Functionally graded materials processed by the thermal spray techniques such as electron beam physical vapor deposition and plasma spray forming are known to have an orthotropic structure with reduced mechanical properties. Debonding related failures in these types of material systems occur due to embedded cracks that are perpendicular to the direction of the material property gradation. These cracks are inherently under mixed-mode loading and fracture analysis requires the extraction of the modes I and II stress intensity factors. The present study aims at developing semi-analytical techniques to study embedded crack problems in graded orthotropic media under various boundary conditions. The cracks are assumed to be aligned parallel to one of the principal axes of orthotropy. The problems are formulated using the averaged constants of plane orthotropic elasticity and reduced to two coupled integral equations with Cauchy type dominant singularities. The equations are solved numerically by adopting an expansion - collocation technique. The main results of the analyses are the mixed mode stress intensity factors and the energy release rate as functions of the material nonhomogeneity and orthotropy parameters. The effects of the boundary conditions on the mentioned fracture parameters are also duly discussed.

Doctor of Philosophy (PhD) / This thesis presents a computational method for the inductive inference of explicit and implicit semantic design knowledge from the symbolic-mathematical syntax of design formulations using an unsupervised pattern recognition and extraction approach. Existing research shows that AI / machine learning based design computation approaches either require high levels of knowledge engineering or large training databases to acquire problem reformulation knowledge. The method presented in this thesis addresses these methodological limitations. The thesis develops, tests, and evaluates ways in which the method may be employed for design problem reformulation. The method is based on the linear algebra based factorization method Singular Value Decomposition (SVD), dimensionality reduction and similarity measurement through unsupervised clustering. The method calculates linear approximations of the associative patterns of symbol cooccurrences in a design problem representation to infer induced coupling strengths between variables, constraints and system components. Unsupervised clustering of these approximations is used to identify useful reformulations. These two components of the method automate a range of reformulation tasks that have traditionally required different solution algorithms. Example reformulation tasks that it performs include selection of linked design variables, parameters and constraints, design decomposition, modularity and integrative systems analysis, heuristically aiding design “case” identification, topology modeling and layout planning. The relationship between the syntax of design representation and the encoded semantic meaning is an open design theory research question. Based on the results of the method, the thesis presents a set of theoretical postulates on computable relationships between design syntax and semantics. The postulates relate the performance of the method with empirical findings and theoretical insights provided by cognitive neuroscience and cognitive science on how the human mind engages in symbol processing and the resulting capacities inherent in symbolic representational systems to encode “meaning”. The performance of the method suggests that semantic “meaning” is a higher order, global phenomenon that lies distributed in the design representation in explicit and implicit ways. A one-to-one local mapping between a design symbol and its meaning, a largely prevalent approach adopted by many AI and learning algorithms, may not be sufficient to capture and represent this meaning. By changing the theoretical standpoint on how a “symbol” is defined in design representations, it was possible to use a simple set of mathematical ideas to perform unsupervised inductive inference of knowledge in a knowledge-lean and training-lean manner, for a knowledge domain that traditionally relies on “giving” the system complex design domain and task knowledge for performing the same set of tasks.

Doctor of Philosophy (PhD) / This thesis presents a computational method for the inductive inference of explicit and implicit semantic design knowledge from the symbolic-mathematical syntax of design formulations using an unsupervised pattern recognition and extraction approach. Existing research shows that AI / machine learning based design computation approaches either require high levels of knowledge engineering or large training databases to acquire problem reformulation knowledge. The method presented in this thesis addresses these methodological limitations. The thesis develops, tests, and evaluates ways in which the method may be employed for design problem reformulation. The method is based on the linear algebra based factorization method Singular Value Decomposition (SVD), dimensionality reduction and similarity measurement through unsupervised clustering. The method calculates linear approximations of the associative patterns of symbol cooccurrences in a design problem representation to infer induced coupling strengths between variables, constraints and system components. Unsupervised clustering of these approximations is used to identify useful reformulations. These two components of the method automate a range of reformulation tasks that have traditionally required different solution algorithms. Example reformulation tasks that it performs include selection of linked design variables, parameters and constraints, design decomposition, modularity and integrative systems analysis, heuristically aiding design “case” identification, topology modeling and layout planning. The relationship between the syntax of design representation and the encoded semantic meaning is an open design theory research question. Based on the results of the method, the thesis presents a set of theoretical postulates on computable relationships between design syntax and semantics. The postulates relate the performance of the method with empirical findings and theoretical insights provided by cognitive neuroscience and cognitive science on how the human mind engages in symbol processing and the resulting capacities inherent in symbolic representational systems to encode “meaning”. The performance of the method suggests that semantic “meaning” is a higher order, global phenomenon that lies distributed in the design representation in explicit and implicit ways. A one-to-one local mapping between a design symbol and its meaning, a largely prevalent approach adopted by many AI and learning algorithms, may not be sufficient to capture and represent this meaning. By changing the theoretical standpoint on how a “symbol” is defined in design representations, it was possible to use a simple set of mathematical ideas to perform unsupervised inductive inference of knowledge in a knowledge-lean and training-lean manner, for a knowledge domain that traditionally relies on “giving” the system complex design domain and task knowledge for performing the same set of tasks.

