Дисертації: "Polynomial chao"

1

Xiaochen, Liu. "Statistical Analysis of Integrated Circuits Using Decoupled Polynomial Chaos." Thesis, Université d'Ottawa / University of Ottawa, 2016. http://hdl.handle.net/10393/34836.

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One of the major tasks in electronic circuit design is the ability to predict the performance of general circuits in the presence of uncertainty in key design parameters. In the mathematical literature, such a task is referred to as uncertainty quantification. Uncertainty about the key design parameters arises mainly from the difficulty of controlling the physical or geometrical features of the underlying design, especially at the nanometer level. With the constant trend to scale down the process feature size, uncertainty quantification becomes crucial in shortening the design time. To achieve the uncertainty quantification, this thesis presents a new approach based on the concept of generalized Polynomial Chaos (gPC) to perform variability analysis of general nonlinear circuits. The proposed approach is built upon a decoupling formulation of the Galerkin projection (GP) technique, where the large matrix is transformed into a block-diagonal whose diagonal blocks can be factorized independently. The proposed methodology provides a general framework for decoupling the GP formulation based on a general system of orthogonal polynomials. Moreover, it provides a new insight into the error level that is caused by the decoupling procedure, enabling an assessment of the performance of a wide variety of orthogonal polynomials. For example, it is shown that, for the same order, the Chebyshev polynomials outperforms other commonly used gPC polynomials.

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2

Yorke, Rory. "Chaos control using local polynomial approximation." Master's thesis, University of Cape Town, 2001. http://hdl.handle.net/11427/5075.

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Includes bibliographical references.
Chaotic systems may be defined as those whose behaviour is sensitively dependent on initial conditions. Such systems may be made periodic using small input perturbations, as proposed in [OGY90]; this is called Ott-Grebogi-Yorke (OGY) chaos control. The original method used a linear model for controller design; a later development of chaos control was [CCdF99], in which a polynomial model is used. This dissertation proposes using local Taylor polynomial models as a basis for chaos control.

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3

Templeton, Brian Andrew. "A Polynomial Chaos Approach to Control Design." Diss., Virginia Tech, 2009. http://hdl.handle.net/10919/28840.

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A method utilizing H2 control concepts and the numerical method of Polynomial Chaos was developed in order to create a novel robust probabilistically optimal control approach. This method was created for the practical reason that uncertainty in parameters tends to be inherent in system models. As such, the development of new methods utilizing probability density functions (PDFs) was desired. From a more theoretical viewpoint, the utilization of Polynomial Chaos for studying and designing control systems has not been very thoroughly investigated. The current work looks at expanding the H2 and related Linear Quadratic Regulator (LQR) control problems for systems with parametric uncertainty. This allows solving deterministic linear equations that represent probabilistic linear differential equations. The application of common LTI (Linear Time Invariant) tools to these expanded systems are theoretically justified and investigated. Examples demonstrating the utilized optimization process for minimizing the H2 norm and parallels to LQR design are presented. The dissertation begins with a thorough background section that reviews necessary probability theory. Also, the connection between Polynomial Chaos and dynamic systems is explained. Next, an overview of related control methods, as well as an in-depth review of current Polynomial Chaos literature is given. Following, formal analysis, related to the use of Polynomial Chaos, is provided. This lays the ground for the general method of control design using Polynomial Chaos and H2. Then an experimental section is included that demonstrates controller synthesis for a constructed probabilistic system. The experimental results lend support to the method.
Ph. D.

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4

Whittle, Lisa. "Stochastic Optimal Trajectory Generation via Multivariate Polynomial Chaos." Thesis, Luleå tekniska universitet, Institutionen för system- och rymdteknik, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:ltu:diva-65746.

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This thesis presents a framework that has been developed in order to compute stochastic optimal trajectories. This is achieved by transforming the initial set of stochastic ordinary differential equations into their deterministic equivalent by application of Multivariate Polynomial Chaos. Via Galerkin projection, it is possible to include stochastic information in the optimal-trajectory generation process, and to solve the corresponding optimal-control problem using pseudospectral methods. The resultant trajectory is therefore less sensitive to the uncertainties included in the analysis, e.g., those present in system parameters, initial conditions or path constraints. The accurate, yet computationally efficient manner in which solutions are obtained is presented and a comparison with deterministic results show the benefits of the proposed approach for a variety of numerical examples.

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5

Szepietowska, Katarzyna. "POLYNOMIAL CHAOS EXPANSION IN BIO- AND STRUCTURAL MECHANICS." Thesis, Bourges, INSA Centre Val de Loire, 2018. http://www.theses.fr/2018ISAB0004/document.

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Cette thèse présente une approche probabiliste de la modélisation de la mécanique des matériaux et des structures. Le dimensionnement est influencé par l'incertitude des paramètres d'entrée. Le travail est interdisciplinaire et les méthodes décrites sont appliquées à des exemples de biomécanique et de génie civil. La motivation de ce travail était le besoin d'approches basées sur la mécanique dans la modélisation et la simulation des implants utilisés dans la réparation des hernies ventrales. De nombreuses incertitudes apparaissent dans la modélisation du système implant-paroi abdominale. L'approche probabiliste proposée dans cette thèse permet de propager ces incertitudes et d’étudier leurs influences respectives. La méthode du chaos polynomial basée sur la régression est utilisée dans ce travail. L'exactitude de ce type de méthodes non intrusives dépend du nombre et de l'emplacement des points de calcul choisis. Trouver une méthode universelle pour atteindre un bon équilibre entre l'exactitude et le coût de calcul est encore une question ouverte. Différentes approches sont étudiées dans cette thèse afin de choisir une méthode efficace et adaptée au cas d’étude. L'analyse de sensibilité globale est utilisée pour étudier les influences des incertitudes d'entrée sur les variations des sorties de différents modèles. Les incertitudes sont propagées aux modèles implant-paroi abdominale. Elle permet de tirer des conclusions importantes pour les pratiques chirurgicales. À l'aide de l'expertise acquise à partir de ces modèles biomécaniques, la méthodologie développée est utilisée pour la modélisation de joints de bois historiques et la simulation de leur comportement mécanique. Ce type d’étude facilite en effet la planification efficace des réparations et de la rénovation des bâtiments ayant une valeur historique
This thesis presents a probabilistic approach to modelling the mechanics of materials and structures where the modelled performance is influenced by uncertainty in the input parameters. The work is interdisciplinary and the methods described are applied to medical and civil engineering problems. The motivation for this work was the necessity of mechanics-based approaches in the modelling and simulation of implants used in the repair of ventral hernias. Many uncertainties appear in the modelling of the implant-abdominal wall system. The probabilistic approach proposed in this thesis enables these uncertainties to be propagated to the output of the model and the investigation of their respective influences. The regression-based polynomial chaos expansion method is used here. However, the accuracy of such non-intrusive methods depends on the number and location of sampling points. Finding a universal method to achieve a good balance between accuracy and computational cost is still an open question so different approaches are investigated in this thesis in order to choose an efficient method. Global sensitivity analysis is used to investigate the respective influences of input uncertainties on the variation of the outputs of different models. The uncertainties are propagated to the implant-abdominal wall models in order to draw some conclusions important for further research. Using the expertise acquired from biomechanical models, modelling of historic timber joints and simulations of their mechanical behaviour is undertaken. Such an investigation is important owing to the need for efficient planning of repairs and renovation of buildings of historical value

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6

Nydestedt, Robin. "Application of Polynomial Chaos Expansion for Climate Economy Assessment." Thesis, KTH, Optimeringslära och systemteori, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-223985.

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In climate economics integrated assessment models (IAMs) are used to predict economic impacts resulting from climate change. These IAMs attempt to model complex interactions between human and geophysical systems to provide quantifications of economic impact, typically using the Social Cost of Carbon (SCC) which represents the economic cost of a one ton increase in carbon dioxide. Another difficulty that arises in modeling a climate economics system is that both the geophysical and economic submodules are inherently stochastic. Even in frequently cited IAMs, such as DICE and PAGE, there exists a lot of variation in the predictions of the SCC. These differences stem both from the models of the climate and economic modules used, as well as from the choice of probability distributions used for the random variables. Seeing as IAMs often take the form of optimization problems these nondeterministic elements potentially result in heavy computational costs. In this thesis a new IAM, FAIR/DICE, is introduced. FAIR/DICE is a discrete time hybrid of DICE and FAIR providing a potential improvement to DICE as the climate and carbon modules in FAIR take into account feedback coming from the climate module to the carbon module. Additionally uncertainty propagation in FAIR/DICE is analyzed using Polynomial Chaos Expansions (PCEs) which is an alternative to Monte Carlo sampling where the stochastic variables are projected onto stochastic polynomial spaces. PCEs provide better computational efficiency compared to Monte Carlo sampling at the expense of storage requirements as a lot of computations can be stored from the first simulation of the system, and conveniently statistics can be computed from the PCE coefficients without the need for sampling. A PCE overloading of FAIR/DICE is investigated where the equilibrium climate sensitivity, modeled as a four parameter Beta distribution, introduces an uncertainty to the dynamical system. Finally, results in the mean and variance obtained from the PCEs are compared to a Monte Carlo reference and avenues into future work are suggested.
Inom klimatekonomi används integrated assessment models (IAMs) för att förutspå hur klimatförändringar påverkar ekonomin. Dessa IAMs modellerar komplexa interaktioner mellan geofysiska och mänskliga system för att kunna kvantifiera till exempel kostnaden för den ökade koldioxidhalten på planeten, i.e. Social Cost of Carbon (SCC). Detta representerar den ekonomiska kostnaden som motsvaras av utsläppet av ett ton koldioxid. Faktumet att både de geofysiska och ekonomiska submodulerna är stokastiska gör att SCC-uppskattningar varierar mycket även inom väletablerade IAMs som PAGE och DICE. Variationen grundar sig i skillnader inom modellerna men också från att val av sannolikhetsfördelningar för de stokastiska variablerna skiljer sig. Eftersom IAMs ofta är formulerade som optimeringsproblem leder dessutom osäkerheterna till höga beräkningskostnader. I denna uppsats introduceras en ny IAM, FAIR/DICE, som är en diskret tids hybrid av DICE och FAIR. Den utgör en potentiell förbättring av DICE eftersom klimat- och kolmodulerna i FAIR även behandlar återkoppling från klimatmodulen till kolmodulen. FAIR/DICE är analyserad med hjälp av Polynomial Chaos Expansions (PCEs), ett alternativ till Monte Carlo-metoder. Med hjälp av PCEs kan de osäkerheter projiceras på stokastiska polynomrum vilket har fördelen att beräkningskostnader reduceras men nackdelen att lagringskraven ökar. Detta eftersom många av beräkningarna kan sparas från första simuleringen av systemet, dessutom kan statistik extraheras direkt från PCE koefficienterna utan behov av sampling. FAIR/DICE systemet projiceras med hjälp av PCEs där en osäkerhet är introducerad via equilibrium climate sensitivity (ECS), vilket i sig är ett värde på hur känsligt klimatet är för koldioxidförändringar. ECS modelleras med hjälp av en fyra-parameters Beta sannolikhetsfördelning. Avslutningsvis jämförs resultat i medelvärde och varians mellan PCE implementationen av FAIR/DICE och en Monte Carlo-baserad referens, därefter ges förslag på framtida utvecklingsområden.

