A direct decoupling approach for efficient reliability-based design optimization

This paper develops an efficient methodology to perform reliability-based design optimization (RBDO) by decoupling the optimization and reliability analysis iterations that are nested in traditional formulations. This is achieved by approximating the reliability constraints based on the reliability analysis results. The proposed approach does not use inverse first-order reliability analysis as other existing decoupled approaches, but uses direct reliability analysis. This strategy allows a modular approach and the use of more accurate methods, including Monte-Carlo-simulation (MCS)-based methods for highly nonlinear reliability constraints where first-order reliability approximation may not be accurate. The use of simulation-based methods also enables system-level reliability estimates to be included in the RBDO formulation. The efficiency of the proposed RBDO approach is further improved by identifying the potentially active reliability constraints at the beginning of each reliability analysis. A vehicle side impact problem is used to examine the proposed method, and the results show the usefulness of the proposed method.     

A multi-stage particle swarm for optimum design of truss structures

The contribution of this study is to propose a multi-stage particle swarm optimization (MSPSO) for structural optimization. In this paper, three auxiliary improving mechanisms are added to the standard particle swarm optimization (PSO) in order to enhance its efficiency and reliability dealing with optimum design of truss structures. These mechanisms effectively accelerate the convergence rate of the PSO and also make it robust to attain better optimum solutions during various runs of the algorithm. The effectiveness of the MSPSO is illustrated by several benchmark structural optimization problems. Results demonstrate the efficiency and robustness of the proposed MSPSO algorithm compared to the standard version of the PSO.     

Bayesian reliability-based design optimization using eigenvector dimension reduction (EDR) method

In practical engineering design, most data sets for system uncertainties are insufficiently sampled from unknown statistical distributions, known as epistemic uncertainty. Existing methods in uncertainty-based design optimization have difficulty in handling both aleatory and epistemic uncertainties. To tackle design problems engaging both epistemic and aleatory uncertainties, reliability-based design optimization (RBDO) is integrated with Bayes theorem. It is referred to as Bayesian RBDO. However, Bayesian RBDO becomes extremely expensive when employing the first- or second-order reliability method (FORM/SORM) for reliability predictions. Thus, this paper proposes development of Bayesian RBDO methodology and its integration to a numerical solver, the eigenvector dimension reduction (EDR) method, for Bayesian reliability analysis. The EDR method takes a sensitivity-free approach for reliability analysis so that it is very efficient and accurate compared with other reliability methods such as FORM/SORM. Efficiency and accuracy of the Bayesian RBDO process are substantially improved after this integration.     

A local adaptive sampling method for reliability-based design optimization using Kriging model

Reliability-based design optimization (RBDO) in practical applications is hindered by its huge computational cost during structure reliability evaluating process. Kriging-model-based RBDO is an effective method to overcome this difficulty. However, the accuracy of Kriging model depends directly on how to select the sample points. In this paper, the local adaptive sampling (LAS) is proposed to enhance the efficiency of constructing Kriging models for RBDO problems. In LAS, after initialization, new samples for probabilistic constraints are mainly selected within the local region around the current design point from each optimization iteration, and in the local sampling region, sample points are first considered to be located on the limit state constraint boundaries. The size of the LAS region is adaptively defined according to the nonlinearity of the performance functions. The computation capability of the proposed method is demonstrated using three mathematical RBDO problems and a honeycomb crash-worthiness design application. The comparison results show that the proposed method is very efficient.     

基于均匀设计的粒子群算法及其在飞控系统中的应用

将粒子群算法应用于飞行控制系统的优化设计中,需要解决两个问题：如何选择目标函数和如何确定初始种群和算法运行参数。针对这两个问题,分别提出了基于参考模型的飞行控制系统优化策略和基于均匀设计的粒子群算法初始种群和运行参数的选择方法。仿真结果表明,本文所提出的优化策略能够有效地解决飞行控制系统的优化设计问题,粒子群初始种群分布均匀,收敛速度快。     

An adaptive local range sampling method for reliability-based design optimization using support vector machine and Kriging model

