Second‐order reliability method‐based inverse reliability analysis using Hessian update for accurate and efficient reliability‐based design optimization

First‐order reliability method (FORM) has been mostly utilized for solving reliability‐based design optimization (RBDO) problems efficiently. However, second‐order reliability method (SORM) is required in order to estimate a probability of failure accurately in highly nonlinear performance functions. Despite accuracy of SORM, its application to RBDO is quite challenging due to unaffordable numerical burden incurred by a Hessian calculation. For reducing the numerical efforts, a quasi‐Newton approach to approximate the Hessian is introduced in this study instead of calculating the true Hessian. The proposed SORM with the approximated Hessian requires computations only used in FORM, leading to very efficient and accurate reliability analysis. The proposed SORM also utilizes a generalized chi‐squared distribution in order to achieve better accuracy. Furthermore, SORM‐based inverse reliability method is proposed in this study. An accurate reliability index corresponding to a target probability of failure is updated using the proposed SORM. Two approaches in terms of finding an accurate most probable point using the updated reliability index are proposed. The proposed SORM‐based inverse analysis is then extended to RBDO in order to obtain a reliability‐based optimum design satisfying probabilistic constraints more accurately even for a highly nonlinear system. The numerical study results show that the proposed reliability analysis and RBDO achieve efficiency of FORM and accuracy of SORM at the same time. Copyright © 2014 John Wiley & Sons, Ltd.     

Tolerance‐based reliability and optimization design of switched‐mode power supply

Due to its high efficiency and low power consumption, switched‐mode power supply (SMPS) represents the development trend of the stabilized voltage power supply. However, tolerance has become one of the key factors in the design of SMPS because of the process fluctuation of electronic components, unstable input parameters of the circuit system, influence of working conditions and environment, and the effect of aging. In order to improve the reliability of SMPS and reduce the manufacturing cost, this paper proposes a reliability analysis and optimization design method based on the tolerance and sensitivity analysis. Finally, this method is applied to the tolerance design for the positive switching power supply of the SMPS circuits, and the optimal tolerance design scheme is obtained. Furthermore, the reliability and probability density curves are evaluated.     

Mooring system design optimization using a surrogate assisted multi-objective genetic algorithm

This article presents a novel framework for the multi-objective optimization of offshore renewable energy mooring systems using a random forest based surrogate model coupled to a genetic algorithm. This framework is demonstrated for the optimization of the mooring system for a floating offshore wind turbine highlighting how this approach can aid in the strategic design decision making for real-world problems faced by the offshore renewable energy sector. This framework utilizes validated numerical models of the mooring system to train a surrogate model, which leads to a computationally efficient optimization routine, allowing the search space to be more thoroughly searched. Minimizing both the cost and cumulative fatigue damage of the mooring system, this framework presents a range of optimal solutions characterizing how design changes impact the trade-off between these two competing objectives.     

Post optimization for accurate and efficient reliability‐based design optimization using second‐order reliability method based on importance sampling and its stochastic sensitivity analysis

In this study, a post optimization technique for a correction of inaccurate optimum obtained using first‐order reliability method (FORM) is proposed for accurate reliability‐based design optimization (RBDO). In the proposed method, RBDO using FORM is first performed, and then the proposed second‐order reliability method (SORM) is performed at the optimum obtained using FORM for more accurate reliability assessment and its sensitivity analysis. In the proposed SORM, the Hessian of a performance function is approximated by reusing derivatives information accumulated during previous RBDO iterations using FORM, indicating that additional functional evaluations are not required in the proposed SORM. The proposed SORM calculates a probability of failure and its first‐order and second‐order stochastic sensitivity by applying the importance sampling to a complete second‐order Taylor series of the performance function. The proposed post optimization constructs a second‐order Taylor expansion of the probability of failure using results of the proposed SORM. Because the constructed Taylor expansion is based on the reliability method more accurate than FORM, the corrected optimum using this Taylor expansion can satisfy the target reliability more accurately. In this way, the proposed method simultaneously achieves both efficiency of FORM and accuracy of SORM. Copyright © 2015 John Wiley & Sons, Ltd.     

Collaborative optimization with inverse reliability for multidisciplinary systems uncertainty analysis

This article introduces a method which combines the collaborative optimization framework and the inverse reliability strategy to assess the uncertainty encountered in the multidisciplinary design process. This method conducts the sub-system analysis and optimization concurrently and then improves the process of searching for the most probable point (MPP). It reduces the load of the system-level optimizer significantly. This advantage is specifically more prominent for large-scale engineering system design. Meanwhile, because the disciplinary analyses are treated as the equality constraints in the disciplinary optimization, the computation load can be further reduced. Examples are used to illustrate the accuracy and efficiency of the proposed method.     

