基于自由曲线建模方法的翼梢小翼优化设计
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摘要
翼梢小翼具有减少诱导阻力的作用,对于运输机而言,安装小翼可以提高其气动性能。但是,翼梢小翼又会导致翼根弯矩增加,影响机翼结构强度。多参数控制的NURBS曲线具有很强的成形能力,适合于构建新型翼梢小翼参数模型。小翼翼型对小翼气动性能影响较大,设计中需要选择合适的翼型。此外,气动计算方法和优化策略的选择直接影响优化结果的好坏。
本文以小翼减阻最大化为设计目标,以翼根弯矩增加量小于5%为约束,以商用飞机的巡航状态为设计状态,对所设计的新型小翼进行优化。本文首先根据小翼特点,确定小翼翼型的选择标准,然后通过叠加Hicks-Henne函数方法优选得到合适的翼型。其次,利用基于NURBS原理的自由曲线构造小翼的前后缘线,最后利用曲面扫掠形成模型。第三,考虑到涡格法计算速度快,CFX计算精度高,全局优化范围广,分级优化能降低优化规模,本文将它们组合成两种策略:基于涡格法的全局优化、基于CFX的分级优化。同时,加入基于涡格法的分级优化用于对比,全局优化分为后缘有约束和无约束两种设计。最后,对优化结果进行分析,将其与简单增展机翼的减阻效果和机理进行对比。结果表明:本文所设计的翼梢小翼通过不同优化策略得到的小翼都有良好的减阻效果,其中后缘有约束的全局优化得到的小翼和基于CFX的分级优化得到的小翼分别减阻力7.02%和6.7%。在近场,简单增展机翼能够使翼尖涡远离机翼,而翼梢小翼不仅如此,还能将较强翼尖涡分散成强度降低的涡。在远场,小翼和简单增展机翼都没能加快翼尖涡消散。
Application of winglet on transport will lead to improvement of aerodynamic characteristics for it is able to reduce induced drag of wing. However, Wing Root Bending Moment (WRBM) will also increase due to the winglet, and then the structure of wing has to be adapted. NURBS curve controlled by a lot of number is able to from various forms, thus it is suit for modeling a novel winglet. Airfoil used in the model of winglet should be selected particularly for it has large effect on the performance of winglet. Besides, the calculating method and strategies of optimization used in optimal design have direct influence on the results.
The design target of the optimization of winglet modeled in this paper is to reduce the drag of wing with winglet, which subjects to the constraint that the percent of increase of WRBM should be less than 5%, and the design status is cruise phase of commercial aircraft. Firstly, the criteria of choosing suitable airfoil is setup according to the feature of winglet, and then a good airfoil is carried out by Hicks-Henne function method. Secondly, surface model of winglet is established by couple of freestyle curve based on NURBS and airfoil chose in CATIA. Thirdly, considering that Vortex Lattice Method (VLM) is a fast method, CFX is of high fidelity, global optimization is able to search in large scale; stratified optimization can reduce the amount of calculation, this paper put them into two strategies: Global Optimization based on VLM and Stratified Optimization based on CFX. To make some comparison, stratified optimization base on VLM is also adopted in Optimization of winglet; Global Optimization based on VLM with the constraint of trailing edge in small range is also conducted. Finally, the flow field of winglet and simple extension of wing with same span are analyzed and compared to figure out the mechanism of winglet reducing drag. The results indicate that:Winglets designed in this paper via different methods and strategies achieve large reduction of drag, particularly winglet designed by global optimization with constraint and the one by stratified optimization based on CFX reduce drag by 7.02% and 6.7% respectively. Winglet can reduce drag by scattering the strong tip vortex into several vortex of less strength in the neighborhood of wingtip and moving the tip vortex away while simple extension of wing can only move vortex away. However, in far field, all of winglets and extension of wing have little effect on scattering wingtip vortex.
