Reliability-based design optimization for problems with interval distribution parameters

Traditional reliability-based design optimization (RBDO) generally describes uncertain variables using random distributions, while some crucial distribution parameters in practical engineering problems can only be given intervals rather than precise values due to the limited information. Then, an important probability-interval hybrid reliability problem emerged. For uncertain problems in which interval variables are included in probability distribution functions of the random parameters, this paper establishes a hybrid reliability optimization design model and the corresponding efficient decoupling algorithm, which aims to provide an effective computational tool for reliability design of many complex structures. The reliability of an inner constraint is an interval since the interval distribution parameters are involved; this paper thus establishes the probability constraint using the lower bound of the reliability degree which ensures a safety design of the structure. An approximate reliability analysis method is given to avoid the time-consuming multivariable optimization of the inner hybrid reliability analysis. By using an incremental shifting vector (ISV) technique, the nested optimization problem involved in RBDO is converted into an efficient sequential iterative process of the deterministic design optimization and the hybrid reliability analysis. Three numerical examples are presented to verify the proposed method, which include one simple problem with explicit expression and two complex practical applications.     

An efficient reliability-based optimization scheme for uncertain linear systems subject to general Gaussian excitation

A very efficient methodology to carry out reliability-based optimization of linear systems with random structural parameters and random excitation is presented. The reliability-based optimization problem is formulated as the minimization of an objective function for a specified reliability. The probability that design conditions are satisfied within a given time interval is used as a measure of the system reliability. Approximation concepts are used to construct high quality approximations of dynamic responses in terms of the design variables and uncertain structural parameters during the design process. The approximations are combined with an efficient simulation technique to generate explicit approximations of the reliability measures with respect to the design variables. In particular, an efficient importance sampling technique is used to estimate the failure probabilities. The number of dynamic analyses as well as reliability estimations required during the optimization process are reduced dramatically. Several example problems are presented to illustrate the effectiveness and feasibility of the suggested approach.     

Genetic algorithm based multi-objective reliability optimization in interval environment

In most of the real world design or decision making problems involving reliability optimization, there are simultaneous optimization of multiple objectives such as the maximization of system reliability and the minimization of system cost, weight and volume. In this paper, our goal is to solve the constrained multi-objective reliability optimization problem of a system with interval valued reliability of each component by maximizing the system reliability and minimizing the system cost under several constraints. For this purpose, four different multi-objective optimization problems have been formulated with the help of interval mathematics and our newly proposed order relations of interval valued numbers. Then these optimization problems have been solved by advanced genetic algorithm and the concept of Pareto optimality. Finally, to illustrate and also to compare the results, a numerical example has been solved.     

不确定可靠性优化问题的多目标粒子群优化算法

针对元件可靠性为区间值的系统可靠性优化问题, 提出一种区间多目标粒子群优化方法. 首先, 建立问题的区间多目标优化模型; 然后, 利用粒子群算法优化该模型, 定义一种不精确Pareto 支配关系, 并给出编码、约束处理、外部存储器更新、领导粒子选择等关键问题的解决方法; 最后, 将该方法应用于可靠性优化问题实例, 验证了方法的有效性.

     

Structural optimization of linear dynamical systems under stochastic excitation: a moving reliability database approach

A method to carry out structural synthesis of deterministic linear dynamical systems under stochastic excitation is introduced. The structural optimization problem is written as a nonlinear mathematical programming problem with reliability constraints. Probability that design conditions are satisfied within a given time period is used as a measure of system reliability. The solution of the original optimization problem is replaced by the solution of a sequence of approximate sub-optimization problems. An explicit approximation of the system reliability in terms of the design variables is constructed in each sub-optimization problem. The approximations are locally adjusted to a reliability database, which is obtained by an efficient importance sampling technique. Each approximate optimization problem is solved in an efficient manner due to the availability of the system reliability in explicit form. Numerical examples are presented to illustrate the performance and efficiency of the proposed methodology.     

Reliability evaluation and optimal design in heterogeneous multi-state series-parallel systems

This paper addresses the heterogeneous redundancy allocation problem in multi-state series-parallel reliability structures with the objective to minimize the total cost of system design satisfying the given reliability constraint and the consumer load demand. The demand distribution is presented as a piecewise cumulative load curve and each subsystem is allowed to consist of parallel redundant components of not more than three types. The system uses binary capacitated components chosen from a list of available products to provide redundancy so as to increase system performance and reliability. The components are characterized by their feeding capacity, reliability and cost. A system that consists of elements with different reliability and productivity parameters has the capacity strongly dependent upon the selection of constituent components. A binomial probability based method to compute exact system reliability index is suggested. To analyze the problem and suggest an optimal/near-optimal system structure, an ant colony optimization algorithm has been presented. The solution approach consists of a series of simple steps as used in early ant colony optimization algorithms dealing with other optimization problems and offers straightforward analysis. Four multi-state system design problems have been solved for illustration. Two problems are taken from the literature and solved to compare the algorithm with the other existing methods. The other two problems are based upon randomly generated data. The results show that the method can be appealing to many researchers with regard to the time efficiency and yet without compromising over the solution quality.     

