利用曲线网格有限差分方法研究三维倾斜断层的破裂动力学
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摘要
利用曲线网格有限差分方法,研究了三维倾斜断层的破裂传播过程.基于断层面生成贴体曲线网格,并通过坐标变换将含曲线网格的物理空间转换到含均匀直角网格的计算空间,实现了有限差分方法对复杂界面的处理.通过模拟地震断层的自相似破裂和自发破裂,并与已有发表的结果对比,发现拟合程度较高,验证了本方法的有效性和精确性.重点研究了不同倾角的倾斜断层破裂,最后展望了今后用本方法对非均匀介质中和任意起伏地表下的任意非平面断层破裂动力学的进一步研究.
We use curvilinear-grid finite difference method (FDM) to study 3-D rupture dynamics of dipping faults.By generating boundary-conforming curvilinear grids based on fault plane and using coordinate transformation,physical space with curvilinear grids can be mapped to computational space with uniform orthogonal grids,so FDM can be implemented on complex surfaces.In order to verify the validity and accuracy of this method,we model self-similar and spontaneous rupture of earthquake faults and compare our results with previously published ones,and find them well fitted.We focus on discussing rupture dynamics of the faults with different dip angles.We are expected to do further research on arbitrarily "non-planar" rupture dynamics in heterogeneous media and under arbitrarily curved surface using this method.
We use curvilinear-grid finite difference method (FDM) to study 3-D rupture dynamics of dipping faults.By generating boundary-conforming curvilinear grids based on fault plane and using coordinate transformation,physical space with curvilinear grids can be mapped to computational space with uniform orthogonal grids,so FDM can be implemented on complex surfaces.In order to verify the validity and accuracy of this method,we model self-similar and spontaneous rupture of earthquake faults and compare our results with previously published ones,and find them well fitted.We focus on discussing rupture dynamics of the faults with different dip angles.We are expected to do further research on arbitrarily "non-planar" rupture dynamics in heterogeneous media and under arbitrarily curved surface using this method.
引文
罗扬.2007.有限差分方法求解断层动力学问题[D].北京:北京大学地球物理系:12--43.
张海明.2004.半无限空间中平面断层的三维自发破裂传播的理论研究[D].北京:北京大学地球物理系:19--36.
张伟.2006.含起伏地形的三维非均匀介质中地震波传播的有限差分算法及其在强地面震动模拟中的应用[D].北京:北京大学地球物理系:10--75.
祝贺君.2008.使用有限差分方法研究地震波传播和震源动力学问题[D].北京:北京大学地球物理系:7--38.
Aagaard B T,Heaton T H,Hall J F.2001.Dynamic earthquake ruptures in the presence of lithostatic normal stressesI mplications for friction models and heat production[J].Bull Seism Soc Amer,91(6):1765--1796.
Aagaard B T,Heaton T H.2004.Near-source ground motions fromsi mulations of sustained intersonic and supersonicfault ruptures[J].Bull Seism Soc Amer,94(6):2064--2078.
Andrews D.1976a.Rupture propagation with finite stress in antiplane strain[J].J Geophys Res,81(20):3575--3582
Andrews D.1976b.Rupture velocity of plane strain shear cracks[J].J Geophys Res,81(32):5679--5687.
Andrews DJ.1999.Test of two methods for faultingin finite difference calculations[J].Bull Seism Soc Amer,89(4):931--937.
Aochi H.1999.Theoretical Studies on Dynamic Rupture Propagation Along a3D Non-Planar Fault System[D].To-kyo:University of Tokyo:25--45.
Aochi H,Fukuyama E.2002.Three-di mensional nonplannar si mulation of the1992Landers earthquake[J].J GeophysRes,107(B2):2035,doi:10.1029/2000JB000061.
Aochi H,Fukuyama E,Matsu’ura M.2000a.Spontaneous rupture propagation on a non-plannar fault in3-Delastic me-dium[J].Pure Appl Geophys,157:2003--2027.
Aochi H,Fukuyama E,Matsu’ura M.2000b.Selectivity of spontaneous rupture propagation on a branched fault[J].Geophys Res Lett,27(22):3635--3638.
Bayliss A,Jordan K E,Le Mesurier BJ,Turkel E.1986.Afourth-order accurate finite-difference scheme for the compu-tation of elastic waves[J].Bull Seism Soc Amer,76(4):1115--1132.
Chen X F,Zhang H M.2006.Modelling rupture dynamics of a planar fault in3-D half space by boundaryintegral equa-tion method:An overview[J].Pure Appl Geophys,163:267--299.
Cruz-Atienza V,Virieux J.2004.Dynamic rupture si mulation of nonplannar faults with a finite-difference approach[J].Geophys J Int,158(3):939--954.
