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.     

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.     

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.     

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.     

多源不确定性条件下气动弹性系统颤振可靠性分析方法

传统的气动弹性系统颤振分析模型大多是在确定性参数条件下建立的,当系统中存在不确定因素时,按确定性方法设计的气动弹性系统存在颤振失效风险.以概率和非概率区间模型为基础,建立了单源不确定性条件下颤振可靠性分析模型;在此基础上,针对含随机和区间多源不确定参数的气动弹性系统颤振可靠性分析问题,提出一种基于分步求解策略的新型混合...     

Non-convex multiobjective optimization under uncertainty: a descent algorithm. Application to sandwich plate design and reliability

In this paper a novel algorithm for solving multiobjective design optimization problems with non-smooth objective functions and uncertain parameters is presented. The algorithm is based on the existence of a common descent vector for each sample of the random objective functions and on an extension of the stochastic gradient algorithm. The proposed algorithm is applied to the optimal design of sandwich material. Comparisons with the genetic algorithm NSGA-II and the DMS solver are given and show that it is numerically more efficient due to the fact that it does not necessitate the objective function expectation evaluation. It can moreover be entirely parallelizable. Another simple illustration highlights its potential for solving general reliability problems, replacing each probability constraint by a new objective written in terms of an expectation. Moreover, for this last application, the proposed algorithm does not necessitate the computation of the (small) probability of failure.     

Integrated multi-objective robust optimization and sensitivity analysis with irreducible and reducible interval uncertainty

Uncertainty considered in robust optimization is usually treated as irreducible since it is not reduced during an optimization procedure. In contrast, uncertainty considered in sensitivity analysis is treated as partially or fully reducible in order to quantify the effect of input uncertainty on the outputs of the system. Considering this, and the usual existence of both reducible and irreducible uncertainty, an approach that can perform robust optimization and sensitivity analysis simultaneously is of much interest. This article presents such an integrated optimization model that can be used for both robust optimization and sensitivity analysis for problems that have irreducible and reducible interval uncertainty, multiple objective functions and mixed continuous-discrete design variables. The proposed model is demonstrated by two engineering examples with differing complexity to demonstrate its applicability.     

可靠性优化的一种新的启发式算法

建立了可靠性冗余优化模型,分析了各种优化方法的优缺点。分析了几种常见的启发式算法,根据拉格朗日乘子法和K-T方程,提出了一种新的启发式算法,结果表明该方法比较有效。     

An entropy-based three-stage approach for multi-objective system reliability optimization considering uncertainty

This article presents a three-stage approach for solving multi-objective system reliability optimization problems considering uncertainty. The reliability of each component is considered in the formulation as a component reliability estimate in the form of an interval value and discrete values. Component reliability may vary owing to variations in the usage scenarios. Uncertainty is described by defining a set of usage scenarios. To address this problem, an entropy-based approach to the redundancy allocation problem is proposed in this study to identify the deterministic reliability of each component. In the second stage, a multi-objective evolutionary algorithm (MOEA) is applied to produce a Pareto-optimal solution set. A hybrid algorithm based on k-means and silhouettes is performed to select representative solutions in the third stage. Finally, a numerical example is presented to illustrate the performance of the proposed approach.     

Time-variant reliability global sensitivity analysis with single-loop Kriging model combined with importance sampling

This paper develops a novel failure probability-based global sensitivity index by introducing the Bayes formula into the moment-independent global sensitivity index to approximate the effect of input random variables or stochastic processes on the time-variant reliability. The proposed global sensitivity index can estimate the effect of uncertain inputs on the time-variant reliability by comparing the difference between the unconditional probability density function of input variables and the conditional probability density function in failure state of input variables. Furthermore, a single-loop active learning Kriging method combined with metamodel-based importance sampling is employed to improve the computational efficiency. The accuracy of the results obtained by Kriging model is verified by the reference results provided by the Monte Carlo simulation. Four examples are investigated to demonstrate the significance of the proposed failure probability-based global sensitivity index and the effectiveness of the computational method.     

An effective evidence theory-based reliability analysis algorithm for structures with epistemic uncertainty

The purpose of this article is to develop an effective method to evaluate the reliability of structures with epistemic uncertainty so as to improve the applicability of evidence theory in practical engineering problems. The main contribution of this article is to establish an approximate semianalytic algorithm, which replaces the process of solving the extreme value of performance function and greatly improve the efficiency of solving the belief measure and the plausibility measure. First, the performance function is decomposed as a combination of a series of univariate functions. Second, each univariate function is approximated as a unary quadratic function by the second-order Taylor expansion. Finally, based on the property of the unary quadratic function, the maximum and minimum values of each univariate function are solved, and then the maximum and minimum values of performance function are obtained according to the monotonic relationship between each univariate function and their combination. As long as the first- and second-order partial derivatives of the performance function with respect to each input variable are obtained, the belief measure and plausibility measure of the structure can be estimated effectively without any additional computational cost. Two numerical examples and one engineering application are investigated to demonstrate the accuracy and efficiency of the proposed method.     

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.     

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.     

A robust optimization approach for the capacitated vehicle routing problem with demand uncertainty

In this paper we introduce a robust optimization approach to solve the Vehicle Routing Problem (VRP) with demand uncertainty. This approach yields routes that minimize transportation costs while satisfying all demands in a given bounded uncertainty set. We show that for the Miller-Tucker-Zemlin formulation of the VRP and specific uncertainty sets, solving for the robust solution is no more difficult than solving a single deterministic VRP. Our computational results on benchmark instances and on families of clustered instances show that the robust solution can protect from unmet demand while incurring a small additional cost over deterministic optimal routes. This is most pronounced for clustered instances under moderate uncertainty, where remaining vehicle capacity is used to protect against variations within each cluster at a small additional cost. We compare the robust optimization model with classic stochastic VRP models for this problem to illustrate the differences and similarities between them. We also observe that the robust solution amounts to a clever management of the remaining vehicle capacity compared to uniformly and non-uniformly distributing this slack over the vehicles.     

