基于“有无复合体”的应力约束下桁架和平面膜结构拓扑优化的统一模型

THE UNIFORM MODEL BASED ON THE EXIST-NULL COMBINATION FOR THE TRUSS AND MEMBRANE TOPOLOGICAL OPTIMIZATION WITH STRESS CONSTRAINT

根据独立连续拓扑变量概念 ,建立了桁架和平面膜结构拓扑优化的有无复合体模型 ,从而不引入过滤函数实现拓扑变量在连续型和离散型之间的转换 .推导了有无复合体杆单元的面积与膜单元的厚度同重量、单元刚度阵都是“有单元”和“无单元”相应量的线性组合 ,进而把这一线性关系延拓到许用应力 .借助于有无复合体建立了在应力约束下骨架和连续体结构拓扑优化的统一模型 ,同时提出了求解这一模型的有效算法 ,获得了令人满意的计算结果 .

According to the idea of independent and continuous topological variables, a uniform exist null combined model of the bar element and the membrane element for the topological optimization is constructed to implement the transformation between continuous and discrete topological variables without using the filter function. It is derived that the area of bar and the thickness of membrane with the weight or the stiffness matrix are all the linear combinations of the corresponding quantities of the ‘exist element’ and the ‘null element’. Then this linear relation is extended to the allowable stress. The uniform mathematical model of the topological optimization of skeleton and continuum structure with stress constraint is established in terms of the exist null combination. At the same time an effective algorithm is proposed to solve the uniform mathematical model. In line with the zero approximation of the stress constraint function, a solution of topological optimization with stress constraint of single loading case is obtained from the strength condition. Then, the average value of each topological variable under all of the loading cases is taken as a solution of multiple loading cases. Finally, a self adaptive algorithm gives the transformation of the continuous topological variable to discrete variable according to a doorsill. It can get satisfactory computational results with rapid and stable convergence. This work also shows that the presence of independent and continuous topological variable is valuable to the research of structural topology optimization.