雙曲線熱傳導問題之數值方法探討

Numerical Methods for the Hyperbolic Heat Conduction Problems

陳金治(Chin-Chih Chen)

指導教授 ： 陳澤明

國立臺北科技大學/機電學院/機電科技研究所/博士(2012年)

摘要

為解決雙曲線熱傳導方程式數值結果振蘯的問題，本研究提出Collection method、混合格林函數法與混合積分轉換法用以解決雙曲線熱傳導問題中銳緣波前的現象。以Collection method求解一具曲率因素之實心體的一維雙曲線熱傳導問題，在研究中以拉氏轉換移除統御方程式中的時間項並結合雙曲線形狀函數用以求解雙曲線熱傳導問題，最後提出四個不同例子加以分析。
以混合格林函數法求解一維至三維的直角、圓柱與球狀座標之雙曲線熱傳導問題，在研究中使用拉氏轉換移除統御方程式的時間項，以格林函數求解空間域之溫度函數。在研究中分別提出四到六個例子加以分析。
以混合積分轉換法求解一維至三維的直角、圓柱與球狀座標之雙曲線熱傳導問題，在研究中使用拉氏轉換移除統御方程式的時間項，以積分轉換法求解空間域之溫度函數。在研究中分別提出五到六個例子加以分析。
經研究發現，三種方法之數值結果與Kao所求得的解析解比較均可獲致良好的一致性，且可有效的解決雙曲線熱傳導問題的數值振蘯的狀況。

關鍵字

Collection method ； 混合格林函數法 ； 積分轉換法 ； 雙曲線熱傳導

並列摘要

The difficulty encountered in the numerical solutions of hyperbolic heat conduction problems (HHC) is the numerical oscillation in vicinity of sharp discontinuities. In the present study, we have proposed collection method, hybrid Green function and hybrid integral transform method to overcome the numerical oscillation on HHC.
Using the collection method investigates the effect of the surface curvature of a solid body on HHC. The present method combined the Laplace transform and the hyperbolic shape function to solve time dependent HHC equation; four different examples have been analyzed by the current method.
The hybrid Green function is developed to solve HHC problems in Cartesian, cylindrical and spherical coordinates system. The present method combines with the Laplace transform for the time domain, Green function for the space domain. For one- two-, and three- dimensional problems, four to six different examples have been analyzed.
A hybrid Integral transform method is applied in Cartesian, cylindrical and spherical coordinates of HHC problems. The present method combines with the Laplace transform for the time domain, integral transform scheme for the space domain. For one- two-, and three- dimensional problems, five to six different examples have been analyzed.
It is found from these examples that the three methods are in good agreement with the analytical solutions [19] and do not exhibit numerical oscillations at the wave front.

並列關鍵字

Collection method ； Hybrid Green function method ； Integral transforms method ； hyperbolic heat conduction

參考文獻

[1] H.D.Weymann, “Finite speed of propagation in heat conduction, diffusion, and viscous shear motion,” American J. Phys., vol.35, 1967, pp.488-496.

[2] H. S. Carslaw and J.C.Jaeger, Conduction of Heat in Solids, second ed., New York: Oxford University, 1959.

[3] D.C. Kelly, “Diffusion: a relativistic appraisal,” American J. Phys., vol. 36, 1968, pp. 585-591.

[4] J. C. Maxwell, “On the dynamic theory of gases,”Phios Trans. Soc. London, vol.157, 1867, pp.49-88.

[5] C. Cattaneo, “Sulla conduzione de calore,” Atti del semin. Mat. e Fis. Univ. Modena, vol. 3, 1948, p.3.