基于无单元Galerkin法电流场数值模拟

Numerical Simulation of Current Field Based on Element-Free Galerkin Method

摘要: 直流电场勘探中，一般采用点电源通过傅氏变换来进行电流场二维正演，相对于利用线源进行二维电流场正演来说，其计算较为复杂；同时，无单元法解决了传统有限元方法的缺点，无单元Galerkin法只对节点进行处理，不受网格的影响，前期处理十分简单，具备精确的计算值、能够求解二次连续的优势。本文通过推导线源直流电的微分方程，采用强加边界条件即第一类边界条件，基于无单元Galerkin法建立了电位场目标函数。然后，由经典的均匀地下半空间模型，通过无单元Galerkin法正演模拟计算与理论值计算结果进行比较分析，验证了该方法的有效性及准确性，最后，在均匀地下半空间模型的基础上，建立了高阻和低阻异常体，通过正演计算，得到了其电位分布与电荷密度分布结果。结果表明该方法适用于对地下构造的正演模拟分析，其能够准确的找出地下含异常的位置，为实际中直流电场反演地下介质信息奠定了基础。

Abstract: In direct current field exploration, two-dimensional current field forward modeling is generally carried out by point power source through Fourier transform. Compared with two-dimensional current field forward modeling by line source, its calculation is more complicated. At the same time, the element-free method solves the shortcomings of the traditional finite element method. The element-free Galerkin method only deals with nodes and is not affected by the grid. The preliminary processing is very simple, and it has the advantages of accurate calculation values and quadratic continuity. In this paper, the objective function of potential field is established based on the element-free Galerkin method by deriving the differential equation of linear source direct current and applying the imposed boundary condition (the first kind of boundary condition). Then, based on the classical underground half-space model, the effectiveness and accuracy of this method are verified by comparing the forward simulation results of element-free Galerkin method with the theoretical calculation results. Finally, on the basis of the uniform underground half-space model, the high and low resistance abnormal bodies are established. Through forward calculation, the potential distribution and charge density distribution were obtained. The results show that this method is suitable for forward modeling and analysis of underground structure, and it can accurately find the location of underground anomaly, which lays a foundation for inversion of underground media information by DIRECT current electric field in practice.