瞬变电磁三维FDTD正演多分辨网格方法
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摘要
瞬变电磁三维时域有限差分(FDTD)正演的网格剖分受最小网格尺寸、时间步长、边界条件、目标尺寸、模型尺寸等的影响,结构化网格一直存在最小网格尺寸受限于异常目标尺寸的矛盾;尽管非均匀网格能够在保证模型尺寸的前提下尽可能的降低网格数量,但由于Yee网格结构的限制,非均匀网格不能无限制的扩大单一方向的尺寸,这是为了避免边界网格区域出现长宽比过大的畸形网格,影响计算精度甚至导致结果发散.在非均匀网格剖分的基础上,本文提出了瞬变电磁三维FDTD正演的多尺度网格方法,即首先使用较大尺寸的粗网格进行第一次剖分,然后在希望加密的区域进行二次剖分,使计算域中包含粗、细两套网格.尽管细网格包含在粗网格内部,但其具有Yee网格的全部属性,因而可以在网格中设置不同的电性参数模拟不同形状的目标.基于Maxwell方程组推导了细网格内电场和磁场的迭代公式,基于泰勒展开给出了设置粗、细网格后产生的内部边界条件,使电磁场的传播在粗、细网格和时间步进上得到统一.采用均匀半空间中包含三维低阻异常的经典模型和三维接触带复杂模型进行精度验证,发现多分辨网格方法计算结果满足精度要求.使用"L"型异常模型计算采用多分辨网格方法和不采用多分辨网格的传统FDTD方法对比计算效率,发现多分辨网格算法能够显著提高计算效率,并能够保证计算精度.
The grid meshing in three-dimensional transient electromagnetic using finite difference time-domain(FDTD)is usually limited by boundary conditions,time step,target size,model size,etc.The smallest grid size is often determined by the abnormal body size,which is a common contradiction for structured grids.The non-uniform grid is an effective method to reduce the number of total grids.However,the aspect ratio cannot be too large according to thelimitations of Yee cell to keep the computation accuracy.A multi-scale grid scheme is established based on the non-uniform meshing.A coarse grid is meshed in advance and a fine grid is meshed only in the focused areas.Then the modeling area include at least two types of meshes.The fine grid have all the Yee properties although they are inside of the coarse one.The fine grid can also apply resistivity parameters to form different targets.The electromagnetic iterative formula are derived according to the Maxwell equations.The newly generated inside boundary conditions are given based on the Taylor expansion.Then the coarse and fine grids can be calculated in space and time domain.The typical model with a three dimensional body in a homogeneous half space and a complex conductor at a vertical contact are used for the accuracy verification.The modeling result are in good agreement with that of integral equation method,the normal FDTD method,FEM method and MFVM(Mimetic Finite Volume Method)method.An L shaped model is used to compare the modeling efficiency.The computation time is obviously reduced with the new multiscale grid scheme.
The grid meshing in three-dimensional transient electromagnetic using finite difference time-domain(FDTD)is usually limited by boundary conditions,time step,target size,model size,etc.The smallest grid size is often determined by the abnormal body size,which is a common contradiction for structured grids.The non-uniform grid is an effective method to reduce the number of total grids.However,the aspect ratio cannot be too large according to thelimitations of Yee cell to keep the computation accuracy.A multi-scale grid scheme is established based on the non-uniform meshing.A coarse grid is meshed in advance and a fine grid is meshed only in the focused areas.Then the modeling area include at least two types of meshes.The fine grid have all the Yee properties although they are inside of the coarse one.The fine grid can also apply resistivity parameters to form different targets.The electromagnetic iterative formula are derived according to the Maxwell equations.The newly generated inside boundary conditions are given based on the Taylor expansion.Then the coarse and fine grids can be calculated in space and time domain.The typical model with a three dimensional body in a homogeneous half space and a complex conductor at a vertical contact are used for the accuracy verification.The modeling result are in good agreement with that of integral equation method,the normal FDTD method,FEM method and MFVM(Mimetic Finite Volume Method)method.An L shaped model is used to compare the modeling efficiency.The computation time is obviously reduced with the new multiscale grid scheme.
引文
Buselli G,McCracken K G,Rutter H.1992.Manual for Sirotem Field Procedures and Data Interpretation(in Chinese).Jiang B YTrans.Beijing:Geological Publishing House.
Cai H Z,Xiong B,Han M R,et al.2014.3Dcontrolled-source electromagnetic modeling in anisotropic medium using edgebased finite element method.Computer&Geosciences,73:164-176.
