thesis (Ph.D.)--Marquette University, 2009. / D. Lyle Dabney, Ralph Del Colle, Deirdre Dempsey, Philip J. Rossi, S.J., Advisors.

Thesis (M.S. in Applied Mathematics) Naval Postgraduate School, June 1997. / Thesis advisors, Christopher Frenzen, Philip Beaver. Includes bibliographical references (p. 103-104). Also available online.

Les matériaux composites à matrice organique sont de plus en plus utilisés dans divers domaines tels que l'aérospatiale ou les énergies marines renouvelables en raison de leurs excellentes propriétés spécifiques. Cependant, les procédés de fabrication des structures composites sont complexes et peuvent conduire à l'apparition de défauts, en particulier de plissement des plis et de porosité, qui affectent les propriétés mécaniques de la structure. Les pièces composites sont ainsi systématiquement soumises à des contrôles CND long et coûteux. En cas de résultats négatifs par rapport à des critères conservatifs, celles-ci peuvent être rejetées, avec des conséquences économiques non négligeables. L'objectif de cette étude est de quantifier l'impact des défauts observés et des incertitudes associées sur le comportement de pièce composite. Dans ce travail, nous adoptons une vision paramétrique des incertitudes consistant à représenter le contenu probabiliste à travers d’un ensemble fini de variables aléatoires. Nous nous concentrons sur la propagation des incertitudes basée sur des méthodes stochastiques spectrales. L'étude portant sur le défaut de porosités se fait à l’échelle microscopique puis macroscopique. Les paramètres aléatoires d'entrée sont liés à la géométrie des porosités et à leur taux. L'étude du défaut plissements à l'échelle mésoscopique est basée sur une représentation paramétrique de la géométrie du plissement. Les paramètres aléatoires d'entrée représentent alors la forme et la taille de ces défauts. Il est donc possible d'analyser l'impact de ces défauts à l'échelle structurelle par des grandeurs mécaniques classiques et des critères de rupture. / Composite materials are increasingly used in various fields such as aerospace or renewable marine energies due to their excellent specific properties. However, the manufacturing processes of the composite structures are complex, which can lead to the appearance of defects, particularly wrinkles and porosities, which affect the mechanical properties of the structure. Based on conservative criteria, a system of non-destructive testing of composite parts thus makes it possible to judge their conformity. In case of non-conformity, those components are rejected, with non-negligible economic consequences. The objective of this study is to quantify the impact of the defects and associated uncertainties on the behavior of composite parts. In this work, we adopt a parametric vision of the uncertainties consisting in representing the probabilistic content through a finite set of random variables. We focus on the propagation of uncertainties based on spectral stochastic methods. The study involving porosity is done at the micro-scale and then at the macro-scale. The random input parameters are related to the geometry of the porosities and their rates. The study of the wrinkle defect, done at the mesoscopic scale, is based on a parametric representation of the geometry of the wrinkle. The random input parameters then represent the shape and size of this defect. It is therefore possible to analyze the impact of these two manufacturing defects at a structural scale through classical mechanical quantities and check the failure of the structure with failure criteria.

The thesis is in its first part focused on finding a link between the relatively young mathematical discipline theory of chaos, and theoretical and practical work of theatre director. The first part explains the principles of chaos theory, fractal geometry, understanding chaos in ancient Greek culture, in postmodern and in current state of culture. The thesis in the second part applies the named apparatus to the practice and concludes with are several essential characteristics, functions and tasks of directing in the theatre. Third part is two case studies based on stage praxis: directing of performances Punk Rock and Otcizeni in a DAMU school theatre DISK.

No description available.

Discrete dynamical systems can exhibit rich and interesting dynamics at lower dimensions (and co-dimensions) than that of ODE models. Classically, the minimal dimension to observe chaotic behavior in an ODE model is three; whereas it can be achieved in a one-dimensional discrete map. It is often the choice of mathematical biologists to use discrete systems as it fills many roles such as sparse data, incorporation of life cycle stages and noisy measurements. This work is analyzes a discrete time model of an infected salmon population. It provides an in-depth analysis of non-smooth bifurcations for alternate functional forms for density dependence in the growth function of a given model. These demonstrate interesting structures and chaotic behaviors with biologically feasible interpretations such as intrinsic growth rate and probability of death. The choice of density dependence function, as well as parameterization, leads to whether chaos occurs or not. / Master of Science / Often times biological processes do not happen in a continuous streamlined chain of events. We observe discrete life stages, ages, and morphological differences. Similarly, data is generally collected in discrete (and often fixed) time intervals. This work focuses on the role that population density has on the behavior of these systems. We dive into a case study for a viral infection in a salmon population. We show chaotic behavior can be observed as low as a single dimension model and discuss the biological implications. Additionally, we show that the choice of density dependence in a given infectious disease model directly impacts disease dynamics and can allow or prohibit chaotic behavior.

