多孔弹性波方程的多尺度波场模拟
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摘要
本文研究了二维多孔弹性波方程的多尺度波场数值模拟方法.该多尺度方法可采用较粗的网格计算,同时又能反映细尺度上物性参数的变化信息.文中详细阐述了多尺度模拟方法与算法,并推导了相应的计算格式.基本思想是建立粗细两套网格,在粗网格上,基于有限体积方法计算更新波场;在细网格上,计算多尺度基函数,这基于有限元方法通过求解一个局部化问题得到.对含有随机分布散射体的多孔介质模型进行了数值计算,计算中应用了完全匹配层(PML)吸收边界条件,数值结果验证了本文方法和算法的正确性和有效性.
In this paper,a multiscale method of wave simulation for solving two-dimensional poroelastic wave equations numerically is developed.The multiscale method allows us to apply coarser grids in computations while the information of physical parameter variations in small scale still can be captured.In this paper,the theoretical method and algorithm of the multiscale method are expounded in detail.The corresponding computational schemes are also derived.The basic idea is to construct two sets of meshes,i.e.,coarse grids and fine grids.The computations for wavefield updating on coarse grids are implemented based on the finite volume method while the multiscale basis functions are computed on fine grids with the finite element method by solving a local problem.Numerical computations with the perfectly matched layer(PML)absorbing boundary conditions are implemented for a poroelastic model with randomly distributed scatterers and the numerical results verify the correctness and effectiveness of the method and algorithm in this paper.
In this paper,a multiscale method of wave simulation for solving two-dimensional poroelastic wave equations numerically is developed.The multiscale method allows us to apply coarser grids in computations while the information of physical parameter variations in small scale still can be captured.In this paper,the theoretical method and algorithm of the multiscale method are expounded in detail.The corresponding computational schemes are also derived.The basic idea is to construct two sets of meshes,i.e.,coarse grids and fine grids.The computations for wavefield updating on coarse grids are implemented based on the finite volume method while the multiscale basis functions are computed on fine grids with the finite element method by solving a local problem.Numerical computations with the perfectly matched layer(PML)absorbing boundary conditions are implemented for a poroelastic model with randomly distributed scatterers and the numerical results verify the correctness and effectiveness of the method and algorithm in this paper.
引文
Abdulle A,Grote M J.2011.Finite element heterogeneous multiscale method for the wave equation.Multiscale Modeling&Simulation,9(2):766-792.
Allaire G.1992.Homogenization and two-scale convergence.SIAMJournal on Mathematical Analysis,23(6):1482-1518.
Atalla N,Panneton R,Debergue P.1996.A mixed displacementpressure formulation for Biot′s poroelastic equations.J.Acoust.Soc.Am.,99(4):2487-2500.
Bensoussan A,Lions J L,Papanicolaou G.1978.Asymptotic Analysis for Periodic Structures.Amsterdam:North-Holland Pub.Co.
Berenger J P.1994.A perfectly matched layer for the absorption of electromagnetic waves.J.Comput.Phys.,114(2):185-200.
Biot M A.1956a.Theory of propagation of elastic waves in a fluidsaturated porous solid.I.Low-frequency range.J.Acoust.Soc.Am.,28(2):168-178.
Biot M A.1956b.Theory of propagation of elastic waves in a fluidsaturated porous solid.II.Higher-frequency range.J.Acoust.Soc.Am.,28(2):179-191.
Biot M A.1962a.Generalized theory of acoustic propagation in porous dissipative media.J.Acoust.Socie.Am.,34(9A):1254-1264.
Biot M A.1962b.Mechanics of deformation and acoustic propagation in porous media.J.Acoust.Soc.Am.,33(4):1482-1498.
Chung E T,Engquist B.2009.Optimal discontinuous Galerkin methods for the acoustic wave equation in higher dimensions.SIAM Journal on Numerical Analysis,47(5):3820-3848.
Clayton R,Engquist B.1977.Absorbing boundary conditions for acoustic and elastic wave equations.Bull.Seism.Soc.Am.,67(6):1529-1540.
