基于辛方法的二维非稳态热传导问题研究

摘要

Based on the two-dimensional heat conduction theory, the Hamiltonianm system is introduced to solve the unsteady heat conduction, and the original problems come down to solve the eigensolutions of zero eigen-value and non-zeros eigenvalue with the methods of separation of variables and eigenfunction expansion. The sym-plectic concept makes no hypothesis of deformation along the thickness direction and shows a rational derivation. Thus, the current method can precisely analyze thermal conduction with arbitrary depth-to-length ratio and time step. Numerical examples in comparison with other methods are given to illustrate the accuracy of the present sym-plectic approach and the effect of boundaries of temperature and heat flux under different thickness ratios and time are studied. The results show that effect of boundaries will be more obvious increasing of thickness ratios, while temperature will attenuate quickly with the time increasing, and the heat flux changed little. The symplectic method can provide a new idea for the problem of transient heat conduction.%基于二维热传导理论，通过引入对偶变量，推导了非稳态热传导温度场问题的辛对偶方程组。采用分离变量法和本征展开方法，建立起一种本征值和本征解的直接求解方法，得到了适用于任意跨厚比的平面非稳态问题的解析解。由于在求解过程中不需要事先人为地选取试函数，而是从基本方程出发，直接利用数学方法求出问题的解，使得问题的求解更加合理化。探讨不同跨厚比、不同时间步长情况下温度和热流密度的分布规律，并与已有解进行比较。结果表明，辛方法是一类可行的研究非稳态热传导的方法。考虑到非零本征值本征解具有局部性特点，进一步讨论不同跨厚比、不同时间情况下温度和热流密度分布的端部效应问题。为非稳态问题的理论及实际应用研究提供了新的途径。