Kriging model is an effective method to overcome huge computational cost for reliability-based design optimization (RBDO) problems. However, the results of RBDO usually depend on constraint boundaries within the local range that contains the RBDO optimum. Determining this local range and building adaptive response surfaces within it can avoid selecting samples in unrelated areas. In this research, a new RBDO process is proposed. In the first phase, Kriging models of constraints are built based on Latin Hypercube sampling method, and updated by two new samples in each iteration. One of these two samples is selected based on SVM and mean squared error to make sure it is located near constraint boundaries. Another one is the deterministic optimum point (DOP) of current Kriging models, which is obtained based on the deterministic optimization and specifies the direction to the RBDO optimum. And the RBDO design point is obtained by SORA. When consecutive RBDO design points are close enough to each other, the local range is determined based on the current RBDO design point and the current DOP. In the second phase, new samples are located on constraint boundaries within the local range to refine Kriging models. The location and the size of the local range is adaptively defined by the RBDO design point and the DOP during each iteration. Several optimization examples are selected to test the computation capability of the proposed method. The results indicate that the new method is more efficient and more accurate.     

An efficient strategy for reliability-based multidisciplinary design optimization using BLISS

This paper presents an efficient reliability-based multidisciplinary design optimization (RBMDO) strategy. The conventional RBMDO has tri-level loops: the first level is an optimization in the deterministic space, the second one is a reliability analysis in the probabilistic space, and the third one is the multidisciplinary analysis. Since it is computationally inefficient when high-fidelity simulation methods are involved, an efficient strategy is proposed. The strategy [named probabilistic bi-level integrated system synthesis (ProBLISS)] utilizes a single-level reliability-based design optimization (RBDO) approach, in which the reliability analysis and optimization are conducted in a sequential manner by approximating limit state functions. The single-level RBDO is associated with the BLISS formulation to solve RBMDO problems. Since both the single-level RBDO and BLISS are mainly driven by approximate models, the accuracy of models can be a critical issue for convergence. The convergence of the strategy is guaranteed by employing the trust region–sequential quadratic programming framework, which validates approximation models in the trust region radius. Two multidisciplinary problems are tested to verify the strategy. ProBLISS significantly reduces the computational cost and shows stable convergence while maintaining accuracy.     

Reliability-based structural design optimization: hybridized conjugate mean value approach

The efficiency and robustness of reliability techniques are important in reliability-based design optimization (RBDO). Commonly, advanced mean value (AMV) is utilized in reliability loop of RBDO but unstable solutions using AMV may be obtained for highly concave performance functions. Owing to the challenges of commonly reliability methods, the conjugate gradient analysis (CGA) is proposed as a robust methodology but it shows inefficient results for convex constraints. In this research, hybrid conjugate mean value (HCMV) method is proposed using sufficient condition for the enhancement of efficiency and robustness of RBDO. The CGA and AMV are dynamically utilized for simple solution of convex/concave constraints using sufficient descent criterion in HCMV. The HCMV is used to evaluate the convergence performances and is compared with numerous existing reliability methods through three reliability problems (two concave/convex mathematical examples and one applicable structure) and four RBDO problems. From the numerical results, the HCMV exhibited the better efficiency, and robustness compared to other studied formulations in reliability and RBDO problems.

     

An effective co-evolutionary particle swarm optimization for constrained engineering design problems

Many engineering design problems can be formulated as constrained optimization problems. So far, penalty function methods have been the most popular methods for constrained optimization due to their simplicity and easy implementation. However, it is often not easy to set suitable penalty factors or to design adaptive mechanism. By employing the notion of co-evolution to adapt penalty factors, this paper proposes a co-evolutionary particle swarm optimization approach (CPSO) for constrained optimization problems, where PSO is applied with two kinds of swarms for evolutionary exploration and exploitation in spaces of both solutions and penalty factors. The proposed CPSO is population based and easy to implement in parallel. Especially, penalty factors also evolve using PSO in a self-tuning way. Simulation results based on well-known constrained engineering design problems demonstrate the effectiveness, efficiency and robustness on initial populations of the proposed method. Moreover, the CPSO obtains some solutions better than those previously reported in the literature.     

