拓扑优化变密度法的灰度单元分层双重惩罚方法  

A Hierarchical Double Penalty Method of Gray-Scale Elements for SIMP in Topology Optimization

作　　者：廉睿超 敬石开[1] 何志军 史泽芳 Lian Ruichao;Jing Shikai;He Zhijun;Shi Zefang(School of Mechanical Engineering,Beijing Institute of Technology,Beijing 100081)

机构地区：[1]北京理工大学机械与车辆学院,北京100081

出　　处：《计算机辅助设计与图形学学报》2020年第8期1349-1356,1227,共9页Journal of Computer-Aided Design & Computer Graphics

基　　金：国家重点研发计划(2017YFB1102804)。

摘　　要：在连续体结构拓扑优化中,应用敏度过滤法可有效地去除数值不稳定问题,但易出现优化结构边界灰度扩散现象.为了获得边界清晰的拓扑结果,提出一种变密度法的灰度单元分层双重惩罚方法.该方法通过调节不含敏度过滤的SIMP优化算法中的惩罚因子,对过滤后单元敏度进行修正,加速中间密度单元向0或1的离散状态逼近.为了加快这个过程,将该方法与分层网格细分策略相结合,优化从一个粗的有限元网格开始,利用单元密度等效映射方法将粗网格求解优化问题的结果映射为同一问题具有更细网格的初始输入,通过减少优化过程中的计算消耗,在取得具有清晰边界拓扑结构的同时提升优化过程的收敛速率.采用不同方法求解MBB梁,对最终优化结构中所含的中间密度单元数量和优化所需时间消耗进行对比;利用不同网格划分下的悬臂梁算例验证该方法的网格依赖性.结果表明,结合分层双重惩罚的SIMP算法在保留原始求解稳定性的同时,能获得具有清晰边界的拓扑构型,并提升收敛速率.In the topology optimization for continuum structures,the sensitivity filtering methods can effectively eliminate the problem of numerical instability,but it is easy to cause the gray-scale diffusion phenomenon at the boundary of the final optimized structure.In order to obtain a crisp topological configuration,a hierarchical double penalty method of gray-scale elements for SIMP(solid isotropic microstructures with penalization)is proposed.The method applies a penalty factor in the SIMP method without sensitivity filter to modify sensitivity of the element from the standard SIMP method,which accelerates the approximation of the intermediate density units to the discrete state of 0 or 1.To further speed up the process,the method is implemented in a hierarchical mesh subdivision strategy.Starting from a coarse finite element mesh and using the element density equivalent mapping method,the solution with the coarse mesh as a starting input for the same problem but with a refined mesh.By reducing the computational cost of the units in the optimization process,the convergence rate of the optimization process is improved while achieving the distinct topology structure.Different methods are used to solve the MBB beam problem.The number of intermediate density elements contained in the optimized structures obtained by different methods and their time consumption in the entire optimization process were compared,respectively.The mesh dependency of this method is verified by using cantilever beam examples under different mesh divisions.The results show that the SIMP algorithm combined with hierarchical double penalty obtains a topological configuration with clear boundaries and improves the convergence rate while retaining the stability of the original solution.

关 键 词：拓扑优化 变密度法 敏度过滤 灰度单元 惩罚因子 

分 类 号：TP11[自动化与计算机技术—控制理论与控制工程]