Review and Case Study of Structural Optimization Design of Large-aperture Mirrors

doi:10.3901/JME.260619

Abstract

Abstract: With the increasing demand for improving the performance of large-aperture mirrors, it becomes crucial to efficiently achieve collaborative optimization of multiple components of the mirror system under the coupling effect of multiple physical fields. The optimization design of large-aperture mirrors and their supporting structures is reviewed in this study. First, the existing design and optimization analysis methods of large-aperture mirror components are reviewed, including the first-order design stage, topology optimization, sensitivity analysis, parameter optimization, and traditional surrogate model optimization methods, with a summary provided on the current optimization design framework for mirrors and their supporting structures. Secondly, the surrogate model-assisted optimization technology is applied to supplement the defects of the existing framework in solving high-dimensional collaborative optimization problems. Finally, an engineering example is used to demonstrate the effectiveness of surrogate model-assisted optimization in the optimization design of high-dimensional multi-objective problems of mirrors and their supporting components. The optimization results show that compared with the first-order design, this method can achieve the coordinated optimization of 45 parameters under dual-temperature conditions, and the RMS values are reduced by 73.32% and 82.28% respectively; compared with the traditional optimization method, this method reduces the optimization time by two orders of magnitude; compared with the traditional proxy model optimization method, this method requires less training data and can quickly converge to the Pareto solution set that meets the design requirements, has better HV indicators, and ensures the richness and accuracy of the solution. New ideas and methods are provided for the design and analysis of modern optomechanical systems through this research, which has been demonstrated to hold significant engineering application value.

Key words: large-aperture mirrors, supporting structures, design optimization, data-driven, surrogate model

CLC Number:

TH122

GUO Guanchen, LIANG Pengwei, PANG Yong, LI Yanting, SONG Xueguan. Review and Case Study of Structural Optimization Design of Large-aperture Mirrors[J]. Journal of Mechanical Engineering, 2026, 62(15): 195-220.

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References

[1] RAYLEIGH L V. Investigations in optics，with special reference to the spectroscope[J]. The London，Edinburgh，and Dublin Philosophical Magazine and Journal of Science，1880，9(53)：40-55.
[2] YODER P，VUKOBRATOVICH D. Opto-mechanical systems design：Two volume set[M]. 4th ed. Boca Raton：CRC Press，2015.
[3] TOMIOKA K，王向阳. H-ⅱ运载火箭的振动､冲击和声振研究情况[J]. 国外导弹与航天运载器，1990(12)：58-65. TOMIOKA K，WANG Xiangyang. Research on vibration，shock and acoustic vibration of H-Ⅱ launch vehicle[J]. Foreign Missiles and Space Launch Vehicles，1990(12)：58-65.
[4] 鲍贤栋. 国外运载火箭低频振动环境条件分析[J]. 国外导弹技术，1982(4)：83-92. BAO Xiandong. Analysis of low-frequency vibration environmental conditions for foreign launch vehicles[J]. Foreign Missile Technology，1982(4)：83-92.
[5] 黎明，吴清文，江帆，等. 三线阵立体测绘相机热控系统的设计[J]. 光学精密工程，2010，18(6)：1367-1373. LI Ming，WU Qingwen，JIANG Fan，et al. Design of thermal control system for three-line array stereoscopic mapping camera[J]. Optics and Precision Engineering，2010，18(6)：1367-1373.
[6] BURGETT W，BERNSTEIN R，ASHBY D，et al. The giant magellan telescope project in 2024：Status and look ahead[C]//Ground-based and Airborne Telescopes X. Bellingham：SPIE-the International Society for Optical Engineering，2024：515-538.
[7] TRUBEY M，BOYER C，WANG L，et al. Thirty meter telescope laser guide star facility preliminary design completion[C]//Adaptive Optics Systems IX. Bellingham：SPIE-the International Society for Optical Engineering，2024：1268-1283.
[8] TEAM J，LIU Chengze，ZU Ying，et al. The jiao tong university spectroscopic telescope project[J]. Astronomical Techniques and Instrument，2024，1(1)：16-30.
[9] CHENG Jingquan. The principles of astronomical telescope design：Vol. 360[M]. New York：Springer New York，2009.
[10] 屈艳军. 空间大口径反射镜拓扑优化设计及运动学支撑技术研究[D]. 西安：中国科学院大学(中国科学院西安光学精密机械研究所)，2020. QU Yanjun. Research on topology optimization design and kinematic support technology of space large-aperture mirror[D]. Xi'an：University of Chinese Academy of Sciences (Xi'an Institute of Optics and Precision Mechanics，Chinese Academy of Sciences)，2020.
[11] 秦涛. 空间轻量化反射镜设计及支撑技术研究[D]. 成都：中国科学院大学(中国科学院光电技术研究所)，2023. QIN Tao. Research on design and support technology of space lightweight mirror[D]. Chengdu：University of Chinese Academy of Sciences (Institute of Optics and Electronics，Chinese Academy of Sciences)，2023.
[12] 邱成波. 低温大口径反射镜支撑装调系统研究[D]. 哈尔滨：哈尔滨工业大学，2017. QIU Chengbo. Research on support and alignment system of cryogenic large-aperture mirror[D]. Harbin：Harbin Institute of Technology，2017.
