本文應用線性最小均方根誤差法(linear least-squares error method)結合逆向矩陣法(reversed matrix method)及虛擬區域(virtual area)的概念，來求解不同類型爐壁系統的穩態熱傳導逆向問題(inverse heat conduction problem, IHCP)。其過程係藉由爐壁內少數量測點的溫度來估測外壁表面的溫度分布及內壁邊界的幾何形狀。
首先，應用有限差分法(finite-difference method)將擴張後之虛擬爐壁系統的統御方程式及邊界條件離散化(discretization)並建構成線性矩陣方程式，藉由排列矩陣方程式使未知的虛擬邊界溫度及爐壁外表面溫度可用一行矩陣(column matrix)明確地表示出來。其次，利用線性最小均方根誤差法將問題最佳化，再以爐壁厚度內少數量測點的溫度代入此線性逆模型中，以求得虛擬邊界及爐壁外表面的溫度。最後，藉由直接解(direct solution)的計算方式，即可推估內壁邊界等溫線分布的位置，此曲線即為內壁邊界的幾何形狀。
本方法最大的優點在於分析熱傳問題時，可以避開傳統方法必須迭代的運算程序，不需初始猜測值的狀況下即可直接解出未知條件。所以計算上較一般傳統方法快速與精確。本文中利用直接問題(direct problem)所求得的溫度值來模擬實際的溫度量測，並考慮量測誤差大小對於估測結果的影響。由逆解結果顯示，預測的幾何形狀隨著量測點遠離預測邊界、減少量測點數目及增加量測誤差，將導致預測偏差有增加的趨勢。不過，在合理的量測誤差下，本方法於靠近外壁處以少數的量測點資料皆可順利推估爐壁的幾何形狀。
本文為了減少逆運算過程所需要的量測點數目，特別使用灰預測理論的決定性灰色動態預測模型(deterministic grey dynamic model)。該方法的主要目的係藉由少數點數的直接量測溫度值，推估得到更多點的間接量測溫度值。因此，雖然實際上僅執行少數點數的直接量測，但卻能達到增加量測點的實際效果，藉以提供更多有效的資料提高估測值的準確度。本文所提出的逆向矩陣法合併灰預測模型可應用於一維、二維甚至三維之問題，故此方法可成為研究逆問題的有效方法。

In this study, the inverse heat conduction problem (IHCP) for various furnace wall systems is solved by using the linear least-squares error and the reversed matrix method combined with a virtual area. The temperature distribution of the outer surface and the geometry of inner surface of the furnace wall were estimated from a few of measured temperature in the furnace wall.
In the estimation process, the finite-difference method is first used for the furnace wall system containing the virtual area to discretize the governing equation and boundary condition and to establish a linear matrix equation. The temperatures of unknown virtual boundary and outer surface of furnace wall can be represented by a column matrix in the linear matrix equation. Then the problem is computed by the linear least-squares error method and a few of measured temperature in furnace wall are put into the linear inverse model to obtain the temperatures of the virtual area boundary and the temperature of the outer surface of furnace wall. Finally, the location of isothermal curve for inner boundary of furnace wall is estimated by a direct process. The isothermal curve is represented the geometry of inner surface of furnace wall.
The advantage of this approach is that the unknown condition can be directly solved without the initially guessed temperatures and the iteration process for traditional method. Therefore, the calculation in this work is more efficient and accurate than traditional method. In this work, the temperatures obtained from direct problem are used to simulate the measured temperatures and the effect of measurement error on the estimated result is also considered. The result shows that the estimation error of geometry increased with increase in distance between measured points and inner surface and with increase in preset error, but with decrease in the number of measured points. However, the geometry of furnace inner surface could be successfully estimated by only the temperatures of a small number of measured points within and near the outer surface under reasonable preset error.
For reducing the number of measured points, a deterministic grey dynamic model in grey prediction theory is used in this study. It is used to obtain more values of temperature from the presented measurements. Therefore, the accuracy for estimation can promote due to more effective data. The inverse matrix method combined with grey prediction model proposed in this study can be applied to multi-dimensional problem. It is indeed a effective method for the study of inverse problems.

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