The objectives of this study are (1) to develop an accurate and efficient fatigue analysis procedure that can be used in reliability analysis and reliability-based design optimization (RBDO) of composite wind turbine blades; (2) to develop a wind load uncertainty model that provides realistic uncertain wind load for the reliability analysis and the RBDO process; and (3) to obtain an optimal composite wind turbine blade that satisfies target reliability for durability under the uncertain wind load. The current research effort involves: (1) developing an aerodynamic analysis method that can effectively calculate detailed wind pressure on the blade surface for stress analysis; (2) developing a fatigue failure criterion that can cope with non-proportional multi-axial stress states in composite wind turbine blades; (3) developing a wind load uncertainty model that represents realistic uncertain wind load for fatigue reliability of wind turbine systems; (4) applying the wind load uncertainty model into a composite wind turbine blade and obtaining an RBDO optimum design that satisfies a target probability of failure for a lifespan of 20 years under wind load uncertainty. In blade fatigue analysis, resultant aerodynamic forces are usually applied at the aerodynamic centers of the airfoils of a blade to calculate stress/strain. However, in reality the wind pressures are applied on the blade surface. A wind turbine blade is often treated as a typical beam-like structure for which fatigue life calculations are limited in the edge-wise and/or flap-wise direction(s). Using the beam-like structure, existing fatigue analysis methods for composite wind turbine blades cannot cope with the non-proportional multi-axial stress states that are endured by wind turbine blades during operation. Therefore, it is desirable to develop a fatigue analysis procedure that utilizes detailed wind pressures as wind loads and considers non-proportional multi-axial stress states in fatigue damage calculation. In this study, a 10-minute wind field realization, determined by a 10-minute mean wind speed V10 and a 10-minute turbulence intensity I10, is first simulated using Veers’ method. The simulated wind field is used for aerodynamic analysis. An aerodynamic analysis method, which could efficiently generate detailed quasi-physical blade surface pressures, has been developed. The generated pressures are then applied on a high-fidelity 3-D finite element blade model for stress and fatigue analysis. The fatigue damage calculation considers the non-proportional multi-axial complex stress states. A detailed fatigue damage contour, which indicates the fatigue failure locally, can be obtained using the developed fatigue analysis procedure. As the 10-minute fatigue analysis procedure is deterministic in this study, the calculated 10-minute fatigue damage is determined by V10 and I10. It is necessary to clarify that the rotational speed of the wind turbine blade is assumed to be constant (12.1 rpm) and the pitch angle is fixed to be 0 degree for different wind conditions, since the rotational speed control and pitch angle control have not been considered in this study. For predicting the fatigue life of a wind turbine, a fixed Weibull distribution is widely used to determine the percentage of time the wind turbine experiences different mean wind speeds during its life-cycle. Meanwhile, fixed turbulence intensities are often used based on the designed wind turbine types. These simplifications, i.e., fixed Weibull distribution and fixed turbulence intensities, ignore the realistic uncertain wind load when designing a reliable wind turbine system. In the real world, both the mean wind speed and turbulence intensity vary constantly over one year, and their annual distributions are different at different locations and in different years. Thus, it is necessary to develop a wind load uncertainty model that can provide a realistic uncertain wind load for designing reliable wind turbine systems. In this study, 249 groups of measured wind data, collected at different locations and in different years, are used to develop a dynamic wind load uncertainty model. The dynamic wind load uncertainty model consists of annual wind load variation and wind load variation in a large spatiotemporal range, i.e., at different locations and in different years. The annual wind load variation is represented by the joint probability density function of V10 and I10. The wind load variation in a large spatiotemporal range is represented by the probability density functions of five parameters, C, k, a, b, and τ, which determine the joint probability density function of V10 and I10. In order to obtain the RBDO optimum design