Commer M,Newman G.2004.A parallel finite-difference approach for 3D transient electromagnetic modeling with galvanic sources.Geophysics,69(5):1192-1202.
Commer M,Newman G A.2006.An accelerated time domain finite difference simulation scheme for three-dimensional transient electromagnetic modeling using geometric multigrid concepts.Radio Science,41:RS3007,doi:10.1029/2005RS003413.
Ge D B,Yan Y B.2005.Finite Difference Time Domain Method of Electromagnetic Wave(in Chinese).Xi′an:Xidian University Press.
Li J H,Hu X Y,Zeng S H,et al.2013.Three-dimensional forward calculation for loop source transient electromagnetic method based on electric field Helmholtz equation.Chinese Journal of Geophysics(in Chinese),56(12):4256-4267,doi:10.6038/cjg20131228.
Li J H,Farquharson C G,Hu X Y.2017.3Dvector finite-element electromagnetic forward modeling for large loop sources using a total-field algorithm and unstructured tetrahedral grids.Geophysics,82(1):E1-E16.
Li X.2002.Theory and Application of Transient Electromagnetic Sounding(in Chinese).Xi′an:Shaanxi Science&Technology Press.
Li Z,Huang Q.2015.Three dimensional TEM forward modeling using FDTD accelerated by GPU.∥Presented at the Fall Meeting.AGU.
Newman G A,Hohmann G W,Anderson W L.1986.Transient electromagnetic response of a three-dimensional body in a layered earth.Geophysics,51(8):1608-1627.
Newman G A,Commer M.2005.New advances in three dimensional transient electromagnetic inversion.Geophysical Journal International,160(1):5-32.
Oristaglio M L,Hohmann G W.1984.Diffusion of electromagnetic fields into a two-dimensional earth:A finite-difference approach.Geophysics,49(7):870-894.
Piao H R.1990.Theory of Electromagnetic Sounding(in Chinese).Beijing:Geological Publishing House.
Qiu Z P,Li Z H,Li D Z,et al.2013.Non-orthogonal-Grid-based three dimensional modeling of transient electromagnetic field with topography.Chinese Journal of Geophysics(in Chinese),56(12):4245-4255,doi:10.6038/cjg20131227.
Sun H F.2013.Three-dimensional transient electromagnetic responses of water bearing structures in tunnels and prediction of water inrush sources[Ph.D.thesis](in Chinese).Ji′nan:Shandong University.
Sun H F,Li X,Li S C,et al.2013.Three-dimensional FDTDmodeling of TEM excited by loop source considering ramp time.Chinese Journal of Geophysics(in Chinese),56(3):1049-1064,doi:10.6038/cjg20130333.
Wang C Q,Zhu X L.1994.Finite Difference Time Domain Method in Computational Electromagnetic(in Chinese).Beijing:Peking University Press.
Wang T,Hohmann G W.1993.A finite-difference,time-domain solution for three-dimensional electromagnetic modeling.Geophysics,58(6):797-809.
Xu Y C,Lin J,Li S Y,et al.2012.Calculation of full-waveform airborne electromagnetic response with three-dimension finitedifference solution in time-domain.Chinese Journal of Geophysics(in Chinese),55(6):2105-2114,doi:10.6038/j.issn.0001-5733.2012.06.032.
Xue G Q,Li X,Di Q Y.2008.Research progress in TEM forward modeling and inversion calculation.Progress in Geophysics(in Chinese),23(4):1165-1172.
Yan S,Chen M S,Fu J M.2002.Direct time-domain numerical analysis of transient electromagnetic fields.Chinese Journal of Geophysics(in Chinese),45(2):275-284.
Yu W H,Su T,Mittra R,et al.2005.Parallel Finite Difference Time Domain Method(in Chinese).Beijing:Communication University of China Press.
Yu X,Wang X B,Li X J,et al.2017.Three-dimensional finite difference forward modeling of the transient electromagnetic method in the time domain.Chinese Journal of Geophysics(in Chinese),60(2):810-819,doi:10.6038/cjg20170231.
Zhdanov M S.2010.Electromagnetic geophysics:Notes from the past and the road ahead.Geophysics,75(5):75A49.
Zhou J M,Liu W T,Li X,et al.2018.Research on the 3Dmimetic finite volume method for loop-source TEM response in biaxial
anisotropic formation.ChineseJournalofGeophysics(in Chinese),61(1):368-378,doi:10.6038/cjg2018K0598.
Buselli G,McCracken K G,Rutter H.1992.瞬变场法野外工作方法和数据解释手册.蒋邦远译.北京:地质出版社.