Particle Swarm Optimization with Chaos-based Initialization for Numerical Optimization

Particle swarm optimization (PSO) is a population based swarm intelligence algorithm that has been deeply studied and widely applied to a variety of problems. However, it is easily trapped into the local optima and premature convergence appears when solving complex multimodal problems. To address these issues, we present a new particle swarm optimization by introducing chaotic maps (Tent and Logistic) and Gaussian mutation mechanism as well as a local re-initialization strategy into the standard PSO algorithm. On one hand, the chaotic map is utilized to generate uniformly distributed particles to improve the quality of the initial population. On the other hand, Gaussian mutation as well as the local re-initialization strategy based on the maximal focus distance is exploited to help the algorithm escape from the local optima and make the particles proceed with searching in other regions of the solution space. In addition, an auxiliary velocity-position update strategy is exclusively used for the global best particle, which can effectively guarantee the convergence of the proposed particle swarm optimization. Extensive experiments on eight well-known benchmark functions with different dimensions demonstrate that the proposed PSO is superior or highly competitive to several state-of-the-art PSO variants in dealing with complex multimodal problems.     

Application of reliability-based optimization to earth-moving machine: hydraulic cylinder components design process

Reliability-based design optimization (RBDO) incorporates probabilistic analysis into optimization process so that an optimum design has a great chance of staying in the feasible design space when the inevitable variability in design variables/parameters is considered. One of the biggest drawbacks of applying RBDO to practical problem is its high computational cost that is often impractical to industries. In search of the most suitable RBDO method for industrial applications, we first evaluated several existing RBDO approaches in details such as the double-loop RBDO, the sequential optimization and reliability assessment, and the response surface method. Then, based on industry needs, a platform incorporating/integrating the existing algorithm of optimization and reliability analysis is built for a practical RBDO problem. Effectiveness of the proposed RBDO approach is demonstrated using a simple cantilever beam problem and a more complicated industry problem.     

Relaxed performance measure approach for reliability-based design optimization

The efficiency and robustness of reliability analysis methods are important factors to evaluate the probabilistic constraints in reliability-based design optimization (RBDO). In this paper, a relaxed mean value (RMV) approach is proposed in order to evaluate probabilistic constraints including convex and concave functions in RBDO using the performance measure approach (PMA). A relaxed factor is adaptively determined in the range from 0 to 2 using an inequality criterion to improve the efficiency and robustness of the inverse first-order reliability methods. The performance of the proposed RMV is compared with six existing reliability methods, including the advanced mean value (AMV), conjugate mean value (CMV), hybrid mean value (HMV), chaos control (CC), modified chaos control (MCC), and conjugate gradient analysis (CGA) methods, through four nonlinear concave and convex performance functions and three RBDO problems. The results demonstrate that the proposed RMV is more robust than the AMV, CMV, and HMV for highly concave problems, and slightly more efficient than the CC, MCC, and CGA methods. Furthermore, the proposed relaxed mean value guarantees robust and efficient convergence for RBDO problems with highly nonlinear performance functions.     

Reliability-based design optimization using kriging surrogates and subset simulation

The aim of the present paper is to develop a strategy for solving reliability-based design optimization (RBDO) problems that remains applicable when the performance models are expensive to evaluate. Starting with the premise that simulation-based approaches are not affordable for such problems, and that the most-probable-failure-point-based approaches do not permit to quantify the error on the estimation of the failure probability, an approach based on both metamodels and advanced simulation techniques is explored. The kriging metamodeling technique is chosen in order to surrogate the performance functions because it allows one to genuinely quantify the surrogate error. The surrogate error onto the limit-state surfaces is propagated to the failure probabilities estimates in order to provide an empirical error measure. This error is then sequentially reduced by means of a population-based adaptive refinement technique until the kriging surrogates are accurate enough for reliability analysis. This original refinement strategy makes it possible to add several observations in the design of experiments at the same time. Reliability and reliability sensitivity analyses are performed by means of the subset simulation technique for the sake of numerical efficiency. The adaptive surrogate-based strategy for reliability estimation is finally involved into a classical gradient-based optimization algorithm in order to solve the RBDO problem. The kriging surrogates are built in a so-called augmented reliability space thus making them reusable from one nested RBDO iteration to the other. The strategy is compared to other approaches available in the literature on three academic examples in the field of structural mechanics.     

