指尖密封型线主动控制优化研究
|
摘要
20世纪80年代后,发展新型密封技术已成为下一代高性能航空发动机研制的“前线部队”。指尖密封技术即是在这一情况下继篦齿密封和刷式密封技术之后发展起来的一种先进密封技术,它对于提高航空发动机性能和寿命具有良好的应用前景,开展指尖密封技术研究是一项迫切的任务,具有重要科学意义。本论文研究的目的就是通过优化设计来选择合理的指尖密封曲梁型线,以获得满足要求的指尖密封结构,从而为进一步的研究奠定基础。
为此,在指尖密封国内外研究现状分析的基础上,本文基于影响指尖密封工作性能和寿命的两个关键因素——迟滞特性和径向刚度特性,着重对指尖密封结构的型线进行了优化选择。
本文首先以迟滞程度最小为目标,根据有关优化理论采用拉格朗日乘子法对指尖曲梁型线进行了优化,并采用有限元技术对优化型线的指尖密封和对数螺线型线指尖密封进行了分析,以比较两者的迟滞性能之间的区别。研究工作表明,较之对数螺线型线本文提出的优化型线指尖密封的迟滞性能获得了很大的改善。在此基础上,针对不同密封位置对迟滞和径向刚度的不同匹配需求,通过对指尖密封曲梁型线优化技术的研究,提出了指尖密封型线主动控制优化设计的方法。其后的优化设计实例及其有限元仿真分析证明了所提出的指尖密封型线主动控制优化方法的可行性和有效性。最后,针对指尖密封型线主动控制优化问题中在要求迟滞比较小的情况下目标函数病态化的现象,提出了相应的处理方法,实现了小迟滞比状态下指尖密封型线的主动控制优化设计,使本文提出的指尖密封型线主动控制优化设计方法具有较大适用性。
The research of the new-type seal technology has been the frontier of the development in the next generation aero-engine with high performance after 1980's~[1]. The finger seal is a new innovative seal after labyrinth seal and brush seal. The purpose of this project is to get the appropriate shape-curve of finger seal for the special work-conditions' seal demand and establish the base for farther research.With summarized a great deal of datum in home and overseas, based on the finger seal's hysteresis and stiffness, the paper will discuss the optimized selection of shape-curve in finger seals.Firstly the shape-curve of finger seal is optimized aiming at farthest lowering hysteresis effect with optimization theory. Base on results, the hysteresis effect of finger seal with optimized shape-curve is analyzed with FEM (Finite Element Method), and is compared with hysteresis effect of finger seal with logarithen spiral I shape-curve. The research indicates that compared with finger seal with logarithen spiral I shape-curve, the finger seal with optimized shape-curve is put forward for improvements of hysteresis effect. Furthermore, considering the purpose of reasonable finger seal construction, we present the active control optimization method across the shape-curve optimization design research. The results obtained by FEM show that active control optimization method is reasonable. Finally, according to the morbidity of design objective lead to difficult to get the finger shape-curve of lower hysteresis with the active control optimization method, we bring forward the corresponding design method to achieve the active control optimization of finger shape-curve of lower hysteresis. This make the shape-curve optimization design method by active control in finger seals have more uses.
