基于多辛结构谱元法的保结构地震波场模拟
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摘要
近年来构造高精度、高效且具有长时程跟踪能力的保结构算法已逐渐成为地震波模拟算法发展的重要方向之一.本文基于谱元法(SEM)进行空间域离散结合新推导的三阶辛算法(NTSTO)进行时间域离散,构造了一种具有时-空保结构特性的新算法.本文给出的多组数值试验对比结果表明,本算法无论在内存消耗、稳定性及计算耗时,还是长时程跟踪能力方面都有上佳的表现;另外,本文给出的起伏地表多层介质模型的数值算例验证了该算法处理复杂几何形状和复杂介质时的有效性.该多辛结构谱元法的发展将为长时程地震波传播的计算及模拟提供更为广泛而有效的选择.
In recent years,structure-preserving algorithms effective for treating problems of high-precision and long-time tracing have emerged as one attractive topic of the numerical simulation of seismic waves.In this paper,a new kind of time-space structure-preserving algorithm,which combines the spectral element method(SEM) in spatial discretization with a new three-stage third-order symplectical algorithm(NTSTO) in temporal discretization,is designed to simulate seismic wave propagation.The comparisons of numerical experiments show that the NTSTO-SEM algorithm is much superior to the other methods in terms of efficiency,long-time tracing ability,stability,and low storage requirement.Finally,the numerical experiments of a multilayered model with irregular topography are given to illustrate the effectiveness of NTSTO-SEM for handling complex geometry and complex media.These appealing features of the multisymplectic algorithm would make it effective to model the large-scale and long-time seismic wave propagation.
In recent years,structure-preserving algorithms effective for treating problems of high-precision and long-time tracing have emerged as one attractive topic of the numerical simulation of seismic waves.In this paper,a new kind of time-space structure-preserving algorithm,which combines the spectral element method(SEM) in spatial discretization with a new three-stage third-order symplectical algorithm(NTSTO) in temporal discretization,is designed to simulate seismic wave propagation.The comparisons of numerical experiments show that the NTSTO-SEM algorithm is much superior to the other methods in terms of efficiency,long-time tracing ability,stability,and low storage requirement.Finally,the numerical experiments of a multilayered model with irregular topography are given to illustrate the effectiveness of NTSTO-SEM for handling complex geometry and complex media.These appealing features of the multisymplectic algorithm would make it effective to model the large-scale and long-time seismic wave propagation.
引文
[1]Alterman Z,Karal F C Jr.Propagation of elastic waves inlayered media by finite difference methods.Bull.Seism.Soc.Am.,1968,58(1):367-398.
[2]Boore D M.Finite-difference methods for seismic wavepropagation in heterogeneous materials.//Bolt B A ed.Methods in Computational Physics 11,New York:AcademicPress.Inc.,1972.
[3]Madariaga R.Dynamics of an expanding circular fault.Bull.Seismol.Soc.Am.,1976,66(3):639-669.
[4]Virieus J.P-SV wave propagation in heterogeneous media:velocity-stress finite-difference method.Geophysics,1986,51(4):889-901.
[5]Mora P.Modeling anisotropic seismic waves in 3-D.59thAnn.Internat.Mtg.Soc.Expl.Geophys.,ExpandedAbstracts,1989,2:1039-1043.
[6]Frankel A,Vidale J.A three-dimensional simulation ofseismic waves in the Santa Clara valley,California,from theLoma Prieta aftershock.Bull.Seismol.Soc.Am.,1992,82(5):2045-2074.
[7]Frankel A.Three-dimensional simulations of ground motionsin the San Bernardino valley,California,for hypotheticalearthquakes on the San Andreas fault.Bull.Seismol.Soc.Am.,1993,83(4):1020-1041.
[8]Pitarka A.3D elastic finite-difference modeling of seismicmotion using staggered grids with nonuniform spacing.Bull.Seism.Soc.Am.,1999,89(1):54-68.
