New Approaches for Identifying the Heat Source in Heat Conduction Problems

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Project title

Code/計畫編號

NSC99-2221-E019-053-MY3

Translated Name/計畫中文名

熱源估算在熱傳導問題上的新計算方法

Project Coordinator/計畫主持人

Jiang-Ren Chang

Funding Organization/主管機關

National Science and Technology Council

Department/Unit

Department of Systems Engineering and Naval Architecture

Website

https://www.grb.gov.tw/search/planDetail?id=2385527

Year

2012

Start date/計畫起

01-08-2012

Expected Completion/計畫迄

31-07-2013

Bugetid/研究經費

759千元

ResearchField/研究領域

機械工程

本計劃擬提出一修正型的多項式展開法求解與時間相關的熱源估測問題，此一反算問題的困難在於反算問題的解若存在，卻不連續依賴於給定的資料，當量測資料有微小擾動時，會造成很大的計算誤差。針對此一問題，本計劃擬透過變數變換，消去未知的熱源函數部分，將原本的兩點邊界值問題，轉換為三點邊界值問題，亦即邊界條件並非單獨給在兩個邊界的函數值，而是三個點函數值的耦合。相較於傳統的兩點邊界值問題，一般方法很難處理此類邊界條件，消去未知的非齊性項後，本計劃擬使用多項式展開法同時做時間與空間部分的離散，將計算域內的解寫成多項式函數的線性疊加，並提出特徵長度的概念來改善傳統多項式展開法產生的劣性矩陣問題。多項式展開法乃屬於無網格法的一種，其優勢在於對於各種邊界條件都能適應，並能直接進行微分、積分的操作。將此三點邊界值的解經過微分操作後，便可得到所要求的熱源函數。對於離散後的線性代數方程系統，本計劃也提出新的純量同倫法來進行求解。相較於傳統的向量同倫法方程式與未知數數目必須相同的限制，純量同倫法也可以處理過定系統與欠定系統，這讓我們可以用較低階數的多項式配上較多的配點數來進行數值計算，避免高階基底造成的矩陣劣化，配合適當特徵長度的選取，相信可以提高求解的穩定性與精度。本計劃擬對於一維、二維的熱源估測問題，進行深入的研究與探討，包括數值計算的穩定性與精度，量測噪音擾動下對計算結果的影響，討論多項式展開階數、配點點數、特徵長度等參數的選取對於純量同倫法收斂速度與計算精度的影響，解不連續或高度振盪時數值結果的穩定性，並設計數個數值算例來驗證本計劃所提出之方法的正確性與可行性。"In this research, a modified polynomial expansion method will be developed to solve problems of identifying the heat source with time dependence, in which an inverse problem will be encountered. The difficulty of problems lies on that if the solution of this inverse problem exists, it will not be continuously dependent on the given data. If there are small disturbances existing in the measured data, huge amounts of errors will arise in the numerical calculation. Aimed at this problem, the variation of variables will be adopted to eliminate the unknown heat source function and an original two-point boundary value problem will be changed into a three-point boundary value problem. It means boundary conditions are not only given by functions at two boundaries but at a combination of three different points functions. As compared with the conventional two-point boundary value problem, the three-point boundary value problem is quite hard to be dealt with. After those unknown non-homogeneous terms being eliminated, the polynomial expansion method will be introduced to discretize the time and space fields, respectively. Then, the solution inside the field will be expressed as a linear superposition of polynomial functions. After that, a characteristic length concept will be adopted to resolve ill-posed matrix problems arising in those conventional polynomial expansion methods. The polynomial expansion method is, basically, a kind of meshless methods and its merits are: easily uses in various boundary conditions, can be directly put into differential and integral operations. After putting the solution of the three-point boundary value into differential operations, the desired heat source function can be obtained. As for the discretized linear algebra equation system, a new Scalar Homotopy Method (SHM) will be developed to solve. As compared with the conventional Vector Homotopy Method (VHM), the SHM can avoid the limit of the same equation and unknown numbers and can directly deal with under- or over- determined equation system. Thus, it makes us to adopt the lower order polynomials with more allocation points for numerical calculation and to avoid dealing with the ill-posed matrix owing to the higher order bases. By way of choosing an appropriate characteristic length, one can ensure the solution’s stability and accuracy. This three-year project will aim at problems of identifying the heat source of one- and two- dimensional heat conduction to study. Discussions and analyses on numerical stability and accuracy, influences on numerical results owing to the measured noise data disturbance, and the order of the expanded polynomials the number of allocated points will be conducted. Besides, influences of choosing characteristic length parameters on the numerical convergence speed and calculation accuracy for the SHM and the numerical instability due to the solution discontinuity or highly oscillation will also be discussed. Some numerical experiments with designed testing examples will be included in this project to validate the accuracy and effectiveness of the proposed approach."

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New Approaches for Identifying the Heat Source in Heat Conduction Problems

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