Optimal design of aeroengine turbine disc based on kriging surrogate models

A design optimization method based on kriging surrogate models is proposed and applied to the shape optimization of an aeroengine turbine disc. The kriging surrogate model is built to provide rapid approximations of time-consuming computations. For improving the accuracy of surrogate models without significantly increasing computational cost, a rigorous sample selection is employed to reduce additional design samples based on design of experiments over a sequential trust region. The minimum-mass shape design of turbine discs under thermal and mechanical loads has demonstrated the effectiveness and efficiency of the presented optimization approach.     

一种求解无约束优化问题的非单调信赖域算法

提出了非单调信赖域算法求解基于锥模型的无约束优化问题,该算法在求解信赖域子问题时充分利用了当前迭代点的一阶梯度信息。提出了一个新的信赖域半径的选取机制,并和经典的信赖域方法作比较分析。设定了一些条件,在这些假设条件下证明了算法是整体收敛的。数值实验结果表明,该算法对基于锥模型的无约束优化问题的求解是行之有效的,拓展了非单调信赖域算法的应用领域。     

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.     

On stochastic gradient and subgradient methods with adaptive steplength sequences

Traditionally, stochastic approximation (SA) schemes have been popular choices for solving stochastic optimization problems. However, the performance of standard SA implementations can vary significantly based on the choice of the steplength sequence, and in general, little guidance is provided about good choices. Motivated by this gap, we present two adaptive steplength schemes for strongly convex differentiable stochastic optimization problems, equipped with convergence theory, that aim to overcome some of the reliance on user-specific parameters. The first scheme, referred to as a recursive steplength stochastic approximation (RSA) scheme, optimizes the error bounds to derive a rule that expresses the steplength at a given iteration as a simple function of the steplength at the previous iteration and certain problem parameters. The second scheme, termed as a cascading steplength stochastic approximation (CSA) scheme, maintains the steplength sequence as a piecewise-constant decreasing function with the reduction in the steplength occurring when a suitable error threshold is met. Then, we allow for nondifferentiable objectives but with bounded subgradients over a certain domain. In such a regime, we propose a local smoothing technique, based on random local perturbations of the objective function, that leads to a differentiable approximation of the function. Assuming a uniform distribution on the local randomness, we establish a Lipschitzian property for the gradient of the approximation and prove that the obtained Lipschitz bound grows at a modest rate with problem size. This facilitates the development of an adaptive steplength stochastic approximation framework, which now requires sampling in the product space of the original measure and the artificially introduced distribution.     

An efficient multi-objective optimization method for black-box functions using sequential approximate technique

Multi-objective optimization problems in practical engineering usually involve expensive black-box functions. How to reduce the number of function evaluations at a good approximation of Pareto frontier has been a crucial issue. To this aim, an efficient multi-objective optimization method based on a sequential approximate technique is suggested in this paper. In each iteration, according to the prediction of radial basis function with a micro multi-objective genetic algorithm, an extended trust region updating strategy is adopted to adjust the design region, a sample inheriting strategy is presented to reduce the number of new function evaluations, and then a local-densifying strategy is proposed to improve the accuracy of approximations in concerned regions. At the end of each iteration, the obtained actual Pareto optimal points are stored in an external archive and are updated as the iteration process. The effect of the present method is demonstrated by eight test functions. Finally, it is employed to perform the structure optimization of a vehicle door.     

Sequential integer programming methods for stress constrained topology optimization

This paper deals with topology optimization of load carrying structures defined on a discretized design domain where binary design variables are used to indicate material or void in the various finite elements. The main contribution is the development of two iterative methods which are guaranteed to find a local optimum with respect to a 1-neighbourhood. Each new iteration point is obtained as the optimal solution to an integer linear programming problem which is an approximation of the original problem at the previous iteration point. The proposed methods are quite general and can be applied to a variety of topology optimization problems defined by 0-1 design variables. Most of the presented numerical examples are devoted to problems involving stresses which can be handled in a natural way since the design variables are kept binary in the subproblems.     

Metamodel-based collaborative optimization framework

This paper focuses on the metamodel-based collaborative optimization (CO). The objective is to improve the computational efficiency of CO in order to handle multidisciplinary design optimization problems utilising high fidelity models. To address these issues, two levels of metamodel building techniques are proposed: metamodels in the disciplinary optimization are based on multi-fidelity modelling (the interaction of low and high fidelity models) and for the system level optimization a combination of a global metamodel based on the moving least squares method and trust region strategy is introduced. The proposed method is demonstrated on a continuous fiber-reinforced composite beam test problem. Results show that methods introduced in this paper provide an effective way of improving computational efficiency of CO based on high fidelity simulation models.     

