紧支Shepard近似在拓扑优化中的应用研究  

Application of compactly supported Shepard approximation in topology optimization

作　　者：杜义贤[1,2] 付君健[1] 严双桥[1] 陶然[1] 

机构地区：[1]三峡大学机械与材料学院,湖北宜昌443002 [2]三峡大学水电机械设备设计与维护湖北省重点实验室,湖北宜昌443002

出　　处：《华中科技大学学报（自然科学版）》2013年第8期101-105,共5页Journal of Huazhong University of Science and Technology(Natural Science Edition)

基　　金：国家自然科学基金资助项目(51105229);湖北省自然科学基金资助项目(2010CDB10805);三峡大学水电机械设备设计与维护湖北省重点实验室开放基金资助项目(2010KJX04)

摘　　要：为了解决有限元拓扑优化的数值不稳定性问题,以紧支径向基函数作为Shepard插值函数的权函数,建立了紧支Shepard近似函数,用近似函数替代原有离散的密度场量和敏度场量,构造了光滑的密度场和敏度场,提出了用于拓扑优化的密度近似法和敏度近似法.数值算例表明:密度近似法和敏度近似法能有效解决棋盘格和网格依赖性等数值不稳定性问题,随着网格密度的增加,密度近似法得到的拓扑图形的灰度单元成正比增加,但敏度近似法能有效地抑制灰度单元.To examine the numerical instabilities in the topology optimization by finite element method, compactly supported radial basis function was employed as the weight function of Shepard method to construct a compactly supported Shepard approximation function. Continuous density field and sensitivity field were established by replacing the original discrete density variables and sensitivity variables with the approximations based on Shepard function. Density approximation and sensitivity approximation methods were thus proposed for topology optimization. Numerical examples show that checkerboard pattern and mesh dependency problems can be solved by the density approximation and sensitivity approximation methods. As the refinement of mesh size increases, the gray scale elements of topology optimization results obtained from the density approximation method increase proportionally. However, the sensitivity approximation method can suppress the grey scale elements effectively.

关 键 词：拓扑优化 数值不稳定性 Shepard近似函数 密度近似 敏度近似 

分 类 号：TH122[机械工程—机械设计及理论]