Abstract：Accuracy analysis is the foundation for nume-rical simulation of seismic waves. With regard to the numerical stability, dispersion, and dissipation of the discontinuous Galerkin finite element me-thod (DGFEM) based on triangular meshes, a trian-gular periodic mesh model is constructed, which can be used to study the effects of different triangular elements on simulation accuracy. The theoretical and numerical results show that the stability condition of the DGFEM based on the Runge-Kutta time scheme is related to the shape of triangle elements. The maximum time step for stable mode-ling has a linear relationship with the radius of the inscribed circle of the element, and the equilateral triangle element has the least rigorous stability condition. Meanwhile, the wave field from DGFEM simulation based on the local Lax-Friedrichs flux shows weak dispersion but strong dissipation, and both dispersion and dissipation present directivity in the periodic mesh. In addition, the lo-garithm of the modeling error has a linear relationship with that of the mesh size. The numerical experiments compare the influence of different mesh shapes on the wave field and verify the theoretical directional difference. The results of this paper can provide a theoretical basis for the triangular mesh division, parameter setting, and selection of nume-rical flow in DGFEM.
韩德超, 刘卫华, 司文朋. 三角网格间断有限元法弹性波模拟精度分析[J]. 石油地球物理勘探, 2021, 56(4): 758-770.
HAN Dechao, LIU Weihua, SI Wenpeng. Analysis of elastic wave simulation accuracy with discontinuous Galerkin finite element method based on triangular meshes. Oil Geophysical Prospecting, 2021, 56(4): 758-770.
裴正林,王尚旭,狄帮让,等.无组合检波采集方法的三维地震波数值模拟研究[J].石油地球物理勘探,2005,40(2):138-142.PEI Zhenglin,WANG Shangxu,DI Bangrang,et al.Study on acquisition methods without geophone array by numeric modeling of 3D seismic wavefield[J].Oil Geophysical Prospecting,2005,40(2):138-142.
[2]
崔宏良,王瑞贞,陈敬国,等.可控震源地震勘探中的数值模拟法应用[J].石油地球物理勘探,2017,52(2):209-213.CUI Hongliang,WANG Ruizhen,CHEN Jingguo,et al.Application of numerical simulation in vibroseis seismic acquisition[J].Oil Geophysical Prospecting,2017,52(2):209-213.
[3]
何宝庆,谢小碧.宽线地震数据采集和处理的数值模拟[J].石油地球物理勘探,2019,54(3):500-511.HE Baoqing,XIE Xiaobi.A numerical investigating on wide-line seismic data acquisition and processing[J].Oil Geophysical Prospecting,2019,54(3):500-511.
[4]
马国旗,曹丹平,尹教建,等.分布式声传感井中地震信号检测数值模拟方法[J].石油地球物理勘探,2020,55(2):311-320.MA Guoqi,CAO Danping,YIN Jiaojian,et al.Nume-rical simulation of detecting seismic signals in DAS wells[J].Oil Geophysical Prospecting,2020,55(2):311-320.
[5]
王文枫,岳大力,赵继勇,等.利用地震正演模拟方法研究地层结构——以鄂尔多斯盆地合水地区延长组三段为例[J]. 石油地球物理勘探,2020,55(2):411-418.WANG Wenfeng, YUE Dali, ZHAO Jiyong, et al.Research on stratigraphic structure based on seismic forward modeling:A case study of the third member of the Yanchang Formation in Heshui area, Ordos Basin[J].Oil Geophysical Prospecting, 2020, 55(2):411-418.
[6]
Virieux J,Calandra H,Plessix R É.A review of the spectral,pseudo-spectral,finite-difference and finite-element modelling techniques for geophysical imaging[J].Geophysical Prospecting,2011,59(5):794-813.
[7]
Kelly K R,Ward R W,Treitel S,et al.Synthetic seismograms:A finite-difference approach[J].Geophy-sics,1976,41(1):2-27.
[8]
Dablain M A.The application of high-order differencing to the scalar wave equation[J].Geophysics,1986,51(1):54-66.
[9]
董良国,马在田,曹景忠.一阶弹性波方程交错网格高阶差分解法[J].地球物理学报,2000,43(3):411-419.DONG Liangguo,MA Zaitian,CAO Jingzhong,et al.A staggered-grid high-order difference method of one-order elastic wave equation[J].Chinese Journal of Geophysics,2000,43(3):411-419.
