耦合有限单元法扩边的直流电阻率勘探无单元Galerkin法正演
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摘要
本文分析了目前直流电阻率正演模拟中的无单元Galerkin法(EFGM)和有限单元法(FEM)的优缺点,针对采用第一类边界条件需要足够大的计算域时EFGM计算成本高的问题,在计算域外围区域采用FEM扩边,提出了直流电阻率的无单元Galerkin-有限单元耦合法(EFG-FE).采用具有Kronecker delta函数性质的径向基点插值法(RPIM)构造EFGM形函数,在外围区域将EFGM与FEM直接耦合,无需其他处理手段,消除了传统EFGM与FEM耦合中存在的界面耦合困难.EFG-FE将模型计算域分割为EFGM区域和FEM区域,模型核心区域采用EFGM计算,发挥EFGM灵活性、适应性强和高精度的优点,使得模型建立简单方便,对任意复杂地电模型适应性强,同时获得高精度模拟结果.在模型计算域外围采用快速扩展的FEM单元网格进行剖分,利用其数值稳定性和高效性,使用少量FEM节点和单元网格将计算域大范围扩大满足第一类边界条件,同时不大幅增加计算成本,进而提高计算效率.最后,通过不同正演方法的模型算例的模拟结果对比,验证了本文提出的EFG-FE有效可行,其模拟结果具有很高的模拟精度,且相比于采用第三类边界条件的EFGM提高了计算效率,具有更好的模拟性能.
In this paper,we analyze the advantages and disadvantages of the Element-Free Galerkin Method(EFGM)and Finite Element Method(FEM)in forward simulation of DirectCurrent(DC)resistivity.A large enough computational domain is required when the first class boundary condition is used,which will significantly increase the computational cost of EFGM.To solve this problem,we propose a Element-Free Galerkin-Finite Element(EFG-FE)coupling method for forward modeling of DC resistivity.The FEM is used in the peripheral region of the calculation domain in the EFG-FE.In order to eliminate the difficulties in the traditional EFGM and FEM coupling method on the interface,the Radial Point Interpolation Method(RPIM)is used to construct the element-free shape function.Due to the RPIM shape function has the Kronecker delta function property,EFGM and FEM can be coupled directly without any other processing technology.EFG-FE divides the model calculation domain into a EFGM region and FEM region.In order to exert the flexibility,adaptability and high precision of EFGM,the core area of the model is calculated using EFGM,which results in the simplicity and convenience of model establishment,the adaptability of any complex geoelectric model and high precision of simulation results.In order to reduce the calculation time and expand the computational domain so that the natural boundary conditions are satisfied,a rapidly expanding FEM grid is used in the periphery of the EFGM region,which results in a small number of nodes and FEM grid cells.Finally,the simulation results of different forward modeling methods are compared.The results show that the proposed EFG-FE is feasible.And compared to the EFGM with the third kind of boundary conditions,the proposed EFG-FE method can improve the computational efficiency.In conclusion,the proposed EFGFE has a better simulation performance.
In this paper,we analyze the advantages and disadvantages of the Element-Free Galerkin Method(EFGM)and Finite Element Method(FEM)in forward simulation of DirectCurrent(DC)resistivity.A large enough computational domain is required when the first class boundary condition is used,which will significantly increase the computational cost of EFGM.To solve this problem,we propose a Element-Free Galerkin-Finite Element(EFG-FE)coupling method for forward modeling of DC resistivity.The FEM is used in the peripheral region of the calculation domain in the EFG-FE.In order to eliminate the difficulties in the traditional EFGM and FEM coupling method on the interface,the Radial Point Interpolation Method(RPIM)is used to construct the element-free shape function.Due to the RPIM shape function has the Kronecker delta function property,EFGM and FEM can be coupled directly without any other processing technology.EFG-FE divides the model calculation domain into a EFGM region and FEM region.In order to exert the flexibility,adaptability and high precision of EFGM,the core area of the model is calculated using EFGM,which results in the simplicity and convenience of model establishment,the adaptability of any complex geoelectric model and high precision of simulation results.In order to reduce the calculation time and expand the computational domain so that the natural boundary conditions are satisfied,a rapidly expanding FEM grid is used in the periphery of the EFGM region,which results in a small number of nodes and FEM grid cells.Finally,the simulation results of different forward modeling methods are compared.The results show that the proposed EFG-FE is feasible.And compared to the EFGM with the third kind of boundary conditions,the proposed EFG-FE method can improve the computational efficiency.In conclusion,the proposed EFGFE has a better simulation performance.