[9]Olsen K B.Site amplification in the Los Angeles basin fromthree-dimensional modeling of ground motion.Bull.Seismol.Soc.Am.,2000,90(6B):S77-S94.
[10]Zingg D W.Comparison of high-accuracy finite-differencemethods for linear wave propagation.SIAM Journal onScientific Computing,2000,22(2):476-502.
[11]Chaljub E,Komatitsch D,Vilotte J P,et al.Spectral-element analysis in seismology.Advances in Geophysics,2007,48:365-419.
[12]Marfurt K J.Accuracy of finite-difference and finite-elementmodeling of the scalar and elastic wave equations.Geophysics,1984,49(5):533-549.
[13]Komatitsch D,Vilotte J P.The spectral element method:Anefficient tool to simulate the seismic response of 2Dand 3Dgeological structures.Bull Seism.Soc.Am.,1998,88(2):368-392.
[14]Tessmer E,Kosloff D.3-D elastic modeling with surfacetopography by a Chebyshev spectral method.Geophysics,1994,59(3):464-473.
[15]Carcione J M.The wave equation in generalized coordinates.Geophysics,1994,59(12):1911-1919.
[16]Furumura M,Kennett B L N,Furumura T.Seismicwavefield calculation for laterally heterogeneous Earthmodels-II:the influence of upper mantle heterogeneity.Geophys.J.Int.,1999,139(3):623-644.
[17]Furumura T,Kennett B L N,Furumura M.Seismicwavefield calculation for laterally heterogeneous whole Earthmodels using the pseudospectral method.Geophys.J.Int.,1998,135(3):845-860.
[18]Carcione J M,Wang P J.A Chebyshev collocation methodfor the wave equation in generalized coordinates.Comp.Fluid Dyn.J.,1993,2:269-290.
[19]Shen J,Tang T.Spectral and High-Order Methods withApplications.Beijing:Science Press,2006.
[20]Komatitsch D,Martin R,Tromp J,et al.Wave propagationin 2-D elastic media using a spectral element method withtriangles and quadrangles.J.Comput.Acoust.,2001,9(2):703-718.
[21]Chaljub E,Valette B.Spectral element modelling of three-dimensional wave propagation in a self-gravitating Earth withan arbitrarily stratified outer core.Geophys.J.Int.,2004,158(1):131-141.
[22]Chaljub E,Capdeville Y,Vilotte J P.Solving elastodynamicsin a fluid-solid heterogeneous sphere:a parallel spectralelement approximation on non-conforming grids.J.Comput.Phys.,2003,187(2):457-491.
[23]Komatitsch D,Ritsema J,Tromp J.The spectral-elementmethod,Beowulf computing,and global seismology.Science,2002,298(5599):1737-1742.
[24]Komatitsch D,Barnes C,Tromp J.Simulation of anisotropicwave propagation based upon a spectral element method.Geophysics,2000,65(4):1251-1260.
[25]Komatitsch D,Tromp J.Spectral-element simulations ofglobal seismic wave propagation-I.Validation.Geophys.J.Int.,2002,149(2):390-412.
[26]Komatitsch D,Tromp J.Spectral-element simulations ofglobal seismic wave propagation-II.Three-dimensional models,oceans,rotation,and self-gravitation.Geophys.J.Int.,2002,150(1):303-318.
[27]Komatitsch D,Tromp J.Introduction to the spectral-elementmethod for three-dimensional seismic wave propagation.Geophys.J.Int.,1999,139(3):806-822.
[28]Komatitsch D,Barnes C,Tromp J.Wave propagation near afluid-solid interface:a spectral-element approach.Geophysics,2000,65(2):623-631.
[29]Liu H P,Anderson D L,Kanamori H.Velocity dispersiondue to anelasticity:implications for seismology and mantlecomposition.Geophys.J.R.Astron.Soc.,1976,47(1):41-58.