为此,在指尖密封国内外研究现状分析的基础上,本文基于影响指尖密封工作性能和寿命的两个关键因素——迟滞特性和径向刚度特性,着重对指尖密封结构的型线进行了优化选择。
本文首先以迟滞程度最小为目标,根据有关优化理论采用拉格朗日乘子法对指尖曲梁型线进行了优化,并采用有限元技术对优化型线的指尖密封和对数螺线型线指尖密封进行了分析,以比较两者的迟滞性能之间的区别。研究工作表明,较之对数螺线型线本文提出的优化型线指尖密封的迟滞性能获得了很大的改善。在此基础上,针对不同密封位置对迟滞和径向刚度的不同匹配需求,通过对指尖密封曲梁型线优化技术的研究,提出了指尖密封型线主动控制优化设计的方法。其后的优化设计实例及其有限元仿真分析证明了所提出的指尖密封型线主动控制优化方法的可行性和有效性。最后,针对指尖密封型线主动控制优化问题中在要求迟滞比较小的情况下目标函数病态化的现象,提出了相应的处理方法,实现了小迟滞比状态下指尖密封型线的主动控制优化设计,使本文提出的指尖密封型线主动控制优化设计方法具有较大适用性。
The research of the new-type seal technology has been the frontier of the development in the next generation aero-engine with high performance after 1980's~[1]. The finger seal is a new innovative seal after labyrinth seal and brush seal. The purpose of this project is to get the appropriate shape-curve of finger seal for the special work-conditions' seal demand and establish the base for farther research.With summarized a great deal of datum in home and overseas, based on the finger seal's hysteresis and stiffness, the paper will discuss the optimized selection of shape-curve in finger seals.Firstly the shape-curve of finger seal is optimized aiming at farthest lowering hysteresis effect with optimization theory. Base on results, the hysteresis effect of finger seal with optimized shape-curve is analyzed with FEM (Finite Element Method), and is compared with hysteresis effect of finger seal with logarithen spiral I shape-curve. The research indicates that compared with finger seal with logarithen spiral I shape-curve, the finger seal with optimized shape-curve is put forward for improvements of hysteresis effect. Furthermore, considering the purpose of reasonable finger seal construction, we present the active control optimization method across the shape-curve optimization design research. The results obtained by FEM show that active control optimization method is reasonable. Finally, according to the morbidity of design objective lead to difficult to get the finger shape-curve of lower hysteresis with the active control optimization method, we bring forward the corresponding design method to achieve the active control optimization of finger shape-curve of lower hysteresis. This make the shape-curve optimization design method by active control in finger seals have more uses.
引文
1.林基恕等.21世纪航空发动机动力传输系统的展望[J].航空动力学报,第16卷第2期,2001年4月
2. Gul K. Arora, Margaret P. Proctor, Bruce M. Steinetz, et al. Pressure balanced, low hysteresis, finger seal test results[A]. The 35th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit[C]. Los Angeles, USA. 1999
3. Bruce M. Steinefz, Robert C. Hendricks. Advanced Seal Technology in Meeting Next Generation Turbine Engine Goal[R]. NASA/TM—1998—206961
4. M. J. Braun B. M. Steinetz. Two-And Three Dimensional Numerical Experiments Representing Two Limiting Cases Of An In-Line Pair Of Finger Seal Compone—nts[A]. The 9th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery[C], Honolulu, Hawaii, February 10-14, 2002
5. Bruce M. Steinefz, Robert C. Hendricks. Engine Seal Technology Requirements to Meet NASAs Advanced Subsonic Technology Program Goals[R]. AIAA94—2698
6. G. J. Stargess. Application of CAD to Gas Turbine Engine Secondary Flow System— The Labyrinth Seal[R]. AIAA—SS—3203 1988
7. Frederic Mahler and Esther Boyes.THE APPLICATION OF BRUSH SEALS IN LARGE COMMERCIAL JET ENGINES[A]. The 31th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit[C]. July 10-12, 1995/San Diego, CA
8. Gul K. Arora, Margaret P. Proctor. JTAGG Ⅱ Brush Seal Test Result [A]. The 33rd AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit [C]. Seattle, Washington, July 6-9, 1997
9. M. J. Braun, V. V. Kudriavtsev, R. C. Hendricks: A Three Dimensional Navier-Stocks Simulation of Flow in a Passive-Adaptive Finger Seal[A]. The 9th Internation Symposium on Transport Phenomena and Dynamics of Rotating Machinery[C], Honolulu, Hawaii, February 10-14, 2002.
10.陈国定,徐华,虞烈等.指尖密封型线的力学性能的有限元分析[J].西北工业大学学报,第20卷第2期,2002年5月
11.陈国定,徐华,虞烈等.指尖密封的迟滞特性分析[J].机械工程学报,第39卷第5期,2003年5月
12.陈国定,徐华等.对数螺线指尖密封的装配变形分析[J].机械科学与技术,第 21卷第6期
13.陈国定,徐华,虞烈.基于粗糙接触理论的橡胶金属摩擦副的摩擦分析[J].西安交通大学学报,第36卷第3期,2002年3月
14.陈国定,徐华等.轴向尺寸对指尖密封性能影响的分析[J].燃起涡轮试验与研究,第16卷第1期,2003年
15.陈国定,苏华,张永红.指尖密封轴向布局的变尺寸结构研究[J].航空动力学报,第18卷第4期,2003年8月
16.陈国定,吴昊天.后遮流板结构尺寸对指尖密封性能影响的研究[J].中国机械工程,第15卷第1期,2004年1月上半月
17.苏华,陈国定.指尖密封型线形成的研究[J].机械科学与技术,第22卷第6期,2003年11月
18.苏华,陈国定.低迟滞渐开线型指尖密封的结构参数优化[J].机械科学与技术,第23卷第9期,2004年9月
19.桂普国.非渐扩形结构指尖密封特性研究[D].西北工业大学硕士论文,2003年
20.雷燕妮,陈国定.背压腔结构对压力平衡型指尖密封性能的影响分析[J].机械科学与技术,第23卷第4期,2004年4月
21.雷燕妮,陈国定.压力平衡型指尖密封结构影响研究[J].机械科学与技术,第23卷第10期,2004年10月
22.李芳,凌道盛.工程结构优化设计发展综述[J].工程设计学报,第9卷第5期,2002年12月
23.徐颖,周华.现代结构优化设计理论浅析[J].矿山机械,2003年12月
24.汪树玉,刘国华,包志仁.结构优化设计的现状与进展[J].基建优化,第20卷第4期,1999年
25. Fleury C. Structural weight optimization by dual methods of convex programming[J]. Int J Num Meths Engrg, 1979, 14: 1761-1783.
