轻质夹芯结构的有限元分析和多目标优化
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摘要
蜂窝夹芯结构已经广泛应用于航空航天领域,特别是在现代卫星结构中,它已经成为主要的承力结构。周期桁架类夹芯结构则是近年来随着材料制备和成形加工技术的发展而出现的一类新型多功能材料,具有超轻、高比强、高比刚度、高强韧、高能量吸收等优良机械性能,以及减震、散热、吸声、电磁屏蔽、渗透性优等特殊性质,既可作为许多应用的结构材料,也可作为一些场合的功能材料,是一种备受瞩目的性能优异的多功能性工程材料。
本文采用有限元软件对蜂窝夹芯结构和周期性空间桁架夹芯结构进行建模和计算,并利用结果,以结构重量和其他功能指标为设计目标,用多目标遗传算法进行多目标优化。
本文共分五章,主要内容如下:
第一章为绪论。简要介绍了选题的工程背景和意义,概述了蜂窝夹芯结构及周期性桁架夹芯结构的研究现状,主要包括:理论分析方法、静态实验、动态实验、多功能特性以及优化设计等方面,并对有限元法及ANSYS软件进行简介,最后总结本文的主要工作。
第二章讨论了多目标优化的基本概念和常规解法,并给出了本文拟使用的非支配排序遗传算法(NSGA)和改进型非支配排序遗传算法(NSGA-Ⅱ)的算法流程。
第三章对蜂窝夹芯结构、周期性桁架结构从特点、制造技术、有限元计算方法等方面进行研究,并给出算例。
第四章在蜂窝夹芯结构、周期性桁架结构有限元分析的基础上,采用遗传算法实现了多目标优化,并给出了算例。
第五章是对全文的总结,也提出了几个可以参考本文内容进一步展开的工作方向。
本文研究工作得到国家自然科学基金重大项目(10590354)和国家自然科学基金项目(10572031)的资助。
Honeycomb sandwich structures have been widely used in the fields of astronautics and aerospace, especially in Satellite structure. Periodic truss-core sandwich plates have recently been investigated for their lightweight and multi-functional characteristics such as ultra-light, high comparing stiff and strength, shock resistance, thermal resistance, sound insulation and electromagnetic shielding. Recent developments in the manufacturing techniques appear to heighten the interest in such panels, and these are considered as the most perspective and novel ultra-light materials combined with specific strength and multi-functions.
In this paper the finite element models of honeycomb sandwich structures and Periodic truss-core sandwich plates were built, and were calculated using the finite element software. Then solve the multi-objective optimization using the result with the weight and other functional target as the object.
This paper includes five chapters, as briefly reviewed follow:
In chapter 1, background of this paper, with the review of literature is introduced. The emphasis is put on introducing some representative jobs in this field in the foreign countries and domestic research, including, theoretical methods, static and dynamic experiment, multi-functional characteristic and the optimization of the Honeycomb sandwich structures and Periodic truss-core sandwich plates, after that, FEM and ANSYS software are introduced too. Finally, the research of this paper is proposed based on the above.
In chapter 2, some general multi-objectives optimization problems and solutions are discussed. And then, classical multi-objective optimization and general genetic algorithm are introduced. Nondominated Sorting Genetic Algorithm(NSGA) and Nondominated Sorting Genetic Algorithm II (NSGA- II) are introduced especially.
Chapter 3, the manufacturing, structural mechanical properties and computational method of honeycomb sandwich structures and periodic truss-core sandwich plates are presented. Then the simple examples are presented.
Chapter 4, the numerical examples of multi-objectives optimization are presented.
Chapter 5 gives a summary of this thesis and some possible future work with reference to this paper.
This thesis is supported by the National Natural Science Foundation of China under Grant 10590354,10572031.
