变物性非线性热传导问题的泛函存在条件及变分定理

用有限元法分析热弹性问题时,需要导出热传导问题控制方程的泛函,并建立变分定理.当温度变化范围较大时,热传导的物性系数随温度的变化不能忽略.考虑到这种情况,本文给出非线性的热传导问题的控制方程组,证明了这一非线性问题的泛函仅在一定条件下才能存在,而这些条件不完全具备,不能建立变分定理.为此,本文将控制方程组分解成为两个线性方程组,第一个方程组(或主方程组)是物性系数为常数时的非定常热传导方程及其边界条件和初始条件,它的泛函和变分定理已为一些著作所给出,虽然缺乏严格的证明;第二个方程组(或辅方程组)是物性系数变化引起的温度场增量部分所构成的变系数非齐次的线性方程及相应的边界条件和初始条件,本文证明了它的泛函存在性并建立了变分定理.

The Existence of Functional and Variational Theorem in Nonlinear Heat Conduction with Variable Thermal Property

The nonlinear equations of non-stationary heat conduction with variable thermal property are giv- en.It has been proved that the functional of this nonlinear problem doesn't exist. The nonlinear equations set is divided into two sets of linear equations.The 1st set is that the mate- rial coefficients are constants.Its functional and variational theorem were given already.The 2 nd set consists of the temperature increments due to the fact that the material coefficients are variables.The existence of functional of this set is proved and corresponding varational theorem is deduced in this pa- per.