非均匀介质中流动和传热问题的有限分析法

数值求解非均匀介质中的输运问题广泛应用于科学计算和工程领域．介质的强非均匀性给相关问题的准确求解带来极大的困难．近年来，本课题组将有限分析法拓展到该领域，建立了非均匀介质中输运问题的有限分析法．该算法基于网格奇点邻域内类拉普拉斯方程局部解析解构建，算法具有很高的精度，且不依赖于介质的非均匀性强度．不管相邻网格传导率差异如何，仅需对原始网格进行很少地细分就可以获得非常准确的计算结果，因此与其他传统数值算法相比，可以大幅提高计算精度和效率．该算法可广泛应用于求解非均匀多孔介质中的渗流、复合材料中的热传导及电场分布等问题．

Finite Analytic Numerical Method for the Fluid Flows and Heat Transfer in Heterogeneous Media

Numerical solution of the transport problem in heterogeneous media is widely applied in scientific and engineering fields. For the strong heterogeneous case, it is a challenge and long-standing problem to perform an accurate numerical simulation. We find out that this difficulty is caused by the appearance of the singularity when the conductivity is heterogeneous. A finite analytic numerical scheme is proposed to deal with this problem. Numerical examples show that the proposed numerical scheme makes the convergences much quicker than the traditional methods. Using few grid refinements, the proposed numerical scheme can provide very accurate solutions. Especially, the convergent speed of the numerical scheme is independent of the conductivity heterogeneity. In contrast, when using the traditional numerical schemes to simulate the heat conduction or fluid flows in a strong heterogeneous medium, the grid cell needs to be refined dramatically to get an accurate result. The proposed method is applicable to solving the problems of seepage in heterogeneous porous media, heat conduction in composite materials or electric field distribution, etc.