26. FLEURY C, SANDER G. Dual methods for optimizing finite element flexural systems[J]. Computer Methods in Applied Mechanics and Engineering, 1983, 37(3): 249-275.
27. ARORA J S, HAUG E J. Methods of design sensitivity analysis instructural optimization[J]. AIAA Journal, 1979, 19(9): 970-974.
28. HAFTKA R T. Second order sensitivity derivatives in structural optimization[J]. AIAA Journal, 1982,20:1765-1766.
29. BENDSOE M P,KIKUCHI N.Generating optimal topologies in structural design using a homogenization method[J].Comput App/ Mech Engrg, 1988,71:197-224.
30. XIE Y M,STEVEN G P.A simple evolutionary procedure for structural optimization[J].Comput&Struct,1993,49(5):885-896.
31. XIE Y M,STEVEN G P.A simple approach to structural frequency optimization[J].Comput&Struct,1994,53(6):1487-1491.
32. XIE Y M,STEVEN G P.Evolutionary structural optimization for dynamic problems[J].Comput&Struct,1996,58(6):1067-1073.
33. MANICKARAJAH D,XIE Y M,STEVEN G P.Optimization of columns and frames against buckling[J].Comput&Struct,2000,75:45-54.
34. RONG J H,XIE Y M,YANG X Y.An improved for evolutionary structural optimization against buckling[J].Comput Struct,2001,79:253-263.
35. YOUNG V, QUERIN O M, STEVEN G P. 3D and multiple load case bi-directional evolutionary structural optimization(BESO)[J]. Structural Optimization, 1999,18:183-192.
36. KIM H, QUERIN O M. A method for varying the number of cavities in an optimized topology using evolutionary structural optimization[J]. Struct Multidisc Optim,2000,19:113-121.
37. GRANT Steven,OSVALDO Querin,MIKE Xie.Evolutionary structural optimization(ESO) for combined topology and size optimization of discrete structures agains tbuckling[J].Comput Meth Appl Mech Engrg,2000,188:743 - 754 .
38. DEMS K.Multi-parameter shape optimization of elastic bars intorsion[J]. Int J Numer Meth Engng, 1980,13:247-263.
39. KIKUCHI N,SUZUKI K.A homogenization method for shape and topology optimization[J].Comput Meth Appl Mech Engrg, 1991,93:291-318.
40. BELEGUNDU A D,ARORA J S.A study of mathematical programming methods for structural optimization.Part Ⅱ:numerical results [J]. Int J Numer Meth Engng,1985,19(8):1601-1623.
41. CHEN Jian-jun,CHE Jian-wen(陈建军,车建文 ).A review on structural dynamic optimum design[J].Advances in Mechanics(力学进展),2001,31(2):181-191.(in Chinese)
42. LUO Zhi-jun, QIAO Xin(罗志军,乔新). Optimization design of radome by genetical gorithm[J]. Chinese Journal of Computational Mechanics(计算力学学报), 1997, 14(3): 361-365. (in Chinese)
43. ZHANG Wen-yuan, WU Zhi-feng(张文元,吴知丰). Genetical gorithm for the design of structural optimization[J]. Journal of Harbin University of CE&Architecture(哈尔滨建筑大学学报), 1999, 32(4): 18-23. (in Chinese)
44. BAUMAL A E, MACPHEE J J, CALAMAI P H. Application of genetical gorithms to the design optimization of anactive vehicle suspension system homogenization[J]. Comput Meth Appl Mech Engrg, 1998, 163: 87-94.
