结构动力学拓扑优化局部模态现象分析

Investigation of Localized Modes in Topology Optimization of Dynamic Structures

使用拓扑优化技术进行结构自然频率最大化设计时的主要问题是结构空洞区域可能出现的局部模态现象。采用刚度矩阵和质量矩阵求解动力学特征方程,通过算例分析了结构密度差异、惩罚因子对自然频率和模态的影响以及局部模态现象产生的原因。结果表明,由于实体各向同性材料惩罚函数法(SIMP)使用了指数惩罚因子,空洞区域的密度与刚度差异远远大于实体区域,导致空洞区域局部模态的产生。分析了局部模态的产生过程,给出了结构自然频率和模态相对实体-孔洞区域密度的比值以及惩罚因子的变化趋势。同时,基于传统体胞微结构均匀化等效模型方法,提出了描述结构空洞区域的多种体胞微结构,计算证实这些微结构形式均可有效地避免局部模态的发生。

One of the main problems in the frequency maximization of dynamic structures by topology optimization is the appearance of localized modes in low-density areas. This paper aims at investigating the inherent cause by means of numerical examples. Based on the stiffness and mass matrices of the structure, the dynamic characteristic equation is solved to figure out how the density difference and penalty factor affect the natural frequencies and modal vectors. It is found that due to the penalty factor of the SIMP method, the ratio of the density to stiffness is greater in the solid region than in the void region. Such a difference consequently results in the appearance of localized modes in the void region. Furthermore, natural frequencies and modal vectors are studied to show their variations versus the density ratio between the solid and the void. In the meantime, the microstructure based on homogenization method is discussed. Based on the existing unit cell, a variety of new unit cells is proposed and numerically validated to avoid localized modes.