本研究的主題在建立逆解模式，藉以探討金屬成形加工時工件與模具接觸面間摩擦係數及工件材料熱傳導係數變化的歷程。本研究分為五個研究主題，分別概述如下：
第一個主題：鍛粗加工逆解摩擦係數之彈塑性大變形有限元素模式。本研究主題的目的主要是先假設材料為片斷線性的情形下，針對鍛粗成形加工發展一以矩陣表示線性最小平方差之逆解金屬成形彈塑性大變形有限元素模式，並配合鍛粗加工之實驗，取得在鍛粗成形的過程中鍛粗負荷的歷程，再利用本研究所提出之線性修正實驗鍛粗負荷基準的方法，藉以修正在假設材料為片斷線性的情形下所造成鍛粗負荷之誤差。以此修正後之鍛粗加工負荷作為逆解材料與模具接觸邊界摩擦係數的依據，並應用cubic spline 綴合得到鍛粗加工摩擦係數的歷程，進而了解在塑性成形加工製程中模具與工件間之摩擦現象、工件等效應力以及等效應變場之分佈。
第二個主題：鍛粗加工逆解摩擦係數之彈塑性大變形有限元素結合Tikhonov 正則化法。第一個研究主題所推導的逆解模式是以鍛粗加工實驗之量測負荷作基準，而此量測負荷具有不可避免之量測誤差，因此若直接使用此逆解模式逆解每一個步驟之工件與模具接觸面間之摩擦係數會導致所得之鍛粗加工摩擦係數變動的歷程產生嚴重不穩定之變動。本研究主題的目的主要是針對鍛粗成形加工，結合第一個研究主題所推導的逆解模式與正則化的Tikhonov method ，並配合鍛粗加工實驗，取得在塑性成形的過程中加工負荷的歷程，以此鍛粗加工負荷作為逆解材料與模具接觸邊界摩擦係數的依據，得到鍛粗加工摩擦係數的歷程。由於於鍛粗加工過程中，摩擦係數的變化是連續且在有限元素加工步驟間其變化是相當小的，因此本研究以此步驟與前一步驟間摩擦係數之差的絕對值為正則化的Tikhonov method中的鎮定泛函，藉以穩定因量測誤差而產生的震盪現象。進而了解在塑性成形加工製程中模具與工件間之摩擦現象、工件應力、應變場及接觸應力之分佈。
第三個主題：擠製加工工件與模具斷面縮減接觸面間摩擦係數之逆解模式。本研究主題的目的主要是針對擠製成形加工發展一以矩陣表示線性最小平方差之逆解金屬擠製彈塑性大變形有限元素模式，經由擠製加工實驗，取得在擠製成形的過程中擠製負荷的歷程，將此擠製負荷分配至衝頭與工件接觸節點，得到衝頭與工件接觸節點之節點力增量，以此擠製加工衝頭與工件接觸節點之節點力增量作為逆解工件與模具斷面縮減部分接觸邊界摩擦係數的依據，並配合本文所提出之正則化的方法，得到擠製加工模具斷面縮減部分與工件接觸面間摩擦係數的變化的歷程，進而了解在擠製成形加工製程中模具與工件間之摩擦現象、工件應力以及應變場之分佈。
第四個主題：鍛粗加工逆解熱傳導係數之熱傳有限元素分析模式。本研究的目的是針對鍛粗加工建立一逆解方法，藉以逆解鍛粗加工時工件材料之熱傳導係數隨工件平均溫度的改變而變化的歷程。
第五個主題：鍛粗加工之摩擦係數與工件熱傳導係數之整合逆解模式。本研究主題之主要目的是經由鍛粗加工實驗取得鍛粗加工負荷以及溫度量測點溫度變化的歷程，基於此實驗鍛粗負荷及量測溫度建立一矩陣表示線性最小平方差逆解熱彈塑性有限元素模式與逆解熱傳導有限元素模式之整合逆解模式，利用此整合逆解模式可以同時逆解出鍛粗加工工件與模具接觸面間摩擦係數變化的歷程以及工件材料熱傳導係數變化的歷程。本研究進一步將所建立之整合逆解模式結合 Tikhonov 正則法以改善逆解所得之工件與模具接觸面間摩擦係數及工件材料熱傳導係數變化的歷程之穩定性。同時，基於此工件與模具接觸面間摩擦係數及工件材料熱傳導係數變動的歷程，得到鍛粗加工時工件之應力場、應變場及溫度場的分佈。
期望本研究之成果提供金屬成形加工產業參考，提昇塑性加工製品的品質與產量。

The main purpose of this study is to establish the inverse model, by using which to investigate the history of friction coefficient between the workpiece and the die interface and the history of thermal conductivity of the material of workpiece during metal forming. There are five topics included in this study, as describe following:
The first topic is: matrix presentation linear least square errors method inverse elastic-plastic large deformation finite element model for upsetting. The purpose of this topic is to establish the matrix presentation linear least square errors method inverse elastic-plastic large deformation finite element model for upsetting under linear material assumption. By the upsetting experiment, the history of upsetting loading during upsetting process would be obtained. Then, the history of experimental upsetting loading could be modified using the linear modified upsetting loading standard as proposed in this study to modify the errors from the assumption of linear material. Based on the modified experimental upsetting loading, the history of friction coefficient could be derived by using the inverse model proposed in this study and by the curve fitting of cubic spline. Furthermore, it can understand the friction phenomenon between the interface between the workpiece and the die, the equivalent stress distribution and the equivalent strain distribution of the workpiece.
