面向目标自适应海洋可控源电磁三维矢量有限元正演
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摘要
本文实现了一种面向目标自适应海洋可控源电磁三维矢量有限元方法.为满足三维复杂电性结构模拟的需求,网格剖分采用非结构化六面体.在组装刚度矩阵之后,形成的大型复数线性方程组分解为等价的实数形式,利用带预条件的广义最小残差法进行求解.在获得微分方程的解之后,为提高解的准确性,通过面向目标的自适应误差估计来指示网格细化,重点加密能使观测点数值模拟精度提高的网格.对于大规模三维数据,为了使模型空间的并行计算达到均衡负载的效果,我们使用METIS函数库来进行网格计算任务量的划分.最后,通过对比一维解析解与三维自适应矢量有限元计算结果,验证了程序的正确性;通过自适应过程中误差指示子的分布,验证了面向目标自适应的有效性;通过对三维复杂模型进行均衡负载下的并行计算,测试了程序的可扩展性.
We have developed a parallel goal-oriented adaptive finite element method for accurate and efficient electromagnetic modeling.To accommodate arbitrarily complex 3D conductivity variations,we use the total-field solution approaches of unstructured hexahedral grids.After assembling the local element matrices into a global system,one can obtain a sparse linear system of equations.We employed the FGMRES method with a very efficient block-diagonal pre-conditioner for the large conductivity contrasts system.Accuracy of the finite element solution could be achieved through adaptive mesh refinement that is performed iteratively until the solution converges to the desired accuracy tolerance.Refinement is guided by agoal-oriented error estimator that uses a dual-weighted residual method to optimize the mesh for improving observation numerical simulation accuracy.To further improve the computational efficiency,our algorithm is parallelized over meshpartitioning using METIS.In this paper,We validate the newly developed algorithm by comparison with controlled source EM solutions for a 1D layered model.Afterwards,we demonstrate that the estimated error for each element for a variable problem sizes for the effectiveness of the goal-oriented adaptive.In order to verify scalable numerical scheme,we have implemented the dynamic load balancing algorithms used for parallel computations.
We have developed a parallel goal-oriented adaptive finite element method for accurate and efficient electromagnetic modeling.To accommodate arbitrarily complex 3D conductivity variations,we use the total-field solution approaches of unstructured hexahedral grids.After assembling the local element matrices into a global system,one can obtain a sparse linear system of equations.We employed the FGMRES method with a very efficient block-diagonal pre-conditioner for the large conductivity contrasts system.Accuracy of the finite element solution could be achieved through adaptive mesh refinement that is performed iteratively until the solution converges to the desired accuracy tolerance.Refinement is guided by agoal-oriented error estimator that uses a dual-weighted residual method to optimize the mesh for improving observation numerical simulation accuracy.To further improve the computational efficiency,our algorithm is parallelized over meshpartitioning using METIS.In this paper,We validate the newly developed algorithm by comparison with controlled source EM solutions for a 1D layered model.Afterwards,we demonstrate that the estimated error for each element for a variable problem sizes for the effectiveness of the goal-oriented adaptive.In order to verify scalable numerical scheme,we have implemented the dynamic load balancing algorithms used for parallel computations.
引文
Ansari S,Farquharson C G.2014.3Dfinite-element forward modeling of electromagnetic data using vector and scalar potentials and unstructured grids.Geophysics,79,E149-E165.
Baker A,Falgout R,Kolev T,et al.2012.Scaling hypre′s multigrid solvers to 100,000cores,in Berry M W,Gallivan KA,Gallopoulos E,Grama A,Philippe B,Saad Y,and Saied F,eds.,High Performance Scientific Computing:Algorithms and Applications:Springer,261-279.
Chen J Q,Chen Z M,Cui T,et al.2010.An adaptive finite element method for the eddy current model with circuit/field couplings.SIAM Journal on Scientific Computing,32(2):1020-1042.
Chen Z M,Wang L,Zheng W Y.2007.An adaptive multilevel method for time-harmonic Maxwell equations with singularities.SIAM Journal on Scientific Computing,29(1):118-138.
Colombo D,McNeice G,Curiel E S,et al.2013.Full tensor CSEMand MT for subsalt structural imaging in the Red Sea:Implications for seismic and electromagnetic integration.The Leading Edge,32(4):436-449.
Constable S.2010.Ten years of marine CSEM for hydrocarbon exploration.Geophysics,75(5):75A67-75A81.
