Construction of nested maximin designs based on successive local enumeration and modified novel global harmony search algorithm

Engineering design often involves different types of simulation, which results in expensive computational costs. Variable fidelity approximation-based design optimization approaches can realize effective simulation and efficiency optimization of the design space using approximation models with different levels of fidelity and have been widely used in different fields. As the foundations of variable fidelity approximation models, the selection of sample points of variable-fidelity approximation, called nested designs, is essential. In this article a novel nested maximin Latin hypercube design is constructed based on successive local enumeration and a modified novel global harmony search algorithm. In the proposed nested designs, successive local enumeration is employed to select sample points for a low-fidelity model, whereas the modified novel global harmony search algorithm is employed to select sample points for a high-fidelity model. A comparative study with multiple criteria and an engineering application are employed to verify the efficiency of the proposed nested designs approach.     

Hybrid metamodel-based design space management method for expensive problems

Metamodel-based global optimization methods have been extensively studied for their great potential in solving expensive problems. In this work, a design space management strategy is proposed to improve the accuracy and efficiency of metamodel-based optimization methods. In this strategy, the whole design space is divided into two parts: the important region constructed using several expensive points and the other region. Combined with a previously developed hybrid metamodel strategy, a hybrid metamodel-based design space management method (HMDSM) is developed. In this method, three representative metamodels are used simultaneously in the search of the global optimum in both the important region and the other region. In the search process, the important region is iteratively reduced and the global optimum is soon captured. Tests using a series of benchmark mathematical functions and a practical expensive problem demonstrate the excellent performance of the proposed method.     

Sequential latin hypercube design with both space‐filling and projective properties

Space‐filling and projective properties are probably the two most important features in computer experiment. The existing research works have tried to develop different kinds of sequential Latin hypercube design (LHD) to meet these two properties. However, most if not all of them cannot simultaneously ensure these two properties in their versions of sequential LHD. In this paper, we propose a novel sequential LHD that can simultaneously meet the space‐filling and the projective properties at each stage. A search algorithm is employed to find how many design points should be added in each stage to ensure the projective property; and the “Maximin" criterion is used to meet the space‐filling property. Two kinds of examples for low dimension and higher dimension are presented to illustrate how these sequential sampling processes are realized. The proposed method can be applied to the areas where computationally expensive simulations are involved.     

An adaptive metamodel-based global optimization algorithm for black-box type problems

In this article, an adaptive metamodel-based global optimization (AMGO) algorithm is presented to solve unconstrained black-box problems. In the AMGO algorithm, a type of hybrid model composed of kriging and augmented radial basis function (RBF) is used as the surrogate model. The weight factors of hybrid model are adaptively selected in the optimization process. To balance the local and global search, a sub-optimization problem is constructed during each iteration to determine the new iterative points. As numerical experiments, six standard two-dimensional test functions are selected to show the distributions of iterative points. The AMGO algorithm is also tested on seven well-known benchmark optimization problems and contrasted with three representative metamodel-based optimization methods: efficient global optimization (EGO), GutmannRBF and hybrid and adaptive metamodel (HAM). The test results demonstrate the efficiency and robustness of the proposed method. The AMGO algorithm is finally applied to the structural design of the import and export chamber of a cycloid gear pump, achieving satisfactory results.     

An adaptive sampling method for variable-fidelity surrogate models using improved hierarchical kriging

Variable-fidelity (VF) modelling methods have been widely used in complex engineering system design to mitigate the computational burden. Building a VF model generally includes two parts: design of experiments and metamodel construction. In this article, an adaptive sampling method based on improved hierarchical kriging (ASM-IHK) is proposed to refine the improved VF model. First, an improved hierarchical kriging model is developed as the metamodel, in which the low-fidelity model is varied through a polynomial response surface function to capture the characteristics of a high-fidelity model. Secondly, to reduce local approximation errors, an active learning strategy based on a sequential sampling method is introduced to make full use of the already required information on the current sampling points and to guide the sampling process of the high-fidelity model. Finally, two numerical examples and the modelling of the aerodynamic coefficient for an aircraft are provided to demonstrate the approximation capability of the proposed approach, as well as three other metamodelling methods and two sequential sampling methods. The results show that ASM-IHK provides a more accurate metamodel at the same simulation cost, which is very important in metamodel-based engineering design problems.     

