U型拱式渡槽的优化设计研究
|
摘要
渡槽是跨越河流、道路、山冲、山谷等的架空输水交叉建筑物,是渠系建筑物中应用最广的交叉建筑物之一。对于渡槽的结构造型优化国内外研究较少。随着一些大型输水工程的建设,在保证渡槽设计的合理性和实用性的前提下,如何降低工程投资,是我们急需解决的问题。结构拓扑优化可使复杂结构在初步选型阶段即可灵活理性地进行方案优选,近年来已成为国际结构优化研究领域最热点的问题之一。
为此,本文在阅读大量国内外文献的基础上,研究了MATLAB和拓扑优化的基本理论,分别从水力学和结构力学角度入手,采用差分进化法和均匀化方法,初步探讨了渡槽形式结构的优化设计和比选,主要研究工作有:
(1)从水力学的角度,基于MATLAB求解,研究了渡槽槽身断面的优化比选设计。以边坡系数为设计变量,采用差分进化法对实用经济断面进行了分析。计算结果表明,底部为半圆形的U型渡槽在满足过流量等基本要求下,能有效地减少截面尺寸,减轻支撑承受的荷载,对于尺寸较大的渡槽宜采用U形断面。
(2)从结构优化设计出发,介绍了结构优化设计的基本概念、建模步骤和结构拓扑优化理论的历史背景。在此基础上,对均匀化方法的基本原理、数学模型和基于ANSYS的结构拓扑优化实现进行了研究。
(3)选用工程实例,以ANSYS的APDL语言二次开发连续体结构拓扑优化程序为基础,对渡槽支撑结构建立拓扑优化模型,获得了渡槽拱架较合理的初始结构方案,其支撑在均布荷载作用下结构呈拱形,符合经典抛物线拱理论,证明了抛物线作为合理拱轴线的正确性。研究结果表明了拓扑优化方法在渡槽设计上的可行性和优越性,应用前景较为广阔。
The aqueduct is a cross construction which is used to transport water. And it span river, road, mountain, valley etc. It is one of the cross construction which is the most widely used to the canal system construction. There is less study about the framework optimization of the aqueduct in domestic or overseas. With the big transpot water project being bulit, how to reduce investment is a problem which need to resolve in the future. Because topology optimization with better schemes can be chosen in introduction design stage. Recently, it has become the hottest problems in the international structure optimization filed.
Therefor, after reading numbers of foreign documents and domestic relative papers, this text further studied the basic theory of MATLAB and topology optimization and discussed the structure’s optimal design and select of aqueduct at the point of views of hydraulics and structure mechanics which separately adopted the method of the evolution and homogenization. The most works are:
(1) At the point of hydraulics , this thesis studied the aqueduct body’s optimal design based on MATLAB. It adopted the method of the evolution and analyzed practical economical section plane with the variable of slope modulus. The result indicated that the U-style aqueduct with hemicycle bottom can effectively decrease the section size and lighten load based on the basic need of certain flux. The biggish aqueduct should adopt U-style section plane.
(2) Starting with the structure optimization design, the thesis introduced basic concept, modeling process and history background of the topology optimization theory. And it studied the basic theory of homogenization method, mathematic model and the realization of the topology optimization based on ANSYS.
(3) Using actual project, the thesis established the geometry model for the underprop of aqueduct base on the topology optimization program with the APDL language of ANSYS and gained reasonable original frame project. Under the even load, it came into being in arch and accorded with the cast curve theory. It proved that the cast curve is correct as the reasonable arch axes. The result indicate that the feasibility and advantage of the topology optimization for the design. The outcome also show that the topology optimization is better and it can be used widely.