[13] 张坤. 高精度大视场空间目标测量光学系统研究[D]. 长春：中国科学院大学(中国科学院长春光学精密机械与物理研究所)，2021. ZHANG Kun. Researh on high-precision wide-field optical system for space target measurement[D]. Changchun：University of Chinese Academy of Sciences (Changchun Institute of Optics，Fine Mechanics and Physics，Chinese Academy of Sciences)，2021.
[14] 郭喜庆，王悦勇. 大口径反射镜几种轻量化孔结构形式的分析[J]. 光学精密工程，2000(6)：518-521. GUO Xiqing，WANG Yueyong. Analysis of several lightweight hole structures for large-aperture mirrors[J]. Optics and Precision Engineering，2000(6)：518-521.
[15] 陈洪达，陈永和，史婷婷，等. 空间反射镜的轻量化及支撑设计研究[J]. 红外与激光工程，2014，43(2)：535-540. CHEN Hongda，CHEN Yonghe，SHI Tingting，et al. Research on lightweight design and support design of space mirror[J]. Infrared and Laser Engineering，2014，43(2)：535-540.
[16] MONTAGNINO L A. Test and evaluation of the hubble space telescope 2.4-meter primary mirror[C]//29th Annual Technical Symposium. Bellingham：SPIE-the International Society for Optical Engineering，1986：0571.
[17] LAMPTON M L，AKERLOF C W，ALDERING G，et al. SNAP telescope[C]//Astronomical Telescopes and Instrumentation. Bellingham：SPIE-the International Society for Optical Engineering，2002，4849：215-226.
[18] LAMPTON M L，SHOLL M J，KRIM M H，et al. SNAP telescope：An update[C]//Optical Science and Technology，SPIE’s 48th Annual Meeting. Bellingham：SPIE-the International Society for Optical Engineering，2004，5166：113-123.
[19] BITTNER H，ERDMANN M，ERHARD M，et al. The optical system of the SOFIA telescope[C]//SPIE Astronomical Telescopes + Instrumentation. Bellingham：SPIE-the International Society for Optical Engineering，2004，5489：805-816.
[20] KRABBE A. SOFIA telescope[C]//Astronomical Telescopes and Instrumentation. Bellingham：SPIE-the International Society for Optical Engineering，2002，4014：276-281.
[21] PETROVSKY G T，TOLSTOY M N，LJUBARSKY S V，et al. 2.7-meter-diameter silicon carbide primary mirror for the SOFIA telescope[C]//Symposium on Astronomical Telescopes & Instrumentation for the 21st Century. Bellingham：SPIE-the International Society for Optical Engineering，1994，2199：263-270.
[22] SEIN E，TOULEMONT Y，SAFA F，et al. A Φ 3.5m diameter sic telescope for herschel mission[C]// Astronomical Telescopes and Instrumentation. Bellingham：SPIE-the International Society for Optical Engineering，2003，4850：606-618.
[23] PILBRATT G L，RIEDINGER J R，PASSVOGEL T，et al. Herschel space observatory：An ESA facility for far-infrared and submillimetre astronomy[J]. Astronomy and Astrophysics，2010，518：L1.
[24] TOULEMONT Y，BREYSSE J，PIEROT D，et al. The 3.5-m all-SiC telescope for SPICA[C]//SPIE Astronomical Telescopes + Instrumentation. Bellingham：SPIE-the International Society for Optical Engineering，2004，5487：1001-1012.
[25] SABELHAUS P A，DECKER J E. An overview of the James Webb space telescope (JWST) project[C]//SPIE Astronomical Telescopes + Instrumentation. Bellingham：SPIE-the International Society for Optical Engineering，2004：550.
[26] GARDNER J P，MATHER J C，CLAMPIN M，et al. The James Webb space telescope[J]. Space Science Reviews，2006，123(4)：485-606.
[27] LIGHTSEY P A. James Webb space telescope：Large deployable cryogenic telescope in space[J]. Optical Engineering，2012，51(1)：011003.
[28] 刘辉，于清华，裴云天. 大口径反射镜轻量化结构比较及面形分析[J]. 光电工程，2009，36(11)：70-74. LIU Hui，YU Qinghua，PEI Yuntian. Comparison of lightweight structures and surface figure analysis for large-aperture mirrors[J]. Opto-Electronic Engineering，2009，36(11)：70-74.
[29] VUKOBRATOVICH D，IRANINEJAD B，RICHARD R M，et al. Optimum shapes for lightweighted mirrors[C]//Astronomy Conferences. Bellingham：SPIE-the International Society for Optical Engineering，1982：419-423.
[30] 王书新. 2m量级空间反射镜组件设计与优化[D]. 长春：中国科学院大学(中国科学院长春光学精密机械与物理研究所)，2013. WANG Shuxin. Design and optimization of 2m-class space mirror assembly[D]. Changchun：University of Chinese Academy of Sciences (Changchun Institute of Optics，Fine Mechanics and Physics，Chinese Academy of Sciences)，2013.
