Improving multi-objective genetic algorithms with adaptive design of experiments and online metamodeling

Applications of multi-objective genetic algorithms (MOGAs) in engineering optimization problems often require numerous function calls. One way to reduce the number of function calls is to use an approximation in lieu of function calls. An approximation involves two steps: design of experiments (DOE) and metamodeling. This paper presents a new approach where both DOE and metamodeling are integrated with a MOGA. In particular, the DOE method reduces the number of generations in a MOGA, while the metamodeling reduces the number of function calls in each generation. In the present approach, the DOE locates a subset of design points that is estimated to better sample the design space, while the metamodeling assists in estimating the fitness of design points. Several numerical and engineering examples are used to demonstrate the applicability of this new approach. The results from these examples show that the proposed improved approach requires significantly fewer function calls and obtains similar solutions compared to a conventional MOGA and a recently developed metamodeling-assisted MOGA.     

Data driven surrogate-based optimization in the problem solving environment WBCSim

Large scale, multidisciplinary, engineering designs are always difficult due to the complexity and dimensionality of these problems. Direct coupling between the analysis codes and the optimization routines can be prohibitively time consuming due to the complexity of the underlying simulation codes. One way of tackling this problem is by constructing computationally cheap(er) approximations of the expensive simulations that mimic the behavior of the simulation model as closely as possible. This paper presents a data driven, surrogate-based optimization algorithm that uses a trust region-based sequential approximate optimization (SAO) framework and a statistical sampling approach based on design of experiment (DOE) arrays. The algorithm is implemented using techniques from two packages—SURFPACK and SHEPPACK that provide a collection of approximation algorithms to build the surrogates and three different DOE techniques—full factorial (FF), Latin hypercube sampling, and central composite design—are used to train the surrogates. The results are compared with the optimization results obtained by directly coupling an optimizer with the simulation code. The biggest concern in using the SAO framework based on statistical sampling is the generation of the required database. As the number of design variables grows, the computational cost of generating the required database grows rapidly. A data driven approach is proposed to tackle this situation, where the trick is to run the expensive simulation if and only if a nearby data point does not exist in the cumulatively growing database. Over time the database matures and is enriched as more and more optimizations are performed. Results show that the proposed methodology dramatically reduces the total number of calls to the expensive simulation runs during the optimization process.     

Cross-validation based single response adaptive design of experiments for Kriging metamodeling of deterministic computer simulations

A new approach for single response adaptive design of deterministic computer experiments is presented. The approach is called SFCVT, for Space-Filling Cross-Validation Tradeoff. SFCVT uses metamodeling to obtain an estimate of cross-validation errors, which are maximized subject to a constraint on space filling to determine sample points in the design space. The proposed method is compared, using a test suite of forty four numerical examples, with three DOE methods from the literature. The numerical test examples can be classified into symmetric and asymmetric functions. Symmetric examples refer to functions for which the extreme points are located symmetrically in the design space and asymmetric examples are those for which the extreme regions are not located in a symmetric fashion in the design space. Based upon the comparison results for the numerical examples, it is shown that SFCVT performs better than an existing adaptive and a non-adaptive DOE method for asymmetric multimodal functions with high nonlinearity near the boundary, and is comparable for symmetric multimodal functions and other test problems. The proposed approach is integrated with a multi-scale heat exchanger optimization tool to reduce the computational effort involved in the design of novel air-to-water heat exchangers. The resulting designs are shown to be significantly more compact than mainstream heat exchanger designs.     

A discrete post-processing method for structural optimization

This paper introduces a discrete variable post-processing method for structural design optimization. The motivation behind the method is to find a good discrete solution at manageable cost while the traditional discrete optimization algorithms are regarded as impractical for large-scale structural design problems. In this paper, the Design of Experiments (DOE) and Conservative Discrete Design (CDD) approaches have been proposed to deal with discrete variables with limited computational cost. Both methods work on the explicit approximate discreteproblem to explore the discrete design. These two approaches, together with engineering rounded-off methods, can be used to process discrete variables at any specified continuous design optimization cycle for structural design problems. Brief background and a theoretical discussion about these approaches are given in this paper. Finally, the methods that have been implemented in MSC.Nastran are demonstrated by academic and real engineering examples.     