为此,本文在阅读大量国内外文献的基础上,研究了MATLAB和拓扑优化的基本理论,分别从水力学和结构力学角度入手,采用差分进化法和均匀化方法,初步探讨了渡槽形式结构的优化设计和比选,主要研究工作有:
(1)从水力学的角度,基于MATLAB求解,研究了渡槽槽身断面的优化比选设计。以边坡系数为设计变量,采用差分进化法对实用经济断面进行了分析。计算结果表明,底部为半圆形的U型渡槽在满足过流量等基本要求下,能有效地减少截面尺寸,减轻支撑承受的荷载,对于尺寸较大的渡槽宜采用U形断面。
(2)从结构优化设计出发,介绍了结构优化设计的基本概念、建模步骤和结构拓扑优化理论的历史背景。在此基础上,对均匀化方法的基本原理、数学模型和基于ANSYS的结构拓扑优化实现进行了研究。
(3)选用工程实例,以ANSYS的APDL语言二次开发连续体结构拓扑优化程序为基础,对渡槽支撑结构建立拓扑优化模型,获得了渡槽拱架较合理的初始结构方案,其支撑在均布荷载作用下结构呈拱形,符合经典抛物线拱理论,证明了抛物线作为合理拱轴线的正确性。研究结果表明了拓扑优化方法在渡槽设计上的可行性和优越性,应用前景较为广阔。
The aqueduct is a cross construction which is used to transport water. And it span river, road, mountain, valley etc. It is one of the cross construction which is the most widely used to the canal system construction. There is less study about the framework optimization of the aqueduct in domestic or overseas. With the big transpot water project being bulit, how to reduce investment is a problem which need to resolve in the future. Because topology optimization with better schemes can be chosen in introduction design stage. Recently, it has become the hottest problems in the international structure optimization filed.
Therefor, after reading numbers of foreign documents and domestic relative papers, this text further studied the basic theory of MATLAB and topology optimization and discussed the structure’s optimal design and select of aqueduct at the point of views of hydraulics and structure mechanics which separately adopted the method of the evolution and homogenization. The most works are:
(1) At the point of hydraulics , this thesis studied the aqueduct body’s optimal design based on MATLAB. It adopted the method of the evolution and analyzed practical economical section plane with the variable of slope modulus. The result indicated that the U-style aqueduct with hemicycle bottom can effectively decrease the section size and lighten load based on the basic need of certain flux. The biggish aqueduct should adopt U-style section plane.
(2) Starting with the structure optimization design, the thesis introduced basic concept, modeling process and history background of the topology optimization theory. And it studied the basic theory of homogenization method, mathematic model and the realization of the topology optimization based on ANSYS.
(3) Using actual project, the thesis established the geometry model for the underprop of aqueduct base on the topology optimization program with the APDL language of ANSYS and gained reasonable original frame project. Under the even load, it came into being in arch and accorded with the cast curve theory. It proved that the cast curve is correct as the reasonable arch axes. The result indicate that the feasibility and advantage of the topology optimization for the design. The outcome also show that the topology optimization is better and it can be used widely.
引文
白新理,黄和法,刘琰玲,赵新铭. 2001.连续刚构预应力U型薄壳渡槽有限元分析.华北水利水电学院学报, 1:1~4
博嘉科技. 2002.有限元分析软件——ANSYS融汇与贯通.北京:中国水利水电出版社:352~355
蔡坤,张洪武,陈飙松. 2006.基于wolff法则的连续体结构拓扑优化方法.力学学报,38(4):514～521.
长江勘测规划设计研究院. 2000.东深供水改造工程——U型薄壳预应力渡槽槽身结构研究报告.武汉
长江水利委员会信息研究中心. 1990.工程设计与施工技术汇编——国内外渡槽.武汉
陈文义,赵顺波,李树瑶. 1996.南水北调工程大型预应力混凝土波槽结构选型研究.华北水利水电学院学报, 17(4):9~15
董桂西,王藏柱. 2000.结构优化设计的现状及展望.电力情报, 1: 5~7
段进,倪栋,王国业. 2006. ANSYS10.0结构分析从入门到精通.北京:兵器工业出版社,北京科海电子出版社:6~8
飞思科技产品研发中心. 2003. MATLAB6.5辅助优化计算与设计.北京:电子工业出版社:1~3
傅志敏. 2003.斜拉式大型渡槽槽身优化设计研究. [硕士学位论文].南京:河海大学
高小翠,彭琳. 2003.大型预应力混凝土箱式渡槽有限元分析.中国农村水利水电, (5):53~55
郭旭,赵康. 2004.基于拓扑描述函数的连续体结构拓扑优化方法.力学学报,36(5):520～526.
