带有边界吸收的热介质温度场数值模拟算法

论文总字数：28943字

摘 要

热传导过程遍及自然生物和人类生活几乎各个方面，热传导问题在很多领域成为了最普遍的问题之一，通过对它们的求解，我们可以解决很多生活、工作的实际问题。而在大多数情况下，难以求得热传导方程的解析解，进而需要研究它的数值解。本文研究了带有边界吸收的热介质传导问题，分别运用差分格式和边界积分方程两种方法对问题进行了求解。本文首先构造向后差分格式和紧差分格式，对两种格式的数值精度进行了比较，得出紧差分格式的数值精度要比向后差分格式高，本文中运用的紧差分格式在空间上是三阶精度，在时间上是二阶精度。运用边界积分方程方法同样对问题进行求解，得出的结果是当介质的介质的热扩散系数较大时，或者高维情况当边界形状不规则时，运用边界积分方程方法求解的精度往往要比运用差分法得到的结果的精度更高，而且求解速度更快。最后，通过实例说明了热传导问题中的介质的热扩散系数、比较点、边界吸收率与数值解误差的关系：比较点、边界吸收率不同，数值解与真实解的误差基本无变化，但是介质的热扩散系数越大，数值解与真实解的误差也越大。

关键词：热传导方程；差分格式；边界积分方程方法

Abstract

The heat conduction problem in current scientific research and engineering applications are one of the most common problems. The importance of studying it is that it can solve many practical problems in life and work by solving all kinds of heat conduction problems. In most cases, it is difficult to find the analytical solution of the heat conduction equation, and then need to study its numerical solution. In this paper, the heat medium conduction problem with boundary absorption is studied, and the problem is solved by using the difference scheme and the boundary integral equation respectively. In this paper, the backward difference scheme and the compact difference scheme are constructed, and the numerical precision of the two schemes is compared. The numerical accuracy of the compact difference scheme is higher than that of the backward difference scheme. The numerical accuracy of the compact difference scheme is .Using the boundary integral equation method to solve the problem, the result is that the boundary variable variable gradient is larger, or the boundary variable at the singularity of the crack problem, the use of boundary integral equation method is often more accurate and faster to solve than the use of the difference method. Finally, the relationship between the thermal conductivity, the observation point, the boundary conduction coefficient and the numerical solution error are explained. When the observation point and the boundary conduction coefficient in the heat conduction problem are changed, the error of the numerical solution and the real solution is basically unchanged. But the thermal conductivity is larger , The greater the error between the numerical solution and the real solution.

KEY WORDS: The heat conduction problem, the difference scheme, the boundary integral equation

目 录

摘要 I

Abstract I

第一章 绪论 1

1.1 引言 1

1.2 热传导问题数值解法的研究现状 1

1.3 热传导方程的常用数值解法 2

1.4 本文的研究内容 2

第二章 带有边界吸收的热传导问题的差分格式 3

2.1 差分格式求解热传导问题的基本步骤 3

2.2 追赶法（Thomas算法）求解方程组的具体步骤 3

2.3 带有边界吸收热传导问题的向后差分格式的构造 5

2.4 带有边界吸收热传导问题的紧差分格式的构造 6

2.5 本章小结 9

第三章 边界积分方程方法求解带有边界吸收的热传导问题 9

3.1 边界积分方程方法的基本原理和发展过程 9

3.2 边界积分方程方法求解混合边界条件下热传导问题的具体步骤 10

3.3 差分格式和边界积分方程方法求解对比 19

3.4 本章小结 20

四，数值实验 20

五，结论 27

参考文献 28

致 谢 29

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