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標題: 以逆算法預測細長圓柱之時變熱源
Application of the inverse method to estimate the time-varying heat sources on infinite cylinders.
作者: 王俊雄
Wang, Chun-Hsiung
關鍵字: inverse problem
heat conduction
infinite cylinders
heat source
出版社: 應用數學系所
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摘要: 本論文係以逆算法去預測細長圓柱在不同時間及位置的熱源。在方程式中,先將時間及位置做等距離切割,再以二階有限差分方法將其離散化,把已知的熱傳導係數、邊界條件、初始條件及各位置各時間點量測溫度值代入方程式中,將方程式矩陣化再逆算未知的熱源。其中各位置各時間點的量測溫度是有誤差的,文中量測誤差以解析解加上1%、3%、5%的隨機誤差當作量測溫度,再以此量測溫度去逆算熱源,發現此預測值比熱源正解有較大誤差,故使用數位濾波法分別以七點及十一點加權平均將溫度量測值平滑化,再代入方程式逆算未知熱源,在同一時間點的條件下,與之前未平滑化得到之預測熱源值比較,發現經平滑後的熱源值比未平滑化之熱源值更精準。 另外,研究也發現當量測點數越多時,可發現搭配平滑點數十一點的準確度會比七點的準確度較好。如果量測點數少時,平滑點數選取七點去平滑即可。
This paper estimates the heat source of infinite cylinders at different time and in various locations by inverse method. First, equally divide time and space in the formula and discretize them using the second-order finite difference method. Next, substitute into the quantities of the known thermal conductivity, boundary conditions, initial conditions, as well as the measured temperatures of every location and time. However, there are errors in the measured temperatures of every location and time. In this paper, the measured temperatures are the results of adding random errors of 1%, 3%, and 5% to the analytical solution, respectively. Using these noisy measured temperatures to inverse calculate the heat sources results in more error in the estimated values than the exact heat source solution. Consequently, Savitzky-Gollay digital filter method is adopted to smooth the measured temperatures by weight averaging seven or eleven measured points, respectively, prior to substituting the values into the discretized equations for the inverse calculation. The result demonstrates that comparing to the unsmoothed estimated heat source value, the smoothed heat source value is much more precise. In addition, the research finds that Savitzky-Gollay digital filter method is more effective to estimate the heat source when there are multiple measured points. In the digital filter method, inverse calculation gives more accuracy after smoothing eleven measured points than seven measured points. Nevertheless, if the amount of measured points is not large, smoothing seven measured points would be accurate enough.
其他識別: U0005-1507201114055600
Appears in Collections:應用數學系所



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