本文利用混合逆算法來探討逆向熱傳導問題。本文首先利用結合了拉氏轉換( Laplace transform )、有限差分( finite difference )及最小平方法( least-squares methods )之混合逆算法搭配試件內部之量測溫度來預測線性與非線性逆向熱傳導問題之未知表面條件( surface conditions )；求解非線性逆向熱傳導問題時，先以泰勒級數近似法( Taylor’s series approximation )對統制方程式中之非線性項進行線性化處理；本文之逆算法求解時，試件之未知表面條件的函數形式不需事先已知，在進行逆運算前先將整個時間域區分成多個小時間域，而後每個小時間域的預測值便可以利用本文之逆算法求得。
本文並以上述之混合逆算法結合三次仿樣曲線( cubic spline )來預測二維穩態與暫態逆向熱傳導問題之系統邊界上的熱傳係數( heat transfer coefficient )；本文之逆算法求解時，未知熱傳係數的函數形式不需事先已知，在進行逆運算前先將整個空間域區分成多個小區域，而後再以三次仿樣曲線來模擬穩態熱傳係數隨位置的變化情形；求解暫態逆向熱傳導問題時，本文將以多個連續之位置三次多項式函數與時間線性函數來模擬未知表面條件隨位置與時間的變化情形。
為了驗證本文混合逆算法的精確度與可靠度，預測結果將與正確值、參考文獻之估算結果及實驗數據比較，同時也將討論未知係數之起始猜測值、量測位置及量測誤差對預測結果的影響；結果顯示應用本文之逆算法確實可以求得良好的表面條件與熱傳係數預測結果，且起始猜測值與量測位置對預測結果的影響並不是很明顯，即使有量測誤差的影響，仍然可以獲得良好且穩定的預測結果；在求解二維逆向熱傳導問題時，由於本文之逆算法結合了三次仿樣曲線，所以不僅可以減少測溫點的數目，同時還可以增加預測結果的精確度。

This study applies the hybrid inverse method to investigate the inverse heat conduction problems. A hybrid inverse scheme involving the Laplace transform, finite difference and least-squares methods in conjunction with experimental data inside the test material is proposed to estimate the unknown surface conditions for the linear and nonlinear inverse heat conduction problems. The nonlinear terms in the differential equations of the nonlinear inverse heat conduction problem are linearized using the Taylor’s series approximation. The functional form of the surface conditions is unknown a priori. The whole time domain is divided into several analysis sub-time intervals and then the unknown estimates on each sub-time interval can be predicted.
The above hybrid scheme in conjunction with the cubic spline is applied to predict the distribution of the heat transfer coefficient on a surface exposed to a moving fluid in the two-dimensional steady and transient inverse heat conduction problems. The functional form of the heat transfer coefficient is unknown a priori. The whole spatial domain of the unknown heat transfer coefficient can be divided into several analysis sub-intervals. Later, a cubic spline is introduced to estimate the distribution of the steady heat transfer coefficient, a series of connected cubic polynomial function in space and a linear function in time are introduced to estimate the distribution of the unknown surface conditions for the transient inverse heat conduction problem.
In order to show the accuracy and reliability of the present inverse scheme, comparisons among the present estimates, exact solution, previous results and experimental data are made. The effects of the initial guesses, measurement locations and measurement errors on the estimated results are also investigated. The results show that good estimation on the surface conditions and heat transfer coefficient can be obtained. The estimated results are not very sensitive to the initial guesses and measurement locations. The present estimates exhibit stable behavior and slightly deviate from the exact solution in the cases with measurement errors. Due to the application of the cubic spline, the present inverse scheme not only can reduce the number of the measurement locations but also can increase the accuracy of the estimated results for the two-dimensional inverse heat conduction problems.

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