三角网格有限元法波动模拟的数值频散及稳定性研究
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摘要
三角网格有限元法能够准确模拟复杂构造和复杂介质条件下的地震波场,数值频散和稳定性条件是地震波数值模拟中参数选择的主要依据.基于均匀的线性三角网格单元,根据结构刚度矩阵的组装原理以及平面波理论,推导了集中质量矩阵下两种网格结构的声波频散函数以及稳定性条件,并对数值频散特性以及稳定性进行了详细研究:三角网格单元中波动的数值频散除了受到空间采样间隔、单元网格纵横比和波传播方向等常规因素的影响外,还受到网格布局的影响,过锐或过钝的三角单元会对波动数值频散产生不良的影响,不同类型的单元网格、单元纵横比对应着不同的稳定性条件,正三角单元中的波动具有较好的数值频散特性,其数值各向异性(频散随波传播方向的变化)效应最弱,稳定性条件也较为宽松.最后通过数值模拟直观地验证了以上分析结果,为有限元正演三角网格的剖分和参数的设置提供一定的理论依据.
The forward modeling of wave equation plays a very important role in seismic data acquisition,processing and interpretation.In order to accurately simulate the propagation of seismic wave in underground medium,not only requiring geophysical model is consistent with the actual formation,but also needing high precision numerical simulation method.Finite element method(FEM)with triangular grid can divide arbitrary complex boundary effectively,so it can accurately simulate the propagation of seismic wave field in complex medium.The spatial discretization of continuous medium by FEM introduces dispersion errors to the solution of wave equation,which causes phase velocity(numerical phase velocity)varies with the frequency of seismic wave that do not actually exist in the continuum,and this artefact is called numerical dispersion.In general,numerical dispersion not only has no practical significance,but also affects the understanding of the real fluctuation phenomenon.Therefore,it is very necessary to clarify the influence factors of the numerical dispersion,which can help to improve the accuracyof the numerical simulation.This paper focuses on studying the numerical dispersion and stability of wave motion with triangle-based finite element algorithm.Considering two kinds of commonly used grid structures(denoted by I mesh structure and II mesh structure,respectively),according to the assemble principle of structural stiffness matrix and plane wave theory,we obtain dispersion functions under the assumption that computational area is uniform and without borders.The stability conditions are obtained in the condition of second order middle difference,to provide reference for the selection of time step.In order to fully understand the characteristics of numerical dispersion in triangular meshes,we analysed quantitatively the effect of spatial sampling interval,the propagation direction of seismic wave,the ratio of vertical to horizontal(mesh shape)on numerical dispersion.To obtain optimal mesh generation,the influences of mesh shape are studied mainly.In a practical simulation,we should consider the numerical calculation accuracy.Therefore,the maximum dispersion errors of the two kinds of grid structures are analyzed comparatively.Finally,we verify our conclusions by wave-field simulations.The authors hope that the study in this paper can provide some useful suggestions for the division of triangular mesh and parameter setting.From the theoretical analyses and numerical experiments,the following results can be gained:(1)The numerical phase velocity varies significantly with the propagation direction,and this variation is periodic,the period of the numerical dispersion isπexcept regular triangle mesh,the period of which isπ/3,because the mesh structure is invariant underπ/3rotation.(2)When the ratio of vertical to horizontal is equal to 1,the maximum dispersion error and minimum dispersion error of I mesh structure appear in the direction ofθ=0°andθ=45°,respectively;and the maximum dispersion error and minimum dispersion error of II mesh structure appear in the direction ofθ=90°andθ=0°,respectively.(3)In terms of I mesh structure,the dispersion error in horizontal direction has nothing to do with the ratio of vertical to horizontal,which is also the maximum dispersion error,in other words,by reducing the ratio of vertical to horizontal to suppress the numerical dispersion is not an effective method.(4)In terms of II mesh structure,the maximum dispersion error can be suppressed effectively by reducing the ratio of vertical to horizontal as long as it does not appear obtuse triangle mesh;it is worth pointing out that regular triangle mesh almost removes the directional dependence of the phase velocity,which means the numerical anisotropy is the weakest.(5)In the case of no visible numerical dispersion,the number of sampling point in the wavelength corresponding to peak frequency of source wavelet of II mesh structure is less than that of I mesh structure,and this can help improve the calculation efficiency and reduce the occupation of memory.(6)The stability factor of II mesh structure is also larger than that of I mesh structure.The dispersion characteristics analysed in the above can provide some theoretical guidances for mesh generation.The arrangements of elements should be chosen with care;unreasonable meshes will reduce the precision of numerical simulation,and even lead to erroneous results.The best arrangement of triangles appears to be hexagonal,which has the further benefit that it almost removes the directional dependence of the phase velocity,which can be useful in maintaining wave front definition.
引文
Abboud N N,Pinsky P M.1992.Finite element dispersion analysisfor the three-dimensional second-order scalar wave equation.Int.J.Numer.Methods Eng.,35(6):1183-1218.
    Ainsworth M,Monk P,Muniz W.2006.Dispersive and dissipativeproperties of discontinuous Galerkin finite element methods forthe second-order wave equation.J.Sci.Comput.,27(1-3):5-40.
