弱形式时域完美匹配层——滞弹性近场波动数值模拟
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摘要
本文旨在构建适用于滞弹性近场时域波动有限元模拟的高精度人工边界条件:完美匹配层(Perfectly Matched Layer:PML),其中阻尼介质时域本构基于广义标准线性体建立.与以往研究不同,本文采用复坐标延拓技术变换弱形式波动方程构建了可直接用有限元离散的弱形式时域PML,规避以往独立对无限域内波动方程及界面条件进行延拓可导致的PML场方程和界面条件匹配不合理引发数值失稳、计算精度低下等问题.其次,针对PML中多极点有理分式与频域函数乘积的傅里叶反变换难以计算的问题,利用PML精度对复坐标延拓函数中延拓参数微调不敏感这一特点,明确给出了参数微调准则以规避多重极点,进而利用有理分式分解给出了一种普适、简便的计算方法,极大地简化了PML计算.基于该方法可实现任意高阶PML.最后,将本文构建滞弹性PML与高阶勒让德谱元(高精度集中质量有限元)结合得到滞弹性近场波动谱元离散方案.基于算例验证了滞弹性PML的计算效率、精度及新离散方案的长持时稳定特性.新离散方案可应用于计入实际介质阻尼的地震波动正、反问题数值模拟,提高波形模拟的精度以及地下波速结构反演的精度和可靠性.
Taking into account of the viscoelastic effect of a damping medium in the infinite domain,a novel viscoelastic PML(perfectly matched layer)for truncation of infinite-domain has been constructed,inside which the constitutive relation of the damping medium is established based upon the generalized standard linear body.Being different from previous studies,the PML in a weak-form is derived by complex stretching the weak-form viscoelastic wave equation in the infinite domain.Compared with the strong-form PML derived in a traditional way,inside which the field equation of PML together with the boundary and/or interface condition of PML are independently obtained by complex stretching their counterparts in infinite domain,the mismatch between the obtained field equation and boundary and/or interface condition can be naturally avoided.Secondly,a clean and simple approach for time-domain PML implementation has been proposed to avoid the difficult computation for inverse Fourier transform of the complex multipole rational fraction kernels.To accomplish this,we utilize the feature that the accuracy of PML is insensitive to slightly tuning of the parameters in the complex stretching function to decompose such kernels into the sum of single-pole fraction kernels,of which the corresponding convolution integral can be localized and computed in an iterative way.The new approach facilitates the implementation of arbitrary high-order PML.Finally,utilizing the novel viscoelastic PML for infinite domain truncation,a new scheme for infinite-domain viscoelastic wave simulation has been derived using high-order spectral element methods for space discretization.The computational efficiency and accuracy of the novel viscoelastic PML and the long-term stability of the new scheme have been verified by numerical tests.The new scheme can also be used in full waveform inversion taking into account of the damping effect of the real medium to improve the accuracy and robustness of inversion for the underground wave velocity structure.
Taking into account of the viscoelastic effect of a damping medium in the infinite domain,a novel viscoelastic PML(perfectly matched layer)for truncation of infinite-domain has been constructed,inside which the constitutive relation of the damping medium is established based upon the generalized standard linear body.Being different from previous studies,the PML in a weak-form is derived by complex stretching the weak-form viscoelastic wave equation in the infinite domain.Compared with the strong-form PML derived in a traditional way,inside which the field equation of PML together with the boundary and/or interface condition of PML are independently obtained by complex stretching their counterparts in infinite domain,the mismatch between the obtained field equation and boundary and/or interface condition can be naturally avoided.Secondly,a clean and simple approach for time-domain PML implementation has been proposed to avoid the difficult computation for inverse Fourier transform of the complex multipole rational fraction kernels.To accomplish this,we utilize the feature that the accuracy of PML is insensitive to slightly tuning of the parameters in the complex stretching function to decompose such kernels into the sum of single-pole fraction kernels,of which the corresponding convolution integral can be localized and computed in an iterative way.The new approach facilitates the implementation of arbitrary high-order PML.Finally,utilizing the novel viscoelastic PML for infinite domain truncation,a new scheme for infinite-domain viscoelastic wave simulation has been derived using high-order spectral element methods for space discretization.The computational efficiency and accuracy of the novel viscoelastic PML and the long-term stability of the new scheme have been verified by numerical tests.The new scheme can also be used in full waveform inversion taking into account of the damping effect of the real medium to improve the accuracy and robustness of inversion for the underground wave velocity structure.