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7

Perez, Rafael A. "Uncertainty Analysis of Computational Fluid Dynamics Via Polynomial Chaos." Diss., Virginia Tech, 2008. http://hdl.handle.net/10919/28984.

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The main limitations in performing uncertainty analysis of CFD models using conventional methods are associated with cost and effort. For these reasons, there is a need for the development and implementation of efficient stochastic CFD tools for performing uncertainty analysis. One of the main contributions of this research is the development and implementation of Intrusive and Non-Intrusive methods using polynomial chaos for uncertainty representation and propagation. In addition, a methodology was developed to address and quantify turbulence model uncertainty. In this methodology, a complex perturbation is applied to the incoming turbulence and closure coefficients of a turbulence model to obtain the sensitivity derivatives, which are used in concert with the polynomial chaos method for uncertainty propagation of the turbulence model outputs.
Ph. D.

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8

Ishak, Hassoun. "Étude stochastique de l'impact des défauts de porosités et de plissements dans les matériaux composites." Thesis, Nantes, 2017. http://www.theses.fr/2017NANT4090/document.

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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

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9

Koehring, Andrew. "The application of polynomial response surface and polynomial chaos expansion metamodels within an augmented reality conceptual design environment." [Ames, Iowa : Iowa State University], 2008.

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10

Fisher, James Robert. "Stability analysis and control of stochastic dynamic systems using polynomial chaos." [College Station, Tex. : Texas A&M University, 2008. http://hdl.handle.net/1969.1/ETD-TAMU-2853.

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11

Ayres, Daniel. "Uncertainty quantification in nuclear criticality modelling using methods of polynomial chaos." Thesis, Imperial College London, 2015. http://hdl.handle.net/10044/1/26993.

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In this thesis we use polynomial chaos expansions to represent the response of criticality calculations when they are subject to large numbers (many hundreds) of correlated nuclear data uncertainties. An adaptive high dimensional model representation (HDMR) is used to decompose the response parameter keff into a superposition of lower dimensional subspaces which are in-turn projected on to a polynomial basis. These projections are evaluated using an adaptive quadrature scheme which is used to infer the polynomial orders of the basis. The combination of adaptive HDMR and adaptive quadrature techniques results in a sparse polynomial expansion which has been optimised to represent the variance of the response with the minimum number of polynomials. The combined application of these techniques is illustrated using UOX and MOX pin cell problems with evaluated nuclear covariance data. We show that this approach to calculating the variance in keff is an order of magnitude more efficient when compared to Latin hypercube sampling with the same number of samples for problems involving up to 988 random dimensions. In the final chapter of this thesis, the adaptive HDMR and quadrature methods combined with polynomial chaos are applied to an industrially relevant problem; the computation of keff uncertainties due to evaluated covariance data. Uncertainties and first order sensitivities are computed from the polynomial chaos expansion which are compared to the results from the first order sensitivity method implemented in the Monte Carlo code MONK. We found that the local sensitivities and uncertainties derived from the PCE compare well with the MONK sensitivity method. These uncertainty quantification approaches were applied to fast spectrum uranium, plutonium and americium-241 critical assemblies. Comparisons between the uranium/plutonium and americium-241 uncertainties were made in the context of the 0.95 sub-critical limit. Suggestions for new sub-critical limits based on differing numbers of standard deviations below the mean values were proposed.

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12

Blatman, Géraud. "Adaptive sparse polynomial chaos expansions for uncertainty propagation and sensitivity analysis." Clermont-Ferrand 2, 2009. https://tel.archives-ouvertes.fr/tel-00440197.

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Cette thèse s'insère dans le contexte générale de la propagation d'incertitudes et de l'analyse de sensibilité de modèles de simulation numérique, en vue d'applications industrielles. Son objectif est d'effectuer de telles études en minimisant le nombre d'évaluations du modèle, potentiellement coûteuses. Le présent travail repose sur une approximation de la réponse du modèle sur la base du chaos polynomial (CP), qui permet de réaliser des post-traitements à un coût de calcul négligeable. Toutefois, l'ajustement du CP peut nécessiter un nombre conséquent d'appels au modèle si ce dernier dépend d'un nombre élevé de paramètres (e. G. Supérieur à 10). Pour contourner ce problème, on propose deux algorithmes pour ne sélectionner qu'un faible nombre de termes importants dans la représentation par CP, à savoir une procédure de régression pas-à-pas et une procédure basée sur la méthode de Least Angle Regression (LAR). Le faible nombre de coefficients associés aux CP creux obtenus peuvent ainsi être déterminés à partir d'un nombre réduit d'évaluations du modèle. Les méthodes sont validées sur des cas-tests académiques de mécanique, puis appliquées sur le cas industriel de l'analyse d'intégrité d'une cuve de réacteur à eau pressurisée. Les résultats obtenus confirment l'efficacité des méthodes proposées pour traiter les problèmes de propagation d'incertitudes et d'analyse de sensibilité en grande dimension

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13

Price, Darryl Brian. "Estimation of Uncertain Vehicle Center of Gravity using Polynomial Chaos Expansions." Thesis, Virginia Tech, 2008. http://hdl.handle.net/10919/33625.

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The main goal of this study is the use of polynomial chaos expansion (PCE) to analyze the uncertainty in calculating the lateral and longitudinal center of gravity for a vehicle from static load cell measurements. A secondary goal is to use experimental testing as a source of uncertainty and as a method to confirm the results from the PCE simulation. While PCE has often been used as an alternative to Monte Carlo, PCE models have rarely been based on experimental data. The 8-post test rig at the Virginia Institute for Performance Engineering and Research facility at Virginia International Raceway is the experimental test bed used to implement the PCE model. Experimental tests are conducted to define the true distribution for the load measurement systemsâ uncertainty. A method that does not require a new uncertainty distribution experiment for multiple tests with different goals is presented. Moved mass tests confirm the uncertainty analysis using portable scales that provide accurate results. The polynomial chaos model used to find the uncertainty in the center of gravity calculation is derived. Karhunen-Loeve expansions, similar to Fourier series, are used to define the uncertainties to allow for the polynomial chaos expansion. PCE models are typically computed via the collocation method or the Galerkin method. The Galerkin method is chosen as the PCE method in order to formulate a more accurate analytical result. The derivation systematically increases from one uncertain load cell to all four uncertain load cells noting the differences and increased complexity as the uncertainty dimensions increase. For each derivation the PCE model is shown and the solution to the simulation is given. Results are presented comparing the polynomial chaos simulation to the Monte Carlo simulation and to the accurate scales. It is shown that the PCE simulations closely match the Monte Carlo simulations.
Master of Science

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14

Shimp, Samuel Kline III. "Vehicle Sprung Mass Parameter Estimation Using an Adaptive Polynomial-Chaos Method." Thesis, Virginia Tech, 2008. http://hdl.handle.net/10919/32056.

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The polynomial-chaos expansion (PCE) approach to modeling provides an estimate of the probabilistic response of a dynamic system with uncertainty in the system parameters. A novel adaptive parameter estimation method exploiting the polynomial-chaos representation of a general quarter-car model is presented. Because the uncertainty was assumed to be concentrated in the sprung mass parameter, a novel pseudo mass matrix was developed for generating the state-space PCE model. In order to implement the PCE model in a real-time adaptation routine, a novel technique for representing PCE output equations was also developed. A simple parameter estimation law based on the output error between measured accelerations and PCE acceleration estimates was developed and evaluated through simulation and experiment. Simulation results of the novel adaptation algorithm demonstrate the estimation convergence properties as well as its limitations. The simulation results are further verified by a real-time experimental implementation on a quarter-car test rig. This work presents the first truly real-time implementation of a PCE model. The experimental real-time implementation of the novel adaptive PCE estimation method shows promising results by its ability to converge and maintain a stable estimate of the unknown parameter.
Master of Science

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15

SOLANO, ALEJANDRA CAMACHO. "UNCERTAINTY QUANTIFICATION IN OIL RESERVOIR SIMULATION VIA GENETIC PROGRAMMING AND CHAOS POLYNOMIAL." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2015. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=26216@1.