A cell evolution method for reliability-based design optimization

The reliability-based design optimization (RBDO) presents to be a systematic and powerful approach for process designs under uncertainties. The traditional double-loop methods for solving RBDO problems can be computationally inefficient because the inner reliability analysis loop has to be iteratively performed for each probabilistic constraint. To solve RBDOs in an alternative and more effective way, Deb et al. [1] proposed recently the use of evolutionary algorithms with an incorporated fastPMA. Since the imbedded fastPMA needs the gradient calculations and the initial guesses of the most probable points (MPPs), their proposed algorithm would encounter difficulties in dealing with non-differentiable constraints and the effectiveness could be degraded significantly as the initial guesses are far from the true MPPs. In this paper, a novel population-based evolutionary algorithm, named cell evolution method, is proposed to improve the computational efficiency and effectiveness of solving the RBDO problems. By using the proposed cell evolution method, a family of test cells is generated based on the target reliability index and with these reliability test cells the determination of the MPPs for probabilistic constraints becomes a simple parallel calculation task, without the needs of gradient calculations and any initial guesses. Having determined the MPPs, a modified real-coded genetic algorithm is applied to evolve these cells into a final one that satisfies all the constraints and has the best objective function value for the RBDO. Especially, the nucleus of the final cell contains the reliable solution to the RBDO problem. Illustrative examples are provided to demonstrate the effectiveness and applicability of the proposed cell evolution method in solving RBDOs. Simulation results reveal that the proposed cell evolution method outperforms comparative methods in both the computational efficiency and solution accuracy, especially for multi-modal RBDO problems.     

A fuzzy particle swarm optimization algorithm for computer communication network topology design

Particle swarm optimization (PSO) is a powerful optimization technique that has been applied to solve a number of complex optimization problems. One such optimization problem is topology design of distributed local area networks (DLANs). The problem is defined as a multi-objective optimization problem requiring simultaneous optimization of monetary cost, average network delay, hop count between communicating nodes, and reliability under a set of constraints. This paper presents a multi-objective particle swarm optimization algorithm to efficiently solve the DLAN topology design problem. Fuzzy logic is incorporated in the PSO algorithm to handle the multi-objective nature of the problem. Specifically, a recently proposed fuzzy aggregation operator, namely the unified And-Or operator (Khan and Engelbrecht in Inf. Sci. 177: 2692–2711, 2007), is used to aggregate the objectives. The proposed fuzzy PSO (FPSO) algorithm is empirically evaluated through a preliminary sensitivity analysis of the PSO parameters. FPSO is also compared with fuzzy simulated annealing and fuzzy ant colony optimization algorithms. Results suggest that the fuzzy PSO is a suitable algorithm for solving the DLAN topology design problem.     

Quadratic interpolation based orthogonal learning particle swarm optimization algorithm

Particle swarm optimization (PSO) is a population based algorithm for solving global optimization problems. Owing to its efficiency and simplicity, PSO has attracted many researchers’ attention and developed many variants. Orthogonal learning particle swarm optimization (OLPSO) is proposed as a new variant of PSO that relies on a new learning strategy called orthogonal learning strategy. The OLPSO differs in the utilization of the information of experience from the standard PSO, in which each particle utilizes its historical best experience and globally best experience through linear summation. In OLPSO, particles can fly in better directions by constructing an efficient exemplar through orthogonal experimental design. However, the global version based orthogonal learning PSO (OLPSO-G) still have some drawbacks in solving some complex multimodal function optimization. In this paper, we proposed a quadratic interpolation based OLPSO-G (QIOLPSO-G), in which, a quadratic interpolation based construction strategy for the personal historical best experience is applied. Meanwhile, opposition-based learning, and Gaussian mutation are also introduced into this paper to increase the diversity of the population and discourage the premature convergence. Experiments are conducted on 16 benchmark problems to validate the effectiveness of the QIOLPSO-G, and comparisons are made with four typical PSO algorithms. The results show that the introduction of the three strategies does enhance the effectiveness of the algorithm.     