引文
Aki K,Richards P G.2002.Quantitative Seismology.2nd ed.California:University Science Books.
Askan A,Akcelik V,Bielak J,et al.2007.Full waveform inversion for seismic velocity and anelastic losses in heterogeneous structures.Bulletin of the Seismological Society of America,97(6):1990-2008.
Assimaki D,Kallivokas L F,Kang J W,et al.2012.Time-domain forward and inverse modeling of lossy soils with frequencyindependent Qfor near-surface applications.Soil Dynamics and Earthquake Engineering,43:139-159.
Basu U,Chopra A K.2004.Perfectly matched layers for transient elastodynamics of unbounded domains.International Journal for Numerical Methods in Engineering,59(8):1039-1074.
Bielak J,Karaoglu H,Taborda R.2011.Memory-efficient displacementbased internal friction for wave propagation simulation.Geophysics,76(6):T131-T145.
Blanc E,Komatitsch D,Chaljub E,et al.2016.Highly accurate stability-preserving optimization of the Zener viscoelastic model,with application to wave propagation in the presence of strong attenuation.Geophysical Journal International,205(1):427-439.
Carcione J M,Gei D,Treitel S.2010.The velocity of energy through a dissipative medium.Geophysics,75(2):T37-T47.
Carcione J M,Kosloff D,Kosloff R.1988.Wave propagation simulation in a linear viscoelastic medium.Geophysical Journal International,95(3):597-611.
Chew W C,Jin J M,Michielssen E.1997.Complex coordinate stretching as a generalized absorbing boundary condition.Microwave and Optical Technology Letters,15(6):363-369.
Collino F,Monk P.1998.The perfectly matched layer in curvilinear coordinates.SIAM Journal on Scientific Computing,19(6):2061-2090.
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.
Connolly D P,Giannopoulos A,Forde M C.2014.A higher order perfectly matched layer formulation for finite-difference timedomain seismic wave modeling.Geophysics,80(1):T1-T16.
Correia D,Jin J M.2005.On the development of a higher-order PML.IEEE Transactions on Antennas and Propagation,53(12):4157-4163.
Duru K,Kreiss G.2014.Numerical interaction of boundary waves with perfectly matched layers in two space dimensional elastic waveguides.Wave Motion,51(3):445-465.
Ely G P,Day S M,Minster J B.2008.A support-operator method for viscoelastic wave modelling in 3-D heterogeneous media.Geophysical Journal International,172(1):331-344.
Emmerich H,Korn M.1987.Incorporation of attenuation into time-domain computations of seismic wave fields.Geophysics,52(9):1252-1264.
Feng H K,Zhang W,Zhang J,et al.2017.Importance of doublepole CFS-PML for broad-band seismic wave simulation and optimal parameters selection.Geophysical Journal International,209(2):1148-1167.
Festa G,Vilotte J P.2005.The Newmark scheme as velocitystress time-staggering:An efficient PML implementation for spectral element simulations of elastodynamics.Geophysical Journal International,161(3):789-812.
Fichtner A.2011.Full Seismic Waveform Modelling and Inversion.Berlin,Heidelberg:Springer Science&Business Media.
Givoli D.1991.Non-reflecting boundary conditions.Journal of Computational Physics,94(1):1-29.
Givoli D.2004.High-order local non-reflecting boundary conditions:Areview.Wave Motion,39(4):319-326.
Groby J P,Tsogka C.2006.A time domain method for modeling viscoacoustic wave propagation.Journal of Computational Acoustics,14(2):201-236.