过迟. 1994.南水北调穿黄工程总体布置研究.人民长江, 25(1):12~17
韩亦方. 2001.南水北调工程的重大科技问题.水利水电技术, 32(1):35
华东水利学院. 1979.水力学.北京:科学出版社:123~134
华东水利学院. 1987.水工设计手册.北京:水利电力出版社:16~18
姜冬菊. 2008.结构拓扑和布局优化及工程应用研究. [博士学位论文].南京:河海大学
姜新佩. 1998. U型薄壳渡槽槽身设计的研究.水利水电技术, 8(2):13~204
李廉锟. 1979.结构力学.北京:高等教育出版社:131~137
林继镛. 2006.水工建筑物.第四版.北京:中国水利水电出版社:497
凌均忆. 1992.斜拉输水结构布置及其技术特性.水利水电技术, 6(2):20~24
刘东常,白新理,刘宪亮. 1995.南水北调中线穿黄渡槽工程中预应力U形薄壳渡槽结构分析(预应力U形薄壳渡槽研究报告之一).郑州
刘国纬. 2000.关于中国南水北调的思考.水科学进展, 11(4):346~350
刘宁. 2006.南水北调中线一期工程穿黄方案的论证与选择.水利学报, 37(1):1~9
刘蓉华. 1994.拱型结构分析的拓扑变化法.西安交通大学学报: 29(4):557~563
刘儒勋,舒其望. 2003.计算流体力学的若干新方法.北京:科学出版社, 2003:1~4l
刘晓文. 2001.渡槽结构优化研究及实现.武汉大学学报(工学版): (3):47~50.
罗震,陈立平,黄玉盈,张云清. 2004.连续体结构的拓扑优化设计.力学进展, 34(4):463~476
梅来彬,王波. 2008.浅析ANSYS在桥梁拓扑优化设计中的应用.科技创新导报, 27:104~105
荣见华,姜节胜,颜东煌. 2002.多约束的桥梁结构拓扑优化.工程力学学报, 19(4):160~165
隋允康,叶红玲,彭细荣. 2006.应力约束全局化策略下的连续体结构拓扑优化.力学学报,38(3):364-370.
王海波,张伯艳,周杰. 2004.南水北调工程的抗震安全性研究.南水北调与水利科技, 2(5):l~3
王呼佳,陈洪军. 2006. ANSYS工程分析进阶实例.北京:中国水利水电出版社:2~3
王沫然. 2003.MATLAB与科学计算.北京:电子工业出版社:3~5
王书亭,左孔天. 2004.一种基于均匀化理论的拓扑优化准则法.华中科技大学学报:自然科学版,32(10):28~30
王晓明,刘震宇. 1999.基于均匀化理论的微小型柔性结构拓扑优化的敏度分析.中国机械工程,10(11):1264~1266.
王正林,龚纯,何倩.2009.精通MATLAB科学计算.北京:电子工业出版社:1~5
吴长征,张治平,阎红梅. 2003.南水北调中线穿黄工程渡槽设计研究.人民黄河, 25(2)42~44
吴持恭.2008.水力学.第四版.北京:高等教育出版社:196
吴秀松. 1999.为可持续发展提供水资源保障.中国水利, 6:16~20
武汉水利电力学院. 1981.水工建筑物.北京:水利电力出版社:127~128
于萍萍,张进忠,林存刚. 2006.水资源现状分析及保护对策的研究.资源环境与发展, (2):21~27
俞澄生. 1994.南水北调的资源、环境和社会效应.长江流域资源与环境, 3(3):265~270
袁振,吴长春. 2003.复合材料周期性线弹性微结构拓扑优化设计.固体力学学报, 24(1):40~45
袁振,吴长春. 2003.复合材料扭转截面微结构拓扑优化设计.力学学报, 35(1):39~42
张黎明,李玉圭,傅章吉,等. 2003.东深供水改造工程设计技术总结.广东:广东省水利电力勘测设计院:61~81
张东旭. 1992.连续体结构拓扑优化和形状若干问题. [博士学位论文].大连:大连理工大学
张修真. 1999.南水北调-中国可持续发展的支撑工程.北京:中国水利水电出版社:4
赵顺波,李晓克,赵平. 1999.大型钢筋混凝土多纵梁渡槽结构设计方法的研究.水利学报, (4):6~8
赵文华,陈德亮,颜其照,管枫年. 1989.渡槽.第二版.北京:中国水利水电出版社:1
赵瑜,陈长胜,刘宪亮,白新理. 2002.东深供水工程矩形渡槽优化设计及受力分析.长江科学院报, 5:31~37
中国水利现代化研究课题组. 2004.国家现代化与水利现代化.中国水情分析研究报告, (6):15~17
周厚贵,马吉明,马金刚. 2007.国内外渡槽的发展与南水北调中线漕河渡槽.南水北调与水利科技, 5(3):12~15
周开利,邓春晖.2007. MATLAB基础及其应用教程.北京:北京大学出版社:3
周克民,李俊峰,李霞. 2005.结构拓扑优化研究方法综述.力学进展, 35(1):69~76
朱伯芳,黎展眉,张壁城. 1984.结构优化设计原理与应用.北京:水利电力出版社:94~107
朱召泉. 1999.斜拉渡槽结构的方案优选和有限元分析.河海大学学报, 27(6):86~89
竺慧珠,陈德亮,管枫年. 2005.渡槽.北京:水利水电出版社:56~57,196~272.
