近二十年來所發展的多點搜尋最佳化解法，雖可得逼近的全域解，但普遍需要相當長的計算時間。傳統的單點搜尋法，雖求解計算時間較少，但卻無法確定得到全域解。尤其是應用最佳化於工程結構設計上時，需使用有限元素的分析，導致計算量遽增，需更多的計算時間，降低最佳化的實用性。本文即為探討多點搜尋的最佳化程序結合近似函數的可行性，藉由近似法的特性可減少有限元素分析的次數，減少最佳化程序的整體計算時間。
本文的多點搜尋最佳化方法是採用粒子群最佳化(PSO)，以改良飛回策略為處理限制的機制，合成含限制的粒子群最佳化(CPSO)。選用稱為Tana3的近似函數，建構該近似函數的C++程式，發展成可求解全域值的粒子群多點搜尋最佳化解法，稱為ACPSO。文中以十桿、二十五桿桁架及三維工具機床台等結構設計題目為例，建立有限元素分析模型，及應用ACPSO求解。同時與單點搜尋最佳化結果比較，並討論近似法的差異。比較討論的關鍵項目為全域最佳解的精確性有限元分析的次數與電腦計算時間，結果顯示ACPSO確實能有效率及得到全域解，適合應用於大型結構工程之最佳化設計。

In recent 20 years, multiple-points search optimization methods have been developed for possible global solutions. However, it is recognized that the computation is time consuming. In traditional single-point search optimization methods spend less computational time, but is easy falling into local optimum. In modern optimal engineering structural design, very often it requires the finite element (FE) analysis. This results in large amount computation time, and losing the efficiency. This thesis explores an selected multiple-points search optimization method including an approximation technique that can reduce the work of FE analysis as well as the computation time.
The proposed multiple-points search optimization method in this thesis is particle swarm optimization (PSO). The improve flying back strategy for constraints handling developed in our research group had been combined in PSO resulted in CPSO. An approximation technique called Tana3 is adopted and developed in C++ programming. The CPSO combined Tana3 herein is named ACPSO. The structures of 10-bar truss, 25-bar truss and a two-pieces platform assembly for machine tool are utilized as examples to show the research work and performances. The mechanical analysis applied FE software named ANSYS that connected to ACPSO. The results are compares each other with or without Tana3 in multiple-points or single-point search optimization method. The final result shows that the presented ACPSO indeed can promote the computational efficiency and simultaneously the global design can be obtained. It is concluded that ACPSO is suitable for general large-scale engineering structural design optimization.

目  錄
誌謝.......................................................I
中文摘要...................................................II
英文摘要...................................................IV
目錄......................................................VI
圖目錄...................................................VIII
表目錄......................................................X
符號表....................................................XII
第一章	緒論...............................................1
1.1 動機與目的..............................................1
1.2 文獻回顧................................................2
1.3本文架構.................................................6
			
第二章	TANA3建構及最佳化(AMFDM).............................7
2.1 TANA3的程式化及驗證......................................7
2.2 TANA3近似法最佳化………....................................12
2.3 應用例題………………………………...................................15
		
第三章	粒子群的TANA3近似法最佳化(ACPSO).....................26
3.1最佳化演算程序...........................................26
3.2例題應用................................................34
3.3效率及全域解探討..........................................61
		
第四章   ACPSO的結構設計應用.................................67
4.1結構模型的建立及驗證部分結構................................67
4.1.1題目敘述及求解.........................................67
4.1.2 建構有限元模型並驗證...................................74
4.2最佳化設計模型及程序......................................80
4.3最佳化結果..............................................85
4.4比較與討論..............................................91
第五章   結論..............................................96
5.1結論...................................................96
5.2未來展望................................................98
	