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PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO
COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR
PROGRAMA DE EXCELENCIA ACADEMICA
Os modelos de simulação de reservatórios estão sujeitos à incerteza presente em uma grande variedade de seus parâmetros de entrada. Esta incerteza é o resultado da heterogeneidade das formações geológicas, erros nas medições dos dados e da modelagem petrofísica, estrutural e do transporte dos fluidos no meio poroso. Uma quantificação precisa da incerteza requer, na maioria dos casos, uma quantidade elevada de simulações, o que é usualmente inviável se considerarmos o tempo consumido para simular modelos de grande escala. Por outro lado, uma avaliação adequada da incerteza aumenta a qualidade e robustez das decisões tomadas para o gerenciamento dos campos de petróleo. Com esta motivação, foi investigado o método das Expansões por Caos Polinomial (PCE, por suas siglas em inglês). PCE é uma técnica de convergência rápida utilizada para analisar como se propaga, na saída de um modelo, a incerteza presente nos parâmetros de entrada. Mediante PCE, pode-se representar a resposta aleatória de um modelo de simulação de reservatórios de petróleo como um polinômio, construído a partir de uma base de funções que dependem da distribuição de probabilidade das variáveis incertas de entrada. Por outro lado, quando a relação entre os parâmetros de entrada e a saída do modelo têm um componente não polinomial, o algoritmo de Programação Genética (PG) pode ser utilizado para representar esta dependência utilizando funções ou operadores mais complexos. PG é um algoritmo de regressão simbólica capaz de encontrar uma expressão aleatória explícita, que aproxime a saída de um modelo de simulação de reservatórios de petróleo, conhecendo-se a priori a distribuição de probabilidade dos parâmetros de entrada. Neste trabalho foram aplicadas as duas técnicas, antes mencionadas, num modelo de simulação de reservatórios baseado no campo PUNQ-S3, considerando até vinte e três parâmetros incertos durante um período de produção de 13 anos. Foi feita uma análise de incerteza, calculando-se a distribuição de probabilidade completa da saída do simulador. Os resultados foram comparados com o método de Monte Carlo, indicando um alto desempenho em termos de custo computacional e acurácia. Ambas as técnicas conseguem níveis de ajuste superiores a 80 porcento com uma quantidade de simulações consideravelmente baixa.
Reservoir simulation models are subject to uncertainty in a wide variety of its inputs. This uncertainty is a result of the heterogeneity of the geological formations, data measurement errors, and petrophysical, structural, and fluid transport in porous media modelling. An accurate uncertainty quantification requires, in most cases, a large number of simulations, which is unviable considering the time it takes to simulate large scale models. On the other hand, a proper uncertainty assessment, increases the robustness of the decision making process for the oil field management. To this end, the method of Polynomial Chaos Expansions (PCE) was studied. PCE is a fast paced convergence technique, used to analyze the uncertainty propagation of the input parameters all the way to the output of the model. Through PCE is possible to represent the response of an oil reservoir simulation model as a polynomial, built from a function basis, that depend on the probability distribution of the uncertain input variables. Furthermore, when the relationship between the input and output parameters of the model has a non-polynomial component, the algorithm of Genetic Programming (GP) can be used to represent this dependency by more elaborate functions or operators. GP is a symbolic regression algorithm, capable of finding an explicit expression that approximates the output of a reservoir simulation model, with prior knowledge of the probability distribution of the input parameters. In this work, the two previously mentioned techniques were applied in a reservoir simulation model, based on the oil field PUNQ-S3, considering up to twenty three uncertain parameters during a simulation period of 13 years. An uncertainty analysis of the output of the simulator was conducted, calculating the entire probability distribution. The results were compared to the Monte Carlo simulation method, presenting a satisfactory performance in terms of accuracy and computational cost. Both techniques show adjustment levels higher than 80 percent, with a considerable small amount simulations.

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16

Lee, Hyunwook. "A Polynomial Chaos Approach for Stochastic Modeling of Dynamic Wheel-Rail Friction." Diss., Virginia Tech, 2010. http://hdl.handle.net/10919/77195.

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Accurate estimation of the coefficient of friction (CoF) is essential to accurately modeling railroad dynamics, reducing maintenance costs, and increasing safety factors in rail operations. The assumption of a constant CoF is popularly used in simulation studies for ease of implementation, however many evidences demonstrated that CoF depends on various dynamic parameters and instantaneous conditions. In the real world, accurately estimating the CoF is difficult due to effects of various uncertain parameters, such as wheel and rail materials, rail roughness, contact patch, and so on. In this study, the newly developed 3-D nonlinear CoF model for the dry rail condition is introduced and the CoF variation is tested using this model with dynamic parameters estimated from the wheel-rail simulation model. In order to account for uncertain parameters, a stochastic analysis using the polynomial chaos (poly-chaos) theory is performed using the CoF and wheel-rail dynamics models. The wheel-rail system at a right traction wheel is modeled as a mass-spring-damper system to simulate the basic wheel-rail dynamics and the CoF variation. The wheel-rail model accounts for wheel-rail contact, creepage effect, and creep force, among others. Simulations are performed at train speed of 20 m/s for 4 sec using rail roughness as a unique excitation source. The dynamic simulation has been performed for the deterministic model and for the stochastic model. The dynamics results of the deterministic model provide the starting point for the uncertainty analysis. Six uncertain parameters have been studied with an assumption of 50% uncertainty, intentionally imposed for testing extreme conditions. These parameters are: the maximum amplitude of rail roughness (MARR), the wheel lateral displacement, the track stiffness and damping coefficient, the sleeper distance, and semi-elliptical contact lengths. A symmetric beta distribution is assumed for these six uncertain parameters. The PDF of the CoF has been obtained for each uncertain parameter study, for combinations of two different uncertain parameters, and also for combinations of three different uncertain parameters. The results from the deterministic model show acceptable vibration results for the body, the wheel, and the rail. The introduced CoF model demonstrates the nonlinear variation of the total CoF, the stick component, and the slip component. In addition, it captures the maximum CoF value (initial peak) successfully. The stochastic analysis results show that the total CoF PDF before 1 sec is dominantly affected by the stick phenomenon, while the slip dominantly influences the total CoF PDF after 1 sec. Although a symmetric distribution has been used for the uncertain parameters considered, the uncertainty in the response obtained displayed a skewed distribution for some of the situations investigated. The CoF PDFs obtained from simulations with combinations of two and three uncertain parameters have wider PDF ranges than those obtained for only one uncertain parameter. FFT analysis using the rail displacement has been performed for the qualitative validation of the stochastic simulation result due to the absence of the experimental data. The FFT analysis of the deterministic rail displacement and of the stochastic rail displacement with uncertainties demonstrates consistent trends commensurate with loss of tractive efficiency, such as the bandwidth broadening, peak frequency shifts, and side band occurrence. Thus, the FFT analysis validates qualitatively that the stochastic modeling with various uncertainties is well executed and is reflecting observable, real-world results. In conclusions, the development of an effective model which helps to understand the nonlinear nature of wheel-rail friction is critical to the progress of railroad component technology and rail safety. In the real world, accurate estimation of the CoF at the wheel-rail interface is very difficult since it is influenced by several uncertain parameters as illustrated in this study. Using the deterministic CoF value can cause underestimation or overestimation of CoF values leading to inaccurate decisions in the design of the wheel-rail system. Thus, the possible PDF ranges of the CoF according to key uncertain parameters must be considered in the design of the wheel-rail system.
Ph. D.

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17

Beddek, Karim. "Propagation d’incertitudes dans les modèles éléments finis en électromagnétisme : application au contrôle non destructif par courants de Foucault." Thesis, Lille 1, 2012. http://www.theses.fr/2012LIL10175.

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La quantification d’incertitudes est une démarche consistant à prendre en compte les incertitudes des coefficients caractéristiques (matériaux, géométries, sources ...) d’un modèle mathématique en vue d’estimer l’effet de ces méconnaissances sur les grandeurs physiques recherchées. Dans ce travail de thèse, nous nous sommes intéressés aux approches probabilistes de propagation d’incertitudes portées par les lois de comportement (perméabilités et conductivités) aux sein de modèles éléments finis de l’électromagnétisme quasi-statique de taille industrielle. Cette thèse vise à comparer les deux approches spectrales NISP et SSFEM qui sont basées sur une représentation fonctionnelle dans le chaos polynomial des grandeurs d’intérêt aléatoires. Cette étude de comparaison est effectuée en terme de précision numérique et de coût de calcul, et pour des grandeurs d’intérêt scalaires et vectorielles complexes. Les applications numériques nous ont montré que la SSFEM peut être assez compétitive par rapport à la NISP pour des problèmes probabilistes à grandes dimensions stochastiques. Il en résulte que celle-ci est la méthode de prédilection pour l’étude des systèmes électromagnétiques dont les lois de comportement des matériaux sont aléatoires. Enfin, les deux méthodes spectrales ont été appliquées sur un problème de détection de bouchage par la magnétite des plaques entretoises des générateurs de vapeur d’une centrale nucléaire. Dans cette étude probabiliste, nous nous sommes attelés à quantifier la contribution des incertitudes, subsistant dans les conductivités et perméabilités de la magnétite et de la plaque, à la variabilité des signaux et du ratio SAX
The uncertainty quantification technique aims to quantify the effect of uncertainties of input parameters of numerical models, e.g. material, geometry, source terms, on the quantity of interest. In this thesis, we focus on probabilistic approaches in order to spread uncertainties of magnetic and electric behavior laws over large scale electromagnetic finite element models. The main objective of this work is to compare two spectral stochastic methods (Non Intrusive Spectral Projection (NISP) and Spectral Stochastic Finite Element Method (SSFEM)), which are based on chaos polynomial representation of the random quantities. The comparison between the NISP and the SSFEM is carried out by confronting the computational costs and the precision when scalar and vector complex quantities of interest are computed. The numerical applications show that the SSFEM method become as competitive as the NISP method in terms of computational cost when solving probabilistic problems with large number of random parameters. Thus, the SSFEM method is chosen as the best adapted to solve electromagnetic problems when the behavior laws are random. In fact, the NISP method is inappropriate to compute vector complex quantities when equipped with adaptive sparse grid procedures. Finally, the NISP and SSFEM methods are used to study the clogging of the Tube Support Plate (TSP) of steam generators of nuclear power plants. The effect of uncertainties of the permeability and the conductivity of the TSP and the magnetite (clogging product) on the control signal and the SAX ratio is investigated

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18

Blatman, Géraud. "Chaos polynomial creux et adaptatif pour la propagation d'incertitudes et l'analyse de sensibilité." Phd thesis, Université Blaise Pascal - Clermont-Ferrand II, 2009. http://tel.archives-ouvertes.fr/tel-00440197.

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Cette thèse s'insère dans le contexte général de la propagation d'incertitudes et de l'analyse de sensibilité de modèles de simulation numérique, en vue d'applications industrielles. Son objectif est d'effectuer de telles études en minimisant le nombre d'évaluations du modèle, potentiellement coûteuses. Le présent travail repose sur une approximation de la réponse du modèle sur la base du chaos polynomial(CP), qui permet de réaliser des post-traitements à un coût de calcul négligeable. Toutefois, l'ajustement du CP peut nécessiter un nombre conséquent d'appels au modèle si ce dernier dépend d'un nombre élevé de paramètres (e.g. supérieur à 10). Pour contourner ce problème, on propose deux algorithmes pour ne sélectionner qu'un faible nombre de termes importants dans la représentation par CP, à savoir une procédure de régression pas-à-pas et une procédure basée sur la méthode de Least Angle Regression (LAR). Le faible nombre de coefficients associés aux CP creux obtenus peuvent ainsi être déterminés à partir d'un nombre réduit d'évaluations du modèle. Les méthodes sont validées sur des cas-tests académiques de mécanique, puis appliquées sur le cas industriel de l'analyse d'intégrité d'une cuve de réacteur à eau pressurisée. Les résultats obtenus confirment l'efficacité des méthodes proposées pour traiter les problèmes de propagation d'incertitudes et d'analyse de sensibilité en grande dimension.

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19

Li, Lin. "Treatment of Uncertainties in Vehicle and Terramechanics Systems Using a Polynomial Chaos Approach." Diss., Virginia Tech, 2008. http://hdl.handle.net/10919/29030.

