在凝固過程中會有相變化，其中所涉及的潛熱效應為重要的物理現象，本文使用有限元素法搭配不同處理潛熱的數值方法，來分析相變化凝固熱傳問題的溫度場分佈。
本文主要探討的凝固問題為一維史帝芬問題及一維紐曼問題，使用四種處理潛熱釋放效應的數值方法，分別為等效比熱法、等效比熱/熱焓法、可變時間步伐之等效比熱法及可變時間步伐之等效比熱/熱焓法，來分析比較計算溫度場的準確度、潛熱釋放量、計算時間，其中溫度場的準確度以總誤差(total-error)做為比較之數據。
本文使用有限元素法透過多種不同的參數來提升溫度的準確度，例如: 求解區域以不同的元素形狀組成、單個元素形狀使用不同的節點數、高斯積分法使用不同的高斯積分點求解積分方程式。本文有限元素法中，使用三角形及四邊形之二維元素，各種元素可分為一個元素中包含不同節點數，三角形元素使用三節點、四邊形元素使用四節點及九節點，本文將以上三種元素組合及搭配的潛熱計算數值方法分析比較總誤差、潛熱釋放及計算時間，判斷何種方法準確度最好、潛熱釋放最多及計算時間最少。

A solidification process has phase change, which involving latent heat effect is important physical phenomenon. FORTRAN programs are written to simulate solidification heat transfer problems such as one-dimensional Stefan problem and one-dimensional Neumann problem with the finite element method and adaptive time step scheme. In this paper, a variety of parameters are utilized to solve the two problems. The elements used are three-node triangular element, four-node quadrilateral element and night-node quadrilateral element. The simulation of latent heat release uses the effective specific heat method, the enthalpy/specific heat method, the adaptive time step of effective specific heat method and the adaptive time step of enthalpy/specific heat method. The Gaussian method is employed to solve the integrations in the finite element equations based on the different Gaussian points. To judge which method is a proper one, the accuracy, the release amount of latent heat and the computation time are used. For the Stefan problem and all numerical methods of handling the latent heat, the quadrilateral element is superior to the triangular element. The adaptive time step of effective specific heat method has improved accuracy and decreased quite substantially on computation time. Compared to the equivalent specific heat method, the enthalpy/specific heat method has improved accuracy significantly at the result of slight increase on the computation time. For the Neumann problem and the effective specific method, the triangular element is superior to the quadrilateral element.

[1] R. Courant, "Variational methods for the solution of problems of equilibrium and vibrations."
[2] J. Turner, L. J. Topp, H. C. Martin, and R. W. Clough, Stiffness and Deflection Analysis of Complex Structures, 1956.
[3] J. Greenstadt, "On the reduction of continuous problems to discrete form," IBM Journal of Research and Development, vol. 3, pp. 355-363, 1959.
[4] R. W. Clough, The Finite Element Method in Plane Stress Analysis: American Society of Civil Engineers, 1960.
[5] Y. Cheung, "Finite Elements in the Solution of," Engineer, 1965.
[6] L. Goodrich, "Efficient numerical technique for one-dimensional thermal problems with phase change," International Journal of Heat and Mass Transfer, vol. 21, pp. 615-621, 1978.
[7] J. Hsiao, "An efficient algorithm for finite-difference analyses of heat transfer with melting and solidification," Numerical Heat Transfer, vol. 8, pp. 653-666, 1985.
[8] A. Date, "A strong enthalpy formulation for the Stefan problem," International journal of heat and mass transfer, vol. 34, pp. 2231-2235, 1991.
[9] C. Swaminathan and V. Voller, "General source-based method for solidification phase change," Numerical Heat Transfer, pp. 175-189, 1992.
[10] T. Tszeng, Y. Im, and S. Kobayashi, "Thermal analysis of solidification by the temperature recovery method," International Journal of Machine Tools and Manufacture, vol. 29, pp. 107-120, 1989.
[11] J. Stefan, "Über die Theorie der Eisbildung, insbesondere über die Eisbildung im Polarmeere," Annalen der Physik, vol. 278, pp. 269-286, 1891.
[12] Y.-C. Liu and L.-S. Chao, "Modified effective specific heat method of solidification problems," Materials transactions, vol. 47, pp. 2737-2744, 2006.
[13] 張宇良, "利用有限元素法求解相變化熱傳問題," 成功大學工程科學系學位論文, pp. 1-107, 2012.
[14] 曾相文, "以有限元素法分析凝固熱傳問題," 成功大學工程科學系學位論文, pp. 1-105, 2013.
[15] 羅浩哲, "凝固熱傳問題之使用高斯積分法的有限元素分析," 成功大學工程科學系學位論文, pp. 1-109, 2014.
[16] 彭勳章, "從澆注到凝固之鑄造過程的可變時間步伐數值分析," 2010.
[17] K. H. Huebner, A. Thornton, and T. G. Byrom, The Finite Element Method for Engineers: Wiley, 1994.
[18] A. Baker, "Finite element computational fluid mechanics," NASA STI/Recon Technical Report A, vol. 83, p. 41874, 1983.
[19] R. D. Cook, Concepts and applications of finite element analysis: John Wiley & Sons, 2007.
[20] O. C. Zienkiewicz, R. L. Taylor, O. C. Zienkiewicz, and R. L. Taylor, The finite element method vol. 3: McGraw-hill London, 1977.
[21] C. A. Felippa and R. W. Clough, The finite element method in solid mechanics: American Mathematical Society, 1970.
[22] P. Davis and P. Rabinowitz, "Abscissas and weights for Gaussian quadratures of high order," J. Res. Nat. Bur. Standards, vol. 56, pp. 35-37, 1956.
[23] H. RATHOD, K. NAGARAJA, B. VENKATESUDU, and N. RAMESH, "Gauss Legendre quadrature over a triangle," J. Indian Inst. Sci, vol. 84, pp. 183-188, 2004.
[24] P. Dunne, "Complete polynomial displacement fields for finite element method(Polynomial displacement fields in membrane and bending plate elements, discussing existence conditions)," Aeronautical Journal, vol. 72, p. 245, 1968.
[25] F. Hussain, M. Karim, and R. Ahamad, "Appropriate Gaussian quadrature formulae for triangles," International Journal of Applied Mathematics and Computation, vol. 4, pp. 023-038, 2012.
[26] H. Rathod, K. Nagaraja, and B. Venkatesudu, "Symmetric Gauss Legendre quadrature formulas for composite numerical integration over a triangular surface," Applied mathematics and computation, vol. 188, pp. 865-876, 2007.
[27] C. Reddy and D. Shippy, "Alternative integration formulae for triangular finite elements," International journal for numerical methods in engineering, vol. 17, pp. 133-139, 1981.
[28] P. C. Hammer and A. H. Stroud, "Numerical evaluation of multiple integrals II," Mathematical Tables and other Aids to Computation, pp. 272-280, 1958.
[29] C. Reddy, "Improved three point integration schemes for triangular finite elements," International Journal for Numerical Methods in Engineering, vol. 12, pp. 1890-1896, 1978.