Scale‐related topology optimization of cellular materials and structures

The integrated optimization of lightweight cellular materials and structures are discussed in this paper. By analysing the basic features of such a two‐scale problem, it is shown that the optimal solution strongly depends upon the scale effect modelling of the periodic microstructure of material unit cell (MUC), i.e. the so‐called representative volume element (RVE). However, with the asymptotic homogenization method used widely in actual topology optimization procedure, effective material properties predicted can give rise to limit values depending upon only volume fractions of solid phases, properties and spatial distribution of constituents in the microstructure regardless of scale effect. From this consideration, we propose the design element (DE) concept being able to deal with conventional designs of materials and structures in a unified way. By changing the scale and aspect ratio of the DE, scale‐related effects of materials and structures are well revealed and distinguished in the final results of optimal design patterns. To illustrate the proposed approach, numerical design problems of 2D layered structures with cellular core are investigated. Copyright © 2006 John Wiley & Sons, Ltd.     

基于ICM法的高层建筑大型支撑体系拓扑优化

提出应用连续体结构拓扑优化ICM法对高层建筑大型支撑体系进行拓扑优化。针对高层建筑规范对结构刚度限值是以层间相对位移差形式给出、并结合结构拓扑优化特点,推导了相对位移差敏度分析的伴随法公式,有效提高了计算效率。应用ICM法建立位移约束下结构重量极小化的优化模型,与高层建筑规范对结构刚度限值要求的提法更符合,得到的最优拓扑完全满足规范要求。所提方法应用在概念设计阶段,提供了一种自动化的分析计算及优化设计工具,可以有效地弥补基于经验设计的不足。     

稳态热传导结构非概率可靠性拓扑优化设计

研究具有区间参数的稳态热传导结构在散热弱度非概率可靠性约束下的拓扑优化设计问题。建立了以单元相对导热系数为设计变量,导热材料体积极小化为目标函数,满足散热弱度非概率可靠性为约束条件的稳态热传导结构的拓扑优化设计数学模型。基于区间因子法,推导出散热弱度的均值及离差的计算表达式。采用渐进结构优化法的求解策略与方法,并利用过滤技术消除优化过程中的数值不稳定性现象。通过算例验证文中模型及求解策略、方法的合理性和有效性。     

Global optimization of robust truss topology via mixed integer semidefinite programming

This paper discusses a global optimization method of robust truss topology under the load uncertainties and slenderness constraints of the member cross-sectional areas. We consider a non-stochastic uncertainty of the external load, and attempt to minimize the maximum compliance corresponding to the most critical load. A design-dependent uncertainty model in the external load is proposed in order to consider the variation of truss topology rigorously. It is shown that this optimization problem can be formulated as a 0–1 mixed integer semidefinite programming (0–1MISDP) problem. We propose a branch-and-bound method for computing the global optimal solution of the 0–1MISDP. Numerical examples illustrate that the topology of robust optimal truss depends on the magnitude of uncertainty. The presented method can provide global optimal solutions for benchmark examples, which can be used for evaluating the performance of any other local optimization method for robust structural optimization.     

A moving superimposed finite element method for structural topology optimization

Level set methods are becoming an attractive design tool in shape and topology optimization for obtaining efficient and lighter structures. In this paper, a dynamic implicit boundary‐based moving superimposed finite element method (s‐version FEM or S‐FEM) is developed for structural topology optimization using the level set methods, in which the variational interior and exterior boundaries are represented by the zero level set. Both a global mesh and an overlaying local mesh are integrated into the moving S‐FEM analysis model. A relatively coarse fixed Eulerian mesh consisting of bilinear rectangular elements is used as a global mesh. The local mesh consisting of flexible linear triangular elements is constructed to match the dynamic implicit boundary captured from nodal values of the implicit level set function. In numerical integration using the Gauss quadrature rule, the practical difficulty due to the discontinuities is overcome by the coincidence of the global and local meshes. A double mapping technique is developed to perform the numerical integration for the global and coupling matrices of the overlapped elements with two different co‐ordinate systems. An element killing strategy is presented to reduce the total number of degrees of freedom to improve the computational efficiency. A simple constraint handling approach is proposed to perform minimum compliance design with a volume constraint. A physically meaningful and numerically efficient velocity extension method is developed to avoid the complicated PDE solving procedure. The proposed moving S‐FEM is applied to structural topology optimization using the level set methods as an effective tool for the numerical analysis of the linear elasticity topology optimization problems. For the classical elasticity problems in the literature, the present S‐FEM can achieve numerical results in good agreement with those from the theoretical solutions and/or numerical results from the standard FEM. For the minimum compliance topology optimization problems in structural optimization, the present approach significantly outperforms the well‐recognized ‘ersatz material’ approach as expected in the accuracy of the strain field, numerical stability, and representation fidelity at the expense of increased computational time. It is also shown that the present approach is able to produce structures near the theoretical optimum. It is suggested that the present S‐FEM can be a promising tool for shape and topology optimization using the level set methods. Copyright © 2005 John Wiley & Sons, Ltd.     

