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Applications of the Lie-Group Shooting Method on Solving Inverse Heat Conduction Problems
|Contributors: ||NTOU:Department of Systems Engineering and Naval Architecture|
Inverse heat conduction problem;Lie-group shooting method;ill-posed problem;semi-discretization
|Issue Date: ||2011-06-28T08:19:58Z
|Abstract: ||摘要:本研究計畫擬發展出一套數值方法求解由待定邊界熱傳方程式所統御的熱傳導反算問題。該問題係屬病態的，乃是因為其解若存在，而其解卻不連續。雖有其他學者提出許多做為改善此種病態問題之方式，卻未能有效改善。本研究首先擬針對此病態問題考慮採用半離散化方法的穩定性來分析，然後再將常微分方程的時間部份以差分方式離散，並進一步應用李群打靶法做為空間方向的數值積分來找到未知初始條件，此方法主要的重點是基於一步李群元素G(T)的建立及廣義中點李群元素G(r)的組成，然後由加入G(T) = G(r)條件，吾人可藉由權重因子1) ,0(∈r的極小解來搜尋缺少的初始條件。由於這個新的計算方法其數值邏輯是建立在堅實的理論基礎之上，預期將可使吾人對此反算熱傳導問題有一更深入的了解。而本法與其它數值方法對該問題解的準確性將進行比較。最後，吾人將設計數個數值算例，特別是針對帶有噪訊值干擾的情況對解的穩定性、量測誤差與量測位置不同對預測邊界之影響等主題，進行深入之探討與解析，也將就該方法之穩定性進行分析討論，藉此來驗證我們所提出方法是否可有效地解決上述諸多的問題。|
abstract:In this research, the inverse heat conduction problem governed by sideways heat equation will be investigated numerically. The problem is ill-posed because the solution, if it exists, does not depend continuously on the data. Even many researchers have proposed lots of methods to overcome the ill-posed problem; however, an effective numerical scheme to tackle the problem is still not available. To begin with, this ill-posed problem is analyzed by considering the semi-discretization numerical schemes and then, the resulting ordinary differential equations at the discretized times are numerically integrated towards the spatial direction by using the Lie-group shooting method to find the unknown initial conditions. The key point is based on the erection of a one-step Lie group element G(T) and the formation of a generalized mid-point Lie group element G(r). Then, by imposing G(T) = G(r) we can seek the missing initial conditions through a minimum discrepancy of the target in terms of the weighting factor .1) ,0(∈r Since this new computational algorithm is based on a concrete theoretical foundation, it can lead to a deeper understanding of the inverse heat conduction problem. Comparisons of the proposed method with other available numerical methods will be made. Several numerical examples for discussing how measurement errors or different measuring positions influence the predicted boundary data will be designed. Besides, some discussions of noise data on the accuracy of the numerical solution will be included. Finally, the numerical instability of the proposed method will also be investigated such that it can further verify the wideness and effectiveness of the present method.
|Appears in Collections:||[系統工程暨造船學系] 研究計畫|
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