本文采用有限元软件对蜂窝夹芯结构和周期性空间桁架夹芯结构进行建模和计算,并利用结果,以结构重量和其他功能指标为设计目标,用多目标遗传算法进行多目标优化。
本文共分五章,主要内容如下:
第一章为绪论。简要介绍了选题的工程背景和意义,概述了蜂窝夹芯结构及周期性桁架夹芯结构的研究现状,主要包括:理论分析方法、静态实验、动态实验、多功能特性以及优化设计等方面,并对有限元法及ANSYS软件进行简介,最后总结本文的主要工作。
第二章讨论了多目标优化的基本概念和常规解法,并给出了本文拟使用的非支配排序遗传算法(NSGA)和改进型非支配排序遗传算法(NSGA-Ⅱ)的算法流程。
第三章对蜂窝夹芯结构、周期性桁架结构从特点、制造技术、有限元计算方法等方面进行研究,并给出算例。
第四章在蜂窝夹芯结构、周期性桁架结构有限元分析的基础上,采用遗传算法实现了多目标优化,并给出了算例。
第五章是对全文的总结,也提出了几个可以参考本文内容进一步展开的工作方向。
本文研究工作得到国家自然科学基金重大项目(10590354)和国家自然科学基金项目(10572031)的资助。
Honeycomb sandwich structures have been widely used in the fields of astronautics and aerospace, especially in Satellite structure. Periodic truss-core sandwich plates have recently been investigated for their lightweight and multi-functional characteristics such as ultra-light, high comparing stiff and strength, shock resistance, thermal resistance, sound insulation and electromagnetic shielding. Recent developments in the manufacturing techniques appear to heighten the interest in such panels, and these are considered as the most perspective and novel ultra-light materials combined with specific strength and multi-functions.
In this paper the finite element models of honeycomb sandwich structures and Periodic truss-core sandwich plates were built, and were calculated using the finite element software. Then solve the multi-objective optimization using the result with the weight and other functional target as the object.
This paper includes five chapters, as briefly reviewed follow:
In chapter 1, background of this paper, with the review of literature is introduced. The emphasis is put on introducing some representative jobs in this field in the foreign countries and domestic research, including, theoretical methods, static and dynamic experiment, multi-functional characteristic and the optimization of the Honeycomb sandwich structures and Periodic truss-core sandwich plates, after that, FEM and ANSYS software are introduced too. Finally, the research of this paper is proposed based on the above.
In chapter 2, some general multi-objectives optimization problems and solutions are discussed. And then, classical multi-objective optimization and general genetic algorithm are introduced. Nondominated Sorting Genetic Algorithm(NSGA) and Nondominated Sorting Genetic Algorithm II (NSGA- II) are introduced especially.
Chapter 3, the manufacturing, structural mechanical properties and computational method of honeycomb sandwich structures and periodic truss-core sandwich plates are presented. Then the simple examples are presented.
Chapter 4, the numerical examples of multi-objectives optimization are presented.
Chapter 5 gives a summary of this thesis and some possible future work with reference to this paper.
This thesis is supported by the National Natural Science Foundation of China under Grant 10590354,10572031.
引文
[1]Gibson L J,Ashby M F著.刘培生译.多孔固体结构与性能(第二版).北京:清华大学出版社,2003.
[2]Evans A G,Hutchinson J W,Fleck N A et al.The topological design of multifunctional cellular metals.Progress in Materials Science,2001,46:309-327.
[3]Evan A G,Hutchinson J W,Ashby M F.Multifunctionality of cellular metal systems.Progress in Materials Science,1999,43:171-221.
[4]Evan A G,Hutchinson J W,Fleck N A,Ashby M F.The topological design of multifunctional cellular metals.Progress in Materials Science,2001,46:309-327.
[5]Wicks Nathan,Hutchinson J W.Optimal truss plates.International Journal of Solids and Structure,2001,38:5165-5183.
[6]Lu T J,Hutchinson J W,Evans A G.Optimal design of a flexural actuator.Journal of the Mechanics and Physics of Solids,2001,49:2071-2093.
[7]Deshpande V S,Fleck N A,Ashby M F.Effective properties of the octet-truss lattice material.Journal of the Mechanics and Physics of Solids,2001,49:1747-1769.
[8]超轻多孔材料和结构创新构型的多功能化基础研究项目组.超轻多孔材料的国内外研究现状.973计划“超轻多孔材料和结构创新构型的多功能化基础研究”项目网站.
