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

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.

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

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.