复合材料扭转轴截面微结构拓扑优化设计

提出复合材料扭转轴截面微结构拓扑优化设计新模型，模型的优化目标是获得具有最大宏观剪切特性加权和的单胞形式．通过模型和均匀化方法及优化技术可以获得优化的微结构单胞，进而改善或者得到最优宏观弹性特性的复合材料．为了便于制造和应用，胞体材料用来获得复合材料的极值剪切模量．最后的优化结果表明，该模型连同数值处理技巧可以非常有效地实现微结构的拓扑优化设计．     

考虑时变刚度特性的复合材料微结构拓扑优化设计方法

理想的骨折内固定植入物在组织愈合或修复的过程中, 其结构性能需要满足不同愈合阶段对生物力学的需求. 提出一种对生物可降解复合材料微结构的时变刚度特性进行调控设计的拓扑优化方法, 以达到理想的骨折内固定植入物特殊的时变刚度特性需求. 使用具有不同降解速率和刚度的两种可降解材料, 以相对密度作为设计变量来描述不同材料的分布, 以特定降解时间步中间结构的刚度之和最大为优化目标, 对复合材料微结构的构型进行拓扑优化设计, 使其具有符合骨愈合规律的时变刚度特性. 使用均匀腐蚀方法, 利用与时间相关的材料残留率描述结构的降解过程, 建立考虑时间维度材料降解的有限元模型, 基于Heaviside函数和Kreisselmeier-Steinhauser函数建立降解更新的连续方程, 利用均匀化方法得到不同降解时间步中间结构的力学性能, 并计算优化目标对于设计变量的灵敏度. 通过与仅使用单材料的结构和无时变刚度特性调控的拓扑优化结构进行对比, 验证了所提出设计方法的有效性, 并研究了不同参数对单胞优化构型和时变刚度特性的影响.      

考虑微结构连接性的双尺度结构自然频率拓扑优化

多孔材料因具有轻量化、高孔隙率和减振/散热等优良多物理特性,在航空航天等领域具有广阔应用前景。采用拓扑优化方法对含多种多孔材料的结构进行结构与材料微结构构型一体化设计,有助于获得具有优良力学性能的结构设计。然而,传统逆均匀化微结构设计方法无法确保不同多孔材料微结构之间的连接性,设计结果不具备可制造性。本文面向含多种多孔材料的双尺度结构基频最大化设计问题,考虑不同微结构之间的连接性,协同设计多孔材料的微结构构型及其在宏观尺度下的布局。采用均匀化方法计算多孔材料的宏观等效力学性能,通过对不同多孔材料微结构单胞的边界区域采用相同的拓扑描述确保双尺度优化过程中任意空间排布下不同微结构的连接性,并通过优化算法确定微结构间的连接形式及微结构拓扑。在宏观尺度,提出结合离散材料插值模型和RAMP插值模型RAMP (Rational Approximation of Material Properties)的多孔材料各向异性宏观等效刚度及质量插值模型,获得清晰的多孔材料宏观尺度布局并减轻优化过程中伪振动模态的影响。建立以双尺度结构基频最大化为目标,以材料用量为约束的优化列式,推导灵敏度表达式,并基于梯度优化算法求解双尺度结构拓扑优化问题。数值算例表明,采用本文优化方法能够有效确保基频最大化双尺度结构设计中不同多孔材料微结构之间的连接性,增强优化设计结果的可制造性。     

基于自适应网格的材料渗透系数设计

采用基于单元(结点)密度为设计变量进行结构和材料的拓扑优化设计时,有限元网格的密度对优化设计有很大影响.在以渗透系数为目标进行材料微结构设计时,为了较好地描述单胞中的流固边界,需要将单胞划分为很小的网格,进一步增加了有限元计算和优化分析的规模.为了降低计算规模,研究了基于自适应网格的逆均匀化方法,以最大化各向同性等效渗透系数为目标,进行材料微结构设计.优化迭代过程中,对单胞中流固界面处的网格进行自适应加密,降低优化问题的计算规模.采用这一算法,对不同初始密度分布得到的单胞优化结果虽然不同,但具有相同的材料微结构,一定程度上说明了该方法的有效性.     

