在工程之應用上，組合件通常都是由多種材料結構接合而成，其乃利用各組成材料之不同化學跟物理性質，可以達到其最佳的強度跟性能，最常見此類之結構如:電子封裝、三明治結構、疊層複材…等。目前在分析工程問題中最常被使用的方法，包括有限差分法(FDM)、有限元素法(FEM)和邊界元素法(BEM)…等，然而在使用有限差分法和有限元素法時，其分割元素之長寬比必須符合同一冪次的比例原則。因此如若分析物件為薄層結構時，物件之全域模擬將會產生出大量的網格數量，因而增加電腦在運算過程中的負擔。本研究之理論基礎，乃是以邊界元素法來做分析處理，其特色就是只需對分析物件的邊界做積分，而不需對其定義域之內部做任何處理，因此不管分析物件之結構厚度多麼薄，邊界之模擬並不會增加其生成之網格數量，因而有助於分析問題時之效率。
然而在使用邊界元素法來求解工程問題時，雖然能夠在分析時間和效能處理上會明顯低比使用有限元素法來快的許多，但其邊界網格模型之建立，若以人工來做分析物件之邊界的切割網格處理，將耗時甚多。所以本論文針對邊界元素法，開發其相關二維自動網格生成之程式，讓使用者能夠更快速建立所需要的網格生成資料，來進行其分析與模擬。最後將提供九個數值範例來驗證所編寫之自動網格生成程式的相關應用。

Many engineering applications often have components consisting of different materials with distinct properties in physical or chemical phases to achieve their ultimate strength and performance. Such applications are commonly seen in, for example, electronic packages, sandwich composites…etc. The most popular numerical tools for engineering analysis include the finite difference method (FDM), finite element method (FEM), and the boundary element method (BEM). The FDM and FEM must conform to the limitation that the length to width aspect ratios of domain meshes must be in the same order to yield reliable results. For this reason it shall require tremendous meshes to model very thin layers and therefore increase the burden of computation. The present work is to develop an automesh code serving as preprocess of the BEM, which is featured by boundary mesh only. Therefore, no matter how thin the thickness of the layer is, the boundary modeling will not add the amount of mesh and is ideal for analyzing engineering problems, especially involving thin layers.
However, when applying the BEM to analyze engineering problems, although the modeling itself is more efficient, the process of manual meshing will still take much effort and is time-consuming. For this reason, this thesis is to develop an automeshing code that enables the user to quickly build necessary input data for pertinent two-dimensional BEM analyses. The end of the thesis provide 9 numerical examples to show the efficiency of the developed code.

誌謝 i
摘要 ii
Abstract iv
目錄 v
圖目錄 viii
表目錄 xiii

第一章 導論 1
第1.1節 前言 1
第1.2節 研究目的和動機 2
第1.3節 文獻回顧 4
第二章 基本理論 9
第2.1節 前言 9
第2.2節 邊界元素法處理異向材料熱應力之分析 10
第2.3節 超薄疊層之近乎奇異性問題 14
第三章 2-D自動網格生成之程式開發 16
第3.1節 程式開發目的 16
第3.2節 模型建構理論 17
3.2.1 建模方式 17
3.2.2 理論公式 18
3.2.3 指令流程 19
3.2.4 線段分析 20
第3.3節 模型建構指令說明 21
第3.4節 建構2D模型 23
3.4.1 五邊形 23
3.4.2 任意八邊形 24
3.4.3 兩孔桿件 24
3.4.4 三孔桿件 24
第3.5節 BEM 程式發展說明 25
4.1 範例1 47
4.2 範例2 48
4.3 範例3 49
4.4 範例4 50
4.5 範例5 51
4.6 範例6 51
4.7 範例7 52
4.8 範例8 53
4.9 範例9 54
第五章 結論與未來展望 76
參考文獻 77























