This dissertation presents a sampling-based method that can be used for uncertainty quantification and deterministic or probabilistic optimization. The objective is to simultaneously address several difficulties faced by classical techniques based on response values and their gradients. In particular, this research addresses issues with discontinuous and binary (pass or fail) responses, and multiple failure modes. All methods in this research are developed with the aim of addressing problems that have limited data due to high cost of computation or experiment, e.g. vehicle crashworthiness, fluid-structure interaction etc.The core idea of this research is to construct an explicit boundary separating allowable and unallowable behaviors, based on classification information of responses instead of their actual values. As a result, the proposed method is naturally suited to handle discontinuities and binary states. A machine learning technique referred to as support vector machines (SVMs) is used to construct the explicit boundaries. SVM boundaries can be highly nonlinear, which allows one to use a single SVM for representing multiple failure modes.One of the major concerns in the design and uncertainty quantification communities is to reduce computational costs. To address this issue, several adaptive sampling methods have been developed as part of this dissertation. Specific sampling methods have been developed for reliability assessment, deterministic optimization, and reliability-based design optimization. Adaptive sampling allows the construction of accurate SVMs with limited samples. However, like any approximation method, construction of SVM is subject to errors. A new method to quantify the prediction error of SVMs, based on probabilistic support vector machines (PSVMs) is also developed. It is used to provide a relatively conservative probability of failure to mitigate some of the adverse effects of an inaccurate SVM. In the context of reliability assessment, the proposed method is presented for uncertainties represented by random variables as well as spatially varying random fields.In order to validate the developed methods, analytical problems with known solutions are used. In addition, the approach is applied to some application problems, such as structural impact and tolerance optimization, to demonstrate its strengths in the context of discontinuous responses and multiple failure modes.

Cette thèse porte sur l'analyse de l'endommagement par fatigue et optimisation fiabiliste des structures soumises à des vibrations aléatoires. Le but de l'optimisation fiabiliste est de trouver le compromis entre le coût et la fiabilité. Plusieurs méthodes, telles que la méthode hybride et la méthode OSF ont été développées. Ces méthodes ont été appliquées dans des cas statiques et certains cas dynamiques spécifiques. Dans la réalité les structures sont soumises à des vibrations aléatoires qui peuvent provoquer un endommagement par fatigue. Dans cette thèse on présente la stratégie numérique de calcul de l'endommagement par fatigue dans le domaine fréquentiel et on propose une extension des méthodes RBDO dans le cas des structures soumises à des vibrations aléatoires. Aussi, une méthode RHM est développée. Enfin,une application industrielle qui porte sur la modélisation de la partie mécanique du banc HALT est présenté. / This thesis deals with the fatigue damage analysis and reliability-based design optimization (RBDO) of structures under random vibrations. The purpose of an RBDO method is to find the best compromise between cost and safety. Several methods, such as Hybrid method and OSF method have been developed. These methods have been applied in static cases and some specific dynamic cases. In fact, structures are subject to random vibrations which can cause a fatigue damage. In this thesis we present the strategy of calculation of the fatigue damage based on the Sines criterion in the frequency domain developed in our laboratory. Then, an extension of the RBDO methods in the case of structures subjected to random vibrations is proposed. Also, an RHM method is developed. Finally, we present an industrial application where we propose a model of the mechanical part of the HALT chamber.