The second topic is: matrix presentation linear least square errors method inverse elastic-plastic large deformation finite element model combined with the regularization of Tikhonov method for upsetting. The inverse model of first topic is based on the experimental upsetting loading that has the inevitable measurement errors. Thus, using the inverse model directly, the obtained history of friction coefficient would be unsteady and serious vibration. The purpose of this topic is to combine the inverse model proposed in first topic and the regularization of Tikhonov method. Based on the experimental upsetting loading using this combined procedure the more stable history of friction coefficient between the workpiece and the die would be derived.
The third topic is: a inverse method for calculating the friction coefficient between the section reduction part of the die and billet during extrusion process. The purpose of this topic is to establish a matrix presentation linear least square errors method inverse elastic-plastic large deformation finite element model for extrusion. The extrusion loading could be obtained by extrusion experiment and then distribute to the contact nodes between the billet and the punch. Based on the loading distribution of the contact nodes, by using the inverse model purposed in this topic associated with the regularization of Tikhonov method, the history of friction coefficient between the section reduction part of the die and the billet can be derived. Furthermore the distribution of the stress and the strain of the billet can be also obtained.
The fourth topic is: matrix presentation linear least square errors method inverse heat transfer finite element model for upsetting. The purpose of this topic is to establish an inverse method that could derive the variation of history of thermal conductivity of the workoiece material and also obtain the average temperature of the workpiece during the upsetting process.
The fifth topic is: the combined inverse model of the friction coefficient and the thermal conductivity for upsetting. The purpose of this topic is based on the experiment upsetting loading and the experiment temperatures of the measurement point to establish a combined inverse model for upsetting of matrix presentation linear least square errors method inverse thermal-elastic-plastic large deformation finite element model and matrix presentation linear least square errors method inverse heat transfer finite element model. By using the combination inverse model the history of friction coefficient between the workpiece and the die and the thermal conductivity of the workpiece material can be obtained. Furthermore, in order to reduce the vibration of the history of the friction coefficient and the thermal conductivity, this study combines the combined inverse model and the regularization of Tikhonov method. Beside, the distributions of the stress, strain and temperature also obtained.
It is expected the results of this study can supply to the industry of metal forming, and promote the quality of the product of metal forming.

1. C. Abbjit, and M. Subrata, “A Finite Element Analysis of Metal Forming Problems with an Elastic-Viscoplastic Material Model,” International Journal for Numerical Methods in Engineering, Vol. 20, pp. 1613~1628 (1984).
2. I. Pillinger, P. Hartley, C. E. N. Sturgess, and G. W. Rowe, “Finite Element Modeling of Metal Flow in Three Dimensional Forming,” International Journal for Numerical Methods in Engineering, Vol.25, pp. 87~97 (1988).
3. Z. C. Lin and S. Y. Lin, “An Investigation of a Coupled Analysis of a Thermo-Elastic-Plastic Model During Warm Upsetting,” International Journal of Machine Tools & Manufacture, Vol.30, No.4, pp. 599~612 (1990).