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.     

Batch sequential design to achieve predictive maturity with calibrated computer models

Sequential experiment design strategies have been proposed for efficiently augmenting initial designs to solve many problems of interest to computer experimenters, including optimization, contour and threshold estimation, and global prediction. We focus on batch sequential design strategies for achieving maturity in global prediction of discrepancy inferred from computer model calibration. Predictive maturity focuses on adding field experiments to efficiently improve discrepancy inference. Several design criteria are extended to allow batch augmentation, including integrated and maximum mean square error, maximum entropy, and two expected improvement criteria. In addition, batch versions of maximin distance and weighted distance criteria are developed. Two batch optimization algorithms are considered: modified Fedorov exchange and a binning methodology motivated by optimizing augmented fractional factorial skeleton designs.     

Sheet forming optimization based on least square support vector regression and intelligent sampling approach

In this paper, a metamodel-based optimization method by integration of support vector regression (SVR) and intelligent sampling strategy is applied to optimize sheet forming design. Compared with other popular metamodeling techniques, the SVR is based on the principle of structure risk minimization (SRM) as opposed to the principle of the empirical risk minimization in conventional regression techniques. Thus, the accuracy and robust metamodel can be obtained. The intelligent sampling strategy is a kind of design of experiment (DOE) essentially. The characteristic of this method is to generate new sample automatically by responses of objective functions. Compared with traditional DOE methods, the number of samples isn’t constant according to different cases. Furthermore, the number of samples and size of design space can be well controlled according to the intelligent strategy. To minimize both objective functions of wrinkling, crack and thickness deformation efficiently, the proposed method is employed as a fast analysis tool to surrogate the time-consuming finite-element (FE) procedure in the iterations of optimization algorithm. An example is studied to illustrate the application of the approach proposed, and it is concluded that the proposed method is feasible for sheet forming optimization.     

Sequential algorithms for structural design optimization under tolerance conditions

A deterministic optimization usually ignores the effects of uncertainties in design variables or design parameters on the constraints. In practical applications, it is required that the optimum solution can endure some tolerance so that the constraints are still satisfied when the solution undergoes variations within the tolerance range. An optimization problem under tolerance conditions is formulated in this article. It is a kind of robust design and a special case of a generalized semi-infinite programming (GSIP) problem. To overcome the deficiency of directly solving the double loop optimization, two sequential algorithms are then proposed for obtaining the solution, i.e. the double loop optimization is solved by a sequence of cycles. In each cycle a deterministic optimization and a worst case analysis are performed in succession. In sequential algorithm 1 (SA1), a shifting factor is introduced to adjust the feasible region in the next cycle, while in sequential algorithm 2 (SA2), the shifting factor is replaced by a shifting vector. Several examples are presented to demonstrate the efficiency of the proposed methods. An optimal design result based on the presented method can endure certain variation of design variables without violating the constraints. For GSIP, it is shown that SA1 can obtain a solution with equivalent accuracy and efficiency to a local reduction method (LRM). Nevertheless, the LRM is not applicable to the tolerance design problem studied in this article.     

Metamodel-based inverse method for parameter identification: elastic–plastic damage model

This article proposed a metamodel-based inverse method for material parameter identification and applies it to elastic–plastic damage model parameter identification. An elastic–plastic damage model is presented and implemented in numerical simulation. The metamodel-based inverse method is proposed in order to overcome the disadvantage in computational cost of the inverse method. In the metamodel-based inverse method, a Kriging metamodel is constructed based on the experimental design in order to model the relationship between material parameters and the objective function values in the inverse problem, and then the optimization procedure is executed by the use of a metamodel. The applications of the presented material model and proposed parameter identification method in the standard A 2017-T4 tensile test prove that the presented elastic–plastic damage model is adequate to describe the material's mechanical behaviour and that the proposed metamodel-based inverse method not only enhances the efficiency of parameter identification but also gives reliable results.     

