充气膜结构反射面的形态分析与优化方法研究
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摘要
空间探索对大型星载天线反射面等大型轻型空间结构的需求日益强烈。利用充气膜结构技术构建的大型空间充气展开结构具有重量轻折叠体积小的优势,是未来空间大型展开结构的发展趋势之一。为了利用充气膜结构研制星载大口径高精度抛物面型的天线反射面,需要建立利用充气变形后形状确定充气变形前形状的逆过程分析方法,需要建立多目标优化模型,通过调整充气压力和边界拉力得到同时满足高精度形状要求和应力分布均匀要求的充气膜结构反射面。因此本文开展了高精度膜结构反射面的形态分析和优化方法研究。
为了更为准确地描述在充气压力作用下充气膜结构形状和应力分布状态的非线性耦合关系,针对膜结构的形状描述方法进行了研究,建立了基于非线性几何场理论的充气膜结构反射面分析模型。基于随体坐标系描述法,以变形后的构形为参考基准,构建了变形和局部转动项组成的反射面应变描述模型,得到了比经典理论更为精确的应变-位移几何关系。
针对充气膜结构反射面充气后的高精度抛物面形状要求,建立了理想状态形状模型和均匀应力分布模型。在非线性几何场理论的基础上,建立了充气膜结构反射面的平衡关系,求解了理想状态反射面应力和位移的解析解,获得了充气膜结构反射面变形前和变形后的形状和应力分布状态。建立了均匀应力分布模型,在满足形面精度要求条件下,获得了应力相对均匀的几何构形。在理想状态形状模型的基础上,提出了通过改变膜厚度方式改进反射面应力均匀度的方法,确定了最佳的反射面厚度变化规律。
充气膜结构反射面通常是不可展曲面,需要由多个平面膜片拼接并粘贴成整体曲面。为获得最佳的初始无应力膜片几何形状,提出了高精度反射面初始构形确定的逆迭代加权找形法。利用非线性有限元数值计算工具,以理想状态形状模型的几何构形为起始形状,通过局部加权回缩量的方法修改初始几何构形进行分析。提出了正弦函数形式的权函数模型,确定了权函数表达式中的参数,进行迭代过程中几何形状调整,通过逆迭代分析,最后得到满足反射面形面精度要求的初始几何形状。分析结果表明,本文的逆迭代加权找形法确定的初始无应力平面膜片几何形状能够有效的减小反射面的“W形”误差。
利用高精度非接触数字摄影测量法进行了充气膜结构反射面形面精度测量试验,得到了不同边界绳张力条件下充气天线反射面的形面精度、口径以及边界处天篷和反射面的夹角。试验结果验证了逆迭代加权找形法分析的准确性。通过试验结果分析,确定了边界绳张力、充气压力等影响因素和形面精度的关系。
综合考虑初始几何构形、充气压力和边界拉力等形面精度的影响因素,建立了基于遗传算法的充气膜结构反射面的多参数、多目标形态优化分析模型,对形面精度和应力分布均匀两项目标同时进行优化分析。考虑这两个优化目标的优先级不同,提出了采用分层解法来解决多目标优化问题,实现了在满足形面精度要求范围内,获得应力最均匀的最优解集,为充气膜结构反射面的设计提供参考。
本文建立的充气膜结构反射面形态分析方法和多目标优化方法,将为提高膜结构反射面的形面精度和控制提供理论基础和技术支持。
Space exploration has heightened the need for large space antenna reflectors. Large-scale space inflatable structure made by inflatable membrane structure which has advantages of exceptional packaging efficiency and low stowed volume is one trend in development for future large deployment structures. In order to develop large space high precision paraboloid inflatable membrane antenna reflectors, establishing the inverse process by using the shape after inflation to determine the shape before inflation is needed. For the demands of shape with high precision and uniformed stress distribution, multi-objective optimization methods of inflatable membrane structure are required to research, which can improve the precision by adjusting the inflation pressure and boundary force. Therefore, in this paper analysis and optimization methods of shape and stress for high precision membrane reflector are investigated.
For the sake of describing nonlinear couple relationship of shape and stress distribution of inflatable membrane after inflation well and truly, against the shape description method of membrane structure analysis model of inflatable membrane reflector based on geometry of nonlinear field theories is established in this paper and shape description of membrane structures are studied. Based on body coordinate system description method, the deformed configuration is set to reference standard, strain described model made of deformation and local rotation is built, the strain and displacement relationship is gained more accurate than the classical theory.