[31] ARNOLD L. Optimized axial support topologies for thin telescope mirrors[J]. Optical Engineering，1995，34(2)：567-574.
[32] HALL H D. Problems in adapting small mirror fabrication techniques to large mirrors[J]. NASA Special Publication，1970，233：149.
[33] MALVICK A J，PEARSON E T. Theoretical elastic deformations of a 4-m diameter optical mirror using dynamic relaxation[J]. Applied Optics，1968，7(6)：1207-1212.
[34] WANG Yang. Research on primary mirror lateral support structure of large-aperture telescope[C]//5th International Symposium on Advanced Optical Manufacturing and Testing Technologies. Bellingham：SPIE-the International Society for Optical Engineering，2010，7654：109-115.
[35] SCHIPANI P，D’ORSI S，FIERRO D，et al. Active optics control of VST telescope secondary mirror[J]. Applied Optics，2010，49(16)：3199.
[36] ZINN J W，JONES G W. Kepler primary mirror assembly：FEA surface figure analyses and comparison to metrology[C]//Optical Engineering + Applications. Bellingham：SPIE-the International Society for Optical Engineering，2007：667105.
[37] 徐佳坤. 空间光学遥感器2m大口径反射镜组件结构优化研究[D]. 长春：中国科学院大学(中国科学院长春光学精密机械与物理研究所)，2023. XU Jiakun. Research on structural optimization of 2 m large-aperture mirror assembly for space optical remote sensor[D]. Changchun：University of Chinese Academy of Sciences (Changchun Institute of Optics，Fine Mechanics and Physics，Chinese Academy of Sciences)，2023.
[38] GLEYZES M A，PERRET L，KUBIK P. Pleiades system architecture and main performances[J]. The International Archives of the Photogrammetry，Remote Sensing and Spatial Information Sciences，2012，XXXIX-B1：537-542.
[39] FAPPANI D，DUCOLLET H. Manufacturing and control of the aspherical mirrors for the telescope of the French satellite Pleiades[C]//Optical Engineering + Applications. Bellingham：SPIE-the International Society for Optical Engineering，2007：66870T.
[40] KIHM H，YANG H. Design optimization of a 1-m lightweight mirror for a space telescope[J]. Optical Engineering，2013，52(9)：091806.
[41] KIHM H，YANG H，LEE Y. Bipod flexure for 1-m primary mirror system[J]. Review of Scientific Instruments，2014，85(12)：125101.
[42] PODGORSKI W A，CHEIMETS P N，GOLUB L，et al. Design，performance prediction，and measurements of the interface region imaging spectrograph (IRIS) telescope[C]//SPIE Astronomical Telescopes + Instrumentation. Bellingham：SPIE-the International Society for Optical Engineering，2012：84433D.
[43] SI Yuanhang，BENABOU L，CHALVET V. Bi-directional evolutionary topology optimization of geometrically nonlinear 3D continuum structures with an additional displacement constraint[J]. International Journal of Mechanics and Materials in Design，2025，21(3)：507-523.
[44] BENDSØE M P，KIKUCHI N. Generating optimal topologies in structural design using a homogenization method[J]. Computer methods in applied mechanics and engineering，1988，71(2)：197-224.
[45] ZHOU Ming，ROZVANY G I. The COC algorithm，part II：Topological，geometrical and generalized shape optimization[J]. Computer methods in applied mechanics and engineering，1991，89(1-3)：309-336.
[46] 施胤成，闫怀德，宫鹏，等. 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[J]. 光子学报，2020，49(6)：209-220. SHI Yincheng，YAN Huaide，GONG Peng，et al. Topology optimization design method for optical mirror support structure based on Zernike coefficient optimization model[J]. Acta Photonica Sinica，2020，49(6)：209-220.
[47] QU Yanjun，JIANG Yanru，FENG Liangjie，et al. Lightweight design of multi-objective topology for a large-aperture space mirror[J]. Applied Sciences，2018，8(11)：2259.
[48] XIE Y M，STEVEN G P. A simple evolutionary procedure for structural optimization[J]. Computers & Structures，1993，49(5)：885-896.
[49] ZHAO Fei. A nodal variable ESO (BESO) method for structural topology optimization[J]. Finite Elements in Analysis and Design，2014，86：34-40.
[50] XIONG Yulin，ZHAO Zilong，LU Hongjia，et al. Parallel BESO framework for solving high-resolution topology optimisation problems[J]. Advances in Engineering Software，2023，176：103389.
[51] ALLAIRE G，JOUVE F，TOADER A. Structural optimization using sensitivity analysis and a level-set method[J]. Journal of Computational Physics，2004，194(1)：363-393.
[52] ZHANG Xiao，GAO Liang，XIAO Mi，et al. T-splines-based panel method for aerodynamic topology optimization of engineering shell structures using isogeometric analysis[J]. Computer Methods in Applied Mechanics and Engineering，2025，444：118154.
[53] 孙宇欣，樊延超，董得义，等. 增材制造空间反射镜结构拓扑优化技术发展综述[J]. 激光杂志，2022，43(7)：1-14. SUN Yuxin，FAN Yanchao，DONG Deyi，et al. Review on development of structural topology optimization technology for additively manufactured space mirrors[J]. Laser Journal，2022，43(7)：1-14.