引文
Belytschko T,Lu Y Y,Gu L.1994a.Element-free Galerkin methods.International Journal for Numerical Methods in Engineering,37(1):229-256.
Belytschko T,Lu Y Y,Gu L.1994b.Fracture and crack growth by element free Galerkin methods.Modellingand Simulationin MaterialsScienceandEngineering,2(3A):519-534.
Belytschko T,Organ D,Krongauz Y.1995.A coupled finite element-element-free galerkin method.Computational Mechanics,17(3):186-195.
Dolbow J,Belytschko T.1999.Numerical integration of the Galerkin weak form in meshfree methods.Computational Mechanics,23(3):219-230.
Feng D S,Wang H H,Dai Q W.2013.Forward simulation of ground penetrating radar based on the element-free Galerkin method.Chinese Journal of Geophysics(in Chinese),56(1):298-308,doi:10.6038/cjg20130131.
Gu Y T,Liu G R.2005.Meshless methods coupled with other numerical methods.Tsinghua Science&Technology,10(1):8-15.
Hajiazizi M,Bastan P.2014.The elastoplastic analysis of a tunnel using the EFG method:A comparison of the EFGM with FEMand FDM.Applied Mathematics and Computation,234:82-113.
Hadinia M,Jafari R.2015.An element-free Galerkin forward solver for the complete-electrode model in electrical impedance tomography.Flow Measurement and Instrumentation,45:68-74.
Hadinia M,Jafari R,Soleimani M.2016.EIT image reconstruction based on a hybrid FE-EFG forward method and the completeelectrode model.Physiological Measurement,37(6):863-878.
Hosseini S,Malekan M,Pitangueira R,et al.2017.imposition of dirichlet boundary conditions in element free galerkin method through an object-oriented implementation.Latin American Journal of Solids and Structures,14(6):1017-1039.
Ji Y J,Huang T Z,Huang W Y,et al.2016.2D anisotropic magnetotelluric numerical simulation using meshfree method under undulating terrain.Chinese Journal of Geophysics(in Chinese),59(12):4483-4493,doi:10.6038/cjg20161211.
Jia L,Huang Q Q,Yin Z P.2007.A new and better EFGM-FEMdirect coupling method.Journal of Northwestern Polytechnical University(in Chinese),25(3):337-341.
Jia X F,Hu T Y,Wang R Q.2006.Wave equation modeling and imaging by using element-free method.Progress in Geophysics(in Chinese),21(1):11-17.
Kumar S,Singh I V,Mishra B K.2014.A multigrid coupled(FE-EFG)approach to simulate fatigue crack growth in heterogeneous materials.Theoretical and Applied Fracture Mechanics,72:121-135.
Lancaster P,Salkauskas K.1981.Surfaces generated by moving least squares methods.Mathematics of Computation,37(155):141-158.
Li D M,Bai F N,Cheng Y M,et al.2012.A novel complex variable element-free Galerkin method for two-dimensional large deformation problems.Computer Methods in Applied Mechanics and Engineering,233/236:1-10.
Li J J,Yan J B.2015.Magnetotelluric two-dimensional forward by finite element-radial point interpolation method.The Chinese Journal of Nonferrous Metals(in Chinese),25(5):1314-1324.