Toward paraboloid shape demand of inflatable membrane structure after inflation, the ideal shape model and the uniform stress distribution model are established. Based on nonlinear geometry field theories, mechanical balance equations of inflatable membrane reflector are built, analytical solution of stress and displacement are obtained, the shape and stress distribution of inflatable membrane reflector before and after inflation are achieved. Utilizing the uniform stress model, the shape with the most uniform stress meet the surface precision requirement is gained. Based on the ideal shape model, variable thickness method is used to improve the uniformity of stress distribution of reflector, and the best thickness variation is determined.
Inflatable membrane reflector is usually non-developable surface, made by a number of plain membranes splicing and paste into the overall membrane surface. In order to obtain the stress-free geometry configuration, inverse iteration and weighted form finding method is introduced. Based on FEA method and the initial shape set to the geometry configuration of the ideal shape model, inverse iteration analysis by weighting the local shrinkage and adjusting the initial geometry configuration, finally the obtained initial geometry configuration could meet the surface precision requirement. The results show inverse iteration and weighted form finding method in this paper could eliminate the "W-profile" error effectively in analysis of determined the initial configuration.
High precision digital photogrammetry is used to measure the surface precision of inflatable membrane reflector. Surface precision, diameters, and the angle between upper surface and lower surface of inflatable antenna reflector under different boundary force are gained. The results from experiment verify the veracity of the inverse iteration and weighted form finding method. By analysis the experiment results, the relationship between boundary force and inflation pressure are determined.
Considering the influencing factors such as initial geometry configuration, inflation pressure and the boundary force, the shape and stress distribution multi-parameter and multi-objective optimization model of inflatable membrane reflector is established based on genetic algorithms. This model optimizes the two objectives both the surface precision and uniform stress distribution simultaneously. Considering these two optimization objectives have different priority level, a method of layered solution is introduced to solve this multi-objective optimization problem, This multi-objective model enable to meet the precision requirement with the most uniform stress distribution, which could provide reference for design inflatable membrane reflectors.
In this paper, shape and stress distribution analysis methods and multi-objective optimization method of inflatable membrane reflector are established, which could improve the surface precision of membrane reflector and provide theoretical foundation and technical support.
为了更为准确地描述在充气压力作用下充气膜结构形状和应力分布状态的非线性耦合关系,针对膜结构的形状描述方法进行了研究,建立了基于非线性几何场理论的充气膜结构反射面分析模型。基于随体坐标系描述法,以变形后的构形为参考基准,构建了变形和局部转动项组成的反射面应变描述模型,得到了比经典理论更为精确的应变-位移几何关系。
针对充气膜结构反射面充气后的高精度抛物面形状要求,建立了理想状态形状模型和均匀应力分布模型。在非线性几何场理论的基础上,建立了充气膜结构反射面的平衡关系,求解了理想状态反射面应力和位移的解析解,获得了充气膜结构反射面变形前和变形后的形状和应力分布状态。建立了均匀应力分布模型,在满足形面精度要求条件下,获得了应力相对均匀的几何构形。在理想状态形状模型的基础上,提出了通过改变膜厚度方式改进反射面应力均匀度的方法,确定了最佳的反射面厚度变化规律。
充气膜结构反射面通常是不可展曲面,需要由多个平面膜片拼接并粘贴成整体曲面。为获得最佳的初始无应力膜片几何形状,提出了高精度反射面初始构形确定的逆迭代加权找形法。利用非线性有限元数值计算工具,以理想状态形状模型的几何构形为起始形状,通过局部加权回缩量的方法修改初始几何构形进行分析。提出了正弦函数形式的权函数模型,确定了权函数表达式中的参数,进行迭代过程中几何形状调整,通过逆迭代分析,最后得到满足反射面形面精度要求的初始几何形状。分析结果表明,本文的逆迭代加权找形法确定的初始无应力平面膜片几何形状能够有效的减小反射面的“W形”误差。
利用高精度非接触数字摄影测量法进行了充气膜结构反射面形面精度测量试验,得到了不同边界绳张力条件下充气天线反射面的形面精度、口径以及边界处天篷和反射面的夹角。试验结果验证了逆迭代加权找形法分析的准确性。通过试验结果分析,确定了边界绳张力、充气压力等影响因素和形面精度的关系。
综合考虑初始几何构形、充气压力和边界拉力等形面精度的影响因素,建立了基于遗传算法的充气膜结构反射面的多参数、多目标形态优化分析模型,对形面精度和应力分布均匀两项目标同时进行优化分析。考虑这两个优化目标的优先级不同,提出了采用分层解法来解决多目标优化问题,实现了在满足形面精度要求范围内,获得应力最均匀的最优解集,为充气膜结构反射面的设计提供参考。
本文建立的充气膜结构反射面形态分析方法和多目标优化方法,将为提高膜结构反射面的形面精度和控制提供理论基础和技术支持。
