2.5D有限元素非傅立葉熱傳法則模擬

Abstract

在分析實際工程問題時，所使用的熱傳公式多是基於傅立業熱傳定律，對於常規的熱傳過程，能夠取得不錯的結果。然而，當涉及到一些非常規的熱傳環境，比如極高（低）溫、溫度急劇變化，傳統的熱傳定律將不再適用。因為在傅立業熱傳定律中，熱的傳播速度為無限大，這與物理規律不符。所以學者提出了非傅立業熱傳定律，以期達到更準確的模擬。 本文首先介紹了非傅立業熱傳的一些基本特性。然後通過分離變量法和傅立業轉換，對非傅立業熱傳的控制方程式推導解析解，由此發現其不同於傳統熱傳的一些特性。隨後利用有限元素法模擬其數值解，提出波動無限元素的假設。針對移動熱載重的問題，借鑒了Yang和Hung（2001）在處理行駛的列車對土壤振動的影響時所用的2.5D方法。推導了2.5D有限元素法的控制方程，並和解析解作對比。最後總結了本文的不足之處和未來展望。

The classical Fourier model has often been adopted to analyze the heat conduction problem encountered in various engineering situations, which is quite satisfactory for the majority of problems considered. However, it fails to adequately predict the temperature variations in situations with drastic changes in temperature,, extreme temperature gradients, or with temperatures near absolute zero. Because Fourier’s law implies that the propagation speed of thermal disturbances is infinite, which is a paradox from the physical point of view. Therefore, it was suggested that the conventional Fourier heat equation should be replaced with a non-Fourier heat equation to account for the finite speed of thermal propagation. Some characteristics of the non-Fourier heat conduction is presented in this paper. The analytical solution of the governing equation based on the non-Fourier law is solved by separation of variables and Fourier transform. Comparison of the results obtained by the classical Fourier theory and non-Fourier heat conduction law is carried out, and some discussions are made. Particularly, the dynamic infinite element is employed, along with the finite elements, to get the numerical solution. With the 2.5D finite element method proposed by Yang and Hung (2001), the temperature distribution in a semi-infinite field induced by a moving heat load is studied, and compared with the analytical one. The areas for further improvement or future research are outlined.