时域有限元弹性波模拟中的位移格式完全匹配层吸收边界(英文)
|
摘要
完全匹配层是地震波方程数值模拟中一种高效的吸收边界条件,本文目的在于将这种技术用于时域有限元弹性波数值模拟中。但时域有限元法是基于二阶位移格式的波动方程的数值模拟方法,所以一阶速度-应力格式的完全匹配层不能直接用于该数值模拟方法中。本文推导了二阶位移格式完全匹配层的有限元矩阵方程,实现了完全匹配层在时域有限元弹性波模拟中的应用。在二维均匀弹性介质P-SV波和SH波传播的有限元模拟中,完全匹配层对体波和面波具有近似零反射系数;不规则地表双层介质模型的数值实验验证了完全匹配层在复杂构造非均质地质模型中地震波传播模拟的效果。
The perfectly matched layer(PML) is a highly efficient absorbing boundary condition used for the numerical modeling of seismic wave equation. The article focuses on the application of this technique to finite-element time-domain numerical modeling of elastic wave equation. However, the finite-element time-domain scheme is based on the secondorder wave equation in displacement formulation. Thus, the first-order PML in velocity-stress formulation cannot be directly applied to this scheme. In this article, we derive the finiteelement matrix equations of second-order PML in displacement formulation, and accomplish the implementation of PML in finite-element time-domain modeling of elastic wave equation. The PML has an approximate zero reflection coefficients for bulk and surface waves in the finite-element modeling of P-SV and SH wave propagation in the 2D homogeneous elastic media. The numerical experiments using a two-layer model with irregular topography validate the efficiency of PML in the modeling of seismic wave propagation in geological models with complex structures and heterogeneous media.
The perfectly matched layer(PML) is a highly efficient absorbing boundary condition used for the numerical modeling of seismic wave equation. The article focuses on the application of this technique to finite-element time-domain numerical modeling of elastic wave equation. However, the finite-element time-domain scheme is based on the secondorder wave equation in displacement formulation. Thus, the first-order PML in velocity-stress formulation cannot be directly applied to this scheme. In this article, we derive the finiteelement matrix equations of second-order PML in displacement formulation, and accomplish the implementation of PML in finite-element time-domain modeling of elastic wave equation. The PML has an approximate zero reflection coefficients for bulk and surface waves in the finite-element modeling of P-SV and SH wave propagation in the 2D homogeneous elastic media. The numerical experiments using a two-layer model with irregular topography validate the efficiency of PML in the modeling of seismic wave propagation in geological models with complex structures and heterogeneous media.
引文
Bao,H.,Bielak,J.,Ghattas,O.,Kallivokas,L.F.,O’Hallaron,D.R.,Shewchuk,J.R.,and Xu,J.,1998,Large-scale simulation of elastic wave propagation in heterogeneous media on parallel computers:Computer Methods in Applied Mechanics and Engineering,152,85–102.
Bécache,E.,Fauqueux,S.,and Joly,P.,2003,Stability of perfectly matched layers,group velocities and anisotropic waves:Journal of Computational Physics,188(2),399–433.
Bérenger,J.P.,1994,A perfectly matched layer for the absorption of electromagnetic waves:Journal of Computational Physics,114(2),185–200.
Cerjan,C.,Kosloff,D.,Kosloff,R.,and Reshef,M.,1985,A nonreflecting boundary condition for discrete acoustic and elastic wave equation:Geophysics,50(4),705–708.
Chew,W.C.,and Liu,Q.,1996,Perfectly matched layers for elastodynamics:A new absorbing boundary condition:Journal of Computational Acoustics,4(4),341–359.
Chew,W.C.,and Weedon,W.H.,1994,A 3-D perfectly matched medium from modified Maxwell’s equations with stretched coordinates:Microwave and Optical Technology Letters,7(13),599–604.
Clayton,R.,and Engquist,B.,1977,Absorbing boundary conditions for acoustic and elastic wave equations:Bulletin of the Seismological Society of America,67(6),1529–1540.