竺慧珠. 1991.空腹拱式渡槽主拱圈轴线的优化设计.武汉水利电力学院学报, (5):10~12
左孔天. 2004.连续体结构拓扑优化理论与应用研究. [博士学位论文].武汉:华中科技大学
Anon. 1984. Aqueduct Uses Prestressed Concrete Design. World Construction, 5(2):11~19
Bendsoe M P, Kikuchi N. 1988. Generating Optimal Topologies in Structure Design Using a Homogenization Method. Computer Methods in Applied Mechanics and Engineering,71:197~224
Bendsoe M P, Kikuchi N. 1988. Optimal Topologies in Structural Design Using a Homogenization Method. Comp Meth Appl Mech Engrg, 71(1): 197~224
Bendsoe M P, Sigmund O. 2003. Topology Optimization :Theory,Methods and Applications. NewYork:Springer, 98~104
Diaz A, Kikuchi N. 1992. Solutions to Shape and Topology Eigenvalue Optimization Problems Using a Homogenization Method. Intemational Journal for Numerical Methods in Engineering, 35:1487~1502
Eschenauer H A, Olhoff N. 2001. Topology Optimization of Continuum Structures. Applied Mechanics Review, 54: 331~390
Eugene L, Dennis P E. 1993.Integration of Computer Systems for California Aqueduct Power Plant Systems. IEEE Transactions on Energy Conversion, 8(2):93~113
Fernandes P, Guedes J M, Rodrigues H. 1999. Topology Optimization of Three-dimensional Linear Elastic Structures with a Constrain on“Perimeter”. Computer&Stmctures, 73:583~594
Golani B D. 1978. Rolling Trusses’as False Work for Aqueduct and Bridges. Indian Concreate Journal, 6(2):13~16
Krog L A, Olhoff N. 1999. Optimum Topology and Reinforcement Design of Disk and Plate Structures with MuItiple Stiffness and Eigenfrequency. Computer&Structures, 72:535~563
Min S, Kikuchi N. 1999. Optimal Topology Design of Structures under Dynamic Loads, Structural Optimization, 17:208~218
Min S, Nishiwal S, Kikuchi N. 2000. Unified Topology Design of Static and Vibrating Structures Using Multi-objective Optimization. Structural Optimization, 75:93~116
Shan J J. 1986. Construction of Gomit Aqueduct. Indian Concrete. 10:80~92
Tenek L H, Hagiwara I. 1993. Satic and Vibrational Shape and Topology Optimization Using Homogenization and Mathematical Programming. Computer Methods in Applied Mechanics and Engineering, 109: 143~154
Tenek L H, Hagiwara I. l993. Optimization of Material Distribution within Isotropic and Anisotropic Plates Using Homogenization. Computer Methods in Applied Mechanics and Engineering, 109:155~167
Youn S K, Park S H. 1997. A Study on the Shape Extraction Process in the Structural Topology Optimization Using Homogenized Material. Computer & Structures, 62(3):527~558
博嘉科技. 2002.有限元分析软件——ANSYS融汇与贯通.北京:中国水利水电出版社:352~355
蔡坤,张洪武,陈飙松. 2006.基于wolff法则的连续体结构拓扑优化方法.力学学报,38(4):514～521.
长江勘测规划设计研究院. 2000.东深供水改造工程——U型薄壳预应力渡槽槽身结构研究报告.武汉
长江水利委员会信息研究中心. 1990.工程设计与施工技术汇编——国内外渡槽.武汉
陈文义,赵顺波,李树瑶. 1996.南水北调工程大型预应力混凝土波槽结构选型研究.华北水利水电学院学报, 17(4):9~15
董桂西,王藏柱. 2000.结构优化设计的现状及展望.电力情报, 1: 5~7
段进,倪栋,王国业. 2006. ANSYS10.0结构分析从入门到精通.北京:兵器工业出版社,北京科海电子出版社:6~8
飞思科技产品研发中心. 2003. MATLAB6.5辅助优化计算与设计.北京:电子工业出版社:1~3
傅志敏. 2003.斜拉式大型渡槽槽身优化设计研究. [硕士学位论文].南京:河海大学
高小翠,彭琳. 2003.大型预应力混凝土箱式渡槽有限元分析.中国农村水利水电, (5):53~55
郭旭,赵康. 2004.基于拓扑描述函数的连续体结构拓扑优化方法.力学学报,36(5):520～526.
