We give a concise explication of the theory of level set methods for modeling motion of an interface as well as the numerical implementation of these methods. We then introduce the geometry of a camera and the mathematical models for 3D reconstruction with a few examples both simulated and from a real camera. We finally describe the model for 3D surface reconstruction from n-camera views using level set methods.

Under the right conditions, a moving sphere may successfully enter, and leave, a soap bubble without rupturing that bubble. The physics behind this phenomena are not well understood, nor the limiting factors (such as sphere size, speed, etc.). This work, investigating this phenomenon using high speed photography, has produced several results which are presented. First, several distinct regimes, noted while photographing the interactions between the spheres and the bubbles, are classified and discussed. Next a probabilistic examination of the soap bubbles rupture by the moving spheres is presented. Then a conjecture for the limiting sphere sizes and speeds is presented. And finally some interesting phenomena, noted in the course of this investigation, are presented and discussed.

京都大学 / 新制・課程博士 / 博士(工学) / 甲第24226号 / 工博第5054号 / 新制||工||1789(附属図書館) / 京都大学大学院工学研究科機械理工学専攻 / (主査)教授 平山 朋子, 教授 岩井 裕, 教授 松原 厚 / 学位規則第4条第1項該当 / Doctor of Philosophy (Engineering) / Kyoto University / DFAM

Heteroscedastic metamodeling has gained popularity as an effective tool for analyzing and optimizing complex stochastic systems. A heteroscedastic metamodel provides an accurate approximation of the input-output relationship implied by a stochastic simulation experiment whose output is subject to input-dependent noise variance. Several challenges remain unsolved in this field. First, in-depth investigations into the consistency of heteroscedastic metamodeling techniques, particularly from the sequential prediction perspective, are lacking. Second, sequential heteroscedastic metamodel-based level-set estimation (LSE) methods are scarce. Third, the increasingly high computational cost required by heteroscedastic Gaussian process-based LSE methods in the sequential sampling setting is a concern. Additionally, when constructing a valid uniform bound for a heteroscedastic metamodel, the impact of noise variance estimation is not adequately addressed. This dissertation aims to tackle these challenges and provide promising solutions. First, we investigate the information consistency of a widely used heteroscedastic metamodeling technique, stochastic kriging (SK). Second, we propose SK-based LSE methods leveraging novel uniform bounds for input-point classification. Moreover, we incorporate the Nystrom approximation and a principled budget allocation scheme to improve the computational efficiency of SK-based LSE methods. Lastly, we investigate empirical uniform bounds that take into account the impact of noise variance estimation, ensuring an adequate coverage capability. / Doctor of Philosophy / In real-world engineering problems, understanding and optimizing complex systems can be challenging and prohibitively expensive. Computer simulation is a valuable tool for analyzing and predicting system behaviors, allowing engineers to explore different scenarios without relying on costly physical prototypes. However, the increasing complexity of simulation models leads to a higher computational burden. Metamodeling techniques have emerged to address this issue by accurately approximating the system performance response surface based on limited simulation experiment data to enable real-time decision-making. Heteroscedastic metamodeling goes further by considering varying noise levels inherent in simulation outputs, resulting in more robust and accurate predictions. Among various techniques, stochastic kriging (SK) stands out by striking a good balance between computational efficiency and statistical accuracy. Despite extensive research on SK, challenges persist in its application and methodology. These include little understanding of SK's consistency properties, an absence of sequential SK-based algorithms for level-set estimation (LSE) under heteroscedasticity, and the increasingly low computational efficiency of SK-based LSE methods in implementation. Furthermore, a precise construction of uniform bounds for the SK predictor is also missing. This dissertation aims at addressing these aforementioned challenges. First, the information consistency of SK from a prediction perspective is investigated. Then, sequential SK-based procedures for LSE in stochastic simulation, incorporating novel uniform bounds for accurate input-point classification, are proposed. Furthermore, a popular approximation technique is incorporated to enhance the computational efficiency of the SK-based LSE methods. Lastly, empirical uniform bounds are investigated considering the impact of noise variance estimation.

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This dissertation addresses path planning for an autonomous vehicle navigating in a two dimensional environment for which an a priori map is inaccurate and for which the environment is sensed in real-time. For this class of application, planning decisions must be made in real-time. This work is motivated by the need for fast autonomous vehicles that require planning algorithms to operate as quickly as possible. In this dissertation, we first study the case in which there are only static obstacles in the environment. We propose a hybrid receding horizon control path planning algorithm that is based on level-set methods. The hybrid method uses global or local level sets in the formulation of the receding horizon control problem. The decision to select a new level set is made based on certain matching conditions that guarantee the optimality of the path. We rigorously prove sufficient conditions that guarantee that the vehicle will converge to the goal as long as a path to the goal exists. We then extend the proposed receding horizon formulation to the case when the environment possesses moving obstacles. Since all of the results in this dissertation are based on level-set methods, we rigorously investigate how level sets change in response to new information locally sensed by a vehicle. The result is a dynamic fast marching algorithm that usually requires significantly less computation that would otherwise be the case. We demonstrate the proposed dynamic fast marching method in a successful field trial for which an autonomous surface vehicle navigated four kilometers through a riverine environment. / Ph. D.

