現今的可靠度設計最佳化(Reliability-based design optimization, RBDO)中主流的方法以一次可靠度方法(First-order reliability method, FORM)之理論分析設計可靠度，以求效率，但對於強烈非線性、非正常分布或多變數的問題則易失去準確度，而難以找出確實符合所需可靠度的設計解。為能使用可靠度分析的準確性最可信賴，但計算量龐大的Monte-Carlo模擬法(Monte-Carlo simulation, MCS)，且希望計算效率與主流方法相當，本研究提出典型分布趨近法(Typical distribution approach, TDA)，統計設計之機率分布，以典型的、具數學式的分布型態之遞增密度函數(Cumulated density function, CDF)近似實際分布，結合性能函數之平均值與變異量計算法推算於各設計點之可靠度，再依MCS所分析出的實際可靠度調整以找出滿足實際所需可靠度的設計解。
  近幾年來的可靠度強健設計最佳化(Reliability-based robust design optimization, RBRDO)方面，有維度縮減法(Dimension reduce method, DRM)與性能矩積分法(Performance moment integration, PMI)等準確的性能函數之平均值與變異量推算法可用於TDA中，以使近似的CDF可隨設計值變化，推算各設計情況時的可靠度。但是由於兩方法皆需大量的計算程序，為了縮減程序，本文針對其中的PMI進行簡易化改良，稱為單純PMI (Single PMI, SPMI)。
  本研究採用數個RBDO問題以測試TDA，並與主流方法之一的連續最佳化與可靠度評估法(Sequential optimization and reliability assessment, SORA)作比較，驗證TDA之準確性。另外再以TDA結合SPMI處理結構的RBRDO問題，並將SPMI之性能變異量計算結果與DRM和PMI比較，再次驗證TDA之準確性與SPMI之適用性。

The method of sequential optimization and reliability assessment (SORA) is one of the most popular and convenient method to solve reliability-based design optimization (RBDO) problems in recent 10 years.  As compared with the First-order reliability method (FORM), SORA shows a better manipulation in efficiency.  However, SORA may not results in an enough accurate solution when it confronts a problem containing multi-variable performance function with highly non-linearity and non-normal distribution.  In this case, the final result by SORA also is not obviously to satisfy required reliability in a RBDO problem.  
An alternative RBDO method named typical distribution approach (TDA) has been proposed in this thesis.  The TDA utilizes the Monte-Carlo Simulation (MCS) which is recognized as the most reliable method for reliability analysis to improve the RBDO accuracy. The probability density function can be approximately represented by a typical formulation and solved by using cumulative probability function (CDF) , with the methods for measuring mean value and variance of performance function to compute the reliability at any design point, then adjust the approximation from analyzed data of MCS to reach the real RBDO solution.  The numerical solution process shows a similar efficiency, as compared the method of SORA.  In recent reliability-based robust design optimization, two methods can be applied to measure the variance of performance function: dimension reduce method (DRM) and performance moment integration (PMI).  Nevertheless, both methods require a large amount computation.  In this work, a single PMI named SPMI is proposed and it can simplify the procedure in PMI.  The proposed TDA and SPMI are successfully illustrated by several RBDO problems and RBRDO problems.

誌謝...I
中文摘要...II
英文摘要...IV
目錄...VI
圖目錄...XI
表目錄...X
符號說明...XII
第1章 緒論...1
  1. 1 動機與目的...1
  1. 2 文獻回顧...2
  1. 3 論文架構...5
第2章 可靠度設計最佳化...6
  2. 1 可靠度分析法...6
    2.1.1 性能函數(Performance function)...6
    2.1.2 機率分布的可靠度計算...7
    2.1.3 Monte-Carlo模擬法(Monte-Carlo simulation, MCS)...8
    2.1.4 一次可靠度方法(First-order reliability method,  
          FORM)...10
  2. 2 可靠度設計最佳化數學式...12
  2. 3 連續最佳化與可靠度評估法...13
    2. 3. 1 性能測量趨近法...13
    2. 3. 2 連續最佳化與可靠度評估法之可靠度設計最佳化...14
  2.4 結合有限元素法之可靠度設計最佳化...21
    2. 4. 1 反應表面法之2次回歸...21
    2. 4. 2 有限元素分析的連續最佳化與可靠度評估法...23
  2.5 以連續最佳化與可靠度評估法處理可靠度設計問題...24
第3章 典型分布法...38
  3. 1 典型分布法之概念...38
  3. 2 性能函數平均值與標準差計算法...39
    3. 2. 1 採樣統計法...40
    3. 2. 2 機率分布的積分法...40
    3. 2. 3 Taylor一次近似法...41
    3. 2. 4 維度縮減法(Dimension Reduction Method, DRM)...41
    3. 2. 5 性能矩積分法(Performance Moment Integration, 
            PMI)...42
    3. 2. 6 單純性能矩積分法(Single PMI, SPMI)...44
  3. 3 近似分布型態之選擇...46
    3. 3. 1 單一隨機變數之分布...46
    3. 3. 2 常態分布型態之近似...47
    3. 3. 3 一般化極值分布之近似...49
  3. 4 典型分布法的可靠度設計最佳化...52
    3. 4. 1 性能函數之分布型態的改變....52
    3. 4. 2 典型分布趨近法之運算流程...52
    3. 4. 3 結合結構有限元素分析的典型分布趨近法...55
    3. 4. 4 例題求解...57
第4章 考量強健性的可靠度設計最佳化...70
  4. 1 強健設計最佳化...70
    4. 1. 1 強健設計最佳化之目標...70
    4. 1. 2 性能函數變異量之計算法...72
    4. 1. 3 考量限制條件之不定性的強計設計...75
  4. 2 可靠度與強健性設計最佳化...76
  4. 3 以典型分布趨近法處理可靠度強建設計最佳化...77
第5章 結論...93
  5. 1 綜合討論與結論...93
  5. 2 未來展望...94
參考文獻...95

