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.     

Multi‐actuated functionally graded piezoelectric micro‐tools design: A multiphysics topology optimization approach

Micro‐tools offer significant promise in a wide range of applications such as cell manipulation, micro‐surgery, and micro/nanotechnology processes. Such special micro‐tools consist of multi‐flexible structures actuated by two or more piezoceramic devices that must generate output displacements and forces at different specified points of the domain and at different directions. The micro‐tool structure acts as a mechanical transformer by amplifying and changing the direction of the piezoceramics output displacements. The design of these micro‐tools involves minimization of the coupling among movements generated by various piezoceramics. To obtain enhanced micro‐tool performance, the concept of multifunctional and functionally graded materials is extended by tailoring elastic and piezoelectric properties of the piezoceramics while simultaneously optimizing the multi‐flexible structural configuration using multiphysics topology optimization. The design process considers the influence of piezoceramic property gradation and also its polarization sign. The method is implemented considering continuum material distribution with special interpolation of fictitious densities in the design domain. As examples, designs of a single piezoactuator, an XY nano‐positioner actuated by two graded piezoceramics, and a micro‐gripper actuated by three graded piezoceramics are considered. The results show that material gradation plays an important role to improve actuator performance, which may also lead to optimal displacements and coupling ratios with reduced amount of piezoelectric material. The present examples are limited to two‐dimensional models because many of the applications for such micro‐tools are planar devices. Copyright © 2008 John Wiley & Sons, Ltd.     

Composite material design of two‐dimensional structures using the homogenization design method

Composite materials of two‐dimensional structures are designed using the homogenization design method. The composite material is made of two or three different material phases. Designing the composite material consists of finding a distribution of material phases that minimizes the mean compliance of the macrostructure subject to volume fraction constraints of the constituent phases, within a unit cell of periodic microstructures. At the start of the computational solution, the material distribution of the microstructure is represented as a pure mixture of the constituent phases. As the iteration procedure unfolds, the component phases separate themselves out to form distinctive interfaces. The effective material properties of the artificially mixed materials are defined by the interpolation of the constituents. The optimization problem is solved using the sequential linear programming method. Both the macrostructure and the microstructures are analysed using the finite element method in each iteration step. Several examples of optimal topology design of composite material are presented to demonstrate the validity of the present numerical algorithm. Copyright © 2001 John Wiley & Sons, Ltd.     

Material cloud method for topology optimization

Material cloud method (MCM), a new approach for topology optimization, is presented. In MCM, an optimal structure can be obtained by manipulating the sizes and positions of material clouds, which are material patches with finite sizes and constant material densities. The optimal distributions of material clouds can be obtained by MCM using fixed background finite element meshes. In the numerical analysis procedure, only active elements, where more than one material cloud is contained, are treated. Optimal material distribution can be element‐wise extracted from the distribution of material clouds. With MCM, an expansion–reduction procedure of design domain can be naturally realized through movements of material clouds, so that a true optimal solution can be found without any significant increase of computational costs. It is also shown that a clear material distribution with narrow region of intermediate density can be obtained with relatively fast convergence. Several numerical examples are shown. Some of the results are compared with those of the traditional density distribution method (DDM). Copyright © 2005 John Wiley & Sons, Ltd.     

Designing materials with prescribed elastic properties using polygonal cells

An extension of the material design problem is presented in which the base cell that characterizes the material microgeometry is polygonal. The setting is the familiar inverse homogenization problem as introduced by Sigmund. Using basic concepts in periodic planar tiling it is shown that base cells of very general geometries can be analysed within the standard topology optimization setting with little additional effort. In particular, the periodic homogenization problem defined on polygonal base cells that tile the plane can be replaced and analysed more efficiently by an equivalent problem that uses simple parallelograms as base cells. Different material layouts can be obtained by varying just two parameters that affect the geometry of the parallelogram, namely, the ratio of the lengths of the sides and the internal angle. This is an efficient way to organize the search of the design space for all possible single‐scale material arrangements and could result in solutions that may be unreachable using a square or rectangular base cell. Examples illustrate the results. Copyright © 2003 John Wiley & Sons, Ltd.     

