基于Zernike系数优化模型的光学反射镜支撑结构拓扑优化设计方法  

Topology Optimization Design Method for Supporting Structures of Optical Reflective Mirrors Based on Zernike Coefficient Optimization Model

作　　者：施胤成 闫怀德 宫鹏 刘韬[1,2] 王强龙 程路超[1,2] 邓健 刘震宇 SHI Yin-cheng;YAN Huai-de;GONG Peng;LIU Tao;WANG Qiang-long;CHENG Lu-chao;DENG Jian;LIU Zhen-yu(Changchun Institute of Optics,Fine Mechanics and Physics,Chinese Academy of Science,Changchun 130033,China;School of Material Science and Opto-Electronic Technology,University of the Chinese of Academy of Science,Beijing 100039,China;China North Industries Corp.,Beijing 100053,China)

机构地区：[1]中国科学院长春光学精密机械与物理研究所,长春130033 [2]中国科学院大学材料科学与光电技术学院,北京100039 [3]中国北方工业有限公司,北京100053

出　　处：《光子学报》2020年第6期203-214,共12页Acta Photonica Sinica

基　　金：国家自然科学基金面上项目(No.51675506);吉林省优秀青年人才基金(No.20190103015JH);科技部国家重点研发计划(No.2018YFF01011503)~。

摘　　要：使用Zernike多项式表征镜面的变形,应用伴随变量法推导Zernike系数对拓扑优化设计变量的敏度,克服了差分法求解敏度时计算量大的问题,实现了基于Zernike系数直接建构具有成千上万设计变量的优化模型的目标函数以及设计约束.同时,在有限元数值离散的理论框架下,采用有限单元基函数以及单元数值积分的程序实现了结构变形以及Zernike系数的求解,简化了计算流程的同时还能保证计算精度.本文算法可以对目标函数或约束为线性组合的Zernike系数的一般结构拓扑优化模型进行优化,具有一定的泛用性.In this paper,the Zernike polynomials are used to describe deformed optical surfaces.Using the adjoint method,the sensitivities of Zernike polynomials to design variables can be derived.This procedure effectively overcomes the bottleneck of computational cost in sensitivity analysis when using the finite difference method.Therefore,the topology optimization models,which usually have thousand or tens of thousands of design variables,can be implemented by using the objectives and design constraints directly based on Zernike coefficients.Meanwhile,within the frame of numerical finite element discretization,adaptive finite element basis functions and element numerical integrals can be implemented to solve structural deformation and Zernike coefficients accurately and efficiently.This algorithm is flexible to be applied to general structural topology optimization models with objectives or constraints being a reasonable linear combination of Zernike coefficients.The numerical examples illustrate that the algorithm can optimize Zernike coefficients effectively.

关 键 词：结构拓扑优化 ZERNIKE多项式 反射镜支撑设计 有限元 光学器件 

分 类 号：O435.2[机械工程—光学工程]