efficiently, a deterministic design optimization (DDO) procedure of a composite wind turbine blade has been first carried out using averaged percentage of time (probability) for each wind condition. A wind condition is specified by two terms: 10-minute mean wind speed and 10-minute turbulence intensity. In this research, a probability table, which consists of averaged probabilities corresponding to different wind conditions, is referred as a mean wind load. The mean wind load is generated using the dynamic wind load uncertainty model. During the DDO process, the laminate thickness design variables are tailored to minimize the total cost of composite materials while satisfying the target fatigue lifespan of 20 years. It is found that, under the mean wind load condition, the fatigue life of the initial design is only 0.0004 year. After the DDO process, even though the cost at the DDO optimum design is increased by 31.5% compared to that at the initial design, the predicted fatigue life at the DDO optimum design is significantly increased to 19.9995 years. Reliability analyses of the initial design and the DDO optimum design have been carried out using the wind load uncertainty model and Monte Carlo simulation. The reliability analysis results show that the DDO procedure reduces the probability of failure from 100% at the initial design to 49.9% at the DDO optimum design considering only wind load uncertainty. In order to satisfy the target 2.275% probability of failure, it is necessary to further improve the fatigue reliability of the composite wind turbine blade by RBDO. Reliability-based design optimization of the composite wind turbine blade has been carried out starting at the DDO optimum design. Fatigue hotspots for RBDO are identified among the laminate section points, which are selected from the DDO optimum design. Local surrogate models for 10-minute fatigue damage have been created at the selected hotspots. Using the local surrogate models, both the wind load uncertainty and manufacturing variability has been included in the RBDO process. It is found that the probability of failure is 50.06% at the RBDO initial design (DDO optimum design) considering both wind load uncertainty and manufacturing variability. During the RBDO process, the normalized laminate thickness design variables are tailored to minimize the total cost of composite materials while satisfying the target 2.275% probability of failure. The obtained RBDO optimum design reduces the probability of failure from 50.06% at the DDO optimum design to 2.28%, while increasing the cost by 3.01%.

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Research in the field of reliability based design is mainly focused on two sub-areas: The computation of the probability of failure and its integration in the reliability based design optimization (RBDO) loop. Four papers are presented in this work, representing a contribution to both sub-areas. In the first paper, a new Second Order Reliability Method (SORM) is presented. As opposed to the most commonly used SORMs, the presented approach is not limited to hyper-parabolic approximation of the performance function at the Most Probable Point (MPP) of failure. Instead, a full quadratic fit is used leading to a better approximation of the real performance function and therefore more accurate values of the probability of failure. The second paper focuses on the integration of the expression for the probability of failure for general quadratic function, presented in the first paper, in RBDO. One important feature of the proposed approach is that it does not involve locating the MPP. In the third paper, the expressions for the probability of failure based on general quadratic limit-state functions presented in the first paper are applied for the special case of a hyper-parabola. The expression is reformulated and simplified so that the probability of failure is only a function of three statistical measures: the Cornell reliability index, the skewness and the kurtosis of the hyper-parabola. These statistical measures are functions of the First-Order Reliability Index and the curvatures at the MPP. In the last paper, an approximate and efficient reliability method is proposed. Focus is on computational efficiency as well as intuitiveness for practicing engineers, especially regarding probabilistic fatigue problems where volume methods are used. The number of function evaluations to compute the probability of failure of the design under different types of uncertainties is a priori known to be 3n+2 in the proposed method, where n is the number of stochastic design variables. / <p>QC 20160317</p>

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.

