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研究生:李鴻昌
研究生(外文):Hung-Chang Lee
論文名稱:修正保群算法在工程上之應用
論文名稱(外文):A Modified Group Preserving Scheme and Its Application to Engineering Problems
指導教授:陳朝光陳朝光引用關係洪振益洪振益引用關係
指導教授(外文):Cha''''o-Kuang ChenChen-I Hung
學位類別:博士
校院名稱:國立成功大學
系所名稱:機械工程學系
學門:工程學門
學類:電資工程學類
論文種類:學術論文
論文出版年:2001
畢業學年度:90
語文別:中文
論文頁數:324
中文關鍵詞:修正保群算法閔氏空間Burgers'''' 方程式Cayley 變換Lie 群Lie 代數
外文關鍵詞:Modified Group Preserving SchemeMinkowski spaceBurgers'''' equationCayley transformationLie groupLie Algebra
相關次數:
  • 被引用被引用:2
  • 點閱點閱:491
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  • 下載下載:0
  • 收藏至我的研究室書目清單書目收藏:1
本文主要在探討修正保群算法在常見的工程問題,包括剛性初始值、邊界值、結構動力、熱傳導問題。修正保群算法主要是由保群算法推導的,而保群算法理論是將一般動態系統先建立在閔氏空間(Minkowski space)中,再應用Cayley映射組成Lie群變換,且同時它具有保持圓錐條件,而得到一個單步顯式時間積分算法。這些算法只能解線性或非線性初始值問題。當狀態變數範數趨近於零,或載荷函數範數(Norm)太大,將造成保群(GP)算法失效或產生求解困難,此時我們引入平移變換,而提出所謂修正保群(MGP)算法。重新去解此類問題以及Burgers’方程式問題的求解與應用。
傳統上的數值分析為了表示所求數值解是否真正的問題解,很重要的是進行一些定性性質分析,必須要具有一致性(Consistency)、穩定性及收斂性的要求,才能確定所求數值解為真正解(True solutions),亦即我們所提出之算法也必須滿足這些準則。應用MGP算法主要必須滿足三個限制條件:調節因數(Adaptive factor)、半步長和圓錐的限制條件。事實上在非常小的固定步長情況下,只需要連續監督調節因數數值大約在1.0附近,代表算法有效;它是在每一時間步驟產生下一步驟解的顯式算法。當正確解未知時,傳統顯式數值方法並沒有較可信賴的訊息,確定解的精度。主要本文研究中,應用兩個增量去估計常微分方程式解的誤差,結果顯示誤差的精度與步長同階;同時經由上述分析結果,也可由調節因數產生大跳躍(Jump)和小跳躍,即可分別預知解的振幅和相位角較大變化的位置。MGP算法對於具有剛性比(Stiffness ratio) 之強剛性初始值問題,顯示計算結果非常好。
結合打靶法可將MGP算法推展至邊界值問題,一般打靶法並無系統化假設猜丟失初始條件的數值。本文將提出有系統對丟失初始值猜值理論(見附錄G)。結合數值線法及MGP算法求解熱傳導及具有 (即雷諾數 =10000)的Burgers’方程式,顯示計算結果與其他數值解非常吻合。理論上,MGP算法較傳統解微分方程式數值方法更具普遍性與實用性。經由許多算例,結果顯示皆非常良好。我們可以確信該算法的計算有效性和精確度。
Abstract
A new numerical method, modified group preserving (MGP) scheme, can be applied to stiff initial-value, boundary value, structural dynamics and heat transfer problems for engineering applications. Generally speaking, MGP scheme stems mainly from GP scheme. The concept of group preserving (GP) scheme preserves the cone structure in the Minkowski space, has been developed by employing the Cayley map to form a Lie group transformation, and providing an explicit one-step time integrator for the general dynamical systems. This scheme has been proposed only for the solution on the initial value problems of linear or nonlinear problems. However, a modified group preserving scheme (MGPS) is proposed by considering the translation of the state variables to resolve the difficulties of small time step problem in solving the initial value problems with stiffness, the norm of state variables is near to the zero point or the norm of forcing function is very large. In this study, Burgers’ equation and its application are proposed.
Since the purpose of traditional numerical analysis is to represent the solution to actual problems, it is important that qualitative properties of the numerical solution should be consisting with consistency, stability and convergence in order to resemble those of the true solutions. Thus the present of qualitative properties of MGP scheme is also satisfy those criteria. In practical applications of MGP scheme are valid to show that three constraints, adaptive factor, half step size and cone constraint, are satisfied. Indicating continuously only monitor the adaptive factor takes about 1.0 for a very small value of fixed step size. It is yield a time forward explicit scheme for the next step solution in each time step. Therefore, each solution is not reliable to have information on the accuracy of solutions in traditional explicit numerical methods when the exact solution cannot be obtained. In the present study, employing two increment quantities to estimate the error of numerical solutions for ordinary differential equations, it shown the same order results of errors and step sizes. The important feature in the above analysis, the value of adaptive factor is large jump or small jump, indicating the location of large change for the amplitude and phase angle of response, respectively. This can further prove that the present scheme is effective in calculating strongly stiff initial value problems with stiffness ratio 10E+6.
