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研究生：	彭勳章 Peng, Hsiun-Chang
論文名稱：	從澆注到凝固之鑄造過程的可變時間步伐數值分析 Numerical Modeling of Casting Process from Mold Filling to Solidification by a Modified Truncation-Error-Based Adaptive Time Stepping Control Strategy
指導教授：	趙隆山 Chao, Long-Sun
學位類別：	博士 Doctor
系所名稱：	工學院 - 工程科學系 Department of Engineering Science
論文出版年：	2010
畢業學年度：	98
語文別：	英文
論文頁數：	119
中文關鍵詞：	鑄造、凝固、可變時間步伐、截尾誤差
外文關鍵詞：	casting, solidification, adaptive time step, truncation error
相關次數：	點閱：97 下載：1
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可變時間步伐方法在數值上可提供以一具效率、高精確度之方式模擬各種暫態問題，然而如何預測適當的時間步伐而有效對凝固過程之暫態特徵進行模式化分析具高困難度，一些常用的可變步伐方法如Backward Euler (BE) 法、Crank-Nicolson (CN) 法及Gresho-Lee-Sani (GLS) 法等，做為時間差分技術配合等效比熱法之模式計算時，均無法有效掌握相變中之潛熱釋放，以致無法獲得具精確效率的模擬結果。
因此，本研究提出以一修正局部時間結尾誤差 (LTE) 為基礎的調步方法，其可在完整凝固過程中整合相變潛熱、純液相及純固相變化預測所需之時間步伐大小，如此潛熱釋放效應可精確地被模式化符合能量守恆，所計算的溫度場結果的準確性亦隨之增進。本研究透過模擬一維與二維典型具不同相變特性及不同邊界條件的凝固熱傳問題，比較各種可變時間步伐方法（含所提出方法）與固定時間步伐方法所計算結果之準確度與效率，相關問題之收斂性分析本文亦有所討論，另透過配合熱焓法之模式計算，其結果進一步確認所提方法對凝固過程進行模式模擬之可行性。
當所提出之方法成功應用於非線性凝固過程時，本研究亦應用類似觀念處理其他具非連續性表面特徵之連續暫態自由液面流之相關問題，如水壩崩流問題與矩形模之澆注問題，透過與其他方法及文獻結果之比較，確認本法之應用彈性。最後，配合其他流場、自由液面、溫度場及凝固模式之設定，在時間計算與求值上，本研究利用此法模擬分析從澆注階段到凝固結束之完整鑄造過程。

Adaptive time step methods provide a computationally efficient means of simulating transient problems with a high degree of numerical accuracy. However, choosing appropriate time steps to model the transient characteristics of solidification processes is difficult. As time discretization techniques, several commonly applied adaptive time step methods, such as Backward Euler (BE) method, Crank-Nicolson (CN) method and Gresho-Lee-Sani (GLS) strategy, fail to bring an accurate and efficient result while the apparent heat capacity method is applied to model the latent heat effects associated with phase change.
The current study develops a modified local time truncation error (LTE)-based strategy designed to adaptively adjust the size of the time step throughout the whole simulated solidification procedure in such a way that the time steps can be adapted in accordance with the local variations in latent heat released during phase change or the effects of pure conduction in a single solid or liquid phase, the effects of latent heat release are more accurately modeled and the precision of the computational solutions is correspondingly improved. The computational accuracy and efficiency of the proposed method are compared with those obtained using other adaptive and uniform time step methods and demonstrated via the simulation of several one-dimensional and two-dimensional classical thermal problems characterized by different phase change phenomena and boundary conditions. Convergence analyses are also discussed. The feasibility of the proposed method for the modeling of solidification processes is further verified via its applications to the enthalpy method.
Since the proposed scheme is successfully applied to non-linear solidification processes, the similar concept is also applied to other continuous transient free-surface flow problems with surface indicator of discontinuous characteristics, such as the broken dam problem and the filling process of a rectangular mold, ensuring the application flexibility for the proposed scheme. Furthermore, a complete casting problem from the filling stage to the end of solidification is modeled using the proposed adaptive time step method.

