Thesis (MSc)--Stellenbosch University, 2015. / ENGLISH ABSTRACT: Recent studies have shown that considering proof test data in a Reliability Based Design Optimization (RBDO) environment can result in design improvement. Proof testing involves the physical testing of each and every component before it enters into service. Considering the proof test data as part of the RBDO process allows for improvement of the original design, such as weight savings, while preserving high reliability levels. Composite Over-Wrapped Pressure Vessels (COPV) is used as an example application of achieving weight savings while maintaining high reliability levels. COPVs are light structures used to store pressurized fluids in space shuttles, the international space station and other applications where they are maintained at high pressure for extended periods of time. Given that each and every COPV used in spacecraft is proof tested before entering service and any weight savings on a spacecraft results in significant cost savings, this thesis put forward an application of RBDO that accounts for proof test data in the design of a COPV. The method developed in this thesis shows that, while maintaining high levels of reliability, significant weight savings can be achieved by including proof test data in the design process. Also, the method enables a designer to have control over the magnitude of the proof test, making it possible to also design the proof test itself depending on the desired level of reliability for passing the proof test. The implementation of the method is discussed in detail. The evaluation of the reliability was based on the First Order Reliability Method (FORM) supported by Monte Carlo Simulation. Also, the method is implemented in a versatile way that allows the use of analytical as well as numerical (in the form of finite element) models. Results show that additional weight savings can be achieved by the inclusion of proof test data in the design process. / AFRIKAANSE OPSOMMING: Onlangse studies het getoon dat die gebruik van ontwerp spesifieke proeftoets data in betroubaarheids gebaseerde optimering (BGO) kan lei tot 'n verbeterde ontwerp. BGO behels vele aspekte in die ontwerpsgebied. Die toevoeging van proeftoets data in ontwerpsoptimering bring te weë; die toetsing van 'n ontwerp en onderdele voor gebruik, die aangepaste en verbeterde ontwerp en gewig-besparing met handhawing van hoë betroubaarsheidsvlakke. 'n Praktiese toepassing van die BGO tegniek behels die ontwerp van drukvatte met saamgestelde materiaal bewapening. Die drukvatontwerp is 'n ligte struktuur wat gebruik word in die berging van hoë druk vloeistowwe in bv. in ruimtetuie, in die internasionale ruimtestasie en in ander toepassings waar hoë druk oor 'n tydperk verlang word. Elke drukvat met saamgestelde materiaal bewapening wat in ruimtevaartstelsels gebruik word, word geproeftoets voor gebruik. In ruimte stelselontwerp lei massa besparing tot 'n toename in loonvrag. Die tesis beskryf 'n optimeringsmetode soos ontwikkel en gebaseer op 'n BGO tegniek. Die metode word toegepas in die ontwerp van drukvatte met saamgestelde materiaal bewapening. Die resultate toon dat die gebruik van proeftoets data in massa besparing optimering onderhewig soos aan hoë betroubaarheidsvlakke moontlik is. Verdermeer, die metode laat ook ontwerpers toe om die proeftoetsvlak aan te pas om sodoende by ander betroubaarheidsvlakke te toets. In die tesis word die ontwikkeling en gebruik van die optimeringsmetode uiteengelê. Die evaluering van betroubaarheidsvlakke is gebaseer op 'n eerste orde betroubaarheids-tegniek wat geverifieer word met talle Monte Carlo simulasie resultate. Die metode is ook so geskep dat beide analitiese sowel as eindige element modelle gebruik kan word. Ten slotte, word 'n toepassing getoon waar resultate wys dat die gebruik van die optimeringsmetode met die insluiting van proeftoets data wel massa besparing kan oplewer.

