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.     

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.     

A multi-constraint failure-pursuing sampling method for reliability-based design optimization using adaptive Kriging

Using surrogate models to substitute the computationally expensive limit state functions is a promising way to decrease the cost of implementing reliability-based design optimization (RBDO). To train the models efficiently, the active learning strategies have been intensively studied. However, the existing learning strategies either do not individually build the models according to importance measurement or do not completely relate to the reliability analysis results. Consequently, some points that are useless to refine the limit state functions or far away from the RBDO solutions are generated. This paper proposes a multi-constraint failure-pursuing sampling method to maximize the reward of adding new training points. A simultaneous learning strategy is employed to sequentially update the Kriging models with the points selected in the current approximate safe region. Moreover, the sensitive Kriging model as well as the sensitive sample point are identified based on the failure-pursuing scheme. A new point that is highly potential to improve the accuracy of reliability analysis and optimization can then be generated near the sensitive sample point and used to update the sensitive model. Besides, numerical examples and engineering application are used to validate the performance of the proposed method.

     

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.     

Reliability-based design optimization for crashworthiness of vehicle side impact

With the advent of powerful computers, vehicle safety issues have recently been addressed using computational methods of vehicle crashworthiness, resulting in reductions in cost and time for new vehicle development. Vehicle design demands multidisciplinary optimization coupled with a computational crashworthiness analysis. However, simulation-based optimization generates deterministic optimum designs, which are frequently pushed to the limits of design constraint boundaries, leaving little or no room for tolerances (uncertainty) in modeling, simulation uncertainties, and/or manufacturing imperfections. Consequently, deterministic optimum designs that are obtained without consideration of uncertainty may result in unreliable designs, indicating the need for Reliability-Based Design Optimization (RBDO).Recent development in RBDO allows evaluations of probabilistic constraints in two alternative ways: using the Reliability Index Approach (RIA) and the Performance Measure Approach (PMA). The PMA using the Hybrid Mean Value (HMV) method is shown to be robust and efficient in the RBDO process, whereas RIA yields instability for some problems. This paper presents an application of PMA and HMV for RBDO for the crashworthiness of a large-scale vehicle side impact. It is shown that the proposed RBDO approach is very effective in obtaining a reliability-based optimum design.     

Extensions of design potential concept for reliability-based design optimization to nonsmooth and extreme cases

The reliability-based design optimization (RBDO) can be described by the design potential concept in a unified system space, where the probabilistic constraint is identified by the design potential surface of the reliability target that is obtained analytically from the first-order reliability method (FORM). This paper extends the design potential concept to treat nonsmooth probabilistic constraints and extreme case design in RBDO. In addition, refinement of the design potential surface, which yields better optimum design, can be obtained using more accurate second-order reliability method (SORM). By integrating performance probability analysis into the iterative design optimization process, the design potential concept leads to a very effective design potential method (DPM) for robust system parameter design. It can also be applied effectively to extreme case design (ECD) by directly representing a probabilistic constraint in terms of the system performance function. Received July 25, 2000     

基于增量SVR模型的改进全局优化算法

SVR(支持向量回归)是一种具有较强稳健性的小样本学习方法,它可有效避免"维数灾难",并被引入到全局优化中。然而现有的基于SVR的全局优化算法存在估值次数多、无法应对高维优化问题等缺点。提出了一种基于增量SVR模型的新的改进全局优化算法DISVR:采用增量SVR方法提高过程响应面的重构效率;采用一种新的增量LHD(Latin Hyper-cube Sampling)方法确保样本集分布均匀;采用DIRECT搜索算法提高全局搜索的稳定性和效率。最后,通过多个测试函数表明,该算法既降低了时间复杂度,也有效减少了源模型的估值次数。     

基于双重权约束期望改进策略的多目标并行代理优化方法

针对多目标仿真优化的高昂成本及黑箱函数难以获取问题,提出基于双重权约束期望改进策略的多目标并行代理优化方法.首先,建立Kriging模型获取未试验点的预测不确定性;其次,构建双重权约束期望改进策略,并利用填充策略矩阵及距离聚合方法实现新改进策略的聚合;然后,最大化聚合双重权约束期望改进策略实现多目标并行优化;最后,达到终止条件,获得Pareto最优解集.选取测试函数及铰接夹芯梁设计案例进行优化验证.验证对比结果表明:所提方法可有效提升多目标问题优化效率,减少昂贵仿真成本;与同类方法相比,低维问题中获取Pareto最优解集的收敛性、多样性及分布性更优.     

