舉凡傳統產業的鑄造以及當今半導體科技，材料凝固過程中潛熱釋放將會影響熱場等等物理場，是重要的物理現象。本文使用有限元素法配合不同數值方法計算潛熱釋放效應，分析凝固熱傳問題的溫度場分布。
此次探討的凝固熱傳問題為史蒂芬問題與紐曼問題。使用的數值方法有等效比熱法、等效比熱法之GLS法(α=2)、等效比熱之修正GLS法(α=2~5)、等效比熱熱焓法、等效比熱熱焓法之GLS法(α=2) 、等效比熱熱焓法之修正GLS法(α=2~5)。藉由比較這些方法的優劣，來判斷使用修正之GLS法與一般GLS法或沒有使用的差別。同時也比較同種方法之間各種元素形狀的差別。在此使用的元素形狀為三邊形三節點、四邊形四節點與四邊形九節點。
判斷標準為凝固過程的總誤差、CPU之運算時間、總潛熱釋放之完整性。

In this study, FORTRAN program is used to simulate the solidification heat transfer problem, such as one-dimensional Stefan problem and Neumann problems with the finite element method and the modified GLS method. Different numerical methods and parameters are used to solve the solidification problems. The employed elements are three-node triangular, four-node quadrilateral and nine-node quadrilateral elements. The calculation of latent heat release uses the effective specific heat method and the enthalpy/specific heat scheme. The adaptive time step scheme employed in the work is the modified GLS method. Guassian quadrature is utilized to calculate the integration in element equation based on the Gaussian points and weight factors. To compare the accuracy of various methods, the total errors of the computed temperature solutions are used as the comparison basis, which are affected by the element shape and the size of time step. By comparing the total error, the latent heat release, and the computation time, the best time step, parameter (α) and numerical methods could be chosen to deal with solidification problems and the results that meet the requirements could be obtained.

[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] V. John and J. Rang, "Adaptive time step control for the incompressible Navier–Stokes equations," Computer Methods in Applied Mechanics and Engineering, vol. 199, pp. 514-524, 2010.

[8] T. Coupez, G. Jannoun, N. Nassif, H. C. Nguyen, H. Digonnet, and E. Hachem, "Adaptive time-step with anisotropic meshing for incompressible flows," Journal of Computational Physics, vol. 241, pp. 195-211, 2013.

[9] Douglas J., Gallie T. M., On the numerical integration of a parabolic differential equation subject to a moving boundary condition, Duke Mathematical Journal 22 (1955)

[10] Goodling J. S., Khader M. S., Inward solidification with radiation-convection boundary condition,
Journal of Heat Transfer-Transactions of the ASME 96 (1974) 114-115.
[11] Gupta R. S., Kumar D., A modified variable time step method for one-dimensional Stefan problem,Computer Methods in Applied Mechanics and Engineering 23 (1980) 101-109.

[12] Gupta R. S., Kumar D., Variable time step methods for one-dimensional Stefan problem with mixed
boundary condition, International Journal of Heat and Mass Transfer 24 (1981) 251-259.

[13] Ouyang T, Tamma K. K., On adaptive time stepping approaches for thermal solidification processes,International Journal of Numerical Methods for Heat and Fluid Flow 6 (1996) 37-50

[14] Gresho P. M., Lee R. L., Sani R. L., Recent Advance in Numerical Methods in Fluids, Pineridge Press,Swansea, 1980

[15] 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.

[16] A. Date, "A strong enthalpy formulation for the Stefan problem," International journal of heat and mass transfer, vol. 34, pp. 2231-2235, 1991.

[17] C. Swaminathan and V. Voller, "General source-based method for solidification phase change," Numerical Heat Transfer, pp. 175-189, 1992.

[18] 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.

[19] J. Stefan, "Über 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, 2006.
[21] 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
[22] 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
[23] 王姿婷, "利用有限元素法及可變時間步伐探討相變化之熱傳問題," 成功大學工程科學系學位論文
[24] 游振暉,"以可變時間步伐分析凝固熱傳問題," 成功大學工程科學系學位論文
[25] 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.
[26] M. M. Joosten, W. G. Dettmer and D. Peric, Analysis of the block Gauss–Seidel solution procedure for a strongly coupled model problem with reference to fluid–structure interaction

[27] L.F. Shampine, Error Estimation and Control for ODEs,2004

[28] ZANARIAH ABDUL MAJID* & MOHAMED SULEIMAN, Predictor-Corrector Block Iteration Method for Solving Ordinary Differential Equations, Sains Malaysiana, vol.40, pp. 659-664

[29] Scott R. Fulton Revised, Semi-Implicit Time Differencing,2004