45.张志涌,徐彦琴.MATLAB教程[M].北京航空航天大学出版社,2002年1月
46.冯玮,涂伟霞等.由潜入深学Maple[M].国防工业出版社,2001年3月
47.刘辉,李海.MAPLE符号处理及应用[M].国防工业出版社,2001年2月
48.马春庭,高萍等.掌握和精通Maple[M].机械工业出版社,2000年9月
49.陈晓霞.Maple指令参考手册[M].国防工业出版社,2002年1月
50.李景湧.有限元法.北京邮电大学出版社[M],2000年4月
51.现代设计方法——最优化设计方法.网上教程http://www.et.uestc.edu.cn/kejian/kejian/chenyi/html/hessian/html/shuxue.html
52.刘涛,杨凤鹏.精通ANSYS[M].清华大学出版社,2002年9月
53.龚曙光.ANSYS基础应用及范例解析[M].机械工业出版社,2003年1月
54.姚新胜,黄洪钟等.基于广义满意度原理的多目标优化理论研究[J].应用科学学报,第20卷第3期,2002年9月
55.罗刚,陈春俊等.多目标优化问题中目标间矛盾性关系的研究[J].西南交通大学学报,第34卷第4期,1999年8月
56.胡义刚,韩云鹏等.对多目标优化方法极小—极大法的探讨[J].山东机械,1994年5月
57.张宪民.柔顺机构拓扑优化设计[J].机械工程学报.第39卷第11期,2003年11月
58. Frecker M I, Ananthasuresh G K, Nishiwaki S, et al. Topological synthesis of compliant mechanisms using multi-criteria optimization[J]. Transaction of the ASME, Journal of Mechanical Design, 1997, 119(2): 238-245
59. Frecker M I, Kota S, Kikuchi N. Use of penalty function in topological synthesis and optimization of strain energy density of compliant mechanisms[A]. In: Proceedings of the 1997 ASME Design Engineering Technical Conferences[C], 1997, DETC97/DAC-3760
60. Frecker M I, Kikuchi N, Kota S. Topology optimization of compliant mechanisms with multiple outputs[J]. Structural Optimization, 1999, 17: 269-278
61. Frecker M I, Canfield S. Design of efficient compliant mechanisms from ground structure based optimal topologies[A]. In: Proceedings of the 2000ASME Design Engineering Technical Conferences[C], 2000, DETC2000/MECH-14142
62. Hetrick J A, Kota S. Topological and Geometric Synthesis of compliant mechanisms[A]. In: Proceedings of the 2000 ASME Design Engineering Technical Conferences[C], 2000, DETC2000/MECH-14140
63.李人宪.有限元法基础[M].国防工业出版社,2002年5月
64.王栋.结构优化设计与渐进移动法[D].西北工业大学博士论文,2003年
65.张铜生,张富德.简明有限元法及其应用[M].地震出版社,1990年8月
2. Gul K. Arora, Margaret P. Proctor, Bruce M. Steinetz, et al. Pressure balanced, low hysteresis, finger seal test results[A]. The 35th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit[C]. Los Angeles, USA. 1999
3. Bruce M. Steinefz, Robert C. Hendricks. Advanced Seal Technology in Meeting Next Generation Turbine Engine Goal[R]. NASA/TM—1998—206961
4. M. J. Braun B. M. Steinetz. Two-And Three Dimensional Numerical Experiments Representing Two Limiting Cases Of An In-Line Pair Of Finger Seal Compone—nts[A]. The 9th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery[C], Honolulu, Hawaii, February 10-14, 2002
5. Bruce M. Steinefz, Robert C. Hendricks. Engine Seal Technology Requirements to Meet NASAs Advanced Subsonic Technology Program Goals[R]. AIAA94—2698
6. G. J. Stargess. Application of CAD to Gas Turbine Engine Secondary Flow System— The Labyrinth Seal[R]. AIAA—SS—3203 1988
7. Frederic Mahler and Esther Boyes.THE APPLICATION OF BRUSH SEALS IN LARGE COMMERCIAL JET ENGINES[A]. The 31th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit[C]. July 10-12, 1995/San Diego, CA
8. Gul K. Arora, Margaret P. Proctor. JTAGG Ⅱ Brush Seal Test Result [A]. The 33rd AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit [C]. Seattle, Washington, July 6-9, 1997
9. M. J. Braun, V. V. Kudriavtsev, R. C. Hendricks: A Three Dimensional Navier-Stocks Simulation of Flow in a Passive-Adaptive Finger Seal[A]. The 9th Internation Symposium on Transport Phenomena and Dynamics of Rotating Machinery[C], Honolulu, Hawaii, February 10-14, 2002.