Cette thèse est consacrée à l’étude du problème de transport de Monge pour le coût quadratique en géométrie sous-Riemannienne et des conditions essentielles à l’obtention des résultats d’existence et et d’unicité de solutions. Ces travaux consistent à étendre ces résultats au cas des structures sous-Riemanniennes admettant des géodésiques minimisantes singulières. Dans une première partie, on développe des techniques inspirées de travaux de Cavalletti et Huesmann pour d’obtenir des résultats significatifs pour des structures de rang 2 en dimension 4. Dans une deuxième partie, on étudie des outils analytiques de la h-semiconcavité de la distance sousriemannienne et on montre comment ce type de régularité peut aboutit à l’obtention d’existence et d’unicité de solutions dans un cas général. / This thesis is devoted to the study of the Monge transport problem for the quadratic cost in sub-Riemannian geometry and the essential conditions to obtain existence and uniqueness of solutions. These works consist in extending these results to the case of sub-Riemannian structures admitting singular minimizing geodesics. In a first part, we develop techniques inspired by works by Cavalletti and Huesmann in order to obtain significant results for structures of rank 2 in dimension 4. In a second part, we study analytical tools of the h-semiconcavity of the sub-Riemannian distance and we show how this type of regularity can lead to the well-posedness of the Monge problem in general cases.

Through-the-wall Imaging (TWI) is crucial for various applications such as law enforcement, rescue missions and defense. TWI methods aim to provide detailed information of spaces that cannot be seen directly. Current state-of-the-art TWI systems utilise ultra-wideband (UWB) signals to simultaneously achieve wall penetration and high resolution. These TWI systems transmit signals and mathematically back-project the reflected signals received to image the scenario of interest. However, these systems are diffraction-limited and encounter problems due to multipath signals in the presence of multiple scatterers. Time reversal (TR) methods have become popular for remote sensing because they can take advantage of multipath signals to achieve superresolution (resolution that beats the diffraction limit). The Decomposition Of the Time-Reversal Operator (DORT in its French acronym) and MUltiple SIgnal Classification (MUSIC) methods are both TR techniques which involve taking the Singular Value Decomposition (SVD) of the Multistatic Data Matrix (MDM) which contains the signals received from the target(s) to be located. The DORT and MUSIC imaging methods have generated a lot of interests due to their robustness and ability to locate multiple targets. However these TR-based methods encounter problems when the targets are behind an obstruction, particularly when the properties of the obstruction is unknown as is often the case in TWI applications. This dissertation introduces a novel total sub-MDM algorithm that uses the highly acclaimed MUSIC method to image targets hidden behind an obstruction and achieve superresolution. The algorithm utilises spatio-temporal windows to divide the full-MDM into sub-MDMs. The summation of all images obtained from each sub-MDM give a clearer image of a scenario than we can obtain using the full-MDM. Furthermore, we propose a total sub-differential MDM algorithm that uses the MUSIC method to obtain images of moving targets that are hiddenbehind an obstructing material.