Design optimization of balloon-expandable coronary stent

To prevent the dogboning effect of stent implantation (i.e. the ends of a stent opening first during expansion), an adaptive optimization method based on the kriging surrogate model is proposed to reduce the absolute value of the dogboning rate. Integrating design of experiment (DOE) methods with the kriging surrogate model can approximate the functional relationship between the dogboning rate and the geometrical design parameters of the stent, replacing the expensive reanalysis of the stent dogboning rate during the optimization process. In this adaptive process, an infilling sampling criterion termed expected improvement (EI) is used to balance local and global search and tends to find the global optimal design. Finite element method is used to analyze stent expansion. As an example, a typical diamond-shaped coronary stent is investigated, where four key geometries are selected to be the design variables. Numerical results demonstrate that the proposed adaptive optimization method can effectively decrease the absolute value of the dogboning rate of stent dilation.     

An adaptive sampling strategy for Kriging metamodel based on Delaunay triangulation and TOPSIS

Metamodels have been widely used in engineering design and optimization. Sampling method plays an important role in the constructing of metamodels. This paper proposes an adaptive sampling strategy for Kriging metamodel based on Delaunay triangulation and TOPSIS (KMDT). In the proposed KMDT, Delaunay triangulation is employed to partition the design space according to current sample points. The area of each partitioned triangle is used to indicate the degree of dispersion of sample points, and the prediction error of Kriging metamodel at each triangle’s centroid is used to represent the local error of each triangle region. By calculating the weight of the area and prediction error for each triangle region using the entropy method and TOPSIS, the degree of dispersion of sample points and local errors of metamodel are taken into consideration to make a trade-off between global exploration and local exploitation during the sequential sampling process. As a demonstration, the proposed approach is compared to other three sampling methods using several numerical cases and the modeling of the aerodynamic coefficient for a three-dimensional aircraft. The result reveals that the proposed approach provides more accurate metamodel at the same simulation cost, which is very important in metamodel-based engineering design problems.     

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.     

Unified uncertainty representation and quantification based on insufficient input data

Uncertainty quantification accuracy of system performance has an important influence on the results of reliability-based design optimization (RBDO). A new uncertain identification and quantification methodology is proposed considering the strong statistical variables, sparse variables, and interval variables simultaneously. Maximum likelihood function and Akaike information criterion (AIC) methods are used to identify the best-fitted distribution types and distribution parameters of sparse variables. The interval variables are represented with evidence theory. Finally, a unified uncertainty quantification framework considering the three types of uncertain design variables is put forward, and then the failure probability of system performance is quantified with belief and plausibility measures. The Kriging metamodel and random sampling method are used to reduce the computational complexity. Three examples are illustrated to verify the effectiveness of the proposed methodology.     

Metamodeling development for reliability-based design optimization of automotive body structure

Metamodels are commonly used in reliability-based design optimization (RBDO) due to the enormously expensive computation cost of numerical simulations. However, for large-scale design optimization of automotive body structure, with the increasing number of design variable and enhanced nonlinearity degree of structural performance, polynomial response surface which is commonly used for vehicle design optimization often suffers exponentially increased computation burden and serious loss of approximation accuracy. In this paper, support vector regression, along with other four complex metamodeling techniques including moving least square, artificial neural network, radial basis function and Kriging, is investigated for approximating frontal crashworthiness performance which is one of the most highly nonlinear performances. It aims at testing support vector regression and providing advanced metamodeling technique for RBDO of automotive body structure. Approximation results are compared in both accuracy and computational efficiency. Based on the frontal crashworthiness example, it is found that support vector regression and moving least square are preferable techniques to approximate structural performances with good accuracy. But support vector regression is recommended for its computational efficiency and better approximation potential. Moreover, the ensemble of support vector regression, moving least square, Kriging and artificial neural network is an effective alternative and is proved, in the RBDO example for the lightweight design of front body structure, to outperform any other single metamodel. The remarkable predominance indicates that the ensemble of support vector regression, moving least square, Kriging and artificial neural network holds great potential in approximating highly nonlinear performances for RBDO of automotive body structure.     