圖目錄
圖1.1 傳統全域解法與邊界元素法之比較 7
圖1.2 疊層之複合材料 8
圖1.3 晶圓封裝內填充膠之脫層 8
圖2.1 二維超薄物件之邊界關係 15
圖2.2 源點與被積元素的關係 15
圖3.1 邊界元素法模型建構之路徑表示 26
圖3.2 網格切割方式 26
圖3.3 網格切割之理論 27
圖3.4 自動化之網格生成圖 27
圖3.5 扇形之指令輸入流程 28
圖3.6 扇形之網格生成 28
圖3.7 使用者定義指令流程圖 29
圖3.8 Auto-mesh指令流程圖 30
圖3.9 控制疏密指令(a,r) 31
圖3.10 控制疏密指令(n,r) 31
圖3.11使用者定義之網格生成圖 32
圖3.12 直線(straight line) 32
圖3.13 弧線(circular arc) 33
圖3.14 圓(circular) 33
圖3.15 銳角(sharp corner) 33
圖3.16 邊界元素法之自動網格生成程式 34
圖3.17 自動網格程式建構指令(一) 34
圖3.18 自動網格程式建構指令（二） 35
圖3.19 自動生成網格之程式輸出之相關檔案 35
圖3.20 邊界元素法之自動網格生成圖 36
圖3.21 兩線段之夾角數據 36
圖3.22 各線段之長度數據 36
圖3.23 力之網格點座標(卡式座標) 37
圖3.24 力之網格點座標(極座標) 38
圖3.25 熱之網格點座標(卡式座標) 39
圖3.26 熱之網格點座標(極座標) 40
圖3.27 薄層結構之網格生成 41
圖3.28 使用者定義之網格生成 41
圖3.29 疊層物件網格點要求圖 42
圖3.30 對稱五邊形 42
圖3.31 五邊形輸出長度跟角度 43
圖3.32 任意八邊形 43
圖3.33 八邊形輸出長度跟角度 44
圖3.34 兩孔桿件-Solid Works建模 44
圖3.35 兩孔桿件-自動生成網格點建模 45
圖3.36 三孔桿件-Solid Works建模 45
圖3.37 三孔桿件-自動生成網格點建模 46
圖4.1 受熱負載之多層複合薄板 55
圖4.2 多層複合薄板之自動生成網格圖 55
圖4.3 ANSYS之網格生成圖 56
圖4.4 受熱負載之三明治疊層板 56
圖4.5 三明治疊層板之自動生成網格圖 57
圖4.6 在 Core 和 Adhesive 之層間應力分佈圖 57
圖4.7 在 Face 和 Adhesive 之層間應力分佈圖 58
圖4.8 受熱負載之合成板 58
圖4.9 合成板之自動生成網格圖 59
圖4.10 (主軸轉角60度)層間應力分佈圖 59
圖4.11 受壓力正方形板中具圓孔之工件 60
圖4.12 具圓孔之工件之自動生成網格圖 60
圖4.13 BEM (Automesh) 與 ANSYS數據比較圖 61
圖4.14 正方形板中具圓孔之工件受溫度變化 61
圖4.15 具圓孔之工件之自動生成網格圖 62
圖4.16 BEM (Automesh) 與 ANSYS數據比較圖 62
圖4.17 受壓力之薄板中具圓孔之工件 63
圖4.18 薄板具圓孔之自動生成網格圖 63
圖4.19 BEM (Automesh) 與 ANSYS數據比較圖 64
圖4.21 軸對稱之建模 65
圖4.22 封裝結構之自動生成網格圖 65
圖4.23 比較圖(CDEF線段- Chipboard) 66
圖4.24 比較圖(CDEF線段- Chipboard) 66
圖4.25 比較圖(CDEF線段- Chipboard) 67
圖4.26 比較圖(CD線段- Silicon) 67
圖4.27 比較圖(CD線段- Silicon) 68
圖4.28 比較圖(CD線段- Silicon) 68
圖4.29 比較圖(GH線段- Silicon) 69
圖4.30 比較圖(GH線段- Silicon) 69
圖4.31 比較圖(GH線段- Silicon) 70
圖4.32 比較圖(GHDE線段- Epoxy) 70
圖4.33 比較圖(GHDE線段- Epoxy) 71
圖4.34 比較圖(GHDE線段- Epoxy) 71
圖4.35 掛鉤型之工件受溫度變化 72
圖4.36 掛鉤型之自動生成網格圖 72
圖4.37 BEM (Automesh) 與 ANSYS數據比較圖 73
圖4.38 受壓力具兩孔之工件 73
圖4.39 具兩孔工件之自動生成網格圖 74
圖4.40 BEM (Automesh) 與 ANSYS數據比較圖 74

















表目錄
表4.1 材料性質 75
表4.2 BEM (Automesh) 與 ANSYS數據比較圖 75

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