An improvement of Kriging based sequential approximate optimization method via extended use of design of experiments

When Kriging is used as a meta-model for an inequality constrained function, approximate optimal solutions are sometimes infeasible in the case where they are active at the constraint boundary. This article explores the development of a Kriging-based meta-model that enhances the constraint feasibility of an approximate optimal solution. The trust region management scheme is used to ensure the convergence of the approximate optimal solution. The present study proposes a method of enhancing the constraint feasibility in which the currently infeasible design is replaced by the most feasible-usable design during the sequential approximate optimization process. An additional convergence condition is also included to reinforce the design accuracy and feasibility. Latin hypercube design and (2n+1) design are used as tools for design of experiments. The proposed approach is verified through a constrained mathematical function problem and a number of engineering optimization problems to support the proposed strategies.     

Sequential sampling designs based on space reduction

In the field of engineering design and optimization, metamodels are widely used to replace expensive simulation models in order to reduce computing costs. To improve the accuracy of metamodels effectively and efficiently, sequential sampling designs have been developed. In this article, a sequential sampling design using the Monte Carlo method and space reduction strategy (MCSR) is implemented and discussed in detail. The space reduction strategy not only maintains good sampling properties but also improves the efficiency of the sampling process. Furthermore, a local boundary search (LBS) algorithm is proposed to efficiently improve the performance of MCSR, which is called LBS-MCSR. Comparative results with several sequential sampling approaches from low to high dimensions indicate that the space reduction strategy generates samples with better sampling properties (and thus better metamodel accuracy) in less computing time.     

Efficient infill sampling for unconstrained robust optimization problems

A novel infill sampling criterion is proposed for efficient estimation of the global robust optimum of expensive computer simulation based problems. The algorithm is especially geared towards addressing problems that are affected by uncertainties in design variables and problem parameters. The method is based on constructing metamodels using Kriging and adaptively sampling the response surface via a principle of expected improvement adapted for robust optimization. Several numerical examples and an engineering case study are used to demonstrate the ability of the algorithm to estimate the global robust optimum using a limited number of expensive function evaluations.     

A new sampling scheme for developing metamodels with the zeros of Chebyshev polynomials

The accuracy of metamodelling is determined by both the sampling and approximation. This article proposes a new sampling method based on the zeros of Chebyshev polynomials to capture the sampling information effectively. First, the zeros of one-dimensional Chebyshev polynomials are applied to construct Chebyshev tensor product (CTP) sampling, and the CTP is then used to construct high-order multi-dimensional metamodels using the ‘hypercube’ polynomials. Secondly, the CTP sampling is further enhanced to develop Chebyshev collocation method (CCM) sampling, to construct the ‘simplex’ polynomials. The samples of CCM are randomly and directly chosen from the CTP samples. Two widely studied sampling methods, namely the Smolyak sparse grid and Hammersley, are used to demonstrate the effectiveness of the proposed sampling method. Several numerical examples are utilized to validate the approximation accuracy of the proposed metamodel under different dimensions.     

Optimizing Latin hypercube design for sequential sampling of computer experiments

Space-filling and projective properties are desired features in the design of computer experiments to create global metamodels to replace expensive computer simulations in engineering design. The goal in this article is to develop an efficient and effective sequential Quasi-LHD (Latin Hypercube design) sampling method to maintain and balance the two aforementioned properties. The sequential sampling is formulated as an optimization problem, with the objective being the Maximin Distance, a space-filling criterion, and the constraints based on a set of pre-specified minimum one-dimensional distances to achieve the approximate one-dimensional projective property. Through comparative studies on sampling property and metamodel accuracy, the new approach is shown to outperform other sequential sampling methods for global metamodelling and is comparable to the one-stage sampling method while providing more flexibility in a sequential metamodelling procedure.     