[54] 沙巍，陈长征，张星祥，等. 空间反射镜轻量化结构的拓扑优化设计[J]. 光电工程，2009，36(4)：35-39. SHA Wei，CHEN Changzheng，ZHANG Xingxiang，et al. Topology optimization design of lightweight structure for space mirror[J]. Opto-Electronic Engineering，2009，36(4)：35-39.
[55] 刘磊，高明辉，李丽富，等. 空间相机主反射镜结构拓扑优化设计[J]. 红外与激光工程，2010，39(6)：1066-1069. LIU Lei，GAO Minghui，LI Lifu，et al. Structural topology optimization design of primary mirror for space camera[J]. Infrared and Laser Engineering，2010，39(6)：1066-1069.
[56] 刘书田，胡瑞，周平，等. 基于筋板式基结构的大口径空间反射镜构型设计的拓扑优化方法[J]. 光学精密工程，2013，21(7)：1803-1810. LIU Shutian，HU Rui，ZHOU Ping，et al. Topology optimization method for configuration design of large-aperture space mirror based on rib-plate ground structure[J]. Optics and Precision Engineering，2013，21(7)：1803-1810.
[57] 韦德静. 空间大口径反射镜设计及其支撑技术研究[D]. 西安：中国科学院大学(中国科学院西安光学精密机械研究所)，2024. WEI Dejing. Research on design and support technology of space large-aperture mirror[D]. Xi’an：University of Chinese Academy of Sciences (Xi'an Institute of Optics and Precision Mechanics，Chinese Academy of Sciences)，2024.
[58] PARK K，LEE J，YOUN S. Lightweight mirror design method using topology optimization[J]. Optical Engineering，2005，44(5)：053002.
[59] 刘秀敏，何斌，沙巍，等. 空间相机主反射镜的拓扑优化设计[J]. 中国光学与应用光学，2010，3(3)：239-244. LIU Xiumin，HE Bin，SHA Wei，et al. Topology optimization design of primary mirror for space camera[J]. China Optics and Applied Optics，2010，3(3)：239-244.
[60] 齐光，李景林，王书新，等. 长条形SiC空间反射镜轻量化结构优化设计[J]. 光电工程，2012，39(2)：43-47. QI Guang，LI Jinglin，WANG Shuxin，et al. Optimization design of lightweight structure for strip-shaped SiC space mirror[J]. Opto-Electronic Engineering，2012，39(2)：43-47.
[61] 罗致帮. 临近空间大口径望远镜光机结构设计研究[D]. 长春：中国科学院大学(中国科学院长春光学精密机械与物理研究所)，2021. LUO Zhibang. Research on opto-mechanical structure design of large-aperture telescope in near space[D]. Changchun：University of Chinese Academy of Sciences (Changchun Institute of Optics，Fine Mechanics and Physics，Chinese Academy of Sciences)，2021.
[62] 周平伟. 空间2m级主镜稳健性设计及支撑关键技术研究[D]. 长春：中国科学院大学(中国科学院长春光学精密机械与物理研究所)，2020. ZHOU Pingwei. Research on robust design and key support technologies of space 2-meter class primary mirror[D]. Changchun：University of Chinese Academy of Sciences (Changchun Institute of Optics，Fine Mechanics and Physics，Chinese Academy of Sciences)，2020.
[63] LIU Guang，GUO Liang，WANG Xintong，et al. Topology and parametric optimization based lightweight design of a space reflective mirror[J]. Optical Engineering，2018，57(7)：075101.
[64] HILPERT E，HARTUNG J，VON LUKOWICZ H，et al. Design，additive manufacturing，processing，and characterization of metal mirror made of aluminum silicon alloy for space applications[J]. Optical Engineering，2019，58(9)：092613.
[65] MICI J，ROTHENBERG B，BRISSON E，et al. Optomechanical performance of 3D-printed mirrors with embedded cooling channels and substructures[C]//SPIE Optical Engineering + Applications. Bellingham：SPIE-the International Society for Optical Engineering，2015：957306.
[66] HORVATH N，DAVIES M. Advancing lightweight mirror design：A paradigm shift in mirror preforms by utilizing design for additive manufacturing[J]. Applied Optics，2021，60(3)：681.
[67] EBERLE S，REUTLINGER A，CURZADD B，et al. Additive manufacturing of an AlSi40 mirror coated with electroless nickel for cryogenic space applications[C]// International Conference on Space Optics — ICSO 2018. Bellingham：SPIE-the International Society for Optical Engineering，2019：40.
[68] HUANG Xiaodong，LI Weibai. Three-field floating projection topology optimization of continuum structures[J]. Computer Methods in Applied Mechanics and Engineering，2022，399：115444.
[69] CAI Shouyu，ZHANG Weihong. An adaptive bubble method for structural shape and topology optimization[J]. Computer Methods in Applied Mechanics and Engineering，2020，360：112778.
[70] NEOFYTOU A，HUANG T，KAMBAMPATI S，et al. Level set topology optimization with nodally integrated reproducing kernel particle method[J]. Computer Methods in Applied Mechanics and Engineering，2021，385：114016.