Dai N,Vafidis A,Kanasewich E R.1995.Wave-propagation in heterogeneous,porous-media-a velocity-stress,finite-difference method[J].Geophysics,60(2):327--340.
Dalguer L.2002.A Full Dynamic Shear and Tensile Crack Propagation During an Earthquake Using a3D Discrete Ele-ment Method[D].Kyoto:University of Kyoto:34--37.
Dalguer L,Day S.2007.Staggered-grid split-node method for spontaneous rupture si mulation[J].J Geophys Res,112(B02302),doi:10.1029/2006JB004467.
Das S,Aki K.1977.Anumerical study of two-di mensional spontaneous rupture propagation[J].Geophys J Rastr Soc,50(3):643--668.
Day S.1977.Finite Element Analysis of Seismic Scattering Problems[D].San Diego:University of California San Die-go:4--50.
Day S.1982a.Three-di mensional finite difference si mulation of fault dynamics:Rectangular faults withfixed rupture ve-locity[J].Bull Seism Soc Amer,72(3):705--727.
Day S.1982b.Three-di mensional si mulation of spontaneous rupture:The effect of nonuniformprestress[J].Bull SeismSoc Amer,72(6):1881--1902.
Day S,Dalguer L,Lapusta N,Liu Y.2005.Comparison of finite difference and boundaryintegral solutions to three-di-mensional spontaneous rupture[J].J Geophys Res,110(B12307),doi:10.1029/2005JB003813.
Duan B,Oglesby D.2006.Heterogeneous faults stresses fromprevious earthquakes andthe effect on dynamics of parallelstrike-slip faults[J].J Geophys Res,111(B05309),doi:10.1029/2005JB004138.
Duan B,Oglesby D.2007.Nonuniformprestress fromprior earthquakes andthe effect on dynamics of branchedfault sys-tems[J].J Geophys Res,112(B05308),doi:10.1029/2006JB004443.
Hixon R.1997.Onincreasingthe accuracy of MacCormack schemes for aeroacoustic applications[J].AIAAPaper,(97--1586):29--39.
Ida Y.1972.Cohesive force across the tip of a longitudinal-shear crack and Griffith’s specific surface energy[J].J Geo-phys Res,77(20):3796--3805.
Kostrov B.1964.Self-si milar problems propagation of shear cracks[J].J Appl Math Mech,28(5):1077--1087.
Kostrov B.1966.Unsteady propagation of longitudinal shear cracks[J].J Appl Math Mech,30(6):1241--1248.
MacCormack R W.1969.The effect of viscosity in hypervelocity i mpact cratering[J].AIAA Paper,(69--354):1--7.
Madariaga R.1976.Dynamic of an expanding circular fault[J].Bull Seism Soc Amer,66(3):639--667.
Madariaga R,Olsen K,Archuleta R.1998.Modeling dynamic rupture in a3D earthquake fault model[J].Bull SeismSoc Amer,88(5):1182--1197.
Miyatake T,Ki mura T.2006.I mprovement in the boundary conditions for a staggered grid finite-difference method[J].Pure Appl Geophys,163:1977--1990.
Oglesby D D,Archuleta RJ,Nielsen S B.1998.Earthquakes on dippingfaults:the effect of broken symmetry[J].Sci-ence,280:1055--1059.
Oglesby D D,Archuleta RJ,Nielsen S B.2000a.Dynamics of dip-slip faulting:Explorations in two di mensions[J].JGeophys Res,105(B6):13643--13653.
Oglesby D D,Archuleta RJ,Nielsen S B.2000b.The three-di mensional dynamics of dipping faults[J].Bull Seism SocAmer,90(3):616--628.
Olsen K,Archuleta R.1996.Three-di mensional si mulation of earthquakes on the Los Angeles fault system[J].BullSeism Soc Amer,86(3):575--596.
Pal mer A C,Rice J R.1973.The growth of slip surfacein the progressive failure of over-consolidated clay[J].Proc RoySoc Lond A,332(1591):527--548.
Pitarka A.1999.3Delastic finite-difference modeling of seismic motion using staggered grids with nonuniforming spaci[J].Bull Seism Soc Amer,89(1):54--68.
Thompson J F,Warsi Z U A,Mastin C W.1985.Numerical Grid Generation-Foundations and Applications[M].NewYork:North Hollad Pulishing Co,New York:31--35.
Tsingas C,Vafidis A,Kanasewich E R.1990.Elastic wave-propagationintransverselyisotropic media usingfinite-differ-ence[J].Geophys Prospecting,38(8):933--949.
Virieux J,Madariaga R.1982.Dynamic faulting studied by a finite difference method[J].Bull SeismSoc Amer,72(2):345--369.