[9]Liu J S,Lu T J.Multi-objective and multi-loading optimization of ultralightweight truss materials.International Journal of Solids and Structure,2004,41:619-635.
[10]李兴斯,张琦,谭涛.多目标结构优化设计的中心法.力学学报,2005,37(5):606-610.
[11]Wallach J C,Gibson L J.Mechanical behavior of a three-dimensional truss material.International Journal of Solids and Structures,2001,38:7181-7196.
[12]Deshpande V S,Fleck N A.Collapse of truss core sandwich beams in 3-point bending.International Journal of Solids and Structure,2001,38:6275-6305.
[13]唐焕文,秦学志.实用最优化方法(第三版).大连:大连理工大学出版社,2005.
[14]田志刚.智能多目标优化理论及工程应用研究:(硕士学位论文).大连:大连理工大学,2003.
[15]崔逊学.多目标进化算法及其应用.北京:国防工业出版,2006.
[16]Lekhnitskii S G.Theory of Elasticity of an Anisotropic Elastic Body.Holden-Day,Inc.,San Francisco,Calif,1963.
[17]张卫红,吴琼,高彤.周期性金属桁架夹芯板的力学性能研究进展.昆明理工大学学报(理工版),2005,6(30):24-28.
[18]Hayes A M,Wang A J,Dempsey B M,McDowell D L.Mechanics of linear cellular alloys.Mechanics of Materials,2004,36:691-713.
[19]Hyun S,Karlsson A M,Torquato S et al.Simulated properties of Kagome and tetragonal truss core panels.International Journal of Solids and Structure,2003,40:6989-6998.
[20]Beomkeun Kim,Richard M,Christensen.Basic two-dimensional core types for sandwich structures.International Journal of Mechanical Science,2002,42:657-676.
[21]Onck P R,Andrews E W,Gibson L J.Size Effects in Ductile Cellular Solids.Part Ⅰ:Modeling.International Journal of Mechanical Science,2001,43:681-699.
[22]Paik J.K,Thayamballi A.K,Kim G.S.The strength characteristics of Aluminum honeycomb sandwich panels.Thin-Walled Structures,1999,35:205-231.
[23]富明慧,尹久仁,蜂窝芯层的等效弹性参数.力学学报.1999,31(1):113-118.
[24]Jack R V,Optimum design of composite honeycomb sandwich panels subjected to uniaxial compression.AIAA,Journal,1986,24(10):1690-1696.
[25]Kodiyalam S,Nagendra S,Destefano J.Composite sandwich structure optimization with application to satellite components.AIAA,Journal,1996,34(3):614-621.
[26]Kam T Y,Lai F M.Minimum weight design of laminated composite plates Subject to strength constraint.AIAA,Journal,1996,35:1699-1708.
[27]Kam T Y,Lai F M.Optimum design of laminated composite foam-filled sandwich plates subject to strength constraint.International Journal of Solids and Structure,1999,36:2865-2889.
[28]Hutchinson J E,He M Y Buckling of cylindrical sandwich shells with metal foam cores.International Journal of Solids and Structure,2000,37:6777-6794
[29]Nayfeh A H,Hefzy M S.Continuum Modeling of Three-Dimensional Truss-like Space Structures.AIAA,1978,16(8):779-787.
[30]徐胜利,王博,程耿东.多层桁架类夹层结构与蜂窝夹层结构的比较及优化.首届全国航空航天领域中的力学问题学术讨论会,2004,中国成都.
[31]Sigmund O.Materials with prescribed constitutive parameters:an inverse homogenization problem.International Journal of Solids and Structure,1994,31:2313-2329.
[32]阎军,程耿东,刘书田,刘岭.周期性点阵类桁架材料等效弹性性能预测及尺度效应.固体力学学报,2005,26(4):421-428.
[33]方岱宁,郭海成,A K Soh,卢天健.轻质点阵材料的力学行为分析.首届全国航空航天领域中的力学问题学术讨论会,2004,中国成都.
