谱元法数值模拟地震波传播
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摘要
谱元法基于弹性力学方程弱形式基础之上,在有限单元上进行谱展开,从而使该方法具备了有限元适应任意复杂介质模型的韧性和伪谱法的精度,该方法又被称为高阶有限元方法或伪谱法的域分解,同时为地震波传播的数值模拟提供一种方法。本文阐述了基于Legendre多项式的谱元法的理论和推导过程,相对于Chebychev谱元法,Legendre谱元法可以形成对角的全局质量矩阵,这样可以形成显式的时间差分算法,提高运算效率。最后,进行了复杂介质的各向同性地震波数值模拟,结果表明这是数值模拟地震波传播的有效方法。
The spectral element method is based on a weak elastic mechanism equation and combines the geometrical flexibility of the finite element method with the accuracy of the pseudospectral method.Hence the method is called the high-order FEM or the domain decomposition of pseudospectral method.It is a new tool for numerical modeeling of elastic wave propagation.Here,the theory and the deducing process of the SEM based on the Legendre polynomials are shown.The Legendre SEM can get diagonal mass matrix relative to the Chebychev SEM,which leads to a very simple explicit time difference scheme with high efficiency.A few of numerical tests are simulated and approved that the SEM is a effective method for modelling of seismic wave propagation.
The spectral element method is based on a weak elastic mechanism equation and combines the geometrical flexibility of the finite element method with the accuracy of the pseudospectral method.Hence the method is called the high-order FEM or the domain decomposition of pseudospectral method.It is a new tool for numerical modeeling of elastic wave propagation.Here,the theory and the deducing process of the SEM based on the Legendre polynomials are shown.The Legendre SEM can get diagonal mass matrix relative to the Chebychev SEM,which leads to a very simple explicit time difference scheme with high efficiency.A few of numerical tests are simulated and approved that the SEM is a effective method for modelling of seismic wave propagation.
引文
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[2]Virieux J.P-SV wave propagation in heterogeneousmedia:Velocity-stress finite-difference method[J].Geophysics,1986,51:889-901
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[6]Robertsson J O A.A numerical free-surface conditionfor elastic/viso elastic finite-difference modeling in thepresence of topography[J].Geophysics,1996,61:1921-1934
[7]王秀明,张海澜.用于具有不规则起伏自由表面的介质中弹性波模拟的有限差分算法[J].中国科学(G辑),2004,34(5):481-493Wang X M,Zhang H L.Modeling the seismic wave inthe media with irregular free interface by the finite-difference method[J].Science in China(Ser.G)2004,34(5):481-493
[8]Graves R.Simulation seismic wave propagation in 3Delastic media using staggered-grid finite difference[J].Bull Seismological Society of America,1996,86:1091-1106
[9]Marfurt K J.Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equa-tions[J].Geophysics,1984,49:533-549
[10]Komatitsch D,Vilotte J P.The spectral elementmethod:an efficient tool to simulate the seismic re-sponse of 2D and 3D geological structures[J].BullSeismological Society of America,1998,88:368-392
[11]Carcione J M.Domain decomposition for wave propa-gation problems[J].Journal of Scientific Computing,1991,6:453-472
[12]Carcione J M,Wang P J.A Chebyshev collocationmethod for the wave equation in generalized coordi-nates[J].Computational Fluid Dynamies Journal,1993,2:269-290
[13]Kosloff D,Tal-Ezer H.A modified Chebyshev pse-dospectral method with an O(N-1)time step restriction[J].Journal of Computational Physics,1993,104:457-469
[14]Patera A T.A spectral element method for fluid dy-namics:laminar flow in a channel expansion[J].Jour-nal of Computational Acoustics,2(4),1994,371-422
[15]Priolo E,Seriani G.A numerical investigation ofChebyshev spectral element method for acoustic wavepropagation[A].Dublin:Proceedings of the 13th I-MACS Conference on Comparative Applied Mathe-matics[C].Dublin:Ireland,1991,551-556