Zhang H M,Chen X F.2006a.Dynamic rupture on a planar fault in three-di mensional half space--Ⅰ.Theory[J].Geo-phy J Int,164(3):633--652.
Zhang H M,Chen X F.2006b.Dynamic rupture on a planar fault inthree-di mensional half space--Ⅱ.Validations and nu-merical experi ments[J].Geophy J Int,167(2):917--932.
Zhang W,Chen X F.2006c.Tractioni mage methodfor irregular free surface boundariesinfinite difference seismic wavesi mulation[J].Geophy J Int,167(1):337--353.
Zhang H M,Chen X F.2009.Dynamic rupture process of the1999Chi-Chi,Tai wan,earthquake[J].Earthq Sci,(1):3--12.
Zhang WB,Iwata T,Irikura K.2006.Dynamic si mulation of a dipping fault using a three-di mensional finite differencemethod with nonuniformgrid spacing[J].J Geophys Res,111(B05301),doi:10.1029/2005JB003725.
张海明.2004.半无限空间中平面断层的三维自发破裂传播的理论研究[D].北京:北京大学地球物理系:19--36.
张伟.2006.含起伏地形的三维非均匀介质中地震波传播的有限差分算法及其在强地面震动模拟中的应用[D].北京:北京大学地球物理系:10--75.
祝贺君.2008.使用有限差分方法研究地震波传播和震源动力学问题[D].北京:北京大学地球物理系:7--38.
Aagaard B T,Heaton T H,Hall J F.2001.Dynamic earthquake ruptures in the presence of lithostatic normal stressesI mplications for friction models and heat production[J].Bull Seism Soc Amer,91(6):1765--1796.
Aagaard B T,Heaton T H.2004.Near-source ground motions fromsi mulations of sustained intersonic and supersonicfault ruptures[J].Bull Seism Soc Amer,94(6):2064--2078.
Andrews D.1976a.Rupture propagation with finite stress in antiplane strain[J].J Geophys Res,81(20):3575--3582
Andrews D.1976b.Rupture velocity of plane strain shear cracks[J].J Geophys Res,81(32):5679--5687.
Andrews DJ.1999.Test of two methods for faultingin finite difference calculations[J].Bull Seism Soc Amer,89(4):931--937.
Aochi H.1999.Theoretical Studies on Dynamic Rupture Propagation Along a3D Non-Planar Fault System[D].To-kyo:University of Tokyo:25--45.
Aochi H,Fukuyama E.2002.Three-di mensional nonplannar si mulation of the1992Landers earthquake[J].J GeophysRes,107(B2):2035,doi:10.1029/2000JB000061.
Aochi H,Fukuyama E,Matsu’ura M.2000a.Spontaneous rupture propagation on a non-plannar fault in3-Delastic me-dium[J].Pure Appl Geophys,157:2003--2027.
Aochi H,Fukuyama E,Matsu’ura M.2000b.Selectivity of spontaneous rupture propagation on a branched fault[J].Geophys Res Lett,27(22):3635--3638.
Bayliss A,Jordan K E,Le Mesurier BJ,Turkel E.1986.Afourth-order accurate finite-difference scheme for the compu-tation of elastic waves[J].Bull Seism Soc Amer,76(4):1115--1132.
Chen X F,Zhang H M.2006.Modelling rupture dynamics of a planar fault in3-D half space by boundaryintegral equa-tion method:An overview[J].Pure Appl Geophys,163:267--299.
Cruz-Atienza V,Virieux J.2004.Dynamic rupture si mulation of nonplannar faults with a finite-difference approach[J].Geophys J Int,158(3):939--954.
Dai N,Vafidis A,Kanasewich E R.1995.Wave-propagation in heterogeneous,porous-media-a velocity-stress,finite-difference method[J].Geophysics,60(2):327--340.
Dalguer L.2002.A Full Dynamic Shear and Tensile Crack Propagation During an Earthquake Using a3D Discrete Ele-ment Method[D].Kyoto:University of Kyoto:34--37.
Dalguer L,Day S.2007.Staggered-grid split-node method for spontaneous rupture si mulation[J].J Geophys Res,112(B02302),doi:10.1029/2006JB004467.
Das S,Aki K.1977.Anumerical study of two-di mensional spontaneous rupture propagation[J].Geophys J Rastr Soc,50(3):643--668.
Day S.1977.Finite Element Analysis of Seismic Scattering Problems[D].San Diego:University of California San Die-go:4--50.
Day S.1982a.Three-di mensional finite difference si mulation of fault dynamics:Rectangular faults withfixed rupture ve-locity[J].Bull Seism Soc Amer,72(3):705--727.
