一种变位移约束限的结构拓扑优化方法

针对仅有位移约束和重量最小的结构拓扑优化问题, 基于ICM(独立、连续、映射)方法思路,提出了一种变位移约束限的结构拓扑优化方法. 在每一轮子循环迭代求解开始时,为了控制拓扑设计变量的变化量, 形成和引进了新的位移约束限. 另外,建立了单元删除阈值和几轮迭代循环的单元删除策略. 为了确保优化迭代中结构非奇异和方法具有增添单元的功能,在结构孔洞和边界周围引入了一层人工材料单元,并建立了一套有效结构信息到结构最大设计域信息的映射转换方法. 结合对偶求解方法,形成了一种新的连续体结构的拓扑优化方法. 给出的算例表明该方法没有目标函数的振荡现象,且验证了该方法的正确性和有效性. 关键词:拓扑优化; 位移约束; 连续体结构; ICM方法; 渐进结构优化;      

基于位移和应力灵敏度的结构拓扑优化设计

针对仅有应力约束以及含位移和应力约束的重量最小结构拓扑优化问题,基于ICM(独立、连续、映射)方法和渐进结构优化方法的思路,提出了一种结构拓扑优化方法. 在优化迭代循环的每一轮子循环迭代求解开始时,通过形成和引进新的位移和应力约束限, 自动构建设计变量移动限.将结构应力约束归并为几个最可能的有效应力约束,大大减少了应力灵敏度的分析量. 另外,建立了单元删除阈值和几轮迭代循环的单元删除策略. 为了确保优化迭代中结构非奇异和方法具有增添单元的功能,在结构孔洞和边界周围引入了一层人工材料单元,构建了其等效的优化模型. 结合位移和应力灵敏度分析,形成了一种新的连续体结构的拓扑优化方法. 给出的算例验证了该方法的正确性和有效性.      

基于蚁群算法的桁架结构布局离散变量优化方法

提出的布局优化方法是将桁架结构的截面变量、拓扑变量及形状变量统一为离散变量.将离散变量转化为适应于蚁群算法求解TSP问题的离散变量,应用MATLAB语言编写求解桁架结构布局优化程序,最终实现对问题的分析与求解.通过对几个经典的平面、空间桁架结构布局优化算例的验算表明:本文设计的基于蚁群算法的桁架结构布局离散变量优化方法较单独处理截面优化、拓扑优化及形状优化问题具有更大的效益,相对于其他布局优化方法也展现出更好的优化效果.“基于蚁群算法的桁架结构布局离散变量优化方法”在程序设计、求解速度、求解空间及其方法通用性等方面都表现出良好的性能,并且简单、实用,适应于实际工程应用.     

连续纤维增强复合材料结构基频最大化设计

与传统的金属材料相比, 纤维增强复合材料在强度、刚度、抗断裂等诸多方面具备更优良的性能, 目前纤维增强复合材料已在汽车、航空航天等工业领域得到了广泛应用. 本文提出一种求解连续纤维增强复合材料结构无阻尼自由振动下的基频最大化问题的拓扑优化方法. 为了实现结构拓扑构型与纤维角度的同步优化, 建立了以准许的材料用量体积分数为约束、以结构的一阶特征值为目标函数的动力学拓扑优化模型, 该模型包括表征结构拓扑构型的密度设计变量和表征纤维方向的角度设计变量. 详细推导了特征值目标函数关于密度设计变量和角度设计变量的解析灵敏度列式, 并采用移动渐进线方法 (method of moving asymptotes, MMA) 进行了优化求解; 最后通过3个数值算例验证本文方法的有效性, 其中包括一个以刚度最大化为目标的静力学优化算例, 和两个以一阶特征值为目标的动力学优化算例. 结果表明, 所提方法优化迭代过程稳健, 收敛快, 能够在实现结构拓扑构型与纤维角度的一体化优化的同时, 有效提高结构的频率.      