[71] SOUSA L，OLIVEIRA S，VIDAL C，et al. A truly meshless approach to structural topology optimization based on the direct meshless local petrov–galerkin (DMLPG) method[J]. Structural and Multidisciplinary Optimization，2024，67(7)：110.
[72] LI Zhao，XU Hongyu，ZHANG Shuai. A comprehensive review of explicit topology optimization based on moving morphable components (MMC) method[J]. Archives of Computational Methods in Engineering，2024，31(5)：2507-2536.
[73] DE VAUCORBEIL A，NGUYEN V P，MANDAL T K. Mesh objective simulations of large strain ductile fracture：A new nonlocal johnson-cook damage formulation for the total lagrangian material point method[J]. Computer Methods in Applied Mechanics and Engineering，2022，389：114388.
[74] DE VAUCORBEIL A，NGUYEN V P，HUTCHINSON C R，et al. Total lagrangian material point method simulation of the scratching of high purity coppers[J]. International Journal of Solids and Structures，2022，239-240：111432.
[75] YAMASAKI S，KAWAMOTO A，NOMURA T. Compliant mechanism design based on the level set and arbitrary lagrangian eulerian methods[J]. Structural and Multidisciplinary Optimization，2012，46(3)：343-354.
[76] LI Yue，LI Xuan，LI Minchen，et al. Lagrangian–eulerian multidensity topology optimization with the material point method[J]. International Journal for Numerical Methods in Engineering，2021，122(14)：3400-3424.
[77] LIN Daji，GAO Liang，GAO Jie. The lagrangian-eulerian described particle flow topology optimization (PFTO) approach with isogeometric material point method[J]. Computer Methods in Applied Mechanics and Engineering，2025，440：117892.
[78] ZHANG Xiao，XIAO Mi，GAO Liang，et al. A T-splines-oriented isogeometric topology optimization for plate and shell structures with arbitrary geometries using bézier extraction[J]. Computer Methods in Applied Mechanics and Engineering，2024，425：116929.
[79] LIU Yang，WANG Yongzhen，REN Hongyuan，et al. Ultrastiff metamaterials generated through a multilayer strategy and topology optimization[J]. Nature Communications，2024，15(1)：2984.
[80] YVONNET J，DA D. Topology optimization to fracture resistance：A review and recent developments[J]. Archives of Computational Methods in Engineering，2024，31(4)：2295-2315.
[81] ZHANG Liu，WANG Tailei，ZHANG Fan，et al. Design and optimization of integrated flexure mounts for unloading lateral gravity of a lightweight mirror for space application[J]. Applied Optics，2021，60(2)：417.
[82] 李钰鹏，王智，沙巍，等. Bipod反射镜支撑结构的柔度计算及分析[J]. 光学精密工程，2018，26(7)：1691-1697. LI Yupeng，WANG Zhi，SHA Wei，et al. Compliance calculation and analysis of Bipod mirror support structure[J]. Optics and Precision Engineering，2018，26(7)：1691-1697.
[83] 翟岩. 大型空间红外相机多反光学系统桁架杆框架的设计[J]. 光学精密工程，2017，25(11)：2923-2930. ZHAI Yan. Design of truss frame for multi-mirror optical system of large space infrared camera[J]. Optics and Precision Engineering，2017，25(11)：2923-2930.
[84] 翟岩. 面向空间高分辨率红外相机的超大口径反射镜设计及支撑技术研究[D]. 长春：长春理工大学，2020. ZHAI Yan. Research on design and support technology of ultra-large-aperture mirror for space high-resolution infrared camera[D]. Changchun：Changchun University of Science and Technology，2020.
[85] 王晓迪，曹玉岩，王富国，等. 大口径透镜混合柔性支撑结构优化设计[J]. 红外与激光工程，2022，51(6)：280-290. WANG Xiaodi，CAO Yuyan，WANG Fugu，et al. Optimization design of hybrid flexible support structure for large-aperture lens[J]. Infrared and Laser Engineering，2022，51(6)：280-290.
[86] 王晓迪. 大口径透镜组柔性支撑结构设计与分析[D]. 长春：中国科学院大学(中国科学院长春光学精密机械与物理研究所)，2022. WANG Xiaodi. Design and analysis of flexible support structure for large-aperture lens group[D]. Changchun：University of Chinese Academy of Sciences (Changchun Institute of Optics，Fine Mechanics and Physics，Chinese Academy of Sciences)，2022.
[87] 韩金邗. 空间大口径反射镜柔性支撑设计与分析[D]. 秦皇岛：燕山大学，2023. HAN Jinhan. Design and analysis of flexible support for space large-aperture mirror[D]. Qinhuangdao：Yanshan University，2023.
[88] 刘贝. 空间大口径反射镜组件优化设计及可调节支撑技术研究[D]. 西安：中国科学院大学(中国科学院西安光学精密机械研究所)，2020. LIU Bei. Research on optimal design of space large-aperture mirror assembly and adjustable support technology[D]. Xi'an：University of Chinese Academy of Sciences (Xi'an Institute of Optics and Precision Mechanics，Chinese Academy of Sciences)，2020.
