本文提出一種混合拉氏轉換法（Laplace transform technique）和有限差分法（Finite difference method）的數值方法，並配合最小平方法（Least squares scheme）及時間接踵而至的觀念(Sequential-in-time concept)來預測二維逆向熱傳導問題之未知的邊界熱通量和熱傳遞係數。首先利用拉氏轉換法處理系統之時間域而後再以有限差分法處理系統轉換後之空間域，最後再以數值逆拉氏轉換法求取系統之溫度值。本文之估算值(Estimates)的函數型態先前是未知的，為了欲求得較精確的估算值，整個時間域被分割成數個小時間區域，而後再利用反算法求出小時間區間的估算值。拉氏轉換法的優點是可以求得在某一特定時間的溫度值，而不需要由初始時間慢慢地求解。最小平方法的應用在於使數值結果能較快地收斂。本文將探討溫度量測誤差、熱電偶（Thermocouple）的安裝位置及其數量對預測結果的影響。預測結果顯示本文之數值方法能夠有效地預測出較精確的估算值。預測結果也顯示量測誤差對估算值的影響並不是很敏感。因此本文之數值逆算法可成功地被應用來解析二維之逆向熱傳導問題。

　　The present study introduces a hybrid numerical method to analyze the inverse heat conduction problem concerning the prediction of the surface behavior. This algorithm combines the Laplace transform technique and the finite-difference method in conjunction with the least-squares scheme and sequential-in-time concept. Time-dependent terms in the governing equation are removed by using the Laplace transform technique, and then the resulting differential equation is solved by using the finite-difference method. Temperature distributions in the domain are obtained by using the numerical inversion of Laplace transform. The functional type of estimates is unknown, for predicting more precise estimates, Whole time domain separated off several small time domain , and then using the inverse scheme to estimate the estimates of the small time domain. Due to the application of the Laplace transform technique, the temperature can be calculated at a specific time without step-by-step computation in the time domain. By the least-squares scheme, the convergence of iteration can become fast and stable. In this thesis, various examples are illustrated to show the applicability and efficiency of the present numerical method. The effect of time-step, measurement error and thermocouple location is investigated. It can be seen from various illustrated examples that the present numerical method can accurately and efficiently estimate the estimates, even though the thermocouple is located far from the estimated surface. Results also show that the estimations are not very sensitive to the measurement error. Thus, it can be concluded that the present numerical method can successfully be applied to analyze the two-dimensional inverse heat conduction problems.

1.Shumakoy, N. V., “A method for the experimental study of the process of heating a solid body,” Soviet Physics-Technical Physics (Translated by Institute of Physics), Vol. 2, pp. 771, 1957.

2.Stolz, G. Tr., “Numerical solution to an inverse problem of heat condition for simple shapes,” ASME J. Heat Transfer, Vol. 82, pp. 20-26, 1960.

3.Beck, J. V., “Calculation of surface heat flux from an integral temperature history,” ASME J. Heat Transfer, 62-HT-46, 1962.

4.Sparrow, E. M., Haji-Sheikh A. and Lundgern, T. S. ,“The inverse problem in transient heat conduction, “ J. Appl. Mech., Vol. 86e, pp. 369-375, 1964.

5.Beck, J. V., “Surface heat flux determination using an integral method,” Nucl. Eng. Des., Vol. 7, pp. 170-178, 1968.

6.Beck, J. V., “Nonlinear estimation applied to the nonlinear inverse heat. conduction problem,” Int. J. Heat Mass Transfer, Vol. 13, pp. 713-71, 1970.

7.Alifanov, O. M., “Solution of an inverse problem of heat conduction by iteration method,” J. of Eng. Phys., Vol. 26, No. 4, pp. 471-476, 1974.

8.Beck, J. V., Litkouhi B., and St. Clair C. R., “Efficient sequential solution of nonlinear inverse heat conduction problem,” Numer. Heat Transfer, Vol. 5, pp. 275-286, 1982.

9.Alnajem, N. M., and Özisik, M. N., “A direct analytical approach for solving linear inverse heat conduction problems,” ASME J. Heat Transfer, Vol. 107, pp. 700-703, 1985.

10.Scott, E. P., and Beck, J. V., “Analysis of order of the sequential regulation solutions of heat conduction problem,” ASME J. Heat Transfer, Vol. 111, pp.218-224, 1989.

11.Beck, J. V., and Murio, D. A., “Combined function specification regularization procedure for solution of inverse heat conduction problem,” AIAA J., Vol. 24, No. 180-185, 1986.