The primary objective of this study is to develop new computational methods for robust design optimization (RDO) and reliability-based design optimization (RBDO) of high-dimensional, complex engineering systems. Four major research directions, all anchored in polynomial dimensional decomposition (PDD), have been defined to meet the objective. They involve: (1) development of new sensitivity analysis methods for RDO and RBDO; (2) development of novel optimization methods for solving RDO problems; (3) development of novel optimization methods for solving RBDO problems; and (4) development of a novel scheme and formulation to solve stochastic design optimization problems with both distributional and structural design parameters. The major achievements are as follows. Firstly, three new computational methods were developed for calculating design sensitivities of statistical moments and reliability of high-dimensional complex systems subject to random inputs. The first method represents a novel integration of PDD of a multivariate stochastic response function and score functions, leading to analytical expressions of design sensitivities of the first two moments. The second and third methods, relevant to probability distribution or reliability analysis, exploit two distinct combinations built on PDD: the PDD-SPA method, entailing the saddlepoint approximation (SPA) and score functions; and the PDD-MCS method, utilizing the embedded Monte Carlo simulation (MCS) of the PDD approximation and score functions. For all three methods developed, both the statistical moments or failure probabilities and their design sensitivities are both determined concurrently from a single stochastic analysis or simulation. Secondly, four new methods were developed for RDO of complex engineering systems. The methods involve PDD of a high-dimensional stochastic response for statistical moment analysis, a novel integration of PDD and score functions for calculating the second-moment sensitivities with respect to the design variables, and standard gradient-based optimization algorithms. The methods, depending on how statistical moment and sensitivity analyses are dovetailed with an optimization algorithm, encompass direct, single-step, sequential, and multi-point single-step design processes. Thirdly, two new methods were developed for RBDO of complex engineering systems. The methods involve an adaptive-sparse polynomial dimensional decomposition (AS-PDD) of a high-dimensional stochastic response for reliability analysis, a novel integration of AS-PDD and score functions for calculating the sensitivities of the failure probability with respect to design variables, and standard gradient-based optimization algorithms, resulting in a multi-point, single-step design process. The two methods, depending on how the failure probability and its design sensitivities are evaluated, exploit two distinct combinations built on AS-PDD: the AS-PDD-SPA method, entailing SPA and score functions; and the AS-PDD-MCS method, utilizing the embedded MCS of the AS-PDD approximation and score functions. In addition, a new method, named as the augmented PDD method, was developed for RDO and RBDO subject to mixed design variables, comprising both distributional and structural design variables. The method comprises a new augmented PDD of a high-dimensional stochastic response for statistical moment and reliability analyses; an integration of the augmented PDD, score functions, and finite-difference approximation for calculating the sensitivities of the first two moments and the failure probability with respect to distributional and structural design variables; and standard gradient-based optimization algorithms, leading to a multi-point, single-step design process. The innovative formulations of statistical moment and reliability analysis, design sensitivity analysis, and optimization algorithms have achieved not only highly accurate but also computationally efficient design solutions. Therefore, these new methods are capable of performing industrial-scale design optimization with numerous design variables.

Conventional reliability analysis methods assume that a simulation model is able to represent the real physics accurately. However, this assumption may not always hold as the simulation model could be biased due to simplifications and idealizations. Simulation models are approximate mathematical representations of real-world systems and thus cannot exactly imitate the real-world systems. The accuracy of a simulation model is especially critical when it is used for the reliability calculation. Therefore, a simulation model should be validated using prototype testing results for reliability analysis. However, in practical engineering situation, experimental output data for the purpose of model validation is limited due to the significant cost of a large number of physical testing. Thus, the model validation needs to be carried out to account for the uncertainty induced by insufficient experimental output data as well as the inherent variability existing in the physical system and hence in the experimental test results. Therefore, in this study, a confidence-based model validation method that captures the variability and the uncertainty, and that corrects model bias at a user-specified target confidence level, has been developed. Reliability assessment using the confidence-based model validation can provide conservative estimation of the reliability of a system with confidence when only insufficient experimental output data are available. Without confidence-based model validation, the designed product obtained using the conventional reliability-based design optimization (RBDO) optimum could either not satisfy the target reliability or be overly conservative. Therefore, simulation model validation is necessary to obtain a reliable optimum product using the RBDO process. In this study, the developed confidence-based model validation is integrated in the RBDO process to provide