近幾年來隨著電腦計算能力與圖形處理的進步，拓樸最佳化開始引起學界與業界的注意。目前文獻上的相關研究多集中在探討最小化總應變能之結構拓樸形式，然而多數結構在實務的使用上都會對於某些需求應加以限制，例如：應力、位移或頻率等。本文提出以懲罰函數法（penalty function method）用以探討含有限制條件之結構拓樸最佳化設計，並推導其對應的數學模型。
懲罰函數法中虛擬目標函數的計算同時包含了原目標函數及限制方程式。前者之目的是為得到最小化應變能之拓樸最佳化；後者則是對給定之限制條件計算對應的懲罰項，目的是為讓設計變數朝可行解區域的邊界移動。兩者之間以懲罰參數加以結合並可調整二者之間的權重，以便得到不同的結構最佳化拓樸形狀。透過計算虛擬目標函數的敏感度做為判斷元素材料（或移除與否）的準則。
數值實例顯示無論是含位移限制式或是頻率限制式之拓樸最佳化問題，本文所提之懲罰函數法均可以在滿足限制條件下並最小化結構物應變能，並對不同形式的懲罰函數參數值提出建議。此外，本研究亦發現不同的限制式容許值會產生不同的拓樸形狀，而文獻上的其他方法無法反映出不同的限制式容許值對拓樸形狀的影響。

Owing to the rapid development of powerful and cost-efficient computers with graphic capabilities, multi-windows and fast computation, topology optimization has received intensive attention in industry and academia in recent years. In the literature, a lot of researches focus on the topology optimization with minimum total strain energy; however, most structures in practical designs have to meet some requirements, such as stress, displacements, and frequencies in the specifications. In this paper, the structural topology optimizations with different constraints are discussed through the penalty function, and the corresponding mathematical models are derived.
The penalty function transforms the original constrained optimization problem to the unconstrained optimization problem by the pseudo objective function which includes the original objective function and different constraints. The object is to obtain the minimum total strain energy of the structures and force the design variables moving to the boundary of the feasible domain by the penalty term resulting from different constraints. The weighting factors of the objective function and the penalty terms can be adjusted to obtain different optimal structural topologies for different requirements. In this paper, the sensitivities of the pseudo objective function are derived and used to be the removing criterion. The optimal topology is achieved by the evolutionary switching algorithm which changes element material properties according to the sensitivity results.
Through the parametric study, the appropriate values of penalty parameters are suggested in this research. Numerical examples demonstrate that the proposed method can solve the structural topology optimization problems with displacement or frequency constraints. The obtained optimal structural topologies not only have minimum strain energies but also satisfy different constraints. In addition, numerical results reveal that the allowable values of constraints will affect the optimal topologies, which are never discussed in other methods.

中文摘要................................................................................................................I
英文摘要..............................................................................................................II
誌謝.....................................................................................................................III
目錄.....................................................................................................................IV
圖目錄.................................................................................................................VI
第一章 緒論......................................................................................................1
1.1 研究背景...............................................................................................1
1.2 研究目的...............................................................................................1
1.3 文獻回顧...............................................................................................2
1.4 研究內容...............................................................................................4
第二章 結構拓撲最佳化..................................................................................7
2.1 傳統結構演進式最佳化（ESO）.......................................................7
2.1.1 平面等參元素..............................................................................7
2.1.2 應力移除準則............................................................................12
2.1.3 應變能移除準則........................................................................14
2.1.4 停止準則....................................................................................16
2.2 位移敏感度拓樸最佳化.....................................................................18
2.2.1 位移敏感度................................................................................19
2.2.2 位移約束條件之最佳化演進流程............................................20
2.3 頻率敏感度拓樸最佳化.....................................................................21
2.3.1 頻率敏感度................................................................................22
2.3.2 演進流程....................................................................................24
2.4 小結.....................................................................................................25
第三章 以懲罰函數考慮限制條件之拓樸最佳化...........................................27
3.1 懲罰函數簡介.....................................................................................27
3.1.1 懲罰函數法的數學形式............................................................29
3.1.2 懲罰函數法數值實例................................................................32
3.2 含位移限制式之拓樸最佳化.............................................................35
3.3 含頻率限制式之拓樸最佳化.............................................................38
3.4 交換式結構最佳化演進.....................................................................41
3.5 小結.....................................................................................................43
第四章 位移限制式之結構靜力拓樸最佳化................................................45
4.1 懲罰函數法中不同參數的影響.........................................................45
4.2 使用不同移除準則的影響.................................................................54
4.3 小結.....................................................................................................61
第五章 頻率限制式之結構動力拓樸最佳化................................................65
5.1 懲罰函數法中不同參數於最大化振動頻率分析的影響.................65
5.2 懲罰函數法中不同參數於最小化振動頻率分析的影響.................73
5.3 使用不同移除準則於最大化頻率分析的影響.................................77
5.4 使用不同移除準則於最小化頻率分析的影響.................................83
5.5 小結.....................................................................................................87
第六章 結論與展望........................................................................................89
6.1 結論.....................................................................................................89
6.2 未來展望.............................................................................................91
參考文獻.............................................................................................................93