An approximate optimization strategy using refined hybrid metamodel

An approximate model called metamodel or surrogate model is a mathematical model that numerically approximates response of a system during an engineering simulation process or test. The introduction of a metamodel makes it possible to express response defined in the design problem as a simple mathematical function of design variables. A metamodel can be built with response surface method (RSM), kriging, neural network, radial basis function, and so on. Each method has its advantages and disadvantages. A combined metamodel called hybrid model, ensemble model, or multiple surrogates has been developed to maximize each metamodel's strength. The hybrid model of this research includes RSM and kriging. Besides, a strategy to refine the hybrid metamodel is implemented by reducing design space. In this process, information related to Hessian is utilized for an unconstrained optimization problem, on the contrary feasibility for a constrained optimization problem. This research presents a new hybrid metamodel-based optimization strategy called refined hybrid metamodel. Five mathematical test problems, two-bar design, spring design, and propeller shaft design problems are solved with the suggested method, verifying its usefulness. Most of the optimal results with the proposed method are closer to exact solutions with smaller function evaluations than existing methods.     

Fast simulation‐driven antenna design using response‐feature surrogates

In this article, a computationally efficient procedure for electromagnetic (EM)‐simulation‐driven design of antennas is presented. Our methodology is based on local approximation models of the antenna response, established using a set of suitably selected characteristic features rather than the entire response (such as reflection versus frequency). The approximation model is utilized to verify the level of satisfying/violating given performance requirements, and to guide the optimization process towards a better design. By exploiting the fact that the dependence of the response features on the designable parameters of the antenna of interest is simple (close to linear or quadratic), the feature‐based optimization converges faster than conventional optimization of frequency‐based EM‐simulated responses. In order to further speed up the design, coarse‐discretization simulations are utilized to estimate the feature gradients with respect to adjustable parameters of the problem at hand. The optimization algorithm is embedded in the trust‐region framework for safeguarding convergence. The proposed technique is demonstrated using two antenna examples. In both the cases, the optimum design is obtained at the computational cost corresponding to a few high‐fidelity EM antenna simulations. © 2014 Wiley Periodicals, Inc. Int J RF and Microwave CAE 25:394–402, 2015.     

一种求解约束优化问题的混合算法

提出一种基于修改增广Lagrange函数和PSO的混合算法用于求解约束优化问题。将约束优化问题转化为界约束优化问题,混合算法由两层迭代结构组成,在内层迭代中,利用改进PSO算法求解界约束优化问题得到下一个迭代点。外层迭代主要修正Lagrange乘子和罚参数,检查收敛准则是否满足,重构下次迭代的界约束优化子问题,检查收敛准则是否满足。数值实验结果表明该混合算法的有效性。     

基于信任迭代的P2P网络信任管理模型

多数P2P网络信任管理模型无法准确计算节点间的推荐信任值,且节点交易过程中不能有效防止恶意推荐.为此,提出一种基于信任迭代的信任管理模型,通过引入信任迭代、推荐可信度和迭代信任值的概念,根据节点间的直接交易经验计算节点间的推荐信任值,将推荐链划分为主链和副链,从而更全面地参考推荐信息,减小因推荐链的取舍对推荐信任值造成的影响,并给出一种新的推荐信任值迭代计算方法,使计算结果更合理.仿真实验结果表明,该模型能够准确地计算推荐信任值,抑制恶意推荐行为.     

Multiple meta-models based design space differentiation method for expensive problems

Meta-models and meta-models based global optimization methods have been commonly used in design optimizations of expensive problems. In this work, a multiple meta-models based design space differentiation (MDSD) method is proposed. In the proposed method, an important region will be constructed using the expensive points inside the whole design space. Then, quadratic function (QF) will be employed in the search of the constructed important region. To avoid the local optima, kriging is employed in the search of the whole design space simultaneously. The MDSD method employs different meta-models in the different design space instead of space reduction, which preserves the advantages of high efficiency of the space reduction methods and avoids their shortcomings of removing the global optimum by mistake in theory. Through extensive test and comparison with three meta-model based algorithms, efficient global optimization (EGO), Mode-pursuing sampling method (MPS) and hybrid and adaptive meta-modeling method (HAM) using several benchmark math functions and an engineering problem involving finite element analysis (FEA), the proposed method shows excellent performance in search efficiency and accuracy.     

Accuracy,Speed and Robustness of Policy Function Iteration

Policy function iteration methods for solving and analyzing dynamic stochastic general equilibrium models are powerful from a theoretical and computational perspective. Despite obvious theoretical appeal, significant startup costs and a reliance on grid-based methods have limited the use of policy function iteration as a solution algorithm. We reduce these costs by providing a user-friendly suite of MATLAB functions that introduce multi-core processing and Fortran via MATLAB’s executable function. Within the class of policy function iteration methods, we advocate using time iteration with linear interpolation. We examine a canonical real business cycle model and a new Keynesian model that features regime switching in policy parameters, Epstein–Zin preferences, and monetary policy that occasionally hits the zero-lower bound on the nominal interest rate to highlight the attractiveness of our methodology. We compare our advocated approach to other familiar iteration and approximation methods, highlighting the tradeoffs between accuracy, speed and robustness.     