10.陈国定,徐华,虞烈等.指尖密封型线的力学性能的有限元分析[J].西北工业大学学报,第20卷第2期,2002年5月
11.陈国定,徐华,虞烈等.指尖密封的迟滞特性分析[J].机械工程学报,第39卷第5期,2003年5月
12.陈国定,徐华等.对数螺线指尖密封的装配变形分析[J].机械科学与技术,第 21卷第6期
13.陈国定,徐华,虞烈.基于粗糙接触理论的橡胶金属摩擦副的摩擦分析[J].西安交通大学学报,第36卷第3期,2002年3月
14.陈国定,徐华等.轴向尺寸对指尖密封性能影响的分析[J].燃起涡轮试验与研究,第16卷第1期,2003年
15.陈国定,苏华,张永红.指尖密封轴向布局的变尺寸结构研究[J].航空动力学报,第18卷第4期,2003年8月
16.陈国定,吴昊天.后遮流板结构尺寸对指尖密封性能影响的研究[J].中国机械工程,第15卷第1期,2004年1月上半月
17.苏华,陈国定.指尖密封型线形成的研究[J].机械科学与技术,第22卷第6期,2003年11月
18.苏华,陈国定.低迟滞渐开线型指尖密封的结构参数优化[J].机械科学与技术,第23卷第9期,2004年9月
19.桂普国.非渐扩形结构指尖密封特性研究[D].西北工业大学硕士论文,2003年
20.雷燕妮,陈国定.背压腔结构对压力平衡型指尖密封性能的影响分析[J].机械科学与技术,第23卷第4期,2004年4月
21.雷燕妮,陈国定.压力平衡型指尖密封结构影响研究[J].机械科学与技术,第23卷第10期,2004年10月
22.李芳,凌道盛.工程结构优化设计发展综述[J].工程设计学报,第9卷第5期,2002年12月
23.徐颖,周华.现代结构优化设计理论浅析[J].矿山机械,2003年12月
24.汪树玉,刘国华,包志仁.结构优化设计的现状与进展[J].基建优化,第20卷第4期,1999年
25. Fleury C. Structural weight optimization by dual methods of convex programming[J]. Int J Num Meths Engrg, 1979, 14: 1761-1783.
26. FLEURY C, SANDER G. Dual methods for optimizing finite element flexural systems[J]. Computer Methods in Applied Mechanics and Engineering, 1983, 37(3): 249-275.
27. ARORA J S, HAUG E J. Methods of design sensitivity analysis instructural optimization[J]. AIAA Journal, 1979, 19(9): 970-974.
28. HAFTKA R T. Second order sensitivity derivatives in structural optimization[J]. AIAA Journal, 1982,20:1765-1766.
29. BENDSOE M P,KIKUCHI N.Generating optimal topologies in structural design using a homogenization method[J].Comput App/ Mech Engrg, 1988,71:197-224.
30. XIE Y M,STEVEN G P.A simple evolutionary procedure for structural optimization[J].Comput&Struct,1993,49(5):885-896.
31. XIE Y M,STEVEN G P.A simple approach to structural frequency optimization[J].Comput&Struct,1994,53(6):1487-1491.
32. XIE Y M,STEVEN G P.Evolutionary structural optimization for dynamic problems[J].Comput&Struct,1996,58(6):1067-1073.
33. MANICKARAJAH D,XIE Y M,STEVEN G P.Optimization of columns and frames against buckling[J].Comput&Struct,2000,75:45-54.
34. RONG J H,XIE Y M,YANG X Y.An improved for evolutionary structural optimization against buckling[J].Comput Struct,2001,79:253-263.
35. YOUNG V, QUERIN O M, STEVEN G P. 3D and multiple load case bi-directional evolutionary structural optimization(BESO)[J]. Structural Optimization, 1999,18:183-192.
36. KIM H, QUERIN O M. A method for varying the number of cavities in an optimized topology using evolutionary structural optimization[J]. Struct Multidisc Optim,2000,19:113-121.
37. GRANT Steven,OSVALDO Querin,MIKE Xie.Evolutionary structural optimization(ESO) for combined topology and size optimization of discrete structures agains tbuckling[J].Comput Meth Appl Mech Engrg,2000,188:743 - 754 .