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Mechanical systems always operate under some degree of uncertainty, which can be due to the inherent properties of the system parameters, to random inputs or external excitations, to poorly known parameters in the interface between different systems, or to inadequate knowledge of the dynamic process. Also, mechanical systems are large and highly nonlinear, while the magnitude of uncertainties may be very large. This dissertation addresses the critical need for understanding of the stochastic nature of mechanical system, especially vehicle and terramechanics systems, and need for developing efficient computational tools to model mechanical systems in the presence of parametric and external uncertainty. This dissertation investigates the influence of parametric and external uncertainties on vehicle dynamics and terramechanics. The uncertainties studied include parametric uncertainties, stochastic external excitations, and random variables between vehicle-terrain and vehicle-soil/snow interface. The methodology developed has been illustrated on a stochastic vehicle-terrain interaction model, a stochastic vehicle-soil interaction model, two stochastic tire-snow interaction models, and two stochastic tire-force relations. The uncertainties are quantified and propagated through vehicle and terramechanics systems using a polynomial chaos approach. Algorithms which can predict the geometry of the contact patch and the interfacial forces and torques on the vehicle-soil interfaces are developed. All stochastic models and algorithms are simulated for various scenarios and maneuvers. Numerical results are analyzed from the computational effort point of view, or from the angle of vehicle dynamics and terramechanics, and provide a deeper understanding of the evolution of stochastic vehicle and terramechanics systems. They can also be used in guiding vehicle design and development. This dissertation represents a pioneer study on stochastic vehicle dynamics and terramechanics. Moreover, the methodology developed is not limited to such systems. Any mechanical system with uncertainties can be treated using the polynomial chaos approach presented, considering their specific characteristics.
Ph. D.

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20

Cooper, Rachel Gray. "Augmented Neural Network Surrogate Models for Polynomial Chaos Expansions and Reduced Order Modeling." Thesis, Virginia Tech, 2021. http://hdl.handle.net/10919/103423.

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Mathematical models describing real world processes are becoming increasingly complex to better match the dynamics of the true system. While this is a positive step towards more complete knowledge of our world, numerical evaluations of these models become increasingly computationally inefficient, requiring increased resources or time to evaluate. This has led to the need for simplified surrogates to these complex mathematical models. A growing surrogate modeling solution is with the usage of neural networks. Neural networks (NN) are known to generalize an approximation across a diverse dataset and minimize the solution along complex nonlinear boundaries. Additionally, these surrogate models can be found using only incomplete knowledge of the true dynamics. However, NN surrogates often suffer from a lack of interpretability, where the decisions made in the training process are not fully understood, and the roles of individual neurons are not well defined. We present two solutions towards this lack of interpretability. The first focuses on mimicking polynomial chaos (PC) modeling techniques, modifying the structure of a NN to produce polynomial approximations of the underlying dynamics. This methodology allows for an extractable meaning from the network and results in improvement in accuracy over traditional PC methods. Secondly, we examine the construction of a reduced order modeling scheme using NN autoencoders, guiding the decisions of the training process to better match the real dynamics. This guiding process is performed via a physics-informed (PI) penalty, resulting in a speed-up in training convergence, but still results in poor performance compared to traditional schemes.
Master of Science
The world is an elaborate system of relationships between diverse processes. To accurately represent these relationships, increasingly complex models are defined to better match what is physically seen. These complex models can lead to issues when trying to use them to predict a realistic outcome, either requiring immensely powerful computers to run the simulations or long amounts of time to present a solution. To fix this, surrogates or approximations to these complex models are used. These surrogate models aim to reduce the resources needed to calculate a solution while remaining as accurate to the more complex model as possible. One way to make these surrogate models is through neural networks. Neural networks try to simulate a brain, making connections between some input and output given to the network. In the case of surrogate modeling, the input is some current state of the true process, and the output is what is seen later from the same system. But much like the human brain, the reasoning behind why choices are made when connecting the input and outputs is often largely unknown. Within this paper, we seek to add meaning to neural network surrogate models in two different ways. In the first, we change what each piece in a neural network represents to build large polynomials (e.g., $x^5 + 4x^2 + 2$) to approximate the larger complex system. We show that the building of these polynomials via neural networks performs much better than traditional ways to construct them. For the second, we guide the choices made by the neural network by enforcing restrictions in what connections it can make. We do this by using additional information from the larger system to ensure the connections made focus on the most important information first before trying to match the less important patterns. This guiding process leads to more information being captured when the surrogate model is compressed into only a few dimensions compared to traditional methods. Additionally, it allows for a faster learning time compared to similar surrogate models without the information.

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21

Prempraneerach, Pradya 1975. "Uncertainty analysis in a shipboard integrated power system using multi-element polynomial chaos." Thesis, Massachusetts Institute of Technology, 2007. http://hdl.handle.net/1721.1/40364.

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Thesis (Ph. D. in Ocean Engineering)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2007.
Errata, dated Oct. 30, 2007, inserted between pages 3 and 4 of text.
Includes bibliographical references (p. 301-307).
The integrated power system has become increasingly important in electric ships due to the integrated capability of high-power equipment, for example, electromagnetic rail guns, advance radar system, etc. Several parameters of the shipboard power system are uncertain, caused by a measurement difficulty, a temperature dependency, and random fluctuation of its environment. To date, there has been little if any studies which account for these stochastic effects in the large and complex shipboard power system from either an analytical or a numerical perspective. Furthermore, all insensitive parameters must be identified so that the stochastic analysis with the reduced dimensional parameters can accelerate the process. Therefore, this thesis is focused on two main issues - stochastic and sensitivity analysis - on the shipboard power system. The stochastic analysis of the large and complex nonlinear systems with the non-Gaussian random variables or processes, in their initial states or parameters, are prohibited analytically and very time consuming using the brute force Monte Carlo method. As a result, numerical stochastic solutions of these systems can be efficiently solved by the generalized Polynomial Chaos (gPC) and Probabilistic Collocation Method (PCM).
(cont.) In the case of the long-time integration and discontinuity in the stochastic solutions, the multi-element technique of PCM, which refines the solution in random space, can significantly improve the solutions' accuracy. Furthermore, the hybrid gPC+PCM is developed to extend the gPC ability to handle a system with nonlinear non-polynomial functions. Then, we systematically establish the convergence rate and compare the convergence performance among all numerical stochastic algorithms on various systems with both continuous and discontinuous solutions as a function of random dimension and the algorithms' accuracy governing parameters. To identify the most significant parameter in the large-scale complex systems, we propose new sensitivity analysis techniques - Monte Carlo Sampling, Collocation, Variance, and Inverse Variance methods - for static functions and show that they agree well with Morris method, which is one of the existing sensitivity analysis techniques for a function with large input dimensions. In addition, we extend the capability of the Sampling, Collocation, Variance, and the Morris methods to study both the parameters' sensitivity and the interaction of the ordinary differential equations.
(cont.) In each approach, both strength and limitations of the sensitivity ranking accuracy and the convergence performance are emphasized. The convergence rate of the Collocation and Variance methods are more than an order of magnitude faster than that of Morris and Sampling methods for low and medium parameters' dimensions. At last, we successfully apply both stochastic and sensitivity analysis techniques to the integrated shipboard power system, with both open-and close-loop control of the propulsion system, to study a propagation of uncertainties and rank parameters in the order of their importance, respectively.
by Pradya Prempraneerach.
Ph.D.in Ocean Engineering

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22

Xu, Yijun. "Uncertainty Quantification, State and Parameter Estimation in Power Systems Using Polynomial Chaos Based Methods." Diss., Virginia Tech, 2019. http://hdl.handle.net/10919/97876.

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It is a well-known fact that a power system contains many sources of uncertainties. These uncertainties coming from the loads, the renewables, the model and the measurement, etc, are influencing the steady state and dynamic response of the power system. Facing this problem, traditional methods, such as the Monte Carlo method and the Perturbation method, are either too time consuming or suffering from the strong nonlinearity in the system. To solve these, this Dissertation will mainly focus on developing the polynomial chaos based method to replace the traditional ones. Using it, the uncertainties from the model and the measurement are propagated through the polynomial chaos bases at a set of collocation points. The approximated polynomial chaos coefficients contain the statistical information. The method can greatly accelerate the calculation efficiency while not losing the accuracy, even when the system is highly stressed. In this dissertation, both the forward problem and the inverse problem of uncertainty quantification will be discussed. The forward problems will include the probabilistic power flow problem and statistical power system dynamic simulations. The generalized polynomial chaos method, the adaptive polynomial chaos-ANOVA method and the multi-element polynomial chaos method will be introduced and compared. The case studies show that the proposed methods have great performances in the statistical analysis of the large-scale power systems. The inverse problems will include the state and parameter estimation problem. A novel polynomial-chaos-based Kalman filter will be proposed. The comparison studies with other traditional Kalman filter demonstrate the good performances of the proposed Kalman filter. We further explored the area dynamic parameter estimation problem under the Bayesian inference framework. The polynomial-chaos-expansions are treated as the response surface of the full dynamic solver. Combing with hybrid Markov chain Monte Carlo method, the proposed method yields very high estimation accuracy while greatly reducing the computing time. For both the forward problem and the inverse problems, the polynomial chaos based methods haven shown great advantages over the traditional methods. These computational techniques can improve the efficiency and accuracy in power system planning, guarantee the rationality and reliability in power system operations, and, finally, speed up the power system dynamic security assessment.
PHD

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23

Nabavi, Seyed Ghavamoddin. "Statistical Analysis of Steady State Response in RF Circuits via Decoupled Generalized Polynomial Chaos." Thesis, Université d'Ottawa / University of Ottawa, 2016. http://hdl.handle.net/10393/35293.

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One of the major factors in RF circuit design is the ability to predict the performance of these circuits in the presence of uncertainty in the key design parameters. This is referred to as uncertainty quantification in the mathematical literature. Uncertainty about the key design parameters arises mainly from the difficulty of controlling the physical or geometrical features of the underlying design, especially at the nanometer level. With the constant trend to scale down the process feature size, uncertainty quantification becomes crucial in shortening the design time. This thesis presents a new approach to statistically characterize the variability of the Harmonic Balance analysis and its application to Intermodulation distortion analysis in the presence of uncertainty in the design parameters. The new approach is based on the concept of Polynomial Chaos (PC) and Stochastic Galerkin (SG) methods. However, unlike the traditional PC, the proposed approach adopts a new mathematical formulation that decouples the Polynomial Chaos problem into several problems whose sizes are equal to the size of the original Harmonic Balance problem. The proposed algorithm produces significant CPU savings with equivalent accuracy to traditional Monte Carlo and standard PC approaches.