模糊参数平面连续体结构的拓扑优化设计

研究了具有模糊参数的连续体结构在模糊载荷作用下的拓扑优化设计问题。利用信息熵将模糊变量转换为随机变量,构建了随机载荷作用下的随机参数的连续体结构的拓扑优化设计数学模型,以结构的形状拓扑信息为设计变量,结构总质量均值极小化为目标函数,满足单元应力可靠性为约束条件,利用分布函数法对应力可靠性约束进行了等价显式化处理。基于随机因子法,利用代数综合法导出了应力响应的数字特征的计算表达式。采用双方向渐进结构优化(BESO)方法求解。通过两个算例验证了该文模型及求解方法的合理性和有效性。     

考虑瞬态响应的薄板结构阻尼材料层拓扑优化

讨论了敷设阻尼材料的薄板结构考虑瞬态响应时阻尼材料层的最优布局问题。基于SIMP方法构造人工阻尼材料惩罚模型和结构拓扑优化模型,以阻尼材料的相对密度作为设计变量,在给定阻尼材料用量的条件下,最小化结构瞬态位移响应的时间积分。由于结构整体呈现非比例阻尼特性,采用逐步积分法对结构的振动方程进行求解。通过伴随变量法得到目标函数对设计变量的灵敏度表达式,在此基础上采用基于梯度的移动渐近线方法求解。数值算例验证了优化模型与算法的合理性和有效性。     

Voigt–Reuss topology optimization for structures with nonlinear material behaviors

This work is directed toward optimizing concept designs of structures featuring inelastic material behaviours by using topology optimization. In the proposed framework, alternative structural designs are described with the aid of spatial distributions of volume fraction design variables throughout a prescribed design domain. Since two or more materials are permitted to simultaneously occupy local regions of the design domain, small-strain integration algorithms for general two-material mixtures of solids are developed for the Voigt (isostrain) and Reuss (isostress) assumptions, and hybrid combinations thereof. Structural topology optimization problems involving non-linear material behaviours are formulated and algorithms for incremental topology design sensitivity analysis (DSA) of energy type functionals are presented. The consistency between the structural topology design formulation and the developed sensitivity analysis algorithms is established on three small structural topology problems separately involving linear elastic materials, elastoplastic materials, and viscoelastic materials. The good performance of the proposed framework is demonstrated by solving two topology optimization problems to maximize the limit strength of elastoplastic structures. It is demonstrated through the second example that structures optimized for maximal strength can be significantly different than those optimized for minimal elastic compliance. © 1997 John Wiley & Sons, Ltd.     

Topology optimization of structures under multiple loading cases with a new compliance–volume product

This article proposes a new topology optimization method for the design of structures under multiple loading cases. The design is formulated as a multi-objective optimization problem by minimizing a new compliance–volume product, which optimizes the overall stiffness and volume simultaneously to avoid the empirical decision on design constraints and obtain an even lower structural volume. A normalized exponential weighted criterion (NEWC) method is included in the multi-objective optimization problem for the capture of the entire Pareto frontier. A weight evaluation method, in terms of the fuzzy multiple-attribute group decision-making (FMAGDM) theory, is incorporated in the problem to evaluate the weights of the objectives and guarantee the optimal design in an acceptable level. The solid isotropic material with penalty (SIMP) method is used to represent the dependence of elemental densities on material properties. Three typical numerical examples are employed to show the effectiveness of the proposed method.     

Simultaneous partial topology and size optimization of a wing structure using ant colony and gradient based methods

This article presents a methodology and process for a combined wing configuration partial topology and structure size optimization. It is aimed at achieving a minimum structural weight by optimizing the structure layout and structural component size simultaneously. This design optimization process contains two types of design variables and hence was divided into two sub-problems. One is structure layout topology to obtain an optimal number and location of spars with discrete integer design variables. Another is component size optimization with continuous design variables in the structure FE model. A multi city-layer ant colony optimization (MCLACO) method is proposed and applied to the topology sub-problem. A gradient based optimization method (GBOM) built in the MSC.NASTRAN SOL-200 module was employed in the component size optimization sub-problem. For each selected layout of the wing structure, a size optimization process is performed to obtain the optimum result and feedback to the layout topology process. The numerical example shows that the proposed MCLACO method and a combination with the GBOM are effective for solving such a wing structure optimization problem. The results also indicate that significant structural weight saving can be achieved.     