[30]Priolo E,Carcione J M,Seriani G.Numerical simulation ofinterface waves by high-order spectral modeling techniques.J.Acoust.Soc.Am.,1994,95(2):681-693.
[31]冯康,秦孟兆.哈密尔顿系统的辛几何算法.杭州:浙江科技出版社,2003.Feng K,Qin M Z.Symplectic Geometric Algorithm forHamiltonian Systems(in Chinese).Hangzhou:ZhejiangScience and Technology Press,2003.
[32]钟万勰.分析结构力学与有限元.动力学与控制学报,2004,2(4):1-8.Zhong W X.Analytical structural mechanics and finiteelement.Journal of Dynamics and Control(in Chinese),2004,2(4):1-8.
[33]王雨顺,王斌,秦孟兆.偏微分方程的局部保结构算法.中国科学A辑:数学,2008,38(4):377-397.Wang Y S,Wang B,Qin M Z.Local structure-preservingalgorithms for partial differential equations.Science in China(Series A:Mathematics)(in Chinese),2008,38(4):377-397.
[34]Iwatsu R.Two new solutions to the third-order symplecticintegration method.Phys.Lett.A.,2009,373(34):3056-3060.
[35]Li X F,Li Y Q,Zhang M G,et al.Scalar seismic-waveequation modeling by a multisymplectic discrete singularconvolution differentiator method.Bulletin of the SeismologicalSociety of America,2011,101(4):1701-1718.
[36]Li X F,Wang W S,Lu M W,et al.Structure-preservingmodelling of elastic waves:a symplectic discrete singularconvolution differentiator method.Geophys.J.Int.,2012,188(3):1382-1392.
[37]Newmark N M.A method of computation for structuraldynamics.Journal of the Engineering Mechanics Division,ASCE,1959,85(3):67-94.
[38]Kane C,Marsden J E,Ortiz M,et al.Variational integratorsand the Newmark algorithm for conservative and dissipativemechanical systems.International Journal for NumericalMethods in Engineering,2000,49(10):1295-1325.
[39]Negrut D,Rampalli R,Ottarsson G,et al.On the use of theHHT method in the context of index 3differential algebraicequations of multibody dynamics.ASME Conf.Proc.,2005,6:207-218.
[40]Jay L O,Negrut D.Extensions of the HHT-αmethod todifferential-algebraic equations in mechanics.ElectronicTransactions on Numerical Analysis,2007,26:190-208.
[41]Cohen G,Fauqueux S.Mixed finite elements with mass-lumping for the transient wave equation.J.Comput.Acoust.,2000,8(1):171-188.
[42]Mercerat E D,Vilotte J P,Sánchez-Sesma F J.Triangularspectral element simulation of two-dimensional elastic wavepropagation using unstructured triangular grids.Geophys.J.Int.,2006,166(2):679-698.
[43]Giraldo F X,Warburton T.A nodal triangle-based spectralelement method for the shallow water equations on thesphere.J.Comput.Phys.,2005,207(1):129-150.
[44]Bittencourt M L.Fully tensorial nodal and modal shapefunctions for triangles and tetrahedra.Int.J.Numer.Meth.Engng.,2005,63(11):1530-1558.
[45]Taylor M A,Wingate B A,Vincent R E.An algorithm forcomputing Fekete points in the triangle.SIAM Journal onNumerical Analysis,2000,38(5):1707-1720.
[46]邢誉峰,冯伟.李级数算法和显式辛算法的相位分析.计算力学学报,2009,26(2):167-171.Xing Y F,Feng W.Phase analysis of Lie Series algorithmand explicit symplectic algorithm.Chinese Journal ofComputational Mechanics(in Chinese),2009,26(2):167-171.
[47]Hairer E,Lubich C,Wanner G.Geometric NumericalIntegration:Structure-Preserving Algorithms for OrdinaryDifferential Equations.Berlin:Springer-Verlag,2006.
