一般而言，工程上的問題分析可依照輸入(Input Source)、系統模式和輸出(Output response)其三者間之關係可分為下列兩大類。第一類問題探討不同輸入對已知系統模式之作用，對輸出結果與影響為何，此問題即一般傳統上所稱之正算問題(Direct problem)。而實際的工程問題上，仍有許多無法藉著量測或直接計算的物理量，因此我們可以藉著輸出及輸入來反算系統模式，或是已知系統模式作用和輸出反求其輸入為何，此第二類問題即則稱為反算問題或逆向問題(Inverse problem)。而反算問題應用在許多複雜工程問題上為常見之方法，其優點在於所預測之未知量並非只求解某些數學統御方程式，而是利用了“量測資料”來對所求解之問題作最佳化之計算。
未知幾何形狀之預測問題可應用於許多工程問題，例如：利用溫度量測預測高溫爐床及爐身殘厚，預測複合材料形狀之界面熱行為， 及材料結晶成長之形狀預測等。本論文分成兩大主題，皆為對複合材料界面之幾何形狀進行預測。在第二章中，吾人利用共軛梯度法(Conjugate Gradient Method)預測二區複合材料之未知界面幾何形狀 (即一組未知界面形狀，(x))，計算過程中之直接問題及所衍生之子問題以邊界元素法(Boundary Element Method)為基礎來加以解之。在第三章中，吾人亦利用共軛梯度法預測三區複合材料之未知界面幾何形狀(即二組未知界面形狀1(x)及2(x) )，直接問題與各子問題亦同樣以邊界元素法來求解。
　　本研究與之前相關研究不同的是，在進行二區和三區複合材料未知界面形狀之預測時，為了得到梯度方程式(gradient equation)吾人必須利用一重複使用條件(over-utilized condition)方可達成目的。此外，吾人亦假設兩區或三區複合材料之界面為良好接觸(perfect thermal contact)，故求解時需利用兩材料交界面上溫度相同且達到熱平衡之原理來求解。
　　最後在數值實驗中測試了考慮量測誤差與量測點數改變的情況，此外並於第二章中探討量測位置對於預測結果之影響。數值實驗結果皆證明了共軛梯度法於逆向幾何形狀預測問題中能夠正確預測複合材料內部界面形狀。

　　A two-dimensional shape identification problem, i.e. inverse geometry problem, in estimating the interfacial geometry for a multiple region domain is solved in this study by using the Conjugate Gradient Method (CGM) and Boundary Element Method (BEM)-based inverse algorithm. An over-utilized condition should be applied in determining the gradient equation, this differs from our previous relevant studies. Numerical experiments by using different measurement errors, measurement positions and number of sensor were performed to justify the validity of the conjugate gradient method in solving this shape identification problem. Finally it is concluded that the present algorithm can estimate the accurate interfacial configurations.

1. Alifanov, O.M. “Solution of an Inverse Problem of Heat Conduction by Iteration
Methods. J. of Engineering Physics,” Vol. 26, No. 4, 1974, pp. 471-476.
2. Huang, C.H., Yeh, C.Y. and Orlande, H.R.B., “A Non-Linear Inverse Problem in Simultaneously Estimating the Heat and Mass Production Rates for A Chemically Reacting Fluid”, Chemical Engineering Science, Vol. 58, No. 16, 2003. pp. 3741-3752,
3. Huang, C.H. and Yan, J.Y., “An Inverse Problem in Simultaneously Measuring Temperature Dependent Thermal Conductivity and Heat Capacity”, Int. J. Heat and Mass Transfer, Vol. 38, No. 18, 1995, pp. 3433-3441.
4. Terrola, P., “A Method to Determine the Thermal Conductivity from Measured Temperature Profiles, Int. J. Heat Mass Transfer”, Vol. 32, 1989, pp. 1425-1430.
5. Huang, C.H. and Ozisik, M.N., "A Direct Integration Approach for Simultaneously Estimating Temperature Dependent Thermal Conductivity and Heat Capacity", Numerical Heat Transfer, Part A, Vol. 20, No. 1, 1991, pp. 95-110.
6. Huang, C.H. and Chao, B.H., “An Inverse Geometry Problem in Identifying Irregular Boundary Configurations”, Int. J. Heat and Mass Transfer, Vol. 40, No. 9, 1997, pp. 2045-2053.
7. Alifanov, O.M., Inverse Heat Transfer Problems, Springer-Verlag, Berlin Heidelberg, 1994.
8. Huang, C.H. and Tsai, C.C., “A Transient Inverse Two-Dimensional Geometry Problem in Estimating Time-Dependent Irregular Boundary Configurations”, Int. J. Heat and Mass Transfer, Vol. 41, No. 12, 1998, pp. 1707-1718.
9. Huang, C.H., Chiang, C.C. and Chen, H.M., ”A Shape Identification Problem in Estimating the Geometry of Multiple Cavities “, AIAA, J. Thermophysics and Heat Transfer, Vol. 12, No. 2, April-June, 1998, pp. 270-277.
10. Huang, C.H. and Chen, H. M., “An Inverse Geometry Problem of Identifying Growth of Boundary Shapes in A Multiple Region Domain”, Numerical Heat Transfer; Part A, Vol. 35, 1999, pp. 435-450.
11. Park, H.M and Shin, H.J., “Empirical Reduction of Modes for the Shape Identification Problems of Heat Conduction Systems”, Computer Methods in Applied Mechanics and Engineering, Vol. 192, 2003, pp. 1893-1908.
12. Park, H.M and Shin, H.J., “Shape Identification for Natural Convection Problems using the Adjoint Variable Method”, Journal of Computational Physics, Vol. 186, 2003, pp. 198-211.
13. Cheng, C.H. and Chang, M.H., “A Simplified Conjugate-Gradient Method for Shape Identification Based on Thermal Data”, Numerical Heat Transfer; Part B, Vol. 43, 2003, pp. 489-507.
14. Kwag D.S., Park I.S. and Kim W.S., “Inverse Geometry Problem of Estimating the Phase Front Motion of Ice in a Thermal Storage System”, Inverse Problems in Engineering, Vol. 12, No. 1, 2004, pp.1-15.
15. Brebbia, C.A. and Dominguez, J., “Boundary Elements, An Introductory Course. McGraw-Hill, New York,1989.
16. Lasdon, L.S., Mitter, S.K. and Warren, A.D., “The Conjugate Gradient Method for Optimal Control Problem, IEEE Transactions on Automatic Control, AC-12, 1967, pp.132-138.
17. IMSL Library Edition 10.0, User's Manual: Math Library Version 1.0, IMSL, Houston, TX, 1987.