A method for interval 0–1 nonlinear programming problem using a genetic algorithm

In this paper, we formulate an optimal design problem of system reliability as a nonlinear integer programming problem with interval coefficients, transform it into a bicriteria 0–1 nonlinear programming problem without interval coefficients, and solve it directly using GA with holding nonlinear objective functions. Also, we demonstrate the efficiency of this method with a numerical example.     

A genetic algorithm for interval nonlinear integer programming problem

In this paper, we formulate an optimal design of system reliability problem as a nonlinear integer programming problem with interval coefficients, transform it into a single objective nonlinear integer programming problem without interval coefficients, and solve it directly with keeping nonlinearity of the objective function by using Genetic Algorithms (GA). Also, we demonstrate the efficiency of this method with incomplete Fault Detecting and Switching (FDS) for allocating redundant units.     

Comparison of global and local response surface techniques in reliability-based optimization of composite structures

This paper discusses the development and application of two alternative strategies, in the form of global and sequential local response surface (RS) techniques, for the solution of reliability-based optimization (RBO) problems. The problem of a thin-walled composite circular cylinder under axial buckling instability is used as a demonstrative example. In this case, the global technique uses a single second-order RS model to estimate the axial buckling load over the entire feasible design space (FDS), whereas the local technique uses multiple first-order RS models, with each applied to a small subregion of the FDS. Alternative methods for the calculation of unknown coefficients in each RS model are explored prior to the solution of the optimization problem. The example RBO problem is formulated as a function of 23 uncorrelated random variables that include material properties, the thickness and orientation angle of each ply, the diameter and length of the cylinder, as well as the applied load. The mean values of the 8 ply thicknesses are treated as independent design variables. While the coefficients of variation of all random variables are held fixed, the standard deviations of the ply thicknesses can vary during the optimization process as a result of changes in the design variables. The structural reliability analysis is based on the first-order reliability method with the reliability index treated as the design constraint. In addition to the probabilistic sensitivity analysis of the reliability index, the results of the RBO problem are presented for different combinations of cylinder length and diameter and laminate ply patterns. The two strategies are found to produce similar results in terms of accuracy, with the sequential local RS technique having a considerably better computational efficiency.     

Improved cuckoo search for reliability optimization problems

An efficient approach to solve engineering optimization problems is the cuckoo search algorithm. It is a recently developed meta-heuristic optimization algorithm. Normally, the parameters of the cuckoo search are kept constant. This may result in decreasing the efficiency of the algorithm. To cope with this issue, the cuckoo search parameters should be tuned properly. In this paper, an improved cuckoo search algorithm, enhancing the accuracy and convergence rate of the cuckoo search algorithm, is presented. Then, the performance of the proposed algorithm is tested on some complex engineering optimization problems. They are four well-known reliability optimization problems, a large-scale reliability optimization problem as well as a complex system, which is a 15-unit system reliability optimization problem. Finally, the results are compared with those given by several well-known methods. Simulation results demonstrate the effectiveness of the proposed algorithm.     

A gradient-based uncertainty optimization framework utilizing dimensional adaptive polynomial chaos expansion

To improve the efficiency of solving uncertainty design optimization problems, a gradient-based optimization framework is herein proposed, which combines the dimension adaptive polynomial chaos expansion (PCE) and sensitivity analysis. The dimensional adaptive PCE is used to quantify the quantities of interest (e.g., reliability, robustness metrics) and the sensitivity. The dimensional adaptive property is inherited from the dimension adaptive sparse grid, which is used to evaluate the PCE coefficients. Robustness metrics, referred to as statistical moments, and their gradients with respect to design variables are easily derived from the PCE, whereas the evaluation of the reliability and its gradient require integrations. To quantify the reliability, the framework uses the Heaviside step function to eliminate the failure domain and calculates the integration by Monte Carlo simulation with the function replaced by PCE. The PCE is further combined with Taylor’s expansion and the finite difference to compute the reliability sensitivity. Since the design vector may affect the sample set determined by dimension adaptive sparse grid, the update of the sample set is controlled by the norm variations of the design vector. The optimization framework is formed by combining reliability, robustness quantification and sensitivity analysis, and the optimization module. The accuracy and efficiency of the reliability quantification, as well as the reliability sensitivity, are verified through a mathematical example, a system of springs, and a cantilever beam. The effectiveness of the framework in solving optimization problems is validated by multiple limit states example, a truss optimization example, an airfoil optimization example, and an ONERA M6 wing optimization problem. The results demonstrate that the framework can obtain accurate solutions at the expense of a manageable computational cost.