A novel Bayesian approach to reliability modeling: The benefits of uncertainty evaluation in the model selection procedure

This paper proposes a different likelihood formulation within the Bayesian paradigm for parameter estimation of reliability models. Moreover, the assessment of the uncertainties associated with parameters, the goodness of fit, and the model prediction of reliability are included in a systematic framework for better aiding the model selection procedure. Two case studies are appraised to highlight the contributions of the proposed method and demonstrate the differences between the proposed Bayesian formulation and an existing Bayesian formulation.     

Research on multiple-state industrial robot system with epistemic uncertainty reliability allocation method

Reliability allocation of industrial robot (IR) system is one of the important means to improve its whole life cycle, reduce maintenance cost, and characterize weak subsystems. The IR system is not only very complex but also has strong customization; meanwhile, its sample data are small, resulting in unclear degeneration and failure. Based on the above two epistemic uncertainties, a new methodology called multiple-state IR system reliability allocation method with epistemic uncertainty (MIRS-RAM-EU) is proposed. First, the Dempster-Shafer (D-S) evidence theory is used to quantify the epistemic uncertainty. Then, the Kolmogorov differential equations of MIR's subsystems are calculated. The reliability index of MIRS is allocated based on Birnbaum importance degree theory, and the reliability allocation coefficient of each IR subsystem is clearly expressed by this method. Finally, compared with traditional importance allocation method, the MIRS-RAM-EU is more efficient and accurate. This method is usefully directive for reliability evaluation of IR.     

An inverse reliability analysis method for reliability-based design optimization with random and dependent interval variables constrained within ellipsoids

In practical design problems, interval variables exist. Many existing methods can handle only independent interval variables. Some interval variables, however, are dependent. In this work, dependent interval variables constrained within a multi-ellipsoid convex set are considered and incorporated into reliability-based design optimization (RBDO). An efficient RBDO method is proposed by employing the sequential single-loop procedure, which separates the coupled reliability analysis procedure from the deterministic optimization procedure. In the reliability analysis procedure, a single-loop optimization for the inverse reliability analysis is performed, and an efficient inverse reliability analysis method for searching for the worst-case most probable point (WMPP) is developed. The search method contains two stages. The first stage deals the situation where the WMPP is on the boundary of the feasible region, while the second stage accommodates the situation where the WMPP is inside the feasible region by interpolation. Three examples are used for a demonstration.     

New reliability modeling methods for structural systems with hybrid uncertainty

The present study investigates the hybrid reliability modeling of structures in which the inputs contain both random variables and interval variables. Hybrid uncertainty is divided into three categories, including random variables mixed with random variables, interval variables mixed with interval variable, and random variables mixed with interval variables. In order to perform the reliability analysis of structural systems, first, the Bayes method is proposed in the present study to obtain distribution parameters of random variables. Moreover, the self-sample method is introduced to obtain the interval boundaries based on the least available measuring data. Then, the reliability models are established for three situations and the reliability indices are defined and derived accordingly. The abovementioned three types of reliability indices outline the general situation of structural systems. Finally, the specific calculation process is described in details through different examples. Furthermore, the accuracy and efficiency of the proposed method is discussed by comparing the results obtained from the Monte Carlo simulation and those of other methods. The obtained results indicate that the performance of the proposed model in solving reliability modeling problems is better.     

A parallel bi-level multidisciplinary design optimization architecture with convergence proof for general problem

Quite a number of distributed Multidisciplinary Design Optimization (MDO) architectures have been proposed for the optimal design of large-scale multidisciplinary systems. However, just a few of them have available numerical convergence proof. In this article, a parallel bi-level MDO architecture is presented to solve the general MDO problem with shared constraints and a shared objective. The presented architecture decomposes the original MDO problem into one implicit nonlinear equation and multiple concurrent sub-optimization problems, then solves them through a bi-level process. In particular, this architecture allows each sub-optimization problem to be solved in parallel and its solution is proven to converge to the Karush–Kuhn–Tucker (KKT) point of the original MDO problem. Finally, two MDO problems are introduced to perform a comprehensive evaluation and verification of the presented architecture and the results demonstrate that it has a good performance both in convergence and efficiency.     

Demonstration of probabilistic ordinal optimization concepts for continuous-variable optimization under uncertainty

A very general and robust approach to solving optimization problems involving probabilistic uncertainty is through the use of Probabilistic Ordinal Optimization. At each step in the optimization problem, improvement is based only on a relative ranking of the probabilistic merits of local design alternatives, rather than on precise quantification of the alternatives. Thus, we simply ask the question: “Is that alternative better or worse than this one?” to some level of statistical confidence we require, not: “HOW MUCH better or worse is that alternative to this one?”. In this paper we illustrate an elementary application of probabilistic ordinal concepts in a 2-D optimization problem. Two uncertain variables contribute to uncertainty in the response function. We use a simple Coordinate Pattern Search non-gradient-based optimizer to step toward the statistical optimum in the design space. We also discuss more sophisticated implementations, and some of the advantages and disadvantages versus other approaches to optimization under uncertainty.