[34]Wang J,Evans A G,Dharmasena K,et al.On the performance of truss panels with Kagome cores.International Journal of Solids and Structure,2003,40:6981-6998.
[35]Yuki Sugimura.Mechanical response of single-layer tetrahedral trusses under shear loading.Mechanics of Materials,2004,36:715-721.
[36]Kooistra G W,Deshpande V S,Wadley H N G.Compressive behavior of age hardenable tetrahedral lattice truss structure made from aluminum.Acta Materialia,2004,52:4229-4237.
[37]Xue Z Y,Hutchinson J W.Preliminary assessment of sandwich plates subject to blast loads.[J]International Journal of Mechanical Sciences,2003,45:687-705.
[38]Xue Z Y,Hutchinson J W.A comparative study of impulse-resistant metal sandwich plates.International Journal of Impact Engineering 2004,30:1283-1305.
[39]Zok F W,Waltner S A,Wei Z,Rathbun H J,McMeeking R M,Evans A G.A protocol for characterizing the structural performance of metallic sandwich panels:application to pyramidal truss cores.International.Journal of Solids and Structure,2004,41:6249-6271.
[40]Chiras S,Mumm D R,Evans A G,Wicks N,Hutchinson J W,Dharmasena K,Wadley H N G,Fichter.The structural performance of near-optimized truss core Panels,International Journal of solids and Structures,2002,39:4093-4115.
[41]Rathbun H J,Zok F W,Evans A G.Strength optimization of metallic sandwich panels subject to bending.International Journal of Solids and Structures,2005,42:6643-6661.
[42]Ruzzene M.Vibration and sound radiation of sandwich beams with honeycomb truss core.Journal of Sound and Vibration,2004,277:741-763.
[43]Kim T,Hodson H P,Lu T J.Contribution of vortex structures and flow separation to local and overall pressure and heat transfer characteristics in an ultra-lightweight lattice material.International Journal of Heat and Mass Transfer,2005,48:4243-4264.
[44]Lu T J,Valdevitc L,Evans A G.Active cooling by metallic sandwich structures with periodic cores.Progress in Materials Science,2005,50:789-815.
[45]Bo Wang and G D Cheng.The Optimal Design for the Heat Dissipation Efficiency of Cellular Mechanics(ICHMM2004).June 21-26,2004,Chongqing University Press,Chongqing,China,pp322-325.
[46]Liu J S,Hollaway L.Design optimization of composite panel structures with stiffening ribs under multiple loading cases.Comput.Struct.2000,78,637-647.
[47]颜鸣啤.现代飞机主要用材料.航空材料,1980,4:1-7
[48]李德俊.90年代复合材料机体结构工艺的发展.航空制造工程,1992,8:23-25.
[49]王勖成.有限单元法.北京:清华大学出版社,2002.
[50]Kalyanmoy Deb,Amrit Pratap,Sameer Agarwal et al.A Fast and Elitist Multi-objective Genetic Algorithm:NSGA-Ⅱ.IEEE Transactions on Evolutionary Computation.2002,6:182-197.
[51]Goldberg D.E.,Deb K,A comparison of selection schemes used in genetic algorithms.In G.E.Rawlins Ed.,Foundations of Genetic Algorithms,San Mateo,California:Morgan Kaufmann,1991,69-93.
[52]卢天健等.超轻多孔金属材料的多功能特性及应用.力学进展,2007,36:517-535.
[53]Allen H G.Analysis and Design of Structural Panels.Oxford:Pergamon Press,1969.
[54]蔡四维 复合材料结构力学.北京:人民交通出版社,1987.
[55]Gibson LJ,Ashby MF,Schajer GS.The mechanics of two-dimension cellular materials.Proc R Soc A382,1982,25-42.
[56]崔光育.蜂窝夹层壳线性与非线性静动力学分析.清华大学博士学位论文,1995.
[57]代永朝,路阳.微波加热在蜂窝夹芯损伤修补中的应用.粘接,2004,3.
[58]周克印等.蜂窝结构缺陷类型识别的研究.数据处理与采集,2004,1.