Groos L,Schfer M,Forbriger T,et al.2014.The role of attenuation in 2D full-waveform inversion of shallow-seismic body and Rayleigh waves.Geophysics,79(6):R247-R261.
Komatitsch D,Tromp J.1999.Introduction to the spectral element method for three-dimensional seismic wave propagation.Geophysical Journal International,139(3):806-822.
Komatitsch D,Xie Z N,Bozdag E,et al.2016.Anelastic sensitivity kernels with parsimonious storage for adjoint tomography and full waveform inversion.Geophysical Journal International,206(3):1467-1478.
Kreiss H O,Petersson N A,Ystr9m J.2002.Difference approximations for the second order wave equation.SIAM Journal on Numerical Analysis,40(5):1940-1967.
Kristek J,Moczo P.2003.Seismic-wave propagation in viscoelastic media with material discontinuities:A 3Dfourth-order staggeredgrid finite-difference modeling.Bulletin of the Seismological Society of America,93(5):2273-2280.
Kurzmann A,Przebindowska A,K9hn D,et al.2013.Acoustic full waveform tomography in the presence of attenuation:Asensitivity analysis.Geophysical Journal International,195(2):985-1000.
Liao Z P.2002.Introduction to Wave Motion Theories in Engineering(in Chinese).2nd ed.Beijing:Science Press.
Liu Q H.1997.An FDTD algorithm with perfectly matched layers for conductive media.Microwave and Optical Technology Letters,14(2):134-137.
Liu Q H,Tao J P.1997.The perfectly matched layer for acoustic waves in absorptive media.The Journal of the Acoustical Society of America,102(4):2072-2082.
Luebbers R J,Hunsberger F.1992.FDTD for Nth-order dispersive media.IEEE Transactions on Antennas and Propagation,40(11):1297-1301.
Ma S,Liu P C.2006.Modeling of the perfectly matched layer absorbing boundaries and intrinsic attenuation in explicit finiteelement methods.Bulletin of the Seismological Society of America,96(5):1779-1794.
Martin R,Komatitsch D.2009.An unsplit convolutional perfectly matched layer technique improved at grazing incidence for the viscoelastic wave equation.Geophysical Journal International,179(1):333-344.
Meng W J,Fu L Y.2017.Seismic wavefield simulation by a modified finite element method with a perfectly matched layer absorbing boundary.Journal of Geophysics and Engineering,14(4):852-864.
Meza-Fajardo K C,Papageorgiou A S.2008.A nonconvolutional,split-field,perfectly matched layer for wave propagation in isotropic and anisotropic elastic media:Stability analysis.Bulletin of the Seismological Society of America,98(4):1811-1836.
Moczo P,Kristek J,Gális M.2014.The Finite-Difference Modelling of Earthquake Motions:Waves and Ruptures.Cambridge:Cambridge University Press.
O'Connell R J,Budiansky B.1978.Measures of dissipation in viscoelastic media.Geophysical Research Letters,5(1):5-8.
Ping P,Zhang Y,Xu Y X,et al.2016.Efficiency of perfectly matched layers for seismic wave modeling in second-order viscoelastic equations.Geophysical Journal International,207(3):1367-1386.
Ramadan O.2003.Auxiliary differential equation formulation:An efficient implementation of the perfectly matched layer.IEEEMicrowave and Wireless Components Letters,13(2):69-71.
Savage B,Komatitsch D,Tromp J.2010.Effects of 3Dattenuation on seismic wave amplitude and phase measurements.Bulletin of the Seismological Society of America,100(3):1241-1251.
Stekl I,Pratt R G.1998.Accurate viscoelastic modeling by frequencydomain finite differences using rotated operators.Geophysics,63(5):1779-1794.
Teixeira F L,Chew W C.2000.Complex space approach to perfectly matched layers:A review and some new developments.International Journal of Numerical Modelling:Electronic Networks,Devices and Fields,13(5):441-455.