本文以小翼减阻最大化为设计目标,以翼根弯矩增加量小于5%为约束,以商用飞机的巡航状态为设计状态,对所设计的新型小翼进行优化。本文首先根据小翼特点,确定小翼翼型的选择标准,然后通过叠加Hicks-Henne函数方法优选得到合适的翼型。其次,利用基于NURBS原理的自由曲线构造小翼的前后缘线,最后利用曲面扫掠形成模型。第三,考虑到涡格法计算速度快,CFX计算精度高,全局优化范围广,分级优化能降低优化规模,本文将它们组合成两种策略:基于涡格法的全局优化、基于CFX的分级优化。同时,加入基于涡格法的分级优化用于对比,全局优化分为后缘有约束和无约束两种设计。最后,对优化结果进行分析,将其与简单增展机翼的减阻效果和机理进行对比。结果表明:本文所设计的翼梢小翼通过不同优化策略得到的小翼都有良好的减阻效果,其中后缘有约束的全局优化得到的小翼和基于CFX的分级优化得到的小翼分别减阻力7.02%和6.7%。在近场,简单增展机翼能够使翼尖涡远离机翼,而翼梢小翼不仅如此,还能将较强翼尖涡分散成强度降低的涡。在远场,小翼和简单增展机翼都没能加快翼尖涡消散。
Application of winglet on transport will lead to improvement of aerodynamic characteristics for it is able to reduce induced drag of wing. However, Wing Root Bending Moment (WRBM) will also increase due to the winglet, and then the structure of wing has to be adapted. NURBS curve controlled by a lot of number is able to from various forms, thus it is suit for modeling a novel winglet. Airfoil used in the model of winglet should be selected particularly for it has large effect on the performance of winglet. Besides, the calculating method and strategies of optimization used in optimal design have direct influence on the results.
The design target of the optimization of winglet modeled in this paper is to reduce the drag of wing with winglet, which subjects to the constraint that the percent of increase of WRBM should be less than 5%, and the design status is cruise phase of commercial aircraft. Firstly, the criteria of choosing suitable airfoil is setup according to the feature of winglet, and then a good airfoil is carried out by Hicks-Henne function method. Secondly, surface model of winglet is established by couple of freestyle curve based on NURBS and airfoil chose in CATIA. Thirdly, considering that Vortex Lattice Method (VLM) is a fast method, CFX is of high fidelity, global optimization is able to search in large scale; stratified optimization can reduce the amount of calculation, this paper put them into two strategies: Global Optimization based on VLM and Stratified Optimization based on CFX. To make some comparison, stratified optimization base on VLM is also adopted in Optimization of winglet; Global Optimization based on VLM with the constraint of trailing edge in small range is also conducted. Finally, the flow field of winglet and simple extension of wing with same span are analyzed and compared to figure out the mechanism of winglet reducing drag. The results indicate that:Winglets designed in this paper via different methods and strategies achieve large reduction of drag, particularly winglet designed by global optimization with constraint and the one by stratified optimization based on CFX reduce drag by 7.02% and 6.7% respectively. Winglet can reduce drag by scattering the strong tip vortex into several vortex of less strength in the neighborhood of wingtip and moving the tip vortex away while simple extension of wing can only move vortex away. However, in far field, all of winglets and extension of wing have little effect on scattering wingtip vortex.
引文
[1] Jupp, J. Wing Aerodynamics and the Science of Compromise. Aeronautical Journal, Vol.105, No.1053, November, 2001, pp. 633-641
[2] Rudolph P.K.C. Boeing, Seattle, US patent for High Taper Wing Tip Extension. No.5039032, filed 19 May 1989.
[3] Herrick L.L. et al. Boeing, Seattle, US Patent for Blunt Leading Edge Raked Wingtips. No.6089502, filed 12 Jun 1998.
[4] Kroo, Ilan. DRAG DUE TO LIFT: Concepts for Prediction and Reduction [J]. Annu. Rev. Fluid Mech, 2001, 33, 587-617.
[5] Gratzer, L. Blended Winglet. Patent NO. US5348253, 1 February 1993.
[6] Gerd Heller, Manfred Maisel, Peter Kreuzer. Wingtip extension for a wing. Patent NO. US 6722615 B2
[7] Louis B. Gratzer. Spiroid-tipped wing. Patent NO. 5102068
[8] Reneaux, J. Overview on Drag Reduction Technologies for Civil Transport Aircraft. Applied Sciences, 1(18), 2004.
[9] McMasters J., Kroo IM. Advanced Configurations for very Large Transport Airplanes. Aircraft Design, 2008, 1:217-42.
[10] Hugues, C.US Patent for Cylindrical Wing Tip with Helical Slot, No. US 6892988 B2. Filed 17 May, 2005.
[11] Smith S.C. A Computational and Experimental study of nonlinear aspects of Induced Drag. NASA TP 3598, Natl. Aeronaut. Space Admin., Washington, DC.
[12] Hackett J.E. Vortex Drag Reduction by Aft-mounted Diffusing Vanes. ICAS Paper 80-13.4.