Liu G R,Gu Y T.2001.A point interpolation method for twodimensional solids.International Journal for Numerical Methods in Engineering,50(4):937-951.
Liu T X,Liu G,Xie Q,et al.2006.An EFG-FE coupling method for microscale adhesive contacts.Journal of Tribology,128(1):40-48.
Liu Z H,Lu M,Li Q,et al.2016.A direct coupling method of meshless local petrov-galerkin(MLPG)and finite element method(FEM).International Journal of Applied Electromagnetics and Mechanics,51(1):51-59.
Ma C Y,Liu J X,Liu H F,et al.2017.A global weak form element free method for direct current resistivity forward simulation.Chinese Journal of Geophysics(in Chinese),60(2):801-809,doi:10.6038/cjg20170230.
Pan K J,Tang J T,Hu H L,et al.2012.Extrapolation cascadic multigrid method for 2.5Ddirect current resistivity modeling.Chinese Journal of Geophysics(in Chinese),55(8):2769-2778,doi:10.6038/j.issn.0001-5733.2012.08.028.
Pathak H,Singh A,Singh I V,et al.2015.Three-dimensional stochastic quasi-static fatigue crack growth simulations using coupled FE-EFG approach.Computers&Structures,160:1-19.
Peng M J,Li D M,Cheng Y M.2011.The complex variable element-free Galerkin(CVEFG)method for elasto-plasticity problems.Engineering Structures,33(1):127-135.
Ruan B Y.2001.2-D electrical modeling due to a current point by FEM with variation of conductivity within each triangular element.Guangxi Sciences(in Chinese),8(1):1-3.
Wang Y Y.2007.Study of element-free galerkin method in the seismic forward modeling.Progress in Geophysics(in Chinese),22(5):1539-1544.
Xu S Z.1994.Finite Element Method in Geophysics.Beijing:Beijing Science Press.
Yan J B,Li J J.2014.Magnetotelluric forward calculation by meshless method.Journal of Central South University(Science and Technology),45(10):3513-3520.
Zhang Y,Wang J G,Zhang B Y.2008.Coupled FEM and meshless radial point interpolation method.Journal of Tsinghua University(Science and Technology),48(6):951-954.
Zhdanov M S,Cox L H,Endo M.2015.Large-scale 3Dinversion of airborne electromagnetic data based on the hybrid IE-FE method and the moving sensitivity domain approach.∥International Workshop and Gravity,Electrical&Magnetic Methods and their Applications.Chenghu,China,2015:303-306.
冯德山,王洪华,戴前伟.2013.基于无单元Galerkin法探地雷达正演模拟.地球物理学学报,56(1):298-308,doi:10.6038/cjg20130131.
嵇艳鞠,黄廷哲,黄婉玉等.2016.起伏地形下各向异性的2D大地电磁无网格法数值模拟.地球物理学报,59(12):4483-4493,doi:10.6038/cjg20161211.
贾亮,黄其青,殷之平.2007.无网格-有限元直接耦合法.西北工业大学学报,25(3):337-341.
贾晓峰,胡天跃,王润秋.2006.无单元法用于地震波波动方程模拟与成像.地球物理学进展,21(1):11-17.
李俊杰,严家斌.2015.大地电磁二维正演中的有限元-径向基点插值法.中国有色金属学报,25(5):1314-1324.
麻昌英,柳建新,刘海飞等.2017.基于全局弱式无单元法直流电阻率正演模拟.地球物理学学报,60(2):801-809,doi:10.6038/cjg20170230.
潘克家,汤井田,胡宏伶等.2012.直流电阻率法2.5维正演的外推瀑布式多重网格法.地球物理学报,55(8):2769-2778,doi:10.6038/j.issn.0001-5733.2012.08.028.
阮百尧.2001.三角单元部分电导率分块连续变化点源二维电场有限元数值模拟.广西科学,8(1):1-3.
王月英.2007.地震波正演模拟中无单元Galerkin法初探.地球物理学进展,22(5):1539-1544.
