An efficient framework for the reliability-based design optimization of large-scale uncertain and stochastic linear systems

This paper is focused on the development of an efficient reliability-based design optimization algorithm for solving problems posed on uncertain linear dynamic systems characterized by large design variable vectors and driven by non-stationary stochastic excitation. The interest in such problems lies in the desire to define a new generation of tools that can efficiently solve practical problems, such as the design of high-rise buildings in seismic zones, characterized by numerous free parameters in a rigorously probabilistic setting. To this end a novel decoupling approach is developed based on defining and solving a limited sequence of deterministic optimization sub-problems. In particular, each sub-problem is formulated from information pertaining to a single simulation carried out exclusively in the current design point. This characteristic drastically limits the number of simulations necessary to find a solution to the original problem while making the proposed approach practically insensitive to the size of the design variable vector. To demonstrate the efficiency and strong convergence properties of the proposed approach, the structural system of a high-rise building defined by over three hundred free parameters is optimized under non-stationary stochastic earthquake excitation.     

Multi-mode reliability-based design of horizontal curves

Recently, reliability analysis has been advocated as an effective approach to account for uncertainty in the geometric design process and to evaluate the risk associated with a particular design. In this approach, a risk measure (e.g. probability of noncompliance) is calculated to represent the probability that a specific design would not meet standard requirements. The majority of previous applications of reliability analysis in geometric design focused on evaluating the probability of noncompliance for only one mode of noncompliance such as insufficient sight distance. However, in many design situations, more than one mode of noncompliance may be present (e.g. insufficient sight distance and vehicle skidding at horizontal curves). In these situations, utilizing a multi-mode reliability approach that considers more than one failure (noncompliance) mode is required. The main objective of this paper is to demonstrate the application of multi-mode (system) reliability analysis to the design of horizontal curves. The process is demonstrated by a case study of Sea-to-Sky Highway located between Vancouver and Whistler, in southern British Columbia, Canada. Two noncompliance modes were considered: insufficient sight distance and vehicle skidding. The results show the importance of accounting for several noncompliance modes in the reliability model. The system reliability concept could be used in future studies to calibrate the design of various design elements in order to achieve consistent safety levels based on all possible modes of noncompliance.     

An accurate and efficient reliability-based design optimization using the second order reliability method and improved stability transformation method

The first order reliability method has been extensively adopted for reliability-based design optimization (RBDO), but it shows inaccuracy in calculating the failure probability with highly nonlinear performance functions. Thus, the second order reliability method is required to evaluate the reliability accurately. However, its application for RBDO is quite challenge owing to the expensive computational cost incurred by the repeated reliability evaluation and Hessian calculation of probabilistic constraints. In this article, a new improved stability transformation method is proposed to search the most probable point efficiently, and the Hessian matrix is calculated by the symmetric rank-one update. The computational capability of the proposed method is illustrated and compared to the existing RBDO approaches through three mathematical and two engineering examples. The comparison results indicate that the proposed method is very efficient and accurate, providing an alternative tool for RBDO of engineering structures.     

An integrated reliability-based design optimization of offshore towers

After recognizing the uncertainty in the parameters such as material, loading, geometry and so on in contrast with the conventional optimization, the reliability-based design optimization (RBDO) concept has become more meaningful to perform an economical design implementation, which includes a reliability analysis and an optimization algorithm. RBDO procedures include structural analysis, reliability analysis and sensitivity analysis both for optimization and for reliability. The efficiency of the RBDO system depends on the mentioned numerical algorithms. In this work, an integrated algorithms system is proposed to implement the RBDO of the offshore towers, which are subjected to the extreme wave loading. The numerical strategies interacting with each other to fulfill the RBDO of towers are as follows: (a) a structural analysis program, SAPOS, (b) an optimization program, SQP and (c) a reliability analysis program based on FORM. A demonstration of an example tripod tower under the reliability constraints based on limit states of the critical stress, buckling and the natural frequency is presented.     