[89] WANG Kejun，DONG Jihong，ZHAO Yang，et al. Research on high performance support technology of space-based large aperture mirror[J]. Optik，2021，226：165929.
[90] KIHM H，YANG H，MOON I，et al. Adjustable bipod flexures for mounting mirrors in a space telescope[J]. Applied Optics，2012，51(32)：7776-7783.
[91] KIHM H，YANG H，LEE Y. Optomechanical analysis of a 1-m light-weight mirror system[J]. Journal of the Korean Physical Society，2013，62(9)：1239-1246.
[92] LIU Bei，WANG Wei，QU Yanjun，et al. Design of an adjustable bipod flexure for a large-aperture mirror of a space camera[J]. Applied Optics，2018，57(15)：4048-1055.
[93] GUO Junli，QIN Tao，HAN Peixian，et al. Research on the position of the neutral surface of the space lightweight mirror with a two-axis bipod flexible mount[J]. Optics Express，2024，32(23)：42126-42148.
[94] LIANG Pengwei，SONG Xueguan，XIANG Yong，et al. Mathematical modelling for large-aperture mirror systems：Enabling optical-mechanical performance analysis and support structure optimization[J]. Applied Mathematical Modelling，2025，141：115938.
[95] 高明辉. 空间光学遥感器超薄反射镜及其支撑结构研究[D]. 长春：中国科学院研究生院(长春光学精密机械与物理研究所)，2006. GAO Minghui. Research on ultra-thin mirror and its support structure for space optical remote sensor[D]. Changchun：Graduate University of Chinese Academy of Sciences (Changchun Institute of Optics，Fine Mechanics and Physics)，2006.
[96] LIU Shutian，HU Rui，LI Quhao，et al. Topology optimization-based lightweight primary mirror design of a large-aperture space telescope[J]. Applied Optics，2014，53(35)：8318-8325.
[97] HU Rui，CHEN Wenjiong，LI Quhao，et al. Design optimization method for additive manufacturing of the primary mirror of a large-aperture space telescope[J]. Journal of Aerospace Engineering，2017，30(3)：04016093.
[98] YU Funan，XU Shuyan. Flexible support structure based on spring principle for a high precision reflecting mirror[J]. Optik，2020，207：164341.
[99] JIANG Ping，ZHOU Pingwei. Optimization of a lightweight mirror with reduced sensitivity to the mount location[J]. Applied Optics，2020，59(12)：3799-3805.
[100] ZHANG Changhao，LI Zongxuan，TAO Shuping，et al. Optimal design and analysis of a space lightweight mirror with a directionally oriented 2.5D-CVT structure[J]. Applied Optics，2024，63(16)：4386-4395.
[101] SHAO Mengqi，ZHANG Lei，JIA Xuezhi. Optomechanical integrated optimization of a lightweight mirror for space cameras[J]. Applied Optics，2021，60(3)：539.
[102] CHEN Yicheng，HUANG Bokai，YOU Zhenting，et al. Optimization of lightweight structure and supporting bipod flexure for a space mirror[J]. Applied Optics，2016，55(36)：10382.
[103] 王巧霞，郭崇岭. 基于iSIGHT集成平台的次镜支撑结构优化设计[J]. 航天返回与遥感，2014，35(2)：54-61. WANG Qiaoxia，GUO Chongling. Optimal design of secondary mirror support structure based on iSIGHT integrated platform[J]. Spacecraft Recovery & Remote Sensing，2014，35(2)：54-61.
[104] 张美君. 航空相机反射镜支撑结构优化设计及环境适应性分析[D]. 长春：中国科学院大学(中国科学院长春光学精密机械与物理研究所)，2023. ZHANG Meijun. Optimal design and environmental adaptability analysis of mirror support structure for aerial camera[D]. Changchun：University of Chinese Academy of Sciences (Changchun Institute of Optics，Fine Mechanics and Physics，Chinese Academy of Sciences)，2023.
[105] HU Rui，LIU Shutian，LI Quhao. Topology- optimization-based design method of flexures for mounting the primary mirror of a large-aperture space telescope[J]. Applied Optics，2017，56(15)：4551-4560.
[106] 陈卓远，耿少娟，刘帅鹏，等. 轮廓度误差对超声速压气机叶栅气动性能的影响[J]. 中国舰船研究，2024，19(2)：197-206. CHEN Zhuoyuan，GENG Shaojuan，LIU Shuaipeng et al. Influence of profile tolerance error on aerodynamic performance of supersonic compressor cascade[J]. Chinese Journal of Ship Research，2024，19(2)：197-206.
[107] 刘铠烨，楚武利，郭正涛，等. 加工误差对超声速叶栅气动性能影响的不确定性分析[J]. 航空动力学报，2024，39(12)：115-127. LIU Kaiye，CHU Wuli，GUO Zhengtao，et al. Uncertainty analysis of influence of machining errors on aerodynamic performance of supersonic cascades[J]. Journal of Aerospace Power，2024，39(12)：115-127.
[108] 苏春，黄漪. 装配误差传递建模及其精度可靠性评估[J]. 中国机械工程，2017，28(19)：2359-2364. SU Chun，HUANG Yi. Assembly error transfer modeling and its precision reliability evaluation[J]. China Mechanical Engineering，2017，28(19)：2359-2364.
[109] 刘现磊，张效栋，徐航宇，等. 基于光学性能评价自由曲面离轴三反系统加工误差分析[J]. 光学学报，2017，37(8)：191-199. LIU Xianlei，ZHANG Xiaodong，XU Hangyu，et al. Machining error analysis of free-form off-axis three-mirror system based on optical performance evaluation[J]. Acta Optica Sinica，2017，37(8)：191-199.
[110] 郭天宇，杨磊，汪毅，等. 基于多面一体的紧凑型同轴四反光学系统[J]. 光学学报，2024，44(12)：279-292. GUO Tianyu，YANG Lei，WANG Yi，et al. Compact coaxial four-mirror optical system based on multi-surface integration[J]. Acta Optica Sinica，2024，44(12)：279-292.
[111] LIU Xianlei，ZHANG Xiaodong，FANG Fengzhou，et al. Performance-controllable manufacture of optical surfaces by ultra-precision machining[J]. The International Journal of Advanced Manufacturing Technology，2018，94(9)：4289-4299.
[112] 王奇，胡光，李飞，等. 面向不确定量测的鲁棒RSS定位方法[J]. 电信科学，2025，41(2)：111-119. WANG Qi，HU Guang，LI Fei，et al. Robust RSS localization method for uncertain measurement[J]. Telecommunications Science，2025，41(2)：111-119.
[113] 陈建锋，丁召，杨晨. 紧凑型光管色彩混合系统研究[J]. 智能计算机与应用，2023，13(12)：46-48. CHEN Jianfeng，DING Zhao，YANG Chen. Research on compact light tube color mixing system[J]. Intelligent Computer and Applications，2023，13(12)：46-48.
[114] 任成明，陈琪，孟庆宇. 低误差敏感度大视场离轴三反光学系统设计[J]. 光学学报，2024，44(16)：1-10. REN Chengming，CHEN Qi，MENG Qingyu. Design of wide-field off-axis three-mirror optical system with low error sensitivity[J]. Acta Optica Sinica，2024，44(16)：1-10.
[115] 刘成武，李连升，钱林方. 随机与区间不确定下基于近似灵敏度的序列多学科可靠性设计优化[J]. 机械工程学报，2015，51(21)：174-184. LIU Chengwu，LI Liansheng，QIAN Linfang. Sequential multidisciplinary reliability design optimization based on approximate sensitivity under random and interval uncertainties[J]. Journal of Mechanical Engineering，2015，51(21)：174-184.
[116] 楚正凯，胡东方. 基于轴孔间隙配合的机械结构不确定性关系分析[J]. 机械强度，2023，45(4)：850-855. CHU Zhengkai，HU Dongfang. Uncertainty relationship analysis of mechanical structure based on shaft-hole clearance fit[J]. Journal of Mechanical Strength，2023，45(4)：850-855.
[117] LIU Fengchang，LI Wei，ZHAO Weiguo，et al. Fast optimization design of the flexure for a space mirror based on mesh deformation[J]. Photonics，2021，8(12)：567.
[118] LIU Xiaohan，GU Kaihui，LI Meixuan，et al. Optimization design of large-aperture primary mirror for a space remote camera[J]. Sensors，2023，23(12)：5441.
[119] 范磊. 2m级地基望远镜SiC主镜轻量化设计及支撑技术研究[D]. 长春：中国科学院大学(中国科学院长春光学精密机械与物理研究所)，2013. FAN Lei. Lightweight design and supporting technology research of SiC primary mirror for 2m-class ground-based telescope[D]. Changchun：University of Chinese Academy of Sciences (Changchun Institute of Optics，Fine Mechanics and Physics，Chinese Academy of Sciences)，2013.
[120] 范磊，张景旭，赵勇志，等. 中型主镜的柔性半运动学支撑[J]. 光学精密工程，2016，24(8)：1965-1972. FAN Lei，ZHANG Jingxu，ZHAO Yongzhi， et al. Flexible semi-kinematic support for medium-sized primary mirror[J]. Optics and Precision Engineering，2016，24(8)：1965-1972.
[121] 郭疆. 碳化硅大口径空间反射镜设计与制造研究[D]. 长春：吉林大学，2019. GUO Jiang. Research on design and manufacturing of large-aperture silicon carbide space mirror[D]. Changchun：Jilin University，2019.
[122] QUEIPO N V，HAFTKA R T，SHYY W，et al. Surrogate-based analysis and optimization[J]. Progress in Aerospace Sciences，2005，41(1)：1-28.
[123] CAI Xiwen，QIU Haobo，GAO Liang，et al. An enhanced RBF-HDMR integrated with an adaptive sampling method for approximating high dimensional problems in engineering design[J]. Structural and Multidisciplinary Optimization，2016，53(6)：1209-1229.
[124] GORISSEN D，COUCKUYT I，DEMEESTER P，et al. A surrogate modeling and adaptive sampling toolbox for computer based design[J]. Journal of Machine Learning Research，2010，11(1)：2051-2055.
