考虑表面层厚度不确定的稳健性拓扑优化方法  

ROBUST TOPOLOGY OPTIMIZATION OF STRUCTURES CONSIDERING THE UNCERTAINTY OF SURFACE LAYER THICKNESS

作　　者：李冉 刘书田[1] Li Ran;Liu Shutian(State Key Laboratory of Structural Analysis for Industrial Equipment,Dalian University of Technology,Dalian 116024,Liaoning,China)

机构地区：[1]大连理工大学工程力学系,工业装备结构分析国家重点实验室,辽宁大连116024

出　　处：《力学学报》2021年第5期1471-1479,I0004,共10页Chinese Journal of Theoretical and Applied Mechanics

基　　金：国家自然科学基金(U1808215)资助项目。

摘　　要：采用增材制造工艺制备结构件时,较差的成型精度和表面粗糙度会导致结构表面层异质,引起表面层厚度的不确定性.为了研究不确定性对拓扑优化结构性能的影响,进而获得对不确定性具有更低敏感性的结构构型,提出了考虑结构表面层厚度不确定性的稳健性拓扑优化方法.首先,采用一种基于腐蚀操作的表面层识别技术,通过基于Helmholtz偏微分方程的PDE光滑过滤和基于Heaviside过滤、tanh函数的离散映射两个过程实现表面层异质等效模型的建立.其次,将表面层厚度作为服从高斯分布的随机变量,基于摄动有限元方法开展了不确定性传播的分析和系统随机响应的预测;以结构柔顺性均值和标准差的加权和作为优化目标,建立了考虑表面层厚度不确定性的拓扑优化模型,并推导了目标函数关于设计变量的敏度.最后,通过数值算例验证了该方法的有效性.数值结果表明,在设计过程中考虑表面层厚度不确定性对结构性能的影响,可以得到具有更强抵抗不确定性能力的结构构型,有效提升了结构性能的稳健性.For additively manufactured structure,the poor forming precision and surface roughness may cause surface layer heterogeneity,which leads to uncertain material properties and/or uncertain structure geometry.In order to obtain a structure with less sensitive to the uncertainty,a rubost topology optimization method accounting for the uncertain surface thickness of structures is proposed,in which two key problems need to be solved to study the surface layer thickness uncertainty caused by the heterogeneity of the structure surface layer.One is accurately identifying the structure surface layer.The other is to carry out propagation analysis and stochastic response estimation of uncertainty.First of all,an erosion-based surface layer identification method is adopted to establishing the equivalent model of surface layer heterogeneity through smooth filtering based on Helmholtz partial differential equation(PDE)as well as discrete mapping based on Heaviside filtering and tanh function,which is called a two-step filtering/projection process.Secondly,while the thickness of the heterogeneous surface layer is regarded as a random variable subject to Gaussian distribution,the uncertain propagation is analyzed and the system stochastic response is predicted based on the perturbation finite element method.Taking the weighted sum of the mean value and standard deviation of structural compliance as the optimization objective,a robust topology optimization model considering the uncertainty of surface layer thickness is established,and the sensitivities of the objective function with respect to design variables are derived.Finally,several numerical examples are given to demonstrate the effectiveness of the proposed method.The numerical results show that the structural configuration with stronger uncertainty resistance can be obtained by considering the influence of surface thickness uncertainty on the structural performance during the design process,which effectively improves the robustness of the structural performance.Therefore,f

关 键 词：稳健性设计 表面层异质 表面层识别 几何不确定性 摄动有限元 拓扑优化 

分 类 号：O343.1[理学—固体力学]