Collino F,Tsogka C.2001.Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media.Geophysics,66(1):294-307.
Dai N,Vafidis A,Kanasewich E R.1995.Wave propagation in heterogeneous,porous media:A velocity-stress,finite-difference method.Geophysics,60(2):327-340.
de la Puente J,Dumbser M,Kser M,et al.2008.Discontinuous Galerkin methods for wave propagation in poroelastic media.Geophysics,73(5):T77-T97.
Dupuy B,de Barros L,Garambois S,et al.2011.Wave propagation in heterogeneous porous media formulated in the frequencyspace domain using a discontinuous Galerkin method.Geophysics,76(4):N13-N28.
Efendiev Y,Hou T Y.2009.Multiscale Finite Element MethodsTheory and Applications.New York:Springer Science+Business Media,LLC.
Engquist B,Holst H,Runborg O.2011.Multi-scale methods for wave propagation in heterogeneous media.Communications in Mathematical Sciences,9(1):33-56.
Gibson Jr R L,Gao K,Chung E,et al.2014.Multiscale modeling of acoustic wave propagation in 2Dmedia.Geophysics,79(2):T61-T75.
Hou T Y,Wu X H.1997.A multiscale finite element method for elliptic problems in composite materials and porous media.Journal of Computational Physics,134(1):169-189.
Komatitsch D,Tromp J.2003.A perfectly matched layer absorbing boundary condition for the second-order seismic wave equation.Geophys.J.Int.,154(1):146-153.
Korostyshevskaya O,Minkoff S E.2006.A matrix analysis of operator-based upscaling for the wave equation.SIAM Journal on Numerical Analysis,44(2):586-612.
Morency C,Tromp J.2008.Spectral-element simulations of wave propagation in porous media.Geophys.J.Int.,175(1):301-345.
Nguetseng G.1989.A general convergence result for a functional related to the theory of homogenization.SIAM Journal on Mathematical Analysis,20(3):608-623.
O′Brien G S.2010.3Drotated and standard staggered finite-difference solutions to Biot′s poroelastic wave equations:stability condition and dispersion analysis.Geophysics,75(4):T111-T119.
Owhadi H,Zhang L.2008.Numerical homogenization of the acoustic wave equations with a continuum of scales.Computer Methods in Applied Mechanics and Engineering,198(3-4):397-406.
Ozdenvar T,McMechan G A.1997.Algorithms for staggered-grid computations for poroelastic,elastic,acoustic,and scalar wave equations.Geophysical Prospecting,45(3):403-420.
Panneton R,Atalla N.1997.An efficient finite element scheme for solving the three-dimensional poroelasticity problem in acoustics.J.Acoust.Soc.Am.,101(6):3287-3298.
Teng J W.2014.The Research and Development of Geophysics and Continental Dynamics in China.Beijing:Science Press.
Vdovina T,Minkoff S E,Korostyshevskaya O.2005.Operator upscaling for the acoustic wave equation.Multiscale Modeling&Simulation,4(4):1305-1338.
Vdovina T,Minkoff S.2008.An a priori error analysis of operator upscaling for the acoustic wave equation.International Journal of Numerical Analysis and Modeling,5(4):543-569.
Vdovina T,Minkoff S,Griffith S M L.2009.A two-scale solution algorithm for the elastic wave equation.SIAM Journal on Scientific Computing,31(5):3356-3368.
Wang X M,Zhang H L,Wang D.2003.Modelling seismic wave propagation in heterogeneous poroelastic media using a highorder staggered finite-difference method.Chinese Journal of Geophysics(in Chinese),46(6):1206-1217.
Yang D H,Wang M X,Ma X.2014.Symplectic stereomodelling method for solving elastic wave equations in porous media.Geophys.J.Int.,196(1):560-579.
Zeng Y Q,He J Q,Liu Q H.2001.The application of the perfectly matched layer in numerical modeling of wave propagation in poroelastic media.Geophysics,66(4):1258-1266.