圖目錄
圖2.1 四邊形Q4元素........................................................................................8
圖2.2 承受純彎矩之四邊形元素.....................................................................11
圖2.3 四邊形Q8元素......................................................................................12
圖3.1 可行解空間示意圖.................................................................................28
圖3.2 外部懲罰函數示意圖.............................................................................30
圖3.3 內部懲罰函數示意圖.............................................................................31
圖3.4 外部懲罰函數實例函數示意圖.............................................................33
圖3.5 內部懲罰函數實例函數示意圖.............................................................34
圖3.6 交換式結構最佳化演進法流程.............................................................43
圖4.1 懸臂梁初始設計領域幾何尺寸.............................................................46
圖4.2例題一之一：三種懲罰參數分析的拓樸形狀.....................................47
圖4.3 例題一之二：三種懲罰參數分析之拓樸形狀....................................47
圖4.4 簡支梁初始設計領域幾何尺寸.............................................................48
圖4.5例題二之一：三種懲罰參數分析之拓樸形狀.....................................49
圖4.6 例題二之二：三種懲罰參數分析之拓樸形狀....................................49
圖4.7例題三之一：三種懲罰參數分析之拓樸形狀.....................................51
圖4.8 例題三之二：三種懲罰參數分析之拓樸形狀....................................51
圖4.9 例題四：三種懲罰參數分析之拓樸形狀............................................53
圖4.10 例題四之二：三種懲罰參數分析之拓樸形狀..................................53
圖4.11 例題五：三種分析之拓樸形狀...........................................................55
圖4.12 文獻上（Xie etal.1995 and Xie and Steven 1996）之應變能分析與敏感度分析的拓樸形狀...........................................................................55
圖4.13 例題六：三種分析之拓樸形狀...........................................................56
圖4.14 例題七：三種分析之拓樸形狀...........................................................57
圖4.15 使用ESO程式與文獻上（Liang etal. 2000）之應變能分析與敏感度分析的拓樸形狀...............................................................................58
圖4.16 例題八：三種分析之拓樸形狀...........................................................59
圖4.17例題九：三種分析之拓樸形狀............................................................60
圖5.1 例題一：初始設計領域幾何尺寸.........................................................66
圖5.2例題一之一：三種懲罰參數分析之拓樸形狀.....................................67
圖5.3 例題一之二：三種懲罰參數分析之拓樸形狀....................................68
圖5.4 例題二：初始設計領域幾何尺寸.........................................................69
圖5.5例題二之一：三種懲罰參數分析之拓樸形狀.....................................69
圖5.6 例題二之二：三種懲罰參數分析之拓樸形狀....................................70
圖5.7 例題三：初始設計領域幾何尺寸.........................................................71
圖5.8例題三之一：三種懲罰參數分析之拓樸形狀.....................................72
圖5.9 例題三之二：三種懲罰參數分析之拓樸形狀....................................73
圖5.10例題四之一：三種懲罰參數分析之拓樸形狀...................................74
圖5.11 例題四之二：三種懲罰參數分析之拓樸形狀...................................75
圖5.12 例題五：三種懲罰參數分析之拓樸形狀..........................................76
圖5.13 例題五之二：三種懲罰參數分析之拓樸形狀..................................77
圖5.14 例題六：三種移除準則分析之拓樸形狀..........................................79
圖5.15 文獻上（Xie and Steven 1996）之敏感度分析的拓樸形狀.............79
圖5.16 例題七：三種移除準則分析之拓樸形狀..........................................81
圖5.17 文獻上（Xie and Steven 1994）之敏感度分析的拓樸形狀.............81
圖5.18 例題八：三種移除準則分析之拓樸形狀..........................................83
圖5.19 文獻上（Xie and Steven 1996）之敏感度分析的拓樸形狀.............83
圖5.20 例題九：三種移除準則分析之拓樸形狀..........................................84
圖5.21例題十：三種移除準則分析之拓樸形狀...........................................86
圖5.22 文獻上（Xie and Steven 1996）之敏感度分析的拓樸形狀.............86

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