零膨胀材料设计与模拟验证

零膨胀材料对提高航空航天结构和电子设备等的热几何稳定性有重要意义。采用拓扑优化技术设计各相材料在单胞域的分布形式, 以获得零膨胀材料的微结构形式。给出了由二相实体材料和空心构成的各向同性零膨胀材料的设计方案, 讨论了初始设计依赖性问题, 分析了该依赖性的存在原因。采用有限元技术代替实际测试, 分析了所设计材料的试件在均匀温度变化下的变形, 验证了所设计材料的零膨胀(低膨胀) 性质, 说明通过拓扑优化技术设计材料的微结构是设计零膨胀材料的有效方法。     

Design of multi-tubular heat exchangers for optimum efficiency of heat dissipation

The optimum design of compact heat exchangers made of a linear metal cellular material is presented. A novel representation of the cylindrical multi-tubular configuration is used. The aim is to maximize the heat dissipation rate while minimizing the prescribed flow pressure by optimizing the multi-tube configuration. The optimum distribution of cellular material for square-cell morphology (cell density and size over given cylindrical cross-section) is found using a structural topology optimization approach. The optimized thermal performance is compared using numerical analysis including both axial temperature fields and variations within the cross-sectional area. The results for the effects of different cross-section shapes, thermal boundary conditions and flow rates are discussed and compared. Interestingly, the present formulation leads to a non-uniform distribution of cellular structures which mimic natural biomaterials. Based on these results, design guidelines for a compact multi-tubular heat exchanger are presented.     

Optimal design of laminated piezocomposite energy harvesting devices considering stress constraints

Energy harvesting devices are smart structures capable of converting the mechanical energy (generally, in the form of vibrations) that would be wasted otherwise in the environment into usable electrical energy. Laminated piezoelectric plate and shell structures have been largely used in the design of these devices because of their large generation areas. The design of energy harvesting devices is complex, and they can be efficiently designed by using topology optimization methods (TOM). In this work, the design of laminated piezocomposite energy harvesting devices has been studied using TOM. The energy harvesting performance is improved by maximizing the effective electric power generated by the piezoelectric material, measured at a coupled electric resistor, when subjected to a harmonic excitation. However, harmonic vibrations generate mechanical stress distribution that, depending on the frequency and the amplitude of vibration, may lead to piezoceramic failure. This study advocates using a global stress constraint, which accounts for different failure criteria for different types of materials (isotropic, piezoelectric, and orthotropic). Thus, the electric power is maximized by optimally distributing piezoelectric material, by choosing its polarization sign, and by properly choosing the fiber angles of composite materials to satisfy the global stress constraint. In the TOM formulation, the Piezoelectric Material with Penalization and Polarization material model is applied to distribute piezoelectric material and to choose its polarization sign, and the Discrete Material Optimization method is applied to optimize the composite fiber orientation. The finite element method is adopted to model the structure with a piezoelectric multilayered shell element. Numerical examples are presented to illustrate the proposed methodology. Copyright © 2015 John Wiley & Sons, Ltd.     

基于变频率区间约束的结构材料优化设计

针对频率约束的结构材料优化问题,基于结构拓扑优化思想,提出变频率区间约束的结构材料优化方法。借鉴均匀化及ICM(独立、连续、映射)方法,以微观单元拓扑变量倒数为设计变量,导出宏观单元等效质量矩阵及导数,进而获得频率一阶近似展开式。结合变频率区间约束思想,获得以结构质量为目标函数、频率为约束条件的连续体微结构拓扑优化近似模型;采用对偶方法求解。通过算例验证该方法的有效性及可行性,表明考虑质量矩阵变化影响所得优化结果更合理。     

Conceptual design of reinforced concrete structures using topology optimization with elastoplastic material modeling

Design of reinforced concrete structures is governed by the nonlinear behavior of concrete and by its different strengths in tension and compression. The purpose of this article is to present a computational procedure for optimal conceptual design of reinforced concrete structures on the basis of topology optimization with elastoplastic material modeling. Concrete and steel are both considered as elastoplastic materials, including the appropriate yield criteria and post‐yielding response. The same approach can be applied also for topology optimization of other material compositions where nonlinear response must be considered. Optimized distribution of materials is achieved by introducing interpolation rules for both elastic and plastic material properties. Several numerical examples illustrate the capability and potential of the proposed procedure. Copyright © 2012 John Wiley & Sons, Ltd.     