基于模糊设计准则的机械可靠性优化设计研究

按照应力-强度分布干涉和模糊理论,对机械模糊可靠性优化设计方法进行系统的研究,提出了剩余可靠度的概念,建立了数学模型,给出了设计实例。研究表明：仅仅以实现要求的可靠度为目标的一般模糊可靠性设计由于不能考虑其他条件对设计变量取值的约束作用,往往导致设计结果不具可行性而失去实际意义。可靠性优化设计将模糊可靠性条件与其他所有限制条件一起作为设计约束,在设计空间围成一个可行设计域,然后按照设计目标要求用一定方法寻求可行域内的最优设计点,从而保证了设计结果既具可靠性,又具可行性和最优性。     

Failure assessment of the first stage high‐pressure turbine blades in an aero‐engine turbine

In this paper, microscopic analysis and fatigue experiment were conducted to detect the damage degree of turbine blades after a 600 h service and determine the reliable service time. First, several service blades were cut into slices on different cross sections and microscopically examined. It is found that the γ′ phase particles are slightly coarsened and the γ′ phase parameters change with temperature and stress distribution. Then, fatigue test was conducted on service blades simulating the real working condition. The test result shows that the blades after a 600 h service would serve for 3596 h more until failure during actual flight. The ultimate fracture is mainly caused by the interaction of fatigue, creep and oxidation. Besides, the γ′ phase parameters change obviously compared with service blade without fatigue test. It indicates that the γ′ phase parameters could be used to evaluate the microdamage of service blades, which has great significance for service reliability of the turbine machinery.     

Aerodynamic optimization of turbomachinery blades using evolutionary methods and ANN-based surrogate models

A fast, flexible, and robust simulation-based optimization scheme using an ANN-surrogate model was developed, implemented, and validated. The optimization method uses Genetic Algorithm (GA), which is coupled with an Artificial Neural Network (ANN) that uses a back propagation algorithm. The developed optimization scheme was successfully applied to single-point aerodynamic optimization of a transonic turbine stator and multi-point optimization of a NACA65 subsonic compressor rotor in two-dimensional flow, both were represented by 2D linear cascades. High fidelity CFD flow simulations, which solve the Reynolds-Averaged Navier-Stokes equations, were used in generating the data base used in building the ANN low fidelity model. The optimization objective is a weighted sum of the performance objectives and is penalized with the constraints; it was constructed so as to achieve a better aerodynamic performance at the design point or over the full operating range by reshaping the blade profile. The latter is represented using NURBS functions, whose coefficients are used as the design variables. Parallelizing the CFD flow simulations reduced the turn-around computation time at close to 100% efficiency. The ANN model was able to approximate the objective function rather accurately and to reduce the optimization computing time by ten folds. The chosen objective function and optimization methodology result in a significant and consistent improvement in blade performance.     

Fatigue reliability assessment of turbine discs under multi‐source uncertainties

Hot section components of aircraft engines like high pressure turbine (HPT) discs usually operate under complex loadings coupled with multi‐source uncertainties. The effect of these uncertainties on structural response of HPT discs should be accounted for its fatigue life and reliability assessment. In this study, a probabilistic framework for fatigue reliability analysis is established by incorporating FE simulations with Latin hypercube sampling to quantify the influence of material variability and load variations. Particularly, variability in material response is characterized by combining the Chaboche constitutive model with Fatemi‐Socie criterion. Results from fatigue reliability and sensitivity analysis of a HPT disc indicated that dispersions of basic variables must be taken into account for its fatigue reliability analysis. Moreover, the proposed framework based on the strength‐damage interference provides more reasonably correlations with its field number of flights rather than the load‐life interference one.     

A reliability-based robust optimization design for the drum brake using adaptive Kriging surrogate model

To decrease random parameters’ influence on the drum brake reliability, the reliability-based robust optimization design (RBROD) of the electric vehicle brake is proposed. Based on the assumption that the maximum temperature of the brake cannot exceed the allowable temperature, a performance function model of thermal–mechanical coupling reliability of drum brakes is established by the adaptive Kriging method, and the analysis of reliability sensitivity and RBROD are conducted. The accuracy of the proposed model is verified by temperature measurement experiment under emergency braking condition. The robust optimization design improves the drum brake reliability to 0.99998 and reduce the influence of the design parameters on the reliability, with the absolute values of the reliability sensitivity and the weight of the drum brake are significantly smaller. Therefore, the objectives of reliability design, robustness design, and optimization design are simultaneously achieved by the proposed methods. Besides, the relative error of the proposed method is 0.373%, the number of function evaluations is 39, and the comparison with four meta-model methods show that the proposed method holds high-accuracy and high-efficiency. This study provides a high-precision theoretical explanation for the robust optimization design of drum brake.     