徐世浙.1994.地球物理中的有限单元法.北京:科学出版社.
严家斌,李俊杰.2014.无网格法在大地电磁正演计算中的应用.中南大学学报(自然科学版),45(10):3513-3520.
张琰,王建国,张丙印.2008.径向基点插值无网格法与有限元耦合法.清华大学学报(自然科学版),48(6):951-954.
Belytschko T,Lu Y Y,Gu L.1994b.Fracture and crack growth by element free Galerkin methods.Modellingand Simulationin MaterialsScienceandEngineering,2(3A):519-534.
Belytschko T,Organ D,Krongauz Y.1995.A coupled finite element-element-free galerkin method.Computational Mechanics,17(3):186-195.
Dolbow J,Belytschko T.1999.Numerical integration of the Galerkin weak form in meshfree methods.Computational Mechanics,23(3):219-230.
Feng D S,Wang H H,Dai Q W.2013.Forward simulation of ground penetrating radar based on the element-free Galerkin method.Chinese Journal of Geophysics(in Chinese),56(1):298-308,doi:10.6038/cjg20130131.
Gu Y T,Liu G R.2005.Meshless methods coupled with other numerical methods.Tsinghua Science&Technology,10(1):8-15.
Hajiazizi M,Bastan P.2014.The elastoplastic analysis of a tunnel using the EFG method:A comparison of the EFGM with FEMand FDM.Applied Mathematics and Computation,234:82-113.
Hadinia M,Jafari R.2015.An element-free Galerkin forward solver for the complete-electrode model in electrical impedance tomography.Flow Measurement and Instrumentation,45:68-74.
Hadinia M,Jafari R,Soleimani M.2016.EIT image reconstruction based on a hybrid FE-EFG forward method and the completeelectrode model.Physiological Measurement,37(6):863-878.
Hosseini S,Malekan M,Pitangueira R,et al.2017.imposition of dirichlet boundary conditions in element free galerkin method through an object-oriented implementation.Latin American Journal of Solids and Structures,14(6):1017-1039.
Ji Y J,Huang T Z,Huang W Y,et al.2016.2D anisotropic magnetotelluric numerical simulation using meshfree method under undulating terrain.Chinese Journal of Geophysics(in Chinese),59(12):4483-4493,doi:10.6038/cjg20161211.
Jia L,Huang Q Q,Yin Z P.2007.A new and better EFGM-FEMdirect coupling method.Journal of Northwestern Polytechnical University(in Chinese),25(3):337-341.
Jia X F,Hu T Y,Wang R Q.2006.Wave equation modeling and imaging by using element-free method.Progress in Geophysics(in Chinese),21(1):11-17.
Kumar S,Singh I V,Mishra B K.2014.A multigrid coupled(FE-EFG)approach to simulate fatigue crack growth in heterogeneous materials.Theoretical and Applied Fracture Mechanics,72:121-135.
Lancaster P,Salkauskas K.1981.Surfaces generated by moving least squares methods.Mathematics of Computation,37(155):141-158.
Li D M,Bai F N,Cheng Y M,et al.2012.A novel complex variable element-free Galerkin method for two-dimensional large deformation problems.Computer Methods in Applied Mechanics and Engineering,233/236:1-10.
Li J J,Yan J B.2015.Magnetotelluric two-dimensional forward by finite element-radial point interpolation method.The Chinese Journal of Nonferrous Metals(in Chinese),25(5):1314-1324.
Liu G R,Gu Y T.2001.A point interpolation method for twodimensional solids.International Journal for Numerical Methods in Engineering,50(4):937-951.
Liu T X,Liu G,Xie Q,et al.2006.An EFG-FE coupling method for microscale adhesive contacts.Journal of Tribology,128(1):40-48.
Liu Z H,Lu M,Li Q,et al.2016.A direct coupling method of meshless local petrov-galerkin(MLPG)and finite element method(FEM).International Journal of Applied Electromagnetics and Mechanics,51(1):51-59.