[125] VENKATARAMAN S，HAFTKA R T. Structural optimization complexity：What has moore’s law done for us?[J]. Structural and Multidisciplinary Optimization，2004，28(6)：375-387.
[126] FORRESTER A J，SÓBESTER A，KEANE A J. Multi-fidelity optimization via surrogate modelling[J]. Proceedings of the Royal Society A：Mathematical，Physical and Engineering Sciences，2007，463(2088)：3251-3269.
[127] PERI D，CAMPANA E F. High-fidelity models and multiobjective global optimization algorithms in simulation-based design[J]. Journal of Ship Research，2005，49(3)：159-175.
[128] YANG Junqi，ZHAN Zhenfei，ZHENG Kai，et al. Enhanced similarity-based metamodel updating strategy for reliability-based design optimization[J]. Engineering Optimization，2016，48(12)：2026-2045.
[129] JIN R，CHEN W，SIMPSON T W. Comparative studies of metamodelling techniques under multiple modelling criteria[J]. Structural and Multidisciplinary Optimization，2001，23(1)：1-13.
[130] SIMPSON T W，POPLINSKI J D，KOCH P N，et al. Metamodels for computer-based engineering design：Survey and recommendations[J]. Engineering with Computers，2001，17(2)：129-150.
[131] 吕利叶，鲁玉军，王硕，等. 代理模型技术及其应用:现状与展望[J]. 机械工程学报，2024，60(3)：254-281. Lü Liye，LU Yujun，WANG Shuo，et al. Surrogate model technology and its applications：Status and prospect[J]. Journal of Mechanical Engineering，2024，60(3)：254-281.
[132] 吕利叶. 先进代理模型方法与应用研究[D]. 大连：大连理工大学，2020. Lü Liye. Research on advanced surrogate model methods and their applications[D]. Dalian：Dalian University of Technology，2020.
[133] FISHER R A. The design of experiments[M]. Sixth edition. Edinburgh：Oliver and Boyd，1951.
[134] GUNST R F，MASON R L. Fractional factorial design[J]. WIREs Computational Statistics，2009，1(2)：234-244.
[135] OWEN A B. Orthogonal arrays for computer experiments，integration and visualization[J]. Statistica Sinica，1992，2(2)：439-452.
[136] MCKAY M D，BECKMAN R J，CONOVER W J. A comparison of three methods for selecting values of input variables in the analysis of output from a computer code[J]. Technometrics，2000，42(1)：55-61.
[137] YE K Q，LI W，SUDJIANTO Agus. Algorithmic construction of optimal symmetric latin hypercube designs[J]. Journal of Statistical Planning and Inference，2000，90(1)：145-159.
[138] WANG Shuo，LV Liye，DU Li，et al. An improved LHS approach for constrained design space based on successive local enumeration algorithm[C]//2019 IEEE 9th Annual International Conference on CYBER Technology in Automation，Control，and Intelligent Systems (CYBER). Piscataway，NJ，USA：Institute of Electrical and Electronics Engineers，2019：896-899.
[139] PANG Yong，YANG Liangliang，WANG Yitang，et al. A latin hypervolume design for irregular sampling spaces and its application in the analysis of cracks[J]. Engineering with Computers，2023，39(5)：3509-3526.
[140] MYERS R H，MONTGOMERY D C，ANDERSON-COOK C M. Response surface methodology：Process and product optimization using designed experiments[M]. Hoboken：John Wiley & Sons，2016.
[141] OLIVER M A，WEBSTER R. Kriging：A method of interpolation for geographical information systems[J]. International Journal of Geographical Information Systems，1990，4(3)：313-332.
[142] SMOLA Al J，SCHÖLKOPF B. A tutorial on support vector regression[J]. Statistics and Computing，2004，14(3)：199-222.
[143] GUTMANN H M. A radial basis function method for global optimization[J]. Journal of Global Optimization，2001，19(3)：201-227.
[144] 兰斌，杨洪波，吴小霞，等. Ф620mm口径地基反射镜组件优化设计[J]. 红外与激光工程，2017，46(1)：198-206. LAN Bin，YANG Hongbo，WU Xiaoxia，et al. Optimal design of Φ620 mm aperture ground-based mirror assembly[J]. Infrared and Laser Engineering，2017，46(1)：198-206.
[145] 王忠善. 天基大口径单体反射镜结构优化及其支撑技术研究[D]. 长春：中国科学院大学(中国科学院长春光学精密机械与物理研究所)，2019. WANG Zhongshan. Research on structural optimization and supporting technology of space-based large-aperture monolithic mirror[D]. Changchun：University of Chinese Academy of Sciences (Changchun Institute of Optics，Fine Mechanics and Physics，Chinese Academy of Sciences)，2019.
[146] 韩忠华. Kriging模型及代理优化算法研究进展[J]. 航空学报，2016，37(11)：3197-3225. HAN Zhonghua. Research progress of Kriging model and surrogate-based optimization algorithms[J]. Acta Aeronautica et Astronautica Sinica，2016，37(11)：3197-3225.
[147] HAN Zhonghua，ZHANG Keshi. Surrogate-based optimization[M]. London：InTech Book，2012.