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

Modeling and simulations of large-scale systems are used extensively to not only better understand a natural phenomenon, but also to predict future events. Accurate model results are critical for design optimization and policy making. They can be used effectively to reduce the impact of a natural disaster or even prevent it from happening. In reality, model predictions are often affected by uncertainties in input data and model parameters, and by incomplete knowledge of the underlying physics. A deterministic simulation assumes one set of input conditions, and generates one result without considering uncertainties. It is of great interest to include uncertainty information in the simulation. By ``Uncertainty Quantification,'' we denote the ensemble of techniques used to model probabilistically the uncertainty in model inputs, to propagate it through the system, and to represent the resulting uncertainty in the model result. This added information provides a confidence level about the model forecast. For example, in environmental modeling, the model forecast, together with the quantified uncertainty information, can assist the policy makers in interpreting the simulation results and in making decisions accordingly. Another important goal in modeling and simulation is to improve the model accuracy and to increase the model prediction power. By merging real observation data into the dynamic system through the data assimilation (DA) technique, the overall uncertainty in the model is reduced. With the expansion of human knowledge and the development of modeling tools, simulation size and complexity are growing rapidly. This poses great challenges to uncertainty analysis techniques. Many conventional uncertainty quantification algorithms, such as the straightforward Monte Carlo method, become impractical for large-scale simulations. New algorithms need to be developed in order to quantify and reduce uncertainties in large-scale simulations. This research explores novel uncertainty quantification and reduction techniques that are suitable for large-scale simulations. In the uncertainty quantification part, the non-sampling polynomial chaos (PC) method is investigated. An efficient implementation is proposed to reduce the high computational cost for the linear algebra involved in the PC Galerkin approach applied to stiff systems. A collocation least-squares method is proposed to compute the PC coefficients more efficiently. A novel uncertainty apportionment strategy is proposed to attribute the uncertainty in model results to different uncertainty sources. The apportionment results provide guidance for uncertainty reduction efforts. The uncertainty quantification and source apportionment techniques are implemented in the 3-D Sulfur Transport Eulerian Model (STEM-III) predicting pollute concentrations in the northeast region of the United States. Numerical results confirm the efficacy of the proposed techniques for large-scale systems and the potential impact for environmental protection policy making. ``Uncertainty Reduction'' describes the range of systematic techniques used to fuse information from multiple sources in order to increase the confidence one has in model results. Two DA techniques are widely used in current practice: the ensemble Kalman filter (EnKF) and the four-dimensional variational (4D-Var) approach. Each method has its advantages and disadvantages. By exploring the error reduction directions generated in the 4D-Var optimization process, we propose a hybrid approach to construct the error covariance matrix and to improve the static background error covariance matrix used in current 4D-Var practice. The updated covariance matrix between assimilation windows effectively reduces the root mean square error (RMSE) in the solution. The success of the hybrid covariance updates motivates the hybridization of EnKF and 4D-Var to further reduce uncertainties in the simulation results. Numerical tests show that the hybrid method improves the model accuracy and increases the model prediction quality. / Ph. D.

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.