Continuum‐based design sensitivity analysis and optimization of nonlinear shell structures using meshfree method

A continuum‐based shape and configuration design sensitivity analysis (DSA) method for a finite deformation elastoplastic shell structure has been developed. Shell elastoplasticity is treated using the projection method that performs the return mapping on the subspace defined by the zero‐normal stress condition. An incrementally objective integration scheme is used in the context of finite deformation shell analysis, wherein the stress objectivity is preserved for finite rotation increments. The material derivative concept is used to develop a continuum‐based shape and configuration DSA method. Significant computational efficiency is obtained by solving the design sensitivity equation without iteration at each converged load step using the same consistent tangent stiffness matrix. Numerical implementation of the proposed shape and configuration DSA is carried out using the meshfree method. The accuracy and efficiency of the proposed method is illustrated using numerical examples. Copyright © 2006 John Wiley & Sons, Ltd.     

Reliability‐based robust design optimization: A multi‐objective framework using hybrid quality loss function

In this globally competitive business environment, design engineers are constantly striving to establish new and effective tools and techniques to ensure a robust and reliable product design. Robust design (RD) and reliability‐based design approaches have shown the potential to deal with variability in the life cycle of a product. This paper explores the possibilities of combining both approaches into a single model and proposes a hybrid quality loss function‐based multi‐objective optimization model. The model is unique because it uses a hybrid form of quality loss‐based objective function that is defined in terms of desirable as well as undesirable deviations to obtain efficient design points with minimum quality loss. The proposed approach attempts to optimize the product design by addressing quality loss, variability, and life‐cycle issues simultaneously by combining both reliability‐based and RD approaches into a single model with various customer aspirations. The application of the approach is demonstrated using a leaf spring design example. Copyright © 2009 John Wiley & Sons, Ltd.     

The effect of material behaviour on the analysis of single crystal turbine blades: Part II – Component analysis

This paper describes the analysis of a turbine blade component using a slip system model developed for modern single‐crystal superalloys. Structural elasto‐viscoplastic calculations are carried out for the component. The emphasis throughout is on the effect of micromechanisms of deformation, accounted for in the material model, on the predicted overall behaviour of the component. With the recent proliferation in detailed material models that are available, it is prudent to take a step back and investigate the implications of such models for component analysis and design. This effect is manifested through the determination of a stabilised and redistributed stress state throughout the component. While some components are creep‐limited in design, many are fatigue‐limited and it is stabilised stresses which control the cyclic life of these components. The accuracy of the material model, incorporating various micromechanisms as a function of stress and temperature, can significantly effect these stabilised stresses. The effect of the crystallographic orientation on blade behaviour is illustrated and the implications of shakedown simulations for fatigue lifing of turbine blades are discussed.     

Risk evaluation in failure mode and effects analysis of aircraft turbine rotor blades using Dempster–Shafer evidence theory under uncertainty

Rotor blades are the major components of an aircraft turbine. Their reliability seriously affects the overall aircraft turbine security. Failure mode and effects analysis (FMEA), especially, the risk priority order of failure modes, is essential in the design process. The risk priority number (RPN) has been extensively used to determine the risk priority order of failure modes. When multiple experts give different risk evaluations to one failure mode, which may be imprecise and uncertain, the traditional RPN is not a sufficient tool for risk evaluation. In this paper, the modified Dempster–Shafer (D–S) is adopted to aggregate the different evaluation information by considering multiple experts’ evaluation opinions, failure modes and three risk factors respectively. A simplified discernment frame is proposed according to the practical application. Moreover, the mean value of the new RPN is used to determine the risk priority order of multiple failure modes. Finally, this method is used to deal with the risk priority evaluation of the failure modes of rotor blades of an aircraft turbine under multiple sources of different and uncertain evaluation information. The consequence of this method is rational and efficient.     

