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 引用本文: 彭细荣, 隋允康, 郑勇刚. 节点拓扑变量非耦合映射的ICM方法. 力学学报, 待出版.
Peng Xirong, Sui Yunkang, Zheng Yonggang. Icm method with a mapping based on node- uncoupled topology variables. Chinese Journal of Theoretical and Applied Mechanics, in press.
 Citation: Peng Xirong, Sui Yunkang, Zheng Yonggang. Icm method with a mapping based on node- uncoupled topology variables. Chinese Journal of Theoretical and Applied Mechanics, in press.

## ICM METHOD WITH A MAPPING BASED ON NODE- UNCOUPLED TOPOLOGY VARIABLES

• 摘要: 文章提出了节点拓扑变量一种非耦合映射的ICM方法, 对于结构拓扑优化问题予以建模和求解: 首先将基结构划分为由较小单元组成的网格, 取节点独立、连续的拓扑变量, 建立了一种双线性形函数插值的变量非耦合映射, 替代了单元独立连续拓扑变量, 使单元的“有”或“无”连续化近似, 实现了节点拓扑变量过滤识别与物理量的单元内插值, 推导建立了节点拓扑设计变量的优化模型, 采用基于变量可分离的二阶对偶规划算法求解, 并且改进了最优拓扑构型的圆整技术. 接着以常见的位移约束下结构重量(或体积)极小拓扑优化问题为例, 演示了上述建模及求解过程. 最后分别给出了单载荷工况和多载荷工况下的位移约束拓扑优化的算例, 数值计算结果验证了本方法的有效性. 研究有如下优点: 克服了以往基于单元拓扑变量研究的缺陷, 即最优结构边界为锯齿形, 得到的最优结构的拓扑边界光滑清晰; 给出了节点拓扑变量和单元拓扑函数场定义, 提炼出构造该场必须遵循的5点准则, 克服了节点拓扑ICM方法有关研究中存在的不足; 得到节点设计变量不再是耦合关系, 可以方便地求出结构物理量的二阶导数, 从而利用变量可分离对偶优化算法进行高效的寻优; 研究成果不仅丰富了ICM方法的内涵, 推动了其发展, 而且对变密度的节点拓扑方法也有参考的裨益.

Abstract: An ICM (Independent Continuous Mapping) method with the mapping of node-uncoupled topology variables is proposed in this paper, which is applied to model and solve the structural topology optimization problems. Firstly, the ground structure is meshed into nodes and elements. Independent and continuous topology variables are defined on the nodes, and an uncoupled mapping is established based on elemental bilinear interpolation, replacing the element-based independent and continuous topology variables. The existence or absence of elements is continuously approximated, enabling the filtering and identification of node topology variables and the interpolation of physical quantities within elements. An optimization model based on node topology design variables is derived, and a second-order dual programming algorithm based on separable variables is employed for solving. Additionally, an improved rounding technique for the optimal topology is proposed. Secondly, the modeling and solving process is demonstrated by applying the topology optimization to minimize structural weight (or volume) under displacement constraints. Finally, numerical examples of displacement-constrained topology optimization under single-load and multi-load scenarios are presented, and the computational results validate the effectiveness of the proposed method. The research in this paper has the following advantages: It overcomes the defects of previous research based on elemental topology variables, i.e., the optimal structural boundary is zigzagged, and obtains optimal structures with smooth and clear topological boundaries. The definitions of node topology variable and element topology function fields are given. The 5 criteria, which must be followed to construct the topology variable field, are extracted. Errors in the research on the ICM method based on node topology variables are addressed. The node design variables are no longer coupled, allowing for the convenient calculation of the second derivatives of structural quantities. This facilitates efficient optimization using dual optimization algorithms based on separable variables. The research not only enriches the connotation of the ICM method and promotes its development but also provides valuable references for the variable-density method based on node variables.

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