[10]
Yang D.A nearly analytic discrete method for acoustic and elastic wave equations in anisotropic media[J].Bulletin of the Seismological Society of America,2003,93(2):882-890.
[11]
Kosloff D D,Baysal E.Forward modeling by a Fourier method[J].Geophysics,1982,47(10):1402-1412.
[12]
罗文山,陈汉明,王成祥,等.时间域黏滞波动方程及其数值模拟新方法[J].石油地球物理勘探,2016,51(4):707-713.LUO Wenshan,CHEN Hanming,WANG Cheng-xiang,et al.A novel time-domain viscoacoustic wave equation and its numerical simulation[J].Oil Geophysical Prospecting,2016,51(4):707-713.
[13]
Hermeline F.Two coupled particle-finite volume methods using Delaunay-Voronoï meshes for the approximation of Vlasov-Poisson and Vlasov-Maxwell equations[J].Journal of Computational Physics,1993,106(1):1-18.
[14]
Versteeg H K,Malalasekera W.An Introduction to Computational Fluid Dynamics:The Finite Volume Method[M].Addison-Wesley,Harlow,1995.
[15]
贾晓峰,胡天跃,王润秋.复杂介质中地震波模拟的无单元法[J].石油地球物理勘探,2006,41(1):43-48.JIA Xiaofeng,HU Tianyue,WANG Runqiu.A mesh-free algorithm of seismic wave simulation in complex medium[J].Oil Geophysical Prospecting,2006,41(1):43-48.
[16]
Marfurt K J.Accuracy of finite-difference and finite-element modeling of scalar and elastic wave equations[J].Geophysics,1984,49(5):533-549.
[17]
Serón F J,Sanz F J,Kindelán M.Finite-element method for elastic wave propagation[J].Communications in Applied Numerical Methods,1990,6(5):359-368.
[18]
薛东川,王尚旭,焦淑静.起伏地表复杂介质波动方程有限元数值模拟方法[J].地球物理学进展,2007,22(2):522-529.XUE Dongchuan,WANG Shangxu,JIAO Shujing.Wave equation finite-element modeling including rugged topography and complicated medium[J].Progress in Geophysics,2007,22(2):522-529.
[19]
Komatitsch D,Roland M,Tromp J,et al.Wave propagation in 2-D elastic media using a spectral element method with triangles and quadrangles[J].Journal of Computational Acoustics,2001,9(2):703-718.
[20]
Komatitsch D,Tromp J.Spectral-element simulations of global seismic wave propagation,Ⅰ:Validation[J].Geophysical Journal International,2002,149(2):390-412.
[21]
Komatitsch D,Vilotte J P.The spectral-element method:An efficient tool to simulate the seismic response of 2D and 3D geological structures[J].Bulletin of the Seismological Society of America,1998,88(2):368-392.
[22]
Liu Y S,Teng J W,Lan H Q,et al.A comparative study of finite element and spectral element methods in seismic wavefield modeling[J].Geophysics,2014,79(2):T91-T104.
[23]
Pasquetti R,Francesca R.Spectral element methods on triangles and quadrilaterals:comparisons and applications[J].Journal of Computational Physics,2004,198(1):349-362.
[24]
Bernardo C,Shu C W.Runge-Kutta discontinuous Galerkin methods for convection-dominated problems[J].Journal of Scientific Computing,2001,16(3):173-261.
[25]
He X J,Yang D H,Wu H.A weighted Runge-Kutta discontinuous Galerkin method for wavefield modelling[J].Geophysical Journal International,2014,200(3):1389-1410.
[26]
Reed W H,Hill T R.Triangular Mesh Methods for the Neutron Transport Equation[R].Los Alamos Scientific Laboratory,Los Alamos,1973.
[27]
薛昭,董良国,李晓波,等.起伏地表弹性波传播的间断Galerkin有限元数值模拟方法[J].地球物理学报,2014,57(4):1209-1223.XUE Zhao,DONG Liangguo,LI Xiaobo,et al.Discontinuous Galerkin finite-element method for elastic wave modeling including surface topography[J].Chinese Journal of Geophysics,2014,57(4):1209-1223.