Inverse analysis method using MPP-based dimension reduction for reliability-based design optimization of nonlinear and multi-dimensional systems

There are two commonly used analytical reliability analysis methods: linear approximation - first-order reliability method (FORM), and quadratic approximation - second-order reliability method (SORM), of the performance function. The reliability analysis using FORM could be acceptable in accuracy for mildly nonlinear performance functions, whereas the reliability analysis using SORM may be necessary for accuracy of nonlinear and multi-dimensional performance functions. Even though the reliability analysis using SORM may be accurate, it is not as much used for probability of failure calculation since SORM requires the second-order sensitivities. Moreover, the SORM-based inverse reliability analysis is rather difficult to develop.This paper proposes an inverse reliability analysis method that can be used to obtain accurate probability of failure calculation without requiring the second-order sensitivities for reliability-based design optimization (RBDO) of nonlinear and multi-dimensional systems. For the inverse reliability analysis, the most probable point (MPP)-based dimension reduction method (DRM) is developed. Since the FORM-based reliability index (β) is inaccurate for the MPP search of the nonlinear performance function, a three-step computational procedure is proposed to improve accuracy of the inverse reliability analysis: probability of failure calculation using constraint shift, reliability index update, and MPP update. Using the three steps, a new DRM-based MPP is obtained, which estimates the probability of failure of the performance function more accurately than FORM and more efficiently than SORM. The DRM-based MPP is then used for the next design iteration of RBDO to obtain an accurate optimum design even for nonlinear and/or multi-dimensional system. Since the DRM-based RBDO requires more function evaluations, the enriched performance measure approach (PMA+) with new tolerances for constraint activeness and reduced rotation matrix is used to reduce the number of function evaluations.     

A constrained, globalized, and bounded Nelder–Mead method for engineering optimization

One of the fundamental difficulties in engineering design is the multiplicity of local solutions. This has triggered much effort in the development of global search algorithms. Globality, however, often has a prohibitively high numerical cost for real problems. A fixed cost local search, which sequentially becomes global, is developed in this work. Globalization is achieved by probabilistic restarts. A spacial probability of starting a local search is built based on past searches. An improved Nelder–Mead algorithm is the local optimizer. It accounts for variable bounds and nonlinear inequality constraints. It is additionally made more robust by reinitializing degenerated simplexes. The resulting method, called the Globalized Bounded Nelder–Mead (GBNM) algorithm, is particularly adapted to tackling multimodal, discontinuous, constrained optimization problems, for which it is uncertain that a global optimization can be afforded. Numerical experiments are given on two analytical test functions and two composite laminate design problems. The GBNM method compares favorably with an evolutionary algorithm, both in terms of numerical cost and accuracy.     

Experience with approximate reliability-based optimization methods II: an exhaust system problem

Traditional reliability-based design optimization (RBDO) requires a double-loop iteration process. The inner optimization loop is to find the reliability and the outer is the regular optimization loop to optimize the RBDO problem with reliability objectives or constraints. It is known that the computation can be prohibitive when the associated function evaluation is expensive. This situation is even worse when a large number of reliability constraints are present. As a result, many approximate RBDO methods, which convert the double loop to a single loop, have been developed. In this research, an engineering problem with a large number of constraints (144) is designed to test RBDO methods based on the first-order reliability method (FORM), including single- and double-loop methods. In addition to the number of constraints, this problem possesses many local minimums. Some original authors of the RBDO methods are also asked to solve the same problem. The results and the efficiencies for different methods are published and discussed.     