38. DEMS K.Multi-parameter shape optimization of elastic bars intorsion[J]. Int J Numer Meth Engng, 1980,13:247-263.
39. KIKUCHI N,SUZUKI K.A homogenization method for shape and topology optimization[J].Comput Meth Appl Mech Engrg, 1991,93:291-318.
40. BELEGUNDU A D,ARORA J S.A study of mathematical programming methods for structural optimization.Part Ⅱ:numerical results [J]. Int J Numer Meth Engng,1985,19(8):1601-1623.
41. CHEN Jian-jun,CHE Jian-wen(陈建军,车建文 ).A review on structural dynamic optimum design[J].Advances in Mechanics(力学进展),2001,31(2):181-191.(in Chinese)
42. LUO Zhi-jun, QIAO Xin(罗志军,乔新). Optimization design of radome by genetical gorithm[J]. Chinese Journal of Computational Mechanics(计算力学学报), 1997, 14(3): 361-365. (in Chinese)
43. ZHANG Wen-yuan, WU Zhi-feng(张文元,吴知丰). Genetical gorithm for the design of structural optimization[J]. Journal of Harbin University of CE&Architecture(哈尔滨建筑大学学报), 1999, 32(4): 18-23. (in Chinese)
44. BAUMAL A E, MACPHEE J J, CALAMAI P H. Application of genetical gorithms to the design optimization of anactive vehicle suspension system homogenization[J]. Comput Meth Appl Mech Engrg, 1998, 163: 87-94.
45.张志涌,徐彦琴.MATLAB教程[M].北京航空航天大学出版社,2002年1月
46.冯玮,涂伟霞等.由潜入深学Maple[M].国防工业出版社,2001年3月
47.刘辉,李海.MAPLE符号处理及应用[M].国防工业出版社,2001年2月
48.马春庭,高萍等.掌握和精通Maple[M].机械工业出版社,2000年9月
49.陈晓霞.Maple指令参考手册[M].国防工业出版社,2002年1月
50.李景湧.有限元法.北京邮电大学出版社[M],2000年4月
51.现代设计方法——最优化设计方法.网上教程http://www.et.uestc.edu.cn/kejian/kejian/chenyi/html/hessian/html/shuxue.html
52.刘涛,杨凤鹏.精通ANSYS[M].清华大学出版社,2002年9月
53.龚曙光.ANSYS基础应用及范例解析[M].机械工业出版社,2003年1月
54.姚新胜,黄洪钟等.基于广义满意度原理的多目标优化理论研究[J].应用科学学报,第20卷第3期,2002年9月
55.罗刚,陈春俊等.多目标优化问题中目标间矛盾性关系的研究[J].西南交通大学学报,第34卷第4期,1999年8月
56.胡义刚,韩云鹏等.对多目标优化方法极小—极大法的探讨[J].山东机械,1994年5月
57.张宪民.柔顺机构拓扑优化设计[J].机械工程学报.第39卷第11期,2003年11月
58. Frecker M I, Ananthasuresh G K, Nishiwaki S, et al. Topological synthesis of compliant mechanisms using multi-criteria optimization[J]. Transaction of the ASME, Journal of Mechanical Design, 1997, 119(2): 238-245
59. Frecker M I, Kota S, Kikuchi N. Use of penalty function in topological synthesis and optimization of strain energy density of compliant mechanisms[A]. In: Proceedings of the 1997 ASME Design Engineering Technical Conferences[C], 1997, DETC97/DAC-3760
60. Frecker M I, Kikuchi N, Kota S. Topology optimization of compliant mechanisms with multiple outputs[J]. Structural Optimization, 1999, 17: 269-278
61. Frecker M I, Canfield S. Design of efficient compliant mechanisms from ground structure based optimal topologies[A]. In: Proceedings of the 2000ASME Design Engineering Technical Conferences[C], 2000, DETC2000/MECH-14142
62. Hetrick J A, Kota S. Topological and Geometric Synthesis of compliant mechanisms[A]. In: Proceedings of the 2000 ASME Design Engineering Technical Conferences[C], 2000, DETC2000/MECH-14140
63.李人宪.有限元法基础[M].国防工业出版社,2002年5月
64.王栋.结构优化设计与渐进移动法[D].西北工业大学博士论文,2003年
65.张铜生,张富德.简明有限元法及其应用[M].地震出版社,1990年8月