过迟. 1994.南水北调穿黄工程总体布置研究.人民长江, 25(1):12~17
韩亦方. 2001.南水北调工程的重大科技问题.水利水电技术, 32(1):35
华东水利学院. 1979.水力学.北京:科学出版社:123~134
华东水利学院. 1987.水工设计手册.北京:水利电力出版社:16~18
姜冬菊. 2008.结构拓扑和布局优化及工程应用研究. [博士学位论文].南京:河海大学
姜新佩. 1998. U型薄壳渡槽槽身设计的研究.水利水电技术, 8(2):13~204
李廉锟. 1979.结构力学.北京:高等教育出版社:131~137
林继镛. 2006.水工建筑物.第四版.北京:中国水利水电出版社:497
凌均忆. 1992.斜拉输水结构布置及其技术特性.水利水电技术, 6(2):20~24
刘东常,白新理,刘宪亮. 1995.南水北调中线穿黄渡槽工程中预应力U形薄壳渡槽结构分析(预应力U形薄壳渡槽研究报告之一).郑州
刘国纬. 2000.关于中国南水北调的思考.水科学进展, 11(4):346~350
刘宁. 2006.南水北调中线一期工程穿黄方案的论证与选择.水利学报, 37(1):1~9
刘蓉华. 1994.拱型结构分析的拓扑变化法.西安交通大学学报: 29(4):557~563
刘儒勋,舒其望. 2003.计算流体力学的若干新方法.北京:科学出版社, 2003:1~4l
刘晓文. 2001.渡槽结构优化研究及实现.武汉大学学报(工学版): (3):47~50.
罗震,陈立平,黄玉盈,张云清. 2004.连续体结构的拓扑优化设计.力学进展, 34(4):463~476
梅来彬,王波. 2008.浅析ANSYS在桥梁拓扑优化设计中的应用.科技创新导报, 27:104~105
荣见华,姜节胜,颜东煌. 2002.多约束的桥梁结构拓扑优化.工程力学学报, 19(4):160~165
隋允康,叶红玲,彭细荣. 2006.应力约束全局化策略下的连续体结构拓扑优化.力学学报,38(3):364-370.
王海波,张伯艳,周杰. 2004.南水北调工程的抗震安全性研究.南水北调与水利科技, 2(5):l~3
王呼佳,陈洪军. 2006. ANSYS工程分析进阶实例.北京:中国水利水电出版社:2~3
王沫然. 2003.MATLAB与科学计算.北京:电子工业出版社:3~5
王书亭,左孔天. 2004.一种基于均匀化理论的拓扑优化准则法.华中科技大学学报:自然科学版,32(10):28~30
王晓明,刘震宇. 1999.基于均匀化理论的微小型柔性结构拓扑优化的敏度分析.中国机械工程,10(11):1264~1266.