微粒群算法在自动控制系统设计中的应用

提出了将微粒群优化(Particle Swarm Optimization,PSO)算法与控制系统设计相结合的系统设计思路和方法。系统设计过程包括两个部分：首先基于历史输入输出数据,用微粒群算法建立系统的模型,然后基于得到的模型进行控制器的设计,并用微粒群算法进行控制器的参数优化整定。仿真试验结果表明,微粒群算法在控制系统设计的模型建立、控制器参数优化等方面发挥了重要的作用,简化了控制系统设计任务,提高了设计效率。     

Discrete material optimization of general composite shell structures

A novel method for doing material optimization of general composite laminate shell structures is presented and its capabilities are illustrated with three examples. The method is labelled Discrete Material Optimization (DMO) but uses gradient information combined with mathematical programming to solve a discrete optimization problem. The method can be used to solve the orientation problem of orthotropic materials and the material selection problem as well as problems involving both. The method relies on ideas from multiphase topology optimization to achieve a parametrization which is very general and reduces the risk of obtaining a local optimum solution for the tested configurations. The applicability of the DMO method is demonstrated for fibre angle optimization of a cantilever beam and combined fibre angle and material selection optimization of a four‐point beam bending problem and a doubly curved laminated shell. Copyright © 2005 John Wiley & Sons, Ltd.     

简谐激励下阻尼复合结构多尺度拓扑优化设计

目的 为得到抗振性能良好的板壳结构,保证设备的正常工作,文中提出一种板壳阻尼复合结构多尺度优化设计方法。方法 以动柔度为目标,建立频域激励下和固定频率点激励下板壳阻尼复合结构中阻尼材料宏观分布和微结构协同设计的多尺度问题的数学模型,推导目标函数和约束条件对设计变量的灵敏度,并基于移动渐近线法求解优化数学模型。结果 所提多尺度设计方法可以有效获得板壳结构最优阻尼材料宏观布局和最优阻尼复合材料微结构构型,提高了结构的动力学性能,同时结果也表明涂敷阻尼复合材料结构的振动响应相较于仅涂敷单一阻尼材料的振动响应大幅减小。结论 研究表明,不同激励频率下阻尼材料的宏观分布形态不同,阻尼材料主要分布于结构模态振型位移的最大处和支撑端,通过加强结构的刚度,抑制了结构变形,减小了振动响应。微结构构型基本类似,其基本形态都是低刚度、高阻尼材料呈条状分布,条状分布的阻尼复合材料微结构在受弯方向上的刚度较大,可以有效抵制结构的弯曲变形。     

Topology optimization with multiple materials via moving morphable component (MMC) method

With the fast development of additive manufacturing technology, topology optimization involving multiple materials has received ever increasing attention. Traditionally, this kind of optimization problem is solved within the implicit solution framework by using the Solid Isotropic Material with Penalization or level set method. This treatment, however, will inevitably lead to a large number of design variables especially when many types of materials are involved and 3‐dimensional (3D) problems are considered. This is because for each type of material, a corresponding density field/level function defined on the entire design domain must be introduced to describe its distribution. In the present paper, a novel approach for topology optimization with multiple materials is established based on the Moving Morphable Component framework. With use of this approach, topology optimization problems with multiple materials can be solved with much less numbers of design variables and degrees of freedom. Numerical examples provided demonstrate the effectiveness of the proposed approach.     

Continuous approximation of material distribution for topology optimization

In this paper, we propose a checkerboard‐free topology optimization method without introducing any additional constraint parameter. This aim is accomplished by the introduction of finite element approximation for continuous material distribution in a fixed design domain. That is, the continuous distribution of microstructures, or equivalently design variables, is realized in the whole design domain in the context of the homogenization design method (HDM), by the discretization with finite element interpolations. By virtue of this continuous FE approximation of design variables, discontinuous distribution like checkerboard patterns disappear without any filtering schemes. We call this proposed method the method of continuous approximation of material distribution (CAMD) to emphasize the continuity imposed on the ‘material field’. Two representative numerical examples are presented to demonstrate the capability and the efficiency of the proposed approach against some classes of numerical instabilities. Copyright © 2004 John Wiley & Sons, Ltd.     

Level-set method for design of multi-phase elastic and thermoelastic materials

The problem of designing composite materials with desired mechanical properties is to specify the materials microstructures in terms of the topology and distribution of their constituent material phases within a unit cell of periodic microstructures. In this paper we present an approach based on a multi-phase level-set model for the geometric and material representation and for numerical solution of a least squares optimization problem. The level-set model precisely specifies the material regions and their sharp boundaries in contrast to a raster discretization of the conventional homogenization-based approaches. Combined with the classical shape derivatives, the level-set method yields a computational system of partial differential equations. In using the Eulerian computation scheme with a fixed rectilinear grid and a fixed mesh in the unit cell, the gradient descent solution of the optimization captures the interfacial boundaries naturally and performs topological changes accurately. The proposed method is illustrated with several 2D examples for the synthesis of heterogeneous microstructures of elastic and/or thermoelastic composites composed of two and three material phases.     