Ma C Y,Liu J X,Liu H F,et al.2017.A global weak form element free method for direct current resistivity forward simulation.Chinese Journal of Geophysics(in Chinese),60(2):801-809,doi:10.6038/cjg20170230.
Pan K J,Tang J T,Hu H L,et al.2012.Extrapolation cascadic multigrid method for 2.5Ddirect current resistivity modeling.Chinese Journal of Geophysics(in Chinese),55(8):2769-2778,doi:10.6038/j.issn.0001-5733.2012.08.028.
Pathak H,Singh A,Singh I V,et al.2015.Three-dimensional stochastic quasi-static fatigue crack growth simulations using coupled FE-EFG approach.Computers&Structures,160:1-19.
Peng M J,Li D M,Cheng Y M.2011.The complex variable element-free Galerkin(CVEFG)method for elasto-plasticity problems.Engineering Structures,33(1):127-135.
Ruan B Y.2001.2-D electrical modeling due to a current point by FEM with variation of conductivity within each triangular element.Guangxi Sciences(in Chinese),8(1):1-3.
Wang Y Y.2007.Study of element-free galerkin method in the seismic forward modeling.Progress in Geophysics(in Chinese),22(5):1539-1544.
Xu S Z.1994.Finite Element Method in Geophysics.Beijing:Beijing Science Press.
Yan J B,Li J J.2014.Magnetotelluric forward calculation by meshless method.Journal of Central South University(Science and Technology),45(10):3513-3520.
Zhang Y,Wang J G,Zhang B Y.2008.Coupled FEM and meshless radial point interpolation method.Journal of Tsinghua University(Science and Technology),48(6):951-954.
Zhdanov M S,Cox L H,Endo M.2015.Large-scale 3Dinversion of airborne electromagnetic data based on the hybrid IE-FE method and the moving sensitivity domain approach.∥International Workshop and Gravity,Electrical&Magnetic Methods and their Applications.Chenghu,China,2015:303-306.
冯德山,王洪华,戴前伟.2013.基于无单元Galerkin法探地雷达正演模拟.地球物理学学报,56(1):298-308,doi:10.6038/cjg20130131.
嵇艳鞠,黄廷哲,黄婉玉等.2016.起伏地形下各向异性的2D大地电磁无网格法数值模拟.地球物理学报,59(12):4483-4493,doi:10.6038/cjg20161211.
贾亮,黄其青,殷之平.2007.无网格-有限元直接耦合法.西北工业大学学报,25(3):337-341.
贾晓峰,胡天跃,王润秋.2006.无单元法用于地震波波动方程模拟与成像.地球物理学进展,21(1):11-17.
李俊杰,严家斌.2015.大地电磁二维正演中的有限元-径向基点插值法.中国有色金属学报,25(5):1314-1324.
麻昌英,柳建新,刘海飞等.2017.基于全局弱式无单元法直流电阻率正演模拟.地球物理学学报,60(2):801-809,doi:10.6038/cjg20170230.
潘克家,汤井田,胡宏伶等.2012.直流电阻率法2.5维正演的外推瀑布式多重网格法.地球物理学报,55(8):2769-2778,doi:10.6038/j.issn.0001-5733.2012.08.028.
阮百尧.2001.三角单元部分电导率分块连续变化点源二维电场有限元数值模拟.广西科学,8(1):1-3.
王月英.2007.地震波正演模拟中无单元Galerkin法初探.地球物理学进展,22(5):1539-1544.
徐世浙.1994.地球物理中的有限单元法.北京:科学出版社.
严家斌,李俊杰.2014.无网格法在大地电磁正演计算中的应用.中南大学学报(自然科学版),45(10):3513-3520.
张琰,王建国,张丙印.2008.径向基点插值无网格法与有限元耦合法.清华大学学报(自然科学版),48(6):951-954.