奇异热流密度场的数值分析(Ⅰ)——热传导特征解问题

为得到用于分析奇异热流密度场的高效的有限元列式，针对不同材料中界面裂纹尖端的扇形区域，推导出二维热传导特征解问题的基本方程和边界条件的弱形式．利用特征方程展开方法，可获得分析裂纹尖端处二维热传导特征解的一维有限元列式．该列式只需对扇形区域在角度方向上离散，最后得到一个二次特征根矩阵的总体方程．求解该方程可得到二维热传导问题的特征解．数值计算表明，该方法可高效准确地求解奇异热流密度场特征解．

Numerical Analysis of Singular Heat Flux Field (Ⅰ): Eignenproblems of Heat Conduction

In order to get a finite element formulation to analyze singular heat flux fields, the weak form of basic equations and boundary conditions describing the 2D heat conduction eigenproblems is derived for the sectorial domains in the vicinity of the interfacial crack tip. By using the eigenfunction expansion technique, a one-dimension finite element formulation is then developed to determine the eigensolutions to the 2D heat conduction problem at the crack tip, which discretizes the sectorial domain circumferenially. A global equation is finally obtained as a second-order characteristic matrix equation. By solving this second-order matrix equation, the eigensolutions to the 2D heat conduction problems in singular heat flux fields can be obtained. Numerical examples demonstrate the correctness and efficiency of the proposed method.