考虑可靠度可信区间的结构可靠性优化设计

 在贝叶斯统计理论和结构可靠性优化设计方法的基础上,研究了结构在小样本情况下考虑可靠度可信区间的结构可靠性优化设计问题.将结构可靠度作为随机变量,根据先验信息和样本信息,采用贝叶斯推断技术获得结构可靠度的概率分布,给出了可靠度的点估计及区间估计.建立了考虑可靠度可信区间的结构可靠性优化设计模型,提出了考虑可靠度可信区间的结构可靠性优化设计方法.所提出的方法为解决小样本情况下的结构可靠性优化设计问题提供了新的解决方案.数值算例验证了所提出的结构可靠性优化设计方法的有效性和正确性.     

A multiple-design-point approach for reliability-based design optimization

The application of design-point-based reliability-based design optimization (RBDO) methods is hindered by the challenge of multiple-design-point problems. In this article, to improve the commonality of design-point-based RBDO methods, a novel multiple-design-point (MDP) approach is developed. The MDP approach uses the trace of the design points from consequent reliability analysis iterations to identify whether there are multiple design points, then all of the design points are used to calculate shifting vectors for the sequential optimization and reliability assessment method, and the corresponding probabilistic constraints are moved to the feasible region along these multiple shifting vectors at the same time. With multiple shifted probabilistic constraints, the design feasibility associated with this probabilistic constraint will be satisfied. Two mathematical examples, a speed reducer design and a honeycomb crashworthiness design, are presented to validate the effectiveness of the MDP method. The results show that the MDP approach is effective for handling multiple-design-point problems.     

Dynamic reliability-based robust design optimization with time-variant probabilistic constraints

With the increasing complexity of engineering systems, ensuring high system reliability and system performance robustness throughout a product life cycle is of vital importance in practical engineering design. Dynamic reliability analysis, which is generally encountered due to time-variant system random inputs, becomes a primary challenge in reliability-based robust design optimization (RBRDO). This article presents a new approach to efficiently carry out dynamic reliability analysis for RBRDO. The key idea of the proposed approach is to convert time-variant probabilistic constraints to time-invariant ones by efficiently constructing a nested extreme response surface (NERS) and then carry out dynamic reliability analysis using NERS in an iterative RBRDO process. The NERS employs an efficient global optimization technique to identify the extreme time responses that correspond to the worst case scenario of system time-variant limit state functions. With these extreme time samples, a kriging-based time prediction model is built and used to estimate extreme responses for any given arbitrary design in the design space. An adaptive response prediction and model maturation mechanism is developed to guarantee the accuracy and efficiency of the proposed NERS approach. The NERS is integrated with RBRDO with time-variant probabilistic constraints to achieve optimum designs of engineered systems with desired reliability and performance robustness. Two case studies are used to demonstrate the efficacy of the proposed approach.     

多失效模式机械系统可靠性稳健设计方法研究

将机械系统可靠性设计理论和稳健设计方法相结合,讨论了多失效模式机械系统可靠性稳健设计问题,提出了多失效模式机械系统可靠性稳健设计的计算方法．把可靠性灵敏度融入可靠性优化设计模型之中,将机械系统可靠性稳健设计归结为满足可靠性要求的多目标优化问题．在基本随机参数的前二阶矩已知的情况下,可以迅速准确地得到机械系统可靠性稳健设计信息．     

A reliability-based fatigue design for mechanical components under material variability

The mechanical components subjected cyclic load unusually fail due to fatigue. The traditional deterministic design method still has the risk of failure while the safety factor method sometimes is overconservative and uneconomic. In this study, reliability-based design optimization is applied in structural design of components under low cycle fatigue. A constitutive model (Jiang and Sehitoglu model) was written into user-defined material subroutine of finite element software to make simulation more accurate. In addition, an adaptive least squares support vector machines (LS-SVM)-based response surface method is employed to improve the efficiency of design process. After constructing the implicit life model, a hybrid directional step method is employed to implement the performance measure approach. Finally, a simple case (thickness optimization for cantilever tube) is used to demonstrate the whole procedure of proposed design procedure.     

Refined second order semi‐analytical design sensitivities

Accurate and efficient calculation of second order design sensitivities in a finite element context is often difficult. The semi‐analytical (SA) method is efficient and easy to implement but has accuracy problems even for first order shape design sensitivities. To overcome accuracy problems a refined semi‐analytical (RSA) method has been developed for first order sensitivities. The present paper investigates the application of the RSA method to second order design sensitivities. It is found that second order RSA sensitivities are significantly more accurate than their SA counterparts. Copyright © 2002 John Wiley & Sons, Ltd.     