[13] Spillman JJ. The Use of Wing Tip Sails to Reduce Vortex Drag. Aeronaut Journal. 1978, 82:387-395.
[14] Kroo, I. Nonplanar Wing Concepts for Increased Aircraft Efficiency. VKI lecture series on Innovative Configurations and Advanced Concepts for Future Civil Aircraft, June 6-10, 2005.
[15] Patterson J.C. US Patent No.4533101, 1985.
[16] Terry JE. Aerodynamic Characteristics of Ring wings: a bibliography. Redstone Scientific Information Center Report RSIC-285.
[17] Miranda LR. Boxplane Configuration– Conceptual analysis and initial Experimental Verification.Lockheed California Company Report LR 25180.
[18] Wolkovitch J. The Joined Wing: an overview. Journal of Aircraft, 1986, 23:161-178.
[19] J.-L. Hnatrais-Gervois, R. Grenon, A. Mann, A. Buscher. Downward pointing winglet design and assessment within the M-DAW research project [J]. The Aeronautical Journal. Vol. 113, No.1142, 2009,pp. 221-232
[20] Heyson H.H, Roebe G.D, Fulton C.L. Theoretical Parametric Study of the Relative Advantages of Winglets and Wing-tip Extensions. NASA TP 1020.
[21] Jones R.T., Lasinski T.A. Effect of Winglets on the Induced Drag of Ideal Wing Shapes. NASA TM 81230, Natl.
[22] Anderson, J. D. Fundamentals of Aerodynamics, Second Edition, McGraw-Hill, 1911.
[23] A Mann, Ing. E Elsholz. The M-DAW Project Investigations in Novel Wing Tip Device Design [J]. AIAA2005_Mann.
[24] Jones D.R.A Taxonomy of Global Optimization Methods Based on Response Surface. Journal of Global Optimization, Vol.21, No.4, December 2001, pp.345-383.
[25] Himish J. Winglet Shape and Load Optimization with a numerically supported Lifting Line Method. 12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 10-12 Sep. 2008.
[26] Maughmer, M. D. Design of Winglets for High-Performance Sailplanes Introduction [J]. Journal of Aircraft, 2003, 40(6), pp.1099-1106.
[27] Liersch C.M., Wunderlich T.F.A Fast Aerodynamic Tool for Preliminary Aircraft Design.12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 10-12 sep. 2008.
[28] Coiro D. P., Nicolosi F., Scherillo F., Maisto U. Design of Multiple Winglets to improve Turning and Soaring Characteristics of Angelo D’arrigo’s Hang-Glider: Numerical and Experimental investigation. Aerotecnica Missili E Spazio, Vol.87, Issue 1, 2008, pp. 74-85.
[29] Douglas C. Montgomery,实验设计与分析,中国统计出版社,1998
[30]田口玄一,质量土程学导论,北京:中国对外翻译出版公司,1985
[31] Goldberg D E. Genetic Alogorithms in Search, Optimiazation and Machine Learning [J]. Addison-Wesely. Reading MA, 1989.
[32] Borestoel, J.W., Huizing, G. H. Transonic Shock-Free Airfoil Design by an Analytic Hodograph Method [J]. AIAA 74-0539, 974.
[33] Borestoel, J. W. Review of the Application of Hodograph Theory to Transonic Aerofoil Design and Theoretical and Experimental Analysis of Shock-Free Aerofoils. Preceding IUTAM Symposium Transonicum2, Springer Verlag, 1976
[34] Sobieczki, H., Yu, N.J., Fung, K.Y., and Seebass, A.R. New Method for Designing Shock-Free Transonic Configurations [J]. AIAA, Vol. 17, NO.7, July 1979, pp.722-729
[35] Rai, P., Miranda, L.R. and Seebas, A.R. A cosf-Effective Method for Shock-Free Supercritical Wing Design. AIAA paper 81-0383, 1981.
[36] Hicks, R.M., Vanderplaats, G.N. Application of Numerical Optimization to the Design of Supercritical Airfoils without Drag-Creep. SAE-770440, 1977.
[37] Cosentino, G.B., Holst, T.L. Numerical Optimization Design of Advanced Transonic Wing Configurations. AIAA-085-0424,1985
[38] Gregg, R.D., Misegrades, K.P. Transonic Wing Optimization Using Evolution Theory. AIAA-87-0520,1987
[39] Montoya, L.C. KC-135 Winglet Flight Results. NASA Dryden Flight Research Center.