Da Silva N V,Morgan J V,MacGregor L,et al.2012.A finite element multifrontal method for 3D CSEM modeling in the frequency domain.Geophysics,77(2):E101-E115.
Grayver A V,Bürg M.2014.Robust and scalable 3-D geo-electromagnetic modelling approach using the finite element method.Geophysical Journal International,198(1):110-125.
Haber E.2014.Computational Methods in Geophysical Electromagnetics.Philadelphia,PA,USA:Society for Industrial and Applied Mathematics.
Hiptmair R,Xu J C.2007.Nodal auxiliary space preconditioning in H(curl)and H(div)spaces.SIAM Journal on Numerical Analysis,45(6):2483-2509.
Jin J M.2002.The Finite Element Method in Electromagnetics.2nd ed.New York:Wiley.
Jahandari H,Farquharson C G.2015.Finite-volume modelling of geophysical electromagnetic data on unstructured grids using potentials.Geophysical Journal International,202(3):1859-1876.
Karypis G,Kumar V.1998.A parallel algorithm for multilevel graph partitioning and sparse matrix ordering.Journal of Parallel and Distributed Computing,48:71-85.
Key K.2009.1D inversion of multicomponent,multifrequency marine CSEM data:Methodology and synthetic studies for resolving thin resistive layers,Geophysics,74(2):F9-F20.
Key,K,Ovall J.2011.A parallel goal-oriented adaptive finite element method for 2.5Delectro-magnetic modelling.Geophysical Journal International.186:137-154.
Key K.2012.Marine electromagnetic studies of seafloor resources and tectonics.Surveys in Geophysics,33(1):135-167.
Kolev T V,Vassilevski P S.2009.Parallel auxiliary space AMG for H(curl)problems.Journal of Computational Mathematics,27(5):604-623.
Li Y,Wu X P,Lin P R,et al.2017.Three-dimensional modeling of marine controlled-source electromagnetism using the vector finite element method for arbitrary anisotropic media.Chinese Journal of Geophysics(in Chinese),60(5):1955-1978,doi:10.6038/cjg20170528.
Li Y G,Key K.2007.2Dmarine controlled-source electromagnetic modeling:Part 1-An adaptive finite-element algorithm.Geophysics,72(2):WA51-WA62.
Li Y G,Dai S.2011.Finite element modeling of marine controlledsource electromagnetic responses in two-dimensional dipping anisotropic conductivity structures.Geophysical Journal International,185(2):622-636.
Liu Y,Li Y G,Han B.2017.Adaptive edge finite element modeling of the 3D CSEM field on unstructured grids.Chinese Journal of Geophysics(in Chinese),60(12):4874-4886,doi:10.6038/cjg20171227.
Monk P.2003.Finite Element Methods for Maxwell′s Equations.Oxford:Oxford University Press.
Operto S,Virieux J,Amestoy P,et al.2007.3Dfinite-difference frequency-domain modeling of visco-acoustic wave propagation using a massively parallel direct solver:a feasibility study.Geophysics,72(5):SM195-SM211.
Pardo D,Paszynski M,Collier N,et al.2012.A survey on direct solvers for Galerkin methods.SeMA Journal,57(1):107-134.
Plessix R E,Edouard M,Mathieu M W.2007.An approach for 3Dmultisource,multifrequency CSEM modeling.Geophysics,72(5):SM177.
Puzyrev V,Koldan J,De La Puente J,et al.2013.A parallel finiteelement method for three-dimensional controlled-source electromagnetic forward modelling.Geophysical Journal International,193(2):678-693.
Ren Z Y,Kalscheuer T,Greenhalgh S,et al.2013.A goal-oriented adaptive finite-element approach for plane wave 3-D electromagnetic modelling.Geophysical Journal International,194(2):700-718.
Saad Y.2003.Iterative Methods for Sparse Linear Systems.2nd ed.Philadelphia:Society for Industrial and Applied Mathematics.
Schwarzbach C,B9rner R U,Spitzer K.2011.Three-dimensional adaptive higher order finite element simulation for geoelectromagnetics-A marine CSEM example.Geophysical Journal International,187(1):63-74.
Streich R.2009.3Dfinite-difference frequency-domain modeling of controlled-source electromagnetic data:direct solution and optimization for high accuracy.Geophysics,74(5):F95-F105.
Um E S,Commer M,Newman G A.2013.Efficient pre-conditioned iterative solution strategies for the electromagnetic diffusion in the Earth:Finite-element frequency domain approach.Geophysical Journal International,193(3):1460-1473.
