近年來，封裝體為滿足市場需求而不斷走向小型化和高電晶體密度來提升效能，然而隨著時間推移，此趨勢面臨到物理與技術上的極限，人們轉而發展2.5D或3D封裝與異質整合(Heterogeneous Integration)等技術來達成更高的效能。然而此技術的發展亦須面對新的挑戰，例如：在一封裝體中含有多個晶片容易造成散熱困難，另外封裝體整體結構複雜性大幅增加會導致其在設計、製造和測試過程的成本上升，此時引入有限元素法進行模擬有助於解決問題，透過模擬能夠快速分析結構中的熱點或應力集中位置，以使作為依據來更改結構或適當增加散熱元件，此外透過模擬也能大幅減少過往進行可靠度測試實驗所需耗費的時間成本，使封裝體能夠更快速進入市場。
然而，即便模擬能夠減少測試時間並幫助分析破壞成因用以改善設計，但隨著封裝體的複雜度增加，於模擬中建立模型和求解所需的時間大幅上升，為解決此問題需要引入簡化模型的方法。等效理論和全域—局部有限元素法為兩有效簡化模型的方法，然而於模擬中使用此兩種方法仍存在些許不確定性，若欲於複雜封裝體模擬中採用此兩簡化方式，其適用性需要經過驗證。此外，隨著封裝體微小化和功率增加，可能產生在原先設計尺寸下不存在的破壞，例如：基板(Substrate)線路因電子遷移(Electromigration)產生電氣失效。電子遷移會受到多種物理因素影響，例如：應力、溫度和電流密度等，且各因素之間可能產生交互作用。
本研究主要分為兩部分：以多個簡單複合材料模型測試等效理論公式之準確性；以PCB線路中的破壞做為案例，選用結構較為簡單的球柵陣列(Ball Grid Array, BGA)封裝作為載體，測試於一模型中同時採用等效理論和全域—局部有限元素法的情況下，是否能得到合理的模擬結果使我們能夠快速推得容易產生失效位置，藉此驗證簡化模型方法的適用性，此研究結果可推廣至結構更複雜的封裝體中—例如小晶片封裝(Chiplet)，作為其參考依據。

In recent years, to satisfy market demands, packages become smaller and the transistor density become higher in order to enhance performance. However, as time progresses, this trend faces physical and technological limits. To push the limit, advanced packaging techniques such as 2.5D or 3D packaging and heterogeneous integration has developed. However, the development of these package also brings about new challenges. For instance, incorporating multiple chips within a single package can lead to thermal dissipation difficulties, and the increased complexity of package structure leads to rising costs in design, manufacturing, and testing processes. In such cases, finite element method can offer solutions. With FEM simulation, we could rapidly find thermal hotspots or stress concentration areas within the structure, and we could modify the structure or design base on the results. Additionally, simulations significantly reduce the time and cost required for reliability testing experiments, allowing packages to enter the market more quickly.
Even though simulations can reduce testing time and aid in analyzing the causes of failures for design improvement, as the complexity of the packaging increases, the time required to establish models and solve increases significantly. To address this, simplifying methods need to be introduced. Equivalent theory and global-local finite element methods are two effective approaches for simplifying models. However, there still exists some uncertainty when applying these methods in simulations. If these methods are to be applied in complex packaging simulation, their applicability needs to be validated.
Furthermore, with the miniaturization of packaging and an increase in power, the possibility of failures that did not exist within the original design dimensions may arise. For example, in Substrate circuits, electrical failures can occur due to electromigration. Electromigration is influenced by various physical factors such as stress, temperature, and current density. These physical factors may interact with each other.
This research can be primarily divided into two parts: one is testing the accuracy of equivalent theory formulas by using multiple simple composite material models. And the other is taking failure in Substrate circuits as a case study. A relatively simple packaging, the Ball Grid Array (BGA) package, is chosen as the test vehicle. The aim is to test whether reasonable simulation results can be obtained when both equivalent theory and global-local finite element methods are applied in a single model. The simulation results help us to quickly find out areas that prone to failure. Through this process, the applicability of the simplified method can be validated. The findings of this study can be extended to more complex packaging structures, such as chiplets ,serving as a reference.

摘要 I
Abstract II
目錄 IV
圖目錄 VI
表目錄 XI

第一章 緒論 ........................................................................................................................... 12
1.1 研究動機 ................................................................................................................. 12
1.2 文獻回顧 ................................................................................................................. 14
1.2.1 全域－局部有限元素法 ....................................................................................... 14
1.2.2 等效理論分析 ....................................................................................................... 18
1.2.3 熱應力分析 ........................................................................................................... 23
1.2.4 小晶片封裝(Chiplet) ............................................................................................. 23
1.3 研究目標 ................................................................................................................. 25
第二章 基礎理論 ................................................................................................................... 27
2.1 有限元素法基礎理論 ............................................................................................. 27
2.1.1 材料線彈性理論 ................................................................................................... 27
2.1.2 材料非線性理論 ................................................................................................... 31
2.1.3 數值收斂方法與準則 ........................................................................................... 32
2.1.4 材料應變硬化法則（Strain Hardening Rule） ................................................... 33
A. 等向硬化法則(Isotropic Hardening Rule) .......................................................... 34
B. 動態硬化法則(Kinematic Hardening Rule) ....................................................... 35
2.1.5 Chaboche 模型 ...................................................................................................... 36
2.1.6 錫球外型預測 ....................................................................................................... 37
2.2 多點約束方程法 ..................................................................................................... 39
2.2.1 連接不同元素類型 ............................................................................................... 39
2.2.2 連接不同疏密程度之網格 ................................................................................... 40
2.2.3 建立剛性區域 ....................................................................................................... 42
2.3 等效材料參數 ......................................................................................................... 42
2.3.1 混合定律(Rule of Mixture, ROM)與反混合定律(Inverse Rule of Mixture,
IROM) ................................................................................................................... 42
A. 楊氏係數、剪切係數與普松比 ......................................................................... 42
B. 熱膨脹係數 ......................................................................................................... 46
2.3.2 能量法(Energy Principles) .................................................................................... 48
A. 最小勢能原理(Principle of Minimum Potential Energy)與最小補能原理
(Principle of Minimum Complementary Energy) .................................................... 48
第三章 有限元素之模型建立與驗證 ................................................................................... 51
3.1 有限元素模型基本假設 ......................................................................................... 51
3.2 模擬材料選擇 ......................................................................................................... 52
3.3 網格劃分與元素 ..................................................................................................... 56
3.4 模型邊界條件 ......................................................................................................... 56
3.5 模擬條件負載 ......................................................................................................... 58
3.6 有限元素模型建構 ................................................................................................. 58
第四章 研究結果分析與討論 ............................................................................................... 60
4.1 多點約束方程法驗證 ............................................................................................. 60
4.1.1 連接不同元素類型 ............................................................................................... 60
4.1.2 連接不同疏密程度之網格 ................................................................................... 61
4.2 等效材料參數驗證 ................................................................................................. 66
4.2.1 等效熱膨脹係數驗證 ........................................................................................... 66
4.2.2 等效楊氏係數驗證 ............................................................................................... 70
4.3 BGA 模型模擬結果分析 ........................................................................................ 73
結論與未來展望 ..................................................................................................................... 80 參考文獻 ................................................................................................................................. 82

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