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24

Kersaudy, Pierric. "Modélisation statistique de l'exposition humaine aux ondes radiofréquences." Thesis, Paris Est, 2015. http://www.theses.fr/2015PESC1120/document.

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L'objectif de cette thèse est de traiter la problématique de la caractérisation et du traitement de la variabilité de l'exposition humaine aux ondes radio à travers l'utilisation de la dosimétrie numérique. En effet, si les progrès dans le domaine du calcul hautes performances ont contribué à significativement réduire les temps de simulation pour l'évaluation de l'exposition humaine, ce calcul du débit d'absorption spécifique reste un processus coûteux en temps. Avec la grande variabilité des usages, cette contrainte fait que la prise en compte de l'influence de paramètres d'entrée aléatoires sur l'exposition ne peut se faire par des méthodes classiques telles que les simulations de Monte Carlo. Nous proposons dans ces travaux deux approches pour répondre à cette problématique. La première s'appuie sur l'utilisation et l'hybridation de méthodes de construction de modèles de substitution afin d'étudier l'influence globale des paramètres d'entrée. La deuxième vise à l'évaluation efficace et parcimonieuse des quantiles à 95% des distributions de sortie et s'appuie sur le développement d'une méthode de planification d'expériences adaptative et orientée couplée à la construction de modèles de substitution. Les méthodes proposées dans ce manuscrit sont comparées et testées sur des exemples analytiques et ensuite appliquées à des problèmes concrets issus de la dosimétrie numérique
The purpose of this thesis is to deal with the problem of the management and the characterization of the variability of the human exposure to radio frequency waves through the use of the numerical dosimetry. As a matter of fact, if the recent advances in the high performance computing domain led to reduce significantly the simulation duration for the evaluation of the human exposure, this computation of the specific absorption rate remains a time-consuming process. With the variability of the usage, this constraint does not allow the analysis of the influence of random input parameters on the exposure to be achieved with classical approaches such as Monte Carlo simulations. In this work, two approaches are proposed to address this problem. The first one is based on the use and the hybridization of construction methods of surrogate models in order to study the global influence of the input parameters. The second one aims at assessing efficiently the 95th-percentiles of the output distributions in a parcimonous way. It is based on the development of an adaptive and oriented methodology of design of experiments combined with the construction of surrogate models. In this manuscript, the proposed methods are compared and tested on analytical examples and then applicated to full-scale problems from the numerical dosimetry

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25

Holdorf, Lopez Rafael. "Optimisation en présence d’incertitudes." Thesis, Rouen, INSA, 2010. http://www.theses.fr/2010ISAM0009.

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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

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26

Owen, Nathan Edward. "A comparison of polynomial chaos and Gaussian process emulation for uncertainty quantification in computer experiments." Thesis, University of Exeter, 2017. http://hdl.handle.net/10871/29296.

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Computer simulation of real world phenomena is now ubiquitous in science, because experimentation in the field can be expensive, time-consuming, or impossible in practice. Examples include climate science, where future climate is examined under global warming scenarios, and cosmology, where the evolution of galaxies is studied from the beginning of the universe to present day. Combining complex mathematical models and numerical procedures to solve them in a computer program, these simulators are computationally expensive, in that they can take months to complete a single run. The practice of using a simulator to understand reality raises some interesting scientific questions, and there are many sources of uncertainty to consider. For example, the discrepancy between the simulator and the real world process. The field of uncertainty quantification is concerned with the characterisation and reduction of all uncertainties present in computational and real world problems. A key bottleneck in any uncertainty quantification analysis is the cost of evaluating the simulator. The solution is to replace the expensive simulator with a surrogate model, which is computationally faster to run, and can be used in subsequent analyses. Polynomial chaos and Gaussian process emulation are surrogate models developed independently in the engineering and statistics communities respectively over the last 25 years. Despite tackling similar problems in the field, there has been little interaction and collaboration between the two communities. This thesis provides a critical comparison of the two methods for a range of criteria and examples, from simple test functions to simulators used in industry. Particular focus is on the approximation accuracy of the surrogates under changes in the size and type of the experimental design. It is concluded that one method does not unanimously outperform the other, but advantages can be gained in some cases, such that the preferred method depends on the modelling goals of the practitioner. This is the first direct comparison of polynomial chaos and Gaussian process emulation in the literature. This thesis also proposes a novel methodology called probabilistic polynomial chaos, which is a hybrid of polynomial chaos and Gaussian process emulation. The approach draws inspiration from an emerging field in scientific computation known as probabilistic numerics, which treats classical numerical methods as statistical inference problems. In particular, a probabilistic integration technique called Bayesian quadrature, which employs Gaussian process emulators, is applied to a traditional form of polynomial chaos. The result is a probabilistic version of polynomial chaos, providing uncertainty information where the simulator has not yet been run.

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27

Blanchard, Emmanuel. "Polynomial Chaos Approaches to Parameter Estimation and Control Design for Mechanical Systems with Uncertain Parameters." Diss., Virginia Tech, 2010. http://hdl.handle.net/10919/26727.

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Mechanical systems operate under parametric and external excitation uncertainties. The polynomial chaos approach has been shown to be more efficient than Monte Carlo approaches for quantifying the effects of such uncertainties on the system response. This work uses the polynomial chaos framework to develop new methodologies for the simulation, parameter estimation, and control of mechanical systems with uncertainty. This study has led to new computational approaches for parameter estimation in nonlinear mechanical systems. The first approach is a polynomial-chaos based Bayesian approach in which maximum likelihood estimates are obtained by minimizing a cost function derived from the Bayesian theorem. The second approach is based on the Extended Kalman Filter (EKF). The error covariances needed for the EKF approach are computed from polynomial chaos expansions, and the EKF is used to update the polynomial chaos representation of the uncertain states and the uncertain parameters. The advantages and drawbacks of each method have been investigated. This study has demonstrated the effectiveness of the polynomial chaos approach for control systems analysis. For control system design the study has focused on the LQR problem when dealing with parametric uncertainties. The LQR problem was written as an optimality problem using Lagrange multipliers in an extended form associated with the polynomial chaos framework. The solution to the Hâ problem as well as the H2 problem can be seen as extensions of the LQR problem. This method might therefore have the potential of being a first step towards the development of computationally efficient numerical methods for Hâ design with parametric uncertainties. I would like to gratefully acknowledge the support provided for this work under NASA Grant NNL05AA18A.
Ph. D.

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28

Alkhateeb, Osama. "DATA-DRIVEN UNCERTAINTY QUANTIFICATION IN APPLICATIONS OF ELECTROMAGNETICS AND WIRELESS COMMUNICATION VIA ARBITRARY POLYNOMIAL CHAOS." University of Akron / OhioLINK, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=akron1509988525627307.

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29

Cooper, Michele Desiree. "Control Design and Model Validation for Applications in Nonlinear Vessel Dynamics." Diss., Virginia Tech, 2015. http://hdl.handle.net/10919/52905.

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In recent decades, computational models have become critical to how engineers and mathematicians understand nature; as a result they have become an integral part of the design process in most engineering disciplines. Moore's law anticipates computing power doubling every two years; a prediction that has historically been realized. As modern computing power increases, problems that were previously too complex to solve by hand or by previous computing abilities become tractable. This has resulted in the development of increasingly complex computational models simulating increasingly complex dynamics. Unfortunately, this has also resulted in increased challenges in fields related to model development, such as model validation and model based control, which are needed to make models useful in the real world. Much of the validation literature to date has focused on spatial and spatiotemporal simulations; validation approaches are well defined for such models. For most time series simulations, simulated and experimental trajectories can be directly compared negating the need for specialized validation tools. In the study of some ship motion behavior, chaos exists, which results in chaotic time series simulations. This presents novel challenges for validation; direct comparison may not be the most apt approach. For these applications, there is a need to develop appropriate metrics for model validation. A major thrust of the current work seeks to develop a set of validation metrics for such chaotic time series data. A complementary but separate portion of work investigates Non-Intrusive Polynomial Chaos as an approach to reduce the computational costs associated with uncertainty analysis and other stochastic investigations into the behavior of nonlinear, chaotic models. A final major thrust of this work focuses on contributing to the control of nonlinear marine systems, specifically the autonomous recovery of an unmanned surface vehicle utilizing motion prediction information. The same complexity and chaotic nature that makes the validation of ship motion models difficult can also make the development of reliable, robust controllers difficult as well. This body of work seeks to address several facets of this broad need that has developed due to our increased computational abilities by providing validation metrics and robust control laws.
Ph. D.

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30

Yadav, Vaibhav. "Novel Computational Methods for Solving High-Dimensional Random Eigenvalue Problems." Diss., University of Iowa, 2013. https://ir.uiowa.edu/etd/4927.

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The primary objective of this study is to develop new computational methods for solving a general random eigenvalue problem (REP) commonly encountered in modeling and simulation of high-dimensional, complex dynamic systems. Four major research directions, all anchored in polynomial dimensional decomposition (PDD), have been defined to meet the objective. They involve: (1) a rigorous comparison of accuracy, efficiency, and convergence properties of the polynomial chaos expansion (PCE) and PDD methods; (2) development of two novel multiplicative PDD methods for addressing multiplicative structures in REPs; (3) development of a new hybrid PDD method to account for the combined effects of the multiplicative and additive structures in REPs; and (4) development of adaptive and sparse algorithms in conjunction with the PDD methods. The major findings are as follows. First, a rigorous comparison of the PCE and PDD methods indicates that the infinite series from the two expansions are equivalent but their truncations endow contrasting dimensional structures, creating significant difference between the two approximations. When the cooperative effects of input variables on an eigenvalue attenuate rapidly or vanish altogether, the PDD approximation commits smaller error than does the PCE approximation for identical expansion orders. Numerical analysis reveal higher convergence rates and significantly higher efficiency of the PDD approximation than the PCE approximation. Second, two novel multiplicative PDD methods, factorized PDD and logarithmic PDD, were developed to exploit the hidden multiplicative structure of an REP, if it exists. Since a multiplicative PDD recycles the same component functions of the additive PDD, no additional cost is incurred. Numerical results show that indeed both the multiplicative PDD methods are capable of effectively utilizing the multiplicative structure of a random response. Third, a new hybrid PDD method was constructed for uncertainty quantification of high-dimensional complex systems. The method is based on a linear combination of an additive and a multiplicative PDD approximation. Numerical results indicate that the univariate hybrid PDD method, which is slightly more expensive than the univariate additive or multiplicative PDD approximations, yields more accurate stochastic solutions than the latter two methods. Last, two novel adaptive-sparse PDD methods were developed that entail global sensitivity analysis for defining the relevant pruning criteria. Compared with the past developments, the adaptive-sparse PDD methods do not require its truncation parameter(s) to be assigned a priori or arbitrarily. Numerical results reveal that an adaptive-sparse PDD method achieves a desired level of accuracy with considerably fewer coefficients compared with existing PDD approximations.