FUNDAMENTAL FREQUENCY MAXIMIZATION DESIGN FOR CONTINUOUS FIBER-REINFORCED COMPOSITE STRUCTURES 1)

与传统的金属材料相比, 纤维增强复合材料在强度、刚度、抗断裂等诸多方面具备更优良的性能, 目前纤维增强复合材料已在汽车、航空航天等工业领域得到了广泛应用. 本文提出一种求解连续纤维增强复合材料结构无阻尼自由振动下的基频最大化问题的拓扑优化方法. 为了实现结构拓扑构型与纤维角度的同步优化, 建立了以准许的材料用量体积分数为约束、以结构的一阶特征值为目标函数的动力学拓扑优化模型, 该模型包括表征结构拓扑构型的密度设计变量和表征纤维方向的角度设计变量. 详细推导了特征值目标函数关于密度设计变量和角度设计变量的解析灵敏度列式, 并采用移动渐进线方法 (method of moving asymptotes, MMA) 进行了优化求解; 最后通过3个数值算例验证本文方法的有效性, 其中包括一个以刚度最大化为目标的静力学优化算例, 和两个以一阶特征值为目标的动力学优化算例. 结果表明, 所提方法优化迭代过程稳健, 收敛快, 能够在实现结构拓扑构型与纤维角度的一体化优化的同时, 有效提高结构的频率.     

考虑结构自重的基于NURBS插值的3D拓扑描述函数法

在许多如大坝、桥梁等大型土木工程结构中,结构的自重是初始设计阶段必须考虑的重要载荷之一,因此研究自重载荷作用下的结构拓扑优化设计问题具有十分重要的意义.针对考虑自重载荷作用的拓扑优化问题所面临的主要困难,总结了现有处理考虑自重载荷的拓扑优化问题的三类主要方法;提出一种基于非均匀有理B样条(non-uniform rational B-splines,NURBS)基函数插值的拓扑描述函数方法,基于此方法研究了考虑设计依赖自重载荷作用的2D/3D结构优化设计问题.在列式下,高阶NURBS基函数被同时用于三维NURBS实体片中的几何场、位移场及设计变量场插值,实现了几何模型、分析模型和优化模型的有效统一,确保了位移场及设计变量场的高阶连续性;详细推导了基于NURBS基函数插值的考虑自重载荷作用的三维结构拓扑优化模型及其灵敏度列式,并采用移动渐进线方法(method of moving asymptotes,MMA)进行了优化求解;多个算例验证了方法的有效性和稳定性,结果表明,优化迭代过程稳健,收敛快,能够有效地克服自重载荷作用下连续体结构拓扑优化中经常遇到的低密度区域材料的寄生效应及目标函数的非单调性等问题.     

基于拓扑描述函数的特定性能材料设计方法

研究了拓扑描述函数在材料设计中的应用,给出了一种基于拓扑描述函数的特定性能材料设计问题的提法和求解方法.将拓扑描述函数表示成含参数的基函数之和,将材料微结构拓扑优化问题转化为设计基函数描述参数的尺寸优化问题,使问题求解更方便.基于拓扑描述函数的方法可以准确确定设计域上任意点的材料分布,避免了变密度法常出现的棋盘效应、设计变量和有限元单元相关的缺点;与传统的水平集方法相比,其优化模型可以利用现有的优化方法求解,避免了差分法求偏微分方程.具有正泊松比和负泊松比的特定弹性性能材料的设计算例,说明了基于拓扑描述函数的材料设计方法的有效性.     

同时满足刚度和强度约束的框架拓扑优化

基于ICM（Independent Continuous Mapping,即独立、连续、映射）方法．对单元重量、单元许用应力和单元刚度分别引入不同的过滤函数,把0-1型离散拓扑变量转化为[0．1]区间上的连续变量．建立了拓扑变量连续的优化模型。借助满应力准则将应力约束转化为拓扑变量的动态下限．用单位虚载荷法将位移约束显式化．得到拓扑优化的近似显式模型。为了提高模型的求解效率,根据对偶理论求解原问题的对偶模型,通过在对偶空间迭代求解对偶模型得到原模型的解。引入结构非奇异、结构响应不被违背和结构重量不改变三个准则判断迭代收敛。并根据这三个准则自适应的调整折减系数来搜索最佳阁值。然后根据闻值将连续拓扑变量回归为0-1型离散拓扑变量。利用MSC／Nastran的开放性,借助MSC／Patran提供的PCL（Patran Command Language）开发环境．完成了满足刚度和强度的多变量的框架拓扑优化程序。算例结果表明．用ICM方法解决多变量框架拓扑优化问题是快速、有效的。     