Computational micromechanics analysis of carbon nanotube-epoxy nanocomposites, containing aligned nanotubes, is performed using the mesh independent intrinsic extended finite element method (IXFEM). The IXFEM employs a localized intrinsic enrichment strategy to treat arbitrary discontinuities defined through the level-set method separate from the problem domain discretization, i.e. the finite element (FE) mesh. A global domain decomposition identifies local subdomains for building distinct partition of unities that appropriately suit the approximation. Specialized inherently enriched shape functions, constructed using the moving least square method, enhance the approximation space in the vicinity of discontinuity interfaces, maintaining accuracy of the solution, while standard FE shape functions are used elsewhere. Comparison of the IXFEM in solving validation problems with strong and weak discontinuities against a standard finite element method (FEM) and analytic solutions validates the enriched intrinsic bases, and shows anticipated trends in the error convergence rates. Applying the IXFEM to model composite materials, through a representative volume element (RVE), the filler agents are defined as individual weak bimaterial interfaces. Though a series of RVE studies, calculating the effective elastic material properties of carbon nanotube-epoxy nanocomposite systems, the benefits in substituting the conventional mesh dependent FEM with the mesh independent IXFEM when completing micromechanics analysis, investigating effects of high filler count or an evolving microstructure, are demonstrated. / Master of Science

Ce travail de recherche propose un modèle numérique pour simuler les procédés par infusion de résine en utilisant la méthode des éléments finis. Ce modèle permet de représenter l'écoulement d'une résine liquide dans des préformes poreuses subissant de grandes déformations. Dans cette étude, une modélisation macroscopique est utilisée. Au niveau du procédé, une zone de résine liquide est déposée sur les préformes. Ces dernières étant considérées comme un milieu poreux. Les équations de Stokes et de Darcy sont utilisées pour modéliser l'écoulement de la résine respectivement dans le drainant et dans les préformes. L'originalité du modèle réside dans le fait qu'un seul maillage est utilisé pour les deux milieux. La discrétisation est réalisée avec des éléments mixtes : dans Stokes, des éléments P1+/P1 sont utilisés et dans Darcy, des éléments P1/P1 stabilisés avec une formulation multi-échelle sont employés. Des fonctions distances signées sont utilisées pour représenter l'interface entre Stokes-Darcy et pour représenter le front de résine. Concernant la déformation des préformes, une formulation Lagrangienne réactualisée est utilisée. Dans cette formulation Lagrangienne, le comportement des préformes humides est représenté à l'aide du modèle de Terzaghi dans lequel les préformes sèches ont un comportement élastique non-linéaire. La perméabilité est reliée à la porosité via la relation de Carman-Kozeny. Celle-ci est déterminée à partir de l'équation de conservation de la masse. Ce modèle a été implémenté dans ZéBuLoN. Plusieurs simulations numériques d'infusion de résine sont présentées à la fin de ce manuscrit. / This work proposes a numerical model to simulate the manufacturing processes by resin infusion using the finite element method. This model allows to represent the resin flow into porous preforms, which are themselves subject to large deformations. In this study, a macroscopic description is used. The preforms are considered as a porous medium. The Stokes and the Darcy equations are used respectively to describe the resin flow into the liquid zone and into the preforms.The originality of the model consists in using one single unstructured mesh. The discretization is ensured by using a mixed velocity-pressure formulation. Indeed, a P1/P1 formulation is employed throughout the entire discretized domain, stabilized in the Darcy region with a multi-scale formulation and in the Stokes subdomain with a hierarchical-based bubble, i.e. a P1+/P1 finite element. Signed distance functions are used both to represent the Stokes-Darcy interface and to capture the moving flow front. Concerning the deformations of the preforms, an updated Lagrangian scheme is used. In the Lagrangian formulation, the behavior of the wet preforms is represented by using the Terzaghi model in which the dry preforms have a non-linear elastic behavior. The permeability depends on the porosity through the Carman-Kozeny relationship. This model has been implemented in Zset. Several numerical simulations of manufacturing processes by resin infusion are presented at the end of this manuscript.