參考文獻..................................................99

圖目錄
圖2.1 TANA3近似函數與原函數相對誤差曲線圖.......................11
圖2.2 AMFDM最佳化例題一迭代結果...............................17
圖2.3 MFDM最佳化例題一迭代結果................................17
圖2.4 AMFDM最佳化例題二迭代結果...............................19
圖2.5 MFDM最佳化例題二迭代結果................................19
圖2.6 AMFDM最佳化例題三迭代結果...............................21
圖2.7 MFDM最佳化例題三迭代結果................................21
圖2.8 AMFDM最佳化例題四迭代結果...............................23
圖2.9 MFDM最佳化例題四迭代結果................................23
圖3.1 十桿桁架結構及受力圖....................................36
圖3.2 MFDM最佳化最小全域值目標函數迭代結果......................39
圖3.3 AMFDM最佳化最小全域值目標函數迭代結果.....................42
圖3.4 CPSO最佳化最小全域值目標函數迭代結果......................43
圖3.5 ACPSO最佳化最小全域值目標函數迭代結果.....................46
圖3.6 二十五桿桁架結構及受力圖................................49
圖3.7 MFDM最佳化最小全域值目標函數迭代結果......................52
圖3.8 AMFDM最佳化最小全域值目標函數迭代結果.....................55
圖3.9 CPSO最佳化最小全域值目標函數迭代結果......................56
圖3.10 ACPSO最佳化最小全域值目標函數迭代結果....................59
圖4.1 工具機床台結構尺寸圖....................................68
圖4.2 工具機床台邊界條件及受力情形.............................68
圖4.3 元素幾何形狀及節點位置..................................70
圖4.4 抗振材料結構設計變數....................................71
圖4.5 中碳鋼結構設計變數.....................................71
圖4.6 工具機床台原始結構網格模型...............................74
圖4.7 1/4工具機床台結構自由體圖...............................77
圖4.8 1/4工具機床台結構受力及邊界條件..........................78
圖4.9 ACPSO最佳化最小全域值目標函數迭代結果.....................93
圖4.10 AMFDM最佳化最小全域值目標函數迭代結果....................93
圖4.11 AMFDM最佳化最小全域值目標函數迭代結果....................94
圖4.12 MFDM最佳化最小全域值目標函數迭代結果.....................94

表目錄
表2.1 TANA3近似函數程式化數值結果..............................9
表2.2 TANA3程式化計算誤差分析結果.............................10
表3.1 五次MFDM十桿桁架結構目標函數結果.........................38
表3.2 五次AMFDM十桿桁架結構目標函數結果........................41
表3.3 五次ACPSO十桿桁架結構目標函數結果........................45
表3.4 四種最佳化方法十桿桁架結構求解時間........................46
表3.5 四種最佳化方法十桿桁架結構有限元求解次數...................47
表3.6 二十五桿桁架結構兩種負載情形.............................49
表3.7 五次MFDM二十五桿桁架結構目標函數結果......................51
表3.8 五次AMFDM二十五桿桁架結構目標函數結果.....................54
表3.9 五次ACPSO二十五桿桁架結構目標函數結果.....................58
表3.10 四種最佳化方法二十五桿桁架結構求解時間(sec)...............59
表3.11 四種最佳化方法二十五桿桁架結構有限元求解次數...............60
表4.1 工具機床台結構材料性質..................................70
表4.2 仿做與書籍中模態分析比較結果.............................75
表4.3 工具機床台結構與1/4工具機床台結構模態分析..................79
表4.4 ACPSO最佳設計變數及目標函數結果..........................85
表4.5 AMFDM最佳設計變數及目標函數結果..........................87
表4.6 MFDM最佳設計變數及目標函數結果...........................89
表4.7 四種最佳化方法工具機床台結構求解時間(sec)..................90
表4.8 四種最佳化方法工具機床台結構有限元求解次數..................90
表4.9 四種最佳化與文獻工具機床台結構分析結果.....................92

參考文獻

[1]林俊毅，TANA-TDQA的兩點混合近似法最佳化探討及應用，淡江大學機械與機電工程學系碩士班，碩士論文，2013.

[2]Schmit, L. A., and Farshi, B., 1974, “Some approximation concepts for structural synthesis,” AIAA Journal, Vol. 12,No. 5, pp. 692-699.

[3]Storaasli, O. O., and Sobieszczanski-Sobieski, J., 1974, “On the accuracy of the Taylor approximation for structure resizing,” AIAA Journal, Vol. 12, pp. 231-233.

[4]Haftka, R. T., and Shore, C. P., 1979, “Approximation method for combined thermal structural design,” NASA TP-1428.