[59]王颖坚.蜂窝结构在面内剪力作用下的变形模式北京大学学报(自然科学版).1991,27(3):301-30.
[60]中科院力学研究所.夹层板壳的弯曲、稳定和振动.北京:科学出版社,1977.
[61]飞机设计手册(第三册).北京:国防工业出版社,1990:719-722.
[62]曹志远.厚板动力学理论及其应用.北京:科学出版社,1983.
[63]徐胜今,宋宇,王本利,等.正交异性蜂窝夹层板的动力学分析.复合材料学报,1998,15(4):74-80.
[64]Wadley H N G,Fleck N A,Evans A G.Fabrication and structural performance of periodic cellular metal sandwich structures.Composites Science and Technology,2003,63:2331-2343.
[65]Wallach J C,Gibson L J.Defect sensitivity of a 3D truss material.Scripta Materialia.2001,45:639-644.
[66]Liu T,Deng Z C,Lu T J,Design optimization of truss-cored sandwiches with homogenization,International Journal of Solids and Structures.2006,43:7891-7918.
[67]胡海昌.各向同性夹层板反对称小挠度的若干问题.力学学报.1963,6(1):53-60.
[68]刘习军,贾启芬.工程振动理论与测试技术.北京:高等教育出版社,2004.
[69]Leissa A W.The free vibration of rectangular plates.Journal of Sound and Vibration.1973,31(3):257-293.
[70]王展光,单建,何德坪.金字塔栅格夹心夹层板动力响应分析.力学季刊.2006,27(4):707-713.
[2]Evans A G,Hutchinson J W,Fleck N A et al.The topological design of multifunctional cellular metals.Progress in Materials Science,2001,46:309-327.
[3]Evan A G,Hutchinson J W,Ashby M F.Multifunctionality of cellular metal systems.Progress in Materials Science,1999,43:171-221.
[4]Evan A G,Hutchinson J W,Fleck N A,Ashby M F.The topological design of multifunctional cellular metals.Progress in Materials Science,2001,46:309-327.
[5]Wicks Nathan,Hutchinson J W.Optimal truss plates.International Journal of Solids and Structure,2001,38:5165-5183.
[6]Lu T J,Hutchinson J W,Evans A G.Optimal design of a flexural actuator.Journal of the Mechanics and Physics of Solids,2001,49:2071-2093.
[7]Deshpande V S,Fleck N A,Ashby M F.Effective properties of the octet-truss lattice material.Journal of the Mechanics and Physics of Solids,2001,49:1747-1769.
[8]超轻多孔材料和结构创新构型的多功能化基础研究项目组.超轻多孔材料的国内外研究现状.973计划“超轻多孔材料和结构创新构型的多功能化基础研究”项目网站.
[9]Liu J S,Lu T J.Multi-objective and multi-loading optimization of ultralightweight truss materials.International Journal of Solids and Structure,2004,41:619-635.
[10]李兴斯,张琦,谭涛.多目标结构优化设计的中心法.力学学报,2005,37(5):606-610.
[11]Wallach J C,Gibson L J.Mechanical behavior of a three-dimensional truss material.International Journal of Solids and Structures,2001,38:7181-7196.
[12]Deshpande V S,Fleck N A.Collapse of truss core sandwich beams in 3-point bending.International Journal of Solids and Structure,2001,38:6275-6305.
[13]唐焕文,秦学志.实用最优化方法(第三版).大连:大连理工大学出版社,2005.
[14]田志刚.智能多目标优化理论及工程应用研究:(硕士学位论文).大连:大连理工大学,2003.
[15]崔逊学.多目标进化算法及其应用.北京:国防工业出版,2006.
[16]Lekhnitskii S G.Theory of Elasticity of an Anisotropic Elastic Body.Holden-Day,Inc.,San Francisco,Calif,1963.
[17]张卫红,吴琼,高彤.周期性金属桁架夹芯板的力学性能研究进展.昆明理工大学学报(理工版),2005,6(30):24-28.
[18]Hayes A M,Wang A J,Dempsey B M,McDowell D L.Mechanics of linear cellular alloys.Mechanics of Materials,2004,36:691-713.
