大地电磁二维正演中的有限元-径向基点插值法

摘要

径向基点插值法(RPIM)作为一种插值型无网格方法，为改善无网格点插值法(PIM)在形函数构造过程中可能出现的矩阵奇异性问题而提出的一种方法，该算法支持域无量纲尺寸的选择区间大，能更好地处理各类工程与科学计算问题。介绍了RPIM的近似原理，给出了径向基函数形状参数的推荐值；从大地电磁二维变分问题出发利用Galerkin法结合高斯积分公式推导出相应的系统矩阵离散表达式；为提高RPIM的计算效率，将RPIM与有限元法(FEM)耦合，提出了有限元－径向基点插值法(FE-RPIM)，多个模型的数值计算验证了RPIM精度高、处理复杂模型便利及耦合法计算复杂模型高效的特点。%Polynomial basis interpolation method (RPIM), as a kind of typical interpolation meshfree method, was proposed to overcome the defects of point interpolation method (PIM) that the construction process of the shape function involves the matrix inverse operation. This method overcomes the matrix inverse problem, and supports the wider domain dimensionless size interval to better deal with all kinds of engineering and scientific computing problems. The approximate principle of RPIM was introduced in detail, and the discrete system matrix expression corresponding to the magnetotelluric two-dimensional variational problem by combining the Galerkin method and the gauss integral formula was deduced. In order to overcome the defects of low computational efficiency of RPIM, the finite element−radial point interpolation method (FE−RPIM) based on coupling the FEM and RPIM was proposed. The conclusions were verified by the numerical calculation of several models. The results show that RPIM has the advantage of high precision and convenience to calculate complex models, and FE-RPIM has the characteristics of high calculation efficiency for complex models.