Day S.1982b.Three-di mensional si mulation of spontaneous rupture:The effect of nonuniformprestress[J].Bull SeismSoc Amer,72(6):1881--1902.
Day S,Dalguer L,Lapusta N,Liu Y.2005.Comparison of finite difference and boundaryintegral solutions to three-di-mensional spontaneous rupture[J].J Geophys Res,110(B12307),doi:10.1029/2005JB003813.
Duan B,Oglesby D.2006.Heterogeneous faults stresses fromprevious earthquakes andthe effect on dynamics of parallelstrike-slip faults[J].J Geophys Res,111(B05309),doi:10.1029/2005JB004138.
Duan B,Oglesby D.2007.Nonuniformprestress fromprior earthquakes andthe effect on dynamics of branchedfault sys-tems[J].J Geophys Res,112(B05308),doi:10.1029/2006JB004443.
Hixon R.1997.Onincreasingthe accuracy of MacCormack schemes for aeroacoustic applications[J].AIAAPaper,(97--1586):29--39.
Ida Y.1972.Cohesive force across the tip of a longitudinal-shear crack and Griffith’s specific surface energy[J].J Geo-phys Res,77(20):3796--3805.
Kostrov B.1964.Self-si milar problems propagation of shear cracks[J].J Appl Math Mech,28(5):1077--1087.
Kostrov B.1966.Unsteady propagation of longitudinal shear cracks[J].J Appl Math Mech,30(6):1241--1248.
MacCormack R W.1969.The effect of viscosity in hypervelocity i mpact cratering[J].AIAA Paper,(69--354):1--7.
Madariaga R.1976.Dynamic of an expanding circular fault[J].Bull Seism Soc Amer,66(3):639--667.
Madariaga R,Olsen K,Archuleta R.1998.Modeling dynamic rupture in a3D earthquake fault model[J].Bull SeismSoc Amer,88(5):1182--1197.
Miyatake T,Ki mura T.2006.I mprovement in the boundary conditions for a staggered grid finite-difference method[J].Pure Appl Geophys,163:1977--1990.
Oglesby D D,Archuleta RJ,Nielsen S B.1998.Earthquakes on dippingfaults:the effect of broken symmetry[J].Sci-ence,280:1055--1059.
Oglesby D D,Archuleta RJ,Nielsen S B.2000a.Dynamics of dip-slip faulting:Explorations in two di mensions[J].JGeophys Res,105(B6):13643--13653.
Oglesby D D,Archuleta RJ,Nielsen S B.2000b.The three-di mensional dynamics of dipping faults[J].Bull Seism SocAmer,90(3):616--628.
Olsen K,Archuleta R.1996.Three-di mensional si mulation of earthquakes on the Los Angeles fault system[J].BullSeism Soc Amer,86(3):575--596.
Pal mer A C,Rice J R.1973.The growth of slip surfacein the progressive failure of over-consolidated clay[J].Proc RoySoc Lond A,332(1591):527--548.
Pitarka A.1999.3Delastic finite-difference modeling of seismic motion using staggered grids with nonuniforming spaci[J].Bull Seism Soc Amer,89(1):54--68.
Thompson J F,Warsi Z U A,Mastin C W.1985.Numerical Grid Generation-Foundations and Applications[M].NewYork:North Hollad Pulishing Co,New York:31--35.
Tsingas C,Vafidis A,Kanasewich E R.1990.Elastic wave-propagationintransverselyisotropic media usingfinite-differ-ence[J].Geophys Prospecting,38(8):933--949.
Virieux J,Madariaga R.1982.Dynamic faulting studied by a finite difference method[J].Bull SeismSoc Amer,72(2):345--369.
Zhang H M,Chen X F.2006a.Dynamic rupture on a planar fault in three-di mensional half space--Ⅰ.Theory[J].Geo-phy J Int,164(3):633--652.
Zhang H M,Chen X F.2006b.Dynamic rupture on a planar fault inthree-di mensional half space--Ⅱ.Validations and nu-merical experi ments[J].Geophy J Int,167(2):917--932.
Zhang W,Chen X F.2006c.Tractioni mage methodfor irregular free surface boundariesinfinite difference seismic wavesi mulation[J].Geophy J Int,167(1):337--353.
Zhang H M,Chen X F.2009.Dynamic rupture process of the1999Chi-Chi,Tai wan,earthquake[J].Earthq Sci,(1):3--12.
Zhang WB,Iwata T,Irikura K.2006.Dynamic si mulation of a dipping fault using a three-di mensional finite differencemethod with nonuniformgrid spacing[J].J Geophys Res,111(B05301),doi:10.1029/2005JB003725.
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