Finite analytic numerical method for three-dimensional quasi-laplace equation with conductivity in tensor form
Finite analytic numerical method for three-dimensional quasi-laplace equation with conductivity in tensor form
复制标题
张量电导率三维拟拉普拉斯方程的有限解析数值方法
DOI：
10.1002/num.22148
复制
发表时间：
2017
影响因子：
3.9
通讯作者：
Cao Wei-Dong
中科院分区：
文献类型：
--
作者：
Wang Min;Wang Yan-Feng;Liu Zhi-Feng;Wang Xiao-Hong;Wang Yong;Cao Wei-Dong
构造了三维准拉普拉斯方程的全张量形式的有限解析数值方法。对于立方网格系统，势变量的梯度在趋向四个不同电导率网格的公共边时会发散。然而，沿切向方向的电势梯度沿着是有限的值。因此，三维准拉普拉斯方程将表现为准二维方程。可以在公共边缘周围找到3D准拉普拉斯方程的近似解析解，其表示为幂律函数和线性函数的组合。利用该近似解析解，构造了三维有限解析数值格式。数值算例表明，本文提出的数值格式仅用或剖分就能给出较精确的解。更重要的是，数值格式的收敛速度与电导率非均匀性无关。相比之下，当使用传统的数值方法时，通常如MPFA方法，网格单元的细化比率需要显着增加，以获得强非均匀情况下的准确结果。Numer Methods Partial Differential Eq 33：1475-1492，2017
The finite analytic numerical method for 3D quasi‐Laplace equation with conductivity in full tensor form is constructed in this article. For cubic grid system, the gradient of the potential variable will diverge when tending to the common edge joining the four grids with different conductivities. However, the potential gradient along the tangential direction is of limited value. As a consequence, the 3D quasi‐Laplace equations will behave as a quasi‐2D one. An approximate analytical solution of the 3D quasi‐Laplace equation can be found around the common edge, which is expressed as a combination of a power‐law function and a linear function. With the help of this approximate analytical solution, a 3D finite analytical numerical scheme is then constructed. Numerical examples show that the proposed numerical scheme can provide rather accurate solutions only with or subdivisions. More important, the convergent speed of the numerical scheme is independent of the conductivity heterogeneity. In contrast, when using the traditional numerical schemes, typically such as the MPFA method, the refinement ratio for the grid cell needs to increase dramatically to get an accurate result for the strong heterogeneous case.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1475–1492, 2017