[40] Bento S. de Mattos, Antonini P. Macedo, Durval H. da Silva Filho. Conisderation about Winglet Design [J]. 21st Applied Aerodynamics Conference, AIAA, 2003-3502.
[41] ISHIMITSU K. K. Aerodynamic design and analysis of winglets [J]. AIAA 76-940, 1976
[42]刘周,朱自强.高升阻比翼型的设计[J].空气动力学报2004, 22(4):410:415
[43]王晓鹏,高正红.遗传算法及其在启动优化设计中的应用研究[D].西安:西北工业大学。2000.12
[44] Hicks R. Henne P. Wing Design by Numerical Optimization [J]. Journal of Aircraft, 1978,15(7): 407-413.
[45] Whitcomb R. T., A Design Approach and Selected Wind-Tunnel Results at High subsonic Speed for Wing-Tip Mounted Winglets. NASA TN D-8260, July 1976.
[46] J.-L. Hnatrais-Gervois, R. Grenon, A. Mann, A. Buscher. Downward pointing winglet design and assessment within the M-DAW research project [J]. The Aeronautical Journal. Vol. 113, No.1142, 2009,pp. 221-232
[47] Brakhage Karl-heinz, Lamby Philipp. Modeling of Airplane Wings with Winglets.
[48] Barry S. Lazos, Kenneth D. Visser. Aerodynamic Comparison of Hyper-Elliptic Cambered Span (HECS) Wings with Conventional Configurations [J]. AIAA 2004-2120, 2004.
[49] Jack Moran. An introduction to Theoretical and Computational Aerodynamics. 1984.
[50]安继光.亚音速空气动力学中的优先基本解方法[J].气动研究与发展,第一期, 1978.
[51] Himish J. Winglet Shape and Load Optimization with a numerically supported Lifting Line Method. 12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 10-12 Sep. 2008.
[52] Keizo Takenaka, Keita Hatanaka. Multidisciplinary Design Exploration for a Winglet [J]. Journalof Aircraft, Vol. 45, No. 5, September-October 2008.
[53] Liersch C.M., Wunderlich T.F.A Fast Aerodynamic Tool for Preliminary Aircraft Design.12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 10-12 sep. 2008.
[54] Wenbin Song, Andy J. Keane. Surrogate-Based Aerodynamic Shape Optimization of a civil Aircraft Engine Nacelle [J]. AIAA Journal, Vol.45, NO.10, 2007, pp. 2565-2574
[55]王希诚,钱令希.多层次联合的结构优化设计.计算结构力学及其应用, 1988,7(4):2-8.
[56] Kirsch U. Synthesis of Structural Geometry Using Approximation Concepts. Comp. Struct., 1982, 15(3):358-365.
[57] Topping B H V. Shape Optimization of Shkeletal Structures, a review. J. Struct. Engrg, 1983, ASCE, 109(8):53-69.
[58] Vanderplaus C N, Moses P. Automated Design of Trusses for Optimum Geometry. Struct. Div., ASCE, 1972, 98(ST3):671-690.
[59]随允康,由衷.具有两类变量的空间桁架分层优化方法.计算结构力学及其应用,1990,5(4):82-93.
[60] B.S. Rosen. Transonic Analysis of Canted Winglets. AIAA 84-0302, 1984.
[61] Ryan Babigian, Shigeo Hayashibara. Computational Study of the Vortex Wake Generated by a Three-Dimensional Wing with Dihedral, Taper, and Sweep [J]. 27th AIAA Applied Aerodynamics Conference, AIAA 2009-4107
[62] Les Diegl, Wayne Tiller. The NURBS Book [M]. German: Springer, 1997.
[2] Rudolph P.K.C. Boeing, Seattle, US patent for High Taper Wing Tip Extension. No.5039032, filed 19 May 1989.
[3] Herrick L.L. et al. Boeing, Seattle, US Patent for Blunt Leading Edge Raked Wingtips. No.6089502, filed 12 Jun 1998.
[4] Kroo, Ilan. DRAG DUE TO LIFT: Concepts for Prediction and Reduction [J]. Annu. Rev. Fluid Mech, 2001, 33, 587-617.
[5] Gratzer, L. Blended Winglet. Patent NO. US5348253, 1 February 1993.