Ward S H,Hohmann G W.1988.Electromagnetic theory for geophysical applications.∥Nabighian M N ed.Electromagnetic Methods in Applied Geophysics.Oklahoma:Society of Exploration Geophysicists,1:131-311.
Yang J,Liu Y,Wu X P.2015.3Dsimulation of marine CSEMusing vector finite element method on unstructured grids.Chinese Journal of Geophysics(in Chinese),58(8):2827-2838,doi:10.6038/cjg20150817.
Yin C C,Zhang B,Liu Y H,et al.2017.A goal-oriented adaptive algorithm for 3D magnetotelluric forward modeling.Chinese Journal of Geophysics(in Chinese),60(1):327-336,doi:10.6038/cjg20170127.
Zhang Y X,Key K,Holst M,et al.2015.Parallel goal-oriented adaptive finite element modeling for 3Delectromagnetic exploration.∥85th Ann.Internat Mtg.,Soc.Expi.Geophys..Expanded Abstracts.
李勇,吴小平,林品荣等.2017.电导率任意各向异性海洋可控源电磁三维矢量有限元数值模拟.地球物理学报,60(5):1955-1978,doi:10.6038/cjg20170528.
刘颖,李予国,韩波.2017.可控源电磁场三维自适应矢量有限元正演模拟.地球物理学报,60(12):4874-4886,doi:10.6038/cjg20171227.
杨军,刘颖,吴小平.2015.海洋可控源电磁三维非结构矢量有限元数值模拟.地球物理学报,58(8):2827-2838,doi:10.6038/cjg20150817.
殷长春,张博,刘云鹤等.2017.面向目标自适应三维大地电磁正演模拟.地球物理学报,60(1):327-336,doi:10.6038/cjg20170127.
Baker A,Falgout R,Kolev T,et al.2012.Scaling hypre′s multigrid solvers to 100,000cores,in Berry M W,Gallivan KA,Gallopoulos E,Grama A,Philippe B,Saad Y,and Saied F,eds.,High Performance Scientific Computing:Algorithms and Applications:Springer,261-279.
Chen J Q,Chen Z M,Cui T,et al.2010.An adaptive finite element method for the eddy current model with circuit/field couplings.SIAM Journal on Scientific Computing,32(2):1020-1042.
Chen Z M,Wang L,Zheng W Y.2007.An adaptive multilevel method for time-harmonic Maxwell equations with singularities.SIAM Journal on Scientific Computing,29(1):118-138.
Colombo D,McNeice G,Curiel E S,et al.2013.Full tensor CSEMand MT for subsalt structural imaging in the Red Sea:Implications for seismic and electromagnetic integration.The Leading Edge,32(4):436-449.
Constable S.2010.Ten years of marine CSEM for hydrocarbon exploration.Geophysics,75(5):75A67-75A81.
Da Silva N V,Morgan J V,MacGregor L,et al.2012.A finite element multifrontal method for 3D CSEM modeling in the frequency domain.Geophysics,77(2):E101-E115.
Grayver A V,Bürg M.2014.Robust and scalable 3-D geo-electromagnetic modelling approach using the finite element method.Geophysical Journal International,198(1):110-125.
Haber E.2014.Computational Methods in Geophysical Electromagnetics.Philadelphia,PA,USA:Society for Industrial and Applied Mathematics.
Hiptmair R,Xu J C.2007.Nodal auxiliary space preconditioning in H(curl)and H(div)spaces.SIAM Journal on Numerical Analysis,45(6):2483-2509.
Jin J M.2002.The Finite Element Method in Electromagnetics.2nd ed.New York:Wiley.
Jahandari H,Farquharson C G.2015.Finite-volume modelling of geophysical electromagnetic data on unstructured grids using potentials.Geophysical Journal International,202(3):1859-1876.
Karypis G,Kumar V.1998.A parallel algorithm for multilevel graph partitioning and sparse matrix ordering.Journal of Parallel and Distributed Computing,48:71-85.
Key K.2009.1D inversion of multicomponent,multifrequency marine CSEM data:Methodology and synthetic studies for resolving thin resistive layers,Geophysics,74(2):F9-F20.
Key,K,Ovall J.2011.A parallel goal-oriented adaptive finite element method for 2.5Delectro-magnetic modelling.Geophysical Journal International.186:137-154.
Key K.2012.Marine electromagnetic studies of seafloor resources and tectonics.Surveys in Geophysics,33(1):135-167.
Kolev T V,Vassilevski P S.2009.Parallel auxiliary space AMG for H(curl)problems.Journal of Computational Mathematics,27(5):604-623.