Cohen,G.C.,2002,Higher-order Numerical Methods for Transient Wave Equations:Springer:Berlin.
Collino,F.,and Tsogka,C.,2001,Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media:Geophysics,66(1),294–307.
Courant,R.,Friedrichs,K.O.,and Lewy,H.,1928,über die partiellen Differenzengleichungen dermathematischen Physik:Mathematische Annalen,100,32 –74.
Drake,L.A.,1972,Rayleigh wave at continental boundary by the finite element method:Bulletin of the Seismological Society of America,62(5),1259-1268.
Engquist,B.,and Majda,A.,1977,Absorbing boundary conditions for the numerical simulation of waves:Proceedings of the National Academy of Science,74(5),1765–1766.
Hastings,F.D.,Schneider,J.B.,and Broschat,S.L.,1996,Application of the perfectly matched layer PML absorbing boundary condition to elastic wave propagation:Journal of the Acoustical Society of America,100(5),3061–3069.
Higdon,R.L.,1991,Absorbing boundary conditions for elastic waves:Geophysics,56(2),231–241.
Katsibas,T.K.,and Antonopoulos,C.S.,2002,An efficient PML absorbing medium in FDTD simulations of acoustic scattering in lossy media:IEEE Proceeding on Ultrasonic Symposium,1,551–554.
Komatitsch,D.,and Tromp,J.,2003,A perfectly matched layer absorbing boundary condition for the secondorder seismic wave equation:Geophysical Journal International,154(1),146–153.
Liu,Q.,and Tao,J.,1997,The perfectly matched layer for acoustic waves in absorptive media:Journal of the Acoustical Society of America,102(4),2072–2082.
Liu,Q.H.,1999,Perfectly matched layers for elastic waves in cylindrical and spherical coordinates:Journal of the Acoustical Society of America,105(4),2075–2084.
Madariaga,R.,2007,Seismic source theory.Treatise on Geophysics,Amsterdam:Elsevier,59–82.
Marfurt,K.J.,1984,Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equation:Geophysics,49(5),533–549.
Meza-Fajardo,K.C.,and 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.
Newmark,N.M.,1959,A method of computation for structural dynamics:ASCE Journal of the Engineering Mechanics Division,85(3),67–94.
Quarteroni,A.,Tagliani,A.,and Zampieri,E.,1998,Generalized Galerkin approximations of elastic waves with absorbing boundary conditions:Computer Methods in Applied Mechanics and Engineering,163,323–341.
Shi,R.,Wang,S.,and Zhao,J.,2012,An unsplit complexfrequency-shifted PML based on matched Z-transform for FDTD modeling of seismic wave equations:Journal of Geophysics and Engineering,9,218–229.
Sochacki,J.,Kubichek,R.,George,J.,Fletcher,W.R.,and Smithson,S.,1987,Absorbing boundary conditions and surface waves:Geophysics,52(1),60–71.
Stacey,R.,1988,Improved transparent boundary formulations for the elastic wave equation:Bulletin of the Seismological Society of America,78(6),2089–2097.
Zeng,Y.Q.,He,J.Q.,and Liu,Q.H.,2001,The applicationof the perfectly matched layer in numerical modelingof wave propagation in poroelastic media:Geophysics,66 (4),1258–1266.
Zienkiewicz,O.C.,Taylor,R.L.,and Zhu,J.,2005,The Finite Element Method:its Basis and Fundamentals(6th edition):Elsevier Butterworth-Heinemann,London.
Bécache,E.,Fauqueux,S.,and Joly,P.,2003,Stability of perfectly matched layers,group velocities and anisotropic waves:Journal of Computational Physics,188(2),399–433.
Bérenger,J.P.,1994,A perfectly matched layer for the absorption of electromagnetic waves:Journal of Computational Physics,114(2),185–200.
Cerjan,C.,Kosloff,D.,Kosloff,R.,and Reshef,M.,1985,A nonreflecting boundary condition for discrete acoustic and elastic wave equation:Geophysics,50(4),705–708.
