An efficient approach to reliability-based design optimization within the enhanced sequential optimization and reliability assessment framework

Reliability based design optimization (RBDO) has been widely implemented in engineering practices for high safety and reliability. It is an important challenge to improve computational efficiency. Sequential optimization and reliability assessment (SORA) has made great efforts to improve computational efficiency by decoupling a RBDO problem into sequential deterministic optimization and reliability analysis as a single-loop method. In this paper, in order to further improve computational efficiency and extend the application of the current SORA method, an enhanced SORA (ESORA) is proposed by considering constant and varying variances of random design variables while keeping the sequential framework. Some mathematical examples and an engineering case are given to illustrate the proposed method and validate the efficiency.     

Reliability-based design optimization using convex approximations and sequential optimization and reliability assessment method

In this study, an effective method for reliability-based design optimization (RBDO) is proposed enhancing sequential optimization and reliability assessment (SORA) method by convex approximations. In SORA, reliability estimation and deterministic optimization are performed sequentially. The sensitivity and function value of probabilistic constraint at the most probable point (MPP) are obtained in the reliability analysis loop. In this study, the convex approximations for probabilistic constraint are constructed by utilizing the sensitivity and function value of the probabilistic constraint at the MPP. Hence, the proposed method requires much less function evaluations of probabilistic constraints in the deterministic optimization than the original SORA method. The efficiency and accuracy of the proposed method were verified through numerical examples.     

Reliability-based design optimization of slider air bearings

This paper presents a design methodology for determining configurations of slider air bearings considering the randomness of the air-bearing surface (ABS) geometry by using the iSIGHT. A reliability-based design optimization (RBDO) problem is formulated to minimize the variations in the mean values of the flying heights from a target value while satisfying the desired probabilistic constraints keeping the pitch and roll angles within a suitable range. The reliability analysis is employed to estimate how the fabrication tolerances of individual slider parameters affect the final flying attitude tolerances. The proposed approach first solves the deterministic optimization problem. Then, beginning with this solution, the RBDO is continued with the reliability constraints affected by the random variables. Reliability constraints overriding the constraints of the deterministic optimization attempt to drive the design to a reliability solution with minimum increase in the objective. The simulation results of the RBDO are listed in comparison with the values of the initial design and the results of the deterministic optimization, respectively. To show the effectiveness of the proposed approach, the reliability analyses are simply carried out by using the mean value first-order second-moment (MVFO) method. The Monte Carlo simulation of the RBDO’s results is also performed to estimate the efficiency of the proposed approach. Those results are demonstrated to satisfy all the desired probabilistic constraints, where the target reliability level for constraints is defined as 0.8.     

Reliability-based design optimization via high order response surface method

To reduce the computational effort of reliability-based design optimization (RBDO), the response surface method (RSM) has been widely used to evaluate reliability constraints. We propose an efficient methodology for solving RBDO problems based on an improved high order response surface method (HORSM) that takes advantage of an efficient sampling method, Hermite polynomials and uncertainty contribution concept to construct a high order response surface function with cross terms for reliability analysis. The sampling method generates supporting points from Gauss-Hermite quadrature points, which can be used to approximate response surface function without cross terms, to identify the highest order of each random variable and to determine the significant variables connected with point estimate method. The cross terms between two significant random variables are added to the response surface function to improve the approximation accuracy. Integrating the nested strategy, the improved HORSM is explored in solving RBDO problems. Additionally, a sampling based reliability sensitivity analysis method is employed to reduce the computational effort further when design variables are distributional parameters of input random variables. The proposed methodology is applied on two test problems to validate its accuracy and efficiency. The proposed methodology is more efficient than first order reliability method based RBDO and Monte Carlo simulation based RBDO, and enables the use of RBDO as a practical design tool.     

Multidisciplinary wing design optimization considering global sensitivity and uncertainty of approximation models

In recent years, high-fidelity analysis tools, such as computational fluid dynamics and finite element method, have been widely used in multidisciplinary design optimization (MDO) to enhance the accuracy of design results. However, complex MDO problems have many design variables and require long computation times. Global sensitivity analysis (GSA) is proposed to assuage the complexity of design problems by reducing dimensionality where variables that have low impact on the objective function are neglected. This avoids wasting computational effort and time on low-priority variables. Additionally, uncertainty introduced by the fidelity of the analysis tools is considered in design optimization to increase the reliability of design results. Reliability-based design optimization (RBDO) and possibility-based design optimization (PBDO) methods are proposed to handle uncertainty in design optimization. In this paper, the extended Fourier amplitude sensitivity test was used for GSA, whereas a collaborative optimization-based framework with RBDO and PBDO was used to consider uncertainty introduced by approximation models. The proposed method was applied to an aero-structural design optimization of an aircraft wing to demonstrate the feasibility and efficiency of the developed method. The objective function was to maximize the lift-to-drag ratio. The proposed process reduced calculation efforts by reducing the number of design variables and achieved the target probability of failure when it considered uncertainty. Moreover, this work evaluated previous research in RBDO with MDO for the wing design by comparing it with the PBDO result.     

