Implicit / Explicit Type Lie-Group Scheme of High Order for Solving the Nonlinear Backward Heat Conduction Problems

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Project title

Code/計畫編號

MOST105-2221-E019-040

Translated Name/計畫中文名

高階顯式與隱式李群積分法求解非線性反算熱傳導問題

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=11898071

Year

2016

Start date/計畫起

01-08-2016

Expected Completion/計畫迄

31-07-2017

Bugetid/研究經費

427千元

ResearchField/研究領域

機械工程

"本計劃擬提出高階顯式與隱式李群積分法求解非線性反算熱傳導問題。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，本計劃擬透過變數轉換，引入擬時間變數產生獨立原熱傳導方程式之擬時間軸座標系統，將原本非線性與非齊次熱傳方程式轉變為演化型式之熱傳方程式。本計晝相較於反算熱傳問題於閩式空間計算，保群算法為了滿足限制條件:光錐構造、李代數與李群特性，求解的過程中變數轉換之參數，擬時間步長與黏性阻尼係數在計算過程中屬於非線性行為，造成使用之參數無法將其線性化，導致解的收斂情況高度相依於黏性阻尼與擬時間步長而無法收斂。因此，針對閔式空間下之無法克服之情況，本計畫擬先將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。此外，本計畫將利用相同群的特性降低隱格式李群積分法疊代次數。最重要的是，利用兩種方法可以將困難之非線性反算熱傳導問題變成線性問題，不用挑選黏性阻尼、擬時間步長與擬時間終止等參數。最後本計晝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 scheme for solving multidimensional nonlinear backward heat conduction problems. If the existence of a solution for an inverse heat transfer problem can be assured based on physical reasoning, the requirement of uniqueness can only be formally proved for some special cases. Also, the inverse problem solution is highly dependent on input data. Small perturbations in the input data, like random errors inherent to the measurements used in the analysis, can cause large oscillations on the solution. In order to this problem, this project will use the transformation formula and obtain a fictitious time coordinate system by introducing a fictitious time variable. The original nonlinear and nonhomogeneous heat conduction equation can be transformed into a new heat conduction equation of an evolution type. Because the group preserving scheme (GPS) in Minkowski space must satisfy the constraint of the cone structure, Lie group and Lie algebra, the constraint ones for the fictitious time and viscosity-damping coefficient cause nonlinear behaviour and cannot use linear technique to avoid occurrence. The solutions highly depended on the fictitious time and viscous damp cannot easy convergence. Therefore, a new heat conduction equation will be calculated in Euclidean space, and the high-order implicit / explicit type Lie-group scheme will be developed that use a general linear group to combine with Runge-Kutta method. 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 additional, this project will use the same property of the group to reduce iterative number of the explicit type Lie-group scheme. More importantly, for this very difficult BHCP, the non-linear problems by using both schemes become linear problem, without choosing the parameters such as the viscosity-damping coefficient, fictitious time step and the fictitious terminal time. 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
疊代正則法
Heat conduction equation
Inverse problem
Ill-posed problem
Iterative regularization method

DSpace CRIS

Implicit / Explicit Type Lie-Group Scheme of High Order for Solving the Nonlinear Backward Heat Conduction Problems

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