[19]Hyun S,Karlsson A M,Torquato S et al.Simulated properties of Kagome and tetragonal truss core panels.International Journal of Solids and Structure,2003,40:6989-6998.
[20]Beomkeun Kim,Richard M,Christensen.Basic two-dimensional core types for sandwich structures.International Journal of Mechanical Science,2002,42:657-676.
[21]Onck P R,Andrews E W,Gibson L J.Size Effects in Ductile Cellular Solids.Part Ⅰ:Modeling.International Journal of Mechanical Science,2001,43:681-699.
[22]Paik J.K,Thayamballi A.K,Kim G.S.The strength characteristics of Aluminum honeycomb sandwich panels.Thin-Walled Structures,1999,35:205-231.
[23]富明慧,尹久仁,蜂窝芯层的等效弹性参数.力学学报.1999,31(1):113-118.
[24]Jack R V,Optimum design of composite honeycomb sandwich panels subjected to uniaxial compression.AIAA,Journal,1986,24(10):1690-1696.
[25]Kodiyalam S,Nagendra S,Destefano J.Composite sandwich structure optimization with application to satellite components.AIAA,Journal,1996,34(3):614-621.
[26]Kam T Y,Lai F M.Minimum weight design of laminated composite plates Subject to strength constraint.AIAA,Journal,1996,35:1699-1708.
[27]Kam T Y,Lai F M.Optimum design of laminated composite foam-filled sandwich plates subject to strength constraint.International Journal of Solids and Structure,1999,36:2865-2889.
[28]Hutchinson J E,He M Y Buckling of cylindrical sandwich shells with metal foam cores.International Journal of Solids and Structure,2000,37:6777-6794
[29]Nayfeh A H,Hefzy M S.Continuum Modeling of Three-Dimensional Truss-like Space Structures.AIAA,1978,16(8):779-787.
[30]徐胜利,王博,程耿东.多层桁架类夹层结构与蜂窝夹层结构的比较及优化.首届全国航空航天领域中的力学问题学术讨论会,2004,中国成都.
[31]Sigmund O.Materials with prescribed constitutive parameters:an inverse homogenization problem.International Journal of Solids and Structure,1994,31:2313-2329.
[32]阎军,程耿东,刘书田,刘岭.周期性点阵类桁架材料等效弹性性能预测及尺度效应.固体力学学报,2005,26(4):421-428.
[33]方岱宁,郭海成,A K Soh,卢天健.轻质点阵材料的力学行为分析.首届全国航空航天领域中的力学问题学术讨论会,2004,中国成都.
[34]Wang J,Evans A G,Dharmasena K,et al.On the performance of truss panels with Kagome cores.International Journal of Solids and Structure,2003,40:6981-6998.
[35]Yuki Sugimura.Mechanical response of single-layer tetrahedral trusses under shear loading.Mechanics of Materials,2004,36:715-721.
[36]Kooistra G W,Deshpande V S,Wadley H N G.Compressive behavior of age hardenable tetrahedral lattice truss structure made from aluminum.Acta Materialia,2004,52:4229-4237.
[37]Xue Z Y,Hutchinson J W.Preliminary assessment of sandwich plates subject to blast loads.[J]International Journal of Mechanical Sciences,2003,45:687-705.
[38]Xue Z Y,Hutchinson J W.A comparative study of impulse-resistant metal sandwich plates.International Journal of Impact Engineering 2004,30:1283-1305.
[39]Zok F W,Waltner S A,Wei Z,Rathbun H J,McMeeking R M,Evans A G.A protocol for characterizing the structural performance of metallic sandwich panels:application to pyramidal truss cores.International.Journal of Solids and Structure,2004,41:6249-6271.
[40]Chiras S,Mumm D R,Evans A G,Wicks N,Hutchinson J W,Dharmasena K,Wadley H N G,Fichter.The structural performance of near-optimized truss core Panels,International Journal of solids and Structures,2002,39:4093-4115.
[41]Rathbun H J,Zok F W,Evans A G.Strength optimization of metallic sandwich panels subject to bending.International Journal of Solids and Structures,2005,42:6643-6661.