[28]
De Basabe J D,Sen Mrinal K.Grid dispersion and stability criteria of some common finite-element methods for acoustic and elastic wave equations[J].Geophy-sics,2007,72(6):T81-T95.
[29]
Mulder W A.Spurious modes in finite-element discre-tizations of the wave equation may not be all that bad[J].Applied Numerical Mathematics,1999,30(4):425-445.
[30]
Sármány D,Botchev M A,Van Der Vegt J J W.Dispersion and dissipation error in high-order Runge-Kutta discontinuous Galerkin discretisations of the Maxwell equations[J].Journal of Scientific Computing,2007,33(1):47-74.
[31]
Seriani G,Oliveira S P.Dispersion analysis of spectral element methods for elastic wave propagation[J].Wave Motion,2008,45(6):729-744.
[32]
Zyserman F I,Santos J E.Analysis of the numerical dispersion of waves in saturated poroelastic media[J].Computer Methods in Applied Mechanics and Engineering,2007,196(45-48):4644-4655.
[33]
刘少林,李小凡,刘有山,等.三角网格有限元法声波与弹性波模拟频散分析[J].地球物理学报,2014,57(8):2620-2630.LIU Shaolin,LI Xiaofan,LIU Youshan,et al.Dispersion analysis of triangle-based finite element method for acoustic and elastic wave simulations[J].Chinese Journal of Geophysics,2014,57(8):2620-2630.
[34]
曹丹平,周建科,印兴耀.三角网格有限元法波动模拟的数值频散及稳定性研究[J].地球物理学报,2015,58(5):1717-1730.CAO Danping,ZHOU Jianke,YIN Xingyao.The study for numerical dispersion and stability of wave motion with triangle-based finite element algorithm[J].Chinese Journal of Geophysics,2015,58(5):1717-1730.
[35]
Liu T,Sen Mrinal K,Hu T Y,et al.Dispersion analysis of the spectral element method using a triangular mesh[J].Wave Motion,2012,49(4):474-483.
[36]
Mazzieri I,Francesca R.Dispersion analysis of triangle-based spectral element methods for elastic wave propagation[J].Numerical Algorithms,2012,60(4):631-650.
[37]
李琳,刘韬,胡天跃.三角谱元法及其在地震正演模拟中的应用[J].地球物理学报,2014,57(4):1224-1234.LI Lin,LIU Tao,HU Tianyue.Spectral element method with triangular mesh and its application in seismic modeling[J].Chinese Journal of Geophysics,2014,57(4):1224-1234.
[38]
De Basabe J D,Sen Mrinal K,Wheeler M F.The interior penalty discontinuous Galerkin method for elastic wave propagation:Grid dispersion[J].Geophysical Journal International,2008,175(1):83-93.
[39]
贺茜君,杨顶辉,吴昊.间断有限元方法的数值频散分析及其波场模拟[J].地球物理学报,2014,57(3):906-917.HE Xijun,YANG Dinghui,WU Hao.Numerical dispersion and wave-field simulation of the Runge-Kutta discontinuous Galerkin method[J].Chinese Journal of Geophysics,2014,57(3):906-917.
[40]
Meng W J,Fu L Y.Numerical dispersion analysis of discontinuous Galerkin method with different basis functions for acoustic and elastic wave equations[J].Geophysics,2018,83(3):T87-T101.
[41]
Hu F Q,Hussaini M Y,Rasetarinera P.An analysis of the discontinuous Galerkin method for wave propagation problems.Journal of Computational Physics,1999,151(2):921-946.
[42]
Käser M,Michael D.An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes,I:The two-dimensional isotropic case with external source terms[J].Geophysical Journal International,2006,166(2):855-877.
[43]
Delcourte S,Fezoui L,Glinsky-Olivier N.A high-order discontinuous Galerkin method for the seismic wave propagation[J].ESAIM:Proceedings,2009,27:70-89.
[44]
He X J,Yang D H,Ma X,et al.Dispersion-dissipation analysis of the triangle-based discontinuous Galerkin method for scalar wave equation[J].Geophysical Journal International,2019,218(2):1174-1198.
[45]
Hermann V,Käser M, Castro C E.Non-conforming hybrid meshes for efficient 2-D wave propagation using the discontinuous Galerkin method[J].Geophy-sical Journal International,2011,184(2):746-758.
