双层球对称热传导问题的解及数值验证

The Solution and Numerical Verification of the Two-Layer Spherical Symmetric Heat Conduction Problem

摘要: 具有两种材料双层热传导问题，由于具有两种传热性质不同的材料，标准的三角函数正交系难以满足两种材料界面处温度和热流量的连续条件，问题难以求解。本文基于三角函数在正交性上良好的性质，根据函数系正交性的定义，用连续条件构造了一个间断的三角函数正交系，并且该正交函数系的系数中含有待定函数。利用该正交函数系级数展开的方法可以满足连续条件、边界条件和任意的初始条件，并且其系数中的待定函数由连续条件和初始条件共同确定，而在一般情况下系数中的待定函数是由初始条件决定的。该解析解与数值模拟对比一致性较好。文中提出的求解方法构造了一种新的间断的正交函数系，为多相介质的热传导问题提供了解析解。也为其他多相介质求解析解或者求解不光滑的解析解提供了一种新的思路。

Abstract: Because there are two kinds of materials with different heat transfer properties, the standard trigonometric orthogonal system makes it difficult to meet the continuous conditions of temperature and heat flow at the interface of the two materials, and the problem is difficult to solve. In this paper, a discontinuous orthogonal system of trigonometric functions is constructed based on the good properties of orthogonality of trigonometric functions and the definition of orthogonality of function systems, and its coefficients contain undetermined functions. The continuous condition, boundary condition and any initial condition can be satisfied by the method of series expansion of the orthogonal system of functions, and the undetermined function in the coefficient is determined by the continuous condition and the initial condition together. In contrast, the initial condition, in general, determines the undetermined function in the coefficient. The analytical solution is in good agreement with the numerical simulation. The method proposed in this paper constructs a new discontinuous orthogonal function system, which provides an analytical solution for the heat conduction problem in multiphase media. It also provides a new way of thinking for solving analytic solutions of other polyphase media or solving non-smooth analytical solutions.