[42]Ruzzene M.Vibration and sound radiation of sandwich beams with honeycomb truss core.Journal of Sound and Vibration,2004,277:741-763.
[43]Kim T,Hodson H P,Lu T J.Contribution of vortex structures and flow separation to local and overall pressure and heat transfer characteristics in an ultra-lightweight lattice material.International Journal of Heat and Mass Transfer,2005,48:4243-4264.
[44]Lu T J,Valdevitc L,Evans A G.Active cooling by metallic sandwich structures with periodic cores.Progress in Materials Science,2005,50:789-815.
[45]Bo Wang and G D Cheng.The Optimal Design for the Heat Dissipation Efficiency of Cellular Mechanics(ICHMM2004).June 21-26,2004,Chongqing University Press,Chongqing,China,pp322-325.
[46]Liu J S,Hollaway L.Design optimization of composite panel structures with stiffening ribs under multiple loading cases.Comput.Struct.2000,78,637-647.
[47]颜鸣啤.现代飞机主要用材料.航空材料,1980,4:1-7
[48]李德俊.90年代复合材料机体结构工艺的发展.航空制造工程,1992,8:23-25.
[49]王勖成.有限单元法.北京:清华大学出版社,2002.
[50]Kalyanmoy Deb,Amrit Pratap,Sameer Agarwal et al.A Fast and Elitist Multi-objective Genetic Algorithm:NSGA-Ⅱ.IEEE Transactions on Evolutionary Computation.2002,6:182-197.
[51]Goldberg D.E.,Deb K,A comparison of selection schemes used in genetic algorithms.In G.E.Rawlins Ed.,Foundations of Genetic Algorithms,San Mateo,California:Morgan Kaufmann,1991,69-93.
[52]卢天健等.超轻多孔金属材料的多功能特性及应用.力学进展,2007,36:517-535.
[53]Allen H G.Analysis and Design of Structural Panels.Oxford:Pergamon Press,1969.
[54]蔡四维 复合材料结构力学.北京:人民交通出版社,1987.
[55]Gibson LJ,Ashby MF,Schajer GS.The mechanics of two-dimension cellular materials.Proc R Soc A382,1982,25-42.
[56]崔光育.蜂窝夹层壳线性与非线性静动力学分析.清华大学博士学位论文,1995.
[57]代永朝,路阳.微波加热在蜂窝夹芯损伤修补中的应用.粘接,2004,3.
[58]周克印等.蜂窝结构缺陷类型识别的研究.数据处理与采集,2004,1.
[59]王颖坚.蜂窝结构在面内剪力作用下的变形模式北京大学学报(自然科学版).1991,27(3):301-30.
[60]中科院力学研究所.夹层板壳的弯曲、稳定和振动.北京:科学出版社,1977.
[61]飞机设计手册(第三册).北京:国防工业出版社,1990:719-722.
[62]曹志远.厚板动力学理论及其应用.北京:科学出版社,1983.
[63]徐胜今,宋宇,王本利,等.正交异性蜂窝夹层板的动力学分析.复合材料学报,1998,15(4):74-80.
[64]Wadley H N G,Fleck N A,Evans A G.Fabrication and structural performance of periodic cellular metal sandwich structures.Composites Science and Technology,2003,63:2331-2343.
[65]Wallach J C,Gibson L J.Defect sensitivity of a 3D truss material.Scripta Materialia.2001,45:639-644.
[66]Liu T,Deng Z C,Lu T J,Design optimization of truss-cored sandwiches with homogenization,International Journal of Solids and Structures.2006,43:7891-7918.
[67]胡海昌.各向同性夹层板反对称小挠度的若干问题.力学学报.1963,6(1):53-60.
[68]刘习军,贾启芬.工程振动理论与测试技术.北京:高等教育出版社,2004.
[69]Leissa A W.The free vibration of rectangular plates.Journal of Sound and Vibration.1973,31(3):257-293.
[70]王展光,单建,何德坪.金字塔栅格夹心夹层板动力响应分析.力学季刊.2006,27(4):707-713.