Application of High-Order Implicit / Explicit Lie-Group Adaptive Method on Solving the Nonlinear Backward Heat Conduction Problems

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Project title

Code/計畫編號

MOST106-2221-E019-056

Translated Name/計畫中文名

應用高階顯式與隱式李群自適法求解時間反向非線性熱傳導問題

Project Coordinator/計畫主持人

Yung-Wei Chen

Funding Organization/主管機關

National Science and Technology Council

Department/Unit

Department of Marine Engineering

Website

https://www.grb.gov.tw/search/planDetail?id=12226566

Year

2017

Start date/計畫起

01-08-2017

Expected Completion/計畫迄

31-07-2018

Bugetid/研究經費

518千元

ResearchField/研究領域

機械工程

"本計劃擬提出高階顯式與隱式李群自適法求解時間反向非線性熱傳導問題。此一反算問題的困難在於反算問題的解若存在，會高度依賴給定的資料，特別是當量測資料有微小擾動時，將造成很大的震盪在於求解上。針對此一問題，本計劃將初始條假設為一未知函數，將求解此未知函數透過變數變換求解空間相依熱源 H(x)在T (x, t) T (x, t) H(x) t xx   。然後透過高階顯式與隱式保群算法積分，求解一線性系統。本計畫相較於時間反算熱傳問題於閩式空間計算，保群算法必須滿足限制條件:光錐構造、李代數與李群特性，閩適空間下微分流形無法高精度且穩定投影在其積分路徑上；導致一步李群元素G(t) 不相等廣義中點李群元素G(r)，所以閩氏空間保群算法結合李群自適法LGAM 無法任意選取積分時間步長及增加空間離散點問題。因此，針對閔式空間下之無法克服之情況，本計畫擬先將時間反算熱傳導問題轉換至歐式空間下計算，再利用廣義線性群結合 Runge-Kutta 法，發展高階顯式與隱式保群算法。如此一來，可讓有限維度李群作用於局部座標系統時，每一時間步長滿足於微分流行上，同時，歐式空間下李群元素相等廣義中值原理點李群元素G(t)= G(r)。最重要的是，本計畫將進一步利用相同群的特性降低隱格式李群積分法迭代次數在「時間」與「空間」方向積分。最後本計畫將與相關文獻上的方法做比較，測試演算法計算之穩定性與精度，特別是在有量測噪音擾動下對計算結果的影響，來驗證本計劃所提出之方法的正確性與可行性。""This project will develop the high-order implicit / explicit type Lie-group adaptive method (LGAM) for solving nonlinear backward heat conduction problems. If the existence of a solution for an inverse heat transfer problem can be assured owing to physical reason, the requirement of uniqueness can only be formally proved for some special cases. Also, the solution of the inverse problem is highly dependent on the input data. Small perturbations in the input data, like random errors inherent to the measurements in the analysis, can cause large oscillations on the solution. In order to solve this problem, this project assumes a function for the unknown initial condition and resolves this problem for estimates a spatially-dependent heat source H(x) in Tt(x, t) = Txx(x, t) + H(x). The high-order implicit / explicit LGAM is applied to find H(x) by integrations or by solving a linear system. Because the conventional group preserving scheme (GPS) in Minkowski space must satisfy the constraints of the cone structure, Lie-group and Lie algebra, inverse problems by the conventional GPS cannot easily satisfy the he constraint ones especially when the time step or mesh-grid in spatial is chosen; the conventional GPS in Minkowski space cannot easily preserve the integrated path on the manifold and cause the numerical instability. Further, a one-step Lie group element G(t) cannot be equal to the formation of a generalized mid-point Lie group element G(r) by the LGAM. Therefore, a new heat conduction equation will be conducted in Euclidean space and the high-order implicit / explicit type Lie-group scheme which uses a general linear group combining with Runge-Kutta method will be developed. It is very important that one wants to establish a local coordinate on a smooth manifold on which a finite dimensional Lie-group is acting. In addition, a Lie group element G(t) in Euclidean space must be equal to G(r) by the LGAM. More importantly, for the different integral direction of BHCP, this project will use the same property of the group to reduce iterative number of the explicit type Lie-group scheme. The accuracy and efficiency of the present method will be validated by comparing the estimating results with previous literatures even under noisy measurement data."

Keyword(s)

熱傳導方程式
逆問題
病態問題
疊代法
Heat conduction equation
Inverse problem
Ill-posed problem
Iterative regularization method

DSpace CRIS

Application of High-Order Implicit / Explicit Lie-Group Adaptive Method on Solving the Nonlinear Backward Heat Conduction Problems

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