Ns-kriging based microstructural optimization applied to minimizing stochastic variation of homogenized elasticity of fiber reinforced composites

This paper describes an approximate microstructural optimization using the ns-kriging (noise-resistant smoothed kriging) method for minimizing the maximum stochastic variation of homogenized elastic properties of a composite material caused by microscopic uncertainties of component materials. Since evaluation of a stochastic characteristic of a homogenized material property such as expectation or variance will involve a high computational cost and its results include inaccuracy in using the Monte Carlo simulation, an approximation-based optimization technique is useful for solving the optimization problem considering the multiscale stochastic problem. Especially, the ns-kriging will work well in case of using inaccurate data for an unknown objective function. In order to investigate applicability and effectiveness of the proposed ns-kriging based approach to the optimization problem, it is applied to the cross-sectional shape optimization of fiber in a unidirectional FRP. From the numerical results, validity and effectiveness of the proposed approach are confirmed.     

机翼柱面梳状接头设计和参数优化

为解决工程实际中常规机翼梳状接头应力严重、结构偏重的问题,设计一种用于机翼结构对接的柱面梳状接头,并借助试验设计和Kriging代理模型技术提出其细节参数优化方法,应用试验设计法选取样本点,通过非线性有限元接触分析得到该样本点的响应,以此建立Kriging代理模型,并采用更新技术提高Kriging代理模型的精度;应用多岛遗传算法优化该代理模型并获得最优解.算例表明:采用该方法,机翼减重效果明显,优化效率提高.     

Approximate optimization of high-speed train nose shape for reducing micropressure wave

The paper deals with the nose shape design of high-speed railways to minimize the maximum micropressure wave, which is known to be mainly affected by train speed, train-to-tunnel area ratio, slenderness and shape of train nose, etc. It is advantageous to develop a proper approximate metamodel for replacing the real analysis code in the context of approximate design optimization. The study has adopted a newly introduced regression technique; the central of the paper is to develop and examine the support vector machine (SVM) for use in the sequential approximate optimization process. In the sequential approximate optimization process, Owen’s random orthogonal arrays and D-optimal design are used to generate training data for building approximate models. The paper describes how SVM works and how efficiently SVM is compared with an existing Kriging model. As a design result, the present study suggests an optimal nose shape that is an improvement over current design in terms of micropressure wave.     

基于遗传算法的Kriging模型构造与优化

相关模型参数的确定是Kriging模型构造的关键,讨论了利用传统数值优化方法,如模式搜索方法,确定相关参数存在依赖搜索起始点等缺点;利用遗传算法获得满足目标函数全局最小情况下的相关模型参数,解决了模型的构造对起始点依赖的问题;将遗传算法与改进后的Kriging模型结合,基于近似模型对系统进行全局最优化.     

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.     

A Kriging-based algorithm to optimize production systems approximated by analytical models

Production system optimization still remains a difficult problem even if fast analytical methods are used to estimate their mean performance measures. This paper addresses optimization problems in which the system performance measures are obtained from analytical methods implemented in computer codes that are usually time expensive. A global search algorithm is proposed to solve the addressed optimization problem. A Kriging metamodel is built to approximate the system performance function on the basis of the deterministic output values provided by the analytical model. Then a standard optimization method is applied on the explicit metamodel expression. The main advantages of the proposed method are its generality and ease of use. Indeed, the algorithm can be applied to optimize any production system assessable by an analytical method. Also, the Kriging technique allows contemporarily building the approximation of the unknown function and assessing its quality. Numerical results are satisfactory and prove the applicability of the method to real problems.     

Parameter optimization of the sheet metal forming process using an iterative parallel Kriging algorithm

Different numerical optimization strategies were used to find an optimized parameter setting for the sheet metal forming process. A parameterization of a time-dependent blank-holder force was used to control the deep-drawing simulation. Besides the already well-established gradient and direct search algorithms and the response surface method the novel Kriging approach was used as an optimization strategy. Results for two analytical and two sheet metal forming test problems reveal that the new Kriging approach leads to a fast and stable convergence of the optimization process. Parallel simulation is perfectly supported by this method.     

An investigation of structural optimization in crashworthiness design using a stochastic approach

In this paper the response surface methodology (RSM) and stochastic optimization (SO) are compared with regard to their efficiency and applicability in crashworthiness design. Optimization of simple analytic expressions and optimization of a front rail structure are the applications used to assess the respective qualities of both methods. A low detail vehicle structure is optimized to demonstrate the applicability of the methods in engineering practice. The investigations reveal that RSM is better compared to SO for fewer than 10–15 design variables. The convergence behaviour of SO improves compared to RSM when the number of design variables is increased. A novel zooming method is proposed which improves the convergence behaviour. A combination of both the RSM and the SO is efficient, stochastic optimization could be used in order to determine appropriate starting points for an RSM optimization, which continues the optimization. Two examples are investigated using this combined method.     