基于D-P准则的压力相关材料结构拓扑优化

基于描述材料力学行为的Drucker-Prager(D-P)屈服准则, 研究了压力相关材料连续体结构拓扑优化设计问题的数学模型和数值算法. 以单元材料人工密度为设计变量, 结合SIMP惩罚模型和多孔微结构局部应力插值模型, 建立了以材料体积最小化为目标、考虑材料D-P屈服条件约束的优化问题数学模型. 利用\varepsilon-松弛方法消除奇异解现象, 采用伴随法有效推导约束函数灵敏度计算公式, 运用基于梯度的连续变量优化算法迭代求解优化问题. 数值算例验证了优化模型的正确性及数值算法的有效性, 并通过与von Mises应力约束优化结果的比较, 说明了材料的压力相关特性会对结构最优拓扑产生重要影响. 该方法设计出的最优拓扑由于充分利用了压力相关材料的抗压能力, 因而更为合理和实际.      

连续体结构的拓扑优化设计

对基于有限元数值求解技术的连续体结构的拓扑优化设计技术进行了综述. 利用密度-刚度插值格式和优化准则方法, 以结构的柔度最小化作为优化的目标函数, 论述并建立线弹性结构的静力学拓扑优化设计的数学模型和设计变量显示的迭代格式; 基于数学规划方法中的一种凸规划方法----移动渐近线方法和密度方法, 以结构的频率最大化作为优化的目标函数, 论述并建立了特征值问题拓扑优化设计的数学模型和设计变量隐式的更新方法. 对多目标拓扑优化问题、柔性机构的拓扑优化问题以及多物理场拓扑优化设计问题进行了讨论. 对优化结构中出现的棋盘格式和网格依赖性等数值计算问题进行剖析和讨论, 介绍和分析了目前解决数值计算问题常见的方法, 在此基础上对边界扩散现象进行了讨论. 给出了连续体结构拓扑优化设计的程序流程, 并用Matlab程序实现了算法, 通过几个典型的算例证明所综述方法的有效性.      

Optimal internal architectures of femoral bone based on relaxation by homogenization and isotropic material design

The influence of the loading conditions on the trabecular architecture of a femur is investigated by using topology optimization methods. The response of the bone to physiological loads results in changes of the internal architecture of bone, reflected by a modification of internal effective density and mechanical properties. The homogenization based optimization model is developed for predicting optimal bone density distribution, wherein bone tissue is assumed to be a composite material consisting of a mixture of material and void. The homogenization scheme treats the geometric parameters of the microstructures and their orientation as design variables and homogenizes the properties in that microstructure, which is generally anisotropic. The penalization of the optimal material density then leads to a classical optimal structure which consists of regions with bone material and regions without bone material. The IMD (Isotropic Material Design) approach is next applied to determine the optimal elasticity tensor in terms of the bulk and shear moduli for the present loading applied to the femoral bone sample. IMD is able to provide both the external shape and topology together with the optimal layout of the isotropic moduli. Both topology optimization methods appear to be complementary. Simulations of the internal bone architecture of the human proximal femur results in a density distribution pattern with good consistency with that of the real bone.     

特定方向"零膨胀"的最小柔顺性结构优化设计

工程中很多承载结构必须面对苛刻的温度变化工作环境,如卫星天线、太空照相机和电子器件等。剧烈的温度变化引起较大的热变形,造成仪器信号失真,精度下降;同时温度应力也会造成结构破坏甚至失效,因此零膨胀材料的研制备受关注。近年来国内外很多学者对此进行了研究,设计出具有特定等效膨胀系数的微结构,但考虑到制备工艺的限制,这类具有复杂微结构的材料制备起来比较困难,成本较高;同时这类材料一般不具备足够的刚度,难以满足承载性能的要求。本文基于结构优化设计技术,采用拓扑优化方法直接设计出具备较高的承载性能和特定方向变形较少受热载荷影响的结构。本文提出采用多目标优化的方法设计圆环结构,使其具有较高的刚度和在热载荷下圆环内表面具有较好的热几何稳定性。由于用单相材料无法同时满足高刚度和低热膨胀的要求,因此假设结构由两种不同的材料构成,用连续体拓扑优化的方法设计三相材料(两种实体材料MAT-I、MAT-II和空材料)在设计域上的最优分布,使结构满足设计要求。由对称性,设计域取为圆环的一个扇面,将设计域离散成有限元网格,每个单元具有两个设计变量:实体材料的体分比和MAT-I在实体材料中所占的体分比,采用伴随法进行灵敏度分析,用GCMMA方法求解此问题,采用体积守恒的Heaviside密度过滤函数保证获得清晰的最优拓扑构型以及避免棋盘格式的出现。通过两个数值算例,表明使用本文提出的多目标优化模型能够得到特定方向"零膨胀"同时具有一定刚度的结构设计,且这种宏观结构尺度上的两种材料组成的拓扑构型相对易于制造。     