12.Huang, C. H., Özisik, M. N., “Inverse problem of determining the unknown wall heat flux in laminar flow through a parallel plate duct,” Numer. Heat Transfer, Part A, Vol. 21, pp. 55-70, 1992.

13.Huang, C. H., Özisik, M. N., “Inverse problem of determining the unknown strength of an internal plate heat source,” J. Franklin Institute, Vol. 329, No. 4, pp. 751-764, 1992.

14.Arledge, R. G., and Haji-Sheikh, A., “A iterative approach to the solution of inverse heat conduction problem,” Numer. Heat Transfer, Vol. 1, pp. 365-376, 1978.

15.Woo, K. W., and Chow, L. C., “Inverse heat conduction by direct inverse laplace transform,” Numer. Heat Transfer, Vol. 4, pp. 499-504, 1981.

16.Raynaud, M., and Beck, J. V., “Methodology for comparison of inverse heat conduction methods,” ASME J. Heat Transfer, Vol. 110, pp. 30-37, 1988.

17.Chen, H. T., and Lin, J. Y., “Hybrid Laplace transform technique for non-linear transient thermal problems,” Int. J. Heat Mass Transfer, Vol. 34, pp. 1301-1308, 1991.

18.Chen, H. T., and Lin, J. Y., “ Numerical analysis for hyperbolic heat conduction,” Int. J. Mass Transfer, Vol.
36, pp. 2891-2898, 1993.

19.Chen, H. T., and Lin, J. Y., “Analysis of two-dimensional hyperbolic heat conduction problems,” Int. J. Heat Mass Transfer, Vol. 37, pp. 153-164, 1994.

20.Beck, J. V., Blackwell, B., and Chair, C. R. St., Jr, Inverse Heat Conduction：Ill-posed problem, Wiley, New York, 1985.

21.吳朝煌, ”逆向熱傳導問題之研究 - 熱傳導係數及未知邊界溫度之預測,” 國立成功大學機械工程研究所, 碩士論文, 1994.

22.Honig, G., and Hirdes, U., “A method for the numerical inversion of Laplace transforms,” J. Comp. Appl. Math., Vol. 9, pp. 113-132, 1984.

23.Hills, R. G., Mulholland G. P., and Matthews L. K., “The application of the Backus-Gilbert method to the inverse heat conduction problem in composite meadia,” ASME Paper No. 82-HT-26, 1982.

24.Frank, I., “An application of least-squares method to the solution of the inverse problem of heat conduction,” J. Heat Transfer, 85C, pp. 378-379, 1963.

25.Mulholland, G. P., Gupta B. P., and San Martin R. L., “Inverse problem of heat conduction in composite meadia,” ASME Paper No. 75-WA/HT-83, 1975.

26.Deverall, L. I., And Channapragada R. S., “ An new integral equation for heat flux in inverse heat conduction,” J. Heat Transfer, 88C, pp. 327-328, 1966.

27.Burggraf, O. R., “An exact solution of the inverse problem in heat conduction theory and applications,” ASME J. Heat Transfer, Vol. 84, pp. 373-382, 1964.

28.Yeung, W. K. and Lam, T. T., “ Second-order finite difference approximation for inverse determination of thermal conductivity,” Int. J. Heat Mass Transfer, Vol.
39, No. 17, pp. 3685-3693, 1996.

29.Bass, B. R. and Ott, L. J., “ A finite elment formulation of the two-dimensional inverse heat conduction problem,” Adv. Comput. Technol., Vol. 2, pp. 238-248, 1980.

30.Liu, Y. and Murio, D. A., “Numerical experiments in 2-D IHCP on bounded domains part I: the ‘interio’ cube problem,” Computer Math. Appl, Vol. 31, No. 1, pp. 15-32, 1996.

31.Imber, M., “A temperature extrapolation method for hollow cylinders,” AIAA. II. pp. 117-118, 1973.

32.Busby, H. R. and Trujillo, D. M.,“Numerical solution to a two-dimensional inverse heat conduction problem,” Int. J. Numer. Meth. Eng., Vol. 21, pp. 349-359, 1985.

33.Zabaras, N. and Liu, J. C., “An analysis of two-dimensional linear heat transfer problems using an integral method,” Numer. Heat Transfer, Vol. 13, pp. 527-533, 1988.

34.Lawson, C. L. and Hansen, R. J., Solving Least Square Problems, Prentice-Hall, NJ, 1974.

35. Chen, H. T. and Lin, S. Y., “Estimation of surface temperature in two-dimensional inverse heat conduction problems,” Int. J. Heat Mass Transfer, Vol. 44, pp. 1455-1463, 2001.