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研究生(外文):Chiun-Chuan Su
論文名稱(外文):Solving Groundwater Seepage Problems Using the Fictitious Time Integration Method
指導教授(外文):Cheng-Yu KuShuh-Gi Chern
外文關鍵詞:SeepageFictitious Time Integration methodFinite difference methodGroundwaterNumerical method
  • 被引用被引用:5
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本研究主要是以加入一個虛擬時間維度的新方法,由劉進賢教授與Satya N. Atluri教授於2008年所發展的擬時間積分法,並配合由顧承宇教授等人於2009年所發展之新的虛擬時間函數做數值運算,此方法不但較傳統常用之牛頓法有效率,尤其在解病態問題方面更是有效,因為此方法不需計算Jacobian 矩陣,使得數值方法對於求解非線性問題上有更好的處理空間,並結合有限差分法及時間項之差分應用於地下水滲流的推導,藉此能更精確的求出各種地下水滲流問題之現象與描述。
In recent years, due to the rapid development of the computer technology and academic studies, the knowledge for engineering applications is improved significantly. Therefore, the use of latest technology and knowledge to solve engineering problems is an important issue.
The purpose of this study is to solve groundwater seepage problems using the latest developed method, named the Fictitious Time Integration Method (FTIM). The FTIM was first used to solve a nonlinear system of algebraic equations by introducing fictitious time (Liu and S. N. Atluri, 2008), such that it is a mathematically equivalent system in the augmented n+1-dimensional space as the original algebraic equation system is in the original n- dimensional space. The fixed point of these evolution equations, which is the root for the original algebraic equation, is obtained by applying numerical integrations on the resultant ordinary differential equations, which do not require the information of derivative of nonlinear algebraic equation and their inverse. Since the FTIM has the advantages that it does not need to calculate the Jacobian matrix and its inverse and is thus very time saving, it has great potential for solving groundwater seepage problems.
In this study, the FTIM is incorporated with the finite difference method for solving groundwater seepage problems. Several applications, including two- dimensional confined and unconfined aquifer problems with the consideration of homogenous and non-homogenous, isotropic and anisotropic, and steady-state and transient conditions using the FTIM are conducted. Results obtained demonstrate that with the ease of numerical implementation, the FTIM can easily deal with groundwater seepage problems and has high efficiency as well as high accuracy. In addition, the numerical method developed in this study can also correctly calculate groundwater seepage problems with very large difference of material permeability without any special numerical treatment.

摘要 Ⅰ
Abstract Ⅱ
目錄 Ⅳ
表目錄 Ⅶ
圖目錄 Ⅷ
符號表 XI
第一章 緒論
1-1 前言 1
1-2 研究動機與目的 2
1-3 研究內容 3
第二章 文獻回顧
2-1地下水滲流數值方法發展 5
2-2擬時間積分法發展 8
2-3擬時間積分法應用 9
第三章 地下水滲流之理論介紹
3-1 地下水滲流之理論介紹 11
3-1.1 地下水滲流之控制方程式 14
3-1.2 侷限含水層 20
3-1.3 非侷限含水層 21
3-2 地下水滲流之有限差分理論推導 22
3-3 地下水滲流控制方程式時間項之差分推導 24
第四章 地下水滲流之數值方法
4-1 擬時間積分法 28
4-1.1 擬時間積分法之推導 29
4-1.2 新的擬時間函數 32
4-2擬時間積分法結合有限差分應用於地下水滲流數值方法推導 34
4-2.1 非均質之地下水滲流數值方法推導 34
4-2.2 非均質非等向之地下水滲流數值方法推導 36
4-2.3 非侷限含水層之地下水滲流數值方法推導 41
4-2.4 暫態之地下水滲流數值方法推導 43
第五章 地下水滲流數值模式之驗證與應用
5-1 非均質之地下水滲流模式應用 56
5-1.1 穩態之均質等向侷限含水層數值案例 56
5-1.2 穩態之非均質等向侷限含水層數值案例 59
5-1.3 穩態之非均質等向有限差分法數值案例-以深開挖為例 63
5-1.4 穩態之非均質等向數值案例-以深開挖為例 66
5-2 非等向之地下水滲流模式應用 70
5-2.1 穩態之非均質非等向侷限含水層數值案例 70
5-3 非侷限含水層之地下水滲流模式應用 75
5-3.1 穩態之均質等向非侷限含水層數值案例 75
5-3.2 穩態之非均質非等向非侷限含水層數值案例 78
5-4暫態之地下水滲流模式應用 82
5-4.1 暫態之均質等向侷限含水層數值案例 82
5-4.2 暫態之非均質非等向侷限含水層數值案例 85
5-4.3 暫態之均質等向非侷限含水層數值案例 89
5-4.4 暫態之非均質非等向非侷限含水層數值案例 92
5-5 時間項差分之案例驗證 97
5-5.1 暫態之隱式法熱傳導數值案例 97
5-5.2 暫態之Crank-Nicolson法熱傳導數值案例 101
5-6擬時間函數之案例驗證 104
5-6.1 擬時間積分法之數值案例 104
5-6.2 新擬時間函數之數值案例 107
5-7三維之地下水滲流模式應用 110
5-7.1 穩態之均質等向侷限含水層數值案例 111
5-7.2 穩態之非均質非等向侷限含水層數值案例 115
5-7.3 暫態之均質等向侷限含水層數值案例 120
5-7.4 暫態之非均質非等向侷限含水層數值案例 124
第六章 結論與建議
6-1 結論 130
6-2 建議 130
參考文獻 132

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