基于传力路径的结构布局优化方法研究
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摘要
在飞机结构设计中,对于飞机主承力构件的轻量化设计十分关键。布局优化技术是目前开展结构轻量化设计研究十分有效的手段之一。拓扑优化作为布局优化技术的一种重要方法,正被更多的结构设计者所关注。实际上,无论是拓扑优化还是其它布局优化技术,其本质都是在寻求最直接的传力路径,即传力路径最短。
本文基于研究传力路径的重要意义,首先从研究Michell桁架准则入手,引入结构承载因子概念,以结构的传力路径作为设计变量,通过在传力路径上对结构内力进行积分,得到相应的结构承载因子。由于该承载因子是结构传力路径的泛函,以此本文用变分法导出对应结构最短传力路径的微分方程,并用里兹法求解该微分方程,得到结构的最短传力路径。并和用Michell桁架理论得到的最小质量桁架进行比较,证明了本文提出量化模型的有效性。
接着,通过对飞机机身加强框的传力分析,分别采用基于传力路径的解析方法和数值优化方法,优化得到了最优的刚框缘条轮廓和最佳的刚框内力分配比值。通过对比优化结果,其最大误差在5%以内,证明了量化模型的正确性。
最后,通过对相同外形和载荷作用下的半框有限元模型进行尺寸优化,得到了与解析模型相同的结构承载因子,证明了本文提出解析模型的有效性。同时,优化后的结构最小质量略高于解析解,表明本文提出的解析模型能够估算传力结构的最小质量,也为设计者在结构减重设计时提供了一个参照目标。
In the design of aircraft structures, the lightweight design of aircraft with the main component of force bearing is very critical. Currently, the layout optimization technique is one of very effective mean to carry out the design of structural lightweight. In layout optimization technique, topology optimization is a kind of important method, are being more structural designers attention. In fact, whether topological optimization or other layout optimization technology, its essence is seeking the most direct load-carrying path, namely the shortest load-carrying path.
This paper, based on the research significance of load-carrying path, we first studied Michell truss criterion, introduced concept of structural load-carrying factor, took the structural load-carrying path as the design variables, by the integral to internal force of structure in the load-carrying path, corresponding load-carrying factor is obtained. As the load-carrying factor is the functional of structural load-carrying path, then we derived differential equation of the shortest load path of the corresponding structure with variational method, and solving the differential equation with the Ritz method, obtained the shortest load-carrying path structure. Compared with the result by Michell truss theory, proved that we put forward the quantification model is effective.
Then, the paper analyzed the force transfer of fuselage bulkhead, respectively by analytical method and numerical optimization method based on the load-carrying path, optimized and got the best outline of edge strip and optimum the internal force distribution ratio of bulkhead. Through comparing the optimization results, the maximum error is less than 5%, proved the validity of the model theory.
Finally ,adopt size optimization method optimized the half frame finite element model under the same shape and loads, we obtained the same load-carrying factor with the analytical model, Proved the validity of analytical model is put forward in this paper. Meanwhile through comparing the optimization results and analytical calculations of the structure minimum quality, show that the proposed analytical model can estimate the minimum quality of the force transmission structure also provided a reference goal for designers when they reduced weight in structure design.
本文基于研究传力路径的重要意义,首先从研究Michell桁架准则入手,引入结构承载因子概念,以结构的传力路径作为设计变量,通过在传力路径上对结构内力进行积分,得到相应的结构承载因子。由于该承载因子是结构传力路径的泛函,以此本文用变分法导出对应结构最短传力路径的微分方程,并用里兹法求解该微分方程,得到结构的最短传力路径。并和用Michell桁架理论得到的最小质量桁架进行比较,证明了本文提出量化模型的有效性。
接着,通过对飞机机身加强框的传力分析,分别采用基于传力路径的解析方法和数值优化方法,优化得到了最优的刚框缘条轮廓和最佳的刚框内力分配比值。通过对比优化结果,其最大误差在5%以内,证明了量化模型的正确性。
最后,通过对相同外形和载荷作用下的半框有限元模型进行尺寸优化,得到了与解析模型相同的结构承载因子,证明了本文提出解析模型的有效性。同时,优化后的结构最小质量略高于解析解,表明本文提出的解析模型能够估算传力结构的最小质量,也为设计者在结构减重设计时提供了一个参照目标。
In the design of aircraft structures, the lightweight design of aircraft with the main component of force bearing is very critical. Currently, the layout optimization technique is one of very effective mean to carry out the design of structural lightweight. In layout optimization technique, topology optimization is a kind of important method, are being more structural designers attention. In fact, whether topological optimization or other layout optimization technology, its essence is seeking the most direct load-carrying path, namely the shortest load-carrying path.
