摘要：Meshfree method offers high accuracy and computational capability and constructs the shape function without relying on predefined elements. We comparatively analyze the global weak form meshfree methods, such as element-free Galerkin method （EFGM）, the point interpolation method （PIM）, and the radial point interpolation method （RPIM）. Taking two dimensional Poisson equation as an example, we discuss the support-domain dimensionless size, the field nodes, and background element settings with respect to their effect on calculation accuracy of the meshfree method. RPIM and EFGM are applied to controlled- source two-dimensional electromagnetic modeling with fixed shape parameters. The accuracy of boundary conditions imposed directly and by a penalty function are discussed in the case of forward modeling of two-dimensional magnetotellurics in a homogeneous medium model. The coupling algorithm of EFG-PIM and EFG-RPIM are generated by integrating the PIM or RPIM and EFGM. The results of the numerical modeling suggest the following. First, the proposed meshfree method and corresponding coupled methods are well-suited for electromagnetic numerical modeling. The accuracy of the algorithm is the highest when the support-domain dimensionless size is 1.0 and the distribution of field nodes is consistent with the nodes of background elements. Second, the accuracy of PIM and RPIM are lower than that of EFGM for the Poisson equation but higher than EFGM for the homogeneous medium MT response. Third, RPIM overcomes the matrix inversion problem of PIM and has a wider selection of support-domain dimensionless sizes as compared to RPIM.

摘要：在电磁法节点有限元正演模拟中,需要对主场的有限元解进行数值微分求取辅助场,或者求解位的有限元解,从而得到电磁场分量。针对传统的后处理方法精度较低的问题,引入一种超收敛单元片梯度值恢复(SPR)技术,应用于可控源电磁法节点有限元正演的后处理。首先,基于可控源电磁法的电场二次场双旋度方程,采用结构化的六面体网格和节点伽辽金有限元法求解电场(主场)分量;然后,根据节点有限元法的超收敛性质,以围绕某一节点的所有的单元组成单元片,在单元片上以高斯点作为采样点对电场梯度值进行最小二乘曲面拟合,恢复单元片上节点的电场梯度值;最后,计算高精度磁场,进而获得高精度的视电阻率和相位响应。模型算例分析表明,与常规的单元形函数微分(SFD)法、拉格朗日插值(LI)法和移动最小二乘(MLSI)法相比,超收敛单元片梯度值恢复后处理技术能在极小幅度地增加内存和计算时间的情况下,非常显著地提高磁场分量的精度并且保持良好的稳定性。

摘要：无单元Galerkin法(EFGM)作为一种相对成熟的无网格方法,避免了网格剖分,其精度高,适用于复杂电导率分布和复杂边界形状的计算。本文将EFGM用于大地电磁三维正演,详述了三维EFGM形函数的构造过程,从大地电磁三维变分问题出发,利用Galerkin法结合高斯积分公式推导了相应的系统矩阵离散表达式,简述了边界条件的加载技术,研究了支持域尺寸对EFGM三维正演计算精度的影响,最后通过数值计算验证了EFGM三维算法的正确性。