Detailed design of a lattice composite fuselage structure by a mixed optimization method

In this article, a procedure for designing a lattice fuselage barrel is developed. It comprises three stages: first, topology optimization of an aircraft fuselage barrel is performed with respect to weight and structural performance to obtain the conceptual design. The interpretation of the optimal result is given to demonstrate the development of this new lattice airframe concept for the fuselage barrel. Subsequently, parametric optimization of the lattice aircraft fuselage barrel is carried out using genetic algorithms on metamodels generated with genetic programming from a 101-point optimal Latin hypercube design of experiments. The optimal design is achieved in terms of weight savings subject to stability, global stiffness and strain requirements, and then verified by the fine mesh finite element simulation of the lattice fuselage barrel. Finally, a practical design of the composite skin complying with the aircraft industry lay-up rules is presented. It is concluded that the mixed optimization method, combining topology optimization with the global metamodel-based approach, allows the problem to be solved with sufficient accuracy and provides the designers with a wealth of information on the structural behaviour of the novel anisogrid composite fuselage design.     

Multi-objective aerodynamic shape optimization of small livestock trailers

This article presents a formal optimization study of the design of small livestock trailers, within which the majority of animals are transported to market in the UK. The benefits of employing a headboard fairing to reduce aerodynamic drag without compromising the ventilation of the animals’ microclimate are investigated using a multi-stage process involving computational fluid dynamics (CFD), optimal Latin hypercube (OLH) design of experiments (DoE) and moving least squares (MLS) metamodels. Fairings are parameterized in terms of three design variables and CFD solutions are obtained at 50 permutations of design variables. Both global and local search methods are employed to locate the global minimum from metamodels of the objective functions and a Pareto front is generated. The importance of carefully selecting an objective function is demonstrated and optimal fairing designs, offering drag reductions in excess of 5% without compromising animal ventilation, are presented.     

An adaptive stochastic inverse solver for multiscale characterization of composite materials

We present an adaptive variant of the measure‐theoretic approach for stochastic characterization of micromechanical properties based on the observations of quantities of interest at the coarse (macro) scale. The salient features of the proposed nonintrusive stochastic inverse solver are identification of a nearly optimal sampling domain using enhanced ant colony optimization algorithm for multiscale problems, incremental Latin‐hypercube sampling method, adaptive discretization of the parameter and observation spaces, and adaptive selection of number of samples. A complete test data of the TORAY T700GC‐12K‐31E and epoxy #2510 material system from the National Institute for Aviation Research report is employed to characterize and validate the proposed adaptive nonintrusive stochastic inverse algorithm for various unnotched and open‐hole laminates. Copyright © 2016 John Wiley & Sons, Ltd.     

Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems

The following techniques for uncertainty and sensitivity analysis are briefly summarized: Monte Carlo analysis, differential analysis, response surface methodology, Fourier amplitude sensitivity test, Sobol' variance decomposition, and fast probability integration. Desirable features of Monte Carlo analysis in conjunction with Latin hypercube sampling are described in discussions of the following topics: (i) properties of random, stratified and Latin hypercube sampling, (ii) comparisons of random and Latin hypercube sampling, (iii) operations involving Latin hypercube sampling (i.e. correlation control, reweighting of samples to incorporate changed distributions, replicated sampling to test reproducibility of results), (iv) uncertainty analysis (i.e. cumulative distribution functions, complementary cumulative distribution functions, box plots), (v) sensitivity analysis (i.e. scatterplots, regression analysis, correlation analysis, rank transformations, searches for nonrandom patterns), and (vi) analyses involving stochastic (i.e. aleatory) and subjective (i.e. epistemic) uncertainty.     

Implementation of Discrete Capability into the Enhanced Multipoint Approximation Method for Solving Mixed Integer-Continuous Optimization Problems

Multipoint approximation method (MAM) focuses on the development of metamodels for the objective and constraint functions in solving a mid-range optimization problem within a trust region. To develop an optimization technique applicable to mixed integer-continuous design optimization problems in which the objective and constraint functions are computationally expensive and could be impossible to evaluate at some combinations of design variables, a simple and efficient algorithm, coordinate search, is implemented in the MAM. This discrete optimization capability is examined by the well established benchmark problem and its effectiveness is also evaluated as the discreteness interval for discrete design variables is increased from 0.2 to 1. Furthermore, an application to the optimization of a lattice composite fuselage structure where one of design variables (number of helical ribs) is integer is also presented to demonstrate the efficiency of this capability.     