高层建筑抗风优化设计和风振控制相关问题研究

高层建筑由于自振周期长、阻尼小,其高柔的特征使其对风荷载特别敏感,风荷载是沿海地区超高层建筑的主要水平控制荷载,因此在强/台风作用下,其抗风设计须在满足规范安全要求的前提下,同时又要经济实用和结构性能高效,为此,开展高层建筑抗风优化和风振控制方面的研究具有十分重要的现实意义。该文在对高层建筑抗风优化设计和风振控制研究现状做简要介绍的基础上,首先根据风荷载的特点,着重研究了考虑风速风向联合概率分布和基于可靠度及性能化的高层建筑抗风设计方法,采用最优准则法,以结构的总重或总造价为目标函数,以顶部位移、层间侧移以及顶部风致加速度为约束条件,对高层建筑结构杆件截面抗风优化设计的相关问题进行了研究。同时为提高基因遗传智能优化算法的收敛速度和获得最可能优化解,该文提出了传统基因遗传的改进算法(如基于改进罚函数及分级遗传算法)用于结构抗风优化设计。在结构拓扑抗风优化方面,则主要引入分层优化的概念,对变密度法和改进动态进化率的双向渐进拓扑优化方法,应用于抗风结构的拓扑构型优化算法进行了相关研究。通过实例分析验证了上述结构抗风优化算法的高效和正确性。在风振控制方面,该文结合摩擦摆系统和调谐质量阻尼器各自的优点,提出了摩擦摆调谐质量阻尼器(FPS-TMD)被动控制系统,对其力学和动力特性,以及高层建筑顶部带FPS-TMD系统的风振控制理论,进行了相关研究。以结构控制第三代Benchmark模型为实例,研究顶部带FPS-TMD系统的高层建筑风振控制效果,同时结合该文开发的基于小型电振动台的实时混合实验测试平台,采用风振控制实时混合实验结果与理论模拟计算结果的对比,验证了该文提出的FPS-TMD被动控制系统,应用于高层建筑风振控制的有效性。     

Bidirectional evolutionary structural optimization for nonlinear structures under dynamic loads

This study aims to develop efficient numerical optimization methods for finding the optimal topology of nonlinear structures under dynamic loads. The numerical models are developed using the bidirectional evolutionary structural optimization method for stiffness maximization problems with mass constraints. The mathematical formulation of topology optimization approach is developed based on the element virtual strain energy as the design variable and minimization of compliance as the objective function. The suitability of the proposed method for topology optimization of nonlinear structures is demonstrated through a series of two- and three-dimensional benchmark designs. Several issues relating to the nonlinear structures subjected to dynamic loads such as material, geometric, and contact nonlinearities are addressed in the examples. It is shown that the proposed approach generates more reliable designs for nonlinear structures.     

Eigenfrequecy-based damage identification method for non-destructive testing based on topology optimization

Non-destructive testing (NDT) detects damage according to a difference in a physical phenomenon between a normal structure and damaged structure. As a solution avoiding human errors in NDT, a numerical method based on a dynamical numerical analysis model and a structural optimization algorithm was proposed. This method automatically derives a structure with a response that is equal to that of a damaged structure through an optimization procedure. Among structural optimization methods, topology optimization can optimize the structure fundamentally by changing the topology and not just the shape of a structure. Thus, topology optimization is employed together with eigenfrequency analysis, which is the most fundamental methodology of NDT. The proposed method derives a structure that has the same eigenfrequencies as a damaged structure employing topology optimization. The shape and location of damage can be identified through the optimal shape obtained.     

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.     

Topology optimization of composite material plate with respect to sound radiation

In this paper, topology optimization of composite material plate with respect to minimization of the sound power radiation has been studied. A new low noise design method based on topology optimization is proposed, which provides great guidance for acoustic designers. The structural vibrations are excited by external harmonic mechanical load with prescribed frequency and amplitude. The sound power is calculated using boundary element method. An extended solid isotropic material with penalization (SIMP) model is introduced for acoustic design sensitivity analysis in topology optimization, where the same penalization is applied for the stiffness and mass of the structural volume elements. Volumetric densities of stiffer material are chosen as design variables. Finally, taking a simple supported thin plate as a simulation example, the sound power radiation from structures subjected to forced vibration can be considerably reduced, leading to a reduction of 20 dB. It is shown that the optimal topology is easy to manufacture at low frequency, while as the loading frequency increases, the optimal topology shows a more and more complicated periodicity which makes it difficult to manufacture.     