The dissertation consists of two parts. The first part (Chapter 1 to 4) is on some contributions to the development of a non-linear analysis on the quintessential fractal set Sierpinski gasket and its probabilistic interpretation. The second part (Chapter 5) is on the asymptotic tail decays for suprema of stochastic processes satisfying certain conditional increment controls. Chapters 1, 2 and 3 are devoted to the establishment of a theory of backward problems for non-linear stochastic differential equations on the gasket, and to derive a probabilistic representation to some parabolic type partial differential equations on the gasket. In Chapter 2, using the theory of Markov processes, we derive the existence and uniqueness of solutions to backward stochastic differential equations driven by Brownian motion on the Sierpinski gasket, for which the major technical difficulty is the exponential integrability of quadratic processes of martingale additive functionals. A Feynman-Kac type representation is obtained as an application. In Chapter 3, we study the stochastic optimal control problems for which the system uncertainties come from Brownian motion on the gasket, and derive a stochastic maximum principle. It turns out that the necessary condition for optimal control problems on the gasket consists of two equations, in contrast to the classical result on ℝ^d, where the necessary condition is given by a single equation. The materials in Chapter 2 are based on a joint work with Zhongmin Qian (referenced in Chapter 2). Chapter 4 is devoted to the analytic study of some parabolic PDEs on the gasket. Using a new type of Sobolev inequality which involves singular measures developed in Section 4.2, we establish the existence and uniqueness of solutions to these PDEs, and derive the space-time regularity for solutions. As an interesting application of the results in Chapter 4 and the probabilistic representation developed in Chapter 2, we further study Burgers equations on the gasket, to which the space-time regularity for solutions is deduced. The materials in Chapter 4 are based on a joint work with Zhongmin Qian (referenced in Chapter 4). In Chapter 5, we consider a class of continuous stochastic processes which satisfy the conditional increment control condition. Typical examples include continuous martingales, fractional Brownian motions, and diffusions governed by SDEs. For such processes, we establish a Doob type maximal inequality. Under additional assumptions on the tail decays of their marginal distributions, we derive an estimate for the tail decay of the suprema (Theorem 5.3.2), which states that the suprema decays in a manner similar to the margins of the processes. In Section 5.4, as an application of Theorem 5.3.2, we derive the existence of strong solutions to a class of SDEs. The materials in this chapter is based on the work [44] by the author (Section 5.2 and Section 5.3) and an ongoing joint project with Guangyu Xi (Section 5.4).

Submitted by Santos Davilene (davilenes@ufba.br) on 2017-06-01T19:53:01Z No. of bitstreams: 1 tese_andressa (3).pdf: 1107402 bytes, checksum: 2af37806adf427975c78abffa99bafc4 (MD5) / Approved for entry into archive by Vanessa Reis (vanessa.jamile@ufba.br) on 2017-06-07T11:12:05Z (GMT) No. of bitstreams: 1 tese_andressa (3).pdf: 1107402 bytes, checksum: 2af37806adf427975c78abffa99bafc4 (MD5) / Made available in DSpace on 2017-06-07T11:12:05Z (GMT). No. of bitstreams: 1 tese_andressa (3).pdf: 1107402 bytes, checksum: 2af37806adf427975c78abffa99bafc4 (MD5) / Neste trabalho obtemos uma cota superior para a taxa exponencial de grandes desvios para observáveis contínuos em semiuxos de suspensão sobre uma base unidimensional não-uniformemente expansora com singularidades não flat ou descontinuidades, onde a função teto que define a suspensão se comporta como o logaritmo da distância para o conjunto singular/descontínuo da aplicação base. Para obtermos tal cota, mostramos que a transformação da base apresenta recorrência exponencialmente lenta para o conjunto descontínuo. Os resultados são aplicados, em particular, para semiuxos que modelam sumidouros hiperbólicos singulares em variedades tridimensionais não necessariamente transitivos. Como corolários obtemos taxas de escape de subconjuntos destes sumidouros sem medida total e resultado de existência de medida física para classes de transformações do intervalo expansoras por pedaços com singularidades.