A neural composite dynamic surface control for pure‐feedback systems with unknown control gain signs and full state constraints

This paper investigates a composite neural dynamic surface control (DSC) method for a class of pure‐feedback nonlinear systems in the case of unknown control gain signs and full‐state constraints. Neural networks are utilized to approximate the compound unknown functions, and the approximation errors of neural networks are applied in the design of updated adaptation laws. Comparing the proposed composite approximation method with the conventional ones, a faster and better approximation performance result can be obtained. Combining the composite neural networks approximation with the DSC technique, an improved composite neural adaptive control approach is designed for the considered nonlinear system. Then, together with the Lyapunov stability theory, all the variables of the closed‐loop system are semiglobal uniformly ultimately bounded. The infringements of full state constraints can be avoided in the case of unknown control gain signs as well as unknown disturbances. Finally, two simulation examples show the effectiveness and feasibility of the proposed results.     

Multilevel design of experiments for statistical moment and probability calculation

A procedure for statistical moment estimation and reliability analysis using design of experiment (DOE) is proposed. A numerical method of finding the optimal levels and weights of DOE for statistical moment estimation is established and applied to three- and five-level cases. The four statistical moments of the system response function are then calculated from the full-factorial DOE, and the probability distribution of the system response function is obtained using the empirical distribution systems such as the Pearson system. The proposed method is tested through several examples and compared with other analysis methods, including the previous developments of a three-level full-factorial design. The results show that it relieves much of the difficulties met in the previous method and provides good accuracy compared to other methods for various input distributions.     

Adaptive dynamic surface control using disturbance observer for nonlinear systems with input saturation and output constraints

An adaptive dynamic surface control (DSC) approach using fuzzy approximation and nonlinear disturbance observer (NDO) for uncertain nonlinear systems in the presence of input saturation, output constraint and unknown external disturbances is proposed in this paper. The issue of input saturation is addressed by introducing a lower bound assumption on the approximation function of saturation. The output constraint is handled by introducing an appropriate barried Lyapunov function. The nonlinear disturbance observer (NDO) is employed to estimate the unknown unmatched disturbances. It is manifested that the ultimately bounded convergence of all the variables in the closed-loop system is guaranteed and the tracking error can be made farely small by tuning the design parameters. Finally, two simulation examples illustrate the effectiveness and feasibility of the proposed approach.     

A systematic optimization approach for assembly sequence planning using Taguchi method, DOE, and BPNN

Research in assembly planning can be categorised into three types of approach: graph-based, knowledge-based and artificial intelligence approaches. The main drawbacks of the above approaches are as follows: the first is time-consuming; in the second approach it is difficult to find the optimal solution; and the third approach requires a high computing efficiency. To tackle these problems, this study develops a novel approach integrated with some graph-based heuristic working rules, robust back-propagation neural network (BPNN) engines via Taguchi method and design of experiment (DOE), and a knowledge-based engineering (KBE) system to assist the assembly engineers in promptly predicting a near-optimal assembly sequence. Three real-world examples are dedicated to evaluating the feasibility of the proposed model in terms of the differences in assembly sequences. The results show that the proposed model can efficiently generate BPNN engines, facilitate assembly sequence optimisation and allow the designers to recognise the contact relationships, assembly difficulties and assembly constraints of three-dimensional (3D) components in a virtual environment type.     

ANN inverse analysis based on stochastic small-sample training set simulation

A new approach of inverse analysis is proposed to obtain parameters of a computational model in order to achieve the best agreement with experimental data. The inverse analysis is based on the coupling of a stochastic simulation and an artificial neural network (ANN). The identification parameters play the role of basic random variables with a scatter reflecting the physical range of potential values. A novelty of the approach is the utilization of the efficient small-sample simulation method Latin Hypercube Sampling (LHS) used for the stochastic preparation of the training set utilized in training the artificial neural network. Once the network has been trained, it represents an approximation consequently utilized to solve the key task: To provide the best possible set of model parameters for the given experimental data. The efficiency of the approach is shown using numerical examples from civil engineering computational mechanics.     

基于均值改进控制策略的昂贵约束并行代理优化算法

针对具有黑箱特性的昂贵约束优化问题及工程中计算资源利用率不高问题,提出了新的基于均值改进控制策略的并行代理优化算法.该算法为了减少仿真建模计算负担,选取Kriging近似模型对目标函数和约束函数进行近似估计.在Kriging模型基础上,利用均值改进与新增试验样本间的不等关系构建具有距离特性的控制函数.算法的均值改进控制...     