An optimal shifting vector approach for efficient probabilistic design

The application of reliability-based design optimization (RBDO) is hindered by the unbearable computational cost in the structure reliability evaluating process. This study proposes an optimal shifting vector (OSV) approach to enhance the efficiency of RBDO. In OSV, the idea of using an optimal shifting vector in the decoupled method and the notation of conducting reliability analysis in the super-sphere design space are proposed. The shifted limit state function, instead of the specific performance function, is used to identify the inverse most probable point (IMPP) and derive the optimal shifting vector for accelerating the optimization process. The super-sphere design space is applied to reduce the number of constraints and design variables for the novel reliability analysis model. OSV is very efficient for highly nonlinear problems, especially when the contour lines of the performance functions vary widely. The computation capability of the proposed method is demonstrated and compared to existing RBDO methods using four mathematical and engineering examples. The comparison results show that the proposed OSV approach is very efficient.     

Improved hybrid method as a robust tool for reliability-based design optimization

In the engineering problems, the randomness and the uncertainties of the distribution of the structural parameters are a crucial problem. In the case of reliability-based design optimization (RBDO), it is the objective to play a dominant role in the structural optimization problem introducing the reliability concept. The RBDO problem is often formulated as a minimization of the initial structural cost under constraints imposed on the values of elemental reliability indices corresponding to various limit states. The classical RBDO leads to high computing time and weak convergence, but a Hybrid Method (HM) has been proposed to overcome these two drawbacks. As the hybrid method successfully reduces the computing time, we can increase the number of variables by introducing the standard deviations as optimization variables to minimize the error values in the probabilistic model. The efficiency of the hybrid method has been demonstrated on static and dynamic cases with extension to the variability of the probabilistic model. In this paper, we propose a modification on the formulation of the hybrid method to improve the optimal solutions. The proposed method is called, Improved Hybrid Method (IHM). The main benefit of this method is to improve the structure performance by much more minimizing the objective function than the hybrid method. It is also shown to demonstrate the optimality conditions. The improved hybrid method is next applied to two numerical examples, with consideration of the standard deviations as optimization variables (for linear and nonlinear distributions). When integrating the improved hybrid method within the probabilistic model variability, we minimize the objective function more and more.     

Optimal design of arch dams subjected to earthquake loading by a combination of simultaneous perturbation stochastic approximation and particle swarm algorithms

An efficient optimization procedure is introduced to find the optimal shapes of arch dams considering fluid–structure interaction subject to earthquake loading. The optimization is performed by a combination of simultaneous perturbation stochastic approximation (SPSA) and particle swarm optimization (PSO) algorithms. This serial integration of the two single methods is termed as SPSA–PSO. The operation of SPSA–PSO includes three phases. In the first phase, a preliminary optimization is accomplished using the SPSA. In the second phase, an optimal initial swarm is produced using the first phase results. In the last phase, the PSO is employed to find the optimum design using the optimal initial swarm. The numerical results demonstrate the high performance of the proposed strategy for optimal design of arch dams. The solutions obtained by the SPSA–PSO are compared with those of SPSA and PSO. It is revealed that the SPSA–PSO converges to a superior solution compared to the SPSA and PSO having a lower computation cost.     

Efficient reliability-based design optimization using a hybrid space with application to finite element analysis

The design of high technology structures aims to define the best compromise between cost and safety. The Reliability-Based Design Optimization (RBDO) allows us to design structures which satisfy economical and safety requirements. However, in practical applications, the coupling between the mechanical modelling, the reliability analyses and the optimization methods leads to very high computational time and weak convergence stability. Traditionally, the solution of the RBDO model is achieved by alternating reliability and optimization iterations. This approach leads to low numerical efficiency, which is disadvantageous for engineering applications on real structures. In order to avoid this difficulty, we propose herein a very efficient method based on the simultaneous solution of the reliability and optimization problems. The procedure leads to parallel convergence for both problems in a Hybrid Design Space (HDS). The efficiency of the proposed methodology is demonstrated on the design of a steel hook, where the RBDO is combined with Finite Element Analysis (FEA).