4. Z. C. Lin and W. C. Pan, “A Coupled Thermo-Elastic-Plastic Model with a Modified Direct Iteration Method in Cold Extrusion Process”, Journal of Strain Analysis, Institution of Mechanical Engineering, I. Mech. E. U. K., Vol.28, No.2, pp. 89~100 (1993).
5. A. N. Tikhonov and V. Y. Arsenin, “Solutions of Ill-posed Problem,” V. H. Wistom & Sons, Washington, DC, 1997
6. K. C. Park and A. Felippa Carlos, “A Flexibility-based Inverse Algorithm for Identification of Substructural and Joint Properties,” Computational Methods for Solution of Inverse Problems in Mechanics, AMD, Vol. 228, ASME, pp. 11~22 (1998).
7. E. M. Sparrow, A. Haji-Sheikh and T. S. Lundgern, “The Inverse Problem in Transient Heat Conduction,” Journal of Apply Mechanic, Vol. 86e, pp. 369~375 (1964).
8. O. M. Alifanov, “Solution of an Inverse Problem of Heat Conduction by Iteration Method,” Journal of Engineering Physical, Vol. 26, No.4, pp. 471~476 (1974).
9. D. M. Trujillo, “Application of Dynamic Programming to the General Inverse Problem,” International Journal for Numerical Methods in Engineering, Vol. 12, pp. 613~624 (1978).
10. D. M. Trujillo and H. R. Busby, “Practical Inverse Analysis in Engineering,” CRC Press, Boca Raton, New York (1997).
11. J. V. Beck, “Surface Heat Flux Determination Using an Integral Method,” Nuclear Engineering and Design, Vol. 7, pp. 170~178 (1968).
12. O. M. Alifanov and V. V. Mikhailov, “Solution of the Nonlinear Inverse Thermal Conductivity Problem by the Iteration Method,” High Temperature, Vol.36, No.6, pp. 1501~1506 (1978).
13. J. V. Beck, B. Blackwell and A. Haji-Sheikh, “Comparison of Some Inverse Heat Conduction Methods Using Experimental Data,” International Journal of Heat and Mass Transfer, Vol.39, No.17, pp. 3649~3657 (1996).
14. G. Blanc, J. V. Beck and M. Raynaud, “Solution of the Inverse Heat Conduction Problem with a Time-Variable Number of Future Temperatures,” Numerical Heat Transfer, Part B, Vol.32, pp. 437~451 (1997).
15. C. H. Huang, M. N. Ozisik and B. Sawaf, “Conjugate Gradient Method for Determining Unknown Contact Conductance During Metal Casting,” International Journal of Heat and Mass Transfer, Vol.25, No.7, pp. 1786~1799 (1992).
16. 洪茂人, ”熱鍛製程變形與熱傳的逆向分析”, 國立成功大學機械工程學系碩士論文 (1998).
17. J. C. Gelin and O. Ghouati, “The Inverse Approach for The Determination of Constitutive Equation in Metal Forming,” Annals of the CIRP, Vol.44, pp. 189~192 (1995).
18. Ching-yu Yang and Cha’o-Kuang Chen, “The Boundary Estimation in Two-Dimensional Inverse Heat Conduction Problems,” Journal of Physics D-Applied Physics,Vol. 29, pp. 333~339 (1996).
19. T. G. Faurholdt, “Inverse Modeling of Constitutive Parameters for Elastoplastic Problems,” Journal of Strain Analysis, Vol. 35, No. 6, pp. 471~478 (2000).
20. C. A. Conceição António and N. Magalhães Dourado, “Metal-forming Process Optimization by Inverse Evolutionary Search,” Journal of Materials processing Technology, Vol. 121, pp. 403~413 (2002).
21. Elisabeth Massoni, Béatrice Boyer and Romain Forestier, “Inverse Analysis of Thermomechanical Upsetting Tests Using Gradient Method with Semi-analytical Derivatives,” International Journal of Thermal Sciences, Vol. 41, pp. 557~563 (2002).
22. G. W. pearsall and W. A. Backofen, “Frictional Boundary Conditions in Plastic Compression,” Journal of Engineering for Industry, Transactions of ASME, pp. 68~76, February (1963).
23. A. T. Male and M. G. Cockcroft, “A Method for The Determination of The Coefficient of Friction of Metals Under Conditions of Bulk Plastic Deformation,” J. Inst. Met., Vol. 93, pp.38-46 (1964).