The nonlinear and chaotic dynamics of a shallow fluid layer are investigated numerically using large-scale parallel numerical simulations. Two particular situations are studied in detail. First, a fluid layer is placed between rigid boundaries and heated from below to yield the chaotic dynamics of thermal convection rolls (Rayleigh-Bénard convection). Second is a free-surface fluid layer placed on a shaker to yield nonlinear surface waves (Faraday waves). In both cases the full governing partial differential equations are solved using parallel spectral element methods. Rayleigh-Bénard convection is studied in a cylindrical dish with realistic boundaries. The complete flow field is obtained as well as the spectrum of Lyapunov exponents and Lyapunov vectors. The Lyapunov exponents and their corresponding perturbation fields are used to determine when and where events occur that contribute most to the chaotic dynamics. Roll pinch-off and roll mergers are found to be the largest contributors. Two dimensional and three dimensional Faraday waves are studied with periodic boundary conditions. The full Navier-Stokes equations are solved including the complex dynamics of the free surface waves to gain a better understanding of the interplay between the viscous boundary layers, the nonlinear streaming flow, and the bulk flow. The vortices in the bulk flow are weak compared to the flow in the viscous boundary layers. The surface waves are found to be non-sinusoidal and the time evolution of the waves are explored for both large and small amplitude waves. / Master of Science

Rare events play a crucial role in our society and a great effort has been dedicated to numerically study them in different contexts. This thesis proposes a numerical methodology based on Monte Carlo Metropolis-Hastings algorithm to efficiently sample rare events in chaotic systems. It starts by reviewing the relevance of rare events in chaotic systems, focusing in two types of rare events: states in closed systems with rare chaoticities, characterised by a finite-time Lyapunov exponent on a tail of its distribution, and states in transiently chaotic systems, characterised by a escape time on the tail of its distribution. This thesis argues that these two problems can be interpreted as a traditional problem of statistical physics: sampling exponentially rare states in the phase-space - states in the tail of the density of states - with an increasing parameter - the system size. This is used as the starting point to review Metropolis-Hastings algorithm, a traditional and flexible methodology of importance sampling in statistical physics. By an analytical argument, it is shown that the chaoticity of the system hinders direct application of Metropolis-Hastings techniques to efficiently sample these states because the acceptance is low. It is argued that a crucial step to overcome low acceptance rate is to construct a proposal distribution that uses information about the system to bound the acceptance rate. Using generic properties of chaotic systems, such as exponential divergence of initial conditions and fractals embedded in their phase-spaces, a proposal distribution that guarantees a bounded acceptance rate is derived for each type of rare events. This proposal is numerically tested in simple chaotic systems, and the efficiency of the resulting algorithm is measured in numerous examples in both types of rare events. The results confirm the dramatic improvement of using Monte Carlo importance sampling with the derived proposals against traditional methodologies: the number of samples required to sample an exponentially rare state increases polynomially, as opposed to an exponential increase observed in uniform sampling. This thesis then analyses the sub-optimal (polynomial) efficiency of this algorithm in a simple system and shows analytically how the correlations induced by the proposal distribution can be detrimental to the efficiency of the algorithm. This thesis also analyses the effect of high-dimensional chaos in the proposal distribution and concludes that an anisotropic proposal that takes advantage of the different rates of expansion along the different unstable directions, is able to efficiently find rare states. The applicability of this methodology is also discussed to sample rare states in non-hyperbolic systems, with focus on three systems: the logistic map, the Pomeau-Manneville map, and the standard map. Here, it is argued that the different origins of non-hyperbolicity require different proposal distributions. Overall, the results show that by incorporating specific information about the system in the proposal distribution of Metropolis-Hastings algorithm, it is possible to efficiently find and sample rare events of chaotic systems. This improved methodology should be useful to a large class of problems where the numerical characterisation of rare events is important.