Space exploration has heightened the need for large space antenna reflectors. Large-scale space inflatable structure made by inflatable membrane structure which has advantages of exceptional packaging efficiency and low stowed volume is one trend in development for future large deployment structures. In order to develop large space high precision paraboloid inflatable membrane antenna reflectors, establishing the inverse process by using the shape after inflation to determine the shape before inflation is needed. For the demands of shape with high precision and uniformed stress distribution, multi-objective optimization methods of inflatable membrane structure are required to research, which can improve the precision by adjusting the inflation pressure and boundary force. Therefore, in this paper analysis and optimization methods of shape and stress for high precision membrane reflector are investigated.
For the sake of describing nonlinear couple relationship of shape and stress distribution of inflatable membrane after inflation well and truly, against the shape description method of membrane structure analysis model of inflatable membrane reflector based on geometry of nonlinear field theories is established in this paper and shape description of membrane structures are studied. Based on body coordinate system description method, the deformed configuration is set to reference standard, strain described model made of deformation and local rotation is built, the strain and displacement relationship is gained more accurate than the classical theory.
Toward paraboloid shape demand of inflatable membrane structure after inflation, the ideal shape model and the uniform stress distribution model are established. Based on nonlinear geometry field theories, mechanical balance equations of inflatable membrane reflector are built, analytical solution of stress and displacement are obtained, the shape and stress distribution of inflatable membrane reflector before and after inflation are achieved. Utilizing the uniform stress model, the shape with the most uniform stress meet the surface precision requirement is gained. Based on the ideal shape model, variable thickness method is used to improve the uniformity of stress distribution of reflector, and the best thickness variation is determined.
Inflatable membrane reflector is usually non-developable surface, made by a number of plain membranes splicing and paste into the overall membrane surface. In order to obtain the stress-free geometry configuration, inverse iteration and weighted form finding method is introduced. Based on FEA method and the initial shape set to the geometry configuration of the ideal shape model, inverse iteration analysis by weighting the local shrinkage and adjusting the initial geometry configuration, finally the obtained initial geometry configuration could meet the surface precision requirement. The results show inverse iteration and weighted form finding method in this paper could eliminate the "W-profile" error effectively in analysis of determined the initial configuration.
High precision digital photogrammetry is used to measure the surface precision of inflatable membrane reflector. Surface precision, diameters, and the angle between upper surface and lower surface of inflatable antenna reflector under different boundary force are gained. The results from experiment verify the veracity of the inverse iteration and weighted form finding method. By analysis the experiment results, the relationship between boundary force and inflation pressure are determined.
Considering the influencing factors such as initial geometry configuration, inflation pressure and the boundary force, the shape and stress distribution multi-parameter and multi-objective optimization model of inflatable membrane reflector is established based on genetic algorithms. This model optimizes the two objectives both the surface precision and uniform stress distribution simultaneously. Considering these two optimization objectives have different priority level, a method of layered solution is introduced to solve this multi-objective optimization problem, This multi-objective model enable to meet the precision requirement with the most uniform stress distribution, which could provide reference for design inflatable membrane reflectors.
In this paper, shape and stress distribution analysis methods and multi-objective optimization method of inflatable membrane reflector are established, which could improve the surface precision of membrane reflector and provide theoretical foundation and technical support.