A deterministic sequential maximin Latin hypercube design method using successive local enumeration for metamodel-based optimization

Space-filling and projective properties of design of computer experiments methods are desired features for metamodelling. To enable the production of high-quality sequential samples, this article presents a novel deterministic sequential maximin Latin hypercube design (LHD) method using successive local enumeration, notated as sequential-successive local enumeration (S-SLE). First, a mesh-mapping algorithm is proposed to map the positions of existing points into the new hyper-chessboard to ensure the projective property. According to the maximin distance criterion, new sequential samples are generated through successive local enumeration iterations to improve the space-filling uniformity. Through a number of comparative studies, several appealing merits of S-SLE are demonstrated: (1) S-SLE outperforms several existing LHD methods in terms of sequential sampling quality; (2) it is flexible and robust enough to produce high-quality multiple-stage sequential samples; and (3) the proposed method can improve the overall performance of sequential metamodel-based optimization algorithms. Thus, S-SLE is a promising sequential LHD method for metamodel-based optimization.     

基于神经网络遗传算法的深腔型零件拉深工艺参数优化

目的 解决冲压成形中工艺参数优化难的问题.方法 以一种深腔型零件的冲压成形为例.首先,借助灰度关联分析法对有限元中的工艺参数进行分析,获取该零件冲压成形中影响成形质量的2个主要因素——冲压速度和压边力.其次,借助拉丁超立方抽样法对上述2个因素进行随机取样,并借助DYNAFORM软件对其进行逐一模拟.再次,将冲压速度和压边力作为输入,最大减薄作为输出,训练在MATLAB中建立的BP神经网络,并借助遗传算法对其进行寻优.结果 最优成形压边力为1.372 MN,最优加载速度为1.5366 m/s.结论 与神经网络遗传算法预测相比,有限元结果的相对误差小于2％,零件试制结果的相对误差小于6％,该方法有较高的预测精度.     

基于代理模型的通风盘式制动器制动盘结构优化设计

该文提出一种基于代理模型的复杂结构优化设计方法,并用于通风盘式制动器制动盘结构优化设计。提出的优化设计方法集成了CAE分析、实验设计、代理模型构造及非线性优化几部分,实验设计采用拉丁超立方抽样策略,代理模型选用改进的响应面模型,非线性优化算法采用序列二次规划算法。为了解决传统的响应面模型部分预测值与实验值误差较大问题,改进方法认为只有能够确保在每一个抽样点处的预测值与试验值的相对误差均在一定范围内的响应面模型才是一个可行的模型。在保证制动盘质量不变情况下,以寿命最大化为目标,应用设计的集成优化方法对制动盘进行优化设计,优化设计结果较好,其中制动盘疲劳寿命根据Coffin-Manson方法预测,制动过程中制动盘表面最大热应力及最高温度通过热机耦合的有限元模拟紧急制动过程获得。优化结果表明该文提出的方法是一种有效的复杂结构的优化设计方法。     

基于频响函数奇异值的模型修正方法

为减少实测环境中噪声的干扰,提出了一种基于频响函数奇异值的模型修正方法。利用计算得到的频响函数重构吸引子矩阵,对其进行奇异值分解,并在受噪声影响时根据极值点数量突变原则选择保留主要特征信息的奇异值个数,确定待修正参数;采用拉丁超立方抽样抽取初始样本点,结合修正参数所对应的奇异值响应,用粒子群算法寻得最优相关系数,构建Kriging模型;以奇异值响应差的平方最小构造目标函数,利用布谷鸟算法求解参数修正值。仿真算例表明:以奇异值作为结构响应,构建Kriging模型能获得较高的修正精度;在频响函数中加入不同信噪比的高斯白噪声,仍能得到较满意的修正效果,证明了该方法对噪声具有较强的鲁棒性。     

基于概率-凸集混合模型的汽车正面碰撞结构可靠性优化设计

本文基于概率和凸集模型研究汽车正面碰撞可靠性优化设计问题。根据汽车吸能结构厚度、材料参数等不确定参数类型,分别采用概率和多椭球凸模型进行描述,以汽车正面碰撞安全性可靠性指标为约束,考虑汽车吸能结构质量为优化目标,建立了一种基于混合模型的可靠性优化设计模型。采用拉丁方试验设计构造了目标函数和约束函数的Kriging近似模型,利用功能度量法求解可靠度指标值,通过基于移动因子序列优化与可靠性评定将嵌套优化解耦为单层次优化。实际算例表明算法具有较高的计算效率及精度,对实际设计工作有一定参考价值。     