This paper, based on the research significance of load-carrying path, we first studied Michell truss criterion, introduced concept of structural load-carrying factor, took the structural load-carrying path as the design variables, by the integral to internal force of structure in the load-carrying path, corresponding load-carrying factor is obtained. As the load-carrying factor is the functional of structural load-carrying path, then we derived differential equation of the shortest load path of the corresponding structure with variational method, and solving the differential equation with the Ritz method, obtained the shortest load-carrying path structure. Compared with the result by Michell truss theory, proved that we put forward the quantification model is effective.
Then, the paper analyzed the force transfer of fuselage bulkhead, respectively by analytical method and numerical optimization method based on the load-carrying path, optimized and got the best outline of edge strip and optimum the internal force distribution ratio of bulkhead. Through comparing the optimization results, the maximum error is less than 5%, proved the validity of the model theory.
Finally ,adopt size optimization method optimized the half frame finite element model under the same shape and loads, we obtained the same load-carrying factor with the analytical model, Proved the validity of analytical model is put forward in this paper. Meanwhile through comparing the optimization results and analytical calculations of the structure minimum quality, show that the proposed analytical model can estimate the minimum quality of the force transmission structure also provided a reference goal for designers when they reduced weight in structure design.
引文
[1] Charlton T M. Maxwell-Michell theory of minimum weight of structures [J]. Nature, 1963, 200: 251~252.
[2] Cox H L. The Design of Structures of Least Weight [M]. Oxford: Pergamon, 1965.
[3] Hegemier G A; Prager W. On Michell trusses [J]. International Journal of Mechanical Sciences, 1969, 11(2): 209~215.
[4] Hemp W S. Optimum Structure [M]. Oxford: Clarendon Press, 1973, 70~101.
[5] Rozvany G I N. Some shortcomings in Michell's truss theory [J]. Structural Optimization, 1996, 12 (4): 244~250.
[6] Rozvany G I N. Some shortcomings in Michell's truss theory [J]. Structural Optimization, 1997, 13 (2~3): 203~204.
[7] Rozvany G I N, Gollub W. Michell layouts for various combinations of line support [J]. International Journal of Mechanical Sciences, 1990, 32(12):1021~1043.
[8] Rozvany G I N, Gollub W, Zhou M. Exact Michell layouts for various combinations of line supports—Part II [J],Structural Optimization,1997,14(2~3):146~147.
[9] Chan A S L. The design of Michell optimum structures. Cranfield: College of Aeronautics Report No 142, 1960.
[10] Chan H S Y. Half-plane slip-line fields and Michell structures [J]. Quarterly Journal of Mechanics and Applied Mathematics, 1967, 20,453~469.
[11] Lewinski T, Zhou M, Rozvany GIN. Extended exact solutions for least-weight truss layouts-Paper I: cantilever with a horizontal axis of symmetry [J]. International Journal of Mechanical Sciences, 1994, 36(5)375~398.
[12] Rozvany G I N. Structural Design via Optimality Criteria—The Prager Approach to Structural Optimization [M]. Dordrecht Boston London: Kluwer, Academic Publishers, 1989.
[13]周克民,胡云昌.利用有限元构Michell桁架的一种方法[J].力学学报,2002, 34(6): 935~940.
[14] Dorn W S, Gomory R E, Greenberg H J. Automatic design of optimal structures [J]. J. de Mechanique, 1964, 3(1):25~52.