引文
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72叶继红,周树路.改进动力松弛法在膜结构找形中的应用.工程力学. 2008, 25(12): 194~201
73温世峰,支希哲.膜结构找形分析中的混合法.计算机仿真. 2008,25(7): 341~344
74 Wuchner R, Bletzinger K U. Stress-adapted numerical form finding of pre-stressed surfaces by the updated reference strategy. Internation Journal for Numerical Methods in Engineering. 2005, 64: 143~166
75张明,张海滨.索杆梁膜结构找形方法研究.山西建筑. 2008,34(17): 89~91
76邓新宇.使用非线性有限元法对张力膜结构找形分析.山西建筑. 2007, 33(18): 71~72
77高柏峰,崔振山,黄健.基于非线性有限元法的膜结构找形研究.燕山大学学报. 2002, 26(4): 331~334
78何志军,丁浩民.基于非线性有限元的膜结构初始形态设计.四川建筑科学研究. 2004, 30(3): 15~17
79 Hiraku Sakamoto, Park K C, Yasuyuki Miyazaki. Evaluation of membrane structure designs using boundary web cables for uniform tensioning. Acta astronautica. 2007, 60:846~857
80任涛.索网结构混合法找形分析.低温建筑技术. 2009, 10: 45~47
81卫东,沈世钊.薄膜结构初始形态确定的几种分析方法.哈尔滨建筑大学学报. 2000, 33(4): 16~20
82杨维国,刘智敏.薄膜体系找形设计中二次找形方法的提出及其力学原理.工程力学, 2005, 22(1): 38~42
83夏劲松,丁成云,关富玲.一种结合找形和找力的索膜结构设计方法.空间结构. 2008, 14(2): 48~52
84夏劲松,李建宏,关富玲,徐彦.膜结构找力分析的无矩理论和优化的复位平衡法.计算力学学报. 2006, 23(2): 180~185
85冯星,张拉膜结构形状优化.建筑科学. 1999, 15(4):24~27
86许素强,夏人伟.结构优化方法研究综述.航空学报. 1995, 16(4): 385~396
87钱基宏,宋涛.张拉膜结构的找形分析和形态优化研究.建筑结构学报.2002, 23(3): 84~88
88马明,钱基宏,蓝天.膜结构裁剪分析中考虑预张力释放的计算方法.建筑结构学报. 2002, 23(3): 137~143
89卫东.张拉膜结构形态分析及优化.哈尔滨工业大学博士论文. 2001
90卫东,沈世钊.张拉膜结构的形态优化设计.土木工程学报. 2004, 37(2): 12~18
91周明,孙树栋.遗传算法原理及应用.国防工业出版社. 1999
92翁雁麟,关富玲,徐彦.充气膜结构的优化设计.科技通报. 2006, 22(4): 535~540
93刘淳安.基于进化机制的动态多目标优化方法.微电子学与计算机. 2009, 26(1): 169~173
94李红梅.多目标优化演化算法研究综述.现代计算机. 2009,4:44-46
95许昆,李智勇.改进的量子粒子群多目标优化算法.计算机工程与设计. 2009, 30(1): 164~168
96李方义,李光耀,郑刚.基于区间的不确定多目标优化方法研究.固体力学学报. 2010, 31(1): 86~93
97郭晓东,王丽芳.求解多目标优化问题的分布估计算法.太原科技大学学报. 2010, 31(1): 55~58
98陈民铀,张聪誉,罗辞勇.基于自适应进化粒子群算法的多目标优化方法.系统仿真学报. 2009, 21(22): 7061~7065
99伞冰冰,武岳,卫东.膜结构的多目标形态优化.土木工程学报. 2008, 41(9): 1~7
100董晶.多目标优化的Pareto解的表达与求取.武汉科技大学硕士论文. 2009
101范培蕾,张晓今,杨涛.基于灵敏度分析的Pareto解改进计算方法.系统工程与电子技术. 2009, 31(12): 2977~2981
102陈至达.板、壳有限变形分析.力学进展. 1983, 13(2): 119~124
103李海蛟,伞冰冰,武岳.脊谷式膜结构的形态优化分析.空间结构. 2010, 16(1): 51~56
104叶继红,田珺.张拉膜结构形态确定的试验研究.工程力学. 2010, 27(4): 117~124
105姜忆南,李栋. ETFE气枕式膜结构——以空气作为建筑材料的结构.建筑技术. 2010, 3: 42~44
106刘凯,高维成.给定索矢跨比的张拉式膜结构协同形态分析.工程力学. 2009, 26(1): 181~186
107张建,杨庆山,李波.气枕式充气膜结构形态与受力分析.哈尔滨工业大学学报. 2008, 40(12): 2020~2023
2 Jenkins C H,Schur W W. Gore/seam architectures for gossamer structures. Journal of Spacecraft and Rockets, 2002. 39(5): 669~673
3 Naboulsi S, Investigation of geometric imperfection in inflatable aerospace structures. Journal of Aerospace Engineering 2004. 17(3): 98~105
4 Jenkins C H,Marker D K, Surface precision of inflatable membrane reflectors. Journal of Solar Energy Engineering, Transactions of the ASME, 1998. 120(4): 298~305
5 Greschik G., Mikulas M M, Palisoc A. Approximations and errors in pressurized axisymmetric membrane shape predictions. 1998. Long Beach, CA, USA: AIAA, Reston, VA, USA.