[6] Gerd Heller, Manfred Maisel, Peter Kreuzer. Wingtip extension for a wing. Patent NO. US 6722615 B2
[7] Louis B. Gratzer. Spiroid-tipped wing. Patent NO. 5102068
[8] Reneaux, J. Overview on Drag Reduction Technologies for Civil Transport Aircraft. Applied Sciences, 1(18), 2004.
[9] McMasters J., Kroo IM. Advanced Configurations for very Large Transport Airplanes. Aircraft Design, 2008, 1:217-42.
[10] Hugues, C.US Patent for Cylindrical Wing Tip with Helical Slot, No. US 6892988 B2. Filed 17 May, 2005.
[11] Smith S.C. A Computational and Experimental study of nonlinear aspects of Induced Drag. NASA TP 3598, Natl. Aeronaut. Space Admin., Washington, DC.
[12] Hackett J.E. Vortex Drag Reduction by Aft-mounted Diffusing Vanes. ICAS Paper 80-13.4.
[13] Spillman JJ. The Use of Wing Tip Sails to Reduce Vortex Drag. Aeronaut Journal. 1978, 82:387-395.
[14] Kroo, I. Nonplanar Wing Concepts for Increased Aircraft Efficiency. VKI lecture series on Innovative Configurations and Advanced Concepts for Future Civil Aircraft, June 6-10, 2005.
[15] Patterson J.C. US Patent No.4533101, 1985.
[16] Terry JE. Aerodynamic Characteristics of Ring wings: a bibliography. Redstone Scientific Information Center Report RSIC-285.
[17] Miranda LR. Boxplane Configuration– Conceptual analysis and initial Experimental Verification.Lockheed California Company Report LR 25180.
[18] Wolkovitch J. The Joined Wing: an overview. Journal of Aircraft, 1986, 23:161-178.
[19] J.-L. Hnatrais-Gervois, R. Grenon, A. Mann, A. Buscher. Downward pointing winglet design and assessment within the M-DAW research project [J]. The Aeronautical Journal. Vol. 113, No.1142, 2009,pp. 221-232
[20] Heyson H.H, Roebe G.D, Fulton C.L. Theoretical Parametric Study of the Relative Advantages of Winglets and Wing-tip Extensions. NASA TP 1020.
[21] Jones R.T., Lasinski T.A. Effect of Winglets on the Induced Drag of Ideal Wing Shapes. NASA TM 81230, Natl.
[22] Anderson, J. D. Fundamentals of Aerodynamics, Second Edition, McGraw-Hill, 1911.
[23] A Mann, Ing. E Elsholz. The M-DAW Project Investigations in Novel Wing Tip Device Design [J]. AIAA2005_Mann.
[24] Jones D.R.A Taxonomy of Global Optimization Methods Based on Response Surface. Journal of Global Optimization, Vol.21, No.4, December 2001, pp.345-383.
[25] Himish J. Winglet Shape and Load Optimization with a numerically supported Lifting Line Method. 12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 10-12 Sep. 2008.
[26] Maughmer, M. D. Design of Winglets for High-Performance Sailplanes Introduction [J]. Journal of Aircraft, 2003, 40(6), pp.1099-1106.
[27] Liersch C.M., Wunderlich T.F.A Fast Aerodynamic Tool for Preliminary Aircraft Design.12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 10-12 sep. 2008.
[28] Coiro D. P., Nicolosi F., Scherillo F., Maisto U. Design of Multiple Winglets to improve Turning and Soaring Characteristics of Angelo D’arrigo’s Hang-Glider: Numerical and Experimental investigation. Aerotecnica Missili E Spazio, Vol.87, Issue 1, 2008, pp. 74-85.
[29] Douglas C. Montgomery,实验设计与分析,中国统计出版社,1998
[30]田口玄一,质量土程学导论,北京:中国对外翻译出版公司,1985
[31] Goldberg D E. Genetic Alogorithms in Search, Optimiazation and Machine Learning [J]. Addison-Wesely. Reading MA, 1989.
[32] Borestoel, J.W., Huizing, G. H. Transonic Shock-Free Airfoil Design by an Analytic Hodograph Method [J]. AIAA 74-0539, 974.
[33] Borestoel, J. W. Review of the Application of Hodograph Theory to Transonic Aerofoil Design and Theoretical and Experimental Analysis of Shock-Free Aerofoils. Preceding IUTAM Symposium Transonicum2, Springer Verlag, 1976
[34] Sobieczki, H., Yu, N.J., Fung, K.Y., and Seebass, A.R. New Method for Designing Shock-Free Transonic Configurations [J]. AIAA, Vol. 17, NO.7, July 1979, pp.722-729
[35] Rai, P., Miranda, L.R. and Seebas, A.R. A cosf-Effective Method for Shock-Free Supercritical Wing Design. AIAA paper 81-0383, 1981.