Zeng Y Q,Liu Q H.2001.A staggered-grid finite-difference method with perfectly matched layers for poroelastic wave equations.J.Acoust.Soc.Am.,109(6):2571-2580.
Zhang W S,Tong L.2013.Numerical modeling of wave propagation in poroelastic media with a staggered-grid finite-difference method.AppliedMechanicsandMaterials,275-277:612-617.
Zhang W S,Tong L,Chung E T.2014.Exact nonreflecting boundary conditions for three dimensional poroelastic wave equations.Commun.Math.Sci.,12(1):61-98.
Zhang W S,Tong L,Chung E T.2014.A hybrid finite difference/control volume method for the three dimensional poroelastic wave equations in the spherical coordinate system.J.Comput.Appl.Math.,255:812-824.
Zhang W S,Zhuang Y,Chung E T.2016.A new spectral finite volume method for elastic wave modelling on unstructured meshes.Geophys.J.Int.,206(1):292-307.
滕吉文.2014.中国地球物理学和大陆动力学研究与发展.北京:科学出版社.
王秀明,张海澜,王东.2003.利用高阶交错网格有限差分法模拟地震波在非均匀孔隙介质中的传播.地球物理学报,46(6):842-849.
Allaire G.1992.Homogenization and two-scale convergence.SIAMJournal on Mathematical Analysis,23(6):1482-1518.
Atalla N,Panneton R,Debergue P.1996.A mixed displacementpressure formulation for Biot′s poroelastic equations.J.Acoust.Soc.Am.,99(4):2487-2500.
Bensoussan A,Lions J L,Papanicolaou G.1978.Asymptotic Analysis for Periodic Structures.Amsterdam:North-Holland Pub.Co.
Berenger J P.1994.A perfectly matched layer for the absorption of electromagnetic waves.J.Comput.Phys.,114(2):185-200.
Biot M A.1956a.Theory of propagation of elastic waves in a fluidsaturated porous solid.I.Low-frequency range.J.Acoust.Soc.Am.,28(2):168-178.
Biot M A.1956b.Theory of propagation of elastic waves in a fluidsaturated porous solid.II.Higher-frequency range.J.Acoust.Soc.Am.,28(2):179-191.
Biot M A.1962a.Generalized theory of acoustic propagation in porous dissipative media.J.Acoust.Socie.Am.,34(9A):1254-1264.
Biot M A.1962b.Mechanics of deformation and acoustic propagation in porous media.J.Acoust.Soc.Am.,33(4):1482-1498.
Chung E T,Engquist B.2009.Optimal discontinuous Galerkin methods for the acoustic wave equation in higher dimensions.SIAM Journal on Numerical Analysis,47(5):3820-3848.
Clayton R,Engquist B.1977.Absorbing boundary conditions for acoustic and elastic wave equations.Bull.Seism.Soc.Am.,67(6):1529-1540.
Collino F,Tsogka C.2001.Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media.Geophysics,66(1):294-307.
Dai N,Vafidis A,Kanasewich E R.1995.Wave propagation in heterogeneous,porous media:A velocity-stress,finite-difference method.Geophysics,60(2):327-340.
de la Puente J,Dumbser M,Kser M,et al.2008.Discontinuous Galerkin methods for wave propagation in poroelastic media.Geophysics,73(5):T77-T97.
Dupuy B,de Barros L,Garambois S,et al.2011.Wave propagation in heterogeneous porous media formulated in the frequencyspace domain using a discontinuous Galerkin method.Geophysics,76(4):N13-N28.
Efendiev Y,Hou T Y.2009.Multiscale Finite Element MethodsTheory and Applications.New York:Springer Science+Business Media,LLC.
Engquist B,Holst H,Runborg O.2011.Multi-scale methods for wave propagation in heterogeneous media.Communications in Mathematical Sciences,9(1):33-56.
Gibson Jr R L,Gao K,Chung E,et al.2014.Multiscale modeling of acoustic wave propagation in 2Dmedia.Geophysics,79(2):T61-T75.