Reliability analysis and reliability-based design optimization of roadway horizontal curves using a first-order reliability method

This article presents reliability analysis and reliability-based optimization of roadway minimum radius design based on vehicle dynamics, mainly focusing on exit ramps and interchanges. The performance functions are formulated as failure modes of vehicle rollover and sideslip. To accurately describe the failure modes, analytical models for rollover and sideslip are derived considering nonlinear characteristics of vehicle behaviour using the commercial software TruckSim. The probability of an accident is evaluated using the first-order reliability method and numerical studies are conducted using a single-unit truck model. To propose a practical application for the study, the reliability analysis for the minimum radius recommended by American Association of State Highway and Transportation Officials is conducted. The results show that, even if there are deviations from assumed design conditions of the current design guideline, the proposed design method can guarantee given target margins of safety against rollover and sideslip. Based on the reliability analysis, reliability-based design optimization is carried out and the results indicate new recommendations for minimum radii satisfying given target reliability levels.     

Structural design of high-performance capacitive accelerometers using parametric optimization with uncertainties

Electrostatic or capacitive accelerometers are among the highest volume microelectromechanical systems (MEMS) products nowadays. The design of such devices is a complex task, since they depend on many performance requirements, which are often conflicting. Therefore, optimization techniques are often used in the design stage of these MEMS devices. Because of problems with reliability, the technology of MEMS is not yet well established. Thus, in this work, size optimization is combined with the reliability-based design optimization (RBDO) method to improve the performance of accelerometers. To account for uncertainties in the dimensions and material properties of these devices, the first order reliability method is applied to calculate the probabilities involved in the RBDO formulation. Practical examples of bulk-type capacitive accelerometer designs are presented and discussed to evaluate the potential of the implemented RBDO solver.     

基于非线性本构关系的复合材料风机叶片有限元极限分析与设计

为了研究具有三维复杂构形的复合材料风机叶片的逐次破坏过程和极限承载能力, 将复合材料细观力学非线性本构理论桥联模型与有限元软件ABAQUS通过用户子程序UGENS结合起来对风力发电机叶片结构进行极限强度分析。只需提供纤维和基体的材料性能参数、 纤维体积含量以及蒙皮和增强筋的铺层数据包括铺设角、 层厚和铺层数, 就可预报出复合材料复杂叶片结构的整体承载能力以及叶片破坏所处的位置, 为正确评估和合理设计风机叶片结构提供了一种简便有效的分析方法。以一种20kW风机叶片为例, 用此方法实现了新型复合材料叶片结构的极限分析和合理设计, 提高了叶片的强度和刚度, 有效降低了叶片的重量。本文中的方法同样适用于其它复合材料复杂结构的极限分析与强度设计。      

An integration design optimization framework of robust design,axiomatic design,and reliability‐based design

Robust design, axiomatic design, and reliability‐based design provide effective approaches to deal with quality problems, and their integration will achieve better quality improvement. An integration design optimization framework of robust design, axiomatic design, and reliability‐based design is proposed in this paper. First, the fitted response model of each quality characteristic is obtained by response surface methodology and the mean square error (MSE) estimation is given by a second‐order Taylor series approximation expansion. Then the multiple quality characteristics robust design model is developed by the MSE criteria. Finally, the independence axiom constraints for decoupling and reliability constraints are integrated into the multiple quality characteristics robust design model, and the integration design optimization framework is formulated, where the weighted Tchebycheff approach is adopted to solve the multiple objective programming. An illustrative example is presented at the end, and the results show that the proposed approach can obtain better trade‐offs among conflicting quality characteristics, variability, coupling degree and reliability requirements. Copyright © 2011 John Wiley & Sons, Ltd.     

Sensitivity analyses of FORM‐based and DRM‐based performance measure approach (PMA) for reliability‐based design optimization (RBDO)

In gradient‐based design optimization, the sensitivities of the constraint with respect to the design variables are required. In reliability‐based design optimization (RBDO), the probabilistic constraint is evaluated at the most probable point (MPP), and thus the sensitivities of the probabilistic constraints at MPP are required. This paper presents the rigorous analytic derivation of the sensitivities of the probabilistic constraint at MPP for both first‐order reliability method (FORM)‐based performance measure approach (PMA) and dimension reduction method (DRM)‐based PMA. Numerical examples are used to demonstrate that the analytic sensitivities agree very well with the sensitivities obtained from the finite difference method (FDM). However, as the sensitivity calculation at the true DRM‐based MPP requires the second‐order derivatives and additional MPP search, the sensitivity derivation at the approximated DRM‐based MPP, which does not require the second‐order derivatives and additional MPP search to find the DRM‐based MPP, is proposed in this paper. A convergence study illustrates that the sensitivity at the approximated DRM‐based MPP converges to the sensitivity at the true DRM‐based MPP as the design approaches the optimum design. Hence, the sensitivity at the approximated DRM‐based MPP is proposed to be used for the DRM‐based RBDO to enhance the efficiency of the optimization. Copyright © 2009 John Wiley & Sons, Ltd.