Wang Z Y,Liu J B.2004.Study on wave motion input and artificial boundary for problem of nonlinear wave motion in layered soil.Chinese Journal of Rock Mechanics and Engineering(in Chinese),23(7):1169-1173.
Waters K R,Hughes M S,Brandenburger G H,et al.2000.On a time-domain representation of the Kramers-Kr9nig dispersion relations.The Journal of the Acoustical Society of America,108(5):2114-2119.
Xie Z N,Matzen R,Cristini P,et al.2016.A perfectly matched layer for fluid-solid problems:Application to ocean-acoustics simulations with solid ocean bottoms.The Journal of the Acoustical Society of America,140(1):165-175.
Xie Z N,Zhang X B.2017.Weak-form time-domain perfectly matched layer.Chinese Journal of Geophysics(in Chinese),60(10):3823-3831,doi:10.6038/cjg20171012.
Xie Z N,Zheng Y L,Zhang X B.2018.Optimized approximation for constitution of constant Qviscoelastic media in time domain seismic wave simulation.Chinese Journal of Geophysics(in Chinese),61(10):4007-4020,doi:10.6038/cjg2018L0704.
Yuan S C,Song X H,Zhang X Q,et al.2018.Finite-difference modeling and characteristics analysis of Rayleigh waves in viscoelastic media.Chinese Journal of Geophysics(in Chinese),61(4):1496-1507,doi:10.6038/cjg2018L0532.
Zhang W,Shen Y.2010.Unsplit complex frequency-shifted PMLimplementation using auxiliary differential equations for seismic wave modeling.Geophysics,75(4):T141-T154.
廖振鹏.2002.工程波动理论导论.2版.北京:科学出版社.
王振宇,刘晶波.2004.成层地基非线性波动问题人工边界与波动输入研究.岩石力学与工程学报,23(7):1169-1173.
谢志南,章旭斌.2017.弱形式时域完美匹配层.地球物理学报,60(10):3823-3831,doi:10.6038/cjg20171012.
谢志南,郑永路,章旭斌.2018.常Q滞弹性介质地震波动数值模拟-时域本构优化逼近.地球物理学报,61(10):4007-4020,doi:10.6038/cjg2018L0704.
袁士川,宋先海,张学强等.2018.黏弹性介质瑞雷波有限差分模拟与特性分析.地球物理学报,61(4):1496-1507,doi:10.6038/cjg2018L0532.
Askan A,Akcelik V,Bielak J,et al.2007.Full waveform inversion for seismic velocity and anelastic losses in heterogeneous structures.Bulletin of the Seismological Society of America,97(6):1990-2008.
Assimaki D,Kallivokas L F,Kang J W,et al.2012.Time-domain forward and inverse modeling of lossy soils with frequencyindependent Qfor near-surface applications.Soil Dynamics and Earthquake Engineering,43:139-159.
Basu U,Chopra A K.2004.Perfectly matched layers for transient elastodynamics of unbounded domains.International Journal for Numerical Methods in Engineering,59(8):1039-1074.
Bielak J,Karaoglu H,Taborda R.2011.Memory-efficient displacementbased internal friction for wave propagation simulation.Geophysics,76(6):T131-T145.
Blanc E,Komatitsch D,Chaljub E,et al.2016.Highly accurate stability-preserving optimization of the Zener viscoelastic model,with application to wave propagation in the presence of strong attenuation.Geophysical Journal International,205(1):427-439.
Carcione J M,Gei D,Treitel S.2010.The velocity of energy through a dissipative medium.Geophysics,75(2):T37-T47.
Carcione J M,Kosloff D,Kosloff R.1988.Wave propagation simulation in a linear viscoelastic medium.Geophysical Journal International,95(3):597-611.
Chew W C,Jin J M,Michielssen E.1997.Complex coordinate stretching as a generalized absorbing boundary condition.Microwave and Optical Technology Letters,15(6):363-369.
Collino F,Monk P.1998.The perfectly matched layer in curvilinear coordinates.SIAM Journal on Scientific Computing,19(6):2061-2090.
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.
