本論文以非傅立葉型態熱傳導方程式之反算問題研究為研究主旨。在傳統的熱擴散理論中，傅立葉定律假設熱量傳遞速度為無窮大，然而當具有很大的溫度梯度、溫度趨近於絕對零度時、極短時間的熱變化或極高的熱通量等情況出現時，此一假設必須加以修正。目前所使用的修正模式主要有雙曲線型熱傳方程式(hyperbolic heat conduction equation)、雙相延遲模式(dual-phase-lag)及修正拋物線型熱波方程式(modified parabolic thermal wave equation)。
而在實際的工程問題上，經常存在著許多物理量無法由量測或計算之方法獲得其值，因此為了求得這些物理量，往往必須藉由其他可量測之資料反求之，這類問題稱之為逆向或反算問題(Inverse Problem)。
第一章中，吾人以反算法之共軛梯度法(Conjugate Gradient Method)來進行雙相延遲模式之雙曲線熱傳問題表面熱通量之預測研究。在進行反算預測之前，吾人先將數值模擬正算與既有的文獻為正解作比較，可確認正算問題無誤，並討論溫度量測誤差對表面熱通量預測值準確度的影響。
在第二章中，同樣採用共軛梯度法(CGM)來疊代反算雙曲線熱傳問題，藉由量測模型內部溫度同時估算在邊界二未知熱通量。由結果發現，任意初始熱源猜值，皆可藉反算法獲得答案。

In practical engineering problem, there exist many physical quantities that are very difficult to measure directly. The techniques for “INVERSE PROBLEM” can be used to solve these kinds of problems. In the present thesis the inverse non-Fourier type heat conduction problems are discussed.
In chapter one an inverse problem for hyperbolic heat conduction with a dual-phase-lag model is solved by the Conjugate Gradient Method (CGM) in estimating the unknown heat generation, due to the ultra-short duration laser heating, based on the interior temperature measurements. Results obtained in this inverse problem will be justified based on the numerical experiments where two different heat source distributions are to be estimated. Results show that the inverse solutions can always be obtained with any arbitrary initial guesses of the heat sources. Finally, it is concluded that accurate heat sources can be estimated in this study.
An inverse hyperbolic heat conduction problem is solved in chapter two by an iterative regularization method, i.e. the Conjugate Gradient Method (CGM), to estimate simultaneously two unknown boundary heat fluxes based on the interior temperature measurements. The inverse solutions will be justified based on the numerical experiments where three different boundary heat flux distributions are to be determined. Results show that the inverse solutions can always be obtained with any arbitrary initial guesses of the boundary heat flux. Finally, it is concluded that accurate boundary heat fluxes can be estimated in the present study when large measured errors are considered.

1. Cattaneo, C., Sur une forme de l’equation de la chaleur eliminant le paradoxe d’une propagation instantanee, Computes Rendus, 247, 431–433 (1958).
2. P. Vernotte, Les paradoxes de la theorie continue de l’equation de la chaleur, Computes Rendus, 246, 3154–3155 (1958).
3. Y. Ishii, A. Mori, A. Onodera, S. Kawano and Y. Mori, “ Phonon Measurement of RbCl at 4.9 kbar”, Physica B, 241-243, 409-411 (1998).
4. M. Ikezawa, T. Okuno, and Y. Masumoto, “Complementary Detection of Confined Acoustic Phonons in Quantum Dots by Coherent Phonon Measurement and Raman Scattering”, Physical Review B, 64, 201315(R)(2001).
5. S. I. Anisimov, B. L. Kapeliovich and T. L. Perel’man, Electron emission from metal surfaces exposed to ultra-short laser pulses, Sov. Phys.–JETP 39, 375-377 (1974).
6. H. E. Elsayed-Ali, T. B. Norris, M. A. Pesot and G. A. Mourou, Time-Resolved Observation of Electron-Phonon Relaxation in Copper, Phys. Rev. Lett. 58, 1212-1215 (1987).