圖 2-1 機率分布函數之可靠度與失敗機...8
圖 2-2 兩變數之可靠度指標表示圖...11
圖 2-3 (a) 雙迴圈法之搜尋方式 (b) 去偶合法之搜尋方...13
圖 2-4 PMA之搜尋法...14
圖 2-5 DO設計於RBDO之效果...19
圖 2-6 SORA流程圖...20
圖 2-7 中心組合設計之採樣例...23
圖 2-8 汽車側面衝撞模型(摘自文獻[23])...29
圖 2-9 角板受力情形圖...33
圖 2-10 角板各處的Von-Mise stress分布情形...34
圖 3-1 PMI之近似取段...44
圖 3-2 SPMI與原PMI之計算流程比較...46
圖 3-3 常態分布之統計分布圖...48
圖 3-4 左右不均的統計分布圖...50
圖 3-5 不同形狀參數 的GEV分布圖...51
圖 3-6 性能函數於不同設計值時的分布型態差異例...52
圖 3-7 TDA之流程圖...55
圖 4-1 承受一垂直力 與側向力 之懸臂樑結構...84
圖 4-2 汽車側面衝撞模型(摘自文獻[18])...89

表 2-1 各機率分布型態之標準常態變數關係...10
表 2-2 例題2-1之DO計算結果...25
表 2-3 例題2-1之SORA初次的邊界轉移最佳化結果...27
表 2-4 例題2-1之本研究SORA的RBDO解與文獻[1]比較...28
表 2-5 各隨機變數 之設定...29
表 2-6 例題2-2之DO計算結果...31
表 2-7 例題2-2 SORA之RBDO計算結果...31
表 2-8 例題2-2 文獻[25]之RBDO解...32
表 2-9 例題2-3之DO計算結果...36
表 2-10 例題2-3 SORA之RBDO計算結果...36
表 3-1 例題3-1之DO計算結果...58
表 3-2 例題3-1 SPMI所計算出的 和 與統計值之比較...60
表 3-3 例題3-1之TDA的初回計算結果...61
表 3-4 例題3-1TDA與SORA之RBDO解比較...62
表 3-5 例題3-2之DO計算結果...64
表 3-6 例題3-2 SPMI所計算出的 和 與統計值之比較...65
表 3-7 例題3-2 DRM所計算出的 和 與統計值之比較...65
表 3-8 例題3-2 TDA RBDO與SORA之計算結果比較...66
表 3-9 例題3-3之DO計算結果...68
表 3-10 例題3-3 TDA RBDO與SORA之計算結果比較...69
表 4-1 例題4-1 DRM、PMI與SPMI之DO計算結果比較...80
表 4-2 例題4-1 DRM、PMI與SPMI之計算結果比較...82
表 4-3 例題4-2本研究純RBDO計算結果與文獻[7]之比較...86
表 4-4 例題4-2純強健設計時之計算結果與文獻的比較...86
表 4-5 例題4-2雙方並重設計時之計算結果與文獻結果比較...87
表 4-6 例題4-3純RBDO計算結果與文獻[23]之比較...91
表 4-7 例題4-3純強健設計DRM、PMI與SPMI之結果比較...91
表 4-8 例題4-3雙方並重設計DRM、PMI與SPMI之結果比較...92

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