電子束熔融技術是金屬積層製造成型技術的一種，而會影響成型精密度的原因多與溫度有所關聯，因此使用相變化熱傳問題作為金屬積層製造過程的研究背景。本研究中使用鈦金屬作為研究材料，利用Fortran自行撰寫的三維有限元素法程式進行分析電子束熔融金屬積層製造過程中的相變化熱傳問題，並利用等效比熱法處理潛熱效應。本研究會分為三個階段，於第一階段中會從一維熱傳問題著手，探討史蒂芬問題、紐曼問題及定溫/熱對流兩種邊界條件，確認程式模擬的準確性，第二階段則將簡化後的三維模擬與二維模擬進行固定熱源、移動熱源兩種邊界條件進行近一步的驗證與比較分析，可得知結果十分相近，第三階段則套用積層製造過程的實際情形，對三維模型進行熱源掃描、鋪粉冷卻並進行多次的疊層模擬，最後再與套裝軟體COMSOL進行結果的比對。由研究結果可以發現，潛熱效應的存在對於溫度有顯著的影響，因此分析過程中不可忽略潛熱效應。三維疊層模擬時沿用與二維疊層模擬相同的熱源420W發現其溫度不足以使熔化深度超過鋪粉厚度，其所得到的結果僅為鬆散的片狀結構，而造成三維模擬溫度會較二維模擬來得低是因為三維多了一個維度的熱傳導效應，使得溫度傳導更為快速，之後將熱源修訂為550W才可使得層與層順利熔接成型，並發現三維模擬不僅要確認電子束的功率是否能使得熔化深度達到鋪粉厚度，也要確認熔池範圍有沒有涵蓋鋪粉厚度內在各個軸向的材料，避免層與層之間有部分熔接但同時也存在鬆散結構，而最後與COMSOL的結果比較，在多次熱源掃描階段其誤差皆小於5%，而在鋪粉冷卻階段其誤差皆小於1%，可以確定程式模擬得出的數值解相當準確。

Electron beam melting (EBM) is one of the metal additive manufacturing processes. Most of the forming problems are related to the temperature field. The purpose of this study is analyzing the heat transfer problem of phase change during the manufacturing process. The numerical method adopted in this study is the finite element method. On the basis of the finite element method theory, the heat transfer problem of EBM process is analyzed by the self-writing numerical code in FORTRAN and by the software, COMSOL. The process of this study divided into three stages; first stage, compare the results between numerical solution and exact solution of the 1D Stefan and Neumann problems and discuss the effect of latent heat; second stage, compare the results between the 2D numerical analysis and simplified 3D numerical analysis with fixed heat source and moving heat source; third stage, add the boundary condition of EBM process into 3D numerical code and make sure the adjacent layers can be welded together, compare the results computed by the FORTRAN code and COMSOL. The results from the third stage show that all of the metallic powder can be integrated together in the expected area at 550W beam power. Moreover, the results between calculated by the self-writing numerical code in FORTRAN and simulated by the software, COMSOL. Between these two methods, the relative errors of the maximum temperatures in five times of heat source scan processes are all below 5% and in four times of powder adding and cooling processes are all below 1%. It shows the results that analyzed by numerical code in FORTRAN and simulated by the software, COMSOL are consistent with each other and own the same trend. The results of this study are expected to be helpful to the additive manufacturing researchers.