Topology optimization of a swing arm type actuator using the response surface method

In the frame of topology optimization, the multi-objective ability has to be considered since structural design is usually required to satisfy more than one requirement. A modified topology optimization method based on the response surface method (RSM) is proposed to generate a structure of a small form factor (SFF) swing arm type actuator satisfying maximum compliance and maximum stiffness at the same time using the multi-objective optimization approach. The multi-objective function is defined to maximize the compliance in the direction of focusing as well as the eigen-frequency of the structure. The design of experiments (DOE) is performed to select sensitive variables. Based on DOE results, the response surface functions are formulated to construct the multi-objective function. The weight factors between conflicting objective functions are determined by the Pareto optimum method. By applying the optimal combination of design variables to the design domain, the optimized topology can be obtained.This work was supported by Korea Research Foundation (KRF) Grant KRF-2004-042-D00004.     

Approximate techniques of strctural reanalysis

A study is made of two approximate techniques for structural reanalysis. These include Taylor series expansions for response variables in terms of design variables and the reduced basis method. In addition, modifications to these techniques are proposed to overcome some of their major drawbacks. The modifications include a rational approach for the selection of the reduced basis vectors and the use of Taylor series approximation in an iterative process. For the reduced basis a normalized set of vectors is chosen which consist of the original analyzed design and the first-order sensitivity analysis vectors.The use of the Taylor series approximation as a first (initial) estimate in an iterative process, can lead to significant improvements in accuracy, even with one iteration cycle. Therefore, the range of applicability of the reanalysis technique can be extended.Numerical examples are presented of space truss structures. These examples demonstrate the gain in accuracy obtained by using the proposed modification techniques, for a wide range of variations in the design variables.     

A hybrid surrogate and pattern search optimization method and application to microelectronics

A hybrid method for robust and efficient optimization process is developed by integrating a new response surface method and pattern search algorithm. The method is based on: (1) multipoint approximations of the objective and constraint functions, (2) a multiquadric radial basis function (RBF) for the zeroth-order function approximation and a new RBF plus polynomial-based moving least-squares approximation for the first-order enhanced function approximation, and (3) a pattern search algorithm to impose a descent condition and applied adaptive subregion management strategy. Several numerical examples are presented to illustrate accuracy and computational efficiency of the proposed method for both function approximation and design optimization. To demonstrate the effectiveness of the proposed hybrid method, it is applied to obtain optimum designs of a microelectronic packaging system. A two-stage optimization approach is proposed for the design optimization. The material properties of microelectronic packaging system and the shape parameters of solder ball are selected as design variables. Through design optimization, significant improvements of durability performances are obtained using the proposed hybrid optimization method.     

Likelihood-based representation of epistemic uncertainty and its application in robustness-based design optimization

In this paper, we propose a new likelihood-based methodology to represent epistemic uncertainty described by sparse point and/or interval data for input variables in uncertainty analysis and design optimization problems. A worst-case maximum likelihood-based approach is developed for the representation of epistemic uncertainty, which is able to estimate the distribution parameters of a random variable described by sparse point and/or interval data. This likelihood-based approach is general and is able to estimate the parameters of any known probability distributions. The likelihood-based representation of epistemic uncertainty is then used in the existing framework for robustness-based design optimization to achieve computational efficiency. The proposed uncertainty representation and design optimization methodologies are illustrated with two numerical examples including a mathematical problem and a real engineering problem.     

Decentralized Robust Adaptive Control for the Multiagent System Consensus Problem Using Neural Networks

A robust adaptive control approach is proposed to solve the consensus problem of multiagent systems. Compared with the previous work, the agent's dynamics includes the uncertainties and external disturbances, which is more practical in real-world applications. Due to the approximation capability of neural networks, the uncertain dynamics is compensated by the adaptive neural network scheme. The effects of the approximation error and external disturbances are counteracted by employing the robustness signal. The proposed algorithm is decentralized because the controller for each agent only utilizes the information of its neighbor agents. By the theoretical analysis, it is proved that the consensus error can be reduced as small as desired. The proposed method is then extended to two cases: agents form a prescribed formation, and agents have the higher order dynamics. Finally, simulation examples are given to demonstrate the satisfactory performance of the proposed method.     

