在凝固的過程中，凝固材料之顯微結構與材料之品質與物理特性有著很密切的關係，而溫度場和濃度場之變化又影響到材料之顯微結構。因此，在本文的研究中，希望藉由模擬網狀晶之成長模式，來了解凝固時鑄件顯微結構之變化。首先是以含向前擴散之謝荷方程式為基礎，求得網狀晶之初始形狀，接著進一步建立包含溫度及濃度場、表面能效應及原子附著效應之完整成長模式。在液固界面上，使用直接法調整液固界面之位置，以迭代方式求得自我滿足之形狀。數值方法為有限元素法，而在求解矩陣中，利用LU分解法配合天空線存取模式，可減少記憶空間與增快求解速度。接著利用此模式來探討在不同初始濃度、成長速率、溫度梯度及網狀晶間距對網狀晶形狀之影響。期望本文之分析結果可做為進一步研究之參考。

　　In a process of solidification, the variations of the temperature and concentration fields will directly affect the microstructures of materials, which have very close relations with the qualities of materials. In this paper, a mathematical model is built to study the cellular growth. Firstly, the Scheil equation with forward diffusion is used to calculate the initial cellular shape. Secondly, a complete model is set up, including the temperature and concentration fields and the effects of capillarity and atomic attachment. The numerical method is the finite element method and the skyline storage mode and the LU decomposition method are used to solve the matrix equations. In the proposed model, the shape of the solid/liquid interface of cellular growth is not known a priori, rather it is calculated as part of solution to the field problem. The direct iteration method is utilized to compute the self-consistent cellular shape. In this paper, the effects of different control parameters, which are growth rate, temperature gradient and the initial concentration, are investigated. It is hoped that the results of this study can be referred to for the further study.

1.Flemings, M.C., Solidification Processing, McGraw-Hill Book Company, New York, USA, 1974.

2.Kurz, W., and D.J. Fisher, Fundamentals of Solidification, 4th ed., Trans Tech Publication, Aedermannsdrof, Switzerl and, 1998.

3.Gruzleski, J.E., and B.M. Closset, The Treatment of Liquid Aluminum- Silicon Alloys, The American Foundarymen’s Society, 1990.

4.Bates, C.E.,“Alloy Elements Effects on Gray Iron Properties: Part Ⅱ,”AFS Transactions, Vol. 94, pp. 889-912, 1986.

5.Sachar, H. and J.F. Wallace, Effect of Microstructure and Testing Mode on the Fatigure Properties of Gray Iron, AFS Transactions, Vol. 90, pp. 777-793, 1982.

6.Kasap, S., Principles of Electrical Engineering Material and Devices, revised Edition, McGraw-Hill Book company, New York, USA, 2000.

7.Desbiolles, J.D., J.J. Droux, and M. Rappaz, Simulation of Solidification of Alloys by the Finite-Element Method, Computer Physics Reports, Vol. 6, pp. 371-383, 1987.

8.Gandin, Ch.-A., M. Rappaz, and R. Tinillier, Three-Dimension Probabilistic Simulation of Solidification Grain Structure Application to Superalloy Precision Casting, Metallurgical Transactions A, Vol. 24A, pp. 467-479, 1993.

9.Kurz, W., Microsegregation in Rapidly Solidified Ag-15wt-percent- Cu, Journal of Crystal Growth, Vol. 91, pp. 123-125, 1988.

10.Rappaz, M., and Gandin, Ch-A., Probabilistic Modeling of Micro- structure Formation In Solidification Processes., Acta Metall., Vol. 59, pp.945, 1966.

11.Chao, L.S. and W. C. Du, Macro-Micro Modeling of Solidification, Pro. Natl. Sci. Counc. ROC(A), Vol. 23, NO. 5, pp. 622-629, 1999.

12.W. Oldfield, Trans. Am. Soc. Met., Vol. 59, pp. 945, 1996.

13.D. M. Stefanescu and S. Trufnesu ,Z. Metalkd., Vol. 65, pp.610, 1974.

14.Fisher, J.C., referred to by B.Chalmers in Principles of Solidification, Wiley, New York, USA, pp.105, 1966.

15.Ivantsov, G.P., Doklady Akademii Nauk SSSR, Vol 58, pp.567,1947.

16.Glicksman, M.E., R.J. Schaefer, J.D. Ayers, Dendritic Growth---A Test of Theory, Metallurgical Transactions, Vol. 7A, pp.1747, 1976.

17.Langer, J.S., and H. Müller-Krumbahaar, J. of Crystal Growth, Vol. 42, pp. 11, 1977.

18.Mullins, W.W. and R.F. Sekerka, J. of Applied Physics, Vol. 35, pp. 444, 1974.

19.Dantzig, J.A. and L.S. Chao, Low-Gravity Fluid Dynamics and Tranport Phenomenon, edited by J.N. Koster and R.L Sani, Progress in AIAA, Washington, DC, USA, Vol. 150, pp. 477-436, 1990.

20.Han, Q. and J.D. Hunt, Numerical modeling of the growth of a cellular /dendritic array in multi-component alloys, Metall. and Materials Trans. A, Vol. 238(1), pp. 192-195, 1997.

21.Brown SGR, Simulation of diffusional composite growth using the cellular automaton finite difference (CAFD), Journal of Materials Science, Vol. 33(19), pp. 4769-4776, 1998.

22.Yu Ym M, Yang GC, Zhao DW, and Lu YL, Numerical simulation of dendritic growth in undercooled melt using phase-field approach, ACTA PHYSICA SINICA, Vol. 50(12), pp. 2423-2428, 2001.

23.Bower, T.F., H.D. Brody, and M.C. Fleming, MET. Trans., Vol. 236, pp. 624, 1966.

24.Y. Hasbani and M. Engelman, Out-Core Solution of Linear Equation with Non-symmertric Coefficient Matrix, Comput. and Flu. Vol.7, pp. 13-31, 1979.