24. V. Depiere and F. Gurney, “A Method for Determination of Constant and Varying Friction Factors During Ring Compression Tests,” Trans. ASME. J. Lubr. Technology, Vol. 96, pp. 482~488 (1974).
25. S. M. Hwang and S. Kobayashi, “A Note on Evaluation of Interface Friction in Ring Tests,” Proc. of the 18th MTDR Conference, pp. 193~196 (1977).
26. W. L. Xu, K. P. Rao, T. Watanabe and M. Hua, “Analysis of Ring Compression Using Artificial Neutral Network,” Journal of Material Process Technology, Vol. 44, pp. 301~308 (1994).
27. K. P. Rao and W. L. Xu, “Neural Evaluation of Friction and Flow Stress Adaptive to Ring Geometry,” JSME International Journal Series A-Solid Mechanics and Material Engineering, Vol. 38, No. 4, pp.506~514 (1995).
28. S. Y. Lin, “An Investigation of Die-Workpiece Interface Friction During the Upsetting Process,” Journal of Materials Processing Technology, Vol. 54, pp. 239~248 (1995).
29. 張三木, “預估鍛粗過程中表面摩擦係數的變化”, 國立成功大學碩士學位論文 (1997).
30. S.B. Petersen, P. A. F. Martins and N. Bay, ”Friction in Bulk Metal Forming: A General Friction Model vs. The Law of Constant Friction,” Journal of Materials Processing Technology, Vol. 66 pp. 186~194 (1997).
31. C. Thiébaut, C. Bonnet and J. M. Morey, “Evolution of The Friction Factor of A Molybdenum Workpiece During Upsetting Tests at Different Temperatures,” Journal of Materials Processing Technology, Vol. 77, pp. 240~245 (1998).
32. Hasan Sofuoglu and Jahan Rasty, “On the Measurement of Friction Coefficient Utilizing the Ring Compression Test,” Tribology International, Vol. 32, pp. 327~335 (1999).
33. 陳正堃, “鍛粗成形參數的預估之逆向問題研究”, 國立台灣科技大學機械工程系碩士學位論文 (2001).
34. Chorng-Der Lee, Cheng-I Weng and Jee-Gong Chang, “A Prediction of the Friction Factor for the Forging Process,” Metallurgical and Materials Transactions, Vol. 32B, pp. 137~143 (Feb. 2001).
35. Xincai Tan, “Comparisons of Friction Models in Bulk Metal Forming,” Tribology International, Vol. 35, pp. 385~393 (2002).
36. M. Bakhshi-Jooybari, “A Theoretical and Experimental Study of Friction in Metal Forming by the Use of the Forward Extrusion Process,” Journal of Materials Processing Technology’ Vol. 125, pp. 369~374 (2002).
37. J. P. Wang, “An Investigation into Friction in Dynamic Plane Upsetting,” Journal of Materials Processing Technology, Vol. 123, pp. 323~328 (2002).
38. H. Sofuoglu and H. Gedikli, “Determination of Friction Coefficient Encountered in Large Deformation Processes,” Tribology International, Vol. 35, pp. 27~34 (2002).
39. 陳翰林, “逆解鍛粗中空圓管及複合材圓柱之摩擦係數研究”, 國立台灣科技大學機械工程系碩士學位論文 (2002).
40. Tze-Chi Hsu, Chien-Chin Huang, “The Friction Modeling of Different Tribological Interfaces in Extrusion Process,” Journal of Materials Processing Technology, Vol. 140, pp. 49~53 (2003).
41. Yong-Taek Im, Seong-Hoon Kang and Jae-Seung Cheon, “Finite Element Investigation of Friction Condition in a Backward Extrusion of Aluminum Alloy,” Transactions of the ASME, Vol.125, pp. 378~383, May (2003).
42. S. Malayappan and R. Narayanasamy, “An Experimental Analysis of Upsetting Forging of Aluminium Cylindrical Billets Considering the Dissimilar Friction Conditions at Flat Die Surfaces,” International Journal of Advanced Manufacturing Technology, Vol. 23, pp. 636~643 (2004).
43. X. L. Hu, J. T. Hai and W. M. Chen, “Experimental Study and Numerical Simulation of the Pressure Distribution on the Die Surface During Upsetting,” Journal of Material Processing Technology, Vol. 151, pp. 367~371 (2004).