L’optimisation est un sujet très important dans tous les domaines. Cependant, parmi toutes les applications de l’optimisation, il est difficile de trouver des exemples de systèmes à optimiser qui ne comprennent pas un certain niveau d'incertitude sur les valeurs de quelques paramètres. Le thème central de cette thèse est donc le traitement des différents aspects de l’optimisation en présence d’incertitudes. Nous commençons par présenter un bref état de l’art des méthodes permettant de prendre en compte les incertitudes dans l’optimisation. Cette revue de la littérature a permis de constater une lacune concernant la caractérisation des propriétés probabilistes du point d’optimum de fonctions dépendant de paramètres aléatoires. Donc, la première contribution de cette thèse est le développement de deux méthodes pour approcher la fonction densité de probabilité (FDP) d’un tel point : la méthode basée sur la Simulation de Monte Carlo et la méthode de projection en dimension finie basée sur l’Approximation par polynômes de chaos. Les résultats numériques ont montré que celle-ci est adaptée à l’approximation de la FDP du point optimal du processus d'optimisation dans les situations étudiées. Il a été montré que la méthode numérique est capable d’approcher aussi des moments d'ordre élevé du point optimal, tels que l’aplatissement et l’asymétrie. Ensuite, nous passons au traitement de contraintes probabilistes en utilisant l’optimisation fiabiliste. Dans ce sujet, une nouvelle méthode basée sur des coefficients de sécurité est développée. Les exemples montrent que le principal avantage de cette méthode est son coût de calcul qui est très proche de celui de l’optimisation déterministe conventionnelle, ce qui permet son couplage avec un algorithme d’optimisation globale arbitraire. / The optimization is a very important tool in several domains. However, among its applications, it is hard to find examples of systems to be optimized that do not possess a certain uncertainty level on its parameters. The main goal of this thesis is the treatment of different aspects of the optimization under uncertainty. We present a brief review of the literature on this topic, which shows the lack of methods able to characterize the probabilistic properties of the optimum point of functions that depend on random parameters. Thus, the first main contribution of this thesis is the development of two methods to eliminate this lack: the first is based on Monte Carlo Simulation (MCS) (considered as the reference result) and the second is based on the polynomial chaos expansion (PCE). The validation of the PCE based method was pursued by comparing its results to those provided by the MCS method. The numerical analysis shows that the PCE method is able to approximate the probability density function of the optimal point in all the problems solved. It was also showed that it is able to approximate even high order statistical moments such as the kurtosis and the asymmetry. The second main contribution of this thesis is on the treatment of probabilistic constraints using the reliability based design optimization (RBDO). Here, a new RBDO method based on safety factors was developed. The numerical examples showed that the main advantage of such method is its computational cost, which is very close to the one of the standard deterministic optimization. This fact makes it possible to couple the new method with global optimization algorithms.

Notre projet de recherche repose sur l’étude de la construction par une approche fractale d’un modèle de prévision de la résistance et de conduite du changement. Nous avons tout d’abord mené une exploration théorique en nous appuyant sur une littérature transdisciplinaire, à la confluence des mathématiques et des sciences de gestion. Nous avons ensuite opté pour une exploration hybride, dont la première phase, exploratoire, basée sur une recherche-intervention, nous a permis d’élaborer un modèle en accord avec notre exploration théorique et notre projet de recherche. Nous avons ensuite testé la validité externe de notre modèle en l’appliquant à d’autres organisations. Cette phase de vérification a montré les limites de la modélisation mathématique du changement organisationnel, particulièrement pour l’évaluation précise des variables à partir de données qualitatives. Ce travail se conclut par des préconisations qui nous ont permis de conforter les recommandations de la littérature. / Our research project relies on the study of the construction, using a fractal approach, of a resistance prevision and change conducing model. We started with a theoretical exploration based on a transdisciplinary literature, at the confluence of mathematics and management sciences. Then, we chose a hybrid exploration, of which the first phase, exploratory, based on an interventional research, allowed us to elaborate a model in complete accord with our theoretical exploration and our research project. Then, we have tested the external validity of our model, using it in other organizations. This verification phase showed the limits of the mathematical modeling of organizational change, particularly to evaluate precisely the variables from qualitative data. We concluded this research with recommendations that allowed us to comfort the literature.