Relating cellular structure of open solid food foams to their Young’s modulus: Finite element calculation

This paper investigates the role of structure on Young’s modulus of open cell materials of relative densities between 0.1 and 0.3. The cellular solid is obtained by generating mixture size of spherical voids using the Random Sequential Addition – RSA algorithm. The relative density of the material is controlled by increasing void number and overlap. Structural effects consider mainly a Gaussian distribution of spherical void size of varying width, distribution centre and void overlap distance. Finite element method is used to calculate effective Young’s modulus using a regular meshing scheme of 3D typical cellular solids and Conjugate Gradient solver. It is found that sphere overlap has the largest effect compared to sphere distribution width for a given density. A large scatter in the wall thickness distribution is predicted when overlapping is increased or when the width of sphere size distribution is decreased. Increased rigidity is found to be correlated to particular arrangement of mixture size spheres which is pointed out using the Pair Correlation Function. Experimental evidence of the role of void overlapping is treated in the case of bread crumbs structures determined using X-ray tomography. The scatter of effective Young’s modulus for a given relative density is sensitive to void overlapping.     

A structural topological optimization method for multi-displacement constraints and any initial topology configuration

In density-based topological design, one expects that the final result consists of elements either black （solid material） or white （void）, without any grey areas. Moreover, one also expects that the optimal topology can be obtained by starting from any initial topology configuration. An improved structural topological optimization method for multidisplacement constraints is proposed in this paper. In the proposed method, the whole optimization process is divided into two optimization adjustment phases and a phase transferring step. Firstly, an optimization model is built to deal with the varied displacement limits, design space adjustments, and reasonable relations between the element stiffness matrix and mass and its element topology variable. Secondly, a procedure is proposed to solve the optimization problem formulated in the first optimization adjustment phase, by starting with a small design space and advancing to a larger deign space. The design space adjustments are automatic when the design domain needs expansions, in which the convergence of the proposed method will not be affected. The final topology obtained by the proposed procedure in the first optimization phase, can approach to the vicinity of the optimum topology. Then, a heuristic algorithm is given to improve the efficiency and make the designed structural topology black/white in both the phase transferring step and the second optimization adjustment phase. And the optimum topology can finally be obtained by the second phase optimization adjustments. Two examples are presented to show that the topologies obtained by the proposed method are of very good 0/1 design distribution property, and the computational efficiency is enhanced by reducing the element number of the design structural finite model during two optimization adjustment phases. And the examples also show that this method is robust and practicable.     

Effects of pressure-sensitivity and plastic dilatancy on void growth and interaction

Hydrostatic stress can affect the non-elastic deformation and flow stress of polymeric materials and certain metallic alloys. This sensitivity to hydrostatic stress can also influence the fracture toughness of ductile materials, which fail by void growth and coalescence. These materials typically contain a non-uniform distribution of voids of varying size-scales and void shapes. In this work, the effects of void shape and microvoid interaction in pressure-sensitive materials are examined via a two-prong approach: (i) an axisymmetric unit-cell containing a single ellipsoidal void and (ii) a plane-strain unit-cell consisting of a single large void and a population of discrete microvoids. The representative material volume in both cases is subjected to physical stress states similar to highly stressed regions ahead of a crack. Results show that oblate voids and microvoid cavitation can severely reduce the critical stress of the material. These effects can be compounded under high levels of pressure-sensitivity. In some cases, the critical stress responsible for rapid void growth is reduced to levels comparable to the yield strength of the material. The contribution of void shape and pressure-sensitivity to the thermal- and moisture-induced voiding phenomenon in IC packages is also discussed.     