Li Y,Wu X P,Lin P R,et al.2017.Three-dimensional modeling of marine controlled-source electromagnetism using the vector finite element method for arbitrary anisotropic media.Chinese Journal of Geophysics(in Chinese),60(5):1955-1978,doi:10.6038/cjg20170528.
Li Y G,Key K.2007.2Dmarine controlled-source electromagnetic modeling:Part 1-An adaptive finite-element algorithm.Geophysics,72(2):WA51-WA62.
Li Y G,Dai S.2011.Finite element modeling of marine controlledsource electromagnetic responses in two-dimensional dipping anisotropic conductivity structures.Geophysical Journal International,185(2):622-636.
Liu Y,Li Y G,Han B.2017.Adaptive edge finite element modeling of the 3D CSEM field on unstructured grids.Chinese Journal of Geophysics(in Chinese),60(12):4874-4886,doi:10.6038/cjg20171227.
Monk P.2003.Finite Element Methods for Maxwell′s Equations.Oxford:Oxford University Press.
Operto S,Virieux J,Amestoy P,et al.2007.3Dfinite-difference frequency-domain modeling of visco-acoustic wave propagation using a massively parallel direct solver:a feasibility study.Geophysics,72(5):SM195-SM211.
Pardo D,Paszynski M,Collier N,et al.2012.A survey on direct solvers for Galerkin methods.SeMA Journal,57(1):107-134.
Plessix R E,Edouard M,Mathieu M W.2007.An approach for 3Dmultisource,multifrequency CSEM modeling.Geophysics,72(5):SM177.
Puzyrev V,Koldan J,De La Puente J,et al.2013.A parallel finiteelement method for three-dimensional controlled-source electromagnetic forward modelling.Geophysical Journal International,193(2):678-693.
Ren Z Y,Kalscheuer T,Greenhalgh S,et al.2013.A goal-oriented adaptive finite-element approach for plane wave 3-D electromagnetic modelling.Geophysical Journal International,194(2):700-718.
Saad Y.2003.Iterative Methods for Sparse Linear Systems.2nd ed.Philadelphia:Society for Industrial and Applied Mathematics.
Schwarzbach C,B9rner R U,Spitzer K.2011.Three-dimensional adaptive higher order finite element simulation for geoelectromagnetics-A marine CSEM example.Geophysical Journal International,187(1):63-74.
Streich R.2009.3Dfinite-difference frequency-domain modeling of controlled-source electromagnetic data:direct solution and optimization for high accuracy.Geophysics,74(5):F95-F105.
Um E S,Commer M,Newman G A.2013.Efficient pre-conditioned iterative solution strategies for the electromagnetic diffusion in the Earth:Finite-element frequency domain approach.Geophysical Journal International,193(3):1460-1473.
Ward S H,Hohmann G W.1988.Electromagnetic theory for geophysical applications.∥Nabighian M N ed.Electromagnetic Methods in Applied Geophysics.Oklahoma:Society of Exploration Geophysicists,1:131-311.
Yang J,Liu Y,Wu X P.2015.3Dsimulation of marine CSEMusing vector finite element method on unstructured grids.Chinese Journal of Geophysics(in Chinese),58(8):2827-2838,doi:10.6038/cjg20150817.
Yin C C,Zhang B,Liu Y H,et al.2017.A goal-oriented adaptive algorithm for 3D magnetotelluric forward modeling.Chinese Journal of Geophysics(in Chinese),60(1):327-336,doi:10.6038/cjg20170127.
Zhang Y X,Key K,Holst M,et al.2015.Parallel goal-oriented adaptive finite element modeling for 3Delectromagnetic exploration.∥85th Ann.Internat Mtg.,Soc.Expi.Geophys..Expanded Abstracts.
李勇,吴小平,林品荣等.2017.电导率任意各向异性海洋可控源电磁三维矢量有限元数值模拟.地球物理学报,60(5):1955-1978,doi:10.6038/cjg20170528.
刘颖,李予国,韩波.2017.可控源电磁场三维自适应矢量有限元正演模拟.地球物理学报,60(12):4874-4886,doi:10.6038/cjg20171227.
杨军,刘颖,吴小平.2015.海洋可控源电磁三维非结构矢量有限元数值模拟.地球物理学报,58(8):2827-2838,doi:10.6038/cjg20150817.
殷长春,张博,刘云鹤等.2017.面向目标自适应三维大地电磁正演模拟.地球物理学报,60(1):327-336,doi:10.6038/cjg20170127.