truly confident RBDO optimum design. The developed confidence-based model validation will provide a conservative RBDO optimum design at the target confidence level. However, it is challenging to obtain steady convergence in the RBDO process with confidence-based model validation because the feasible domain changes as the design moves (i.e., a moving-target problem). To resolve this issue, a practical optimization procedure, which terminates the RBDO process once the target reliability is satisfied, is proposed. In addition, the efficiency is achieved by carrying out deterministic design optimization (DDO) and RBDO without model validation, followed by RBDO with the confidence-based model validation. Numerical examples are presented to demonstrate that the proposed RBDO approach obtains a conservative and practical optimum design that satisfies the target reliability of designed product given a limited number of experimental output data. Thus far, while the simulation model might be biased, it is assumed that we have correct distribution models for input variables and parameters. However, in real practical applications, only limited numbers of test data are available (parameter uncertainty) for modeling input distributions of material properties, manufacturing tolerances, operational loads, etc. Also, as before, only a limited number of output test data is used. Therefore, a reliability needs to be estimated by considering parameter uncertainty as well as biased simulation model. Computational methods and a process are developed to obtain confidence-based reliability assessment. The insufficient input and output test data induce uncertainties in input distribution models and output distributions, respectively. These uncertainties, which arise from lack of knowledge – the insufficient test data, are different from the inherent input distributions and corresponding output variabilities, which are natural randomness of the physical system.

Cette thèse s’inscrit dans le cadre des travaux menés par PSA Peugeot Citroën pour l’allègement de ses véhicules. Les optimisations masse multi-prestations réalisées sur le périmètre de la structure contribuent directement à cette démarche en recherchant une allocation d’épaisseurs de tôles à masse minimale qui respectent des spécifications physiques relatives à différentes prestations (choc, vibro-acoustique, etc.). Ces spécifications sont généralement évaluées à travers des modèles numériques à très haute-fidélité qui présentent des temps de restitution particulièrement élevés. Le recours à des fonctions de substitution, connues sous le nom de métamodèles, reste alors la seule alternative pour mener une étude d’optimisation tout en respectant les délais projet. Cependant la prestation qui nous intéresse, à savoir le choc frontal, présente quelques particularités (grande dimensionnalité, fortes non-linéarités, dispersions physique et numérique) qui rendent sa métamodélisation difficile.L’objectif de la thèse est alors de proposer une approche d’optimisation basée sur des métamodèles adaptatifs afin de dégager de nouveaux gains de masse. Cela passe par la prise en compte du choc frontal dont le caractère chaotique est exacerbé par la présence d’incertitudes. Nous proposons ainsi une méthode d’optimisation fiabiliste avec l’introduction de quantiles comme mesure de conservatisme. L’approche est basée sur des modèles de krigeage avec enrichissement adaptatif afin de réduire au mieux le nombre d’appels aux modèles éléments finis. Une application sur un véhicule complet permet de valider la méthode. / One of the most challenging tasks in modern engineering is that of keeping the cost of manufactured goods small. With the advent of computational design, prototyping for instance, a major source of expenses, is reduced to its bare essentials. In fact, through the use of high-fidelity models, engineers can predict the behaviors of the systems they design quite faithfully. To be fully realistic, such models must embed uncertainties that may affect the physical properties or operating conditions of the system. This PhD thesis deals with the constrained optimization of structures under uncertainties in the context of automotive design. The constraints are assessed through expensive finite element models. For practical purposes, such models are conveniently substituted by so-called surrogate models which stand as cheap and easy-to-evaluate proxies. In this PhD thesis, Gaussian process modeling and support vector machines are considered. Upon reviewing state-of-the-art techniques for optimization under uncertainties, we propose a novel formulation for reliability-based design optimization which relies on quantiles. The formal equivalence of this formulation with the traditional ones is proved. This approach is then coupled to surrogate modeling. Kriging is considered thanks to its built-in error estimate which makes it convenient to adaptive sampling strategies. Such an approach allows us to reduce the computational budget by running the true model only in regions that are of interest to optimization. We therefore propose a two-stage enrichment scheme. The first stage is aimed at globally reducing the Kriging epistemic uncertainty in the vicinity of the limit-state surface. The second one is performed within iterations of optimization so as to locally improve the quantile accuracy. The efficiency of this approach is demonstrated through comparison with benchmark results. An industrial application featuring a car under frontal impact is considered. The crash behavior of a car is indeed particularly affected by uncertainties. The proposed approach therefore allows us to find a reliable solution within a reduced number of calls to the true finite element model. For the extreme case where uncertainties trigger various crash scenarios of the car, it is proposed to rely on support vector machines for classification so as to predict the possible scenarios before metamodeling each of them separately.