[15] Bends?e M P, Kikuchi N. Generating optimal topologies in structural design using a homogenization method [J]. Comp. Meth. Appl. Mech. Engrg., 1988, 71(1):197~224.
[16]程耿东.关于桁架结构拓扑优化中的奇异最优解[J].大连理工大学学报,2000,40:379~383.
[17]隋允康.建模·变换·优化—结构综合方法新进展[M].大连:大连理工大学出版社,1996.
[18]邓扬晨,邱克鹏,张卫红,等.基于传力路径下结构优化设计研究[J].机械科学与技术,2003,22(4):622~626.
[19]张翼.基于结构拓扑优化的机翼主传力路径确定的技术研究[硕士学位论文].沈阳:沈阳航空工业学院,2010.
[20] Komarov, A.A. Foundations of Design of Load-Carrying Structures. Kuibyshev, 1965.
[21] Komarov, V.A. On the Rational Load-Carrying Structures of Delta Wings. Kuibyshev, Samara, Russia, 1968, pp. 6~26.
[22] Kelly D W, Tosh M W. A procedure for determining load paths in elastic continua [J]. Engineering Computations, 1995, 12(5):415~424.
[23] Kelly D W, Tosh M W. Interpreting load paths and stress trajectories in elasticity [J]. Engineering Computations, 2000, 17(2): 117~135.
[24] Kelly D W, Hsu P, Asudullah M. Load paths and load flow in finite element analysis [J]. Engineering Computations, 2001, 18(1): 304~313.
[25]隋允康,叶红玲,杜家政.结构拓扑优化的发展及其模型转化为独立层次的迫切性[J].工程力学,2005, 22:110~114.
[26] Gerd Schuhmacher, Martin Stettner, Herbert Hoernlein, Owen O'Leary, Rainer Zotemantel, Markus Wagner. Optimization assisted structural design of a new military transport aircraft [R]. AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 2004, 10:1~9.
[27]钱令希.工程结构优化设计[M].北京:水利电力出版社, 1983,238~244.
[28]李炳威.结构的优化设计[M].北京:科学出版社, 1979,16~17.
[29]段宝岩,叶尚辉.考虑性态约束时多工况桁架结构拓扑优化设计[J].力学学报,1992, 24(1):59~69.
[30] Hirokazu Nishi, Tomohito Okamoto, Keishi Kawamo. Study on the development of general-purpose genetic algorithm engine [A].WCSMO-4 [C], June 4~8, 2001, Dalian, China, 428~431.
[31] Jog C S, Haber R B. Stability of finite element models for distributed parameter optimization and topology design [J]. Computer Methods in Applied Mechanics and Engineering, 1996, 130:203~226.
[32] Sigmund O, Petersson J. Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima [J]. Structural Optimization, 1998, 16:68~75.
[33]吴嘉蒙,杨德庆,金咸定.对MSC.Nastran-OPTISHAPE结构拓扑优化设计几个问题的探讨. MSC.Software中国用户论文集,2002.
[34]袁振,吴长春.采用非协调元的连续体拓扑优化设计[J].力学学报,2003,35(3):176~180.
[35]左孔天.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.2004.
[36] M P Bendsoe , R B Haber. The Michell truss layout problem as a low volume fraction limit of the perforated plate topology optimization problem: an asymptotic study [J]. Structural Optimization, 1993, 6(4):263~267.
[37] Komarov V.A., Weisshaar T.A. New Approach to Improving the Aircraft Structural Design Process [J]. Journal of Aircraft, 2002, 39(2):227~233.
[38]王志瑾,姚卫星.飞机结构设计[M].北京:国防工业出版社,2007.
[39]周克民,李霞.长悬臂桁架受横向集中力的拓扑优化[J].华侨大学学报,2009,30(1):80~84.
[40]周克民,李俊峰. Michell桁架在离散过程中的一些性质[J].工程力学,2003,增刊:322~325.
[41] Rozvany G I N. Exact analytical solutions for some popular benchmark problems in topology optimization [J]. Structural Optimization, 1998, 15:42~48.
