本文研究凝固熱傳現象，主要探討的問題為一維的史帝芬問題及二維的Rathjen問題，本文使用有限元素法求解這兩問題，分析這兩問題的溫度分佈。
當材料於凝固過程中即會有潛熱釋放，而潛熱釋放的模擬對凝固問題極為重要，潛熱釋放的效應算的越準溫度分佈亦會越準，本文使用兩種方法計算潛熱釋放的效應，分別為等效比熱法及等效比熱/熱焓法，本文將比較此兩方法計算溫度場的準確度，也比較此兩方法計算過程所需時間，對於溫度場的準確度比較，本文以總誤差(total-error)做為比較的數值。
本文使用的數值模擬方法為有限元素法做數值運算，使用有限元素法模擬求解物理問題可透過調整多種不同的參數來達到提升準確度的效果，例如:以不同形狀的元素組成求解區域求解、使用不同的數值積分方式求解積分方程式、對於單個元素增加其節點數(element node)、縮小計算的時間步伐等等，以上這些方式皆能對精準度造成影響，本文將分析上述方法去判斷何種方法的使用能提升準確度又能快速計算完成，縮短計算所需耗費時間。
有限元素法中，對於二維問題常使用三角形元素及四邊形元素，而各種元素又可分為一個元素中包含不同節點數(element node)，一般三角形元素多使用三節點而四邊形元素多使用四節點，本文也使用八節點的四邊形元素去做計算，比較以上三種元素的總誤差及計算時間。
有限元素法的求解過程需要計算元素方程式，即求解積分方程式為代數方程式，本文使用高斯積分法求解積分方程式，利用多種不同case的高斯積分點做計算，也比較多種不同積分點計算總誤差及計算間。

Solidification is often encountered in manufacturing processes. Solidification process analysis is very important for manufacturing since traditional industries, high-tech industry and semiconductor industry all need to deal with problems arising from solidification. In the study, FORTRAN programs are written to simulate solidification heat transfer problems via the finite element method, which includes one-dimensional Stefan problem and two-dimensional Rathjen problem. 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 eight-node quadrilateral element. The simulation of latent heat release uses the effective specific heat method and the enthalpy/specific heat method. The Gaussian method is employed to do the integrations in the finite element equations based on the Gaussian points and the corresponding weighting factors. For the Stefan problem, the quadrilateral element is superior to the triangular element. Four-node quadrilateral element with the enthalpy/specific heat method is very close to eight-node quadrilateral element with the effective specific method, but the latter one requires significantly more computation time. More Gauss points do not necessarily lead to a more accurate solution. Compared to the equivalent specific heat method, the enthalpy/specific heat method has improved accuracy significantly at the cost of a slight increase in the computation time.

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