Constrained mesh optimization on boundary

This paper presents a new mesh optimization approach aiming to improve the mesh quality on the boundary. The existing mesh untangling and smoothing algorithms (Vachal et al. in J Comput Phys 196: 627–644, 2004; Knupp in J Numer Methods Eng 48: 1165–1185, 2002), which have been proved to work well to interior mesh optimization, are enhanced by adding constrains of surface and curve shape functions that approximate the boundary geometry from the finite element mesh. The enhanced constrained optimization guarantees that the boundary nodes to be optimized always move on the approximated boundary. A dual-grid hexahedral meshing method is used to generate sample meshes for testing the proposed mesh optimization approach. As complementary treatments to the mesh optimization, appropriate mesh topology modifications, including buffering element insertion and local mesh refinement, are performed in order to eliminate concave and distorted elements on the boundary. Finally, the optimization results of some examples are given to demonstrate the effectivity of the proposed approach.     

Parallel boundary and best neighbor searching sampling algorithm for drawbead design optimization in sheet metal forming

In the present paper, a Kriging-based metamodeling technique is used to minimize the risk of failure in a sheet metal forming process. The Kriging-based models are fitted to data that are obtained for larger experimental areas than the areas used in low-order polynomial regression metamodels. Therefore, computational time and memory requirement can be an obstacle for Kriging for data sets with many observations. To improve the usability of the Kriging-based metamodeling techniques, a parallel intelligent sampling approach: boundary and best neighbor searching (BBNS) (Wang et al., J Mater Process Technol 197(1–3):77–88, 2008a) is suggested. Compared with the serial BBNS version, the sampling procedure is performed synchronously. Thus, larger sample size should be considered for real-life problems when multiple processors are available. Furthermore, the parallel strategy is prone to converge based on more samples. The performance of the parallel approached is verified by means of nonlinear test functions. Moreover, the drawbead design in sheet metal forming is successfully optimized by the parallel BBNS approach and Kriging metamodeling technique. The optimization results demonstrate that the parallel BBNS approach improves the applicability of the Kriging metamodeling technique substantially.     

一种基于Kriging模型的天线高维全局优化算法

传统的天线优化设计需要对大量的参数组合进行电磁仿真后才能得到最优结果,使得天线高维优化设计效率普遍较低。针对该问题,使用在参数空间均匀分布的少量样本及其仿真结果构建初始Kriging模型,优化循环中每代种群由高适应度个体和高离散性个体组成,依据Kriging模型预测的个体响应和不确定性,对进化后的下一代种群进行筛选,选择最优个体执行电磁仿真并更新Kriging模型。利用此方法优化一个6变量E形天线的工作频点,相比同类优化算法,所需的电磁仿真次数可减少80%左右。     

An interior-point sequential approximate optimization methodology

The use of optimization in a simulation-based design environment has become a common trend in industry today. Computer simulation tools are commonplace in many engineering disciplines, providing the designers with tools to evaluate a designs performance without building a physical prototype. This has triggered the development of optimization techniques suitable for dealing with such simulations. One of these approaches is known as sequential approximate optimization. In sequential approximate minimization a sequence of optimizations are performed over local response surface approximations of the system. This paper details the development of an interior-point approach for trust-region-managed sequential approximate optimization. The interior-point approach will ensure that approximate feasibility is maintained throughout the optimization process. This facilitates the delivery of a usable design at each iteration when subject to reduced design cycle time constraints. In order to deal with infeasible starting points, homotopy methods are used to relax constraints and push designs toward feasibility. Results of application studies are presented, illustrating the applicability of the proposed algorithm.     

On simultaneous shape and material layout optimization of shell structures

This work presents a computational method for integrated shape and topology optimization of shell structures. Most research in the last decades considered both optimization techniques separately, seeking an initial optimal topology and refining the shape of the solution later. The method implemented in this work uses a combined approach, were the shape of the shell structure and material distribution are optimized simultaneously. This formulation involves a variable ground structure for topology optimization, since the shape of the shell mid-plane is modified in the course of the process. It was considered a simple type of design problem, where the optimization goal is to minimize the compliance with respect to the variables that control the shape, material fraction and orientation, subjected to a constraint on the total volume of material. The topology design problem has been formulated introducing a second rank layered microestructure, where material properties are computed by a “smear-out” procedure. The method has been implemented into a general optimization software called ODESSY, developed at the Institute of Mechanical Engineering in Aalborg. The computational model was tested in several numerical applications to illustrate and validate the approach.     