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31

Aguirre, Luis Antonio. "Application of global polynomial models in the identification, analysis and control of nonlinear dynamics and chaos." Thesis, University of Sheffield, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.324232.

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32

Schick, Michael [Verfasser], and V. [Akademischer Betreuer] Heuveline. "Uncertainty Quantification for Stochastic Dynamical Systems : Spectral Methods using Generalized Polynomial Chaos / Michael Schick. Betreuer: V. Heuveline." Karlsruhe : KIT-Bibliothek, 2012. http://d-nb.info/101936193X/34.

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33

Song, Chen [Verfasser], and Vincent [Akademischer Betreuer] Heuveline. "Uncertainty Quantification for a Blood Pump Device with Generalized Polynomial Chaos Expansion / Chen Song ; Betreuer: Vincent Heuveline." Heidelberg : Universitätsbibliothek Heidelberg, 2018. http://d-nb.info/1177252406/34.

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34

Langewisch, Dustin R. "Application of the polynomial chaos expansion to multiphase CFD : a study of rising bubbles and slug flow." Thesis, Massachusetts Institute of Technology, 2014. http://hdl.handle.net/1721.1/92097.

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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Nuclear Science and Engineering, 2014.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 157-167).
Part I of this thesis considers subcooled nucleate boiling on the microscale, focusing on the analysis of heat transfer near the Three-Phase (solid, liquid, and vapor) contact Line (TPL) region. A detailed derivation of one representative TPL model is presented. From this work, it was ultimately concluded that heat transfer in the vicinity of the TPL is rather unimportant in the overall quantification of nucleate boiling heat transfer; despite the extremely high heat fluxes that are attainable, it is limited to a very small region so the net heat transfer from this region is comparatively small. It was further concluded that many of the so-called microlayer heat transfer models appearing in the literature are actually models for TPL heat transfer; these models do not model the experimentally observed microlayer. This portion of the project was terminated early, however, in order to focus on the application of advanced computational uncertainty quantification methods to computational multiphase fluid dynamics (Part II). Part II discusses advanced uncertainty quantification (UQ) methods for long-running numerical models, namely computational multiphase fluid dynamics (CMFD) simulations. We consider the problem of how to efficiently propagate uncertainties in the model inputs (e.g., fluid properties, such as density, viscosity, etc.) through a computationally demanding model. The challenge is chiefly a matter of economics-the long run-time of these simulations limits the number of samples that one can reasonably obtain (i.e., the number of times the simulation can be run). Chapter 2 introduces the generalized Polynomial Chaos (gPC) expansion, which has shown promise for reducing the computational cost of performing UQ for a large class of problems, including heat transfer and single phase, incompressible flow simulations; example applications are demonstrated in Chapter 2. One of main objectives of this research was to ascertain whether this promise extends to realm of CMFD applications, and this is the topic of Chapters 3 and 4; Chapter 3 covers the numerical simulation of a single bubble rising in a quiescent liquid bath. The pertinent quantities from these simulations are the terminal velocity of the bubble and terminal bubble shape. the simulations were performed using the open source gerris flow solver. A handful of test cases were performed to validate the simulation results against available experimental data and numerical results from other authors; the results from gerris were found to compare favorably. Following the validation, we considered two uncertainty quantifications problems. In the first problem, the viscosity of the surrounding liquid is modeled as a uniform random variable and we quantify the resultant uncertainty in the bubbles terminal velocity. The second example is similar, except the bubble's size (diameter) is modeled as a log-normal random variable. In this case, the Hermite expansion is seen to converge almost immediately; a first-order Hermite expansion computed using 3 model evaluations is found to capture the terminal velocity distribution almost exactly. Both examples demonstrate that NISP can be successfully used to efficiently propagate uncertainties through CMFD models. Finally, we describe a simple technique to implement a moving reference frame in gerris. Chapter 4 presents an extensive study of the numerical simulation of capillary slug flow. We review existing correlations for the thickness of the liquid film surrounding a Taylor bubble and the pressure drop across the bubble. Bretherton's lubrication analysis, which yields analytical predictions for these quantities when inertial effects are negligible and Ca[beta] --> o, is considered in detail. In addition, a review is provided of film thickness correlations that are applicable for high Cab or when inertial effects are non-negligible. An extensive computational study was undertaken with gerris to simulate capillary slug flow under a variety of flow conditions; in total, more than two hundred simulations were carried out. The simulations were found to compare favorably with simulations performed previously by other authors using finite elements. The data from our simulations have been used to develop a new correlation for the film thickness and bubble velocity that is generally applicable. While similar in structure to existing film thickness correlations, the present correlation does not require the bubble velocity to be known a priori. We conclude with an application of the gPC expansion to quantify the uncertainty in the pressure drop in a channel in slug flow when the bubble size is described by a probability distribution. It is found that, although the gPC expansion fails to adequately quantify the uncertainty in field quantities (pressure and velocity) near the liquid-vapor interface, it is nevertheless capable of representing the uncertainty in other quantities (e.g., channel pressure drop) that do not depend sensitively on the precise location of the interface.
by Dustin R. Langewisch.
Ph. D.

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35

Bazargan, Hamid. "An efficient polynomial chaos-based proxy model for history matching and uncertainty quantification of complex geological structures." Thesis, Heriot-Watt University, 2014. http://hdl.handle.net/10399/2757.

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A novel polynomial chaos proxy-based history matching and uncertainty quantification method is presented that can be employed for complex geological structures in inverse problems. For complex geological structures, when there are many unknown geological parameters with highly nonlinear correlations, typically more than 106 full reservoir simulation runs might be required to accurately probe the posterior probability space given the production history of reservoir. This is not practical for high-resolution geological models. One solution is to use a "proxy model" that replicates the simulation model for selected input parameters. The main advantage of the polynomial chaos proxy compared to other proxy models and response surfaces is that it is generally applicable and converges systematically as the order of the expansion increases. The Cameron and Martin theorem 2.24 states that the convergence rate of the standard polynomial chaos expansions is exponential for Gaussian random variables. To improve the convergence rate for non-Gaussian random variables, the generalized polynomial chaos is implemented that uses an Askey-scheme to choose the optimal basis for polynomial chaos expansions [199]. Additionally, for the non-Gaussian distributions that can be effectively approximated by a mixture of Gaussian distributions, we use the mixture-modeling based clustering approach where under each cluster the polynomial chaos proxy converges exponentially fast and the overall posterior distribution can be estimated more efficiently using different polynomial chaos proxies. The main disadvantage of the polynomial chaos proxy is that for high-dimensional problems, the number of the polynomial chaos terms increases drastically as the order of the polynomial chaos expansions increases. Although different non-intrusive methods have been developed in the literature to address this issue, still a large number of simulation runs is required to compute high-order terms of the polynomial chaos expansions. This work resolves this issue by proposing the reduced-terms polynomial chaos expansion which preserves only the relevant terms in the polynomial chaos representation. We demonstrated that the sparsity pattern in the polynomial chaos expansion, when used with the Karhunen-Loéve decomposition method or kernel PCA, can be systematically captured. A probabilistic framework based on the polynomial chaos proxy is also suggested in the context of the Bayesian model selection to study the plausibility of different geological interpretations of the sedimentary environments. The proposed surrogate-accelerated Bayesian inverse analysis can be coherently used in practical reservoir optimization workflows and uncertainty assessments.

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Dessombz, Olivier. "Analyse dynamique de structures comportant des paramètres incertains." Ecully, Ecole centrale de Lyon, 2000. http://www.theses.fr/2000ECDL0036.

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Dans le cadre de la mobilisation de structures comportant des paramètres incertains, on s'intéresse aux caractéristiques des réponses statiques et dynamiques de systèmes mécaniques. On distingue dans cette étude le cas de paramètres aléatoires à loi de probabilité connue et le cas de variables dont on ne connaît que les bornes. Dans cette optique, on s'applique dans une première partie à décrire les réponses dynamiques, aussi bien les fonctions de transfert que les modes propres, de structures comportant des paramètres modélisés comme des variables aléatoires. Pour ce faire, on utilise une méthode de projection sur une base de polynômes orthogonaux (chaos polynomial), qui permet d'obtenir les caractéristiques principales des réponses. Dans une deuxième partie, on utilise l'arihmétique des intervalles pour résoudre les problèmes statiques et dynamiques. Après avoir proposé une formulation adaptée à la modélisation des systèmes mécaniques, on reformule un algorithme de résolution de systèmes linéaires intervalles, qu'on utilise alors pour trouver les enveloppes des réponses cherchées
We are interested in the modelling of structures with uncertain parameters. We focus on the characteristics of static and dynamic responses of such mechanical systems. We distinguish in this study two cases : first, the case of random parameters with a known probability law and second the case of variables of which only the bounds are known. In a first part, we investigate the case of structures with uncertain parameters modelled as random variables. We are particularly interested in the dynamic responses, as well the frequency response functions as the eigenmodes. An inovative method is carried out, which consists in a projection on orthogonal polynomial (polynomial chaos) that leads to the main stochastic characteristics of the responses. In a second part, we use the interval arithmetic to solve static and dynamic problems. We first propose an adapted formulation of the mathematical problems with respect to the finite element modeling of mechanical systems. We then introduce a new formulation of an iterative algorithm that leads to enveloppes of responses for interval linear systems

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37

Kreuter, Daniel Christopher. "Bestimmung effektiver Materialkennwerte mit Hilfe modaler Ansätze bei unsicheren Eingangsgrößen." Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2016. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-191159.

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In dieser Arbeit wird für Strukturen, die im makroskopischen aufgrund unterschiedlicher Materialeigenschaften oder komplexer Geometrien eine hohe Netzfeinheit für Finite-Elemente-Berechnungen benötigen, eine neue Möglichkeit zur Berechnung effektiver Materialkennwerte vorgestellt. Durch einen modalen Ansatz, bei dem, je nach Struktur analytisch oder numerisch, mit Hilfe der modalen Kennwerte die Formänderungsenergie eines repräsentativen Volumens der Originalstruktur mit der Formänderungsenergie eines äquivalenten homogen Vergleichsvolumens verglichen wird, können effektive Materialkennwerte ermittelt und daran anschließend eine Finite-Elemente-Berechnung mit einem im Vergleich zum Originalmodell sehr viel gröberen Netz durchgeführt werden, was eine enorme Zeiteinsparung mit sich bringt. Weiterhin enthält die vorgestellte Methode die Möglichkeit, unsichere Eingabeparameter wie Geometrieabmessungen oder Materialkennwerte mit Hilfe der polynomialen Chaos Expansion zu approximieren, um Möglichkeiten zur Aussage bzgl. der daraus resultierenden Verteilungen modaler Kenngrößen auf eine schnelle und effektive Weise zu gewinnen.