     

A robust study of reliability-based optimization methods under eigen-frequency

Nowadays, the search in reliability-based design optimization is becoming an important engineering design activity. Traditionally for these problems, the objective function is to minimize a cost function while satisfying the reliability constraints. The reliability constraints are usually formulated as constraints on the probability of failure. This paper focuses on the study of a particular problem with the failure mode on vibration of structure. The difficulty in evaluating reliability constraints comes from the fact that modern reliability analysis methods are themselves formulated as an optimization problem. Solving such nested optimization problems is extremely expensive for large-scale multidisciplinary systems which are likewise computationally intensive. With this in mind research, we propose in this paper a new method to treat reliability-based optimization methods under frequencies constraint. The goal of this development has resolved just one problem of optimization and reduced the cost of computation. Aircraft wing design typically involves multiple disciplines such as aerodynamics and structure; this numerical example demonstrated the different advantages of the proposed method.     

An extended probabilistic method for reliability analysis under mixed aleatory and epistemic uncertainties with flexible intervals

The reliability analysis approach based on combined probability and evidence theory is studied in this paper to address the reliability analysis problem involving both aleatory uncertainties and epistemic uncertainties with flexible intervals (the interval bounds are either fixed or variable as functions of other independent variables). In the standard mathematical formulation of reliability analysis under mixed uncertainties with combined probability and evidence theory, the key is to calculate the failure probability of the upper and lower limits of the system response function as the epistemic uncertainties vary in each focal element. Based on measure theory, in this paper it is proved that the aforementioned upper and lower limits of the system response function are measurable under certain circumstances (the system response function is continuous and the flexible interval bounds satisfy certain conditions), which accordingly can be treated as random variables. Thus the reliability analysis of the system response under mixed uncertainties can be directly treated as probability calculation problems and solved by existing well-developed and efficient probabilistic methods. In this paper the popular probabilistic reliability analysis method FORM (First Order Reliability Method) is taken as an example to illustrate how to extend it to solve the reliability analysis problem in the mixed uncertainty situation. The efficacy of the proposed method is demonstrated with two numerical examples and one practical satellite conceptual design problem.     

Reliability based optimization in aeroelastic stability problems using polynomial chaos based metamodels

In this work, reliability based design optimization (RBDO) of two aeroelastic stability problems is addressed: (i) divergence, which arises in static aeroelasticity, and (ii) flutter, which arises in dynamic aeroelasticity. A set of design variables is considered as random variables, and the mean mass is minimized for a given set of constraints — including the probability of failure by divergence or flutter. The optimization process requires repeated evaluation of reliability, which is a major contributor to the total computational cost. To reduce this cost, a polynomial chaos expansion (PCE)-based metamodel is created over a grid in the parameter space. These precomputed PCEs are then interpolated for reliability calculation at intermediate points in the parameter space, as demanded by the optimization algorithm. Two new modifications are made to this method in this work. First, the Gauss quadrature rule is used — instead of statistical simulation — to estimate the chaos coefficients for higher computational speed. Second, to increase this computational gain further, a non-uniform grid is chosen instead of a uniform one, based on relative importance of the design parameters. This relative importance is found from a global sensitivity analysis. This new modified method is applied on a rectangular unswept cantilever wing model. For both optimization problems, it is observed that the proposed method yields accurate results with a considerable computational cost reduction, when compared to simulation based methods. The effect of grid spacing is also explored to achieve the best computational efficiency.     

Application of non-probabilistic sensitivity analysis in the optimization of aeronautical hydraulic pipelines

This paper investigates the reliability design optimization of an aeronautical hydraulic pipeline system, in which the constraint locations are treated as design parameters. To reduce the size of the optimization problem, two non-probabilistic global sensitivity indices are introduced and modified to screen out those constraint locations which have no or little effect on the optimization target. Considering the rest of constraint locations as design variables, the complexity of the optimization problem is dramatically reduced. The optimization of the pipeline systems demonstrates that the proposed method is superior to the traditional direct optimization method in both the optimization efficiency and results. This work indicates that introduction of sensitivity analysis can greatly improve the efficiency and performance of optimization, especially in those complex engineering problems involving a large number of design variables.     