7. S. D. Brorson, J. G. Fujimoto and E. P. Ippen, Femtosecond Electron Heat-Transport Dynamics in Thin Gold Film, Phys. Rev. Lett. 59 1962-1965 (1987).
8. D. Y. Tzou, A Unified Field Approach for Heat Conduction from Macro-to Micro- Scales, ASME J. Heat Transfer, 117, 8-16 (1995).
9. H. T. Chen, and S.M. Chang, Application of the Hybrid Method to Inverse Heat-Conduction Problem, Int. J Heat Mass Transfer, 33, 621-628 (1990).
10. H. T. Chen, S. Y. Lin, H. R. Wang, and L. C. Fang, Estimation of Surface Temperature in Two-Dimensional Inverse Heat Conduction Problems, Int. J. Heat and Mass Transfer, 44, 1455-1463 (2001).
11. H. T. Chen, S. Y. Peng, P.C. Yang, and L. C. Fang, Numerical Method for Hyperbolic Inverse Heat Conduction Problem,. Int. Comm. Hat Mass Transfer, 28, 847-856 (2001).
12. C. Y. Yang, Direct and Inverse Solutions of Hyperbolic Heat Conduction Problems, J. Thermophysics and Heat Transfer, 19, 217-225 (2005).
13. C. H. Huang and H. H. Wu, An Inverse Hyperbolic Heat Conduction Problem in Estimating Surface Heat Flux by Conjugate Gradient Method, Journal of Physics D: Applied Physics, 39, 4087–4096 (2006).
14. C. H. Huang, M. N. Ozisik and B. Sawaf, Conjugate Gradient Method for Determining Unknown Contact Conductance During Metal Casting, Int. J. of Heat and Mass Transfer, 35, 1779-1786 (1992).
15. C. H. Huang and H. C. Lo, A Three-Dimensional Inverse Problem in Predicting the Heat Fluxes Distribution in the Cutting Tools, Numerical Heat Transfer, part A-Applications, 48, 1009-1034 (2005).
16. C. H. Huang and J. X. Li, A Non-Linear Optimal Control Problem in Obtaining Homogeneous Concentration for Semiconductor Materials, Journal of Physics D: Applied Physics, 39, 2343–2351 (2006).
17. C. H. Huang and H. H. Wu, An Iterative Regularization Method in Estimating the Base Temperature for Non-Fourier Fins, Int. J. Heat and Mass Transfer, 49, 4893–4902 (2006).
18. Alifanov, O.M., Inverse heat transfer problems, Springer-Verlag, Berlin Heidelberg, 1994.
19. D. W. Tang and N. Araki, The wave characteristics of thermal conduction in metallic films irradiated by ultra-short laser pulses, J. Phys. D: Appl. Phys. 29 2527–2533 (1996).
20. D. Y. Tzou, Marco- to Microscale Heat Transfer, Taylor & Francis, Washington, DC, USA, 1997.
21. G. F. Carey and M. Tsai, Hyperbolic heat transfer with reflection, Numerical Heat Transfer, 5, 309–327 (1982).
22. L. S. Lasdon, S. K. Mitter and A. D. Warren, The Conjugate Gradient Method for Optimal Control Problem, IEEE Transactions on Automatic Control, AC-12, 132-138 (1967).
23. IMSL Library Edition 10.0. User's Manual: Math Library Version 1.0, IMSL, Houston, TX, 1987.
1. Cattaneo, C. and Sullaconduzione, de calore, in: Atti del Seminar, Mat. Fis. Univ., Modena, 1948, 3, 83–101.
2. Vernotte, P., Les paradoxes de la theorie continue de l’equation de la chaleur. Compute Rendus, 1958, 246,3145–3155.
3. Ozisik, M. N. and Tzou, D.Y., On the wave theory in heat conduction. ASME J. Heat Transfer, 1994, 116, 526–535.
4. Carey, G. F. and Tsai, M., Hyperbolic heat transfer with reflection, Numerical. Heat Transfer, 1982, 5, 309–327.
5. Glass, D. E., Ozisik, M. N. and Vick, B., Hyperbolic heat conduction with surface radiation, Int. J. Heat Mass Transfer, 1985, 28, 1823–1830.