A hybrid cellular automaton–bi-directional evolutionary optimization algorithm for topological optimization of crashworthiness

This article proposes a new algorithm for topological optimization under dynamic loading which combines cellular automata with bi-directional evolutionary structural optimization (BESO). The local rules of cellular automata are used to update the design variables, which avoids the difficulty of obtaining gradient information under nonlinear collision conditions. The intermediate-density design problem of hybrid cellular automata is solved using the BESO concept of 0–1 binary discrete variables. Some improvement strategies are also proposed for the hybrid algorithm to solve certain problems in nonlinear topological optimization, e.g. numerical oscillation. Some typical examples of crashworthiness problems are provided to illustrate the efficiency of the proposed method and its ability to find the final optimal solution. Finally, numerical results obtained using the proposed algorithms are compared with reference examples taken from the literature. The results show that the hybrid method is computationally efficient and stable.     

Multi-material topology optimization of laminated composite beam cross sections

This paper presents a novel framework for simultaneous optimization of topology and laminate properties in structural design of laminated composite beam cross sections. The structural response of the beam is evaluated using a beam finite element model comprising a cross section analysis tool which is suitable for the analysis of anisotropic and inhomogeneous sections of arbitrary geometry. The optimization framework is based on a multi-material topology optimization model in which the design variables represent the amount of the given materials in the cross section. Existing material interpolation, penalization, and filtering schemes have been extended to accommodate any number of anisotropic materials. The methodology is applied to the optimal design of several laminated composite beams with different cross sections. Solutions are presented for a minimum compliance (maximum stiffness) problem with constraints on the weight, and the shear and mass center positions. The practical applicability of the method is illustrated by performing optimal design of an idealized wind turbine blade subjected to static loading of aerodynamic nature. The numerical results suggest that the proposed framework is suitable for simultaneous optimization of cross section topology and identification of optimal laminate properties in structural design of laminated composite beams.     

A new efficient convergence criterion for reducing computational expense in topology optimization: reducible design variable method

A new efficient convergence criterion, named the reducible design variable method (RDVM), is proposed to save computational expense in topology optimization. There are two types of computational costs: one is to calculate the governing equations, and the other is to update the design variables. In conventional topology optimization, the number of design variables is usually fixed during the optimization procedure. Thus, the computational expense linearly increases with respect to the iteration number. Some design variables, however, quickly converge and some other design variables slowly converge. The idea of the proposed method is to adaptively reduce the number of design variables on the basis of the history of each design variable during optimization. Using the RDVM, those design variables that quickly converge are not considered as design variables for the next iterations. This means that the number of design variables can be reduced to save the computational costs of updating design variables. Then, the iteration will repeat until the number of design variables becomes 0. In addition, the proposed method can lead to faster convergence of the optimization procedure, which indeed is a more significant time saving. It is also revealed that the RDVM gives identical optimal solutions as those by conventional methods. We confirmed the numerical efficiency and solution effectiveness of the RDVM with respect to two types of optimization: static linear elastic minimization, and linear vibration problems with the first eigenvalue as the objective function for maximization. Copyright © 2012 John Wiley & Sons, Ltd.     

Robust concurrent topology optimization of multiscale structure under single or multiple uncertain load cases

Concurrent topology optimization of macrostructure and material microstructure has attracted significant interest in recent years. However, most of the existing works assumed deterministic load conditions, thus the obtained design might have poor performance in practice when uncertainties exist. Therefore, it is necessary to take uncertainty into account in structural design. This article proposes an efficient method for robust concurrent topology optimization of multiscale structure under single or multiple load cases. The weighted sum of the mean and standard deviation of the structural compliance is minimized and constraints are imposed to both the volume fractions of macrostructure and microstructure. The effective properties of the microstructure are calculated via the homogenization method. An efficient sensitivity analysis method is proposed based on the superposition principle and orthogonal similarity transformation of real symmetric matrices. To further reduce the computational cost, an efficient decoupled sensitivity analysis method for microscale design variables is proposed. The bidirectional evolutionary structural optimization method is employed to obtain black and white designs for both macrostructure and microstructure. Several two-dimensional and three-dimensional numerical examples are presented to demonstrate the effectiveness of the proposed approach and the effects of load uncertainty on the optimal design of both macrostructure and microstructure.