[16]Priolo E,Carcione J M,Seriani G.Numerical simula-tion of interface waves by high-order spectral model-ing techniques[J].Journal of Acoustical Society ofAmerica,1994,95(2):681-693
[17]Priolo E.Earthqnake grourd motion simulationthrongh the Z-D spectral elecment mehtod[A].In Pa-per Presented at the International Conference on Com-putational Aconstics[C].Italy:Trieste,1999
[18]Maday Y,Patera A T.Spectral Element Methods forthe Incompressible Navier-stokes Equations[M].York:ASME,1989,71-143
[19]Komatitsch D.Spectral and Spectral-element Methodsfor the 2D and 3D Elastodynamics Equations in Het-erogeneous Media[D].Paris,France:Insti-tut dePhysique du Globe,1997.22-57
[20]Komatitsch D,Tromp J.Introduction to the spectral-element method for 3-D seismic wave propagation[J].Geophysical Journal International,1999,139:806-822
[21]Komatitsch D,Tromp J.A Perfectly Matched Layerabsorbing boundary condition for the second-orderseismic wave equation[J].Geophysical Journal Inter-national,2003,154:146-153
[22]Komatitsch D,Vilotte J P.The spectral-elementmethod:an efficient tool to simulate the seismic re-sponse of 2D and 3D geological structures[J].BullSeismological Society of America,1998,88:368-392
[23]Komatitsch D,Barnes C,Tromp J.Wave propagationnear a fluid-solid interface:a spectral element ap-proach[J].Geophysics,2000,65(2):623-631
[24]Komatitsch D,Ritsema J,Tromp J.The spectral-ele-ment method,Beowulf computing,and global seis-mology[J].Science,2002,298:1737—1742
[25]Komatitsch D,Tromp J.Spectral-element simulationsof global seismic wave propagation,Part II:3-D mod-els,oceans,rotation,and gravity[J].GeophysicalJournal International,2002,150:303-318
[26]林伟军,王秀明,张海澜.用于弹性波方程模拟的基于逐元技术的谱元[J].自然科学进展,15(9):1048-1087Lin W J,Wang X M,Zhang H L.Modeling seismicwave equation with the spectral element method basedon the elemental technology[J].Progress in NatureScience,15(9):1048-1087
[27]Cerjan C,Kosloff R,Reshef M,et al.A nonreflect-ing boundary condition for discrete acoustic and elasticwave equation[J].Geophysics,1985,50:705-708
[28]Kosloff D,Kosloff R.Absorbing boundaries for wavepropagation problems[J].Journal of ComputationalPhysics,1986,63:363-376
[29]Clayton R,Engquist B.Absorbing boundary condi-tions for acoustic and elastic wave equations[J].BullSeismological Society of America,1977,67:1529-1540
[30]MandyY,Ronquist E M.Optional error analysis ofspectral methods with emphasis on non-constant coef-ficients and deformed geometries[J].ComputationalMethods of Application and Mechanical Engineering,1990,(80):91-115
[2]Virieux J.P-SV wave propagation in heterogeneousmedia:Velocity-stress finite-difference method[J].Geophysics,1986,51:889-901
[3]Levander A R.Fourth-order finite-difference P-SVseismograms[J].Geophysics,1988,53:1425-1436
[4]董良国,马在田,曹景忠,等.一阶弹性波方程交错网格高阶差分稳定性研究[J].地球物理学报,2000,43(6):856-864Dong L G,Ma Z T,Cao J Z,et al.The stabilitystudy of the staggered-grid high-order differencemethod of non-order elastic equation[J].ChineseJournal of Geophysics,2000,43(6):856-864
[5]Boore D M.A note on the effect of simple topographyon seismic SHwaves[J].Bull Seismological Society ofAmerica,1972,62,275-284
[6]Robertsson J O A.A numerical free-surface conditionfor elastic/viso elastic finite-difference modeling in thepresence of topography[J].Geophysics,1996,61:1921-1934
[7]王秀明,张海澜.用于具有不规则起伏自由表面的介质中弹性波模拟的有限差分算法[J].中国科学(G辑),2004,34(5):481-493Wang X M,Zhang H L.Modeling the seismic wave inthe media with irregular free interface by the finite-difference method[J].Science in China(Ser.G)2004,34(5):481-493
[8]Graves R.Simulation seismic wave propagation in 3Delastic media using staggered-grid finite difference[J].Bull Seismological Society of America,1996,86:1091-1106