葛德彪,闫玉波.2005.电磁波时域有限差分方法.西安:西安电子科技大学出版社.
李建慧,胡祥云,曾思红等.2013.基于电场Helmholtz方程的回线源瞬变电磁法三维正演.地球物理学报,56(12):4256-4267,doi:10.6038/cjg20131228.
李貅.2002.瞬变电磁测深的理论与应用.西安:陕西科学技术出版社.
朴化荣.1990.电磁测深法原理.北京:地质出版社.
邱稚鹏,李展辉,李墩柱等.2013.基于非正交网格的带地形三维瞬变电磁场模拟.地球物理学报,56(12):4245-4255,doi:10.6038/cjg20131227.
孙怀凤.2013.隧道含水构造三维瞬变电磁场响应特征及突水灾害源预报研究[博士论文].济南:山东大学.
孙怀凤,李貅,李术才等.2013.考虑关断时间的回线源激发TEM三维时域有限差分正演.地球物理学报,56(3):1049-1064,doi:10.6038/cjg20130333.
王长清,祝西里.1994.电磁场计算中的时域有限差分法.北京:北京大学出版社.
许洋铖,林君,李肃义等.2012.全波形时间域航空电磁响应三维有限差分数值计算.地球物理学报,55(6):2105-2114,doi:10.6038/j.issn.0001-5733.2012.06.032.
薛国强,李貅,底青云.2008.瞬变电磁法正反演问题研究进展.地球物理学进展,23(4):1165-1172.
闫述,陈明生,傅君眉.2002.瞬变电磁场的直接时域数值分析.地球物理学报,45(2):275-284.
余文华,苏涛,Mittra R等.2005.并行时域有限差分.北京:中国传媒大学出版社.
余翔,王绪本,李新均等.2017.时域瞬变电磁法三维有限差分正演技术研究.地球物理学报,60(2):810-819,doi:10.6038/cjg20170231.
周建美,刘文韬,李貅等.2018.双轴各向异性介质中回线源瞬变电磁三维拟态有限体积正演算法.地球物理学报,61(1):368-378,doi:10.6038/cjg2018K0598.
Cai H Z,Xiong B,Han M R,et al.2014.3Dcontrolled-source electromagnetic modeling in anisotropic medium using edgebased finite element method.Computer&Geosciences,73:164-176.
Commer M,Newman G.2004.A parallel finite-difference approach for 3D transient electromagnetic modeling with galvanic sources.Geophysics,69(5):1192-1202.
Commer M,Newman G A.2006.An accelerated time domain finite difference simulation scheme for three-dimensional transient electromagnetic modeling using geometric multigrid concepts.Radio Science,41:RS3007,doi:10.1029/2005RS003413.
Ge D B,Yan Y B.2005.Finite Difference Time Domain Method of Electromagnetic Wave(in Chinese).Xi′an:Xidian University Press.
Li J H,Hu X Y,Zeng S H,et al.2013.Three-dimensional forward calculation for loop source transient electromagnetic method based on electric field Helmholtz equation.Chinese Journal of Geophysics(in Chinese),56(12):4256-4267,doi:10.6038/cjg20131228.
Li J H,Farquharson C G,Hu X Y.2017.3Dvector finite-element electromagnetic forward modeling for large loop sources using a total-field algorithm and unstructured tetrahedral grids.Geophysics,82(1):E1-E16.
Li X.2002.Theory and Application of Transient Electromagnetic Sounding(in Chinese).Xi′an:Shaanxi Science&Technology Press.
Li Z,Huang Q.2015.Three dimensional TEM forward modeling using FDTD accelerated by GPU.∥Presented at the Fall Meeting.AGU.
Newman G A,Hohmann G W,Anderson W L.1986.Transient electromagnetic response of a three-dimensional body in a layered earth.Geophysics,51(8):1608-1627.
Newman G A,Commer M.2005.New advances in three dimensional transient electromagnetic inversion.Geophysical Journal International,160(1):5-32.
Oristaglio M L,Hohmann G W.1984.Diffusion of electromagnetic fields into a two-dimensional earth:A finite-difference approach.Geophysics,49(7):870-894.
Piao H R.1990.Theory of Electromagnetic Sounding(in Chinese).Beijing:Geological Publishing House.
Qiu Z P,Li Z H,Li D Z,et al.2013.Non-orthogonal-Grid-based three dimensional modeling of transient electromagnetic field with topography.Chinese Journal of Geophysics(in Chinese),56(12):4245-4255,doi:10.6038/cjg20131227.