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De, La Torre Gerardo. "Autonomous suspended load operations via trajectory optimization and variational integrators." Diss., Georgia Institute of Technology, 2015. http://hdl.handle.net/1853/53932.

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Advances in machine autonomy hold great promise in advancing technology, economic markets, and general societal well-being. For example, the progression of unmanned air systems (UAS) research has demonstrated the effectiveness and reliability of these autonomous systems in performing complex tasks. UAS have shown to not only outperformed human pilots in some tasks, but have also made novel applications not possible for human pilots practical. Nevertheless, human pilots are still favored when performing specific challenging tasks. For example, transportation of suspended (sometimes called slung or sling) loads requires highly skilled pilots and has only been performed by UAS in highly controlled environments. The presented work begins to bridge this autonomy gap by proposing a trajectory optimization framework for operations involving autonomous rotorcraft with suspended loads. The framework generates optimized vehicle trajectories that are used by existing guidance, navigation, and control systems and estimates the state of the non-instrumented load using a downward facing camera. Data collected from several simulation studies and a flight test demonstrates the proposed framework is able to produce effective guidance during autonomous suspended load operations. In addition, variational integrators are extensively studied in this dissertation. The derivation of a stochastic variational integrator is presented. It is shown that the presented stochastic variational integrator significantly improves the performance of the stochastic differential dynamical programming and the extended Kalman filter algorithms. A variational integrator for the propagation of polynomial chaos expansion coefficients is also presented. As a result, the expectation and variance of the trajectory of an uncertain system can be accurately predicted.

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39

Mühlpfordt, Tillmann [Verfasser], and V. [Akademischer Betreuer] Hagenmeyer. "Uncertainty Quantification via Polynomial Chaos Expansion – Methods and Applications for Optimization of Power Systems / Tillmann Mühlpfordt ; Betreuer: V. Hagenmeyer." Karlsruhe : KIT-Bibliothek, 2020. http://d-nb.info/1203211872/34.

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40

Guerine, Ahmed. "Contribution à l'étude du comportement dynamique d'un système d'engrenage en présence d'incertitudes." Thesis, Rouen, INSA, 2016. http://www.theses.fr/2016ISAM0013/document.

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Dans le cadre de la présente thèse, on a procédé à l’étude du comportement dynamique d’un système d’engrenage comportant des paramètres incertains. Une des principales hypothèses faite dans l’utilisation des méthodes de prise en compte des incertitudes, est que le modèle est déterministe, c’est-à-dire que les paramètres utilisés dans le modèle ont une valeur définie et invariante. Par ailleurs, la connaissance du domaine de variation de la réponse dynamique du système dues aux incertitudes qui découle des coefficients d’amortissement, des raideurs d’engrènement, la présence de frottement entre les pièces, les défauts de montage et de fabrication ou l’inertie des pales dans le cas d’éolienne est essentielle. Pour cela, dans la première partie, on s’applique à décrire la réponse dynamique d’une transmission par engrenage comportant des paramètres modélisés par des variables aléatoires. Pour ce faire, nous utilisons la simulation de Monte Carlo, la méthode de perturbation et la méthode de projection sur un chaos polynomial. Dans la seconde partie,deux approches sont utilisées pour analyser le comportement dynamique d’un système d’engrenage d’éolienne : l’approche probabiliste et l’approche ensembliste basée sur la méthode d’analyse par intervalles. L'objectif consiste à comparer les deux approches pour connaitre leurs avantages et inconvénients en termes de précision et temps de calcul
In the present work, the dynamic behavior of a gear system with uncertain parameters is studied. One of the principal hypotheses in the use of methods for taking into account uncertainties is that the model is deterministic, that is to say that parameters used in the model have a defined and fixed value. Furthermore, the knowledge of variation response of a gear system involving damping coefficients, mesh stiffness, friction coefficient, assembly defect, manufacturing defect or the input blades in the case of wind turbine is essential. In the first part, we investigate the dynamic response of a gear system with uncertain parameters modeled as random variables. A Monte Carlo simulation, a perturbation method and a polynomial chaos method are carried out. In the second part, two approaches are used to analyze the dynamic behavior of a wind turbine gear system : the probabilistic approach and the interval analysis method. The objective is to compare the two approaches to define their advantages and disadvantages in terms of precision and computation time

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41

Dammak, Khalil. "Prise en compte des incertitudes des problèmes en vibro-acoustiques (ou interaction fluide-structure)." Thesis, Normandie, 2018. http://www.theses.fr/2018NORMIR19/document.

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Ce travail de thèse porte sur l’analyse robuste et l’optimisation fiabiliste des problèmes vibro-acoustiques (ou en interaction fluide-structure) en tenant en compte des incertitudes des paramètres d’entrée. En phase de conception et de dimensionnement, il parait intéressant de modéliser les systèmes vibro-acoustiques ainsi que leurs variabilités qui peuvent être essentiellement liées à l’imperfection de la géométrie ainsi qu’aux caractéristiques des matériaux. Il est ainsi important, voire indispensable, de tenir compte de la dispersion des lois de ces paramètres incertains afin d’en assurer une conception robuste. Par conséquent, l’objectif est de déterminer les capacités et les limites, en termes de précision et de coûts de calcul, des méthodes basées sur les développements en chaos polynomiaux en comparaison avec la technique référentielle de Monte Carlo pour étudier le comportement mécanique des problèmes vibro-acoustique comportant des paramètres incertains. L’étude de la propagation de ces incertitudes permet leur intégration dans la phase de conception. Le but de l’optimisation fiabiliste Reliability-Based Design Optimization (RBDO) consiste à trouver un compromis entre un coût minimum et une fiabilité accrue. Par conséquent, plusieurs méthodes, telles que la méthode hybride (HM) et la méthode Optimum Safety Factor (OSF), ont été développées pour atteindre cet objectif. Pour remédier à la complexité des systèmes vibro-acoustiques comportant des paramètres incertains, nous avons développé des méthodologies spécifiques à cette problématique, via des méthodes de méta-modèlisation, qui nous ont permis de bâtir un modèle de substitution vibro-acoustique, qui satisfait en même temps l’efficacité et la précision du modèle. L’objectif de cette thèse, est de déterminer la meilleure méthodologie à suivre pour l’optimisation fiabiliste des systèmes vibro-acoustiques comportant des paramètres incertains
This PhD thesis deals with the robust analysis and reliability optimization of vibro-acoustic problems (or fluid-structure interaction) taking into account the uncertainties of the input parameters. In the design and dimensioning phase, it seems interesting to model the vibro-acoustic systems and their variability, which can be mainly related to the imperfection of the geometry as well as the characteristics of the materials. It is therefore important, if not essential, to take into account the dispersion of the laws of these uncertain parameters in order to ensure a robust design. Therefore, the purpose is to determine the capabilities and limitations, in terms of precision and computational costs, of methods based on polynomial chaos developments in comparison with the Monte Carlo referential technique for studying the mechanical behavior of vibro-acoustic problems with uncertain parameters. The study of the propagation of these uncertainties allows their integration into the design phase. The goal of the reliability-Based Design Optimization (RBDO) is to find a compromise between minimum cost and a target reliability. As a result, several methods, such as the hybrid method (HM) and the Optimum Safety Factor (OSF) method, have been developed to achieve this goal. To overcome the complexity of vibro-acoustic systems with uncertain parameters, we have developed methodologies specific to this problem, via meta-modeling methods, which allowed us to build a vibro-acoustic surrogate model, which at the same time satisfies the efficiency and accuracy of the model. The objective of this thesis is to determine the best methodology to follow for the reliability optimization of vibro-acoustic systems with uncertain parameters

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42

Piprek, Patrick [Verfasser], Florian [Akademischer Betreuer] Holzapfel, Sébastien [Gutachter] Gros, and Florian [Gutachter] Holzapfel. "Robust Trajectory Optimization Applying Chance Constraints and Generalized Polynomial Chaos / Patrick Piprek ; Gutachter: Sébastien Gros, Florian Holzapfel ; Betreuer: Florian Holzapfel." München : Universitätsbibliothek der TU München, 2020. http://d-nb.info/1211086992/34.

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43

Kamath, Atul Krishna. "Surrogate-assisted optimisation-based verification & validation." Thesis, University of Exeter, 2014. http://hdl.handle.net/10871/15637.

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This thesis deals with the application of optimisation based Validation and Verification (V&V) analysis on aerospace vehicles in order to determine their worst case performance metrics. To this end, three aerospace models relating to satellite and launcher vehicles provided by European Space Agency (ESA) on various projects are utilised. As a means to quicken the process of optimisation based V&V analysis, surrogate models are developed using polynomial chaos method. Surro- gate models provide a quick way to ascertain the worst case directions as computation time required for evaluating them is very small. A sin- gle evaluation of a surrogate model takes less than a second. Another contribution of this thesis is the evaluation of operational safety margin metric with the help of surrogate models. Operational safety margin is a metric defined in the uncertain parameter space and is related to the distance between the nominal parameter value and the first instance of performance criteria violation. This metric can help to gauge the robustness of the controller but requires the evaluation of the model in the constraint function and hence could be computationally intensive. As surrogate models are computationally very cheap, they are utilised to rapidly compute the operational safety margin metric. But this metric focuses only on finding a safe region around the nominal parameter value and the possibility of other disjoint safe regions are not explored. In order to find other safe or failure regions in the param- eter space, the method of Bernstein expansion method is utilised on surrogate polynomial models to help characterise the uncertain param- eter space into safe and failure regions. Furthermore, Binomial failure analysis is used to assign failure probabilities to failure regions which might help the designer to determine if a re-design of the controller is required or not. The methodologies of optimisation based V&V, surrogate modelling, operational safety margin, Bernstein expansion method and risk assessment have been combined together to form the WCAT-II MATLAB toolbox.

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44

Yu, Hang. "Reliability-based design optimization of structures : methodologies and applications to vibration control." Phd thesis, Ecole Centrale de Lyon, 2011. http://tel.archives-ouvertes.fr/tel-00769937.