基于拓扑描述函数的特定性能复合材料设计

提出了一种基于拓扑描述函数进行特定性能复合材料设计的新方法. 采用拓扑描 述函数作为设计变量，把复合材料微结构的设计问题转化为一个在周期性单胞上的拓扑优 化问题. 拓扑描述函数以及相应正则化机制的引入不仅可以有效消除棋盘格式等数值不稳定 现象，而且能够有效地抑制传统算法处理此类优化问题时所引发的边界扩散效应. 数值结果 表明所提出的方法可以稳定高效地实现具有特定性能的复合材料微结构设计.     

多相复合材料微结构布局优化设计

基于对传统双向渐进结构优化(BESO)方法的改进,通过均匀化方法建立微观尺度多相材料布局分布与宏观结构材料弹性张量之间的联系,集成宏观结构所得到的位移场,推导出带有宏观结构力学特性的微观灵敏度。提出以宏观结构最大刚度为目标、对极小尺度周期性排列的材料微结构胞元进行布局优化设计的方法。本文算例结果表明,该方法在微结构多相材料布局优化设计过程中,不仅克服了拓扑优化领域常见的"棋盘格"现象,得到了边界清晰、受力合理的多相材料合理分布布局优化结果,而且整体优化过程稳定,具有很强的工程实际应用价值。     

多相材料微结构布局及宏观结构拓扑并发优化

在传统双向渐进结构优化(BESO)方法基础上,充分考虑材料和结构的尺度关联性,基于均匀化理论将材料微结构胞元设计和宏观结构拓扑优化相结合,按照材料属性排序引入材料插值函数依次进行灵敏分析,建立周期性多相材料微结构布局及宏观结构拓扑并发优化设计方法。优化过程中,宏观结构受力的特性嵌入微观敏度生成过程,使得新型材料具备了特定宏观结构力学需求的更加轻型、高强的最佳力学性能;同时,微观材料胞元的等效材料属性又是宏观结构优化的基础材料,从而使得材料/结构具有尺度上的统一。相关算例说明该方法在解决多相材料微观分布优化和周期性多相材料微结构布局及宏观结构拓扑并发优化问题时具有边界清晰和收敛快等优点。     

多相材料微结构多目标拓扑优化设计

在采用多尺度均匀化方法求解微结构等效特性的基础上，提出了多相材料 微结构的多目标优化设计模型. 以组分材料用量为约束，采用周长控制消除棋盘格，结合有 限元方法和对偶凸规划求解技术，对两相和三相材料微结构多项等效模量的组合进行了优化 设计. 研究比较了微结构网格粗细、材料组分以及三相材料微结构优化中的两相实体材料弹 性模量相对比例不同对优化结果的影响. 数值算例验证了优化模型和优化算法的有效性，表 明了相关因素对优化结果的影响.     

循环对称结构的多尺度拓扑优化方法

研究了循环对称结构多尺度拓扑优化问题,阐明了均匀化等效性能的基本性质,提出了循环对称单胞的特征参数概念,建立了均匀化映射计算方法,构造了等效性能的三元参数插值模型,有效简化了不同特征参数、微结构构型与体分比下的单胞均匀化等效过程.通过微结构的特征驱动建模与B样条参数化,克服了微结构拓扑优化变量多和变量离散难以保证其光滑连接的问题.给出了典型数值算例,比较了等比例排列单胞与等间距排列单胞对结构优化结果的影响,验证了多尺度优化方法的有效性.     

基于遗传算法的复合材料细观结构拓扑优化设计

利用高精度通用单胞模型将复合材料的细观拓扑结构与宏观力学性能结合起来,采用遗传算法对复合材料的细观结构进行优化,发展了基于遗传算法的复合材料细观结构拓扑优化设计方法.以材料的宏观力学性能为优化目标,从随机的初始细观结构出发,对复合材料纤维体积百分比进行约束,经过迭代获得满足设计要求的代表性体积单元.在优化过程中,对遗传算法的交叉过程作了较大的改进,实现了复合材料细观拓扑结构的任意变化,提高了对可行域的搜索效率.分别以极限剪切模量和泊松比为优化目标,验证了所提出优化方法的正确性和有效性.     

HOMOGENIZATION—BASED TOPOLOGY DESIGN FOR PURE TORSION OF COMPOSITE SHAFTS

In conjunction with the homogenization theory and the finite element method, the mathematical models for designing the corss-section of composite shafts by maximizing the torsion rigidity are developed in this paper. To obtain the extremal torsion rigidity, both the cross-section of the macro scale shaft and the representative microstructure of the composite material are optimized using the new models. The micro scale computational model addresses the problem of finding the periodic microstructures with extreme shear moduli. The optimal microstructure obtained with the new model and the homogenization method can be used to improve and optimize natural or artificial materials. In order to be more practical for engineering applications, cellular materials rather than ranked materials are used in the optimal process in the existence of optimal bounds for the elastic properties. Moreover, the macro scale model is proposed to optimize the cross-section of the torsional shaft based on the tailared composites. The validating optimal results show that the models are very effective in obtaining composites with extreme elastic properties, and the cross-section of the composite shaft with the extremal torsion rigidity. The project supported by the National Natural Science Foundation of China (10172078 and 10102018)     

Virtual improvement of ice cream properties by computational homogenization of microstructures

Optimal shape design of microstructured materials has recently attracted a great deal of attention in materials science. The shape and the topology of the microstructure have a significant impact on the macroscopic properties. This paper presents different computational models of random microstructures, to virtually improve the physical properties of ice cream. Several sensory properties of this heterogeneous material issued from food industry are directly controlled by the elastic and thermal conducting ones. The material effective elastic and thermal conducting properties are obtained through direct large scale numerical simulations. The different formulations address the problem of finding the shape of the representative microstructural element for random heterogeneous media that increase the elastic moduli and thermal conductivity compared to existing products. The computational models are established using finite element method and images of virtual microstructures. In this paper we propose a new model of microstructures. This model is constructed with hexagonal prismatic rods and plates with volume fractions around 0.7 for the hard phase represented by hexagons of ice. A comparison between three two-phase elastic heterogeneous microstructures models is drawn. This illustrates the concept of design of microstructures using computational homogenization tools.     

基于迭代法的非线性弹性均质化研究

微观结构对复合材料的宏观力学性能具有至关重要的影响, 通过合理设计复合材料微观结构可以得到期望的宏观性能. 均质化方法作为一种有效的设计方法, 它从微观结构的角度出发, 利用均匀化的概念, 实现了对复合材料宏观力学性能的预测和设计. 而当考虑非线性因素, 均质化的实现就非常困难. 本文利用双渐近展开方法, 将位移按照宏观位移和微观位移展开, 推导了非线性弹性均质化方程. 通过直接迭代法, 对非线性弹性均质化方程进行了求解, 并给出了具体的迭代方法和实现步骤. 本文基于迭代步骤和非线性弹性均质化方程编写MATLAB 程序, 对3种典型本构关系的周期性多孔材料平面问题进行了计算, 对比细致模型的应变能、最大位移和等效泊松比, 对程序及迭代方法的准确性进行了验证. 之后对一种三元橡胶基复合材料进行多尺度均质化, 将其分为芯丝尺度和层间尺度. 用线弹性的均质化方法得到了芯丝尺度的等效弹性参数, 并将其作为层间尺度的材料参数. 在层间尺度应用非线性弹性均质化方法对结构进行计算, 得到材料的宏观等效性能, 并以实验结果为基准进行评价.      

Multi-level modeling of woven glass/epoxy composite for multilayer printed circuit board applications

The objective of this paper is to develop a hybrid homogenization method to predict the elastic properties of a common woven glass/epoxy composite substrate for multilayer circuit board applications. Comprehensive high resolution 3D finite element (FE) models of a quarter of the repeated unit cell (RUC) for the woven glass/epoxy composite were developed based on different micromechanical schemes. . Specifically, four different micromechanics schemes were investigated: self-consistent, Mori–Tanaka, three-phase approach and composite cylinder assemblage (CCA). The element based strain concentration matrices were determined and used to obtain the homogenized woven glass/epoxy composite properties via a specially developed MATLAB code. Attention was further devoted to the predictions of the homogenized elastic moduli of the multilayer printed circuit board (PCB). The results from our simulations, based on Mori–Tanaka and CCA, are in good agreement with existing experimental results, indicating that the newly proposed homogenization scheme can be used as a design tool to predict the overall properties of woven composite materials typically used in multilayer PCB applications.     

Optimizing multifunctional materials: Design of microstructures for maximized stiffness and fluid permeability

Topology optimization is used to systematically design periodic materials that are optimized for multiple properties and prescribed symmetries. In particular, mechanical stiffness and fluid transport are considered. The base cell of the periodic material serves as the design domain and the goal is to determine the optimal distribution of material phases within this domain. Effective properties of the material are computed from finite element analyses of the base cell using numerical homogenization techniques. The elasticity and fluid flow inverse homogenization design problems are formulated and existing techniques for overcoming associated numerical instabilities and difficulties are discussed. These modules are then combined and solved to maximize bulk modulus and permeability in periodic materials with cubic elastic and isotropic flow symmetries. The multiphysics problem is formulated such that the final design is dependent on the relative importance, or weights, assigned by the designer to the competing stiffness and flow terms in the objective function. This allows the designer to tailor the microstructure according to the materials’ future application, a feature clearly demonstrated by the presented results. The methodology can be extended to incorporate other material properties of interest as well as the design of composite materials.     

Homogenization for wave propagation in periodic fibre-reinforced media with complex microstructure. I—Theory

The effective elastic properties of periodic fibre-reinforced media with complex microstructure are determined by the method of asymptotic homogenization via a novel solution to the cell problem. The solution scheme is ideally suited to materials with many fibres in the periodic cell. In this first part of the paper we discuss the theory for the most general situation—N arbitrarily anisotropic fibres within the periodic cell. For ease of exposition we then restrict attention to isotropic phases which results in a monoclinic composite material with 13 effective moduli and expressions for each of these are determined. In the second part of this paper we shall discuss results for a variety of specific microstructures.     

确定复合材料宏观屈服准则的细观力学方法

运用细观力学中的均匀化方法，分析了含周期性微结构复合材料的宏观屈服准则，并对Hill-Tsai准则进行了修正。从基于复合材料细观结构的代表性胞元入手，运用塑性极限理论中的机动分析以及有限元方法，计算了细观结构的极限载荷域。通过宏细观尺度对应关系，得到复合材料的宏观屈服准则。     

PLASTIC LIMIT ANALYSIS OF DUCTILE COMPOSITE STRUCTURES FROM MICRO-TO MACRO-MECHANICAL ANALYSIS

The load-bearing capacity of ductile composite structures comprised of periodic composites is studied by a combined micro/macromechanicai approach. Firstly, on the microscopic level, a representative volume element （RVE） is selected to reflect the microstructures of the composite materials and the constituents are assumed to be elastic perfectly-plastic. Based on the homogenization theory and the static limit theorem, an optimization formulation to directly calculate the macroscopic strength domain of the RVE is obtained. The finite element modeling of the static limit analysis is formulated as a nonlinear mathematical programming and solved by the sequential quadratic programming method, where the temperature parameter method is used to construct the self-stress field. Secondly, Hill＇s yield criterion is adopted to connect the micromechanicai and macromechanical analyses. And the limit loads of composite structures are worked out on the macroscopic scale. Finally, some examples and comparisons are shown.