Hydraulisk modellering är ett viktigt verktyg vid utvärdering av lämpliga åtgärder för torrfåror. Modelleringen påverkas dock alltid av osäkerheter och om dessa är stora kan en modells simuleringsresultat bli opålitligt. Det kan därför vara viktigt att presentera dess simuleringsresultat tillsammans med osäkerheter. Denna studie utreder olika typer av osäkerheter som kan påverka hydrauliska modellers simuleringsresultat. Dessutom utförs känslighetsanalyser där en andel av osäkerheten i simuleringsresultatet tillskrivs de olika inmatningsvariablerna som beaktas. De parametrar som ingår i analysen är upplösningen i använd terrängmodell, upplösning i den hydrauliska modellens beräkningsnät, inflöde till modellen och råheten genom Mannings tal. Studieobjektet som behandlades i denna studie var en torrfåra som ligger nedströms Sandforsdammen i Skellefteälven och programvaran TELEMAC-MASCARET nyttjades för samtliga hydrauliska simuleringar i denna studie.  För att analysera osäkerheter kopplade till upplösning i en terrängmodell och ett beräkningsnät användes ett kvalitativt tillvägagångsätt. Ett antal simuleringar utfördes där alla parametrar förutom de kopplade till upplösning fixerades. Simuleringsresultaten illustrerades sedan genom profil, sektioner, enskilda raster och raster som visade differensen mellan olika simuleringar. Resultaten för analysen visade att en låg upplösning i terrängmodeller och beräkningsnät kan medföra osäkerheter lokalt där det är högre vattenhastigheter och där det finns stor variation i geometrin. Några signifikanta effekter kunde dock inte skönjas på större skala.  Separat gjordes kvantitativa osäkerhets- och känslighetsanalyser för vattendjup och vattenhastighet i torrfåran. Inmatningsparametrarna inflöde till modellen och råhet genom Mannings tal ansågs medföra störst påverkan och övriga parametrar fixerades således. Genom script skapade i programmeringsspråket Python tillsammans med biblioteket OpenTURNS upprättades ett stort urval av möjliga kombinationer för storlek på inflöde och Mannings tal. Alla kombinationer som skapades antogs till fullo täcka upp för den totala osäkerheten i inmatningsparametrarna. Genom att använda urvalet för simulering kunde osäkerheten i simuleringsresultaten också beskrivas. Osäkerhetsanalyser utfördes både genom klassisk beräkning av statistiska moment och genom Polynomial Chaos Expansion. En känslighetsanalys följde sedan där Polynomial Chaos Expansion användes för att beräkna Sobols känslighetsindex för inflödet och Mannings tal i varje kontrollpunkt. Den kvantitativa osäkerhetsanalysen visade att det fanns relativt stora osäkerheter för både vattendjupet och vattenhastighet vid behandlat studieobjekt. Flödet bidrog med störst påverkan på osäkerheten medan Mannings tals påverkan var insignifikant i jämförelse, bortsett från ett område i modellen där dess påverkan ökade markant. / Hydraulic modelling is an important tool when measures for dry river stretches are assessed. The modelling is however always affected by uncertainties and if these are big the simulation results from the models could become unreliable. It may therefore be important to present its simulation results together with the uncertainties. This study addresses various types of uncertainties that may affect the simulation results from hydraulic models. In addition, sensitivity analysis is conducted where a proportion of the uncertainty in the simulation result is attributed to the various input variables that are included. The parameters included in the analysis are terrain model resolution, hydraulic model mesh resolution, inflow to the model and Manning’s roughness coefficient. The object studied in this paper was a dry river stretch located downstream of Sandforsdammen in the river of Skellefteälven, Sweden. The software TELEMAC-MASCARET was used to perform all hydraulic simulations for this thesis.  To analyze the uncertainties related to the resolution for the terrain model and the mesh a qualitative approach was used. Several simulations were run where all parameters except those linked to the resolution were fixed. The simulation results were illustrated through individual rasters, profiles, sections and rasters that showed the differences between different simulations. The results of the analysis showed that a low resolution for terrain models and meshes can lead to uncertainties locally where there are higher water velocities and where there are big variations in the geometry. However, no significant effects could be discerned on a larger scale.  Separately, quantitative uncertainty and sensitivity analyzes were performed for the simulation results, water depth and water velocity for the dry river stretch. The input parameters that were assumed to have the biggest impact were the inflow to the model and Manning's roughness coefficient. Other model input parameters were fixed. Through scripts created in the programming language Python together with the library OpenTURNS, a large sample of possible combinations for the size of inflow and Manning's roughness coefficient was created. All combinations were assumed to fully cover the uncertainty of the input parameters. After using the sample for simulation, the uncertainty of the simulation results could also be described. Uncertainty analyses were conducted through both classical calculation of statistical moments and through Polynomial Chaos Expansion. A sensitivity analysis was then conducted through Polynomial Chaos Expansion where Sobol's sensitivity indices were calculated for the inflow and Manning's M at each control point. The analysis showed that there were relatively large uncertainties for both the water depth and the water velocity. The inflow had the greatest impact on the uncertainties while Manning's M was insignificant in comparison, apart from one area in the model where its impact increased.