Chew,W.C.,and Liu,Q.,1996,Perfectly matched layers for elastodynamics:A new absorbing boundary condition:Journal of Computational Acoustics,4(4),341–359.
Chew,W.C.,and Weedon,W.H.,1994,A 3-D perfectly matched medium from modified Maxwell’s equations with stretched coordinates:Microwave and Optical Technology Letters,7(13),599–604.
Clayton,R.,and Engquist,B.,1977,Absorbing boundary conditions for acoustic and elastic wave equations:Bulletin of the Seismological Society of America,67(6),1529–1540.
Cohen,G.C.,2002,Higher-order Numerical Methods for Transient Wave Equations:Springer:Berlin.
Collino,F.,and Tsogka,C.,2001,Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media:Geophysics,66(1),294–307.
Courant,R.,Friedrichs,K.O.,and Lewy,H.,1928,über die partiellen Differenzengleichungen dermathematischen Physik:Mathematische Annalen,100,32 –74.
Drake,L.A.,1972,Rayleigh wave at continental boundary by the finite element method:Bulletin of the Seismological Society of America,62(5),1259-1268.
Engquist,B.,and Majda,A.,1977,Absorbing boundary conditions for the numerical simulation of waves:Proceedings of the National Academy of Science,74(5),1765–1766.
Hastings,F.D.,Schneider,J.B.,and Broschat,S.L.,1996,Application of the perfectly matched layer PML absorbing boundary condition to elastic wave propagation:Journal of the Acoustical Society of America,100(5),3061–3069.
Higdon,R.L.,1991,Absorbing boundary conditions for elastic waves:Geophysics,56(2),231–241.
Katsibas,T.K.,and Antonopoulos,C.S.,2002,An efficient PML absorbing medium in FDTD simulations of acoustic scattering in lossy media:IEEE Proceeding on Ultrasonic Symposium,1,551–554.
Komatitsch,D.,and Tromp,J.,2003,A perfectly matched layer absorbing boundary condition for the secondorder seismic wave equation:Geophysical Journal International,154(1),146–153.
Liu,Q.,and Tao,J.,1997,The perfectly matched layer for acoustic waves in absorptive media:Journal of the Acoustical Society of America,102(4),2072–2082.
Liu,Q.H.,1999,Perfectly matched layers for elastic waves in cylindrical and spherical coordinates:Journal of the Acoustical Society of America,105(4),2075–2084.
Madariaga,R.,2007,Seismic source theory.Treatise on Geophysics,Amsterdam:Elsevier,59–82.
Marfurt,K.J.,1984,Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equation:Geophysics,49(5),533–549.
Meza-Fajardo,K.C.,and 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.
Newmark,N.M.,1959,A method of computation for structural dynamics:ASCE Journal of the Engineering Mechanics Division,85(3),67–94.
Quarteroni,A.,Tagliani,A.,and Zampieri,E.,1998,Generalized Galerkin approximations of elastic waves with absorbing boundary conditions:Computer Methods in Applied Mechanics and Engineering,163,323–341.
Shi,R.,Wang,S.,and Zhao,J.,2012,An unsplit complexfrequency-shifted PML based on matched Z-transform for FDTD modeling of seismic wave equations:Journal of Geophysics and Engineering,9,218–229.
Sochacki,J.,Kubichek,R.,George,J.,Fletcher,W.R.,and Smithson,S.,1987,Absorbing boundary conditions and surface waves:Geophysics,52(1),60–71.
Stacey,R.,1988,Improved transparent boundary formulations for the elastic wave equation:Bulletin of the Seismological Society of America,78(6),2089–2097.
Zeng,Y.Q.,He,J.Q.,and Liu,Q.H.,2001,The applicationof the perfectly matched layer in numerical modelingof wave propagation in poroelastic media:Geophysics,66 (4),1258–1266.
Zienkiewicz,O.C.,Taylor,R.L.,and Zhu,J.,2005,The Finite Element Method:its Basis and Fundamentals(6th edition):Elsevier Butterworth-Heinemann,London.
版权所有：© 2023 中国地质图书馆 中国地质调查局地学文献中心