[36] Hicks, R.M., Vanderplaats, G.N. Application of Numerical Optimization to the Design of Supercritical Airfoils without Drag-Creep. SAE-770440, 1977.
[37] Cosentino, G.B., Holst, T.L. Numerical Optimization Design of Advanced Transonic Wing Configurations. AIAA-085-0424,1985
[38] Gregg, R.D., Misegrades, K.P. Transonic Wing Optimization Using Evolution Theory. AIAA-87-0520,1987
[39] Montoya, L.C. KC-135 Winglet Flight Results. NASA Dryden Flight Research Center.
[40] Bento S. de Mattos, Antonini P. Macedo, Durval H. da Silva Filho. Conisderation about Winglet Design [J]. 21st Applied Aerodynamics Conference, AIAA, 2003-3502.
[41] ISHIMITSU K. K. Aerodynamic design and analysis of winglets [J]. AIAA 76-940, 1976
[42]刘周,朱自强.高升阻比翼型的设计[J].空气动力学报2004, 22(4):410:415
[43]王晓鹏,高正红.遗传算法及其在启动优化设计中的应用研究[D].西安:西北工业大学。2000.12
[44] Hicks R. Henne P. Wing Design by Numerical Optimization [J]. Journal of Aircraft, 1978,15(7): 407-413.
[45] Whitcomb R. T., A Design Approach and Selected Wind-Tunnel Results at High subsonic Speed for Wing-Tip Mounted Winglets. NASA TN D-8260, July 1976.
[46] J.-L. Hnatrais-Gervois, R. Grenon, A. Mann, A. Buscher. Downward pointing winglet design and assessment within the M-DAW research project [J]. The Aeronautical Journal. Vol. 113, No.1142, 2009,pp. 221-232
[47] Brakhage Karl-heinz, Lamby Philipp. Modeling of Airplane Wings with Winglets.
[48] Barry S. Lazos, Kenneth D. Visser. Aerodynamic Comparison of Hyper-Elliptic Cambered Span (HECS) Wings with Conventional Configurations [J]. AIAA 2004-2120, 2004.
[49] Jack Moran. An introduction to Theoretical and Computational Aerodynamics. 1984.
[50]安继光.亚音速空气动力学中的优先基本解方法[J].气动研究与发展,第一期, 1978.
[51] Himish J. Winglet Shape and Load Optimization with a numerically supported Lifting Line Method. 12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 10-12 Sep. 2008.
[52] Keizo Takenaka, Keita Hatanaka. Multidisciplinary Design Exploration for a Winglet [J]. Journalof Aircraft, Vol. 45, No. 5, September-October 2008.
[53] Liersch C.M., Wunderlich T.F.A Fast Aerodynamic Tool for Preliminary Aircraft Design.12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 10-12 sep. 2008.
[54] Wenbin Song, Andy J. Keane. Surrogate-Based Aerodynamic Shape Optimization of a civil Aircraft Engine Nacelle [J]. AIAA Journal, Vol.45, NO.10, 2007, pp. 2565-2574
[55]王希诚,钱令希.多层次联合的结构优化设计.计算结构力学及其应用, 1988,7(4):2-8.
[56] Kirsch U. Synthesis of Structural Geometry Using Approximation Concepts. Comp. Struct., 1982, 15(3):358-365.
[57] Topping B H V. Shape Optimization of Shkeletal Structures, a review. J. Struct. Engrg, 1983, ASCE, 109(8):53-69.
[58] Vanderplaus C N, Moses P. Automated Design of Trusses for Optimum Geometry. Struct. Div., ASCE, 1972, 98(ST3):671-690.
[59]随允康,由衷.具有两类变量的空间桁架分层优化方法.计算结构力学及其应用,1990,5(4):82-93.
[60] B.S. Rosen. Transonic Analysis of Canted Winglets. AIAA 84-0302, 1984.
[61] Ryan Babigian, Shigeo Hayashibara. Computational Study of the Vortex Wake Generated by a Three-Dimensional Wing with Dihedral, Taper, and Sweep [J]. 27th AIAA Applied Aerodynamics Conference, AIAA 2009-4107
[62] Les Diegl, Wayne Tiller. The NURBS Book [M]. German: Springer, 1997.