本研究利用移動漸近線法(MMA)來處理定然式最佳化(DO)分析，因移動漸近線法能有效的近似目標函數與限制條件，且搭配移動限制的觀念，故使具有多變數與多限制條件之問題，能夠迅速地求解出最佳值。另外亦運用拓樸技術至結構最佳化中，使其結構成為拓樸最佳化(DTO)分析，並
融入有限元素法分析概念，讓最終結構能更精準與更符合預設目標與限制條件。
    在不考慮設計變數隨機性的定然式最佳化，其最終結果會非常接近設計的限制條件，而產生出一失敗機率很高的結構設計。因此本研究將設計變數隨機特性納入結構拓樸最佳化分析中(RBTO)，使結構成為可靠度最佳化分析，並讓可靠度達到預設值；因此RBTO得執行最佳化分析與可靠度分析，成為一雙迴圈演算分析，這將造成計算時間冗長和效率不佳等問題。為了改善其雙迴圈法問題，本研究以去偶合法與逆可靠度分析來提昇整體運算效率與速度。
    本研究以範例與文獻對照來驗證整體演算流程，並從中介紹逆可靠分析求取破壞點、過濾器配置和結構棋盤現象消除等運用，以使本研究能將結構整體達到預期的最佳結果。

Reliability-Based Topology Optimization (RBTO) incorporates probabilistic analysis into topology optimization process to take randomness of design variables into account. Earlier researches have shown optimal topology obtained from RBTO is different with the one obtained from the deterministic optimization. 
    A typical approach of RBTO is a nested double-loop approach where the outer loop conducts topology optimization and the inner loop deals with probabilistic analysis. The computational cost for this double-loop procedure is prohibitive for most practical problems. Moreover, topology optimization inherently possesses a large number of design variables making the problem more difficult. To overcome the high computational cost, researchers have suggested many alternative approaches such as the single-loop approach, the decoupling approach and the response surface method. 
    In this study, we take advantage of the decoupling method, in which a group of auxiliary design points are found to replace original random design variables. The auxiliary design points are the MPP points obtained from the inverse reliability analysis. Because the MPP is corresponding to the predefined reliability, the auxiliary design points will force optimal design to vicinity of probabilistic boundary. The effectiveness and accuracy of our proposed method is investigated through several numerical problems. Our approach is shown to produce optimal topology reaching reliability requirement with less calculation. Note that in this approach, topology optimization and reliability analysis are conducted separately, and for each analysis, the existed commercial software is well developed and readily to provide a valuable design with least effort.

誌謝	I
中文摘要	II
英文摘要	III
圖目錄	IX
表目錄	XII
第一章  緒論	1
1-1  前言	1
1-2  研究背景	2
1-2-1  最佳化概述	2
1-2-2  可靠度概述	3
1-3  研究動機與目的	5
1-4  論文回顧	5
1-4-1  定然式最化理論	5
1-4-2  定然式最佳化分析過程	8
1-4-3  可靠度最佳化分析	9
1-5  論文架構	11
第二章  定然式拓樸最佳化	12
2-1  結構最佳化分類	12
2-1-1  結構尺寸最佳化	13
2-1-2  結構型態最佳化	13
2-1-3  結構拓樸最佳化	14
2-1-4  結構最佳化分類比較	14
2-2  結構分析概念	16
2-2-1  有限元素法基本概念	17
2-2-2  有限元素法基本要素	17
2-2-3  有限元素法分析理論	21
2-2-4  有限元素法分析步驟	26
2-3  拓樸最佳化方法	27
2-3-1  均質法	27
2-3-2  密度函數法	29
2-4  最佳化演算法	31
2-5  MMA 基礎理論	32
2-6  MMA 最佳化近似函數	36
2-6-1  移動限制觀念	37
2-6-2  移動限制最佳化	42
2-7  MMA 求解步驟	43
2-8  靈敏度分析	45
2-8-1  直接法	46
2-8-2  伴隨法	48
2-9  MMA 最佳化分析流程	52
2-10  本章小結	53
第三章  可靠度分析演算	54
3-1  隨機參數不確定性	55
3-1-1  不確定性因素	55
3-1-2  隨機參數建立	55
3-1-3  可靠度機率函數	56
3-1-4  常態分佈	59
3-2  可靠度演算法	62
3-2-1  一階二次可靠度法	62
3-2-2  蒙地卡羅模擬法	65
3-3  可靠度分析	67
3-3-1  正可靠度分析	67
3-3-2  逆可靠度分析	69
3-3-3  百分比性能逆可靠度分析	70
3-4  去偶合演算法	73
3-4-1  雙迴圈RBDO	73
3-4-2  去偶合技術	75
3-4-3  去偶合演算法	77
3-5  本章小結	82
第四章  例題分析與結果討論	83
4-1  例題一 懸臂梁搭配不同隨機參數	84
4-1-1  隨機參數為外力F	85
4-1-2  隨機參數為外力F與楊氏係數E	88
4-1-3  隨機參數為外力F與波松比 	90
4-2  例題二 懸臂梁搭配兩個限制條件	98
4-3  例題三 C型懸結構最佳化分析	103
4-4  例題四 剪力牆結構最佳化分析	111
4-5  例題三 本章小結	115
第五章  結論與未來展望	116
5-1  研究貢獻	116
5-2  未來展望	117
參考文獻	119
附錄A	125
附錄B	127
附錄C	132
附錄D	133
附錄E	138