Multi-objective reliability-based optimization for design of trapezoidal labyrinth weirs

In the present research, the reliability-based design optimization (RBDO) of labyrinth weirs has been investigated. The optimization problem is formulated such that the optimal shape of trapezoidal labyrinth weir described by a number of variables is found by minimizing the volume of the trapezoidal labyrinth weir and maximizing the reliability index. The constraint conditions are the weir geometric shape and its different ratios. In order to achieve this purpose, a framework is presented whereby non-dominated sorting genetic algorithm (NSGA-II) is integrated with monte carlo simulation (MCS) method to solve the RBDO approach of trapezoidal labyrinth weirs. The proposed method is applied to UTE Dam labyrinth weir, and the results are compared with the real one. The results show the need for design based on reliability in the labyrinth weirs that propose using RBDO for weir design. The results showed that RBDO approach can achieve a more reliable design in addition to reducing the volume of the trapezoidal labyrinth weir. Finally, the sensitivity analysis of the parameters effective on the reliability index revealed that three design variables of weir width, total upstream head and discharge coefficient are the main parameters affecting weir RBDO solution.     

Reliability-based design optimization of axial compressor using uncertainty model for stall margin

Reliability-based design optimization (RBDO) of the NASA stage 37 axial compressor is performed using an uncertainty model for stall margin in order to guarantee stable operation of the compressor. The main characteristics of RBDO for the axial compressor are summarized as follows: First, the values of mass flow rate and pressure ratio in stall margin calculation are defined as statistical models with normal distribution for consideration of the uncertainty in stall margin. Second, Monte Carlo Simulation is used in the RBDO process to calculate failure probability of stall margin accurately. Third, an approximation model that is constructed by an artificial neural network is adopted to reduce the time cost of RBDO. The present method is applied to the NASA stage 37 compressor to improve the reliability of stall margin with both maximized efficiency and minimized weight. The RBDO result is compared with the deterministic optimization (DO) result which does not include an uncertainty model. In the DO case, stall margin is slightly higher than the reference value of the required constraint, but the probability of stall is 43%. This is unacceptable risk for an aircraft engine, which requires absolutely stable operation in flight. However, stall margin obtained in RBDO is 2.7% higher than the reference value, and the probability of success increases to 95% with the improved efficiency and weight. Therefore, RBDO of the axial compressor for aircraft engine can be a reliable design optimization method through consideration of unexpected disturbance of the flow conditions.     

Reliability based design optimization using response surface augmented moment method

Journal of Mechanical Science and Technology - A new approach toward reliability based design optimization (RBDO) is proposed based on the response surface augmented moment method (RSMM). In RSMM,...     

Improved moment-based quadrature rule and its application to reliability-based design optimization

A moment method known as the fourth moment method can perform reliability analysis without optimization using the first four statistical moments. Numerical integration is used to calculate the statistical moments, where a moment-based quadrature rule can be used for integration nodes and weights. However, the moment-based quadrature rule has to solve a system of linear equations that can be numerically unstable. Considering this point, an improved moment-based quadrature rule is proposed and is applied to reliability-based design optimization. Finally, the moment-based RBDO is applied to numerical examples with a variety of random variables and target reliability indexes. From the numerical results, the performance of the improved moment-based quadrature rule can be confirmed and several guidelines are given for the moment-based RBDO.     

基于改进LHS模式的可靠性设计优化

在设计优化中,确定性优化由于没有考虑输入量的不确定性,其优化结果可能不可靠(不安全),因此基于可靠性的设计优化(reliability-based design optimization,RBDO)得到关注.然而可靠性设计优化计算量大,尤其对于高维问题.基于此,提出一种新方法--改进拉丁超立方体取样(Latin hyp...     

Reliable Space Pursuing for Reliability-based Design Optimization with Black-box Performance Functions

Reliability-based design optimization (RBDO) is intrinsically a double-loop procedure since it involves an overall optimization and an iterative reliability assessment at each search point. Due to the double-loop procedure, the computational expense of RBDO is normally very high. Current RBDO research focuses on problems with explicitly expressed performance functions and readily available gradients. This paper addresses a more challenging type of RBDO problem in which the performance functions are computation intensive. These computation intensive functions are often considered as a "black-box" and their gradients are not available or not reliable. On the basis of the reliable design space (RDS) concept proposed earlier by the authors, this paper proposes a Reliable Space Pursuing (RSP) approach, in which RDS is first identified and then gradually refined while optimization is performed. It fundamentally avoids the nested optimization and probabilistic assessment loop. Three well known RBDO problems from the literature are used for testing and demonstrating the effectiveness of the proposed RSP method.     