Adaptive improved response surface method for reliability-based design optimization

Reliability-based design optimization (RBDO) requires the evaluation of probabilistic constraints (or reliability), which can be very time consuming. Therefore, a practical solution for efficient reliability analysis is needed. The response surface method (RSM) and dimension reduction (DR) are two well-known approximation methods that construct the probabilistic limit state functions for reliability analysis. This article proposes a new RSM-based approximation approach, named the adaptive improved response surface method (AIRSM), which uses the moving least-squares method in conjunction with a new weight function. AIRSM is tested with two simplified designs of experiments: saturated design and central composite design. Its performance on reliability analysis is compared with DR in terms of efficiency and accuracy in multiple RBDO test problems.     

A global single-loop deterministic approach for reliability-based design optimization of truss structures with continuous and discrete design variables

A single-loop deterministic method (SLDM) has previously been proposed for solving reliability-based design optimization (RBDO) problems. In SLDM, probabilistic constraints are converted to approximate deterministic constraints. Consequently, RBDO problems can be transformed into approximate deterministic optimization problems, and hence the computational cost of solving such problems is reduced significantly. However, SLDM is limited to continuous design variables, and the obtained solutions are often trapped into local extrema. To overcome these two disadvantages, a global single-loop deterministic approach is developed in this article, and then it is applied to solve the RBDO problems of truss structures with both continuous and discrete design variables. The proposed approach is a combination of SLDM and improved differential evolution (IDE). The IDE algorithm is an improved version of the original differential evolution (DE) algorithm with two improvements: a roulette wheel selection with stochastic acceptance and an elitist selection technique. These improvements are applied to the mutation and selection phases of DE to enhance its convergence rate and accuracy. To demonstrate the reliability, efficiency and applicability of the proposed method, three numerical examples are executed, and the obtained results are compared with those available in the literature.     

Structural reliability analysis and uncertainties‐based collaborative design and optimization of turbine blades using surrogate model

In power and energy systems, both the aerodynamic performance and the structure reliability of turbine equipment are affected by utilized blades. In general, the design process of blade is high dimensional and nonlinear. Different coupled disciplines are also involved during this process. Moreover, unavoidable uncertainties are transported and accumulated between these coupled disciplines, which may cause turbine equipment to be unsafe. In this study, a saddlepoint approximation reliability analysis method is introduced and combined with collaborative optimization method to address the above challenge. During the above reliability analysis and design optimization process, surrogate models are utilized to alleviate the computational burden for uncertainties‐based multidisciplinary design and optimization problems. Smooth response surfaces of the performance of turbine blades are constructed instead of expensively time‐consuming simulations. A turbine blade design problem is solved here to validate the effectiveness and show the utilization of the given approach.     

Reliability-based optimal design using sample average approximations

An algorithm for reliability-based optimal design is developed using sampling techniques for estimating the failure probability. The algorithm applies a new method for sensitivity calculations of the failure probability. Initially, the estimates of the failure probability are coarse. As the algorithm progresses towards an optimal design, the number of sample points is increased in an adaptive way leading to better estimates of the failure probability. The algorithm is proven to converge to an optimal design. The applicability of the algorithm is shown in an example from the area of highway bridge design.     

Thermal–hydraulic passive system reliability-based design approach

In order to enhance the safety of new advanced reactors, optimization of the design of the implemented passive systems is required. Therefore, a reliability-based approach to the design of a thermal–hydraulic passive system is being considered, and a limit state function (LSF)-based approach elicited from mechanical reliability is developed. The concept of functional failure, i.e., the possibility that the load will exceed the capacity in a reliability physics framework, in terms of performance parameter is introduced here for the reliability evaluation of a natural circulation passive system, designed for decay heat removal of innovative light water reactors. Water flow rate circulating through the system is selected as passive system performance characteristic parameter and the related limit state or performance function is defined. The probability of failure of the system is assessed in terms of safety margin, corresponding to the LSF. Results help the designer to determine the allowable limits or set the safety margin for the system operation parameters, to meet the safety and reliability requirements.     