[48]Van Der Houwen P J,Sommeijer B P.Runge-Kutta(-Nystrǒm)methods with reduced phase errors for computingoscillating solutions.SIAM Journal on Numerical Analysis,1987,24(3):595-617.
[49]Ruth R D.A canonical integration technique.IEEETransactions on Nuclear Science,1983,30(4):2669-2671.
[50]Suzuki M.General theory of higher-order decomposition ofexponential operators and symplectic integrators.PhysicsLetters A,1992,165(5-6):387-395.
[51]McLachlan R I,Atela P.The accuracy of symplecticintegrators.Nonlinearity,1992,5(2):541-562.
[52]Festa G,Vilotte J P.The Newmark scheme as velocity-stresstime-staggering:an efficient PML implementation forspectral element simulations of elastodynamics.Geophys.J.Int,.2005,161(3):789-812.
[53]Zampieri E,Pavarino L F.Approximation of acoustic wavesby explicit Newmark′s schemesand spectral element methods.Journal of Computational and Applied Mathematics,2006,185(2):308-325.
[54]Stacey R.Improved transparent boundary formulations for theelastic-wave equation.Bull.Seism.Soc.Am.,1988,78(6):2089-2097.
[55]De Hoop A.A modification of Cagniard′s method for solvingseismic pulse problems.Appl.Sci.Res.,1960,8(1):349-356.
[2]Boore D M.Finite-difference methods for seismic wavepropagation in heterogeneous materials.//Bolt B A ed.Methods in Computational Physics 11,New York:AcademicPress.Inc.,1972.
[3]Madariaga R.Dynamics of an expanding circular fault.Bull.Seismol.Soc.Am.,1976,66(3):639-669.
[4]Virieus J.P-SV wave propagation in heterogeneous media:velocity-stress finite-difference method.Geophysics,1986,51(4):889-901.
[5]Mora P.Modeling anisotropic seismic waves in 3-D.59thAnn.Internat.Mtg.Soc.Expl.Geophys.,ExpandedAbstracts,1989,2:1039-1043.
[6]Frankel A,Vidale J.A three-dimensional simulation ofseismic waves in the Santa Clara valley,California,from theLoma Prieta aftershock.Bull.Seismol.Soc.Am.,1992,82(5):2045-2074.
[7]Frankel A.Three-dimensional simulations of ground motionsin the San Bernardino valley,California,for hypotheticalearthquakes on the San Andreas fault.Bull.Seismol.Soc.Am.,1993,83(4):1020-1041.
[8]Pitarka A.3D elastic finite-difference modeling of seismicmotion using staggered grids with nonuniform spacing.Bull.Seism.Soc.Am.,1999,89(1):54-68.
[9]Olsen K B.Site amplification in the Los Angeles basin fromthree-dimensional modeling of ground motion.Bull.Seismol.Soc.Am.,2000,90(6B):S77-S94.
[10]Zingg D W.Comparison of high-accuracy finite-differencemethods for linear wave propagation.SIAM Journal onScientific Computing,2000,22(2):476-502.
[11]Chaljub E,Komatitsch D,Vilotte J P,et al.Spectral-element analysis in seismology.Advances in Geophysics,2007,48:365-419.
[12]Marfurt K J.Accuracy of finite-difference and finite-elementmodeling of the scalar and elastic wave equations.Geophysics,1984,49(5):533-549.
[13]Komatitsch D,Vilotte J P.The spectral element method:Anefficient tool to simulate the seismic response of 2Dand 3Dgeological structures.Bull Seism.Soc.Am.,1998,88(2):368-392.
[14]Tessmer E,Kosloff D.3-D elastic modeling with surfacetopography by a Chebyshev spectral method.Geophysics,1994,59(3):464-473.
[15]Carcione J M.The wave equation in generalized coordinates.Geophysics,1994,59(12):1911-1919.
[16]Furumura M,Kennett B L N,Furumura T.Seismicwavefield calculation for laterally heterogeneous Earthmodels-II:the influence of upper mantle heterogeneity.Geophys.J.Int.,1999,139(3):623-644.