随机与区间不确定下基于近似灵敏度的序列

为解决复杂系统多学科可靠性设计优化过程中由于存在多源不确定性和多层嵌套而导致的计算效率低的问题,将近似灵敏度技术与两级集成系统综合策略(Bi-level integrated system synthesis,BLISS)和功能测度法集成,提出一种能同时处理随机和区间不确定性的序列化多学科可靠性设计优化方法。基于概率论和凸模型对混合不确定性进行量化,提出一种随机和区间不确定性下的混合可靠性评价指标,并基于功能测度法建立多学科可靠性设计优化模型。采用近似灵敏度信息替代实际灵敏度值,将近似灵敏度技术同时嵌入多级多学科设计优化策略和多学科可靠性分析方法中,避免每轮循环都进行全局灵敏度信息的分析与迭代,提高了计算效率。基于序列化思想同时将四层嵌套的多学科可靠性设计优化循环和三层嵌套的多学科可靠性分析过程进行解耦,形成一个单循环顺序执行的多学科可靠性设计优化过程,避免了每轮循环对整个可靠性分析模型进行迭代分析的过程,减少灵敏度分析和多学科分析次数。以汽车侧撞工程设计为例,验证了该法具有同时处理随机和区间不确定性的能力,并且计算效率较传统方法分别提高了10.98%和23.63%,表明该法具有一定工程实用价值。     

Design for a Crane Metallic Structure Based on Imperialist Competitive Algorithm and Inverse Reliability Strategy

Uncertainties in parameters such as materials, loading, and geometry are inevitable in designing metallic structures for cranes. When considering these uncertainty factors, reliability-based design optimization (RBDO) offers a more reasonable design approach. However, existing RBDO methods for crane metallic structures are prone to low convergence speed and high computational cost. A unilevel RBDO method, combining a discrete imperialist competitive algorithm with an inverse reliability strategy based on the performance measure approach, is developed. Application of the imperialist competitive algorithm at the optimization level significantly improves the convergence speed of this RBDO method. At the reliability analysis level, the inverse reliability strategy is used to determine the feasibility of each probabilistic constraint at each design point by calculating its α-percentile performance, thereby avoiding convergence failure, calculation error, and disproportionate computational effort encountered using conventional moment and simulation methods. Application of the RBDO method to an actual crane structure shows that the developed RBDO realizes a design with the best tradeoff between economy and safety together with about one-third of the convergence speed and the computational cost of the existing method. This paper provides a scientific and effective design approach for the design of metallic structures of cranes.     

Reliability-based design optimization of time-dependent systems with stochastic degradation

A major hurdle in the application of reliability-based design optimization (RBDO) to time-dependent systems is the continual interplay between calculating time-variant reliability (to ensure reliability policies are met) and moving the design point to optimize some objective function, such as cost, weight, size and so forth. In most cases the reliability can be obtained readily using so-called fast integration methods. However, this option is not available when certain stochastic processes are invoked to model gradual damage or deterioration. In this case, sampling methods must be used. This paper provides a novel way to obviate this inefficiency. First, a meta-model is built to relate time-variant system reliability to the entire design space (and noise space if required). A design of experiments paradigm and Monte Carlo simulation using the mechanistic model determines the corresponding system reliability accurately. A moving least-squares meta-model relates the data. Then, the optimization process to find the best design point, accesses the meta-model to quickly evaluate objectives and reliability constraints. Case-studies include a parallel Daniel's system and a series servo control system. The meta-model approach is simple, accurate and very fast, suggesting an attractive means for RBDO of time-dependent systems.

     

Reliability-based design optimization of a pick-up device of a manganese nodule pilot mining robot using the Coandă effect

Design of a pick-up device using the Coandă effect in a deep-sea mining robot is vital to develop a reliable and sustainable deep-sea mining system. One of the crucial performance metrics of this device is the collection efficiency since it affects the mining efficiency of the entire system. However, the collection efficiency is significantly affected by the uncertainties of shape, size and mass of manganese nodules on the seabed. In this study, reliability-based design optimization (RBDO) was performed to improve the reliability of the collection efficiency of the pick-up device under these environmental uncertainties. First, a computational model based on the Coandă effect that predicts the collection efficiency of the pick-up device was developed. Next, RBDO based on the Akaike information criterion method was employed to design the pick-up device by using this model. The results demonstrated that the proposed design methodology significantly improved the design of the pick-up device for the pilot mining robot.