Hybrid surrogate-model-based multi-fidelity efficient global optimization applied to helicopter blade design

A multi-fidelity optimization technique by an efficient global optimization process using a hybrid surrogate model is investigated for solving real-world design problems. The model constructs the local deviation using the kriging method and the global model using a radial basis function. The expected improvement is computed to decide additional samples that can improve the model. The approach was first investigated by solving mathematical test problems. The results were compared with optimization results from an ordinary kriging method and a co-kriging method, and the proposed method produced the best solution. The proposed method was also applied to aerodynamic design optimization of helicopter blades to obtain the maximum blade efficiency. The optimal shape obtained by the proposed method achieved performance almost equivalent to that obtained using the high-fidelity, evaluation-based single-fidelity optimization. Comparing all three methods, the proposed method required the lowest total number of high-fidelity evaluation runs to obtain a converged solution.     

An efficient ensemble of radial basis functions method based on quadratic programming

Radial basis function (RBF) surrogate models have been widely applied in engineering design optimization problems to approximate computationally expensive simulations. Ensemble of radial basis functions (ERBF) using the weighted sum of stand-alone RBFs improves the approximation performance. To achieve a good trade-off between the accuracy and efficiency of the modelling process, this article presents a novel efficient ERBF method to determine the weights through solving a quadratic programming subproblem, denoted ERBF-QP. Several numerical benchmark functions are utilized to test the performance of the proposed ERBF-QP method. The results show that ERBF-QP can significantly improve the modelling efficiency compared with several existing ERBF methods. Moreover, ERBF-QP also provides satisfactory performance in terms of approximation accuracy. Finally, the ERBF-QP method is applied to a satellite multidisciplinary design optimization problem to illustrate its practicality and effectiveness for real-world engineering applications.     

Reduction of springback by intelligent sampling-based LSSVR metamodel-based optimization

This paper deals with the variable blank holder force in sheet metal forming in order to reduce springback effects after forming. A structural risk minimization principle-based metamodeling technique, least square support vector regression (LSSVR) method is applied to optimization. In order to improve the efficiency, an intelligent sampling strategy proposed by Wang et al. (Mater Des 30:1468–1479, 2009a) is integrated with the LSSVR. Therefore, the proposed strategies establish an adaptive metamodeling optimization system. The optimization procedure can be carried out automatically. To valid the flexibility of this system, the presented method is used to optimize the variable blank force parameters of the models from NUMISHEET’96 and torsion rail model. Compared with other popular metamodel-based optimization methods, the test results demonstrate the potential capability for nonlinear engineering problems.     

复杂系统的多学科设计优化综述

从设计和分析的本质出发,结合复杂系统的特点,通过分析传统设计优化流程在面对复杂系统时存在的困难和缺陷,指出多学科设计优化(multidisciplinary design optimization,MDO)方法是解决复杂系统设计优化问题的一种有效措施.在此基础上,介绍了多学科优化方法的基本思想,总结了子系统耦合方式及MDO在处理耦合时的基本方法,归纳了MDO的知识框架和主要研究内容.最后在现有研究成果的基础上,对MDO今后的研究提出了几点参考意见.     

Swarm algorithms for single- and multi-objective optimization problems incorporating sensitivity analysis

Swarm algorithms such as particle swarm optimization (PSO) are non-gradient probabilistic optimization algorithms that have been successfully applied for global searches in complex problems such as multi-peak problems. However, application of these algorithms to structural and mechanical optimization problems still remains a complex matter since local optimization capability is still inferior to general numerical optimization methods. This article discusses new swarm metaphors that incorporate design sensitivities concerning objective and constraint functions and are applicable to structural and mechanical design optimization problems. Single- and multi-objective optimization techniques using swarm algorithms are combined with a gradient-based method. In the proposed techniques, swarm optimization algorithms and a sequential linear programming (SLP) method are conducted simultaneously. Finally, truss structure design optimization problems are solved by the proposed hybrid method to verify the optimization efficiency.     

