基于SIMP法的周期性传热材料拓扑优化

Topology optimization for periodic thermal conductive material using SIMP method

为了实现基于宏观热传导条件的周期性材料微结构设计,建立了基于固体各向同性材料惩罚法的周期性结构拓扑优化模型.模型以体积比为约束,散热弱度最小为优化目标.为了满足周期性约束,将设计域划分为若干相同子区域,并重新分配散热弱度.基于偏微分方程的图像处理方式可以有效地消除棋盘格和网格依赖性现象.讨论并分析了不同子区域个数及不同载荷工况对拓扑优化构型的影响.数值实验结果表明:周期性结构的建模方式可以实现基于宏观稳态热传导条件的周期性材料微结构设计.子区域个数不同时,优化得到不同的微结构构型,这反映了尺寸效应对材料设计的影响.当子区域个数不断增加时,优化结果逐渐趋向收敛于均匀化方法对应的极限值.

In order to obtain periodic material microstructure under macroscopic thermal conduction condition,the optimal topological model of periodic structure was built by solid isotropic material with penalization（ SIMP） method. The volume fraction was referred as constraint and minimized thermal compliance was taken as optimization objective in this model. To satisfy the periodic constraint,the designable domain was divided into a certain number of identical unit cells and the thermal compliance was reallocated. The filtered variable implicitly as a solution of a partial differential equation（ PDE） was applied to eliminate the checkerboard patterns and mesh-dependence problems efficiently. The optimal topological configurations were analyzed and compared with different numbers of unit cells and different load cases. The numerical results indicate that proposed periodic model is valid in design of periodic material microstructure with macroscopic steady state thermal conduction condition. Microstructure configurations are different when number of unit cells changes and it reflects the influences of size effect to periodic material design. With an increasing number of unit cells,the optimal results gradually converge to the results using homogenization method.