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.

连续体结构多工况拓扑优化与轻量化设计研究

汽车轻量化能有效地降低燃油消耗并减少污染物排放,是实现汽车节能减排的重要举措之一。拓扑优化是连续体结构轻量化设计的常用方法,由于实际工况通常较复杂,一般连续体结构的拓扑优化属于多工况拓扑优化问题,此时拓扑优化结果存在较多影响因素。本文引入多工况拓扑优化性能指标PI,通过算例对多工况拓扑优化权重系数的选取方法进行分析,寻找最佳权重系数分配方法,有效的解决了连续体结构多工况拓扑优化权重系数的选取问题。以某重型商用车平衡轴支架为应用对象,开展平衡轴支架多工况拓扑优化与轻量化设计,结果表明在满足性能要求的前提下,平衡轴支架减重了28.1%。

STUDY ON PARAMETERS FOR TOPOLOGICAL VARIABLES FIELD INTERPOLATED BY MOVING LEAST SQUARE APPROXIMATION

This paper presents a new approach to the structural topology optimization of continuum structures. Material-point independent variables are presented to illustrate the existence condition,or inexistence of the material points and their vicinity instead of elements or nodes in popular topology optimization methods. Topological variables field is constructed by moving least square approximation which is used as a shape function in the meshless method. Combined with finite element analyses,not only checkerboard patterns and mesh-dependence phenomena are overcome by this continuous and smooth topological variables field,but also the locations and numbers of topological variables can be arbitrary. Parameters including the number of quadrature points,scaling parameter,weight function and so on upon optimum topological configurations are discussed. Two classic topology optimization problems are solved successfully by the proposed method. The method is found robust and no numerical instabilities are found with proper parameters.

COMPUTER PROGRAM FOR DIRECTED STRUCTURE TOPOLOGY OPTIMIZATION

To compensate for the imperfection of traditional bi-directional evolutionary structural optimization, material interpolation scheme and sensitivity filter functions are introduced. A suitable filter can overcome the checkerboard and mesh-dependency. And the historical information on accurate elemental sensitivity numbers are used to keep the objective function converging steadily. Apart from rational intervals of the relevant important parameters, the concept of distinguishing between active and non-active elements design is proposed, which can be widely used for improving the function and artistry of structures directly, especially for a one whose accurate size is not given. Furthermore, user-friendly software packages are developed to enhance its accessibility for practicing engineers and architects. And to reduce the time cost for large timeconsuming complex structure optimization, parallel computing is built-in in the MATLAB codes. The program is easy to use for engineers who may not be familiar with either FEA or structure optimization. And developers can make a deep research on the algorithm by changing the MATLAB codes. Several classical examples are given to show that the improved BESO method is superior for its handy and utility computer program software.

Parameterized level set method for structural topology optimization based on the Cosserat elasticity

When describing the mechanical behavior of some engineering materials,such as composites,grains,biological materials and cellular solids,the Cosserat continuum theory has more powerful capabilities compared with the classical Cauchy elasticity since an additional local rotation of point and its counterpart(couple stress)are considered in the Cosserat elasticity to represent the material microscale effects.In this paper,a parameterized level set topology optimization method is developed based on the Cosserat elasticity for the minimum compliance problem of the Cosserat solids.The influence of material characteristic length and Cosserat shear modulus on the optimized structure is investigated in detail.It can be found that the microstructural constants in the Cosserat elasticity have a significant impact on the optimized topology configurations.In addition,the minimum feature size and the geometric complexity of the optimized structure can be controlled implicitly by adjusting the parameters of the characteristic length and Cosserat shear modulus easily.Furthermore,the optimized structure obtained by the developed Cosserat elasticity based parameterized level set method will degenerate to the result by using the classical Cauchy elasticity based parameterized level set method when the Cosserat shear modulus approaches zero.

ICM Method Combined with Meshfree Approximation for Continuum Structure

The independent continuous mapping（ICM） method is integrated into element free Galerkin method and a new implementation of topology optimization for continuum structure is presented.To facilitate the enforcement of the essential boundary condition and derivative of various sensitivities,a singular weight function in element free Galerkin method is introduced.Material point variable is defined to illustrate the condition of material point and its vicinity instead of element or node.The topological variables field is constructed by moving least square approximation which inherits the continuity and smoothness of the weight function.Due to reciprocal relationships between the topological variables and design variables,various structural responses sensitivities are derived according to the method for calculating the partial derivatives of compound functions.Numerical examples indicate that checkerboard pattern and mesh-dependence phenomena are overcome without additional restriction methods.