Connolly D P,Giannopoulos A,Forde M C.2014.A higher order perfectly matched layer formulation for finite-difference timedomain seismic wave modeling.Geophysics,80(1):T1-T16.
Correia D,Jin J M.2005.On the development of a higher-order PML.IEEE Transactions on Antennas and Propagation,53(12):4157-4163.
Duru K,Kreiss G.2014.Numerical interaction of boundary waves with perfectly matched layers in two space dimensional elastic waveguides.Wave Motion,51(3):445-465.
Ely G P,Day S M,Minster J B.2008.A support-operator method for viscoelastic wave modelling in 3-D heterogeneous media.Geophysical Journal International,172(1):331-344.
Emmerich H,Korn M.1987.Incorporation of attenuation into time-domain computations of seismic wave fields.Geophysics,52(9):1252-1264.
Feng H K,Zhang W,Zhang J,et al.2017.Importance of doublepole CFS-PML for broad-band seismic wave simulation and optimal parameters selection.Geophysical Journal International,209(2):1148-1167.
Festa G,Vilotte J P.2005.The Newmark scheme as velocitystress time-staggering:An efficient PML implementation for spectral element simulations of elastodynamics.Geophysical Journal International,161(3):789-812.
Fichtner A.2011.Full Seismic Waveform Modelling and Inversion.Berlin,Heidelberg:Springer Science&Business Media.
Givoli D.1991.Non-reflecting boundary conditions.Journal of Computational Physics,94(1):1-29.
Givoli D.2004.High-order local non-reflecting boundary conditions:Areview.Wave Motion,39(4):319-326.
Groby J P,Tsogka C.2006.A time domain method for modeling viscoacoustic wave propagation.Journal of Computational Acoustics,14(2):201-236.
Groos L,Schfer M,Forbriger T,et al.2014.The role of attenuation in 2D full-waveform inversion of shallow-seismic body and Rayleigh waves.Geophysics,79(6):R247-R261.
Komatitsch D,Tromp J.1999.Introduction to the spectral element method for three-dimensional seismic wave propagation.Geophysical Journal International,139(3):806-822.
Komatitsch D,Xie Z N,Bozdag E,et al.2016.Anelastic sensitivity kernels with parsimonious storage for adjoint tomography and full waveform inversion.Geophysical Journal International,206(3):1467-1478.
Kreiss H O,Petersson N A,Ystr9m J.2002.Difference approximations for the second order wave equation.SIAM Journal on Numerical Analysis,40(5):1940-1967.
Kristek J,Moczo P.2003.Seismic-wave propagation in viscoelastic media with material discontinuities:A 3Dfourth-order staggeredgrid finite-difference modeling.Bulletin of the Seismological Society of America,93(5):2273-2280.
Kurzmann A,Przebindowska A,K9hn D,et al.2013.Acoustic full waveform tomography in the presence of attenuation:Asensitivity analysis.Geophysical Journal International,195(2):985-1000.
Liao Z P.2002.Introduction to Wave Motion Theories in Engineering(in Chinese).2nd ed.Beijing:Science Press.
Liu Q H.1997.An FDTD algorithm with perfectly matched layers for conductive media.Microwave and Optical Technology Letters,14(2):134-137.
Liu Q H,Tao J P.1997.The perfectly matched layer for acoustic waves in absorptive media.The Journal of the Acoustical Society of America,102(4):2072-2082.
Luebbers R J,Hunsberger F.1992.FDTD for Nth-order dispersive media.IEEE Transactions on Antennas and Propagation,40(11):1297-1301.
Ma S,Liu P C.2006.Modeling of the perfectly matched layer absorbing boundaries and intrinsic attenuation in explicit finiteelement methods.Bulletin of the Seismological Society of America,96(5):1779-1794.
Martin R,Komatitsch D.2009.An unsplit convolutional perfectly matched layer technique improved at grazing incidence for the viscoelastic wave equation.Geophysical Journal International,179(1):333-344.
Meng W J,Fu L Y.2017.Seismic wavefield simulation by a modified finite element method with a perfectly matched layer absorbing boundary.Journal of Geophysics and Engineering,14(4):852-864.
