本論文旨在於針對反算問題於聲子輻射熱傳上之研究。聲子幅射熱傳方程式(Equation of Phonon Radiative Transfer)主要由波茲曼傳輸理論(Boltzmann Transport Theory)簡化推導而來，具有描述巨觀與微觀熱傳現象之能力，反算方法為共軛梯度法(Conjugate Gradient Method)，本文研究主題將於下列三章中分別介紹。
在第二章中預測薄膜邊界溫度函數，主要的物理模型為鑽石(Diamond)薄膜，在進行反算預測之前，吾人先將數值模擬正解與既有的正解文獻做比較，可確信正算問題無誤。利用反算方法中的共軛梯度法(CGM)來進行反算分析，試著以不同的方法化解聲子幅射熱傳方程式(EPRT)中難以處理的積分項，並與既有文獻的方法做比較。經比較兩方法後可知，無論在無誤差或存在量測誤差情況下兩者皆可得到相同程度準確的預測結果，因此吾人在第二章中提出另ㄧ種利用共軛梯度法處理EPRT的推導方式，可做為日後研究驗證之用。
在第三章中，亦利用共軛梯度法(CGM)進行未知邊界溫度反算分析工作。物理模型主要是探討砷化鎵/砷化鋁超晶格薄膜( GaAs/AlAs Superlattices)邊界溫度函數預測，在雙層薄膜界面中採用散異理論模式(Diffusive mismatch model)來描述微觀尺度下界面熱阻之效應，及比對尺度變化對於界面熱阻的影響。利用數值實驗模擬正確之邊界溫度，同時在考慮量測標準差的情況下，得知使用共軛梯度法預測仍可得到準確的邊界溫度函數結果。
在第四章中主要利用共軛梯度法(CGM)來預測微觀尺度下熱波傳遞為溫度與頻率函數之鬆弛時間，吾人利用文獻中經驗公式所推知的鬆弛時間作為數值實驗模擬之正解，藉由聲子感測器量測邊界聲子強度變化來預測材質內不同頻率與溫度下之鬆弛時間。本章主要選用兩種不同文獻中鬆弛時間推導方式，其分別為鑽石(Diamond)薄膜與砷化鎵(GaAs)，並在使用任意初始猜值下進行反算預測。當無量測誤差時吾人可成功的預測相當準確之鬆弛時間函數，然而在考慮加入量測誤差後發現反算預測對於誤差容忍能力較小，當存在大量量測誤差時會產生鬆弛時間函數預測效果較差之情況。
本論文研之究重點在於將共軛梯度法(CGM)用於預測微觀熱傳上各種現象，並且利用近年來才開始被研究的聲子強度量測技術，使預測微觀尺度下熱通量與材質鬆弛時間變為可能。

The inverse nanoscale heat conduction problems, using Conjugate Gradient Method (CGM), are discussed in this thesis. The nanoscale conduction phenomenon is modeled by Equation of Phonon Radiative Transport (EPRT), and the objectives are to estimate the unknown boundary temperature distributions and the material relaxation time based on phonon intensity measurements. The research topics of this thesis are divided into three portions and are prepared in chapters two, three and four, respectively.
In chapter two an inverse phonon radiative transport problem with an alternative form of adjoint equation is solved by using CGM to estimate the unknown boundary temperature distributions, based on the phonon intensity measurements. Finally it is shown that accurate boundary temperatures can always be obtained with CGM.
In chapter three CGM is also utilized in the inverse phonon radiative transport problem for a double-layer thin-film structure in estimating the unknown boundary temperature distributions, based on the phonon intensity measurements. Again it is shown that accurate boundary temperatures can always be obtained with CGM for this double-layer thin-film structure.
Finally in chapter four an inverse nanoscale phonon radiative transfer problem is solved by using CGM again to estimate the unknown frequency and temperature dependent relaxation time, based on the simulated phonon intensity measurements. Results obtained in this inverse analysis will be justified based on the numerical experiments where two different unknown distributions of relaxation time are to be estimated. Finally it is shown that the reliable relaxation time can be obtained with CGM.

第二章
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10.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).
11.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).
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17.H. W. Engl, ”Discrepancy principle for Tikhonov regularization of ill-posed problems leading to optimal convergence rate”, J. Optim Theory Applic., 52,no2,(1987).
第三章
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8.G. Chen, “Thermal Conductivity and Ballistic-Phonon Transport in the Crossplane Direction of Super-lattices”, Phys. Rev. B,57, 14958–14973, (1998).
9.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).
10.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).
11.M. Alifanov, “ Solution of an Inverse Problem of HeatConduction by Iteration Methods ”, J. of Engineering Physics, 26,471-476. (1974).
12.S. K. Kim and I. M. Daniel, “Gradient method for inverse heat conduction problem in nanoscale”, Int. J. Numer. Meth. Engng , 60, 2165–2181 (2004).
13.IMSL Library Edition 10.0. User's Manual: Math Library Version 1.0, IMSL, Houston, TX, (1987)
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16.O.M. Alifanov and E. A. Artyukhin, “Regularized numerical solution of nonlinear of inverse heat conduction problem”, J. Engng Phys., 35,no.6, 1501-1506,(1978).
17.O.M. Alifanov, ”Application of the regularization principle to the formulation of approximate solution of inverse heat conduction problem”, J. Engng Phys., 23,no.6, 1566-1571,(1972).
18.H. W. Engl, ”Discrepancy principle for Tikhonov regularization of ill-posed problems leading to optimal convergence rate”, J. Optim Theory Applic., 52,no2,(1987).
第四章
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5.Majumdar, “Microscale Heat Conduction in Dielectric Thin Films”, Journal of Heat Transfer, 115, 7-16 (1993).
6.Joshi and A. Majumdar, “Transient Ballistic and Diffusive Phonon Heat Transport in Thin Films”, Journal of Applied Physics, 74, 31-39 (1993).
7.G. Chen, “Size and Interface Effects on Thermal Conductivity of Superlattices and Periodic Thin-Film Structures”, Journal of Heat Transfer, 119, 220-229, (1997).
8.G. Chen, “Thermal Conductivity and Ballistic-Phonon Transport in the Crossplane Direction of Super-lattices”, Phys. Rev. B, 57, 14958-14973, (1998).
9.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).
10.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).
11.IMSL Library Edition 10.0. User's Manual: Math Library Version 1.0,IMSL, Houston, TX, (1987).
12.S. K. Kim and I. M. Daniel, “Gradient method for inverse heat conduction problem in nanoscale”, Int. J. Numer. Meth. Engng , 60, 2165-2181 (2004).
13.M. Alifanov, “ Solution of an Inverse Problem of HeatConduction by Iteration Methods ”, J. of Engineering Physics, 26,471-476,(1974).
14.O.M. Alifanov and E. A. Artyukhin, “Regularized numerical solution of nonlinear of inverse heat conduction problem”, J. Engng Phys., 35,no.6,1501-1506,(1978).
15.O.M. Alifanov, ”Application of the regularization principle to the formulation of approximate solution of inverse heat conduction problem”, J. Engng Phys., 23,no.6, 1566-1571,(1972).
16.H. W. Engl, ”Discrepancy principle for Tikhonov regularization of ill-posed problems leading to optimal convergence rate”, J. Optim Theory Applic., 52,no2,(1987).