    Christon M A.1999.The influence of the mass matrix on thedispersive nature of the semi-discrete,second-order waveequation.Comput.Methods Appl.Mech.Eng.,173(1):147-166.
    Du S T.1982.Finite element numerical solution of wave propagation innon-homogeneous medium with variable velocities.Journal of EastChina Petroleum Institute(in Chinese),6(2):1-20.
    De Basabe J D,Sen M K.2007.Grid dispersion and stability criteriaof some common finite-element methods for acoustic and elasticwave equations.Geophysics,72(6):T81-T95.
    De Basabe J D,Sen M K,Wheeler M F.2008.The interior penaltydiscontinuous Galerkin method for elastic wave propagation:grid dispersion.Geophys.J.Int.,175(1):83-93.
    Guo J.1991.A kind of fast finite element algorithm.GeophysicalProspecting for Petroleum(in Chinese),30(2):36-43.
    Hu F Q,Hussaini M Y,Rasetarinera P.1999.An analysis of thediscontinuous Galerkin method for wave propagation problems.J.Comput.Phys.,151(2):921-946.
    He X J,Yang D H,Wu H.2014.Numerical dispersion and wavefield simulation of the Runge-Kutta discontinuous Galerkinmethod.Chinese J.Geophys.(in Chinese),57(3):906-917,doi:10.6038/cjg20140320.
    Ke B,Zhao B,Cai J,et al.2001.2-D finite element acoustic wavemodeling including rugged topography.71th AnnualInternational Meeting,SEG,Expanded Abstracts,1199-1202.
    Lee R,Cangellaris A C.1992.A study of discretization error in thefinite element approximation of wave solutions.IEEE Trans.Antennas Propag.,40(5):542-549.
    Liu J B,Sharan S K,Yao L.1994.Wave motion and its dispersiveproperties in a finite element model with distortional elements.Comput.Struct.,52(2):205-214.
    Liu Y,Sen M K.2009.A new time-space domain high-order finitedifference method for the acoustic wave equation.J.Comput.Phys.,228(23):8779-8806.
    Liu T,Wei X,De Basabe J D,et al.2012.Grid dispersion andstability of the spectral element method with triangularelements.82th Annual International Meeting,SEG,ExpandedAbstracts,1-5.
    Liu Y S,Teng J W,Liu S L,et al.2013.Explicit finite elementmethod with triangle meshes stored by sparse format and itsperfectly matched layers absorbing boundary condition.ChineseJ.Geophys.(in Chinese),56(9):3085-3099,doi:10.6038/cjg20130921.
    Liu Y S,Teng J W,Xu T,et al.2014.Numerical modeling ofseismic wavefield with the SEM based on Triangles.Progressin Geophysics(in Chinese),29(4):1715-1726,doi:10.6038/pg20140430.
    Lu X,Zhang S Y,Cui X W.2014.Finite element method for 2.5Dresistivity forward modeling based on anomaly electric field.Progress in Geophysics(in Chinese),29(6):2718-2722,doi:10.6038/pg20140637.
    Li L,Liu T,Hu T Y.2014.Spectral element method withtriangular mesh and its application in seismic modeling.ChineseJ.Geophys.(in Chinese),57(4):1224-1234,doi:10.6038/cjg20140419.
    Mullen R,Belytschko T.1982.Dispersion analysis of finite elementsemidiscretizations of the two-dimensional wave equation.Int.J.Numer.Methods Eng.,18(1):11-29.
    Mulder W A.1999.Spurious modes in finite-element discretizationsof the wave equation may not be all that bad.Appl.Numer.Math.,1999,30(4):425-445.
    Saad Y.2000.Iterative Methods for Sparse Linear Systems.Philadelphia:SIAM.
    Sun C Y.2007.Theory and Methods of Seismic Waves.Dongying:China University of Petroleum Press(in Chinese):31-37.
    Sun C Y,Gong T J,Zhang Y L,et al.2009.Analysis on dispersionand alias in finite-difference solution of wave equation.OilGeophysical Prospecting(in Chinese),44(1):43-48.
    Sun C Y,Xiao Y F,Yin X Y,et al.2010.Study on the stability offinite difference solution of visco-elastic wave equations.ActaSeismologica Sinica(in Chinese),32(2):147-156.
    Seriani G,Oliveira S P.2008.Dispersion analysis of spectral elementmethods for elastic wave propagation.Wave Motion,45(6):729-744.
    Wang M C.2003.Finite Element Method.Beijing:Tsinghua Universitypress(in Chinese):472-475.
    Wu G C,Wang H Z.2005.Analysis of numerical dispersion inwave-field simulation.Progress in Geophysics(in Chinese),20(1):58-65.
    Wang W S,Zhang H,Li X F.2013.Review on application of thediscontinuous Galerkin method for modeling of the seismicwavefield.Progress in Geophysics(in Chinese),28(1):171-179,doi:10.6038/pg20130118.