Neste trabalho investigamos, dos pontos de vistas analítico e numérico, o comportamento assintótico da solução da equação do aerofólio, com uma singularidade do tipo Cauchy, de nida sobre um intervalo com uma pequena abertura. Exibimos um modelo matemático com uma solução f" para o intervalo disjunto G" = (−1,−ε) ∪ (ε, 1) e uma solução f0 que corresponde ao limite de f" quando (ε → 0), relacionando esta última com a solução da equação do aerofólio f no intervalo (−1, 1). Além do mais, demonstramos casos particulares de funções ψ = Tm e ψ = Un(onde Tm e Un são os polinômios de Tchebychev do primeiro e segundo tipo respectivamente) em que temos a igualdade f = f0 e conseqüentemente f" ≈ f. Apresentamos e comparamos numericamente as soluções f", f0 e f para diferentes funções ψ e valores de ε no intervalo G". Mostramos ainda soluções quase polinomiais analíticas da equação do aerofólio, e propomos um método espectral para a equação do aerofólio generalizada. Por m, obtemos soluções analíticas das equações do aerofólio para os intervalos G", (−1, 1)\ {0} e (−1, 1) para diferentes funções ψ(t) através da expansão em série da densidade da integral singular com núcleo Cauchy. / In this work we investigate, of the analytical and numerical points of views, the asymptotic behavior of the airfoil equation solution with a singularity of the Cauchy type, de ned over a interval with a small opening. We display a mathematical model with a f" solution to the disjoint interval G" = (−1,−ε)∪(ε, 1) and a f0 solution corresponding to limit of f" when (ε → 0), linking the latter with the solution of the airfoil equation f in the interval (−1, 1). Furthermore, we demonstrate particular cases of functions ψ = Tm and ψ = Un (where Tm and Un are the Chebyshev polynomials of the rst and second type respectively) where we have equality f = f0 and then f" ≈ f. We present and compare numerically the solutions f", f0 and f for di erent functions ψ and values of ε in G". We also show almost polynomial analytical solutions for the airfoil equation, and we propose a spectral method for the generalized airfoil equation. Finally, we obtain analytical solutions of the airfoil equations to the interval G", (−1, 1)\ {0} and (−1, 1) for various functions ψ(t) by expanding in series the density of the Cauchy singular integral.

Ce travail s’intéresse à la phase de montée d’un aéronef civil. Les trajectoires minimisant le temps de montée ainsi que que celles minimisant la consommation de carburant sont étudiées au travers du contrôle optimal géométrique. La dynamique associée à la phase de montée possède un phénomène dit de perturbation singulière. Ce phénomène, présent dans les systèmes multi-échelle, rend difficile la résolution numérique du problème de contrôle associé. La réduction desystème hamiltonien, permettant de s’affranchir de la difficulté numérique introduite par la perturbation singulière, est étudiée d’un point de vue théorique puis numérique. Dans un second temps, le système réduit est étudié géométriquement. L’utilisation des outils du contrôle géométrique combinée à celui des synthèses à temps court permet de déterminer des familles de trajectoires localement temps-optimales pour des temps courts. Cette étude est complétée par une étude des trajectoires temps-optimales en présence de contraintes d’état. D’un point de vue plus numérique, les méthodes directes et indirectes sont utilisées pour résoudre les différents problèmes. Une synthèse locale est alors réalisée en partant des familles de trajectoires déterminées pour des temps courts. Une étude des trajectoires minimisant la consommation de carburant est également réalisée. / This work concerns the climbing phase of a civil aircraft. The trajectories which minimize the climbing time and the one which minimize the fuel consumption are studied throughout geometric optimal control. The climbing phase dynamics presents a characteristics called singular perturbation. This phenomena exists in multi-scale dynamics which makes the numerical study of the associated control problem difficult. Theoretically and numerically we study the reduction of hamiltonian system. This concept allows to remove the numerical complexity induced by the singular perturbation. Secondly, the reduced system is studied geometrically. Families of timeoptimal trajectories in small time are determined thanks to geometric control tools and small time synthesis. A study of time-optimal trajectories with active state constraints completes this work. From a more numerical point of view, direct and indirect methods are used to solve the climbing problems. A local synthesis for time-optimal trajectory is established starting from the families of trajectory determined in small time. A study of minimum fuel consumption trajectories is also realized.

Made available in DSpace on 2016-06-02T20:28:24Z (GMT). No. of bitstreams: 1 2583.pdf: 836539 bytes, checksum: e0a45d9823e5681bd14db3b57f2cdf99 (MD5) Previous issue date: 2009-04-14 / Financiadora de Estudos e Projetos / In this work, we present necessary and suficient conditions for that an operator of Calderón-Zygmund type can be extended to a bounded operator on L2. By using results of interpolation and cancellation conditions on the kernel to obtain the boundeness in Lp, for all 1 < p < 1. / Neste trabalho, apresentamos condições necessárias e suficientes para que um operador de tipo Calderón-Zygmund possa ser estendido a um operador limitado em L2. Usando resultados de interpolação e condições de cancelamentos sobre o núcleo obter a limitação em Lp, para todo 1 < p < 1.