臺灣博碩士論文加值系統

等位函數法（Level set method）是應用於處理界面最廣泛的方法，由於本身為距離界面之函數，界面永遠由平滑、連續之 （zero level set）函數表達之，可輕易獲得界面位置並且能處理極為複雜的界面，當界面彼此靠近等位函數能自動將界面融合（merging）；反之則自動分離（breaking），並且也容易將程式由二維拓展為三維。
本實驗室發展適應性相場模式處理固化問題已有五年時間，且有許多豐碩的成果；在科學界近年來等位函數法也趨於成熟，能模擬樹枝狀晶體甚至合金固化問題[1]，由於不同於相場模式之diffused-interface model，而是利用sharp-interface model處理界面，在粗網格就能夠得到定性之結果。本研究開創將等位函數法運用於適應性有限體積架構上模擬種種固化問題，首先介紹等位函數法於適應性有限體積法之離散，引入等位函數法後如何維持自然之二階收斂，sharp-interface model如何處理不連續之兩相問題，接著測試一些具有解析解之固化問題驗証程式之正確性及精確性，最後嘗試模擬過冷環境中樹枝狀晶體的成長。

The level set method has been widely used in numerics of propagating interfaces. Level set function is close to a signed distance function, and it can be used to exactly locate the interface in order to apply discretizations. Topological changes in the evolving front are handled naturally. The position of the front at time t is given by the zero-level set of a smooth, continuous function. This set needs not be connected and can break and merge as t advances. Furthermore, it can be easily extended to higher dimensions.

Over the last five years, the adaptive phase field model was widely adopted to study solidification problems in our group, while many fruitful results have been reported. However, because the concept of diffusive interface is adopted, the drawback for performing phase field modeling lies on its very awful computational load. Recently, the development of level set method has become mature and was used to simulate cases with complex distortion of interface, such as dendritic growth of an alloy [1]. Since the formulation of sharp-interface model is embedded locally, in principle, the level set method can simulate these problems accurately by using relatively thicker mesh structure. In this report, we have developed an adaptive level set method based on the finite volume method (FVM) to simulate solidification problems. To check its feasibility, we have derived numerical and physical algorithms carefully and discussed the convergence of our present model. Moreover, comparisons with analytical solutions were given by testing several Stefan problems. Finally, we tried to challenge the case dendritic growth under high supercooling.

致 謝……………………………………………………………..…Ⅰ
中文摘要………………………………………………………..………Ⅱ
英文摘要………………………………………………..……………..Ⅲ
目 錄………………………………………………………..………Ⅳ
符號說明………………………………………………………………..Ⅶ
表目錄…………………………………………K………………….….Ⅸ
圖目錄…………………………………………………………………..Ⅹ



第一章 緒論
1-1 研究動機………………………………………………….…1
1-2 文獻回顧……………………………………………….……3
第二章 物理模式與數值方法
2-1 物理模式………………………………………………….12
2-1.1等位函數法（Level set method）…………………12
2-1.2主導方程式…………………………………………13
2-2 AMR簡介………………………………………….……..16
2-3 數值方法……………………………………………….…..20
2-3.1有限體積法（Finite volume method）…………...20
2-3.2等位函數之離散…………………………………….26
2-3.3重新起始化方程式之離散………………………….31
2-3.4能量方程式在界面之處理………………………….34
2-3.5求解界面速度……………………………………..36
2-3.6總結模式之假設………………………………..…41
2-3.7總結計算流程………………………………………41
第三章 結果與討論
3-1測試問題………………………………………………...…42
3-1.1 穩態測試..…..………….…………………………43
3-1.2 一維凝固問題….……………………………….45
3-1.3 二維穩定固化問題（stable solidification）...52
3-2非穩定固化問題（unstable solidification）……….…64
3-2.1 界面曲率之驗証……………………………..64
3-2.2 等向性晶體成長（isotropic）測試網格效應.68
3-2.3樹枝狀晶體成長模擬…………………………74
3-3與相場模式（Phase field method）比較………...78
3-3.1一維測試….………………………………..78
3-3.2樹枝狀晶體成長….………………………..80
第四章 結論………………………………………………………..…..82
參考文獻……………………………………………………………….84

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