Surrogate modeling approximation using a mixture of experts based on EM joint estimation

An automatic method to combine several local surrogate models is presented. This method is intended to build accurate and smooth approximation of discontinuous functions that are to be used in structural optimization problems. It strongly relies on the Expectation−Maximization (EM) algorithm for Gaussian mixture models (GMM). To the end of regression, the inputs are clustered together with their output values by means of parameter estimation of the joint distribution. A local expert is then built (linear, quadratic, artificial neural network, moving least squares) on each cluster. Lastly, the local experts are combined using the Gaussian mixture model parameters found by the EM algorithm to obtain a global model. This method is tested over both mathematical test cases and an engineering optimization problem from aeronautics and is found to improve the accuracy of the approximation.     

A new unconstrained global optimization method based on clustering and parabolic approximation

A new unconstrained global optimization method based on clustering and parabolic approximation (GOBC-PA) is proposed. Although the proposed method is basically similar to other evolutionary and stochastic methods, it represents a significant advancement of global optimization technology for four important reasons. First, it is orders of magnitude faster than existing optimization methods for global optimization of unconstrained problems. Second, it has significantly better repeatability, numerical stability, and robustness than current methods in dealing with high dimensionally and many local minima functions. Third, it can easily and faster find the local minimums using the parabolic approximation instead of gradient descent or crossover operations. Fourth, it can easily adapted to any theoretical or industrial systems which are using the heuristic methods as an intelligent system, such as neural network and neuro-fuzzy inference system training, packing or allocation of objects, game optimization problems. In this study, we assume that the best cluster center gives the position of the possible global optimum. The usage of clustering and curve fitting techniques brings multi-start and local search properties to the proposed method. The experimental studies, such as performed on benchmark functions, a real world optimization problem and tuning the neural network parameters for classification problems, show that the proposed methodology is simple, faster and, it demonstrates a superior performance when compared with some state of the art methods.     

Optimization of the new Saab 9-3 exposed to impact load using a space mapping technique

The aim of this work is to illustrate how a space mapping technique using surrogate models together with response surfaces can be used for structural optimization of crashworthiness problems. To determine the response surfaces, several functional evaluations must be performed and each evaluation can be computationally demanding. The space mapping technique uses surrogate models, i.e. less costly models, to determine these surfaces and their associated gradients. The full model is used to correct the gradients from the surrogate model for the next iteration. Thus, the space mapping technique makes it possible to reduce the total computing time needed to find the optimal solution. First, two analytical functions and one analytical structural optimization problem are presented to exemplify the idea of space mapping and to compare the efficiency of space mapping to traditional response surface optimization. Secondly, a sub-model of a complete vehicle finite element (FE) model is used to study different objective functions in vehicle crashworthiness optimization. Finally, the space mapping technique is applied to a structural optimization problem of a large industrial FE vehicle model, consisting of 350.000 shell elements and a computing time of 100 h. In this problem the intrusion in the passenger compartment area was reduced by 32% without compromising other crashworthiness parameters.     

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.     

基于响应面和空间映射技术的模面设计优化

阐述了响应面近似模型和空间映射技术用于快速模面设计优化的基本原理．为避免数值噪声干扰和求解隐函数敏度,提出通过多项式响应面方法构造真实目标和约束的逼近曲面,以光滑响应进行全局最优;并利用空间映射技术的代理模型确定新的响应面及其梯度,经精细模型修正优化方向和设计子域,使计算成本降低．算法结合有限元模拟实施,通过对金钣拉延成形和回弹补偿的分析证明,该算法具有很高的效率和鲁棒性,适用于模面设计优化．     

Interval-based global optimization in engineering using model reformulation and constraint propagation

This paper deals with the preliminary design problem when the product is modeled as an analytic model. The analytic models based method aims to use mathematical equations to address both multi-physic and economic characteristics of a product. The proposed approach is to convert the preliminary design problem into a global constrained optimization problem. The objective is to develop powerful optimization methods enough to handle complex analytical models. We propose to adapt an approach to solve this problem based on interval analysis, constraint propagation and model reformulation. In order to understand the optimization algorithm used for engineering design problems, some basic definitions and properties of interval analysis are introduced. Then, the basic optimization algorithms for both unconstrained and constrained problems are introduced and illustrated. The next section introduces the reformulation technique as main accelerating device. An application of the reformulation device and its global optimization algorithm on the optimal design of electrical actuators is presented.     

A nonlinear interval-based optimization method with local-densifying approximation technique

In this paper, a new method is proposed to promote the efficiency and accuracy of nonlinear interval-based programming (NIP) based on approximation models and a local-densifying method. In conventional NIP methods, searching for the response bounds of objective and constraints are required at each iteration step, which forms a nested optimization and leads to extremely low efficiency. In order to reduce the computational cost, approximation models based on radial basis functions (RBF) are used to replace the actual computational models. A local-densifying method is suggested to guarantee the accuracy of the approximation models by reconstructing them with densified samples in iterations. Thus, through a sequence of optimization processes, an optimal result with fine accuracy can be finally achieved. Two numerical examples are used to test the effectiveness of the present method, and it is then applied to a practical engineering problem.