Meza-Fajardo K C,Papageorgiou A S.2008.A nonconvolutional,split-field,perfectly matched layer for wave propagation in isotropic and anisotropic elastic media:Stability analysis.Bulletin of the Seismological Society of America,98(4):1811-1836.
Moczo P,Kristek J,Gális M.2014.The Finite-Difference Modelling of Earthquake Motions:Waves and Ruptures.Cambridge:Cambridge University Press.
O'Connell R J,Budiansky B.1978.Measures of dissipation in viscoelastic media.Geophysical Research Letters,5(1):5-8.
Ping P,Zhang Y,Xu Y X,et al.2016.Efficiency of perfectly matched layers for seismic wave modeling in second-order viscoelastic equations.Geophysical Journal International,207(3):1367-1386.
Ramadan O.2003.Auxiliary differential equation formulation:An efficient implementation of the perfectly matched layer.IEEEMicrowave and Wireless Components Letters,13(2):69-71.
Savage B,Komatitsch D,Tromp J.2010.Effects of 3Dattenuation on seismic wave amplitude and phase measurements.Bulletin of the Seismological Society of America,100(3):1241-1251.
Stekl I,Pratt R G.1998.Accurate viscoelastic modeling by frequencydomain finite differences using rotated operators.Geophysics,63(5):1779-1794.
Teixeira F L,Chew W C.2000.Complex space approach to perfectly matched layers:A review and some new developments.International Journal of Numerical Modelling:Electronic Networks,Devices and Fields,13(5):441-455.
Wang Z Y,Liu J B.2004.Study on wave motion input and artificial boundary for problem of nonlinear wave motion in layered soil.Chinese Journal of Rock Mechanics and Engineering(in Chinese),23(7):1169-1173.
Waters K R,Hughes M S,Brandenburger G H,et al.2000.On a time-domain representation of the Kramers-Kr9nig dispersion relations.The Journal of the Acoustical Society of America,108(5):2114-2119.
Xie Z N,Matzen R,Cristini P,et al.2016.A perfectly matched layer for fluid-solid problems:Application to ocean-acoustics simulations with solid ocean bottoms.The Journal of the Acoustical Society of America,140(1):165-175.
Xie Z N,Zhang X B.2017.Weak-form time-domain perfectly matched layer.Chinese Journal of Geophysics(in Chinese),60(10):3823-3831,doi:10.6038/cjg20171012.
Xie Z N,Zheng Y L,Zhang X B.2018.Optimized approximation for constitution of constant Qviscoelastic media in time domain seismic wave simulation.Chinese Journal of Geophysics(in Chinese),61(10):4007-4020,doi:10.6038/cjg2018L0704.
Yuan S C,Song X H,Zhang X Q,et al.2018.Finite-difference modeling and characteristics analysis of Rayleigh waves in viscoelastic media.Chinese Journal of Geophysics(in Chinese),61(4):1496-1507,doi:10.6038/cjg2018L0532.
Zhang W,Shen Y.2010.Unsplit complex frequency-shifted PMLimplementation using auxiliary differential equations for seismic wave modeling.Geophysics,75(4):T141-T154.
廖振鹏.2002.工程波动理论导论.2版.北京:科学出版社.
王振宇,刘晶波.2004.成层地基非线性波动问题人工边界与波动输入研究.岩石力学与工程学报,23(7):1169-1173.
谢志南,章旭斌.2017.弱形式时域完美匹配层.地球物理学报,60(10):3823-3831,doi:10.6038/cjg20171012.
谢志南,郑永路,章旭斌.2018.常Q滞弹性介质地震波动数值模拟-时域本构优化逼近.地球物理学报,61(10):4007-4020,doi:10.6038/cjg2018L0704.
袁士川,宋先海,张学强等.2018.黏弹性介质瑞雷波有限差分模拟与特性分析.地球物理学报,61(4):1496-1507,doi:10.6038/cjg2018L0532.