[42] Rozvany G I N, Bends?e M P,U Kirsch. Layout optimization of structures [J]. American Society of Mechanical Engineers, 1995, 48(2):48~82.
[43] Komarov V.A. Research of morphing wing efficiency. Samara State Aerospace University. 2004, 6:16~17.
[44]徐旭东,徐超,郑晓亚.飞行器结构设计[M].西安:西北工业大学出版社,2010.
[45]郦正能,程小全,方卫国.飞行器结构学(第2版)[M].北京:北京航空航天大学出版社,2010.
[46]邓扬晨,李学军,朱继宏,等.基于敏度阀值拓扑优化方法在飞机加强框设计中的应用[J].力学与实践,2005,27(4):26~31.
[47]《飞机设计手册》总编委会.飞机设计手册[M].北京:航空工业出版社,1996.
[48]邓扬晨,孙聪,王琦.飞机设计中的工程优化方法与建模[M].沈阳:吉林大学出版社,2009.
[2] Cox H L. The Design of Structures of Least Weight [M]. Oxford: Pergamon, 1965.
[3] Hegemier G A; Prager W. On Michell trusses [J]. International Journal of Mechanical Sciences, 1969, 11(2): 209~215.
[4] Hemp W S. Optimum Structure [M]. Oxford: Clarendon Press, 1973, 70~101.
[5] Rozvany G I N. Some shortcomings in Michell's truss theory [J]. Structural Optimization, 1996, 12 (4): 244~250.
[6] Rozvany G I N. Some shortcomings in Michell's truss theory [J]. Structural Optimization, 1997, 13 (2~3): 203~204.
[7] Rozvany G I N, Gollub W. Michell layouts for various combinations of line support [J]. International Journal of Mechanical Sciences, 1990, 32(12):1021~1043.
[8] Rozvany G I N, Gollub W, Zhou M. Exact Michell layouts for various combinations of line supports—Part II [J],Structural Optimization,1997,14(2~3):146~147.
[9] Chan A S L. The design of Michell optimum structures. Cranfield: College of Aeronautics Report No 142, 1960.
[10] Chan H S Y. Half-plane slip-line fields and Michell structures [J]. Quarterly Journal of Mechanics and Applied Mathematics, 1967, 20,453~469.
[11] Lewinski T, Zhou M, Rozvany GIN. Extended exact solutions for least-weight truss layouts-Paper I: cantilever with a horizontal axis of symmetry [J]. International Journal of Mechanical Sciences, 1994, 36(5)375~398.
[12] Rozvany G I N. Structural Design via Optimality Criteria—The Prager Approach to Structural Optimization [M]. Dordrecht Boston London: Kluwer, Academic Publishers, 1989.
[13]周克民,胡云昌.利用有限元构Michell桁架的一种方法[J].力学学报,2002, 34(6): 935~940.
[14] Dorn W S, Gomory R E, Greenberg H J. Automatic design of optimal structures [J]. J. de Mechanique, 1964, 3(1):25~52.
[15] Bends?e M P, Kikuchi N. Generating optimal topologies in structural design using a homogenization method [J]. Comp. Meth. Appl. Mech. Engrg., 1988, 71(1):197~224.
[16]程耿东.关于桁架结构拓扑优化中的奇异最优解[J].大连理工大学学报,2000,40:379~383.
[17]隋允康.建模·变换·优化—结构综合方法新进展[M].大连:大连理工大学出版社,1996.
[18]邓扬晨,邱克鹏,张卫红,等.基于传力路径下结构优化设计研究[J].机械科学与技术,2003,22(4):622~626.
[19]张翼.基于结构拓扑优化的机翼主传力路径确定的技术研究[硕士学位论文].沈阳:沈阳航空工业学院,2010.
[20] Komarov, A.A. Foundations of Design of Load-Carrying Structures. Kuibyshev, 1965.
[21] Komarov, V.A. On the Rational Load-Carrying Structures of Delta Wings. Kuibyshev, Samara, Russia, 1968, pp. 6~26.