6. Sadd, M.H. and Cha, C.Y., Axisymmetric non-Fourier temperature in cylindrically bounded domains, Int. J. Nonlinear Mech. 1982, 17, 129–136.
7. Barletta, A. and Zanchini, E., A thermal potential formulation of hyperbolic heat conduction, ASME J. Heat Transfer. 1999, 121, 166–169.
8. Honner, M., Heat waves simulation, Comput. Math. Appl. 1999, 38, 233–243.
9. Lin, J.Y. and Chen, H.T., Numerical solution of hyperbolic heat conduction in cylindrical and spherical systems, Appl. Math. Modell. 1994, 18, 384–390.
10. Chen, H. T. and Liu, K.C., Analysis of non-Fickian diffusion problem in a composite medium, Comput. Phys. Commun. 2003, 150, 31–42.
11. Chen, H. T. and Chang, S.M, Application of the Hybrid Method to Inverse Heat-Conduction Problem. Int J Heat Mass Transfer 1990, 33, 621-628.
12. Chen, H. T., Lin, S. Y., Wang, H. R. and Fang, L. C., Estimation of Surface Temperature in Two-Dimensional Inverse Heat Conduction Problems,” Int. J. Heat and Mass Transfer, 2001, 44, 1455-1463.
13. Chen, H. T., Peng, S. Y., Yang, P.C. and Fang, L. C., Numerical Method for Hyperbolic Inverse Heat Conduction Problems. Int. Comm. Hat Mass Transfer, 2001, 28, 847-856.
14. Yang, C. Y., Direct and Inverse Solutions of Hyperbolic Heat Conduction Problems. J. Thermophysics and Heat Transfer, 2005, 19, 217-225.
15. Alifanov, O.M., Inverse heat transfer problems, Springer-Verlag, Berlin Heidelberg, 1994.
16. Huang, C. H. and Chen, C. W., A Boundary Element Based Inverse-Problem in Estimating Transient Boundary Conditions with Conjugate Gradient Method. Int. J. Numerical Methods in Engineering. 1998, 42, 943-965.
17. Huang, C. H. and Wang, S. P., A Three-Dimensional Inverse Heat Conduction Problem in Estimating Surface Heat Flux by Conjugate Gradient Method. Int. J. Heat and Mass Transfer, 1999, 42, 3387-3403.
18. Huang, C. H. and Chen, W. C., A Three-Dimensional Inverse Forced Convection Problem in Estimating Surface Heat Flux by Conjugate Gradient Method, Int. J. Heat and Mass Transfer, 2000, 43, 3171-3181.
19. Huang, C. H. and Chin, S. C., A Two-Dimensional Inverse Problem in Imaging the Thermal Conductivity of a Non-homogeneous Medium, Int. J. Heat and Mass Transfer, 2000, 43, 4061-4071.
20. Huang, C. H., A Nonlinear Inverse Vibration Problem of Estimating the External Forces for A System with Displacement-Dependent Parameters, J. Sound and Vibration, 2001, 248, 789-807.
21. C. H. Huang and H. H. Wu, An Inverse Hyperbolic Heat Conduction Problem in Estimating Surface Heat Flux by Conjugate Gradient Method, Journal of Physics D: Applied Physics, 39, 4087–4096 (2006).
22. Lasdon, L. S., Mitter, S. K. and Warren, A. D., The Conjugate Gradient Method for Optimal Control Problem. IEEE Transactions on Automatic Control, 1967, AC-12, 132-138.
23. IMSL Library Edition 10.0. User's Manual: Math Library Version 1.0, IMSL, Houston, TX, 1987.