王正林,龚纯,何倩.2009.精通MATLAB科学计算.北京:电子工业出版社:1~5
吴长征,张治平,阎红梅. 2003.南水北调中线穿黄工程渡槽设计研究.人民黄河, 25(2)42~44
吴持恭.2008.水力学.第四版.北京:高等教育出版社:196
吴秀松. 1999.为可持续发展提供水资源保障.中国水利, 6:16~20
武汉水利电力学院. 1981.水工建筑物.北京:水利电力出版社:127~128
于萍萍,张进忠,林存刚. 2006.水资源现状分析及保护对策的研究.资源环境与发展, (2):21~27
俞澄生. 1994.南水北调的资源、环境和社会效应.长江流域资源与环境, 3(3):265~270
袁振,吴长春. 2003.复合材料周期性线弹性微结构拓扑优化设计.固体力学学报, 24(1):40~45
袁振,吴长春. 2003.复合材料扭转截面微结构拓扑优化设计.力学学报, 35(1):39~42
张黎明,李玉圭,傅章吉,等. 2003.东深供水改造工程设计技术总结.广东:广东省水利电力勘测设计院:61~81
张东旭. 1992.连续体结构拓扑优化和形状若干问题. [博士学位论文].大连:大连理工大学
张修真. 1999.南水北调-中国可持续发展的支撑工程.北京:中国水利水电出版社:4
赵顺波,李晓克,赵平. 1999.大型钢筋混凝土多纵梁渡槽结构设计方法的研究.水利学报, (4):6~8
赵文华,陈德亮,颜其照,管枫年. 1989.渡槽.第二版.北京:中国水利水电出版社:1
赵瑜,陈长胜,刘宪亮,白新理. 2002.东深供水工程矩形渡槽优化设计及受力分析.长江科学院报, 5:31~37
中国水利现代化研究课题组. 2004.国家现代化与水利现代化.中国水情分析研究报告, (6):15~17
周厚贵,马吉明,马金刚. 2007.国内外渡槽的发展与南水北调中线漕河渡槽.南水北调与水利科技, 5(3):12~15
周开利,邓春晖.2007. MATLAB基础及其应用教程.北京:北京大学出版社:3
周克民,李俊峰,李霞. 2005.结构拓扑优化研究方法综述.力学进展, 35(1):69~76
朱伯芳,黎展眉,张壁城. 1984.结构优化设计原理与应用.北京:水利电力出版社:94~107
朱召泉. 1999.斜拉渡槽结构的方案优选和有限元分析.河海大学学报, 27(6):86~89
竺慧珠,陈德亮,管枫年. 2005.渡槽.北京:水利水电出版社:56~57,196~272.
竺慧珠. 1991.空腹拱式渡槽主拱圈轴线的优化设计.武汉水利电力学院学报, (5):10~12
左孔天. 2004.连续体结构拓扑优化理论与应用研究. [博士学位论文].武汉:华中科技大学
Anon. 1984. Aqueduct Uses Prestressed Concrete Design. World Construction, 5(2):11~19
Bendsoe M P, Kikuchi N. 1988. Generating Optimal Topologies in Structure Design Using a Homogenization Method. Computer Methods in Applied Mechanics and Engineering,71:197~224
Bendsoe M P, Kikuchi N. 1988. Optimal Topologies in Structural Design Using a Homogenization Method. Comp Meth Appl Mech Engrg, 71(1): 197~224
Bendsoe M P, Sigmund O. 2003. Topology Optimization :Theory,Methods and Applications. NewYork:Springer, 98~104
Diaz A, Kikuchi N. 1992. Solutions to Shape and Topology Eigenvalue Optimization Problems Using a Homogenization Method. Intemational Journal for Numerical Methods in Engineering, 35:1487~1502
Eschenauer H A, Olhoff N. 2001. Topology Optimization of Continuum Structures. Applied Mechanics Review, 54: 331~390
Eugene L, Dennis P E. 1993.Integration of Computer Systems for California Aqueduct Power Plant Systems. IEEE Transactions on Energy Conversion, 8(2):93~113
Fernandes P, Guedes J M, Rodrigues H. 1999. Topology Optimization of Three-dimensional Linear Elastic Structures with a Constrain on“Perimeter”. Computer&Stmctures, 73:583~594
Golani B D. 1978. Rolling Trusses’as False Work for Aqueduct and Bridges. Indian Concreate Journal, 6(2):13~16
Krog L A, Olhoff N. 1999. Optimum Topology and Reinforcement Design of Disk and Plate Structures with MuItiple Stiffness and Eigenfrequency. Computer&Structures, 72:535~563
Min S, Kikuchi N. 1999. Optimal Topology Design of Structures under Dynamic Loads, Structural Optimization, 17:208~218
Min S, Nishiwal S, Kikuchi N. 2000. Unified Topology Design of Static and Vibrating Structures Using Multi-objective Optimization. Structural Optimization, 75:93~116
Shan J J. 1986. Construction of Gomit Aqueduct. Indian Concrete. 10:80~92
Tenek L H, Hagiwara I. 1993. Satic and Vibrational Shape and Topology Optimization Using Homogenization and Mathematical Programming. Computer Methods in Applied Mechanics and Engineering, 109: 143~154
Tenek L H, Hagiwara I. l993. Optimization of Material Distribution within Isotropic and Anisotropic Plates Using Homogenization. Computer Methods in Applied Mechanics and Engineering, 109:155~167
Youn S K, Park S H. 1997. A Study on the Shape Extraction Process in the Structural Topology Optimization Using Homogenized Material. Computer & Structures, 62(3):527~558
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |