导电媒质中似稳瞬态电磁场响应的直接时域数值模拟
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摘要
随着电磁场理论的发展和计算机性能的不断提高,计算电磁学在最近几年里得到了长足的发展。其中,瞬态电磁学在各种实际工程中得到了广泛应用。本文研究的是具有导电特性的大地,利用瞬态电磁场的似稳特性探测地下目标。模拟电磁场在地下介质中的传播规律是理解地下介质响应的重要手段,而直接应用有限差分法在时间域对二维时域似稳场进行数值分析是一种有效方法。
本文直接从时间域出发,应用有限差分方法(FD)的DuFort-Frankel格式和时域有限差分方法(FDTD)分析线源二维模型地下和地面的瞬变响应。从反映电磁场基本规律的Maxwell方程组出发,导出时域电场的齐次扩散方程,对所研究的空间区域作差分离散,源作为初始条件加入,采用向上延拓边界条件,利用差分方程进行计算,从而得到电场的数值结果,展现瞬变场在地下随时间扩散的全过程。而后,通过引入虚拟位移电流用FDTD算法对同样的地电模型进行了模拟,同时得到了电场分量和磁场分量,由此可对瞬态场的响应有更全面的了解。半空间模型的数值解与解析解的比较结果表明,本文对边界条件、源的处理是正确的,数值计算精度能够满足实际探测工程的需要。
本文对大地和目标体取不同电磁参数以及结构情况下,对电磁场的瞬态响应做了大量仿真。分析结果表明,瞬变电磁场扩散距离随着时间的推进而增加,场强随之衰减。扩散速度受到介质电阻率的影响,在低阻体中扩散速度慢,在高阻体中扩散速度快。因此,当存在低阻覆盖层时,探测同样的深度需要更长的时间,异常幅度减弱,而高阻覆盖层对目标探测几乎没有影响。瞬变场对低阻体有良好的分辨能力,对高阻目标体的探测则需要更多的识别技巧。
With development of electromagnetic field theory and computer performance of growth with rising,computed electromagnetism is getting quiet great progress in recently years.Among them,transient electromagnetic analysis is widely applied in many electronic engineering areas.In this dissertation,we research with the electric earth and use quasi-static characteristic of transient electromagnetic field in surveying for underground objects.To simulate the propagation law of electromagnetic field underground is a very important measure of understanding the respond of underground medium.Moreover,numerical analysis of a two-dimensional time-domain quasi-stationary field is studied by using finite-difference approach directly in time domain which is a effective method.
In this paper,we directly make it in time domain,apply DuFort-Frankel format of the FD method and FDTD method to analysis the transient respond underground and interface of a two-dimensional model of line source.To derive the homogeneous diffusion equation of time-domain electric field from Maxwell's equation and do the difference discretion in the investigated region.The source is added as initial condition and the boundary is dealt with upward-continuation. Using difference equation to computer the result of electric field,and it could be shown the entire process of transient electromagnetic fields diffusing with time in the earth.Then we introduce a fictitious displacement current in order to apply FDTD method to simulate the same model,and it can obtain electric field and magnetic field at the same time.So we can have comprehensive understanding of the respond of transient electromagnetic field.Compare the results of numerical and analytical solutions in a half-space model,it indicates that the method which is dealt with boundary condition and source are right,and the precision of numerical compute can meet the need of the exploration project.
We take different electromagnetism parameters and different structure to earth and targets,so it makes a great deal of simulation to the transient respond of electromagnetic field.Analysis results indicate,diffusion distance of transient electromagnetic field increases but field density attenuates along with time. Diffusion velocity is influenced by the resistivity of medium;it is slow in the low resistance body and rapid in the high resistance body.Therefore,when low resistance overburden exists,a long time is needed to detecting the same depth while high resistance overburden hardly has any affection to target exploration. Transient electromagnetic field have good resolving power to the low resistance body,but as for high resistance body it needs more identifying techniques.