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Deterministic design optimization is widely used to design products or systems. However, due to the inherent uncertainties involved in different model parameters or operation processes, deterministic design optimization without considering uncertainties may result in unreliable designs. In this case, it is necessary to develop and implement optimization under uncertainties. One way to deal with this problem is reliability-based robust design optimization (RBRDO), in which additional uncertainty analysis (UA, including both of reliability analysis and moment evaluations) is required. For most practical applications however, UA is realized by Monte Carlo Simulation (MCS) combined with structural analyses that renders RBRDO computationally prohibitive. Therefore, this work focuses on development of efficient and robust methodologies for RBRDO in the context of MCS. We presented a polynomial chaos expansion (PCE) based MCS method for UA, in which the random response is approximated with the PCE. The efficiency is mainly improved by avoiding repeated structural analyses. Unfortunately, this method is not well suited for high dimensional problems, such as dynamic problems. To tackle this issue, we applied the convolution form to compute the dynamic response, in which the PCE is used to approximate the modal properties (i.e. to solve random eigenvalue problem) so that the dimension of uncertainties is reduced since only structural random parameters are considered in the PCE model. Moreover, to avoid the modal intermixing problem when using MCS to solve the random eigenvalue problem, we adopted the MAC factor to quantify the intermixing, and developed a univariable method to check which variable results in such a problem and thereafter to remove or reduce this issue. We proposed a sequential RBRDO to improve efficiency and to overcome the nonconvergence problem encountered in the framework of nested MCS based RBRDO. In this sequential RBRDO, we extended the conventional sequential strategy, which mainly aims to decouple the reliability analysis from the optimization procedure, to make the moment evaluations independent from the optimization procedure. Locally "first-torder" exponential approximation around the current design was utilized to construct the equivalently deterministic objective functions and probabilistic constraints. In order to efficiently calculate the coefficients, we developed the auxiliary distribution based reliability sensitivity analysis and the PCE based moment sensitivity analysis. We investigated and demonstrated the effectiveness of the proposed methods for UA as well as RBRDO by several numerical examples. At last, RBRDO was applied to design the tuned mass damper (TMD) in the context of passive vibration control, for both deterministic and uncertain structures. The associated optimal designs obtained by RBRDO cannot only reduce the variability of the response, but also control the amplitude by the prescribed threshold.

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45

Nechak, Lyes. "Approches robustes du comportement dynamique des systèmes non linéaires : Application aux systèmes frottants." Phd thesis, Université de Haute Alsace - Mulhouse, 2011. http://tel.archives-ouvertes.fr/tel-00708215.

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Cette thèse traite de l'analyse robuste du comportement dynamique des systèmes frottants. Ces derniers constituent une classe particulière des systèmes non linéaires et sont caractérisés par des comportements dynamiques très sensibles aux variations des paramètres de conception en particulier aux dispersions des lois de frottement. Cette sensibilité se traduit par des variations qualitatives importantes du comportement dynamique (stabilité, niveaux vibratoire) qui peuvent alors affecter négativement les performances des systèmes frottants. Il est ainsi important, voire indispensable, de pouvoir tenir compte de la dispersion des lois de frottement dans l'étude et l'analyse du comportement dynamique des systèmes frottants afin d'en garantir la robustesse et, dans une perspective plus générale, d'asseoir une démarche de conception robuste des systèmes frottants. Des méthodes spectrales basées sur le concept du chaos polynomial sont proposées dans cette thèse pour traiter de l'analyse robuste du comportement dynamique des systèmes frottants. Pouvant modéliser les fonctions et processus stochastiques, ces méthodes sont adaptées au problème en particulier à l'analyse de la stabilité et à la prédiction des niveaux vibratoires en tenant compte de la dispersion des lois de frottement. Différentes procédures sont proposées et développées pour traiter de ces deux questions. Une efficacité importante a été illustrée à travers l'évaluation des différentes méthodes proposées (chaos polynomial généralisé, chaos polynomial multi-éléments, chaos de Wiener-Haar) en les appliquant sur un exemple de système frottant. En effet, il est montré que ces méthodes offrent une alternative très intéressante à la méthode prohibitive, mais référentielle, de Monte Carlo puisque, pour des niveaux de précision et de confiance similaires, le coût en nombre, en volume et nécessairement en temps de calcul occasionné par les méthodes spectrales sur les différentes analyses (de la stabilité et des niveaux vibratoire) est largement inférieur à celui requis par la technique de Monte Carlo.

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46

Pettersson, Per, Alireza Doostan, and Jan Nordström. "On Stability and Monotonicity Requirements of Finite Difference Approximations of Stochastic Conservation Laws with Random Viscosity." Linköpings universitet, Beräkningsmatematik, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-90995.

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The stochastic Galerkin and collocation methods are used to solve an advection-diusion equation with uncertain and spatially varying viscosity. We investigate well-posedness, monotonicity and stability for the extended system resulting from the Galerkin projection of the advection-diusion equation onto the stochastic basis functions. High-order summationby- parts operators and weak imposition of boundary conditions are used to prove stability of the semi-discrete system. It is essential that the eigenvalues of the resulting viscosity matrix of the stochastic Galerkin system are positive and we investigate conditions for this to hold. When the viscosity matrix is diagonalizable, stochastic Galerkin and stochastic collocation are similar in terms of computational cost, and for some cases the accuracy is higher for stochastic Galerkin provided that monotonicity requirements are met. We also investigate the total spatial operator of the semi-discretized system and its impact on the convergence to steadystate

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Janya-anurak, Chettapong [Verfasser]. "Framework for Analysis and Identification of Nonlinear Distributed Parameter Systems using Bayesian Uncertainty Quantification based on Generalized Polynomial Chaos / Chettapong Janya-anurak." Karlsruhe : KIT Scientific Publishing, 2017. http://www.ksp.kit.edu.

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48

Janya-Anurak, Chettapong [Verfasser]. "Framework for Analysis and Identification of Nonlinear Distributed Parameter Systems using Bayesian Uncertainty Quantification based on Generalized Polynomial Chaos / Chettapong Janya-anurak." Karlsruhe : KIT Scientific Publishing, 2017. http://www.ksp.kit.edu.

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49

Scott, Karen Mary Louise. "Practical Analysis Tools for Structures Subjected to Flow-Induced and Non-Stationary Random Loads." Diss., Virginia Tech, 2011. http://hdl.handle.net/10919/38686.

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There is a need to investigate and improve upon existing methods to predict response of sensors due to flow-induced vibrations in a pipe flow. The aim was to develop a tool which would enable an engineer to quickly evaluate the suitability of a particular design for a certain pipe flow application, without sacrificing fidelity. The primary methods, found in guides published by the American Society of Mechanical Engineers (ASME), of simple response prediction of sensors were found to be lacking in several key areas, which prompted development of the tool described herein. A particular limitation of the existing guidelines deals with complex stochastic stationary and non-stationary modeling and required much further study, therefore providing direction for the second portion of this body of work. A tool for response prediction of fluid-induced vibrations of sensors was developed which allowed for analysis of low aspect ratio sensors. Results from the tool were compared to experimental lift and drag data, recorded for a range of flow velocities. The model was found to perform well over the majority of the velocity range showing superiority in prediction of response as compared to ASME guidelines. The tool was then applied to a design problem given by an industrial partner, showing several of their designs to be inadequate for the proposed flow regime. This immediate identification of unsuitable designs no doubt saved significant time in the product development process. Work to investigate stochastic modeling in structural dynamics was undertaken to understand the reasons for the limitations found in fluid-structure interaction models. A particular weakness, non-stationary forcing, was found to be the most lacking in terms of use in the design stage of structures. A method was developed using the Karhunen Loeve expansion as its base to close the gap between prohibitively simple (stationary only) models and those which require too much computation time. Models were developed from SDOF through continuous systems and shown to perform well at each stage. Further work is needed in this area to bring this work full circle such that the lessons learned can improve design level turbulent response calculations.
Ph. D.

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Notin, Alban. "Evaluation à moindre coût de la fiabilité des structures sollicitées en fatigue." Compiègne, 2010. http://www.theses.fr/2010COMP1877.

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Cette thèse s'insère dans le contexte général de l'estimation de la fiabilité des structures sollicitées en fatigue. Dans le cas d'applications industrielles, chaque évaluation est potentiellement coûteuse en temps de calcul et en espace de stockage. De ce fait, seul un nombre fini de calcul peut être réalisé. Cette évaluation à moindre coût de la fiabilité des structures sollicitées en fatigue suppose de travailler sur l'algorithme de fiabilité mais aussi d'accélérer les calculs mécaniques. Cette double problématique constitue la base de ce travail de thèse. Pour la partie fiabilité, la méthode MRCP (Méthode de Rééchantillonnage du Chaos Polynomial) a été développée. Son objectif est de proposer une troncature adaptative du métamodèle par chaos polynomial en estimant l'erreur par les intervalles de confiance sur l'indice de fiabilité. Les résultats montrent que l'approche est efficace pour des états-limites suffisamment réguliers. Une alternative à l'emploi de métamodèles consiste à accélérer les calculs mécaniques. C'est l'objectif de l'approche SLDL T (décomposition LDL T Stochastique) qui se base sur une modification de la décomposition de Cholesky en supposant que les variations de la matrice L sont négligeables dans le domaine de variation des variables aléatoires. L'aléa est alors reporté sur la matrice diagonale D, optimisée de façon à minimiser l'erreur sur la matrice de rigidité. Les résultats montrent un gain en temps de calcul de l'ordre de 180 sur un exemple industriel dont le comportement mécanique est linéaire élastique et le module d'Young modélisé par un champ stochastique
This thesis take place in the context of the estimation of the reliability of structures under fatigue loading. In the case of industrial applications, each model evaluation may be time and storage consuming. This way, only a few number of evaluations can be performed. This efficient estimation of the reliability of structures under fatigue loading implies to word on the reliability algorithm as well as the speeding up of mechanical computations. In this double issue lies the settlement of this thesis. Concerning the reliability part, the RPCM (Resampling Polynomial Chaos Method) method has been developed. The goal is to build the polynomial chaos basis in an adaptative way such that the troncature error is taken into account. This erros is estimad through confidence intervals on the reliability index. Numerical results show a very good behaviour of the proposed method in the case of smooth limit-state functions. However, metamodels are not the only way to speed up computations. Another strategy consists in accelerate the mechanical computations by approximating the closest calculi controlling the error. This is the idea of the SLDL T (Stochastic LDL T decomposition) approach which is based on a slight modification of the Cholesky decomposition assuming that the fluctuations of the lower matrix L are negligible in the domain of variation of the random inputs. The randonmess is put on the digonal matrix D, which is optimized such a way to minimize the error on the stiffness matrix. In the case of a linear elastic mechanical behaviour with the Young’s modulus modeled by a random field, results show a gain factor round to 180

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