Les premières étapes d'une conception fiabiliste impliquent la formulation de critères de performance et de contraintes de fiabilité d'une part, et le choix d'une représentation des incertitudes d'autre part. Force est de constater que, le plus souvent, des aspects de performance ou de fiabilité conditionnant la solution optimale ne seront pas connus ou seront négligés lors des premières phases de conception. De plus, les techniques de réduction des incertitudes telles que les tests additionnels et la reconception ne sont pas pris en compte dans les calculs de fiabilité initiaux. Le travail exposé dans ce manuscrit aborde la conception optimale de systèmes sous deux angles : 1) le compromis entre performance et coût généré par les tests supplémentaires et les reconceptions et, 2) l'identification de multiples solutions optimales (dont certaines locales) en tant que stratégie contre les erreurs initiales de conception. Dans la première partie de notre travail, une méthodologie est proposée pour estimer l'effet sur la performance et le coût d'un produit d'un test supplémentaire et d'une éventuelle reconception. Notre approche se base, d'une part, sur des distributions en probabilité des erreurs de calcul et des erreurs expérimentales et, d'autre part, sur une rêgle de reconception a priori. Ceci permet d'estimer a posteriori la probabilité et le coût d'un produit. Nous montrons comment, à travers le choix de politiques de prochain test et de re-conception, une entreprise est susceptible de contrôler le compromis entre performance et coût de développement.Dans la seconde partie de notre travail, nous proposons une méthode pour l'estimation de plusieurs solutions candidates à un problème de conception où la fonction coût et/ou les contraintes sont coûteuses en calcul. Une approche pour aborder de tels problèmes est d'utiliser un métamodèle, ce qui nécessite des évaluations de points en diverses régions de l'espace de recherche. Il est alors dommage d'utiliser cette connaissance seulement pour estimer un optimum global. Nous proposons une nouvelle approche d'échantillonnage à partir de métamodèles pour trouver plusieurs optima locaux. Cette méthode procède par partitionnement adaptatif de l'espace de recherche et construction de métamodèles au sein de chaque partition. Notre méthode est testée et comparée à d'autres approches d'optimisation globale par métamodèles sur des exemples analytiques en dimensions 2 à 6, ainsi que sur la conception d'un bouclier thermique en 5 dimensions. / The initial stages of reliability-based design optimization involve the formulation of objective functions and constraints, and building a model to estimate the reliability of the design with quantified uncertainties. However, even experienced hands often overlook important objective functions and constraints that affect the design. In addition, uncertainty reduction measures, such as tests and redesign, are often not considered in reliability calculations during the initial stages. This research considers two areas that concern the design of engineering systems: 1) the trade-off of the effect of a test and post-test redesign on reliability and cost and 2) the search for multiple candidate designs as insurance against unforeseen faults in some designs. In this research, a methodology was developed to estimate the effect of a single future test and post-test redesign on reliability and cost. The methodology uses assumed distributions of computational and experimental errors with re-design rules to simulate alternative future test and redesign outcomes to form a probabilistic estimate of the reliability and cost for a given design. Further, it was explored how modeling a future test and redesign provides a company an opportunity to balance development costs versus performance by simultaneously designing the design and the post-test redesign rules during the initial design stage.The second area of this research considers the use of dynamic local surrogates, or surrogate-based agents, to locate multiple candidate designs. Surrogate-based global optimization algorithms often require search in multiple candidate regions of design space, expending most of the computation needed to define multiple alternate designs. Thus, focusing on solely locating the best design may be wasteful. We extended adaptive sampling surrogate techniques to locate multiple optima by building local surrogates in sub-regions of the design space to identify optima. The efficiency of this method was studied, and the method was compared to other surrogate-based optimization methods that aim to locate the global optimum using two two-dimensional test functions, a six-dimensional test function, and a five-dimensional engineering example.

Computational design has become a predominant element of various engineering tasks. However, the ever increasing complexity of numerical models creates the need for efficient methodologies. Specifically, computational design under uncertainties remains sparsely used in engineering settings due to its computational cost. This dissertation proposes a coherent framework for various branches of computational design under uncertainties, including model update, reliability assessment and reliability-based design optimization. Through the use of machine learning techniques, computationally inexpensive approximations of the constraints, limit states, and objective functions are constructed. Specifically, a novel adaptive sampling strategy allowing for the refinement of any approximation only in relevant regions has been developed, referred to as generalized max-min. This technique presents various computational advantages such as ease of parallelization and applicability to any metamodel. Three approaches tailored for computational design under uncertainties are derived from the previous approximation technique. An algorithm for reliability assessment is proposed and its efficiency is demonstrated for different probabilistic settings including dependent variables using copulas. Additionally, the notion of fidelity map is introduced for model update settings with large number of dependent responses to be matched. Finally, a new reliability-based design optimization method with local refinement has been developed. A derivation of sampling-based probability of failure derivatives is also provided along with a discussion on numerical estimates. This derivation brings additional flexibility to the field of computational design. The knowledge acquired and techniques developed during this Ph.D. have been synthesized in an object-oriented MATLAB toolbox. The help and ergonomics of the toolbox have been designed so as to be accessible by a large audience.

The initial stages of reliability-based design optimization involve the formulation of objective functions and constraints, and building a model to estimate the reliability of the design with quantified uncertainties. However, even experienced hands often overlook important objective functions and constraints that affect the design. In addition, uncertainty reduction measures, such as tests and redesign, are often not considered in reliability calculations during the initial stages. This research considers two areas that concern the design of engineering systems: 1) the trade-off of the effect of a test and post-test redesign on reliability and cost and 2) the search for multiple candidate designs as insurance against unforeseen faults in some designs. In this research, a methodology was developed to estimate the effect of a single future test and post-test redesign on reliability and cost. The methodology uses assumed distributions of computational and experimental errors with re-design rules to simulate alternative future test and redesign outcomes to form a probabilistic estimate of the reliability and cost for a given design. Further, it was explored how modeling a future test and redesign provides a company an opportunity to balance development costs versus performance by simultaneously designing the design and the post-test redesign rules during the initial design stage.The second area of this research considers the use of dynamic local surrogates, or surrogate-based agents, to locate multiple candidate designs. Surrogate-based global optimization algorithms often require search in multiple candidate regions of design space, expending most of the computation needed to define multiple alternate designs. Thus, focusing on solely locating the best design may be wasteful. We extended adaptive sampling surrogate techniques to locate multiple optima by building local surrogates in sub-regions of the design space to identify optima. The efficiency of this method was studied, and the method was compared to other surrogate-based optimization methods that aim to locate the global optimum using two two-dimensional test functions, a six-dimensional test function, and a five-dimensional engineering example.