连续体结构的失效-安全拓扑优化设计

在连续体结构拓扑优化中,优化得到的最终构型可能会缺乏合理的冗余度,一旦出现局部失效情况就会造成结构系统无法正常工作。为了获得具有一定冗余度和可制造性的拓扑结构,提出了一种连续体结构的失效-安全拓扑优化设计方法。以传统拓扑优化结果作为参考依据确定局部破损模式,将局部破损模式无缝平铺在设计域上删除无效区域,以应变能最大为目标遍历失效场景确定最危险的局部破损区域,准确模拟最危险的局部破损情况,再进行后续的优化分析以获得具有更多冗余度和传力路径的拓扑优化结构。通过典型算例验证了该方法在受到单个载荷和多个载荷工况时的可行性,可以获得具有适度冗余和可制造性的拓扑结构。

桁架结构拓扑及截面尺寸优化设计方法

为克服传统基结构设计方法对最优解的束缚,实现桁架结构的拓扑布局及尺寸优化,提出了将连续体与离散杆系相结合的桁架结构优化设计方法。从连续体出发,基于SKO连续体拓扑优化方法得到了最优拓扑布局;以二值图像细化算法为基础,提出了基于有限单元8邻域网格模型的骨架提取算法,通过剥离冗余单元,得到了连续体拓扑优化结果的中心传力骨架;以单元主应力为判据,精确找到骨架中的关键点,并连接关键点形成了初始桁架结构;基于拉格朗日乘数法和KuhnTucker条件,以初始桁架中杆件的内外半径为设计变量,结构体积为约束条件,结构柔度为目标函数,建立了桁架结构杆件尺寸优化的数学模型,并推导出其优化迭代准则。最后,以一悬臂结构为例对该优化方法的应用进行了说明,并使用一经典算例与其他文献中的方法进行了对比,结果表明:该优化方法得到的桁架结构具有优化的拓扑构型和力学特性,杆件布局、尺寸合理,应力均匀。

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.

连续体结构拓扑优化应力约束凝聚化的ICM方法

为克服应力约束下拓扑优化问题约束数目多、应力敏度计算量大的困难,提出了应力约束化凝聚化的ICM方法．在利用Mises强度理论将应力约束转换成应变能约束后,提出了应力约束凝聚化的两条途径:其一为应力全局化的方法,其二为应力约束集成化的方法．由此建立了多工况下以重量为目标、以凝聚化应变能为约束的连续体结构优化模型,并利用对偶理论对优化模型进行了求解．4个数值算例表明:该方法具有较高的计算效率,得到的拓扑结构比较合理,不仅适用于二维连续体结构,也适用于三维连续体结构．

结构拓扑优化ICM方法的改善

对结构拓扑优化的ICM(独立、连续、映射)方法进行了深入探讨,通过选取不同的过滤函数可以不进行每步删除而得到清晰的拓扑图形.以位移约束为例阐述了ICM方法建模及求解过程.对位移约束、频率约束、位移及频率约束、简谐载荷激励下动位移幅值约束等拓扑优化进行了研究,计算算例表明ICIM方法在处理静力问题及动力问题的拓扑优化都是可行的.程序算法都在MSC.Nastran及MSC.Patran的二次开发环境下实现,与原软件有机结合在一起.

受应力约束的平面弹性体的拓扑优化

研究在给定外载和支承的条件下平面弹性体的拓朴优化，要求得到厚度均匀、使用材料最省且每一点都满足应力约束的弹性膜．在分析拓扑优化的特殊困难后，提出修改的满应力方法求解这一问题，使得设计空间在迭代过程中不变．文中的数值结果说明这个方法是有效的算法．

多工况应力和位移约束下连续体结构拓扑优化

将文［１］所提出的对拓扑变量的独立连续映射（ＩＣＭ）的拓扑优化方法应用于连续体结构，从而建立了统一的以重量为目标，考虑应力和位移约束的连续体结构拓扑优化模型。通过对位移一应力拓扑解和各工况下应力拓扑解的综合协调，进而对于协调拓扑解按照阈值完成从离散到连续的反演，并且采用分层与加权策略克服了“荷载病态”困难．给出的经典的二维平面问题和三维连续体结构拓扑优化算例表明，这种统一的模型由骨架结构发展到连续体结构的优化也是成功的．

抑制拓扑优化中灰度单元的双重SIMP方法

为了抑制连续体结构拓扑优化结果中的灰度单元,在实体各向同性材料惩罚密度法(Solid isotropic microstructures with penalization,SIMP)的基础上提出一种双重SIMP方法。首先利用含有灵敏度过滤的SIMP方法对设计变量进行迭代和优化,将优化结果作为中间变量,然后用不含有灵敏度过滤的SIMP方法对中间变量进行优化迭代,得到最终优化结果。以简支梁的结构柔顺度最小为例,将双重SIMP方法用于柔性结构的优化设计中并与标准SIMP算法相比;结果表明,双重SIMP方法得到清晰的拓扑分布结构,有效抑制了灰度单元,并且双重SIMP方法可以得到更小的结构柔顺度,证明了该方法的有效性;双重SIMP算法的灰度单元对惩罚因子的依赖性很小,可以选用较小的惩罚因子;针对不同的网格划分模型,双重SIMP算法得到的优化目标和灰度单元数更稳定,体现出了很好的网格依赖性。说明该方法具有很好的适用性和稳定性。

