基于预条件共轭梯度法的直流电阻率三维有限元正演研究  

3D FEM numerical forward modeling of direct current electrical resistivity based on PCG algorithm

作　　者：刘斌[1] 李术才[1] 李树忱[1] 聂利超[1] 

机构地区：[1]山东大学岩土与结构工程研究中心,山东济南250061

出　　处：《岩土工程学报》2010年第12期1846-1853,共8页Chinese Journal of Geotechnical Engineering

基　　金：国家高技术研究发展计划(863)项目(2007AA11Z131);国家自然科学基金专项基金项目(50727904);交通部西部交通建设科技项目(2009318000008)

摘　　要：对直流电阻率法勘探而言,对其反演结果的精度和速度的要求越来越高,这就需要提出一套优化有限元数值正演速度和精度的计算方案。设计了系数矩阵的一维非零元素压缩存储模式,设置了索引数组以便按照行号和列号对元素进行索引,与变带宽存储模式相比,其内存占用量明显减小。为提高正演计算速度,利用预条件共轭梯度法(PCG)求解有限元中的大型稀疏线性方程组。在PCG法中,将雅可比迭代中的对角阵作为预处理矩阵,与其它预处理矩阵相比,其具有求逆方便、无需存储空间的特点,使得大型线性方程组的求解速度大大提高。另外,在直流电阻率三维正演中,采用了异常电位法,提高了电源点附近的解的精度。以二层地层的电阻率勘探为例,初步验证了计算方案的实用性。利用上述方案,重点对隧道含水断层的电阻率法超前探测进行了有限元数值正演,并进行了相应的物理模型试验。对比显示,数值正演结果与试验数据基本一致,且数值正演的速度和精度均显著提高。For the direct current electrical resistivity prospecting,the requirements for the precision and the speed of inversion are higher and higher.Thus,an optimization scheme is proposed to improve the precision and the speed of FEM numerical modeling.1D compression storage model for coefficient matrix is designed,and an index array is set up for matrix element indexing with its row number and column number.Compared with that of the variable bandwidth storage,the memory space of model greatly reduces.In order to improve the speed of numerical modeling,preconditioning conjugate gradient（PCG） algorithm is used to solve large sparse linear system in FEM.For the PCG algorithm,and the diagonal block matrix in Jacobi iteration is used as the preconditioning matrix,the inversion of which is convenient to be solved and doesn＇t occupy memory spacing.In addition,the abnormal potential method is applied to solving the point source field problem and the precision near the source point is greatly improved.The practicality of the calculation scheme is verified by taking electrical resistivity prospecting for two-layer foundation as an example.Using the above calculation scheme,FEM numerical modeling is carried out to simulate advanced detection of water-bearing faults in a tunnel by means of the electrical resistivity method,and the corresponding model tests are performed.The comparison shows that the numerical modeling results are basically consistent with the experimental data,and the precision and the speed of numerical modeling greatly improve.

关 键 词：直流电阻率勘探 3DFEM PCG 预条件矩阵 一维非零元素压缩存储 含水构造超前探测 模型试验 

分 类 号：TU45[建筑科学—岩土工程]