Reliability and cost optimization of electronic devices considering the component failure rate uncertainty

The objective of this paper is to present an efficient computational methodology to obtain the optimal system structure of electronic devices by using either a single or a multiobjective optimization approach, while considering the constraints on reliability and cost. The component failure rate uncertainty is taken under consideration and it is modeled with two alternative probability distribution functions. The Latin hypercube sampling method is used to simulate the probability distributions. An optimization approach was developed using the simulated annealing algorithm because of its flexibility to be applied in various system types with several constraints and its efficiency in computational time. This optimization approach can handle efficiently either the single or the multiobjective optimization modeling of the system design. The developed methodology was applied to a power electronic device and the results were compared with the results of the complete enumeration of the solution space. The stochastic nature of the best solutions for the single objective optimization modeling of the system design was sampled extensively and the robustness of the developed optimization approach was demonstrated.     

Dual-mode nested search method for categorical uncertain multi-objective optimization

Categorical multi-objective optimization is an important issue involved in many matching design problems. Non-numerical variables and their uncertainty are the major challenges of such optimizations. Therefore, this article proposes a dual-mode nested search (DMNS) method. In the outer layer, kriging metamodels are established using standard regular simplex mapping (SRSM) from categorical candidates to numerical values. Assisted by the metamodels, a k-cluster-based intelligent sampling strategy is developed to search Pareto frontier points. The inner layer uses an interval number method to model the uncertainty of categorical candidates. To improve the efficiency, a multi-feature convergent optimization via most-promising-area stochastic search (MFCOMPASS) is proposed to determine the bounds of objectives. Finally, typical numerical examples are employed to demonstrate the effectiveness of the proposed DMNS method.     

Efficient hybrid evolutionary algorithm for optimization of a strip coiling process

This article proposes an efficient metaheuristic based on hybridization of teaching–learning-based optimization and differential evolution for optimization to improve the flatness of a strip during a strip coiling process. Differential evolution operators were integrated into the teaching–learning-based optimization with a Latin hypercube sampling technique for generation of an initial population. The objective function was introduced to reduce axial inhomogeneity of the stress distribution and the maximum compressive stress calculated by Love's elastic solution within the thin strip, which may cause an irregular surface profile of the strip during the strip coiling process. The hybrid optimizer and several well-established evolutionary algorithms (EAs) were used to solve the optimization problem. The comparative studies show that the proposed hybrid algorithm outperformed other EAs in terms of convergence rate and consistency. It was found that the proposed hybrid approach was powerful for process optimization, especially with a large-scale design problem.     

A Tutorial on Latin Hypercube Design of Experiments

The growing power of computers enabled techniques created for design and analysis of simulations to be applied to a large spectrum of problems and to reach high level of acceptance among practitioners. Generally, when simulations are time consuming, a surrogate model replaces the computer code in further studies (e.g., optimization, sensitivity analysis, etc.). The first step for a successful surrogate modeling and statistical analysis is the planning of the input configuration that is used to exercise the simulation code. Among the strategies devised for computer experiments, Latin hypercube designs have become particularly popular. This paper provides a tutorial on Latin hypercube design of experiments, highlighting potential reasons of its widespread use. The discussion starts with the early developments in optimization of the point selection and goes all the way to the pitfalls of the indiscriminate use of Latin hypercube designs. Final thoughts are given on opportunities for future research. Copyright © 2015 John Wiley & Sons, Ltd.     

基于Kriging代理模型的拉压不同模量平面问题的近似求解

该文建议采用Kriging代理模型数值求解拉压不同模量平面问题。通过本构方程光滑化、有限元法及拉丁超立方采样技术,对拉压不同模量桁架与二维平面问题,给出了基于Kriging模型的近似数值解,以代理基于有限元的数值解,并探讨了样本点数目和问题规模对所建Kriging近似模型求解精度/效率的影响。数值算例表明:所提方法可为求解拉压不同模量平面问题提供精度合理的近似数值解。当问题规模较大且正问题需要多次求解时,该方法有望显著减少计算时间,这对于降低拉压不同模量反问题与优化问题的计算开销十分重要。