[17]Furumura T,Kennett B L N,Furumura M.Seismicwavefield calculation for laterally heterogeneous whole Earthmodels using the pseudospectral method.Geophys.J.Int.,1998,135(3):845-860.
[18]Carcione J M,Wang P J.A Chebyshev collocation methodfor the wave equation in generalized coordinates.Comp.Fluid Dyn.J.,1993,2:269-290.
[19]Shen J,Tang T.Spectral and High-Order Methods withApplications.Beijing:Science Press,2006.
[20]Komatitsch D,Martin R,Tromp J,et al.Wave propagationin 2-D elastic media using a spectral element method withtriangles and quadrangles.J.Comput.Acoust.,2001,9(2):703-718.
[21]Chaljub E,Valette B.Spectral element modelling of three-dimensional wave propagation in a self-gravitating Earth withan arbitrarily stratified outer core.Geophys.J.Int.,2004,158(1):131-141.
[22]Chaljub E,Capdeville Y,Vilotte J P.Solving elastodynamicsin a fluid-solid heterogeneous sphere:a parallel spectralelement approximation on non-conforming grids.J.Comput.Phys.,2003,187(2):457-491.
[23]Komatitsch D,Ritsema J,Tromp J.The spectral-elementmethod,Beowulf computing,and global seismology.Science,2002,298(5599):1737-1742.
[24]Komatitsch D,Barnes C,Tromp J.Simulation of anisotropicwave propagation based upon a spectral element method.Geophysics,2000,65(4):1251-1260.
[25]Komatitsch D,Tromp J.Spectral-element simulations ofglobal seismic wave propagation-I.Validation.Geophys.J.Int.,2002,149(2):390-412.
[26]Komatitsch D,Tromp J.Spectral-element simulations ofglobal seismic wave propagation-II.Three-dimensional models,oceans,rotation,and self-gravitation.Geophys.J.Int.,2002,150(1):303-318.
[27]Komatitsch D,Tromp J.Introduction to the spectral-elementmethod for three-dimensional seismic wave propagation.Geophys.J.Int.,1999,139(3):806-822.
[28]Komatitsch D,Barnes C,Tromp J.Wave propagation near afluid-solid interface:a spectral-element approach.Geophysics,2000,65(2):623-631.
[29]Liu H P,Anderson D L,Kanamori H.Velocity dispersiondue to anelasticity:implications for seismology and mantlecomposition.Geophys.J.R.Astron.Soc.,1976,47(1):41-58.
[30]Priolo E,Carcione J M,Seriani G.Numerical simulation ofinterface waves by high-order spectral modeling techniques.J.Acoust.Soc.Am.,1994,95(2):681-693.
[31]冯康,秦孟兆.哈密尔顿系统的辛几何算法.杭州:浙江科技出版社,2003.Feng K,Qin M Z.Symplectic Geometric Algorithm forHamiltonian Systems(in Chinese).Hangzhou:ZhejiangScience and Technology Press,2003.
[32]钟万勰.分析结构力学与有限元.动力学与控制学报,2004,2(4):1-8.Zhong W X.Analytical structural mechanics and finiteelement.Journal of Dynamics and Control(in Chinese),2004,2(4):1-8.
[33]王雨顺,王斌,秦孟兆.偏微分方程的局部保结构算法.中国科学A辑:数学,2008,38(4):377-397.Wang Y S,Wang B,Qin M Z.Local structure-preservingalgorithms for partial differential equations.Science in China(Series A:Mathematics)(in Chinese),2008,38(4):377-397.
[34]Iwatsu R.Two new solutions to the third-order symplecticintegration method.Phys.Lett.A.,2009,373(34):3056-3060.
[35]Li X F,Li Y Q,Zhang M G,et al.Scalar seismic-waveequation modeling by a multisymplectic discrete singularconvolution differentiator method.Bulletin of the SeismologicalSociety of America,2011,101(4):1701-1718.