Shape design optimization of stationary fluid-structure interaction problems with large displacements and turbulence

This paper presents general and efficient methods for analysis and gradient based shape optimization of systems characterized as strongly coupled stationary fluid-structure interaction (FSI) problems. The incompressible fluid flow can be laminar or turbulent and is described using the Reynolds-averaged Navier-Stokes equations (RANS) together with the algebraic Baldwin–Lomax turbulence model. The structure may exhibit large displacements due to the interaction with the fluid domain, resulting in geometrically nonlinear structural behaviour and nonlinear interface coupling conditions. The problem is discretized using Galerkin and Streamline-Upwind/Petrov–Galerkin finite element methods, and the resulting nonlinear equations are solved using Newtons method. Due to the large displacements of the structure, an efficient update algorithm for the fluid mesh must be applied, leading to the use of an approximate Jacobian matrix in the solution routine. Expressions for Design Sensitivity Analysis (DSA) are derived using the direct differentiation approach, and the use of an inexact Jacobian matrix in the analysis leads to an iterative but very efficient scheme for DSA. The potential of gradient based shape optimization of fluid flow and FSI problems is illustrated by several examples.     

Multi-objective optimization of double suction centrifugal pump using Kriging metamodels

This paper proposes a new multi-objective optimization method for a family of double suction centrifugal pumps with various blade shapes, using a Simulation-Kriging model-Experiment (SKE) approach. The Kriging metamodel is established to approximate the characteristic performance functions of a pump, namely, the efficiency and required net positive suction head (NPSHr). Hence, the two objectives are to maximize the efficiency and simultaneously to minimize NPSHr. The Non-dominated Sorting Genetic Algorithm II (NSGA II) and Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D) have been applied to the multi-objective optimization problem, respectively. The Pareto solution set is obtained by a more effective and efficient manner of the two multi-objective optimization algorithms. A tradeoff optimal design point is selected from the Pareto solution set by means of a robust design based on Monte Carlo simulations, and the optimal solution is further compared with the value of the physical prototype test. The results show that the solution of the proposed multi-objective optimization method is in line with the experiment test.     

An integrated approach for shape and topology optimization of shell structures

In this paper an automated approach for simultaneous shape and topology optimization of shell structures is presented. Most research in the last decades considered these optimization techniques separately, seeking an initial optimal material layout and refining the shape of the solution later. The method developed in this work combines both optimization techniques, where the shape of the shell structure and material distribution are optimized simultaneously, with the aim of finding the optimum design that maximizes the stiffness of the shell. This formulation involves a variable ground structure for topology optimization, since the shape of the shell is modified in the course of the process. The method has been implemented into a computational model and the feasibility of the approach is demonstrated using several examples.     

A sequential coupling of optimal topology and multilevel shape design applied to two-dimensional nonlinear magnetostatics

In this paper, a sequential coupling of two-dimensional (2D) optimal topology and shape design is proposed so that a coarsely discretized and optimized topology is the initial guess for the following shape optimization. In between, we approximate the optimized topology by piecewise Bézier shapes via least square fitting. For the topology optimization, we use the steepest descent method. The state problem is a nonlinear Poisson equation discretized by the finite element method and eliminated within Newton iterations, while the particular linear systems are solved using a multigrid preconditioned conjugate gradients method. The shape optimization is also solved in a multilevel fashion, where at each level the sequential quadratic programming is employed. We further propose an adjoint sensitivity analysis method for the nested nonlinear state system. At the end, the machinery is applied to optimal design of a direct electric current electromagnet. The results correspond to physical experiments. This research has been supported by the Austrian Science Fund FWF within the SFB “Numerical and Symbolic Scientific Computing” under the grant SFB F013, subprojects F1309 and F1315, by the Czech Ministry of Education under the grant AVČR 1ET400300415, by the Czech Grant Agency under the grant GAČR 201/05/P008 and by the Slovak Grant Agency under the project VEGA 1/0262/03.     

基于网格实现的汽轮机基础优化设计

工程优化设计往往需要进行大规模的数值计算,拥有大量闲置资源的网格环境为建立这种高性能计算平台提供了可能.但是网格资源的动态性、异构性和分布性的本质特征,阻碍了网格技术在工程应用上的普及.为了利用网格环境中大量的闲置资源来协同解决实际工程中复杂的优化设计问题,建立了一个4层结构的高性能网格计算平台,并利用Kriging近似模型,在该平台上开发了以减轻基础重量和降低基础振幅为目的的多目标汽轮机优化设计的网格算法.使用该算法,在网格平台上对两个汽轮机基础进行了优化设计,与序列线性规划方法的结果比较表明所开发的优化算法有较高的计算精度.还分析了当使用不同数量的计算节点时网格的加速情况,说明所发展的优化方法能够在网格环境中高效地运行,搭建的网格平台也适合于工程优化设计.