[9]Marfurt K J.Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equa-tions[J].Geophysics,1984,49:533-549
[10]Komatitsch D,Vilotte J P.The spectral elementmethod:an efficient tool to simulate the seismic re-sponse of 2D and 3D geological structures[J].BullSeismological Society of America,1998,88:368-392
[11]Carcione J M.Domain decomposition for wave propa-gation problems[J].Journal of Scientific Computing,1991,6:453-472
[12]Carcione J M,Wang P J.A Chebyshev collocationmethod for the wave equation in generalized coordi-nates[J].Computational Fluid Dynamies Journal,1993,2:269-290
[13]Kosloff D,Tal-Ezer H.A modified Chebyshev pse-dospectral method with an O(N-1)time step restriction[J].Journal of Computational Physics,1993,104:457-469
[14]Patera A T.A spectral element method for fluid dy-namics:laminar flow in a channel expansion[J].Jour-nal of Computational Acoustics,2(4),1994,371-422
[15]Priolo E,Seriani G.A numerical investigation ofChebyshev spectral element method for acoustic wavepropagation[A].Dublin:Proceedings of the 13th I-MACS Conference on Comparative Applied Mathe-matics[C].Dublin:Ireland,1991,551-556
[16]Priolo E,Carcione J M,Seriani G.Numerical simula-tion of interface waves by high-order spectral model-ing techniques[J].Journal of Acoustical Society ofAmerica,1994,95(2):681-693
[17]Priolo E.Earthqnake grourd motion simulationthrongh the Z-D spectral elecment mehtod[A].In Pa-per Presented at the International Conference on Com-putational Aconstics[C].Italy:Trieste,1999
[18]Maday Y,Patera A T.Spectral Element Methods forthe Incompressible Navier-stokes Equations[M].York:ASME,1989,71-143
[19]Komatitsch D.Spectral and Spectral-element Methodsfor the 2D and 3D Elastodynamics Equations in Het-erogeneous Media[D].Paris,France:Insti-tut dePhysique du Globe,1997.22-57
[20]Komatitsch D,Tromp J.Introduction to the spectral-element method for 3-D seismic wave propagation[J].Geophysical Journal International,1999,139:806-822
[21]Komatitsch D,Tromp J.A Perfectly Matched Layerabsorbing boundary condition for the second-orderseismic wave equation[J].Geophysical Journal Inter-national,2003,154:146-153
[22]Komatitsch D,Vilotte J P.The spectral-elementmethod:an efficient tool to simulate the seismic re-sponse of 2D and 3D geological structures[J].BullSeismological Society of America,1998,88:368-392
[23]Komatitsch D,Barnes C,Tromp J.Wave propagationnear a fluid-solid interface:a spectral element ap-proach[J].Geophysics,2000,65(2):623-631
[24]Komatitsch D,Ritsema J,Tromp J.The spectral-ele-ment method,Beowulf computing,and global seis-mology[J].Science,2002,298:1737—1742
[25]Komatitsch D,Tromp J.Spectral-element simulationsof global seismic wave propagation,Part II:3-D mod-els,oceans,rotation,and gravity[J].GeophysicalJournal International,2002,150:303-318
[26]林伟军,王秀明,张海澜.用于弹性波方程模拟的基于逐元技术的谱元[J].自然科学进展,15(9):1048-1087Lin W J,Wang X M,Zhang H L.Modeling seismicwave equation with the spectral element method basedon the elemental technology[J].Progress in NatureScience,15(9):1048-1087
[27]Cerjan C,Kosloff R,Reshef M,et al.A nonreflect-ing boundary condition for discrete acoustic and elasticwave equation[J].Geophysics,1985,50:705-708
[28]Kosloff D,Kosloff R.Absorbing boundaries for wavepropagation problems[J].Journal of ComputationalPhysics,1986,63:363-376
[29]Clayton R,Engquist B.Absorbing boundary condi-tions for acoustic and elastic wave equations[J].BullSeismological Society of America,1977,67:1529-1540
[30]MandyY,Ronquist E M.Optional error analysis ofspectral methods with emphasis on non-constant coef-ficients and deformed geometries[J].ComputationalMethods of Application and Mechanical Engineering,1990,(80):91-115
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