在許多工程問題上常使用傳統正算方法來求解其物理量，亦就是探討將已知條件輸入系統模式來分析其輸出為何，這就是正算問題(Direct Problem)。然而在許多實際工程問題中，存在著很多物理量因為客觀條件限制或量測技術不足而無法直接計算或量測其值。因此，為了取得所需之物理量，必須利用反算法藉由其它已知的參數或物理量反求之，這就是逆向或反算問題(Inverse Problem)。吾人將反算法運用在人工關節置換手術的熱傳問題控制上，期望能以數值分析的方式預測出最佳的邊界條件。
本文的目的即在於解決當進行人工關節置換手術時，因骨水泥(Bone Cement)聚合過程所產生之熱量所造成的骨骼組織熱損害問題，藉由預測出邊界上必須被帶走之熱量Q，進而達到對骨水泥聚合過程中所產生的熱量進行控制的目的。值得注意的地方在於問題的數學模式中，由骨水泥產生的熱量其方程式本身為非線性的，且其熱量的產生會隨著溫度而變化。本文的第二章先利用共軛梯度法(Conjugate Gradient Method)針對骨水泥熱源進行反算問題之探討，第三章再利用拉凡格式法(Levenberg-Marguardt Method)針對骨水泥產生熱量而造成高溫，進行最佳化邊界冷卻條件控制問題的討論。

The traditional direct problems are used to solve engineering issues when the initial and boundary conditions, system parameters and all the necessary conditions are given and known. However in practical problems, there always exist many physical quantities that they can not be measured or calculated directly, therefore the technique of inverse problem should be utilized. In this thesis the techniques of inverse problem is used to estimate the heat generation rate for cement and to control the heat removal rate in the prosthesis replacement operations.
The objective of this thesis is to control the rate of heat generation which was produced by cement polymerization and eventually estimate the optimum cooling condition. The heat generation rate for cement is non-linear since it is a function of temperature. This will make the inverse and optimum control problems more difficult to be solved.
The Conjugate Gradient Method (CGM) is applied in chapter two to estimate the heat generation rate for cement. Results show that the heat generation rate can be estimated accurately when considering either exact or error measurements.
In chapter three the Levenberg-Marguardt Method (LMM) is chosen to obtain the optimum boundary cooling function. Finally it is also shown that the optimum control function can be obtained based on different desired temperature distributions at the cement-bone interface.

1. A. G. Butkovskii and A. Y. Lerner, The Optimal Control Systems with Distribution Parameters, Auto. Remote Control 21(1960)472-477.
2. R. K. Cavi Ⅲ and S. C Tandon., Distribute Parameter System Optimal Control Design via Finite Element Discretization, Automatic 13(1977)611-614.
3. R.A. Meric, Finite Element Analysis of Optimal Heating of a Slab with Temperature Dependent Thermal Conductivity, Int. J. Heat and Mass Transfer,22(1979)1347-1353.
4. R.A. Meric, Finite Element and Conjugate Gradient Methods for a Nonlinear Optimal Heat Transfer Control Problem, Int. J. Numer. Meth. Eng, 14(1979)1851-1863.
5. Y. Sakawa, Solution of an Optimal Control Problem in a Distributed-Parameter System, IEEE Trans. Auto Contr., AC-9(1964)420-426.
6. Y. Sakawa, Optimal Control of a Certain Type of Linear Distributed-Parameter System, IEEE Trans. Auto Contr., AC-11(1966)35-41.
7. W. H. Ray, Some Recent Applications of Distributed-Parameter System Theory-A Survey, Automatica, 14(1978)281-287.
8. C.H. Huang, S.P. Wang, A Three Dimensional Inverse Heat Conduction Problem in Estimating Surface Heat Flux by Conjugate Gradient Method, Int. J. Heat and Mass Transfer, 42(1999)3387-3403.
9. C. Li, S. Schmid and J. Mason, Effects of pre-cooling and pre-heating procedures on cement polymerization and thermal osteonecrosis in cemented hip replacements. Medical Engineering and Physics, 25(2003)559-564.
10. A. Maffezzoli, D. Ronca, G. Guida, I. Pochini and L. Nicolais, In-situ polymerization behavior of bone cement. Journal of Materials Science：Materials In Medicine, 8(1997)75-83.
11. C.H. Huang and C.W. Chen, A Three-Dimensional Inverse Forced Convection Problem in Estimating Surface Heat Flux by Conjugate Gradient Method, J. Heat and Mass Transfer, 43(2000)3171-3181.
12. C. Li, S. Kotha, C.H. Huang, and J. Mason, Finite Element Thermal Analysis Of Bone Cement for Joint Replacements, Journal of Biomechanical Engineering, 125(2003)315-322.
13. C. Li, S. Kotha, C.H. Huang, S. Schmid, J. Mason, D. Yakimicki and M. Hawkins, Finite Element Simulation of Thermal Behavior of Prosthesis-Cement-Bone System, Bioengineering Conference BED-50(2001)235-236.