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Design of retaining structures depends upon the load which is transferred from backfill soil as well as external loads and also the resisting capacity of the structure. The traditional safety factor approach of the design of retaining structures does not address the variability of soils and loads. The properties of backfill soil are inherently variable and influence the design decisions considerably. A rational procedure for the design of retaining structures needs to explicitly consider variability, as they may cause significant changes in the performance and stability assessment. Reliability based design enables identification and separation of different variabilities in loading and resistance and recommends reliability indices to ensure the margin of safety based on probability theory. Detailed studies in this area are limited and the work presented in the dissertation on the Optimum design of retaining structures under static and seismic conditions: A reliability based approach is an attempt in this direction. This thesis contains ten chapters including Chapter 1 which provides a general introduction regarding the contents of the thesis and Chapter 2 presents a detailed review of literature regarding static and seismic design of retaining structures and highlights the importance of consideration of variability in the optimum design and leads to scope of the investigation. Targeted stability is formulated as optimization problem in the framework of target reliability based design optimization (TRBDO) and presented in Chapter 3. In Chapter 4, TRBDO approach for cantilever sheet pile walls and anchored cantilever sheet pile walls penetrating sandy and clayey soils is developed. Design penetration depth and section modulus for the various anchor pulls are obtained considering the failure criteria (rotational, sliding, and flexural failure modes) as well as variability in the back fill soil properties, soil-steel pile interface friction angle, depth of the water table, total depth of embedment, yield strength of steel, section modulus of sheet pile and anchor pull. The stability of reinforced concrete gravity, cantilever and L-shaped retaining walls in static conditions is examined in the context of reliability based design optimization and results are presented in Chapter 5 considering failure modes viz. overturning, sliding, eccentricity, bearing, shear and moment failures in the base slab and stem of wall. Optimum wall proportions are proposed for different coefficients of variation of friction angle of the backfill soil and cohesion of the foundation soil corresponding to different values of component as well as lower bounds of system reliability indices. Chapter 6 presents an approach to obtain seismic passive resistance behind gravity walls using composite curved rupture surface considering limit equilibrium method of analysis with the pseudo-dynamic approach. The study is extended to obtain the rotational and sliding displacements of gravity retaining walls under passive condition when subjected to sinusoidal nature of earthquake loading. Chapter 7 focuses on the reliability based design of gravity retaining wall when subjected to passive condition during earthquakes. Reliability analysis is performed for two modes of failure namely rotation of the wall about its heel and sliding of the wall on its base are considering variabilities associated with characteristics of earthquake ground motions, geometric proportions of wall, backfill soil and foundation soil properties. The studies reported in Chapter 8 and Chapter 9 present a method to evaluate reliability for external as well as internal stability of reinforced soil structures (RSS) using reliability based design optimization in the framework of pseudo static and pseudo dynamic methods respectively. The optimum length of reinforcement needed to maintain the stability against four modes of failure (sliding, overturning, eccentricity and bearing) by taking into account the variabilities associated with the properties of reinforced backfill, retained backfill, foundation soil, tensile strength and length of the geosynthetic reinforcement by targeting various component and system reliability indices is computed. Finally, Chapter 10 contains the important conclusions, along with scope for further work in the area. It is hoped that the methodology and conclusions presented in this study will be beneficial to the geotechnical engineering community in particular and society as a whole.