The objective of this study is to develop an integrated multibody dynamics computational framework for the deterministic and reliability-based design optimization of wind turbine drivetrains to obtain an optimal wind turbine gear design that ensures a target reliability under wind load and gear manufacturing uncertainties. Gears in wind turbine drivetrains are subjected to severe cyclic loading due to variable wind loads that are stochastic in nature. Thus, the failure rate of drivetrain systems is reported to be relatively higher than the other wind turbine components. It is known in wind energy industry that improving reliability of drivetrain designs is one of the key issues to make wind energy competitive as compared to fossil fuels. Furthermore, a wind turbine is a multi-physics system involving random wind loads, rotor blade aerodynamics, gear dynamics, electromagnetic generator and control systems. This makes an accurate prediction of product life of drivetrains challenging and very limited studies have been carried out regarding design optimization including the reliability-based design optimization (RBDO) of geared systems considering wind load and manufacturing uncertainties. In order to address these essential and challenging issues on design optimization of wind turbine drivetrains under wind load and gear manufacturing uncertainties, the following issues are discussed in this study: (1) development of an efficient numerical procedure for gear dynamics simulation of complex multibody geared systems based on the multi-variable tabular contact search algorithm to account for detailed gear tooth contact geometry with profile modifications or surface imperfections; (2) development of an integrated multibody dynamics computational framework for deterministic and reliability-based design optimization of wind turbine drivetrains using the gear dynamics simulation software developed in (1) and RAMDO software by incorporating wide spatiotemporal wind load uncertainty model, pitting gear tooth contact fatigue model, and rotor blade aerodynamics model using NREL AeroDyn/FAST; and (3) deterministic and reliability-based design optimization of wind turbine drivetrain to minimize total weight of a drivetrain system while ensuring 20-year reliable service life with wind load and gear manufacturing uncertainties using the numerical procedure developed in this study. To account for the wind load uncertainty, the joint probability density function (PDF) of 10-minute mean wind speed (V₁₀) and 10-minute turbulence intensity (I₁₀) is introduced for wind turbine drivetrain dynamics simulation. To consider wide spatiotemporal wind uncertainty (i.e., wind load uncertainty for different locations and in different years), uncertainties of all the joint PDF parameters of V₁₀, I₁₀ and copula are considered, and PDF for each parameter is identified using 249 sets of wind data. This wind uncertainty model allows for the consideration of a wide range of probabilistic wind loads in the contact fatigue life prediction. For a given V₁₀ and I₁₀ obtained from the stochastic wind model, the random time-domain wind speed data is generated using NREL TurbSim, and then inputted into NREL FAST to perform the aerodynamic simulation of rotor blades to predict the transmitted torque and speed of the main shaft of the drivetrain that are sent to the multibody gear dynamics simulation as an input. In order to predict gear contact fatigue life, a high-fidelity gear dynamics simulation model that considers the detailed gear contact geometry as well as the mesh stiffness variation needs to be developed to find the variability of maximum contact stresses under wind load uncertainty. This, however, leads to a computationally intensive procedure. To eliminate the computationally intensive iterative online collision detection algorithm, a numerical procedure for the multibody gear dynamics simulation based on the tabular contact search algorithm is proposed. Look-up contact tables are generated for a pair of gear tooth profiles by the contact geometry analysis prior to the dynamics simulation and the contact points that fulfill the non-conformal contact condition and mesh stiffness at each contact point are calculated for all pairs of gears in the drivetrain model. This procedure allows for the detection of gear tooth contact in an efficient manner while retaining the precise contact geometry and mesh stiffness variation in the evaluation of mesh forces, thereby leading to a computationally efficient gear dynamics simulation suited for the design optimization procedure considering wind load uncertainty. Furthermore, the accuracy of mesh stiffness model introduced in this study and transmission error of gear tooth with tip relief are discussed, and a wind turbine drivetrain model developed using this approach is validated against test data provided in the literature. The gear contact fatigue life is predicted based on the gear tooth pitting fatigue criteria and is defined by the sum of the number of stress cycles required for the fatigue crack initiation and the number required for the crack to propagate from the initial to the critical crack length based on Paris-Erdogan equation for Mode II fracture. All the above procedures are integrated into the reliability-based design optimization software RAMDO for design optimization and reliability analysis of wind turbine drivetrains under wind load and manufacturing uncertainties. A 750kW GRC wind turbine gearbox model is used to perform the design optimization and the reliability analysis. A deterministic design optimization (DDO) is performed first using an averaged joint PDF of wind load to ensure a 20-year service life. To this end, gear face width and tip relief (profile modification) are selected as design variables and optimized such that 20-year fatigue life is ensured while minimizing the total weight of drivetrains. It is important to notice here that an increase in face width leads to a decrease in the fatigue damage, but an increase in total weight. On the other hand, the tip relief has almost no effect on the total weight, but it has a major impact on the fatigue damage. It is shown in this study that the optimum tip relief allows for lowering the greatest maximum shear stresses on the tooth surface without relying heavily on face width widening to meet the 20-year fatigue life constraint and it leads to reduction of total drivetrain weight by 8.4%. However, if only face width is considered as design variable, total weight needs to be increased by 4.7% to meet the 20-year fatigue life constraint. Furthermore, the reliability analysis at the DDO optimum design is carried out considering the large spatiotemporal wind load uncertainty and gear manufacturing uncertainty. Local surrogate models at DDO optimum design are generated using Dynamic Kriging method in RAMDO software to evaluate the gear contact fatigue damage. 49.5% reliability is obtained at the DDO optimum design, indicating that the probability of failure is 50.5%, which is as expected for the DDO design. RBDO is, therefore, necessary to further improve the reliability of the wind turbine drivetrain. To this end, the sampling-based reliability analysis is carried out to evaluate the probability of failure for each design using the Monte Carlo Simulation (MCS) method. However, the use of a large number of MCS sample points leads to a large number of contact fatigue damage evaluation time using the 10-minute multibody drivetrain dynamics simulation, resulting in the RBDO calculation process being computational very intensive. In order to overcome the computational difficulty resulting from the use of high-fidelity wind turbine drivetrain dynamics simulation, intermediate surrogate models are created prior to the RBDO process using the Dynamic Kriging method in RAMDO and used throughout the entire RBDO iteration process. It is demonstrated that the RBDO optimum obtained ensures the target 97.725 % reliability (two sigma quality level) with only 1.4 % increase in the total weight from the baseline design with 8.3 % reliability. This result clearly indicates the importance of incorporating the tip relief as a design variable that prevents larger increase in the face width causing an increase in weight. This, however, does not mean that a larger tip relief is always preferred since an optimum tip relief amount depends on stochastic wind loads and an optimum tip relief cannot be found deterministically. Furthermore, accuracy of the RBDO optimum obtained using the intermediate surrogate models is verified by the reliability analysis at the RBDO optimum using the local surrogate models. It is demonstrated that the integrated design optimization procedure developed in this study enables the cost effective and reliable design of wind turbine drivetrains.