Void growth in power-law creeping solids: Effect of surface diffusion and surface energy

This paper addresses the growth of a void in a nonlinearly creeping material in the presence of the void-surface energy effect and void-surface diffusion driven by surface curvature gradients. Large strain finite element analysis of the coupled problem indicates that microstructural variables (porosity and void aspect ratio), as well as macroscopic deformation rates are strongly affected by the relative strength of the void-surface energy effect and the void-surface diffusion process vis-a-vis the rate of creep deformation in the bulk of the solid. The phenomenon is characterized by two-dimensionless groups, one measuring the strength of the surface diffusion process with respect to the nonlinear creep deformation in the interior of the solid, and the other the magnitude of the surface energy of the void in relation to the applied load and the size of the void. The computations reveal a rich variety of solutions that reflect a wide range of external load, material, and geometric parameters. Classical void growth studies that ignore both surface diffusion and surface energy effects are shown to recover only one case of this family of solutions. The computations also serve to quantitatively evaluate recent constitutive theories for porous nonlinear materials that account for continuously evolving microstructure, but do not include surface diffusion or surface energy effects.     

Rate effects on toughness in elastic nonlinear viscous solids

A micromechanics-based constitutive relation for void growth in a nonlinear viscous solid is proposed to study rate effects on fracture toughness. This relation is incorporated into a microporous strip of cell elements embedded in a computational model for crack growth. The microporous strip is surrounded by an elastic nonlinear viscous solid referred to as the background material. Under steady-state crack growth, two dissipative processes contribute to the macroscopic fracture toughness—the work of separation in the strip of cell elements and energy dissipation by inelastic deformation in the background material. As the crack velocity increases, voids grow in the strain-rate strengthened microporous strip, thereby elevating the work of separation. In contrast, the energy dissipation in the background material decreases as the crack velocity increases. In the regime where the work of separation dominates energy dissipation, toughness increases with crack velocity. In the regime where energy dissipation is dominant, toughness decreases with crack velocity. Computational simulations show that the two regimes can exist in certain range of crack velocities for a given material. The existence of these regimes is greatly influenced by the rate dependence of the void growth mechanism (and the initial void size) as well as that of the bulk material. This competition between the two dissipative processes produces a U-shaped toughness-crack velocity curve. Our computational simulations predict trends that agree with fracture toughness vs. crack velocity data reported in several experimental studies for glassy polymers and rubber-modified epoxies.     

A physical model for nucleation and early growth of voids in ductile materials under dynamic loading

Spall fracture and other rapid tensile failures in ductile materials are often dominated by the rapid growth of voids. Recent research on the mechanics of void growth clearly shows that void nucleation may be represented as a bifurcation phenomenon, wherein a void forms spontaneously followed by highly localized plastic flow around the new void. Although thermal, viscoplastic, and work hardening effects all play an essential role in the earliest stages of nucleation and growth, the flow becomes dominated by spherical radial inertia, which soon causes all voids to grow asymptotically at the same rate, regardless of differences in initial conditions or constitutive details, provided only that there is the same density of matrix material and the same excess loading history beyond the cavitation stress.These two facts, initiation by bifurcation at a cavitation stress, at which a void first appears, and rapid domination by inertia, are used to postulate a simple, but physically realistic, model for nucleation and early growth of voids in a ductile material under rapid tensile loading. A reasonable statistical distribution for the cavitation stress at various nucleation sites and a simple similarity solution for inertially dominated void growth permit a simple calculation of the initiation and early growth of porosity in the material.Parametric analyses are presented to show the effect that loading rate, peak loading stress, density of nucleation sites, physical properties of the material, etc. have on the applied pressure and distribution of void sizes when a critical porosity is reached.     

Coupling effects of void size and void shape on the growth of prolate ellipsoidal microvoid

The combined effects of void size and void shape on the void growth are studied by using the classical spectrum method. An infinite solid containing an isolated prolate spheroidal void is considered to depict the void shape effect and the Fleck-Hutchinson phenomenological strain gradient plasticity theory is employed to capture the size effects. It is found that the combined effects of void size and void shape are mainly controlled by the remote stress triaxiality. Based on this, a new size-dependent void growth model similar to the Rice-Tracey model is proposed and an important conclusion about the size-dependent void growth is drawn: the growth rate of the void with radius smaller than a critical radius r_c may be ignored. It is interesting that r_c is a material constant independent of the initial void shape and the remote stress triaxiality.The project supported by the National Natural Science Foundation of China (A10102006) and the New Century Excellent Talents in Universities of China. The English text was polished by Keren Wang.