圖目錄

圖1-1  定然式最佳化問題示意圖	8
圖2-1  結構最佳化分類圓	12
圖2-2  結構拓樸最佳化示意圖	15
圖2-3  三維節點自由度示意圖	19
圖2-4  拓樸幾何體關係圖	20
圖2-5  二維結構孔洞細胞	28
圖2-6  三維結構孔洞細胞	28
圖2-7  設計區域材料分佈	28
圖2-8  各式近似法與MMA移動限制比較圖	39
圖2-9  移動限制有無差別示意圖	42
圖2-10  MMA最佳化分析示意圖	44
圖2-11  DTO結構最佳化分析流程圖	52
圖3-1  常用機率密度函數	57
圖3-2  常態分配曲線	59
圖3-3  標準常態分佈關係圖	61
圖3-4  正可靠度分析圖	67
圖3-5  逆可靠度分析圖	69
圖3-6  百分比特性概念圖	71
圖3-7  雙迴圈法示意圖	74
圖3-8  雙迴圈法與去偶合法比較圖	76
圖3-9  懸臂梁最佳化外力為100	79
圖3-10  懸臂梁最佳化外力為180	79
圖3-11  懸臂梁最佳化外力為150	80
圖3-12  去偶合分析流程圖	81
圖4-1  例題1結構最佳化前示意圖	84
圖4-2  例題1-1  DTO，可靠度=50.21%	85
圖4-3  例題1-1  RBTO，可靠度=99.87%	86
圖4-4  例題1-2  DTO，可靠度=50.46%	88
圖4-5  例題1-2  RBTO，可靠度=99.86%	88
圖4-6  例題1-3  DTO，可靠度=50.18%	90
圖4-7  例題1-3  RBTO，可靠度=99.88%	90
圖4-8  第一次最佳化分析各設計點結構分析之MPP值與單一假設
MPP值分佈比較	93
圖4-9  第一次最佳化中外力在各組的MPP與定值的MPP之比較	95
圖4-10 第三次最佳化中外力在各組的MPP與定值的MPP之比較	95
圖4-11 第一次最佳化中波松比在各組的MPP與定值的MPP之比較	97
圖4-12 第三次最佳化中波松比在各組的MPP與定值的MPP之比較	97
圖4-13  例題2 結構最佳化前示意圖	98
圖4-14  例題2  DTO，可靠度=50.37%	99
圖4-15  例題2  RBTO，可靠度=99.88%	99
圖4-16  例題2 懲罰因子比較圖	102
圖4-17  例題3 結構最佳化前示意圖	103
圖4-18  棋盤現象示意圖	104
圖4-19  本文與文獻[42] 過濾器分析比較圖	107
圖4-20  本文與文獻[42] DTO分析較圖	109
圖4-21  本文與文獻[42] RBTO分析較圖	109
圖4-22  例題4 結構最佳化前示意圖	111
圖4-23  例題4  DTO，可靠度=50.23%	112
圖4-24  例題4  RBTO，可靠度=99.87%	113
圖4-25  文獻[9] DTO分析圖	113

表目錄

表2-1  各元素節點數目表	18
表2-2  各維度元素示意表	19
表2-3  各維度網格示意表	20
表4-1  例題1-1  DTO與RBTO比較	87
表4-2  例題1-1  DTO與RBTO比較	89
表4-3  例題1-3  DTO與RBTO比較	91
表4-4  例題2  DTO與RBTO比較	100
表4-5  文獻[3] 過濾器配置	105
表4-6  文獻[42] 過濾器配置	105
表4-7  本文過濾器配置	106
表4-8  本文與文獻[42] 過濾器分析結果比較	108
表4-9  例題3  DTO與RBTO比較	110
表4-10 例題4  DTO與RBTO比較	114

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