[5]Starnes, J. H., and Haftka, R. T.,1979, “Preliminary design of composite wing for buckling strength and displacement constraints,” Journa of Aircraftl, Vol. 16,No. 8, pp. 564-570.

[6]K. Svanberg,1987, “The method of moving asymptotes-a new method for structural optimization,” Int. J. Num. Meth. Engrg., Vol. 24, pp. 359-373.

[7]Haftka, R. T.,Nachlas, J. A., Watson, L. T.,Rizzo, T. and Desai, R.,1987, “Two-point constraint approximation in structural optimization,”Computer methods in applied mechanics and engineering, Vol. 60, pp. 289-301.

[8]Fadel, G. M.,Riley, M. F.,Barthelemy, J. M.,1990, “Two-point exponential approximation method for structural optimization,”Structural Optimization, Vol. 2, pp. 117-124.

[9]Wang, L. P., and Grandhi, R.V.,1994, “Efficient safety index calculation for structural reliability analysis,” Computer and Structues, Vol. 52, No. 1, pp. 103-111.

 
[10]Wang, L. P., and Grandhi, R.V.,1995, “Improved two-point function approximations for design optimization,” AIAA Journal, Vol. 33, No. 9,pp. 1720-1727.

[11]Xu, S., and Grandhi, R. V.,1998, “Effective two point function	  approximation for design optimization,”AIAA Journal,Vol. 36,No. 12,pp. 2269-2275.

[12]J.Kennedy and R.C. Eberhart, “Particle Swarm Optimization,
”International Conference on Neural Networks,IEEE,vol.4, pp. 1942-1948, November 1995.

[13]Y. Shi and R. C. Eberhart, “A Modified Particle Swarm Optimizer,” IEEE, Evolutionary Computation, pp. 69-73, May, 4-9, 1998.

[14]L.Wang, Q.Kang, H.Xiao, and Q.Wu, “A Modified Adaptive Particle Swarm Optimization Algorithm,” Industrial Technology ,IEEE, pp. 209-214, 2005.

[15]陳志忠，含突變機制的粒子群演算法於多目標工程最佳化，淡江大學機械與機電工程學系碩士班 碩士論文，2011.

[16]Michalewicz, Z., 1995, “Genetic Algorithms, Numerical Optimization and Constraints”, L. Eshelman (Ed.), Proceedings of the Sixth International Conference on Genetic Algorithms, Morgan Kauffman, San Mateo, pp. 151-158.

[17]陳建元，兩點近似法於結構最佳化設計之應用，碩士論文，國立台灣大學工學院機械工程學系研究所，台北，台灣，2002。

[18]Singiresu.S.Rao, “Engineering Optimization Theory and Practice”WILEY INTERSCIENCE.pp. 530-531,1996.

[19]江奇鴻，結構最佳化之加強兩點指數近似法，碩士論文，國立臺灣大學工學院機械工程學系研究所，台北，台灣，2013.

[20]簡聰海，數值分析使用C語言，全華科技圖書股份有限公司，2002。

[21]劉惟信，機械最佳化設計，全華科技圖書股份有限公司，1996。

[22]Guo, X.,Yamazaki, K., and Cheng, G. D.,2001, “A new three-point approximation approach for design optimization problems,”
Int.J.Numer.Meth.Engng, Vol. 50,pp. 869-884.

[23]Kim, J. R., Choi, D. H.,2008, “Enhanced two-point diagonal  quadratic approximation methods,” Comput.Methods Appl.Mech. Engrg. Vol. 197,pp. 846-856.

[24]張耀仁，結構最佳化設計之準二次兩點保守近似，碩士論文，國立台灣大學工學院機械工程學系研究所，台北，台灣，2007。

[25]張蘶贏，桁架結構最佳化使用基因演算法與窄化空間技術，碩士論文，國立交通大學土木工程學系研究所，新竹，台灣，2007。

[26]郭純孜，應用移動漸近線法於結構之最佳化設計，碩士論文，淡江大學航空太空工程學系研究所，台北，台灣，2011。

[27]康淵，翁嘉駿，何燾羽，馮威強，ANSYS進階，全華科技圖書股份有限公司，2006。

[28]陳精一，ANSYS7.0電腦輔助工程實務分析，全華科技圖書股份有限公司，2003。