以簡易可變時間步伐暨有限元素法探討鑄造之凝固熱傳問題

Study of Solidification Heat Transfer Problem in Casting Process by Using Simple Adaptive Time Stepping Scheme and Finite Element Method

王姿婷(Tzu-Ting Wang) ； 趙隆山(Long-Sun Chao)

《鑄造工程學刊》 43卷2期 (2017/06) Pp. 11-24

摘要

潛熱釋放在凝固過程中的相變化是極為重要的物理現象，本文以有限元素法搭配等效比熱法及可變時間步伐之等效比熱法來處理潛熱效應，一般而言使用可變時間步伐會使計算效率增加，因此將著重於可變時間步伐對於準確度與計算效率的影響。比較上述方法之所需計算時間、潛熱釋放多寡及溫度分佈準確度，並探討一維史帝芬問題、一維紐曼問題及二維瑞特延問題。其中利用溫度分佈之數值解與正解的總誤差(total-error)來比較各數值方法及節點的準確度，為使溫度分佈更精準，本研究藉由變換所採用之數值方法及調整參數來達到此目的，例如：縮小時間步伐、針對單位元素增加節點數、以不同形狀之元素組成求解區域及增加求解區域之節點數等等。研究結果發現史帝芬問題之等效比熱法使用四邊形元素求解之溫度分佈較精準，而加上可變時間步伐的等效比熱法溫度分佈及潛熱釋放則較固定時間步伐準確，且電腦運算時間皆有減少；紐曼問題加上可變時間步伐雖不能增加準確度，但可提升計算效率；瑞特延問題使用三種元素節點所得液固界面位置皆很接近數值解，其中以四邊形元素平均誤差較小。

關鍵字

有限元素法 ； 可變時間步伐 ； 凝固熱傳 ； 等效比熱法

並列摘要

The phase change involving latent heat effect in a solidification process is an important physical phenomenon because it may affect the accuracy of the temperature distribution. By finite element method and adaptive time scheme, FORTRAN programs are written to simulate the heat transfer problems including one-dimensional Stefan problem, one-dimensional Neumann problem and two-dimensional Rathjen problem. The methods of calculating latent heat releases are the effective specific heat method and the adaptive time step of effective specific heat method with four-node quadrilateral element, nine-node quadrilateral, and three-node triangular element. Generally, utilizing adaptive time stepping scheme could improve the calculation efficiency and the accuracy of temperature. In this study, Gaussian method is employed to solve the integral for element equations. To determine which numerical method or number of nodes in an element is proper, the accuracy of temperature, release of latent heat, and calculation time of CPU are considered. Based on the result of Stefan problem, the better accuracy of temperature is observed in the quadrilateral element than in the triangular element. Further, the adaptive time step of effect specific heat method could obtain greater accuracy of temperature and calculation efficiency than the uniform time stepping scheme. The result of Neumann problem is similar to the Stefan problem with the effective specific heat method. Whereas, the adaptive time step of effective specific heat method could only reduce the calculation time but not improve the accuracy of temperature. For Rathjen problem, using triangular element brings out higher average error than quadrilateral element.

並列關鍵字

finite element method ； adaptive time step ； effective specific heat method ； solidification