abstract: This dissertation is to address product design optimization including reliability-based design optimization (RBDO) and robust design with epistemic uncertainty. It is divided into four major components as outlined below. Firstly, a comprehensive study of uncertainties is performed, in which sources of uncertainty are listed, categorized and the impacts are discussed. Epistemic uncertainty is of interest, which is due to lack of knowledge and can be reduced by taking more observations. In particular, the strategies to address epistemic uncertainties due to implicit constraint function are discussed. Secondly, a sequential sampling strategy to improve RBDO under implicit constraint function is developed. In modern engineering design, an RBDO task is often performed by a computer simulation program, which can be treated as a black box, as its analytical function is implicit. An efficient sampling strategy on learning the probabilistic constraint function under the design optimization framework is presented. The method is a sequential experimentation around the approximate most probable point (MPP) at each step of optimization process. It is compared with the methods of MPP-based sampling, lifted surrogate function, and non-sequential random sampling. Thirdly, a particle splitting-based reliability analysis approach is developed in design optimization. In reliability analysis, traditional simulation methods such as Monte Carlo simulation may provide accurate results, but are often accompanied with high computational cost. To increase the efficiency, particle splitting is integrated into RBDO. It is an improvement of subset simulation with multiple particles to enhance the diversity and stability of simulation samples. This method is further extended to address problems with multiple probabilistic constraints and compared with the MPP-based methods. Finally, a reliability-based robust design optimization (RBRDO) framework is provided to integrate the consideration of design reliability and design robustness simultaneously. The quality loss objective in robust design, considered together with the production cost in RBDO, are used formulate a multi-objective optimization problem. With the epistemic uncertainty from implicit performance function, the sequential sampling strategy is extended to RBRDO, and a combined metamodel is proposed to tackle both controllable variables and uncontrollable variables. The solution is a Pareto frontier, compared with a single optimal solution in RBDO. / Dissertation/Thesis / Ph.D. Industrial Engineering 2012

This dissertation addresses the ultimate strength analysis of nonlinear beam-columns under axial compression, the sensitivity of the ultimate strength, structural optimization and reliability analysis using ultimate strength analysis, and Reliability-Based Design Optimization (RBDO) of the nonlinear beam-columns. The ultimate strength analysis is based on nonlinear beam theory with material and geometric nonlinearities. Nonlinear constitutive law is developed for elastic-perfectly-plastic beam cross-section consisting of base plate and T-bar stiffener. The analysis method is validated using commercial nonlinear finite element analysis. A new direct solving method is developed, which combines the original governing equations with their derivatives with respect to deformation matric and solves for the ultimate strength directly. Structural optimization and reliability analysis use a gradient-based algorithm and need accurate sensitivities of the ultimate strength to design variables. Semi-analytic sensitivity of the ultimate strength is calculated from a linear set of analytical sensitivity equations which use the Jacobian matrix of the direct solving method. The derivatives of the structural residual equations in the sensitivity equation set are calculated using complex step method. The semi-analytic sensitivity is more robust and efficient as compared to finite difference sensitivity. The design variables are the cross-sectional geometric parameters. Random variables include material properties, geometric parameters, initial deflection and nondeterministic load. Failure probabilities calculated by ultimate strength reliability analysis are validated by Monte Carlo Simulation. Double-loop RBDO minimizes structural weight with reliability index constraint. The sensitivity of reliability index with respect to design variables is calculated from the gradient of limit state function at the solution of reliability analysis. By using the ultimate strength direct solving method, semi-analytic sensitivity and gradient-based optimization algorithm, the RBDO method is found to be robust and efficient for nonlinear beam-columns. The ultimate strength direct solving method, semi-analytic sensitivity, structural optimization, reliability analysis, and RBDO method can be applied to more complicated engineering structures including stiffened panels and aerospace/ocean structures. / Ph. D.

京都大学 / 新制・課程博士 / 博士(工学) / 甲第24575号 / 工博第5081号 / 新制||工||1973(附属図書館) / 京都大学大学院工学研究科建築学専攻 / (主査)教授 大崎 純, 教授 池田 芳樹, 准教授 藤田 皓平 / 学位規則第4条第1項該当 / Doctor of Philosophy (Engineering) / Kyoto University / DFAM

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.