[22] Kelly D W, Tosh M W. A procedure for determining load paths in elastic continua [J]. Engineering Computations, 1995, 12(5):415~424.
[23] Kelly D W, Tosh M W. Interpreting load paths and stress trajectories in elasticity [J]. Engineering Computations, 2000, 17(2): 117~135.
[24] Kelly D W, Hsu P, Asudullah M. Load paths and load flow in finite element analysis [J]. Engineering Computations, 2001, 18(1): 304~313.
[25]隋允康,叶红玲,杜家政.结构拓扑优化的发展及其模型转化为独立层次的迫切性[J].工程力学,2005, 22:110~114.
[26] Gerd Schuhmacher, Martin Stettner, Herbert Hoernlein, Owen O'Leary, Rainer Zotemantel, Markus Wagner. Optimization assisted structural design of a new military transport aircraft [R]. AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 2004, 10:1~9.
[27]钱令希.工程结构优化设计[M].北京:水利电力出版社, 1983,238~244.
[28]李炳威.结构的优化设计[M].北京:科学出版社, 1979,16~17.
[29]段宝岩,叶尚辉.考虑性态约束时多工况桁架结构拓扑优化设计[J].力学学报,1992, 24(1):59~69.
[30] Hirokazu Nishi, Tomohito Okamoto, Keishi Kawamo. Study on the development of general-purpose genetic algorithm engine [A].WCSMO-4 [C], June 4~8, 2001, Dalian, China, 428~431.
[31] Jog C S, Haber R B. Stability of finite element models for distributed parameter optimization and topology design [J]. Computer Methods in Applied Mechanics and Engineering, 1996, 130:203~226.
[32] Sigmund O, Petersson J. Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima [J]. Structural Optimization, 1998, 16:68~75.
[33]吴嘉蒙,杨德庆,金咸定.对MSC.Nastran-OPTISHAPE结构拓扑优化设计几个问题的探讨. MSC.Software中国用户论文集,2002.
[34]袁振,吴长春.采用非协调元的连续体拓扑优化设计[J].力学学报,2003,35(3):176~180.
[35]左孔天.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.2004.
[36] M P Bendsoe , R B Haber. The Michell truss layout problem as a low volume fraction limit of the perforated plate topology optimization problem: an asymptotic study [J]. Structural Optimization, 1993, 6(4):263~267.
[37] Komarov V.A., Weisshaar T.A. New Approach to Improving the Aircraft Structural Design Process [J]. Journal of Aircraft, 2002, 39(2):227~233.
[38]王志瑾,姚卫星.飞机结构设计[M].北京:国防工业出版社,2007.
[39]周克民,李霞.长悬臂桁架受横向集中力的拓扑优化[J].华侨大学学报,2009,30(1):80~84.
[40]周克民,李俊峰. Michell桁架在离散过程中的一些性质[J].工程力学,2003,增刊:322~325.
[41] Rozvany G I N. Exact analytical solutions for some popular benchmark problems in topology optimization [J]. Structural Optimization, 1998, 15:42~48.
[42] Rozvany G I N, Bends?e M P,U Kirsch. Layout optimization of structures [J]. American Society of Mechanical Engineers, 1995, 48(2):48~82.
[43] Komarov V.A. Research of morphing wing efficiency. Samara State Aerospace University. 2004, 6:16~17.
[44]徐旭东,徐超,郑晓亚.飞行器结构设计[M].西安:西北工业大学出版社,2010.
[45]郦正能,程小全,方卫国.飞行器结构学(第2版)[M].北京:北京航空航天大学出版社,2010.
[46]邓扬晨,李学军,朱继宏,等.基于敏度阀值拓扑优化方法在飞机加强框设计中的应用[J].力学与实践,2005,27(4):26~31.
[47]《飞机设计手册》总编委会.飞机设计手册[M].北京:航空工业出版社,1996.
[48]邓扬晨,孙聪,王琦.飞机设计中的工程优化方法与建模[M].沈阳:吉林大学出版社,2009.