6 Greschik G. Approximating paraboloids with axisymmetric pressurized membranes. 1998. Long Beach, CA, USA: AIAA, Reston, VA, USA.
7 Lee J H, Lalk T R, Appleby A J. Modeling electrochemical performance in large scale proton exchange membrane fuel cell stacks. Journal of Power Sources, 1998. 70(2): 258~268
8 Marker D K. Optical evaluation of membrane mirrors with curvature. 1998. San Diego, CA, USA: SPIE, Bellingham, WA, USA.
9 Palisoc A. Geometry attained by pressurized membranes. 1998. Kona, HI, USA: Society of Photo-Optical Instrumentation Engineers, Bellingham, WA, USA.
10 Kalanovic V D, K Padmanabhan, Jenkins C H. Discrete cell model for shape control of precision membrane antennae and reflectors. American Society of Mechanical Engineers, Aerospace Division (Publication) AD, 1999. 59: 227~237
11 Rotge J R. Large optically flat membrane mirrors. Proceedings of SPIE - The International Society for Optical Engineering, 1999. 3760: 207~212
12 Greschik G, M M Mikulas. Solar parachute concept for solar power satellitesand solar sails. 2000. Atlanta, GA, USA: American Inst. Aeronautics and Astronautics Inc., Reston, VA, USA.
13 Huang Y, D Yang. Large deformation analysis of plane membrane acted under internal pressure. Hunan Daxue Xuebao/Journal of Hunan University Natural Sciences, 2000. 27(3): 14~18
14 Greschik G. A rule of thumb for the suspension of film sheets without catenaries. 2003. Norfolk, VA, United States: American Inst. Aeronautics and Astronautics Inc.
15 Katsikadelis J T, Tsiatas G C. The analog equation method for large deflection analysis of heterogeneous anisotropic membranes: A boundary-only solution. 2000. Cambridge, United Kingdom: WITPress, Southampton, United Kingdom.
16 Pencer J, Hallett F R. Small-angle neutron scattering from large unilamellar vesicles: An improved method for membrane thickness determination. Physical Review E. Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 2000. 61(3): 3003
17 Weinberg K, Neff P. A geometrically exact thin membrane model - Investigation of large deformations and wrinkling. International Journal for Numerical Methods in Engineering, 2008. 74(6): 871-893
18 Rotge J R. Progress toward large-aperture membrane mirrors. 2000. San Diego, CA, USA: Society of Photo-Optical Instrumentation Engineers, Bellingham, WA, USA.
19 Yan L, Jenkins C H. and R.L. Pendleton, Polyolefin fiber-reinforced concrete composites. Part I. Damping and frequency characteristics. Cement and Concrete Research, 2000. 30(3): 391~401
20 Baier H. Building blocks of large deployable precision membrane reflectors. 2001. Seattle, WA: American Inst. Aeronautics and Astronautics Inc.
21 Freeland R E, Bilyeu G D, Veal G R. Large inflatable deployable antenna flight experiment results.ACTA Astronautica. 1997, 41 (4-10): 267~277
22 Freeland R E, Bilyeu G D, Veal G R. Development of flight hardware for a large, inflatable-deployable antenna experiment.ACTA Astronautica. 1996, 38 (4-8): 251~260
23 Baier H. Building blocks of advanced large stowable precision membrane reflectors. 2001. Noordwijk, Netherlands: European Space Agency.
24 Thomas M, Veal G. Highly accurate inflatable reflectors. AFRPL. 1984, TR-84~021
25 Greschik G, Mikulas M M. Design study of a square solar sail architecture. 2001. Seattle, WA: American Inst. Aeronautics and Astronautics Inc.
26 Konopacki S J, Akbari H. Measured energy savings and demand reduction from a reflective roof membrane on a large retail store in Austin. 2001: United States
27 Wu T Y, Ting E C. Large deflection analysis of 3D membrane structures by a
4-node quadrilateral intrinsic element. Thin-Walled Structures, 2008. 46(3): 261~275
28 Kosawada T, Sanada K, Takano T, Large deformation mechanics of plasma membrane chained vesicles in cells. JSME International Journal, Series C: Mechanical Systems, Machine Elements and Manufacturing, 2001. 44(4): 928~936
29 Nakasuka S. Furoshiki satellite A large membrane structure as a novel space system. Acta Astronautica, 2001. 48(5): 461~468
30 Roylance D, Jenkins C H, Khanna S K. Web modules linking mechanics and materials science. 2001. San Francisco, CA, United States: Materials Research Society, Warrendale, PA 15086, United States.
31 Roylance D, Jenkins C H, Khanna S K. Innovations in teaching mechanics of materials in materials science and engineering departments. 2001. Albuquerque, NM, United States: American Society for Engineering Education, Washington, DC 20036, United States.
32 Salama M, Jenkins C H. Intelligent gossamer structures: A review of recent developments and future trends. 2001. Seattle, WA: American Inst. Aeronautics and Astronautics Inc.
33 Stamper B. Stretched membrane with electrostatic curvature (SMEC) mirrors for extremely large space telescopes. 2001. San Diego, CA, United States: The International Society for Optical Engineering.
34 Andersen G. Large diameter, holographically corrected membrane telescope.2002. Waikoloa, HI, United States: The International Society for Optical Engineering.
35 Fang H. Inflatable structure for a three-meter reflectarray antenna. Journal of Spacecraft and Rockets, 2004. 41(4): 543~550
36 Andersen G. P. Large-aperture holographically corrected membrane telescope. Optical Engineering, 2002. 41(7): 1603~1607
37 Ng T T. Edge effects in pressurized membranes. Journal of Engineering Mechanics, 2002. 128(10): 1101~1105
38 Ng T T. Edge effect in pressurized membranes. Albuquerque, NM, United States: American Society of Civil Engineers. 2002
39 Greschik G, Mikulas M M. Design study of a square solar sail architecture. Journal of Spacecraft and Rockets, 2002. 39(5): 653~661
40 Ash J T. Shape achievement of optical membrane mirrors using coating/substrate intrinsic stresses. Journal of Spacecraft and Rockets, 2004. 41(4): 551~557
41 Pai P Frank, Young Leyland G. Geometrically Exact Modeling and Design of High Precision Membranes. 43rd AIAA/ASME/AHS/ASC Structures, Structural Dynamlcs, ans Materials Conference. Dallas AIAA 2002-1762
42 Jenkins C H, Marker D K. Surface accuracy of precision membrane reflectors. In Proceedings of the 1998 6th International Conference and Exposition on Engineering Construction, and Operations in Space. 1998
43 Jenkins C H, Kalanovic V D, Padmanabhan K. Practical Aspects of Precision Membrane Antennae Shape Control, in Proceedings of the IEEE International Conference on Systems, Man and Cybernetics. 1998, IEEE, Piscataway, NJ, USA: San Diego, CA, USA. 3194~3199
44 Jenkins C H. Up-to-date review of inflatable structures technology for spaced-based applications. 1998. Albuquerque, NM, USA: ASCE, Reston, VA, USA.
45 Jenkins C H, Wilkes J M, Marker D K. Improved surface accuracy of precision membrane reflectors through adaptive rim control. 1998. Long Beach, CA, USA: AIAA, Reston, VA, USA.
46 Jenkins C H, Faisal S M. Thermal load effects on precision membranes.Collection of Technical Papers - AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, 1999, 4: 2562~2572
47 Jenkins C H. In-situ manufacturing of gossamer spacecraft by artificial web-spinning. 2001. Seattle, WA: American Inst. Aeronautics and Astronautics Inc.
48 Khanna S K, Jenkins C H. Linking mechanics and materials in engineering design: A new approach. 2001. Albuquerque, NM, United States: American Society for Engineering Education, Washington, DC 20036, United States
49 Hencky H, Uber den Apannungszustand in kreisrunden Platten. Z.Math.Phys. 1915. 63: 311~317
50 Jenkins C H, Kalanovic V D. Issues in control of space membrane/inflatable structures. 2000. Big Sky, MT.
51 Jenkins C H, Ash J T, Marker D K. Local defect study of membrane antennas and reflectors. 2001. Big Sky, MT.
52 Wilkes J M. Concave membrane mirrors from aspheric to near-parabolic. Proceedings of SPIE - The International Society for Optical Engineering, 1999. 3760: 213-223
53 Ash J T, Jenkins C H, Marker D K. Deployment of a membrane mirror with an axial plunger. 2000. Atlanta, GA, USA: American Inst. Aeronautics and Astronautics Inc., Reston, VA, USA.
54 Jenkins C H, Vinogradov A M. Active polymers for space applications. 2000. Big Sky, MT.
55 Jenkins C H, Schur W W. Gore/seam architectures for gossamer structures. 2001. Seattle, WA: American Inst. Aeronautics and Astronautics Inc.
56 Jenkins C H, Faisal S M. Thermal load effects on precision membranes. Journal of Spacecraft and Rockets, 2001. 38(2): 207-211
57 Greschik G, Mikulas M M, Palisoc A. Approximations and errors in pressurized axisymmetric membrane shape predictions. 1998. Long Beach, CA, USA: AIAA, Reston, VA, USA.
58 Greschik G. Approximating paraboloids with axisymmetric pressurized membranes. 1998. Long Beach, CA, USA: AIAA, Reston, VA, USA.
59 Greschik G, Mikulas M M, Palisoc A. Torus-less inflated membrane reflector with an exact parabolic center. AIAA Journal. 2004, 42(12): 2579~2584
60 Frank Pai, Jiazhu Hu. Camera-based Digital Real-time Static and Dynamic Testing of Deployable/Inflatable Structures. Proc SPIE Int Soc Opt Eng. 2005, 5758(277): 209~218
61 Pappa Richard S, Thomas W, Jones. Photogrammetry Methodology Development for Gossamer Spacecraft Structures. S V Sound and Vibration. 2002, 36(8): 12~21
62 Fang H. Shape control of large membrane reflector with PVDF actuation. American Institute of Aeronautics and Astronautics Inc. United States: Reston, VA. 2007, 20191-4344
63徐彦,关富玲,管瑜.充气可展开天线精度分析和形面调整.空间科学学报. 2006, 26(4): 292~297
64徐彦,关富玲,马燕红.充气可展开天线的反射面设计及精度测量.浙江大学学报. 2007, 41(11): 1921~1926
65吕乃光,王永强,邓文怡.航天器薄膜充气天线面形的视觉测量方法,光电子激光. 2007, 6(6): 714~716
66王君,吕乃光,邓文怡.多约束条件在薄膜充气天线面形测量中的应用.北京机械工业学院学报. 2007, 22(2): 13~16
67陈务军,唐雅芳,任小强. ETFE气囊膜形态、结构特性与材料性能试验.建筑科学与工程学报. 2007, 24(3): 13~18
68周树路,叶继红.膜结构找形方法——改进力密度法.应用力学学报. 2008, 25(3): 421~424
69 Zhang J Y, Ohsaki M. Adaptive force density method for form-finding probem of tensegrity structures. International Journal of Solids and Structures. 2006, 43: 5658~5673
70向新岸,田伟,赵阳.考虑膜面二维变形的改进非线性力密度法.工程力学. 2010,27(4): 251~257
71李士军,马大为,朱孙科.动力松弛方法中Rayleigh阻尼参数取值分析.计算力学学报. 2010, 17(1): 169~172
72叶继红,周树路.改进动力松弛法在膜结构找形中的应用.工程力学. 2008, 25(12): 194~201
73温世峰,支希哲.膜结构找形分析中的混合法.计算机仿真. 2008,25(7): 341~344
74 Wuchner R, Bletzinger K U. Stress-adapted numerical form finding of pre-stressed surfaces by the updated reference strategy. Internation Journal for Numerical Methods in Engineering. 2005, 64: 143~166
75张明,张海滨.索杆梁膜结构找形方法研究.山西建筑. 2008,34(17): 89~91
76邓新宇.使用非线性有限元法对张力膜结构找形分析.山西建筑. 2007, 33(18): 71~72
77高柏峰,崔振山,黄健.基于非线性有限元法的膜结构找形研究.燕山大学学报. 2002, 26(4): 331~334
78何志军,丁浩民.基于非线性有限元的膜结构初始形态设计.四川建筑科学研究. 2004, 30(3): 15~17
79 Hiraku Sakamoto, Park K C, Yasuyuki Miyazaki. Evaluation of membrane structure designs using boundary web cables for uniform tensioning. Acta astronautica. 2007, 60:846~857
80任涛.索网结构混合法找形分析.低温建筑技术. 2009, 10: 45~47
81卫东,沈世钊.薄膜结构初始形态确定的几种分析方法.哈尔滨建筑大学学报. 2000, 33(4): 16~20
82杨维国,刘智敏.薄膜体系找形设计中二次找形方法的提出及其力学原理.工程力学, 2005, 22(1): 38~42
83夏劲松,丁成云,关富玲.一种结合找形和找力的索膜结构设计方法.空间结构. 2008, 14(2): 48~52
84夏劲松,李建宏,关富玲,徐彦.膜结构找力分析的无矩理论和优化的复位平衡法.计算力学学报. 2006, 23(2): 180~185
85冯星,张拉膜结构形状优化.建筑科学. 1999, 15(4):24~27
86许素强,夏人伟.结构优化方法研究综述.航空学报. 1995, 16(4): 385~396
87钱基宏,宋涛.张拉膜结构的找形分析和形态优化研究.建筑结构学报.2002, 23(3): 84~88
88马明,钱基宏,蓝天.膜结构裁剪分析中考虑预张力释放的计算方法.建筑结构学报. 2002, 23(3): 137~143
89卫东.张拉膜结构形态分析及优化.哈尔滨工业大学博士论文. 2001
90卫东,沈世钊.张拉膜结构的形态优化设计.土木工程学报. 2004, 37(2): 12~18
91周明,孙树栋.遗传算法原理及应用.国防工业出版社. 1999
92翁雁麟,关富玲,徐彦.充气膜结构的优化设计.科技通报. 2006, 22(4): 535~540
93刘淳安.基于进化机制的动态多目标优化方法.微电子学与计算机. 2009, 26(1): 169~173
94李红梅.多目标优化演化算法研究综述.现代计算机. 2009,4:44-46
95许昆,李智勇.改进的量子粒子群多目标优化算法.计算机工程与设计. 2009, 30(1): 164~168
96李方义,李光耀,郑刚.基于区间的不确定多目标优化方法研究.固体力学学报. 2010, 31(1): 86~93
97郭晓东,王丽芳.求解多目标优化问题的分布估计算法.太原科技大学学报. 2010, 31(1): 55~58
98陈民铀,张聪誉,罗辞勇.基于自适应进化粒子群算法的多目标优化方法.系统仿真学报. 2009, 21(22): 7061~7065
99伞冰冰,武岳,卫东.膜结构的多目标形态优化.土木工程学报. 2008, 41(9): 1~7
100董晶.多目标优化的Pareto解的表达与求取.武汉科技大学硕士论文. 2009
101范培蕾,张晓今,杨涛.基于灵敏度分析的Pareto解改进计算方法.系统工程与电子技术. 2009, 31(12): 2977~2981
102陈至达.板、壳有限变形分析.力学进展. 1983, 13(2): 119~124
103李海蛟,伞冰冰,武岳.脊谷式膜结构的形态优化分析.空间结构. 2010, 16(1): 51~56
104叶继红,田珺.张拉膜结构形态确定的试验研究.工程力学. 2010, 27(4): 117~124
105姜忆南,李栋. ETFE气枕式膜结构——以空气作为建筑材料的结构.建筑技术. 2010, 3: 42~44
106刘凯,高维成.给定索矢跨比的张拉式膜结构协同形态分析.工程力学. 2009, 26(1): 181~186
107张建,杨庆山,李波.气枕式充气膜结构形态与受力分析.哈尔滨工业大学学报. 2008, 40(12): 2020~2023