摘要	I
ABSCRACT 	II
TABLE OF CONTENTS	III
LIST OF TABLES	VII
LIST OF FIGURES	VIII
LIST OF SYMBOLS	XIII
CHAPTER ONE: OVERVIEW	1
1 Introduction	1
2 Numerical methods for solution of phase change problems	3
2.1 Apparent heat capacity method	3
2.2 Enthalpy method	4
2.3 Source-based method	5
2.4 Temperature recovery method	7
3 Challenge of fixed grid methods and the corresponding treatment	7
4 Motivation and objective	10
CHAPTER TWO: ADAPTIVE TIME STEP METHODS	12
1 Introduction	12
2 Backward Euler (BE) method (first order scheme)	14
2.1 Predictor (FE) and Corrector (BE)	14
2.2 LTE and time step selection	15
3 GLS predictor-corrector strategy (second order scheme)	16
3.1 Predictor (AB) and Corrector (TR)	16
3.2 LTE and time step selection	17
4 Crank-Nicolson method (second order scheme)	18
5 Modified LTE estimate	18
5.1 Novel technique for determining text	20
5.2 Time step selection based on text	21
5.3 Order of accuracy	21
6 Concept of extreme time step	22
CHAPTER THREE: MATHEMATICAL FORMULATIONS OF PHYSICAL PROBLEMS	24
1 One-dimensional linear heat conduction problem without phase change	24
2 One-dimensional nonlinear phase change problems	24
3 Two-dimensional phase change problem	27
4 Two-dimensional free-surface flow problems	30
5 Two-dimensional rectangular casting problem	34
CHAPTER FOUR: IMPLEMENTATION OF PROPOSED ADAPTIVE TIME-STEP CONTROL SCHEME	36
1 Initialization of text	36
2 Controlling the magnitude of text during phase change	36
3 Fully-implicit computation for solidification processes	38
4 Fully-implicit computational scheme for apparent heat capacity method	40
5 Computational process for a casting problem	43
CHAPTER FIVE: ANALYSIS FOR ONE-DIMENSIONAL LINEAR HEAT CONDUCTION AND NON-LINEAR PHASE CHANGE PROBLEMS	45
1 Linear heat conduction problem without phase change	45
1.1 Total error comparison	46
1.2 Investigation for Time-step history and ||LTE||	46
2 Isothermal solidification problem involving latent heat (Stefan problem)	49
2.1 Total error comparison	49
2.2 Investigation for time step history and ||LTE||	50
2.3 Calculation for latent heat released	51
2.4 Convergence analysis	52
2.5Specifying suitable value of f depending on magnitudes of  and Ste	54
2.6 CPU time test	54
2.7 Modeling of solidification processes using enthalpy method	57
3 Isothermal solidification problem involving latent heat (Neumann’s Problem)	59
3.1 Total error comparison	60
3.2 Investigation for time step history and ||LTE||	60
3.3 Calculation for latent heat released	61
3.4 Convergence analysis	61
3.5Specifying suitable value of f/sel depending on magnitudes of 2and Ste	67
3.6 Specification of suitable value of f/adj	67
3.7 CPU time test	67
CHAPTER SIX: ANALYSIS FOR TWO-DIMENSIONAL PHASE CHANGE PROBLEMS	70
1 Isothermal solidification problem with fixed temperature boundary condition (Rathjen’s problem)	70
1.1 Results obtained by DI process and SI process	71
1.2 CPU time test	73
1.3 Modeling of solidification processes using enthalpy method	75
2. Isothermal solidification problem with convective boundary condition	77
CHAPTER SEVEN: ANALYSIS FOR TWO-DIMENSIONAL FREE-SURFACE FLOW PROBLEMS	79
1 Broken dam problem	79
1.1 Results of uniform time step schemes	79
1.2 Results of adaptive time step schemes	80
2 Filling process of a casting problem	86
2.1 Grid-size test	86
2.2 Results of adaptive time step schemes	87
CHAPTER EIGHT: ANALYSIS FOR TWO-DIMENSIONAL SOLIDIFICATION PROCESSES OF RECTANGULAR CASTING PROBLEM	93
1 Results of Fr<1	93
2 Results of Fr>1	97
CHAPTER NINE: CONCLUSIONS AND SUGGESTIONS	106
1 Conclusions	106
2 Suggestions	108
REFERENCE	110
                                    

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