A multiple-design-point approach for reliability-based design optimization

The application of design-point-based reliability-based design optimization (RBDO) methods is hindered by the challenge of multiple-design-point problems. In this article, to improve the commonality of design-point-based RBDO methods, a novel multiple-design-point (MDP) approach is developed. The MDP approach uses the trace of the design points from consequent reliability analysis iterations to identify whether there are multiple design points, then all of the design points are used to calculate shifting vectors for the sequential optimization and reliability assessment method, and the corresponding probabilistic constraints are moved to the feasible region along these multiple shifting vectors at the same time. With multiple shifted probabilistic constraints, the design feasibility associated with this probabilistic constraint will be satisfied. Two mathematical examples, a speed reducer design and a honeycomb crashworthiness design, are presented to validate the effectiveness of the MDP method. The results show that the MDP approach is effective for handling multiple-design-point problems.     

A TWO-LEVEL DECOMPOSITION METHOD FOR DESIGN OPTIMIZATION

In this paper, a two-level decomposition method for design optimization is proposed which is an extension of the model coordination methods. The method couples the global monotonicity analysis of the first-level subproblem(s) with an optimization method (single-level method) or the second-level problem. Three classes of problems are considered where in the first-level they have: (1) one subproblem with one local variable, (2) several subproblems with one local variable, and (3) several subproblems with several local variables. Some test results have been presented which shows the improved performance of the proposed approach over a conventional single-level optimization method.     

Optimal design of multi-response experiments using semi-definite programming

Optimal design of multi-response experiments for estimating the parameters of multi-response linear models is a challenging problem. The main drawback of the existing algorithms is that they require the solution of many optimization problems in the process of generating an optimal design that involve cumbersome manual operations. Furthermore, all the existing methods generate approximate design and no method for multi-response n-exact design has been cited in the literature. This paper presents a unified formulation for multi-response optimal design problem using Semi-Definite Programming (SDP) that can generate D-, A- and E-optimal designs. The proposed method alleviates the difficulties associated with the existing methods. It solves a one-shot optimization model whose solution selects the optimal design points among all possible points in the design space. We generate both approximate and n-exact designs for multi-response models by solving SDP models with integer variables. Another advantage of the proposed method lies in the amount of computation time taken to generate an optimal design for multi-response models. Several test problems have been solved using an existing interior-point based SDP solver. Numerical results show the potentials and efficiency of the proposed formulation as compared with those of other existing methods. The robustness of the generated designs with respect to the variance-covariance matrix is also investigated.     

复合材料层合板屈曲稳定性优化设计及其灵敏度计算方法

笔者在有限元分析基础上研究了以屈曲稳定性作为约束条件或优化目标的复合材料层合板结构优化设计及其灵敏度分析方法,重点讨论了屈曲临界荷载灵敏度对内力场和载荷的依赖关系及其在铺层优化、尺寸优化和形状优化问题中的不同计算方法,并在JIFEX软件中实现了复杂结构复合材料层合板优化设计方法。数值算例验证了本文算法和程序的有效性。     

An inverse reliability analysis method for reliability-based design optimization with random and dependent interval variables constrained within ellipsoids

In practical design problems, interval variables exist. Many existing methods can handle only independent interval variables. Some interval variables, however, are dependent. In this work, dependent interval variables constrained within a multi-ellipsoid convex set are considered and incorporated into reliability-based design optimization (RBDO). An efficient RBDO method is proposed by employing the sequential single-loop procedure, which separates the coupled reliability analysis procedure from the deterministic optimization procedure. In the reliability analysis procedure, a single-loop optimization for the inverse reliability analysis is performed, and an efficient inverse reliability analysis method for searching for the worst-case most probable point (WMPP) is developed. The search method contains two stages. The first stage deals the situation where the WMPP is on the boundary of the feasible region, while the second stage accommodates the situation where the WMPP is inside the feasible region by interpolation. Three examples are used for a demonstration.