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%.

In this thesis, a new computer model (tool) for the reliability-based design of rubble-mound coastal defense structures is developed in which design is carried out in a user frienly way giving outputs on time variant reliability for the predetermined lifetimes and damage levels. The model aims to perform the following steps: 1. Determine the sources of uncertainties in design parameters 2. Evaluate the damage risk of coastal structures which are at design stage and are recently constructed. 3. Study the sensitivity of limit state functions to the design parameters. Different from other reliability studies on coastal projects, a new design computer program is developed that can be easily used by everyone working in coastal engineering field.

Geotechnical engineering is strongly associated with large uncertainties. Exploring a medium (soil) that is almost entirely and completely hidden from us is no easy task. Investigations can be made only at discrete points, and the majority of a specific soil volume is never tested. All soils experience inherent spatial variability, which contributes to some uncertainty in the design process of a geotechnical structure. Furthermore, uncertainties also arise during testing and when design properties are inferred from these tests. To master the art of making decisions in the presence of uncertainties, probabilistic description of soil properties and reliability-based design play vital roles. Historically, the observational method (sometimes referred to as the “learn-as-you-go-approach”), sprung from ideas by Karl Terzaghi and later formulated by Ralph Peck, has been used in projects where the uncertainties are large and difficult to assess. The design approach is still highly suitable for numerous situations and is defined in Eurocode 7 for geotechnical design. In paper I, the Eurocode definition of the observational method is discussed. This paper concluded that further work in the probabilistic description of soil properties is highly needed, and, by extension, reliability-based design should be used in conjunction with the observational method. Although great progress has been made in the field of reliability-based design during the past decade, few geotechnical engineers are familiar with probabilistic approaches to design. In papers II and III, aspects of probabilistic descriptions of soil properties and reliability-based design are discussed. The connection between performing qualitative investigations and potential design savings is discussed in paper III. In the paper, uncertainties are assessed for two sets of investigations, one consisting of more qualitative investigations and hence with less uncertainty. A simplified Bayesian updating technique, referred to as “the multivariate approach”, is used to cross-validate data to reduce the evaluated total uncertainty. Furthermore, reliability-based design was used to compare the two sets of investigations with the calculated penetration depth for a sheet-pile wall. The study is a great example of how a small amount of both time and money (in the pre-investigation phase) can potentially lead to greater savings in the final design. / <p>QC 20160201</p> / TRUST, Transparent Underground Structures

The geotechnical community has since decades been acquainted with the use of statistical approach for design optimizations. This has been approved as an operational method by many practitioners in the field but is yet to see a major full-scale breakthrough and acceptance in practice. The advantage of quantifying the many different sources of uncertainties in a design is already a fairly acknowledged method and is in this report expanded for the use in the case of road embankments founded on soft soil improved by lime-cement columns. Statistical approach was adopted with practice of reliability base design (RBD ) to consider the importance of ingoing variables’ variability with the target of streamlining the result by decreasing uncertainties (by means of increased measurements, careful installation, etc.). By constructing a working model that gives the corresponding area ratio between columns and soil needed to fulfill the different criterion set as input values, weight is put on investigating the effects of different coefficients of variation (COV ). The analyses show that the property variabilities have a significant influence on the requisite area ratio that an active use of RBD is a useful tool for optimizing designs in geotechnical engineering. The methodology favors the contractors own development of the mixing process since higher design values can be utilized when

No description available.

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.