应力约束全局化处理的连续体结构ICM拓扑优化方法

由于应力约束按单元计,加之多工况,使得连续体结构拓扑优化由于约束数目太多,导致应力敏度分析计算量太大而无法接受。基于第四强度理论提出了应力约束条件全局化处理的方法,化为全局替代约束——总应变能约束,用ICM方法对总应变能约束条件下的连续体结构拓扑优化进行建模及求解,其过程分为三步:第一步选择最大应变能对应的工况,在给定重量下求出最小结构总应变能;第二步提出一个数值经验公式,借助第一步的结果,计算出各工况下的许用总应变能;第三步以第二步计算出来的各工况的许用总应变能作为约束,以重量为目标建立模型并求解。顺便指出,第二步的处理方法可以处理载荷相差特别大的情况,即病态载荷情况。数值算例表明:全局性应力约束可以更好地得到传力路径,对于处理多工况问题具有优势。

统一骨架与连续体的结构拓扑优化的ICM理论与方法

描述了基于ICM方法的结构拓扑优化新模型并应用于骨架与连续体结构。ICM方法意指独立、连续变量与映射变换及其反演。新模型将两种结构统一建立了具有重量目标函数和多工况下应力与位移约束下的优化问题。提出的过滤函数是ICM方法的关键技术之一。

疲劳寿命约束下的连续体结构拓扑优化

在连续体结构拓扑优化设计中,优化模型通常考虑强度、刚度、稳定性等力学性能.为研究在结构优化设计中疲劳特性对于结构拓扑的影响,基于独立、连续、映射(independent continuous mapping,ICM)拓扑优化方法,引入了疲劳寿命过滤函数,提出了连续体结构疲劳拓扑优化的模型和求解方法.建立以结构质量最小为目标、疲劳寿命为约束的连续体疲劳拓扑优化模型,通过将疲劳寿命约束转化为应力约束,对优化模型进行了求解.数值算例对比了不同刚度过滤函数、疲劳寿命过滤函数的系数,得到了过滤函数系数对结构质量与结构构型的影响规律;研究了疲劳寿命和应力分别作为约束时拓扑优化结果的差异性,给出了不同的循环外载荷工况情况下的算例,得出了随循环外载荷变化时的结构质量及结构构型.研究结果验证了连续体疲劳拓扑优化理论的有效性与可行性.

基于ANSYS平台的连续体渐进结构优化设计及其应用

基于ANSYS分析软件平台,在传统的渐进结构优化(ESO)思想的基础上,依据对结构位移灵敏度的分析,较严格地导出了结构的应力灵敏度;并根据荷载的最优传递路线概念提出了一种具有应力约束连续体结构拓扑优化的单元删除策略,从而使得在结构拓扑优化的迭代过程中,最大应力的变化尽可能小,且最优结构所对应的最大应力能控制在容许应力范围之内。若干工程算例的优化结果表明,该方法在结构优化目标质量最轻的前提下计算效率相对较高,且优化结果与工程实际要求较吻合,同时也从数值上验证了该方法的可行性和有效性。

静动态多目标下的连续体结构拓扑优化

为实现静动态多目标下的拓扑优化结构设计,以静态柔顺度或节点位移最小化和动态特征值最大化加权函数为目标,提出并建立静动态多目标连续体结构拓扑优化模型。单元形函数插值作用下的节点独立变量场具有零阶连续性,在连续体结构拓扑描述一定程度上避免棋盘格现象。变量独立性在单元刚度阵、单元质量阵和单元体积的建模中更具有灵活性。规一化目标函数避免不同性质目标函数的量级差异。基于过滤法进一步改善目标函数的光滑性,克服了连续体拓扑优化中常见数值不稳定性问题。数值算例结果表明,模型和方法在静动态多目标下连续体结构拓扑优化设计中的可行性和有效性。

利用导重法进行结构拓扑优化

介绍了导重准则法基本原理并将其应用于杆系结构及连续体结构拓扑优化。对于重量约束结构性能最优化和多性态约束结构重量最小化问题的连续结构拓扑优化问题,详细推导了导重法与变密度SIMP(Solid Isotropic Microstructure with Penalization)法相结合的更加规范的全新优化准则公式,并给出了相应的算例。计算结果表明,导重法不仅适用于传统的结构尺寸优化与形状优化,而且可很好地求解结构拓扑优化问题,并具有公式简单、通用性强、收敛速度快及优化效果好的优点。