Novel numerical methods for reliability analysis and optimization in engineering fuzzy heat conduction problem

Engineering fuzzy heat conduction problem with subjective uncertainties in input parameters constitutes a significant challenge. Based on fuzzy and interval theory, this paper presents novel numerical methods to efficiently identify the effect of fuzzy uncertainty on the system reliability analysis and optimization design. Firstly using the interval ranking strategy, the interval safety possibility in the transition state can be precisely quantified, and the eventual fuzzy safety possibility is calculated by integral operation. Then a fuzzy reliability-based optimization model is established with considerable computational cost caused by the two-layer nested loop. In order to improve the computational efficiency, a subinterval perturbation method based on the first-order Taylor series is presented to replace the inner loop. Comparing numerical results with traditional reliability model, two numerical examples are provided to evidence the superiority of proposed model and method for fuzzy reliability analysis and optimization in practical engineering.     

Optimal interval controller design for uncertain systems

The interval controller design is a hot issue for uncertain systems, whereas how to design an optimal interval controller under the premise of ensuring system stability is a difficult problem that needs further study. This paper mainly aims at the single input single output uncertain system to propose an optimal interval controller based on the Kharitonov theorem and an interval optimization algorithm, which can guarantee the stability and optimization of a closed-loop interval system. According to the Kharitonov theorem, the optimal interval controller design can be transformed into an optimal controller synthesis issue of multiple vertex objects. An interval particle swarm optimization (IPSO) algorithm is then used to optimize the quadratic performance index with interval variables for each vertex object to obtain the solution domains of the controller parameters, and the vertex method is utilized to prevent interval width expansion or divergence in the iteration. Finally, the intersections of the solution domains for all vertex objects are obtained as the optimal interval solution of interval controller parameters. In addition, the stability verification approach of the closed-loop system and the empirical rule to select the interval particle width are given. Simulation results for typical examples demonstrate that the designed interval controller not only performs optimally but also can robustly stabilize the interval system.     

Optimization of system reliability: a survey of problems and techniques†

The paper provides a survey of the work performed over the last two decades on the system reliability optimization. The relevant system models are first given and a set of problems, covering most cases, are formulated. Then, the optimization techniques, used for solving these problems, are briefly described, and a number of representative illustrative examples are collected. It is hoped that the paper helps in obtaining a global view of the state of art of the field of optimal reliability design.     

有色冶金过程不确定优化方法探讨

有色冶金过程受原料来源多样、工况条件波动、生产成分变化等因素的影响,存在大量的不确定性,严重影响了冶炼生产的稳定性与可靠性.鉴于此,综述不同类型不确定性优化问题的描述方法,具体包括概率不确定优化问题、模糊不确定优化问题和区间不确定优化问题.通过分析有色冶金生产过程的特点与需求,以3种典型的有色冶金过程不确定优化问题为例,探讨不同类型的有色冶金过程不确定优化方法.针对氧化铝生料浆配料过程的概率不确定优化问题,采用基于Hammersley sequence sampling(HSS)的方法实现不确定模型的确定性转换;针对湿法炼锌除铜过程的模糊不确定优化问题,采用基于模糊规则的方法进行确定性评估;针对锌电解分时供电过程的区间不确定优化问题,采用基于min-max的方法求解鲁棒解.工业运行数据均验证了上述方法的有效性.     

Reliability-Based Optimization Using Evolutionary Algorithms

Uncertainties in design variables and problem parameters are often inevitable and must be considered in an optimization task if reliable optimal solutions are sought. Besides a number of sampling techniques, there exist several mathematical approximations of a solution's reliability. These techniques are coupled in various ways with optimization in the classical reliability-based optimization field. This paper demonstrates how classical reliability-based concepts can be borrowed and modified and, with integrated single and multiobjective evolutionary algorithms, used to enhance their scope in handling uncertainties involved among decision variables and problem parameters. Three different optimization tasks are discussed in which classical reliability-based optimization procedures usually have difficulties, namely (1) reliability-based optimization problems having multiple local optima, (2) finding and revealing reliable solutions for different reliability indices simultaneously by means of a bi-criterion optimization approach, and (3) multiobjective optimization with uncertainty and specified system or component reliability values. Each of these optimization tasks is illustrated by solving a number of test problems and a well-studied automobile design problem. Results are also compared with a classical reliability-based methodology.