[1] 張宇良, "利用有限元素法求解相變化熱傳問題," 國立成功大學工程科學系碩士論文, 2012.
[2] 吳建彣, "以有限元素法分析電子束熔融金屬積層製造之熱傳問題," 國立成功大學工程科學系碩士論文, 2016.
[3] H. Kodama, "Automatic method for fabricating a three‐dimensional plastic model with photo‐hardening polymer," Review of Scientific Instruments, vol. 52, pp. 1770-1773, 1981.
[4] C. W. Hull, "Apparatus for production of three-dimensional objects by stereolithography," ed: Google Patents, 1986.
[5] P. K. Venuvinod and W. Ma, Rapid Prototyping: Springer US, 2004.
[6] J. Schwerdtfeger, R. F. Singer, and C. Körner, "In situ flaw detection by IR‐imaging during electron beam melting," Rapid Prototyping Journal, vol. 18, pp. 259-263, 2012.
[7] H. B. Qi, Y. N. Yan, F. Lin, W. He, and R. J. Zhang, "Direct metal part forming of 316L stainless steel powder by electron beam selective melting," Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, vol. 220, pp. 1845-1853, November 1, 2006 2006.
[8] M. Kahnert, S. Lutzmann, and M. Zaeh, "Layer formations in electron beam sintering."
[9] A. Simchi, "Direct laser sintering of metal powders: Mechanism, kinetics and microstructural features," Materials Science and Engineering: A, vol. 428, pp. 148-158, 7/25/ 2006.
[10] D. Cormier, O. Harrysson, and H. West, "Characterization of H13 steel produced via electron beam melting," Rapid Prototyping Journal, vol. 10, pp. 35-41, 2004.
[11] J. P. Kruth, L. Froyen, J. Van Vaerenbergh, P. Mercelis, M. Rombouts, and B. Lauwers, "Selective laser melting of iron-based powder," Journal of Materials Processing Technology, vol. 149, pp. 616-622, 6/10/ 2004.
[12] 刘海涛, 赵万华, and 唐一平, "Process Investigation of Direct Metal Fabrication Based on Electron Beam Melting," 西安交通大学学报, vol. 41, pp. 1307-1310, 2007-11-10 2007.
[13] 汤慧萍, 王建, 逯圣路, and 杨广宇, "电子束选区熔化成形技术研究进展," 中国材料进展, pp. 225-235, 2015.
[14] 賈文鵬, 湯慧萍, and 劉海彥, "A method for manufacturing metal part by SEBM accompanied by annealing process," 2008.
[15] H. P. Tang, G. Y. Yang, W. P. Jia, W. W. He, S. L. Lu, and M. Qian, "Additive manufacturing of a high niobium-containing titanium aluminide alloy by selective electron beam melting," Materials Science and Engineering: A, vol. 636, pp. 103-107, 6/11/ 2015.
[16] 鎖紅波, 陳哲源, and 李晉煒, "電子束熔融快速制造Ti-6Al-4V的力學性能," presented at the 第13屆全國特種加工學術會議論文集, 2009.
[17] B. Cheng and K. Chou, "Melt pool geometry simulations for powder-based electron beam additive manufacturing."
[18] R. R. Namburu and K. K. Tamma, "Effective modeling/analysis of isothermal phase change problems with emphasis on representative enthalpy architectures and finite elements," International Journal of Heat and Mass Transfer, vol. 36, pp. 4493-4497, 12// 1993.
[19] J. Stefan, "Ueber die Theorie der Eisbildung, insbesondere über die Eisbildung im Polarmeere," Annalen der Physik, vol. 278, pp. 269-286, 1891.
[20] Y. C. Liu and L. S. Chao, "Modified effective specific heat method of solidification problems," Materials Transactions, vol. 47, pp. 2737-2744, Nov 2006.
[21] J. S. Hsiao, "An Efficient Algorithm for Finite-Difference Analyses of Heat-Transfer with Melting and Solidification," Numerical Heat Transfer, vol. 8, pp. 653-666, 1985.
[22] K. H. Huebner, D. L. Dewhirst, D. E. Smith, and T. G. Byrom, The Finite Element Method for Engineers, 4th ed., 1994.

[23] D. L. Collatz, The Numerical Treatment of Differential Equations: Springer Berlin Heidelberg, 1996.
[24] P.Dunne, "Complete polynomial displacement fields for finite element method," Aeronautical Journal, vol. 72, 1968.
[25] P. Davis and P. Rabinowitz, "Abscissas and weights for Gaussian quadratures of high order," National Bureau of Standards, vol. 56, pp. 35-37, 1956.
[26] R. Boyer, G. Welsch, and E. W. Collings, Materials Properties Handbook: Titanium Alloys. ASM International, 1994.