蒸馏与换热协同的约束多目标在线操作优化方法

针对蒸馏装置与换热网络间缺乏协同优化导致的分馏精度差和能耗高的问题,提出了一种基于代理模型的约束多目标在线协同操作优化方法.为了解决蒸馏装置与换热网络操作参数协同优化时存在的计算耗时和约束的问题,构建Kriging代理模型来近似目标函数和约束条件,提出了基于随机欠采样和Adaboost的分类代理模型（RUSBoost）来解决类别不平衡的收敛判定预测问题.提出了基于多阶段自适应约束处理的代理模型的模型管理方法,该方法采用基于参考向量激活状态的最大化改善期望准则和可行概率准则更新机制来平衡优化初始阶段种群的多样性和可行性,采用支配参考点的置信下限准则更新机制加快收敛速度.通过不断与机理模型交互来在线更新代理模型,实现在线操作优化.通过测试函数和仿真实例验证了本文方法的有效性.     

Quantile-based optimization under uncertainties using adaptive Kriging surrogate models

Uncertainties are inherent to real-world systems. Taking them into account is crucial in industrial design problems and this might be achieved through reliability-based design optimization (RBDO) techniques. In this paper, we propose a quantile-based approach to solve RBDO problems. We first transform the safety constraints usually formulated as admissible probabilities of failure into constraints on quantiles of the performance criteria. In this formulation, the quantile level controls the degree of conservatism of the design. Starting with the premise that industrial applications often involve high-fidelity and time-consuming computational models, the proposed approach makes use of Kriging surrogate models (a.k.a. Gaussian process modeling). Thanks to the Kriging variance (a measure of the local accuracy of the surrogate), we derive a procedure with two stages of enrichment of the design of computer experiments (DoE) used to construct the surrogate model. The first stage globally reduces the Kriging epistemic uncertainty and adds points in the vicinity of the limit-state surfaces describing the system performance to be attained. The second stage locally checks, and if necessary, improves the accuracy of the quantiles estimated along the optimization iterations. Applications to three analytical examples and to the optimal design of a car body subsystem (minimal mass under mechanical safety constraints) show the accuracy and the remarkable efficiency brought by the proposed procedure.     

动态调整成长方式的郊狼优化算法及其应用

郊狼优化算法在迭代运行时种群多样性降低,收敛速度变慢,易陷入局部最优,并且在求解约束优化问题时难以获得可行解。提出一种动态调整成长方式的郊狼优化算法（DGCOA）。在狼群进化中引入变异交叉策略,增强种群多样性,基于郊狼成长策略加入全局最优个体指导搜索,使得每个子种群中的个体从不同的方向快速逼近最优解位置,并根据种群中个体相似度对郊狼位置更新方式进行调整,平衡算法的全局探索与局部开发能力。在求解约束优化问题时,利用自适应约束处理方法构建新的适应度函数,协调优化目标和约束违反度。基于CEC2006对22个测试函数和3个工程设计问题进行仿真,结果表明,与COA、ICTLBO、ODPSO等算法相比,DGCOA算法具有较高的收敛精度和稳定性,适用于求解复杂优化问题。     

Performance assessment of a multi-objective parametric optimization algorithm with application to a multi-physical engineering system

The original problem of reliability-based design optimization (RBDO) is mathematically a nested two-level structure that is computationally time consuming for real engineering problems. In order to overcome the computational difficulties, many formulations have been proposed in the literature. These include SORA (sequential optimization and reliability assessment) that decouples the nested problems. SLA (single loop approach) further improves efficiency in that reliability analysis becomes an integrated part of the optimization problem. However, even SLA method can become computationally challenging for real engineering problems involving many reliability constraints. This paper presents an enhanced version of SLA where the first phase is based on approximation at nominal design point. After convergence of first iterative phase is reached the process transitions to a second phase where approximations of reliability constraints are carried out at their respective minimum performance target point (MPTP). The solution is implemented in Altair OptiStruct, where adaptive approximation and constraint screening strategies are utilized in the RBDO process. Examples show that the proposed two-phase approach leads to reduction in finite element analyses while preserving equal solution quality.