    Wu Z Q,Song W J.2013.Resistivity transverse section method andits application in quantified explanation of oblique faults.Progress in Geophysics(in Chinese),28(5):2748-2752,doi:10.6038/pg20130559.
    Wang R,Wang M Y,Di Q Y,et al.2014.3D1CCSAMT modelingusing finite element method.Progressi n Geophysics(in Chinese),29(2):839-845,doi:10.6038/pg20140249.
    Xu S Z.1994.Finite Element Method for Geophysics.Beijing:Science Press(in Chinese):39-42.
    Xue D C,Wang S X,Jiao S J.2007.Wave equation finite elementmodeling including rugged topography and complicated medium.Progress in Geophysics(in Chinese),22(2):522-529.
    Xue D C,Wang S X.2008a.Wave-equation finite element prestackreverse-time migration.Oil Geophysical Prospecting(in Chinese),43(1):17-21.
    Xue D C,Wang S X.2008b.Using combined mass matrix tosuppress numerical dispersion.Oil Geophysical Prospecting(in Chinese),43(3):318-320.
    Xue Z,Dong L G,Li X B,et al.2014.Discontinuous Galerkinfinite-element method for elastic wave modeling includingsurface topography.Chinese J.Geophys.(in Chinese),57(4):1209-1223,doi:10.6038/cjg20140418.
    Yu K Y.1982.Numerical analysis in the construction of syntheticseismograms by the finite element method.Journal of EastChina Petroleum Institute(in Chinese),6(4):11-27.
    Yue B,Guddati M N.2005.Dispersion-reducing finite elements fortransient acoustics.J.Acoust.Soc.Am.,118(4):2132-2141.
    Yin X Y,Zhou J K,Wu G C,et al.2014.Dispersion analysis forthe finite element algorithm in acoustic wave equation numerical
    simulation.Acta Seismologica Sinica(in Chinese),36(5):[40]944-955.
    杜世通.1982.变速不均匀介质中波动方程的有限元法数值解.华东石油学院学报,6(2):1-20.
    郭建.1991.一种有限元快速算法.石油物探,30(2):36-43.
    贺茜君,杨顶辉,吴昊.2014.间断有限元方法的数值频散分析及其波场模拟.地球物理学报,57(3):906-917,doi:10.6038/cjg20140320.
    刘有山,滕吉文,刘少林等.2013.稀疏存储的显式有限元三角网格地震波数值模拟及其PML吸收边界条件.地球物理学报,56(9):3085-3099,doi:10.6038/cjg20130921.
    刘有山,滕吉文,徐涛等.2014.三角网格谱元法地震波场数值模拟.地球物理学进展,29(4):1715-1726,doi:10.6038/pg20140430.
    鲁杏,张胜业,崔先文.2014.基于异常场的2.5维电阻率有限元正演模拟.地球物理学进展,29(6):2718-2722,doi:10.6038/pg20140637.
    李琳,刘韬,胡天跃.2014.三角谱元法及其在地震正演模拟中的应用.地球物理学报,57(4):1224-1234,doi:10.6038/cjg20140419.
    孙成禹.2007.地震波理论与方法.山东东营:中国石油大学出版社:31-37.
    孙成禹,宫同举,张玉亮等.2009.波动方程有限差分法中的频散与假频分析.石油地球物理勘探,44(1):43-48.
    孙成禹,肖云飞,印兴耀等.2010.黏弹介质波动方程有限差分解的稳定性研究.地震学报,32(2):147-156.
    王勖成.2003.有限单元法.北京:清华大学出版社:472-475.
    吴国忱,王华忠.2005.波场模拟中的数值频散分析与校正策略.地球物理学进展,20(1):58-65.
    汪文帅,张怀,李小凡.2013.间断的Galerkin方法在地震波场数值模拟中的应用概述.地球物理学进展,28(1):171-179,doi:10.6038/pg20130118.
    吴子泉,宋文杰.2013.电阻率横向剖面法对倾斜断层的定量化研究及解释.地球物理学进展,28(5):2748-2752,doi:10.6038/pg20130559.
    王若,王妙月,底青云等.2014.CSAMT三维单分量有限元正演.地球物理学进展,29(2):839-845,doi:10.6038/pg20140249.
    徐世浙.1994.地球物理中的有限元法.北京:科技出版社:39-42.
    薛东川,王尚旭,焦淑静.2007.起伏地表复杂介质波动方程有限元数值模拟方法.地球物理学进展,22(2):522-529.
    薛东川,王尚旭.2008a.波动方程有限元叠前逆时偏移.石油地球物理勘探,43(1):17-21.
    薛东川,王尚旭.2008b.利用组合质量矩阵压制数值频散.石油地球物理勘探,43(3):318-320.
    薛昭,董良国,李晓波等.2014.起伏地表弹性波传播的间断Galerkin有限元数值模拟方法.地球物理学报,57(4):1209-1223,doi:10.6038/cjg20140418.
    俞康胤.1982.合成地震记录制作中的有限元法数值分析.华东石油学院学报,6(4):11-27.
    印兴耀,周建科,吴国忱等.2014.有限元算法在声波方程数值模拟中的频散分析.地震学报,36(5):944-955.

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