[36]Li X F,Wang W S,Lu M W,et al.Structure-preservingmodelling of elastic waves:a symplectic discrete singularconvolution differentiator method.Geophys.J.Int.,2012,188(3):1382-1392.
[37]Newmark N M.A method of computation for structuraldynamics.Journal of the Engineering Mechanics Division,ASCE,1959,85(3):67-94.
[38]Kane C,Marsden J E,Ortiz M,et al.Variational integratorsand the Newmark algorithm for conservative and dissipativemechanical systems.International Journal for NumericalMethods in Engineering,2000,49(10):1295-1325.
[39]Negrut D,Rampalli R,Ottarsson G,et al.On the use of theHHT method in the context of index 3differential algebraicequations of multibody dynamics.ASME Conf.Proc.,2005,6:207-218.
[40]Jay L O,Negrut D.Extensions of the HHT-αmethod todifferential-algebraic equations in mechanics.ElectronicTransactions on Numerical Analysis,2007,26:190-208.
[41]Cohen G,Fauqueux S.Mixed finite elements with mass-lumping for the transient wave equation.J.Comput.Acoust.,2000,8(1):171-188.
[42]Mercerat E D,Vilotte J P,Sánchez-Sesma F J.Triangularspectral element simulation of two-dimensional elastic wavepropagation using unstructured triangular grids.Geophys.J.Int.,2006,166(2):679-698.
[43]Giraldo F X,Warburton T.A nodal triangle-based spectralelement method for the shallow water equations on thesphere.J.Comput.Phys.,2005,207(1):129-150.
[44]Bittencourt M L.Fully tensorial nodal and modal shapefunctions for triangles and tetrahedra.Int.J.Numer.Meth.Engng.,2005,63(11):1530-1558.
[45]Taylor M A,Wingate B A,Vincent R E.An algorithm forcomputing Fekete points in the triangle.SIAM Journal onNumerical Analysis,2000,38(5):1707-1720.
[46]邢誉峰,冯伟.李级数算法和显式辛算法的相位分析.计算力学学报,2009,26(2):167-171.Xing Y F,Feng W.Phase analysis of Lie Series algorithmand explicit symplectic algorithm.Chinese Journal ofComputational Mechanics(in Chinese),2009,26(2):167-171.
[47]Hairer E,Lubich C,Wanner G.Geometric NumericalIntegration:Structure-Preserving Algorithms for OrdinaryDifferential Equations.Berlin:Springer-Verlag,2006.
[48]Van Der Houwen P J,Sommeijer B P.Runge-Kutta(-Nystrǒm)methods with reduced phase errors for computingoscillating solutions.SIAM Journal on Numerical Analysis,1987,24(3):595-617.
[49]Ruth R D.A canonical integration technique.IEEETransactions on Nuclear Science,1983,30(4):2669-2671.
[50]Suzuki M.General theory of higher-order decomposition ofexponential operators and symplectic integrators.PhysicsLetters A,1992,165(5-6):387-395.
[51]McLachlan R I,Atela P.The accuracy of symplecticintegrators.Nonlinearity,1992,5(2):541-562.
[52]Festa G,Vilotte J P.The Newmark scheme as velocity-stresstime-staggering:an efficient PML implementation forspectral element simulations of elastodynamics.Geophys.J.Int,.2005,161(3):789-812.
[53]Zampieri E,Pavarino L F.Approximation of acoustic wavesby explicit Newmark′s schemesand spectral element methods.Journal of Computational and Applied Mathematics,2006,185(2):308-325.
[54]Stacey R.Improved transparent boundary formulations for theelastic-wave equation.Bull.Seism.Soc.Am.,1988,78(6):2089-2097.
[55]De Hoop A.A modification of Cagniard′s method for solvingseismic pulse problems.Appl.Sci.Res.,1960,8(1):349-356.
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