44. Zone-Ching Lin and Chun-Kung Chen, “Inverse Calculation of the Friction Coefficient During the Warm Upsetting of Molybdenum,” International Journal of Mechanical Sciences, Vol. 47, pp. 1059~1078 (2005).
45. P. Terrola, “A Method to Determine the Thermal Conductivity from Measured Temperature Profiles,” Int. J. Heat Mass Transfer, Vol. 32, pp. 1425~1430 (1989).
46. N. M. Alnajem and M. N. Özisik, “A Direct Analytical Approach for Solving Linear Inverse Conduction Problems,” Journal of Heat Transfer-Transactions of the ASME, Vol. 107, pp. 700-703 (1985).
47. E. P. Scott and J. V. Beck, “Analysis of Order of the Sequential Regulation Solutions of Heat Conduction Problem,” Journal of Heat Transfer-Transactions of the ASME, Vol. 111, pp. 218-224 (1989).
48. J. V. Beck and D. A. Murio, “Combined Function Specification Regularization Procedure for Solution of Inverse Heat Conduction Problem,” AIAA J., Vol. 24, pp. 180-185, January (1986).
49. C. H. Huang and M. N. Ozisik, “ A Direct Integration Approach for Simulataneously Estimating Spatially Varying Thermal Conductivity and Heat Capacity,” International Journal of Heat and Fluid Flow, Vol. 11, No. 3, pp. 262~268 (1990).
50. C. H. Huang and M. N. Ozisik, ”A Direct Integration Approach for Simulataneously Estimating Temperature Dependent Thermal Conductivity and Heat Capacity,” Numerical Heat Transfer, Part A, Vol. 20, No. 1, pp. 95~100 (1991).
51. Ching-yu Yang, “A Linear Inverse Model for the Temperature Dependent Thermal Conductivity Determineation in One-dimensional Problems,” Applied Mathematical Modeling, Vol.22, pp. 1~9 (1998).
52. Cheng-Hung Huang and Sheng-Chieh Chin, “A Two-dimensional Inverse Problem in Imaging the Thermal Conductivity of a Non-homogeneous Medium,” International Journal of Heat and Mass Transfer, Vol.43, pp. 4061~4071 (2000).
53. P. Kwon, T. Schiemann and R. Kountanya, “An Inverse Estimation Scheme to Measure Steady-state Tool-chip Interface Temperature Using an Infrared,” International Journal of Machine Tools & Manufacture, Vol. 41, pp. 1015~1030 (2001).
54. Sin Kim, Min Chan Kim and Kyung Youn Kim, “Non-Iterative Estimation of Temperature-dependent Thermal Conductivity Without Internal Measurements,” International Journal of Heat and Mass Transfer, Vol. 46, pp. 1801-1810 (2003).
55. Tadeusz Telejko and Zbigniew Malinowski, “Application of an Inverse Solution to the Thermal Conductivity Identification Using the Finite Element Method,” Journal of Materials Processing Technology, Vol. 146, pp. 145-155 (2004).
56. 山田嘉昭, 非線性有限元素法基礎, 亞東書局, (1985).
57. Y. Yamada, “Visco-elasticity plasticity”, Baifukan, Tokyo, Japan (1980).
58. S. S. Rao, “The Finite Element Method in Engineering”, New York, Pergamon Press Inc. pp.352-354 (1989).
59. M. C. Shaw, A. Ber and P. A. Mamin, “Friction Characteristics of Sliding Surface Undergoing Subsurface Plastic Flow,” Trans J Basic Eng, Vol. 82, pp. 342~346 (1960).
60. U. Krause, ”Comparison of Different Methods for Determining Flow Stress in Cold Metal Forming,” Dr.-Ing. Thesis, Technical University, Hannover, (1962).
61. E. Orowan, ”The Calculation of Roll Pressure in Hot and Cold Flat Rolling,” Proc. Instn. Mech. Engrs. Vol. 150, No. 1, pp. 40~67 (1943).
62. A. N. Tihonov (Tikhonov), “Solution of Incorrectly Posed Problems and The Regularization Method,” Soviet Math. Dokl., Vol. 4, No. 4, pp. 1035~1038 (1963).
63. J. Carvill, Mechanical Engineer’s Data Handbook, CRC Press, Boca Raton, New York (1993).
64. J.P. Holman, Heat Transfer, Sixth Edition, McGraw-Hill, New York, (1986).