Sun H F.2013.Three-dimensional transient electromagnetic responses of water bearing structures in tunnels and prediction of water inrush sources[Ph.D.thesis](in Chinese).Ji′nan:Shandong University.
Sun H F,Li X,Li S C,et al.2013.Three-dimensional FDTDmodeling of TEM excited by loop source considering ramp time.Chinese Journal of Geophysics(in Chinese),56(3):1049-1064,doi:10.6038/cjg20130333.
Wang C Q,Zhu X L.1994.Finite Difference Time Domain Method in Computational Electromagnetic(in Chinese).Beijing:Peking University Press.
Wang T,Hohmann G W.1993.A finite-difference,time-domain solution for three-dimensional electromagnetic modeling.Geophysics,58(6):797-809.
Xu Y C,Lin J,Li S Y,et al.2012.Calculation of full-waveform airborne electromagnetic response with three-dimension finitedifference solution in time-domain.Chinese Journal of Geophysics(in Chinese),55(6):2105-2114,doi:10.6038/j.issn.0001-5733.2012.06.032.
Xue G Q,Li X,Di Q Y.2008.Research progress in TEM forward modeling and inversion calculation.Progress in Geophysics(in Chinese),23(4):1165-1172.
Yan S,Chen M S,Fu J M.2002.Direct time-domain numerical analysis of transient electromagnetic fields.Chinese Journal of Geophysics(in Chinese),45(2):275-284.
Yu W H,Su T,Mittra R,et al.2005.Parallel Finite Difference Time Domain Method(in Chinese).Beijing:Communication University of China Press.
Yu X,Wang X B,Li X J,et al.2017.Three-dimensional finite difference forward modeling of the transient electromagnetic method in the time domain.Chinese Journal of Geophysics(in Chinese),60(2):810-819,doi:10.6038/cjg20170231.
Zhdanov M S.2010.Electromagnetic geophysics:Notes from the past and the road ahead.Geophysics,75(5):75A49.
Zhou J M,Liu W T,Li X,et al.2018.Research on the 3Dmimetic finite volume method for loop-source TEM response in biaxial
anisotropic formation.ChineseJournalofGeophysics(in Chinese),61(1):368-378,doi:10.6038/cjg2018K0598.
Buselli G,McCracken K G,Rutter H.1992.瞬变场法野外工作方法和数据解释手册.蒋邦远译.北京:地质出版社.
葛德彪,闫玉波.2005.电磁波时域有限差分方法.西安:西安电子科技大学出版社.
李建慧,胡祥云,曾思红等.2013.基于电场Helmholtz方程的回线源瞬变电磁法三维正演.地球物理学报,56(12):4256-4267,doi:10.6038/cjg20131228.
李貅.2002.瞬变电磁测深的理论与应用.西安:陕西科学技术出版社.
朴化荣.1990.电磁测深法原理.北京:地质出版社.
邱稚鹏,李展辉,李墩柱等.2013.基于非正交网格的带地形三维瞬变电磁场模拟.地球物理学报,56(12):4245-4255,doi:10.6038/cjg20131227.
孙怀凤.2013.隧道含水构造三维瞬变电磁场响应特征及突水灾害源预报研究[博士论文].济南:山东大学.
孙怀凤,李貅,李术才等.2013.考虑关断时间的回线源激发TEM三维时域有限差分正演.地球物理学报,56(3):1049-1064,doi:10.6038/cjg20130333.
王长清,祝西里.1994.电磁场计算中的时域有限差分法.北京:北京大学出版社.
许洋铖,林君,李肃义等.2012.全波形时间域航空电磁响应三维有限差分数值计算.地球物理学报,55(6):2105-2114,doi:10.6038/j.issn.0001-5733.2012.06.032.
薛国强,李貅,底青云.2008.瞬变电磁法正反演问题研究进展.地球物理学进展,23(4):1165-1172.
闫述,陈明生,傅君眉.2002.瞬变电磁场的直接时域数值分析.地球物理学报,45(2):275-284.
余文华,苏涛,Mittra R等.2005.并行时域有限差分.北京:中国传媒大学出版社.
余翔,王绪本,李新均等.2017.时域瞬变电磁法三维有限差分正演技术研究.地球物理学报,60(2):810-819,doi:10.6038/cjg20170231.
周建美,刘文韬,李貅等.2018.双轴各向异性介质中回线源瞬变电磁三维拟态有限体积正演算法.地球物理学报,61(1):368-378,doi:10.6038/cjg2018K0598.