车身结构阻尼材料减振降噪优化设计

本文针对某一乘用车车身结构振动引起的声辐射,建立了车身结构、声学空腔以及声固耦合有限元模型,分析了该乘用车车身的声固耦合特性。通过对车身各板件的贡献度分析,确定了对车内噪声贡献度最大的壁板。针对该壁板的阻尼减振降噪优化设计,建立了拓扑优化模型,采用渐进优化算法（ESO）,计算了阻尼材料的优化布局。研究结果表明：阻尼材料的优化布局使阻尼材料的使用率大大提高,50%的阻尼材料用量能基本达到全覆盖阻尼材料壁板的降噪效果,阻尼结构优化设计对车内噪声控制具有一定的理论指导意义。     

A bi‐value coding parameterization scheme for the discrete optimal orientation design of the composite laminate

The discrete optimal orientation design of the composite laminate can be treated as a material selection problem dealt with by using the concept of continuous topology optimization method. In this work, a new bi‐value coding parameterization (BCP) scheme of closed form is proposed to this aim. The basic idea of the BCP scheme is to ‘code’ each material phase using integer values of +1 and –1 so that each available material phase has one unique ‘code’ consisting of +1 and/or –1 assigned to design variables. Theoretical and numerical comparisons between the proposed BCP scheme and existing schemes show that the BCP has the advantage of an evident reduction of the number of design variables in logarithmic form. The benefit is particularly remarkable when the number of candidate materials becomes important in large‐scale problems. Numerical tests with up to 36 candidate material orientations are illustrated for the first time to indicate the reliability and efficiency of the BCP scheme in solving this kind of problem. It proves that the BCP is an interesting and valuable scheme to achieve the optimal orientations for large‐scale design problems. Besides, a four‐layer laminate example is tested to demonstrate that the proposed BCP scheme can easily be extended to multilayer problems. Copyright © 2012 John Wiley & Sons, Ltd.     

Cellular topology optimization on differentiable Voronoi diagrams

Cellular structures manifest their outstanding mechanical properties in many biological systems. One key challenge for designing and optimizing these geometrically complicated structures lies in devising an effective geometric representation to characterize the system's spatially varying cellular evolution driven by objective sensitivities. A conventional discrete cellular structure, for example, a Voronoi diagram, whose representation relies on discrete Voronoi cells and faces, lacks its differentiability to facilitate large-scale, gradient-based topology optimizations. We propose a topology optimization algorithm based on a differentiable and generalized Voronoi representation that can evolve the cellular structure as a continuous field. The central piece of our method is a hybrid particle-grid representation to encode the previously discrete Voronoi diagram into a continuous density field defined in a Euclidean space. Based on this differentiable representation, we further extend it to tackle anisotropic cells, free boundaries, and functionally-graded cellular structures. Our differentiable Voronoi diagram enables the integration of an effective cellular representation into the state-of-the-art topology optimization pipelines, which defines a novel design space for cellular structures to explore design options effectively that were impractical for previous approaches. We showcase the efficacy of our approach by optimizing cellular structures with up to thousands of anisotropic cells, including femur bone and Odonata wing.     

Topological optimization for the design of microstructures of isotropic cellular materials

The aim of this study was to design isotropic periodic microstructures of cellular materials using the bidirectional evolutionary structural optimization (BESO) technique. The goal was to determine the optimal distribution of material phase within the periodic base cell. Maximizing bulk modulus or shear modulus was selected as the objective of the material design subject to an isotropy constraint and a volume constraint. The effective properties of the material were found using the homogenization method based on finite element analyses of the base cell. The proposed BESO procedure utilizes the gradient-based sensitivity method to impose the isotropy constraint and gradually evolve the microstructures of cellular materials to an optimum. Numerical examples show the computational efficiency of the approach. A series of new and interesting microstructures of isotropic cellular materials that maximize the bulk or shear modulus have been found and presented. The methodology can be extended to incorporate other material properties of interest such as designing isotropic cellular materials with negative Poisson's ratio.