The MGP scheme is extended to solve the boundary value problems by a shooting method. It has not a systematic way to assume the missing initial values. In the present study, a systematic way of finding assumed of the missing value of initial condition is proposed (see appendix G). Combining use of numerical method of lines and MGP scheme to heat transfer problems and Burgers’ equation with Reynolds number of 10000. Results consistently agree with the numerical solution is quite satisfactory. Theoretically, the MGP scheme in more universal and practical than that of traditional numerical methods for solving differential equations. Several numerical results is presented, it shows that the MGP scheme works very well, and the merits of the computational efficiency and accuracy of the method can be confirmed.
目 錄
中文摘要………………………………………………………………………I
英文摘要……………………………………………………………………III
誌謝…………………………………………………………………………V
目錄…………………………………………………………………………VI
表目錄………………………………………………………………………XI
圖目錄………………………………………………………………………XIII
符號說明……………………………………………………………………XX
第一章 緒論…………………………………………………………………1
1-1 研究目的與背…………………………………………………………1
1-2 文獻回顧………………………………………………………………4
1-3 本文架構………………………………………………………………18
第二章 修正保群算…………………………………………………………21
2-1 前言……………………………………………………………………21
2-2 常微分方程式理論的基本觀念………………………………………23
2-3 增廣動態系統…………………………………………………………27
2-4 圓錐與路徑……………………………………………………………30
2-4-1 零圓錐(Null cone)………………………………………………30
2-4-2 類時間路徑(Time-like paths)是不允許的……………………31
2-5 用Cayley變換建立保群算法…………………………………………31
2-6 修正保群算法…………………………………………………………38
2-7 誤差及一致性、收斂性分析…………………………………………39
2-8 修正保群算法的特性…………………………………………………41
第三章 常微分方程式初始值問題之數值計算-剛性問題………………48
3-1 前言……………………………………………………………………48
3-2 剛性常微分方程式……………………………………………………49
3-3 範例計算與討論………………………………………………………53
3-4 結論……………………………………………………………………59
第四章 常微分方程式邊界值問題之數值計算……………………………77
4-1 前言……………………………………………………………………77
4-2 打靶法…………………………………………………………………78
4-2-1 數學模型…………………………………………………………78
4-2-2 非線性的打靶算法…………………………………………………79
4-3 範例計算與討論………………………………………………………81
4-4 結論……………………………………………………………………93
第五章 結構動態響應之應用………………………………………………109
5-1 前言…………………………………………………………………109
5-2 範例計算與討論……………………………………………………113
5-3 結論…………………………………………………………………128
第六章 一維非線性熱傳導問題之應用……………………………………164
6-1 前言…………………………………………………………………164
6-2導數的有限差分近似…………………………………………………166
6-2-1 一階導數的有限差分……………………………………………166
6-2-2 二階導數的有限差分……………………………………………168
6-3 熱傳導方程式………………………………………………………170
6-3-1 空間離散…………………………………………………………170
6-4 保群算法……………………………………………………………172
6-5 修正保群算法………………………………………………………175
6-6 範例計算與討論……………………………………………………176
6-7 結論…………………………………………………………………183
第七章 應用修正保群算法解Burgers’方程式…………………………203
7-1 前言…………………………………………………………………203
7-2 Burgers’方程式的正確解…………………………………………204
7-3 修正保群算法………………………………………………………205
7-3-1 空間座標等間隔離散……………………………………………205
7-3-2 空間座標非等間隔離散…………………………………………208
7-4 範例計算與討論……………………………………………………211
7-5 結論…………………………………………………………………212
第八章 綜合結論與建議……………………………………………………221
8-1 綜合結論……………………………………………………………221
8-2 未來研究建議………………………………………………………227
參考文獻……………………………………………………………………229
附錄A 常微分方程的數值解法……………………………………………248
A-1 初始值問題…………………………………………………………248
A-2 Euler 法……………………………………………………………248
A-2-1 各種改進Euler法………………………………………………255
A-3 Runge-Kutta 法……………………………………………………256
A-4 線性多步法…………………………………………………………257
A-5 各種數值方法的比較………………………………………………258
附錄B 微分幾何、Lie群、Lie代數與閔氏空間…………………………260
B-1 基本概念……………………………………………………………260
B-2 微分幾何……………………………………………………………265
B-2-1 拓撲空間、連續映射、同胚映射………………………………265
B-2-2 上的微分運算……………………………………………………268
B-2-3 流形及微分流形…………………………………………………271
B-3 Lie群和Lie代數……………………………………………………275
B-4 Lie氏定理及反定理………………………………………………280
B-5 閔氏空間……………………………………………………………282
B-5-1 絕對、相對時空觀念……………………………………………282
B-5-2 四維時空幾何學…………………………………………………285
B-5-3 四維閔氏時空空間………………………………………………287
B-5-4 事件的時序與因果關係…………………………………………290
B-5-5 (n + 1)維的閔氏空間 …………………………………………290
附錄C 證明增廣狀態係數矩陣A是Lie代數………………………………295
附錄D 證明正規正時洛倫茲群的三個性質………………………………298
附錄E 正規正時洛倫茲群及其Lie代數的性質……………………………302
附錄F 高階對角線Pade’逼近的保群算法[24]…………………………313
附錄G 打靶法中初始條件猜值理論………………………………………316
自述…………………………………………………………………………324
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