     

Design optimization of a wind turbine gear transmission based on fatigue reliability sensitivity

Fatigue failure of gear transmission is one of the key factors that restrict the performance and service life of wind turbines. One of the major concerns in gear transmission under random loading conditions is the disregard of dynamic fatigue reliability in conventional design methods. Various issues, such as overweight structure or insufficient fatigue reliability, require continuous improvements in the reliability-based design optimization (RBDO) methodology. In this work, a novel gear transmission optimization model based on dynamic fatigue reliability sensitivity is developed to predict the optimal structural parameters of a wind turbine gear transmission. In the model, the dynamic fatigue reliability of the gear transmission is evaluated based on stress–strength interference theory. Design variables are determined based on the reliability sensitivity and correlation coefficient of the initial design parameters. The optimal structural parameters with the minimum volume are identified using the genetic algorithm in consideration of the dynamic fatigue reliability constraints. Comparison of the initial and optimized structures shows that the volume decreases by 3.58% while ensuring fatigue reliability. This work provides new insights into the RBDO of transmission systems from the perspective of reliability sensitivity.     

NEW APPROACH FOR RELIABILITY- BASED DESIGN OPTIMIZATION： MINIMUM ERROR POINT

Conventional reliability-based design optimization (RBDO) requires to use the most probable point (MPP) method for a probabilistic analysis of the reliability constraints. A new approach is presented, called as the minimum error point (MEP) method or the MEP based method, for reliability-based design optimization, whose idea is to minimize the error produced by approximating performance functions. The MEP based method uses the first order Taylor's expansion at MEP instead of MPP. Examples demonstrate that the MEP based design optimization can ensure product reliability at the required level, which is very imperative for many important engineering systems. The MEP based reliability design optimization method is feasible and is considered as an alternative for solving reliability design optimization problems. The MEP based method is more robust than the commonly used MPP based method for some irregular performance functions.     

Reliability-based robust optimization design of automobile components with non-normal distribution parameters

In the reliability designing procedure of the vehicle components, when the distribution styles of the random variables are unknown or non-normal distribution, the result evaluated contains great error or even is wrong if the reliability value R is larger than 1 by using the existent method, in which case the formula is necessary to be revised. This is obviously inconvenient for programming. Combining reliability-based optimization theory, robust designing method and reliability based sensitivity analysis, a new method for reliability robust designing is proposed. Therefore the influence level of the designing parameters’ changing to the reliability of vehicle components can be obtained. The reliability sensitivity with respect to design parameters is viewed as a sub-objective function in the multi-objective optimization problem satisfying reliability constraints. Given the first four moments of basic random variables, a fourth-moment technique and the proposed optimization procedure can obtain reliability-based robust design of automobile components with non-normal distribution parameters accurately and quickly. By using the proposed method, the distribution style of the random parameters is relaxed. Therefore it is much closer to the actual reliability problems. The numerical examples indicate the following: (1) The reliability value obtained by the robust method proposed increases (>0.04%) comparing to the value obtained by the ordinary optimization algorithm; (2) The absolute value of reliability-based sensitivity decreases (>0.01%), and the robustness of the products’ quality is improved accordingly. Utilizing the reliability-based optimization and robust design method in the reliability designing procedure reduces the manufacture cost and provides the theoretical basis for the reliability and robust design of the vehicle components.     

Mean-value first-order saddlepoint approximation based collaborative optimization for multidisciplinary problems under aleatory uncertainty

Reliability-based multidisciplinary design optimization (RBMDO) has received increasing attention in engineering design for achieving high reliability and safety in complex and coupling systems (e.g., multidisciplinary systems). Mean-value first-order saddlepoint approximation (MVFOSA) is introduced in this paper and is combined with the collaborative optimization (CO) method for reliability analysis under aleatory uncertainty in RBMDO. Similar to the mean-value first-order second moment (MVFOSM) method, MVFOSA approximated the performance function with the first-order Taylor expansion at the mean values of random variables. MVFOSA uses saddlepoint approximation rather than the first two moments of the random variables to estimate the probability density and cumulative distribution functions. MVFOSA-based CO (MVFOSA-CO) is also formulated and proposed. Two examples are provided to show the accuracy and efficiency of the MVFOSA-CO method.     

Reliability-based design optimization for box-booms considering strength degradation and random total working time

Box-booms are widely used in some cranes. The strength of box-booms would continuously deteriorate with the use of time. In this paper, gamma process is employed to describe strength degradation when dealing with the overall stability of the box-boom. Furthermore, the total working time is not deterministic but random because of the random arrivals of tasks and the random working time of every task. By accounting for strength degradation and the random total working time, a time-dependent reliability-based design optimization (RBDO) model for the box-boom is proposed. An engineering case is used to illustrate the proposed model.