小波神经网络学习的结构风险最小化方法

针对大噪声、小样本情形下神经网络学习的外推能力弱这一突出的问题，根据统计学习理论中结构风险最小化准则的基本原理，提出了一种基于小波神经基元频率谱分布的小波神经网络阵列结构和基于小波多分辨逼近、综合风险分析的小波网络学习算法．该方法充分发挥了小波神经网络的优点，理论基础可靠，实际意义明确，算法实现简便，自适应性强．仿真实验结果和应用实例说明了该方法对于非线性系统在线辨识的有效性，同时也为统计学习理论的工程应用提供了新的途径．     

A new two-point approximation approach for structural optimization

The objective of this work is to build up a high-quality approximation scheme to realize computational savings for the solution of structural optimization problems. To this end, a newly developed two-point approximation scheme is proposed. This scheme is constructed by the linear combination of Taylor expansions in terms of both original and reciprocal variables. The coefficients of the combination are determined by utilizing both the function and gradient information of two different design points obtained during the process of optimization. Based on this approach, the accuracy of the existing constraint approximation methods can be improved. The effectiveness of the proposed approach is demonstrated on a number of numerical examples. The numerical results are also compared with those of previously published work. Received April 2, 1999     

Analytical moment based approximation for robust design optimization

The role of robust design optimization (RDO) has been eminent, ascertaining optimal configuration of engineering systems in the presence of uncertainties. However, computational aspect of RDO can often get tediously intensive in dealing with large scale systems. To address this issue, hybrid polynomial correlated function expansion (H-PCFE) based RDO framework has been developed for solving computationally expensive problems. H-PCFE performs as a bi-level approximation tool, handling the global model behavior and local functional variation. Analytical formula for the mean and standard deviation of the responses have been proposed, which reduces significant level of computations as no further simulations are required for evaluating the statistical moments within the optimization routine. Implementation of the proposed approaches have been demonstrated with two benchmark examples and two practical engineering problems. The performance of H-PCFE and its analytical version have been assessed by comparison with direct Monte Carlo simulation (MCS). Comparison with popular state-of-the-art techniques has also been presented. Excellent results in terms of accuracy and computational effort obtained makes the proposed methodology potential for further large scale industrial applications.     

基于近似模型的整车气动外形优化

以类车体DrivAer的气动阻力因数为优化目标,选取影响其气动性能的5个形状参数作为设计变量,通过引入网格变形、试验设计(Design of Experiment,DOE)及近似模型等技术搭建自动仿真优化平台,探索气动性能最佳的参数匹配方案.优化后的DrivAer气动阻力因数降低4.5%,表明近似模型方法能够较好地取代实际仿真过程进行寻优.分析DOE的结果,发现影响气动阻力因数和气动升力因数的主要参数分别为行李箱高度与离去角,而多参数变化时的交互效应也会影响整车的气动性能.     

Reliability-based robust assessment for multiobjective optimization design of improving occupant restraint system performance

Optimal performance of vehicle occupant restraint system (ORS) requires an accurate assessment of occupant injury values including head, neck and chest responses, etc. To provide a feasible framework for incorporating occupant injury characteristics into the ORS design schemes, this paper presents a reliability-based robust approach for the development of the ORS. The uncertainties of design variables are addressed and the general formulations of reliable and robust design are given in the optimization process. The ORS optimization is a highly nonlinear and large scale problem. In order to save the computational cost, an optimal sampling strategy is applied to generate sample points at the stage of design of experiment (DOE). Further, to efficiently obtain a robust approximation, the support vector regression (SVR) is suggested to construct the surrogate model in the vehicle ORS design process. The multiobjective particle swarm optimization (MPSO) algorithm is used for obtaining the Pareto optimal set with emphasis on resolving conflicting requirements from some of the objectives and the Monte Carlo simulation (MCS) method is applied to perform the reliability and robustness analysis. The differences of three different Pareto fronts of the deterministic, reliable and robust multiobjective optimization designs are compared and analyzed in this study. Finally, the reliability-based robust optimization result is verified by using sled system test. The result shows that the proposed reliability-based robust optimization design is efficient in solving ORS design optimization problems.