Reliable design space and complete single-loop reliability-based design optimization

Reliability-based design optimization (RBDO) has been intensively studied due to its significance and its conceptual and mathematical complexity. This paper proposes a new method for RBDO on the basis of the concept of reliable design space (RDS), within which any design satisfies the reliability requirements. Therefore, a RBDO problem becomes a simple, deterministic optimization problem constrained by RDS rather than its deterministic feasible space. The RDS is found in this work by using the partial derivatives at the current design point as an approximation of the derivatives at its corresponding most probable point (MPP) on the limit state function. This work completely resolves the double loop in RBDO and turns RBDO into a simple optimization problem. Well-known problems from the literature are selected to illustrate the steps of the approach and for result comparison. Discussions will also be given on the limitation of the proposed method, which is shown to be a common limitation overlooked by the research community on RBDO.     

Multi-objective system reliability-based optimization method for design of a fully parametric concept car body

This article investigates multi-objective optimization under reliability constraints with applications in vehicle structural design. To improve computational efficiency, an improved multi-objective system reliability-based design optimization (MOSRBDO) method is developed, and used to explore the lightweight and high-performance design of a concept car body under uncertainty. A parametric model knowledge base is established, followed by the construction of a fully parametric concept car body of a multi-purpose vehicle (FPCCB-MPV) based on the knowledge base. The structural shape, gauge and topology optimization are then designed on the basis of FPCCB-MPV. The numerical implementation of MOSRBDO employs the double-loop method with design optimization in the outer loop and system reliability analysis in the inner loop. Multi-objective particle swarm optimization is used as the outer loop optimization solver. An improved multi-modal radial-based importance sampling (MRBIS) method is utilized as the system reliability solver for multi-constraint analysis in the inner loop. The accuracy and efficiency of the MRBIS method are demonstrated on three widely used test problems. In conclusion, MOSRBDO has been successfully applied for the design of a full parametric concept car body. The results show that the improved MOSRBDO method is more effective and efficient than the traditional MOSRBDO while achieving the same accuracy, and that the optimized body-in-white structure signifies a noticeable improvement from the baseline model.     

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.     

Response surface approximations for structural optimization

Response surface methodology can be used to construct global and midrange approximations to functions in structural optimization. Since structural optimization requires expensive function evaluations, it is important to construct accurate function approximations so that rapid convergence may be achieved. In this paper techniques to find the region of interest containing the optimal design, and techniques for finding more accurate approximations are reviewed and investigated. Aspects considered are experimental design techniques, the selection of the ‘best’ regression equation, intermediate response functions and the location and size of the region of interest. Standard examples in structural optimization are used to show that the accuracy is largely dependent on the choice of the approximating function with its associated subregion size, while the selection of a larger number of points is not necessarily cost-effective. In a further attempt to improve efficiency, different regression models were investigated. The results indicate that the use of the two methods investigated does not significantly improve the results. Finding an accurate global approximation is challenging, and sufficient accuracy could only be achieved in the example problems by considering a smaller region of the design space. © 1998 John Wiley & Sons, Ltd.     

Reliability–sensitivity analysis using dimension reduction methods and saddlepoint approximations

Reliability–sensitivity, which is considered as an essential component in engineering design under uncertainty, is often of critical importance toward understanding the physical systems underlying failure and modifying the design to mitigate and manage risk. This paper presents a new computational tool for predicting reliability (failure probability) and reliability–sensitivity of mechanical or structural systems subject to random uncertainties in loads, material properties, and geometry. The dimension reduction method is applied to compute response moments and their sensitivities with respect to the distribution parameters (e.g., shape and scale parameters, mean, and standard deviation) of basic random variables. Saddlepoint approximations with truncated cumulant generating functions are employed to estimate failure probability, probability density functions, and cumulative distribution functions. The rigorous analytic derivation of the parameter sensitivities of the failure probability with respect to the distribution parameters of basic random variables is derived. Results of six numerical examples involving hypothetical mathematical functions and solid mechanics problems indicate that the proposed approach provides accurate, convergent, and computationally efficient estimates of the failure probability and reliability–sensitivity. Copyright © 2012 John Wiley & Sons, Ltd.