Hou T Y,Wu X H.1997.A multiscale finite element method for elliptic problems in composite materials and porous media.Journal of Computational Physics,134(1):169-189.
Komatitsch D,Tromp J.2003.A perfectly matched layer absorbing boundary condition for the second-order seismic wave equation.Geophys.J.Int.,154(1):146-153.
Korostyshevskaya O,Minkoff S E.2006.A matrix analysis of operator-based upscaling for the wave equation.SIAM Journal on Numerical Analysis,44(2):586-612.
Morency C,Tromp J.2008.Spectral-element simulations of wave propagation in porous media.Geophys.J.Int.,175(1):301-345.
Nguetseng G.1989.A general convergence result for a functional related to the theory of homogenization.SIAM Journal on Mathematical Analysis,20(3):608-623.
O′Brien G S.2010.3Drotated and standard staggered finite-difference solutions to Biot′s poroelastic wave equations:stability condition and dispersion analysis.Geophysics,75(4):T111-T119.
Owhadi H,Zhang L.2008.Numerical homogenization of the acoustic wave equations with a continuum of scales.Computer Methods in Applied Mechanics and Engineering,198(3-4):397-406.
Ozdenvar T,McMechan G A.1997.Algorithms for staggered-grid computations for poroelastic,elastic,acoustic,and scalar wave equations.Geophysical Prospecting,45(3):403-420.
Panneton R,Atalla N.1997.An efficient finite element scheme for solving the three-dimensional poroelasticity problem in acoustics.J.Acoust.Soc.Am.,101(6):3287-3298.
Teng J W.2014.The Research and Development of Geophysics and Continental Dynamics in China.Beijing:Science Press.
Vdovina T,Minkoff S E,Korostyshevskaya O.2005.Operator upscaling for the acoustic wave equation.Multiscale Modeling&Simulation,4(4):1305-1338.
Vdovina T,Minkoff S.2008.An a priori error analysis of operator upscaling for the acoustic wave equation.International Journal of Numerical Analysis and Modeling,5(4):543-569.
Vdovina T,Minkoff S,Griffith S M L.2009.A two-scale solution algorithm for the elastic wave equation.SIAM Journal on Scientific Computing,31(5):3356-3368.
Wang X M,Zhang H L,Wang D.2003.Modelling seismic wave propagation in heterogeneous poroelastic media using a highorder staggered finite-difference method.Chinese Journal of Geophysics(in Chinese),46(6):1206-1217.
Yang D H,Wang M X,Ma X.2014.Symplectic stereomodelling method for solving elastic wave equations in porous media.Geophys.J.Int.,196(1):560-579.
Zeng Y Q,He J Q,Liu Q H.2001.The application of the perfectly matched layer in numerical modeling of wave propagation in poroelastic media.Geophysics,66(4):1258-1266.
Zeng Y Q,Liu Q H.2001.A staggered-grid finite-difference method with perfectly matched layers for poroelastic wave equations.J.Acoust.Soc.Am.,109(6):2571-2580.
Zhang W S,Tong L.2013.Numerical modeling of wave propagation in poroelastic media with a staggered-grid finite-difference method.AppliedMechanicsandMaterials,275-277:612-617.
Zhang W S,Tong L,Chung E T.2014.Exact nonreflecting boundary conditions for three dimensional poroelastic wave equations.Commun.Math.Sci.,12(1):61-98.
Zhang W S,Tong L,Chung E T.2014.A hybrid finite difference/control volume method for the three dimensional poroelastic wave equations in the spherical coordinate system.J.Comput.Appl.Math.,255:812-824.
Zhang W S,Zhuang Y,Chung E T.2016.A new spectral finite volume method for elastic wave modelling on unstructured meshes.Geophys.J.Int.,206(1):292-307.
滕吉文.2014.中国地球物理学和大陆动力学研究与发展.北京:科学出版社.
王秀明,张海澜,王东.2003.利用高阶交错网格有限差分法模拟地震波在非均匀孔隙介质中的传播.地球物理学报,46(6):842-849.