【作者】 隋允康； 铁军；

【Author】 ~*SUI Yun-kang~1,TIE Jun~2 (1.College of Mechanical Engineering and Applied Electronics Technology,Beijing University of Technology,Beijing 100124,China; 2.College of Science,Dalian Nationalities University,Dalian Development Zone 116600,China)

【机构】 北京工业大学机电学院； 大连民族学院理学院；

【摘要】 对于连续体结构拓扑优化问题,本文指出:其最大求解困难除因0-1离散变量的本质造成,更因为目标函数、约束条件在理论上无法建立同0-1拓扑变量的直接联系。ICM(独立-连续-映射)方法解决了这一困难:用阶跃函数构筑了0-1变量与单元具体物理量或几何量联系的桥梁;将其逆函数(我们称之为跨栏函数)代入具体的优化表达式中,就可以显化拓扑优化模型与0-1变量的关系;其中跨栏函数用其逼近函数——过滤函数代替之,于是,不可微的结构拓扑优化规划就被可微化;进而可调常用的光滑算法进行有效的求解了。在磨光函数和过滤函数分别逼近阶跃函数和跨栏函数的同时,传统的0-1拓扑离散变量扩展为[0,1]区间上的连续变量了。由于许用应力、杨氏模量、密度等材料性质被过滤函数识别,因而产生了相应的单元全程性质的概念,同理定义了单元全程几何参数的概念。基于ICM方法,利用单元全程许用应力,可以便捷地推导出结构拓扑优化应力奇异问题的ε-松弛解法的公式。提出了抛物型凝聚函数,证明了相关的定理,并且用于应力凝聚化处理的拓扑优化问题求解。另外,数值算例表明本文方法是有效的。更多还原

【Abstract】 For the topology optimization problems of continuum structures,this paper points out that its largest difficulty lies in the direct relationship between the objective function and the constraints with 0-1 topology variables could not be established in theory,apart from the nature of 0-1 discrete variables.ICM(Independent, Continuous and Mapping) method resolves this difficulty.According to ICM method,use the step function to construct a bridge between the 0-1 variables and element’s concrete physical quantity or geometric quantity.If the inverse functions of the step function,which is called as the hurdle function by us,are substituted into the specific optimization expressions,the relationship between topology optimization model and 0-1 variables can be displayed.Since the hurdle function is replaced with its approximation function-the filter function. non-differentiable structural topology optimization problem becomes differentiable.Therefore,the common smooth algorithms can be used to solve them effectively.When the polish functions and filtering functions approximate step functions and hurdle function respectively,the traditional 0-1 topology discrete variables are expanded to continuous variables in[0,1].As allowable stress,young’s modulus,density and other material properties are recognized by filter functions,the concepts of the global nature of the element are presented. Similarly we can define the concept of global geometric parameters of the element.Based on ICM method and taking advantage of the global allowable stresses of the element,we can easily deduce the formula of stressε-relaxation method in the singularity problem of topology optimizations.Mathematically,we propose a parabolic aggregate function and prove the related theorem which is used to solve the structural topology optimization problems with stress constraints.Moreover,numerical examples indicate that the method is efficient.更多还原