Nous étudions la propagation et la quantification d'incertitudes paramétriques au travers de modèles hydrologiques pour la simulation des processus d'infiltration et d'érosion en présence de pluie et/ou de ruissellement. Les paramètres incertains sont décrits dans un cadre probabiliste comme des variables aléatoires indépendantes dont la fonction de densité de probabilité est connue. Cette modélisation probabiliste s'appuie sur une revue bibliographique permettant de cerner les plages de variations des paramètres. L'analyse statistique se fait par échantillonage Monte Carlo et par développements en polynômes de chaos. Nos travaux ont pour but de quantifier les incertitudes sur les principales sorties du modèle et de hiérarchiser l'influence des paramètres d'entrée sur la variabilité de ces sorties par une analyse de sensibilité globale. La première application concerne les effets de la variabilité et de la spatialisation de la conductivité hydraulique à saturation du sol dans le modèle d'infiltration de Green--Ampt pour diverses échelles spatiales et temporelles. Notre principale conclusion concerne l'importance de l'état de saturation du sol. La deuxième application porte sur le modèle d'érosion de Hairsine--Rose. Une des conclusions est que les interactions paramétriques sont peu significatives dans le modèle de détachement par la pluie mais s'avèrent importantes dans le modèle de détachement par le ruissellement / We study parametric uncertainty propagation and quantification in hydrological models for the simulation of infiltration and erosion processes in the presence of rainfall and/or runoff. Uncertain input parameters are treated in a probabilistic framework, considering them as independent random variables defined by a fixed probability density function. This probabilistic modeling is based on a literature review to identify the range of variation of input parameters. The output statistical analysis is realized by Monte Carlo sampling and by polynomial chaos expansions. Our analysis aims at quantifying uncertainties in model outputs and establishing a hierarchy within input parameters according to their influence on output variability by means of global sensitivity analysis. The first application concerns the variability and spatial localization of the soil saturated hydraulic conductivity in the Green-Ampt infiltration model at different spatial and temporal scales. Our main conclusion is the importance of the soil saturation state. The second application deals with the Harisine--Rose erosion model. One conclusion is that the parametric interactions are not significant in the rainfall detachment model, but they prove to be important in the runoff detachment model

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.

Conservation laws with uncertain initial and boundary conditions are approximated using a generalized polynomial chaos expansion approach where the solution is represented as a generalized Fourier series of stochastic basis functions, e.g. orthogonal polynomials or wavelets. The stochastic Galerkin method is used to project the governing partial differential equation onto the stochastic basis functions to obtain an extended deterministic system. The stochastic Galerkin and collocation methods are used to solve an advection-diffusion equation with uncertain viscosity. We investigate well-posedness, monotonicity and stability for the stochastic Galerkin system. High-order summation-by-parts operators and weak imposition of boundary conditions are used to prove stability. We investigate the impact of the total spatial operator on the convergence to steady-state.  Next we apply the stochastic Galerkin method to Burgers' equation with uncertain boundary conditions. An analysis of the truncated polynomial chaos system presents a qualitative description of the development of the solution over time. An analytical solution is derived and the true polynomial chaos coefficients are shown to be smooth, while the corresponding coefficients of the truncated stochastic Galerkin formulation are shown to be discontinuous. We discuss the problematic implications of the lack of known boundary data and possible ways of imposing stable and accurate boundary conditions. We present a new fully intrusive method for the Euler equations subject to uncertainty based on a Roe variable transformation. The Roe formulation saves computational cost compared to the formulation based on expansion of conservative variables. Moreover, it is more robust and can handle cases of supersonic flow, for which the conservative variable formulation fails to produce a bounded solution. A multiwavelet basis that can handle  discontinuities in a robust way is used. Finally, we investigate a two-phase flow problem. Based on regularity analysis of the generalized polynomial chaos coefficients, we present a hybrid method where solution regions of varying smoothness are coupled weakly through interfaces. In this way, we couple smooth solutions solved with high-order finite difference methods with non-smooth solutions solved for with shock-capturing methods.

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.

Recently there has been growing interest to characterize and reduce uncertainty in stochastic dynamical systems. This drive arises out of need to manage uncertainty in complex, high dimensional physical systems. Traditional techniques of uncertainty quantification (UQ) use local linearization of dynamics and assumes Gaussian probability evolution. But several difficulties arise when these UQ models are applied to real world problems, which, generally are nonlinear in nature. Hence, to improve performance, robust algorithms, which can work efficiently in a nonlinear non-Gaussian setting are desired. The main focus of this dissertation is to develop UQ algorithms for nonlinear systems, where uncertainty evolves in a non-Gaussian manner. The algorithms developed are then applied to state estimation of real-world systems. The first part of the dissertation focuses on using polynomial chaos (PC) for uncertainty propagation, and then achieving the estimation task by the use of higher order moment updates and Bayes rule. The second part mainly deals with Frobenius-Perron (FP) operator theory, how it can be used to propagate uncertainty in dynamical systems, and then using it to estimate states by the use of Bayesian update. Finally, a method to represent the process noise in a stochastic dynamical system using a nite term Karhunen-Loeve (KL) expansion is proposed. The uncertainty in the resulting approximated system is propagated using FP operator. The performance of the PC based estimation algorithms were compared with extended Kalman filter (EKF) and unscented Kalman filter (UKF), and the FP operator based techniques were compared with particle filters, when applied to a duffing oscillator system and hypersonic reentry of a vehicle in the atmosphere of Mars. It was found that the accuracy of the PC based estimators is higher than EKF or UKF and the FP operator based estimators were computationally superior to the particle filtering algorithms.

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