群表示理论在三维电磁体积分方程高效求解中的应用

Application of the group theory in high-efficient 3D EM modeling using integral equations method

电磁法是矿产资源勘探和地壳结构研究的重要手段.传统电磁法理论局限于一维情况,无法处理复杂地质条件.积分方程法只需对异常体进行剖分,极大减小了内存需求,在三维数值模拟中存在较大优势.但对于大型异常体,积分方程离散涉及大型稠密电磁阻抗矩阵求逆和异常体散射电流线性方程组求解.阻抗矩阵需要计算大量的张量格林函数体积分,显著制约了积分方程法的模拟效率.群表示是对称性的一种有效数学表达.本文的目的是利用异常体对称性对散射电流的影响,减少阻抗矩阵元素的计算量,从而提高三维数值模拟效率.本文首先简要介绍了体积分方程和群表示的基本原理.然后,以C2v群为例给出了基函数和检验函数的对称群表示.在此基础上,推导了电磁阻抗矩阵引入C2v群的简化方程.最后,我们采用改进的积分方程法模拟了均匀半空间中具有两个垂直对称面的三维异常体的电磁响应.结果表明,C2v的群表示可将复阻抗矩阵转化为块对角阵,使矩阵计算时间减少至1/4,总存储量和内存量分别减少至1/4和1/16,线性方程组求解时间缩短至1/12.改进算法适用于非均匀地单元离散和电导率分布.此外,本方法与观测系统的几何参数及矩量法的基函数和权函数集无关,极具灵活性,方便程序的编制和植入.总之,群表示对提高三维体积分方程法的模拟效率具有巨大的应用前景.

The electromagnetic method is an important tool in the exploration of mineral resources and research of crustal structure.The traditional electromagnetic theory is limited to the one-dimensional case and thus cannot handle complex geological conditions.In the integral equation method,only the anomaly body needs to be divided,which greatly reduces the memory requirement and makes the method more advantageous in the three-dimensional numerical simulation.However,for large anomaly bodies,the discretization of integral equations involves the inverse of the large and dense electromagnetic impedance matrix and the resolve of the linear equations relating to the scattering current inside the anomaly body.A large amount of volume integrations of tensor Green’s functions are required in the impedance matrix,which significantly limits the simulation efficiency of integral equation method.The group representation is an effective mathematical expression of symmetry.By utilizing the influence of symmetry of anomaly body on the scattering current,the purpose of this paper is to reduce the calculation amount of impedance matrix elements,thus improving the efficiency of three-dimensional numerical simulation.This paper first briefly introduces the basic principles of the volume integral equation and group representation.Then,the symmetry group representation of the basis function and test function tests are given for the group C2v.On this basis,a simplified equation is derived by introducing the group into the electromagnetic impedance matrix.Finally,the improved integral equation method is applied into simulating the three-dimensional electromagnetic response of the anomaly body with two perpendicular symmetry planes in a uniform half-space.The results show that,due to diagonalization of the impedance matrix by using C2v,the computation time of the matrix decreases to around one fourth of the original one.The total amount of storage and memory are reduced proximately to one fourth and one sixteenth,respectively.Time for solving the linear equations is shortened by a factor of 12.The improved method works for the case of non-uniform cell-size division and inhomogeneous conductivity distribution.In addition,the method is independent of the survey geometry configuration as well as the basis functions and weight functions in the method of moment,which makes the method great flexible and helps with the accomplishment and implant of programming.In conclusion,the group representation has great prospects for improving the efficiency of three-dimensional volume integral equation method.