1. Key Laboratory of Metallogenic Prediction of Nonferrous Metals and Geological Environment Monitoring, Ministry of Education(Central South University), Changsha, Hunan 410083, China; 2. School of Geosciences and Info-Physics, Central South University, Changsha, Hunan 410083, China; 3. China Energy Engineering Group Hunan Electric Power Design Institute Co., Ltd., Changsha, Hunan 410083, China; 4. School of Geophysics and Measurement-control Technology, East China University of Technology, Nanchang, Jiangxi 330013, China; 5. College of Information and Electronic Engineering, Hunan City University, Yiyang, Hunan 413002, China
Abstract：In the nodal finite element forward modeling of geophysical electromagnetic methods, the finite element solution of the main field needs to undergo numerical differentiation to derive the auxiliary field or the finite element calculation of potentials is necessary to yield the components of the electromagnetic field. To tackle the problem of low accuracy of traditional post-processing methods, we propose a method based on the superconvergent patch recovery (SPR) for the post-processing of nodal finite element forward modeling, which is applicable to the controlled-source electromagnetic method. First, on the basis of the curl-curl equation of the secondary electric field, the electric field (primary field) is solved with the Galerkin finite element method involving structured hexahedral grids and nodes. Then, given the superconvergence of the nodal finite element method, all adjacent elements at a certain node are used to form a patch. The electric field gradients are subjected to the least-squares surface fitting with Gauss points as sampling points on the patch so that the electric field gradients of nodes on the patch can be reco-vered. Finally, a high-precision magnetic field is achieved according to the recovered electric field gradients, and the high-precision apparent resistivity and phase responses are then obtained. The results show that the SPR-based post-processing algorithm can improve the accuracy of magnetic field components greatly and maintain good stability with slight increases in the memory and calculation time, compared with the conventional shape function differentiation (SFD), Lagrange interpolation (LI), and moving least-squares interpolation (MLSI).
吴桂桔,胡祥云,刘慧.CSAMT三维正演数值模拟研究进展[J].地球物理学进展, 2010, 25(5):1795-1801.WU Guiju,HU Xiangyun,LIU Hui. Progress in CSAMT three-dimensional forward numerical simulation[J].Progress in Geophysics,2010,25(5):1795-1801.
[2]
陈乐寿,王光锷.大地电磁测深法[M].北京:地质出版社,1990.
[3]
Rodi W L. A technique for improving the accuracy of finite element solutions for magnetotelluric data[J].Geophysical Journal International,1976,44(2):483-506.
[4]
陈乐寿,孙必俊.有限元法在大地电磁场二维正演中的改进[J].石油地球物理勘探,1982,17(3):69-76.CHEN Leshou,SUN Bijun. Improvement on the application of finite element method to 2-D forward solution of magnetotelluric field[J].Oil Geophysical Prospecting,1982,17(3):69-76.
[5]
陈小斌.有限元直接迭代算法[J].物探化探计算技术,1999,21(2):165-171.CHEN Xiaobin. Direct iterative algorithm of the finite element[J].Computing Techniques for Geophy-sical and Geochemical Exploration,1999,21(2):165-171.
[6]
陈小斌,胡文宝.有限元直接迭代算法及其在线源频率域电磁响应计算中的应用[J].地球物理学报,2002,45(1):119-130.CHEN Xiaobin,HU Wenbao. Directly iterative finite element algorithm and its application in line current source electromagnetic response modeling[J].Chinese Journal of Geophysics,2002,45(1):119-130.
[7]
陈小斌,张翔,胡文宝.有限元直接迭代算法在MT二维正演计算中的应用[J].石油地球物理勘探,2000,35(4):487-496.CHEN Xiaobin,ZHANG Xiang,HU Wenbao. Application of finite-element direct iteration algorithm to MT 2-D forward computation[J].Oil Geophysical Prospecting,2000,35(4):487-496.
张林成,汤井田,任政勇,等.基于二次场的可控源电磁法三维有限元-无限元数值模拟[J].地球物理学报,2017,60(9):3655-3666.ZHANG Lincheng,TANG Jingtian,REN Zhengyong,et al. Forward modeling of 3D CSEM with the coupled finite-infinite element method based on the se-cond field[J].Chinese Journal of Geophysics,2017,60(9):3655-3666.
[11]
Badea E A,Everett M E,Newman G A,et al. Finite-element analysis of controlled-source electromagnetic induction using Coulomb-gauged potentials[J].Geophysics,2001,66(3):786-799.
[12]
Vladimir P,Jelena K,Josep D L P,et al. A parallel finite-element method for three-dimensional controlled-source electromagnetic forward modelling[J].Geophysical Journal International,2013,56(2):678-693.
[13]
蔡红柱,熊彬,Michael Zhdanov.电导率各向异性的海洋电磁三维有限单元法正演[J].地球物理学报,2015,58(6):2839-2850.CAI Hongzhu,XIONG Bin,MICHAEL Zhdanov. Three-dimensional marine controlled-source electromagnetic modelling in anisotropic medium using finite element method[J].Chinese Journal of Geophysics,2015,58(6):2839-2850.
[14]
陈汉波,李桐林,熊彬,等.基于Coulomb规范势的电导率呈任意各向异性海洋可控源电磁三维非结构化有限元数值模拟[J].地球物理学报,2017,60(8):3254-3263.CHEN Hanbo,LI Tonglin,XIONG Bin,et al. Finite-element modeling of 3D MCSEM in arbitrarily anisotropic medium using potentials on unstructured grids[J].Chinese Journal of Geophysics,2017,60(8):3254-3263.
[15]
Zienkiewicz O C,Cheung Y K. The Finite Element Method in Structural and Continuum Mechanics:Numerical Solution of Problems in Structural and Continuum Mechanics[M].McGraw-Hill Publishing Company Limited,1967.
[16]
Zienkiewicz O C,Zhu J Z. A simple error estimator and adaptive procedure for practical engineering analy-sis[J].International Journal for Numerical Methods in Engineering,1987,24(2):337-357.
[17]
Zienkiewicz O C,Zhu J Z. The superconvergent patch recovery and a posteriori error estimates, Part Ⅰ:The recovery technique[J].International Journal for Numerical Methods in Engineering,1992,33(7):1331-1364.
[18]
Zienkiewicz O C,Zhu J Z. The superconvergent patch recovery and a posteriori error estimates, Part Ⅱ:Error estimates and adaptivity[J].International Journal for Numerical Methods in Engineering,1992,33(7):1365-1382.
[19]
Zienkiewicz O C,Zhu J Z. The superconvergent patch recovery(SPR) and adaptive finite element refinement[J].Computer Methods in Applied Mechanics & Engineering,1992,101(1):207-224.
[20]
Zienkiewicz O C,Zhu J Z. Superconvergence and the superconvergent patch recovery[J].Finite Elements in Analysis & Design,1995,19(1-2):11-23.
[21]
Ren Z,Tang J. 3D direct current resistivity modeling with unstructured mesh by adaptive finite-element method[J].Geophysics,2010,75(1):H7-H17.
[22]
Tabbara M,Blacker T,Belytschko T. Finite element derivative recovery by moving least square interpolants[J].Computer Methods in Applied Mechanics & Engineering,1994, 117(1-2):211-223.