本文直接从时间域出发,应用有限差分方法(FD)的DuFort-Frankel格式和时域有限差分方法(FDTD)分析线源二维模型地下和地面的瞬变响应。从反映电磁场基本规律的Maxwell方程组出发,导出时域电场的齐次扩散方程,对所研究的空间区域作差分离散,源作为初始条件加入,采用向上延拓边界条件,利用差分方程进行计算,从而得到电场的数值结果,展现瞬变场在地下随时间扩散的全过程。而后,通过引入虚拟位移电流用FDTD算法对同样的地电模型进行了模拟,同时得到了电场分量和磁场分量,由此可对瞬态场的响应有更全面的了解。半空间模型的数值解与解析解的比较结果表明,本文对边界条件、源的处理是正确的,数值计算精度能够满足实际探测工程的需要。
本文对大地和目标体取不同电磁参数以及结构情况下,对电磁场的瞬态响应做了大量仿真。分析结果表明,瞬变电磁场扩散距离随着时间的推进而增加,场强随之衰减。扩散速度受到介质电阻率的影响,在低阻体中扩散速度慢,在高阻体中扩散速度快。因此,当存在低阻覆盖层时,探测同样的深度需要更长的时间,异常幅度减弱,而高阻覆盖层对目标探测几乎没有影响。瞬变场对低阻体有良好的分辨能力,对高阻目标体的探测则需要更多的识别技巧。
With development of electromagnetic field theory and computer performance of growth with rising,computed electromagnetism is getting quiet great progress in recently years.Among them,transient electromagnetic analysis is widely applied in many electronic engineering areas.In this dissertation,we research with the electric earth and use quasi-static characteristic of transient electromagnetic field in surveying for underground objects.To simulate the propagation law of electromagnetic field underground is a very important measure of understanding the respond of underground medium.Moreover,numerical analysis of a two-dimensional time-domain quasi-stationary field is studied by using finite-difference approach directly in time domain which is a effective method.
In this paper,we directly make it in time domain,apply DuFort-Frankel format of the FD method and FDTD method to analysis the transient respond underground and interface of a two-dimensional model of line source.To derive the homogeneous diffusion equation of time-domain electric field from Maxwell's equation and do the difference discretion in the investigated region.The source is added as initial condition and the boundary is dealt with upward-continuation. Using difference equation to computer the result of electric field,and it could be shown the entire process of transient electromagnetic fields diffusing with time in the earth.Then we introduce a fictitious displacement current in order to apply FDTD method to simulate the same model,and it can obtain electric field and magnetic field at the same time.So we can have comprehensive understanding of the respond of transient electromagnetic field.Compare the results of numerical and analytical solutions in a half-space model,it indicates that the method which is dealt with boundary condition and source are right,and the precision of numerical compute can meet the need of the exploration project.
We take different electromagnetism parameters and different structure to earth and targets,so it makes a great deal of simulation to the transient respond of electromagnetic field.Analysis results indicate,diffusion distance of transient electromagnetic field increases but field density attenuates along with time. Diffusion velocity is influenced by the resistivity of medium;it is slow in the low resistance body and rapid in the high resistance body.Therefore,when low resistance overburden exists,a long time is needed to detecting the same depth while high resistance overburden hardly has any affection to target exploration. Transient electromagnetic field have good resolving power to the low resistance body,but as for high resistance body it needs more identifying techniques.
引文
[1]王秉中.计算电磁学[M].北京:科学出版社,2002
[2]R.F Harrington.Time-Harmonic Electromagnetic Fields[M].New York:McGraw-Hill Book Company,1961.
[3]R.F.Harrington.Field Computations by Moment Methods[M].New York:Macmillan,1968.
[4]Yee K.S.Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media[J].IEEE Trans.on AP.302-307年代卷期
[5]A.Taflove.Advances in Computational Electrodynamics:The Finite-Difference Time-domain Method.MA:Artech House,1998
[6]A.Taflove,S.C.Hagness.Computational Electcodynamics:The Finite-Difference Time -Domain Mehtod.MA:Artech House,2000
[7]刘圣民.电磁场的数值方法[M].武汉:华中理工大学出版社,1991
[8]Zeland 公司 IE3D:http://www.zeland.com
[9]Ansoft 公司HFSS:http://www.ansoft.com
[10]Remcom 公司 XFDTD:http://www.remcom.com
[11]胡来平,刘占军.电磁学计算方法的比较.现代电子技术,2003(10):75-78
[12]Yee K S.Numerical solution of initial boundary value problems involving Maxwell equations in isotropic media.IEEE Trans.Antenna Propagat.,May 1966,AP-14(3):302-307
[13]高本庆.时域有限差分法FDTD Method.北京:国防工业出版社,1995
[14]R.Holland.THREDE:A free-field EMP coupling and scattering code.IEEE Trans.Nuclear Science,DEC.1977,NS-24(6):2416-2421
[15]K.S.Kunz,K.M.Lee.A 3-D finite difference solution of the external response of an aircraft to a complex transient EM environment:I-The method and its implementation.IEEE Trans.Electromagn.Compat.,May 1978,EMC-20(2):328-333
[16]A.Taflove,M.E.Brodwin.Numerical solution of steady-state EM scattering problems using the timedependent Maxwell's equation.IEEE Trans.Microwave Theory Tech.,Aug.1975,MTT-23:623-630
[17]Cai Y.,Mias C.Fast finite element time domain-floquet modal absorbing boundary condition modelling of periodic structures using recursive convolution.IEEE Trans.Antenna Propagat.,Sept.2007,vol.55,pp:2550-2555.
[18]Paul P.,Webb J.P.An iterative absorbing boundary condition using varying element orders.IEEE Trans.Magnetics,April 2007,vol.43,pp:1325-1325.
[19]Hwang J.N.,Chen F.C.Stability analysis of the Mur's absorbing boundary condition in the alternating direction implicit finite-difference method.IEEE Trans.Microwaves,Antenna Propagat.,June 2007,vol.1,pp:597-601.
[20]Butrym A.Yu.,Legenkiy M.N.Absorbing boundary condition for FDTD modeling of waveguides with lossy medium.MSMW'07 Symposium Proceedings.Kharkov,Ukraine,June,2007,vol.1,pp:298-300.
[21]Davidson D.B.,Botha M.M.Evaluation of a spherical PML for vector FEM applications.IEEE Trans.Antenna Propagat.,Feb.,2007,vol.55,pp:494-498
[22]Ramadan O.Second-order split-step envelope PML algorithm for 2D FDTD simulations.Electronics Letters.July 2007,vol.43,pp:792-793
[23]Yi Y.,Chen B.,Chen H.L.,Fang D.G.TF/SF boundary and PML-ABC for an unconditionally stable FDTD method,Feb.2007,vol.17,pp:91-93
[24]Ramadan O.Complex envelope ADI-PML algorithm for truncating Lorentz dispersive 2-D-FDTD domains.IEEE Microwave and Wireless Components Letters,Jan.2007,vol.17,pp:4-6
[25]盛新庆.计算电磁学要论.北京:科学出版社,2004
[26]吕英华.计算电磁学的数值方法.北京:清华大学出版社,2004
[27]百科在线全文搜索:http://www.dbk2008.com
[28]DuFort E.C.,Frankel S.P.Stability conditions in the numerical treatment of parabolic differential equations.Math.Tables and Other Aids to Comput.(former title of Mathematics of Computation),1953,v.7,P.135-152
[29]张文生.科学计算中的偏微分方程有限差分法.北京:高等教育出版社,2006
[30]葛德彪,闫玉波.电磁波时域有限差分方法[M].西安:西安电子科技大学出版社,2005
[31]王长清.现代计算电磁学基础[M].北京:北京大学出版社,2005
[32]C.E.Baum.Emerging Technology for Transient and Broad-Band Analysis and Synthesis of Antennas and Scatterers.Pro.IEEE,vol.64,pp.1658-1616,1976.
[33]C.L.Bennett,G.F.Ross.Time-Domain Electromagneties and Its Applications.Pro.IEEE,vol.66,pp.299-318,1978.
[34]汪文秉.瞬态电磁场[M].西安:西安交通大学出版社,1996.6.
[35]L.B.Felsen.Transient Electromagnetic Fields,Springer Verlag,New York,1976.
[36]彭仲秋.瞬变电磁场[M].北京:高等教育出版社,1989
[37]Special Issue on Numerical Modeling of Transient Electromagnetics,Int.J.Nuraer.Model.,vol.12,1999.
[38]Special Issue on Time-Domain EM Methods of Exploration,Geophysics,vol.49,no.7,1984.
[39]吴大正.信号与线性系统分析.北京:高等教育出版社,2005.
[40]冯恩信.电磁场与电磁波.西安:西安交通大学出版社,2005
[41]Oristaglio M L,Hohmann G W.Diffusion of electromagnetic fields into a two-dimensional earth:A finite-difference approach[J].Geophysics.1984.49(7):870-894.
[42]Mur G.Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equation.IEEE Trans.Electromagn.Compat.,Nov.1981,EMC-23(4):377-382
[43]Berenger J P.A perfectly matched layer for the absorption of electromagnetic waves.J.Comput.Phys.,Oct.1994,114(2):185-200
[44]Berenger J P.Three-dimensional perfectly matched layer for the absorption of electromagnetic waves.J.Comput.Phys.,Sept.1996,127(2):363-379
[45]Berenger J P.Perfectly matched layer for the FDTD solution of wave-structure interaction problem.IEEE Trans.Antennas Propagat.,Jan.1996,AP-44(1):110-117
[46]张清河.时域有限差分(FDTD)法中的吸收边界条件.三峡大学学报(自然科学版),2004,26(5):464-466
[47]Wang J,Labrecque D J.Application of PML to FDTD simulation of electromagnetic tomography,1996,65~(th)Annual International Meeting SEG.Extended Abstracts:1066-1069
[48]C.M.Bender,S.A.Orszag.Advanced Mathematical Methods for Engineers and Scientists,New York,McGraw-Hill,1978.
[49]Oristaglio M L.Diffusion of electromagnetic fields into the earth from a line source of current[J].Geophysics,1982,47(11):1585-1592
[50]郭敦仁.数学物理方法[M].北京:人民教育出版社,1991
[51]Wang T,Hommann G W.A finite-difference,time-domain solution for three-dimensional electromagnetic modeling[J].Geophysics,1993,58(6):797-809
[52]M.N.Nabighian.The Analytic Signal of Two-Dimensional Magnetic Bodies with Polygonal CrossSection:Its Properties and Use for Automated Interpretation[J].Geophysics,1972,55,1166-1182
[53]J.C.Macnae.Survey Design for Multicomponent Electromagnetic System[J].Geophysics,1984,49,265-273
[54]程佩青.数字信号处理教程[M].北京:清华大学出版社,2001,8
[55]Jeffrey A.Fessler,Bradley P.Sutton.Nonuniform Fast Fourier Transforms Using Min-Max Interpolation.IEEE Trans.Signal processing,Feb.2003,vol.51(2):560-574
[2]R.F Harrington.Time-Harmonic Electromagnetic Fields[M].New York:McGraw-Hill Book Company,1961.
[3]R.F.Harrington.Field Computations by Moment Methods[M].New York:Macmillan,1968.
[4]Yee K.S.Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media[J].IEEE Trans.on AP.302-307年代卷期
[5]A.Taflove.Advances in Computational Electrodynamics:The Finite-Difference Time-domain Method.MA:Artech House,1998
[6]A.Taflove,S.C.Hagness.Computational Electcodynamics:The Finite-Difference Time -Domain Mehtod.MA:Artech House,2000
[7]刘圣民.电磁场的数值方法[M].武汉:华中理工大学出版社,1991
[8]Zeland 公司 IE3D:http://www.zeland.com
[9]Ansoft 公司HFSS:http://www.ansoft.com
[10]Remcom 公司 XFDTD:http://www.remcom.com
[11]胡来平,刘占军.电磁学计算方法的比较.现代电子技术,2003(10):75-78
[12]Yee K S.Numerical solution of initial boundary value problems involving Maxwell equations in isotropic media.IEEE Trans.Antenna Propagat.,May 1966,AP-14(3):302-307
[13]高本庆.时域有限差分法FDTD Method.北京:国防工业出版社,1995
[14]R.Holland.THREDE:A free-field EMP coupling and scattering code.IEEE Trans.Nuclear Science,DEC.1977,NS-24(6):2416-2421
[15]K.S.Kunz,K.M.Lee.A 3-D finite difference solution of the external response of an aircraft to a complex transient EM environment:I-The method and its implementation.IEEE Trans.Electromagn.Compat.,May 1978,EMC-20(2):328-333
[16]A.Taflove,M.E.Brodwin.Numerical solution of steady-state EM scattering problems using the timedependent Maxwell's equation.IEEE Trans.Microwave Theory Tech.,Aug.1975,MTT-23:623-630
[17]Cai Y.,Mias C.Fast finite element time domain-floquet modal absorbing boundary condition modelling of periodic structures using recursive convolution.IEEE Trans.Antenna Propagat.,Sept.2007,vol.55,pp:2550-2555.
[18]Paul P.,Webb J.P.An iterative absorbing boundary condition using varying element orders.IEEE Trans.Magnetics,April 2007,vol.43,pp:1325-1325.
[19]Hwang J.N.,Chen F.C.Stability analysis of the Mur's absorbing boundary condition in the alternating direction implicit finite-difference method.IEEE Trans.Microwaves,Antenna Propagat.,June 2007,vol.1,pp:597-601.
[20]Butrym A.Yu.,Legenkiy M.N.Absorbing boundary condition for FDTD modeling of waveguides with lossy medium.MSMW'07 Symposium Proceedings.Kharkov,Ukraine,June,2007,vol.1,pp:298-300.
[21]Davidson D.B.,Botha M.M.Evaluation of a spherical PML for vector FEM applications.IEEE Trans.Antenna Propagat.,Feb.,2007,vol.55,pp:494-498
[22]Ramadan O.Second-order split-step envelope PML algorithm for 2D FDTD simulations.Electronics Letters.July 2007,vol.43,pp:792-793
[23]Yi Y.,Chen B.,Chen H.L.,Fang D.G.TF/SF boundary and PML-ABC for an unconditionally stable FDTD method,Feb.2007,vol.17,pp:91-93
[24]Ramadan O.Complex envelope ADI-PML algorithm for truncating Lorentz dispersive 2-D-FDTD domains.IEEE Microwave and Wireless Components Letters,Jan.2007,vol.17,pp:4-6
[25]盛新庆.计算电磁学要论.北京:科学出版社,2004
[26]吕英华.计算电磁学的数值方法.北京:清华大学出版社,2004
[27]百科在线全文搜索:http://www.dbk2008.com
[28]DuFort E.C.,Frankel S.P.Stability conditions in the numerical treatment of parabolic differential equations.Math.Tables and Other Aids to Comput.(former title of Mathematics of Computation),1953,v.7,P.135-152
[29]张文生.科学计算中的偏微分方程有限差分法.北京:高等教育出版社,2006
[30]葛德彪,闫玉波.电磁波时域有限差分方法[M].西安:西安电子科技大学出版社,2005
[31]王长清.现代计算电磁学基础[M].北京:北京大学出版社,2005
[32]C.E.Baum.Emerging Technology for Transient and Broad-Band Analysis and Synthesis of Antennas and Scatterers.Pro.IEEE,vol.64,pp.1658-1616,1976.
[33]C.L.Bennett,G.F.Ross.Time-Domain Electromagneties and Its Applications.Pro.IEEE,vol.66,pp.299-318,1978.
[34]汪文秉.瞬态电磁场[M].西安:西安交通大学出版社,1996.6.
[35]L.B.Felsen.Transient Electromagnetic Fields,Springer Verlag,New York,1976.
[36]彭仲秋.瞬变电磁场[M].北京:高等教育出版社,1989
[37]Special Issue on Numerical Modeling of Transient Electromagnetics,Int.J.Nuraer.Model.,vol.12,1999.
[38]Special Issue on Time-Domain EM Methods of Exploration,Geophysics,vol.49,no.7,1984.
[39]吴大正.信号与线性系统分析.北京:高等教育出版社,2005.
[40]冯恩信.电磁场与电磁波.西安:西安交通大学出版社,2005
[41]Oristaglio M L,Hohmann G W.Diffusion of electromagnetic fields into a two-dimensional earth:A finite-difference approach[J].Geophysics.1984.49(7):870-894.
[42]Mur G.Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equation.IEEE Trans.Electromagn.Compat.,Nov.1981,EMC-23(4):377-382
[43]Berenger J P.A perfectly matched layer for the absorption of electromagnetic waves.J.Comput.Phys.,Oct.1994,114(2):185-200
[44]Berenger J P.Three-dimensional perfectly matched layer for the absorption of electromagnetic waves.J.Comput.Phys.,Sept.1996,127(2):363-379
[45]Berenger J P.Perfectly matched layer for the FDTD solution of wave-structure interaction problem.IEEE Trans.Antennas Propagat.,Jan.1996,AP-44(1):110-117
[46]张清河.时域有限差分(FDTD)法中的吸收边界条件.三峡大学学报(自然科学版),2004,26(5):464-466
[47]Wang J,Labrecque D J.Application of PML to FDTD simulation of electromagnetic tomography,1996,65~(th)Annual International Meeting SEG.Extended Abstracts:1066-1069
[48]C.M.Bender,S.A.Orszag.Advanced Mathematical Methods for Engineers and Scientists,New York,McGraw-Hill,1978.
[49]Oristaglio M L.Diffusion of electromagnetic fields into the earth from a line source of current[J].Geophysics,1982,47(11):1585-1592
[50]郭敦仁.数学物理方法[M].北京:人民教育出版社,1991
[51]Wang T,Hommann G W.A finite-difference,time-domain solution for three-dimensional electromagnetic modeling[J].Geophysics,1993,58(6):797-809
[52]M.N.Nabighian.The Analytic Signal of Two-Dimensional Magnetic Bodies with Polygonal CrossSection:Its Properties and Use for Automated Interpretation[J].Geophysics,1972,55,1166-1182
[53]J.C.Macnae.Survey Design for Multicomponent Electromagnetic System[J].Geophysics,1984,49,265-273
[54]程佩青.数字信号处理教程[M].北京:清华大学出版社,2001,8
[55]Jeffrey A.Fessler,Bradley P.Sutton.Nonuniform Fast Fourier Transforms Using Min-Max Interpolation.IEEE Trans.Signal processing,Feb.2003,vol.51(2):560-574