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研究生:許育賓
研究生(外文):Yu-BinHsu
論文名稱:以修正截尾誤差之可變時間步伐計算法分析相變化熱傳問題
論文名稱(外文):Analysis of Heat Transfer Problem of Phase Change by Using Truncation-Error-Based Adaptive Time Stepping Control ategies
指導教授:趙隆山
指導教授(外文):Long-Sun Chao
學位類別:碩士
校院名稱:國立成功大學
系所名稱:工程科學系專班
學門:工程學門
學類:綜合工程學類
論文種類:學術論文
論文出版年:2010
畢業學年度:98
語文別:中文
論文頁數:88
中文關鍵詞:可變時間步伐計算法GLSMGLS相變化
外文關鍵詞:Adaptive time step methodGLSMGLSPhase changeSolidification
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在凝固熱傳過程中,固態與液態之間的相變化是相當重要的物理現象,能夠節省計算時間的可變時間步伐計算法亦有不少種。本文針對兩個較為普遍的可變時間計算步伐來作探討,一個為GLS法,另一為修正 GLS(MGLS)法,是針對GLS法再進行修正所提出的計算法。本研究以這兩種可變時間步伐計算法搭配數種處理潛熱的數值方法,來求解不同凝固熱傳過程的溫度場分布,並對總時間步伐數進行分析比較。本文將介紹GLS法與MGLS法的計算理論,及處理潛熱效應的數值方法,包含等效比熱與熱焓法。所計算的凝固問題有一維史蒂芬問題、紐曼問題、二維瑞特延問題和二維鑄模之凝固問題,上述問題除了二維鑄模之凝固問題之外,其數值解皆與解析解進行準確性和誤差比較,再與固定步伐進行總步伐數的分析,而求得最合適的步伐數。分析的結果發現,在求解史蒂芬問題與紐曼問題時,熱焓法在處理潛熱效應時能提供最好的準確性,而MGLS法確實能比GLS法來得更有效率,提供較少的總步伐數。從這些分析結果,針對凝固熱傳問題,可發現本文的方法可以有效的提供精簡且合適的時間步伐,以在不犧牲準確度的前提下達到節省計算時間的目的。
Phase change is an important physical phenomenon in a solidification process. There are many adaptive time step methods proposed for saving the computing time. In the study, to analyze nonlinear phase-change or solidification processes with two numerical schemes of handing the latent-heat effect, two adaptive time step methods are chosen, one is the Gresho-Lee-Sani (GLS) predictor-corrector strategy and the other is the modified GLS (MGLS) method. The temperature fields of different solidification problems are solved and their total time step numbers are compared and analyzed. In the thesis, the computing algorithms of GLS and MGLS methods are presented and so are the latent-heat treating methods, including the effective specific heat and enthalpy schemes. The computational accuracy and efficiency of the two adaptive time step methods are demonstrated via the simulation of one-dimensional Stefan and Neumann phase-change problems, the two-dimensional Rathjen phase-change problem and the solidification problem of a casting process. In those problems except the two-dimensional one of casting process, the accuracy is compared by analyzing the differences between the numerical and exact solutions for the adaptive and uniform time step methods and so is the total step number. From the analyzing results, it can be found that the enthalpy scheme could give the more accurate solutions than the effective specific heat method for the Stefan and Neumann problems. The adaptive method could solve the problems more efficiently than the uniform one and the MGLS scheme could do better than the GLS one.
摘要 I
Abstract II
致謝 III
目錄 IV
圖目錄 VII
符號說明 XV
第一章 緒論 1
1.1 文獻回顧 2
1.2 研究方法與目的 4
第二章 可變時間步伐計算法之理論分析 6
2.1 GLS可變時間計算法之預測項與修正項理論分析 6
2.1.1亞當斯-巴希霍斯(Adams-Bashforth)預測項與梯形法則
(trapezoid rule)修正項 6
2.1.2 LTE(local time truncation error)和時間步伐選擇 7
2.2 修改LTE方法之分析 8
2.2.1 計算 9
2.2.2 針對 的時間步伐選擇 10
2.3 應用上的考量 10
2.3.1 初始化 10
2.3.2 調整 11
第三章 凝固熱傳問題數學模式與數值方法分析 12
3.1 暫態線性熱傳問題 13
3.2史帝芬問題(Stefan Problem) 14
3.3 求解史帝芬問題之數值方法 15
3.3.1等效比熱法(Effective specific heat method) 15
3.3.2 熱焓法(Enthalpy method) 18
3.3.3 初始條件設定分析 20
3.4 紐曼問題(Neumann Problem) 21
3.5 求解紐曼問題之數值方法 22
3.5.1 等效比熱法(Effective specific heat meth[od) 22
3.5.2 熱焓法(Enthalpy method) 23
3.5.3 初始條件設定分析 24
3.6 瑞特延(Rathjen)問題及二維鑄模之凝固問題 25
第四章 結果與討論 31
4.1 一維暫態熱傳問題 31
4.1.1 GLS數值解與解析解比較 31
4.1.2 GLS數值解與固定步伐之總步伐數比較 32
4.2 史帝芬問題(Stefan Problem) 32
4.2.1 等效比熱法搭配GLS與MGLS 32
4.2.2 等效比熱法搭配GLS與MGLS針對初始條件之分析 33
4.2.3 熱焓法搭配GLS與MGLS 34
4.2.4 熱焓法搭配GLS與MGLS針對初始條件之分析 35
4.3 紐曼問題(Neumann Problem) 35
4.3.1 等效比熱法搭配GLS與MGLS 35
4.3.2 等效比熱法搭配GLS與MGLS針對初始條件之分析 36
4.3.3 熱焓法搭配GLS與MGLS 37
4.3.4 熱焓法搭配GLS與MGLS針對初始條件之分析 38
4.4瑞特延(Rathjan)問題及二維鑄模之凝固問題 39
4.5 GLS法與MGLS法比較 39
4.5.1 等效比熱法求解史帝芬問題及紐曼問題 40
4.5.2 瑞特延(Rathjen)問題 40
4.5.3 二維鑄模之凝固問題 40
4.5.4 前兩個固定步伐大小之影響 41
4.5.5 人工液固區之影響 42
4.5.6 潛熱釋放分析 42
第五章 結論 83
參考文獻 84
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