【作者】 叶红玲；

【导师】 隋允康；

【作者基本信息】 北京工业大学 ， 机械设计及理论， 2005， 博士

【摘要】 结构拓扑优化是目前结构优化研究领域的热点之一,与尺寸优化和形状优化相比,结构拓扑优化需要确定的参数更多,取得的经济效益更大,对工程设计人员更具吸引力。连续体结构拓扑优化因为数学模型建立困难、设计变量较多,数值计算量巨大而被认为是当前结构优化领域内的难点之一,对其研究具有非常重要的理论意义和工程应用前景。本文基于隋允康教授提出的ICM(Independent Continuous Mapping,即独立、连续、映射)拓扑优化方法,研究了以最小重量为目标、以应力和位移为约束的连续体结构的拓扑优化,建立了应力和位移约束与拓扑设计变量之间的近似显式关系,处理了多工况下的连续体结构的拓扑优化问题,同时将该理论由二维连续体结构推广到了三维连续体结构上。以大型通用软件MSC.Patran& Nastran 为平台,开发了相应的软件模块,从而将理论与应用相结合,进而为工程实际服务。主要研究内容包括: 1.模型的建立(1) 基于ICM 方法,对单元重量、单元许用应力和单元刚度分别引入了不同的过滤函数,把0-1 型离散拓扑变量转化为[0,1]区间上的连续拓扑变量,建立了拓扑变量连续的优化模型。(2) 对局部性应力约束,借助于第四强度理论,提出了一种应力全局化方法,将局部的单元Mises 应力约束转化为结构总应变能约束,从而建立了以结构应变能代替应力约束的多载荷工况下的连续体结构拓扑优化模型,并对多载荷工况下的最佳传力路径进行了权衡。(3) 根据单位虚载荷法,将全局性的位移约束显式化,建立了以重量最小为目标的位移约束下的优化模型,同时也处理了多工况下的最佳传力路径的问题。(4) 对于应力和位移约束下的连续体结构,建立了包含两类约束的无量纲化的优化模型。2.模型的求解(1) 利用对偶理论,将建立的优化模型转化为对偶模型,用序列二次规划法进行了求解,从而减少了设计变量的数目,提高了求解效率,使迭代次数由原来的上百次减少为几十次。(2) 利用最小二乘法原理,采用数值方法,对单元重量、单元许用应力和单元刚度的过滤函数形式进行了探讨;对应力全局化方法中的结构应变能,通过数值实验确定了结构应变能约束的修正系数,使许用应变能更加合理。(3) 基于满应力准则法,采用了自适应的删除率,对原有算法中拓扑结构受删除率影响的问题进行了改进。(4) 连续拓扑变量在优化求解过程中自动向0 或1 两端靠近,只在结构优化分析的最后一步将连续拓扑变量反演为离散拓扑变量。其优点在于结构优化过程中,保持了拓扑结构的连续性,减少了直接将单元删除而可能诱发的结构奇异,并且避免了因误删单元而不能使拓扑结构恢复的问题。3.软件模块开发根据上述模型和解法,构造算法并编制程序,以高性能的有限元软件MSC.Patran&Nastran 为平台,开发了界面友好的软件模块,完成了在应力和位移约束下,以二维和三维连续体结构为研究对象的静力拓扑优化模块,为结构优化理论快速走向工程应用提供了一条新的途径。一些工程实例的数值模拟表明本文更多还原

【Abstract】 Topology optimization that is one of the major subjects of optimization research up to date, usually needs to determine many more parameters than size optimization and shape optimization. However, it can be obtained more benifit so that it is more attractive for design engineer. It is well known that topology optimization of continuum structure is one of the major challenges because of difficulty to establish a good geometric model which comprising a large number of design variables, and complexity of optimization algorithm. In order to gain a reasonable structure, in the present dissertation, the ICM (Independent Continuous Mapping) method proposed by professor Y. K. Sui is used to investigate topology optimization of static structures. An optimal model with the object function of weight, stress constraints and displacement constraints is constructed. Two approximated explicit formulations of stress constraints and displacement constraints with respect to design variables are presented. Furthermore, this novel model can readily not only deal with topology optimization of continuum structures under the consideration of multiple load cases, but can be extended to three-dimensional continuum body. The corresponding numerical procedures are also developed, using the MSC.Patran & Nastran software platform. The main contents of this dissertation can be summarized as follows: 1. Establishment of Optimization Model (1) A suitable set of Filter functions of element with respect to weight, allowable stress and stiffness is introduced by means of ICM method. The discrete topology variables usually to be 0 or 1, are transformed into continuous topology ones varied from 0 to 1. Thus, the optimal model can be established using the concept of continuous variables. (2) A globalization method of stress constraints is proposed by virtue of the von Mises’yield criterion in machanics of materials. Thus, transformation of the local stress constraints of element into the global stain energy constraints of structure is obtained. The optimal model of continuum strucure subjected to strain energy of structure is constructed, under the consideration of multiple load cases. (3) An explicit formulation of displacement that is of global with related to topological variables is presented by using of unit virtual load method. Then the optimal model which refers to weight as objective and subjected to displacement constraints is suggested. In addition, the best path transmitted force in the multiple load cases is selected successfully. (4) The optimal model that normalizes with two types of dimensionless constraints, is further derived, for continuum structure with stress constraints and displacement constraints. 2. Solution of Optimization Model (1) The dual model corresponding to the optimal model of continuum structure is更多还原