地震动模拟中的谱元法
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摘要
本文首先从弹性波动方程及其弱形式、空间离散、单元内插函数及积分、全局聚合计算、时间离散以及算法的精度和稳定性等多个方面,对谱元法的理论进行了详细的总结,然后对其在地震动模拟中的应用现状进行了归纳,最后采用该法简单分析了埋藏基岩凸起地形对地表地震动的影响作用.总的来说,谱元法不仅具有有限单元法处理复杂结构的几何灵活性,还具有伪谱法的高精度性和快速收敛性,用其进行数值模拟计算能大大减少内存需求和计算时间.在合理考虑区域地质构造、局部场地条件、介质各向异性以及地震波传播特性等多种影响因素的基础上,建立正确的计算模型,用该法研究盆地效应、地形效应以及烈度异常现象等是可行的.根据目前掌握的资料,在国内应用谱元法进行地震动模拟的研究不多,对谱元法理论进行详细分析和总结的更少.因此,本文不仅对推动谱元法在国内的应用具有重要的理论意义,还对开展全面的地震动模拟工作具有重要的参考价值.
This paper firstly makes a detailed summary of the theory of spectral element method from elastic wave equation and the weak formulation,spatial discretization,element interpolation function and integral, global aggregate computation,time discretization and the precision and stability of the code.Then,it is also summary the application status of spectral element method in the area of seismic ground motion simulation.Finally,makes a simple analysis of the role of underlying convex bedrock terrain on ground motion.In general,the spectral element method combines the flexibility of a finite element method with the accuracy and convergence of a spectral method, which could greatly reduce the memory requirements and computation times of the numerical simulation.On the basis of considering the regional geological structure,local site conditions,media anisotropy and seismic wave propagation characteristics correctly,this method might be suitable for studying basin effect,terrain effect and abnormal intensity for establishing reasonable model.According to the present information,there is not much study on ground motion simulation of using the spectral element method in domestic,and it is not make a detailed analysis and summary about the theory of spectral element method.This paper not only has a great theoretical significance for promoting the application of the spectral element method in domestic,but also has a crucial reference value to carry out more comprehensive seismic ground motion simulation research work in the future.
This paper firstly makes a detailed summary of the theory of spectral element method from elastic wave equation and the weak formulation,spatial discretization,element interpolation function and integral, global aggregate computation,time discretization and the precision and stability of the code.Then,it is also summary the application status of spectral element method in the area of seismic ground motion simulation.Finally,makes a simple analysis of the role of underlying convex bedrock terrain on ground motion.In general,the spectral element method combines the flexibility of a finite element method with the accuracy and convergence of a spectral method, which could greatly reduce the memory requirements and computation times of the numerical simulation.On the basis of considering the regional geological structure,local site conditions,media anisotropy and seismic wave propagation characteristics correctly,this method might be suitable for studying basin effect,terrain effect and abnormal intensity for establishing reasonable model.According to the present information,there is not much study on ground motion simulation of using the spectral element method in domestic,and it is not make a detailed analysis and summary about the theory of spectral element method.This paper not only has a great theoretical significance for promoting the application of the spectral element method in domestic,but also has a crucial reference value to carry out more comprehensive seismic ground motion simulation research work in the future.
引文
Aki K,Richards P G.1986.Quantitative Seismology-the Theoryand Method(the first volume)(in Chinese)[M].Beijing:Seismological Press:41-130.
Chaljub E,Komatitisch D,Vilotte J P,et al.2007.Spectralelement analysis in seismology[J].Advances in Geophysics,48(7):365-419.
Cupillard P,Delavaud E,Burgos G,et al.2012.RegSEM:aversatile code based on the spectral element method to computeseismic wave propagation at the regional scale[J].Geophys.J.Int.,188(3):1203-1220.
De Martin F.2011.Verification of a spectral-element method codefor the Southern California earthquake center LOH.3viscoelastic case[J].Bull.Seismol.Soc.Am.,101(6):2855-2865.
De Basabe J D,Sen M.2010.Stability of the high-order finiteelements for acoustic or elastic wave propagation with highorder time stepping[J].Geophys.J.Int.,181(1):577-590.
Dezfulian H,Seed H B.1970.Seismic response of soil depositsunderlain by sloping rock boundaries[J].Journal of the SoilMechanics and Foundations Division,96(6):1893-1916.
Dezfulian H,Seed H B.1971.Response of nonuniform soil depositsto travelling seismic waves[J].Journal of the Soil Mechanicsand Foundation Division,97(SM1):27-46.
Faccioli E,Maggio F,Paolucci1P,et al.1997.2Dand 3Delasticwave propagation by apseudo-spectral domain decompositionmethod[J].Journal of Seismology,1(3):237-251.
Hu Y X,Liu X R,Luo J H,et al.2011.Simulation of the threedimensional topographic effects on seismic ground motion inWenchuan earthquake region based upon the spectral-elementmethod[J].Journal of Lanzhou University(Natural Sciences)(in Chinese),47(4):24-32.
Komatitsch D,Barnes C,Tromp J.2000.Simulation of anisotropicwave propagation based upon a spectral element method[J].Geophysics,65(4):1251-1260.
Komatitsch D,Liu Q Y,Tromp J,et al.2004.Simulations ofground motion in the Los Angeles basin based upon the spectralelement method[J].Bull.Seismol.Soc.Am.,94(1):187-206.
Komatitsch D,Martin R,Tromp J,et al.2001.Wave propagationin 2-D elastic media using a spectral element method withtriangles and quadrangles[J].Journal of ComputationalAcoustics,9(2):703-718
Komatitsch D,Tromp J.1999.Introduction to the spectral elementmethod for three-dimensional seismic wave propagation[J].Geophys.J.Int.,139(3):806-822.
Komatitsch D,Tromp J.2002.Spectral-element simulations ofglobal seismic wave propagationⅡ.Three-dimensional models,oceans,rotation and self-gravitation[J].Geophys.J.Int.,150(1):303-318.
Komatitsch D,Vilotte J P.1998.The spectral-element method:anefficient tool to simulate the seismic response of 2D and 3Dgeological structures[J].Bull.Seismol.Soc.Am.,88(2):368-392.
Laurenzano G,Priolo E,Tondi E.2008.2Dnumerical simulationsof earthquake ground motion:examples from the Marcheregion,Italy[J].Journal of Seismology,12(3):395-412.
Lee S J,Chen H W,Liu Q Y,et al.2008.Three dimensionalsimulations of seismic-wave propagation in the Taipei basin withrealistic topography based upon the spectral-element method[J].Bull.Seismol.Soc.Am.,98(1):253-264.
Lee S J,Chan Y C,Komatitsch D,et al.2009.Effects of realisticsurface topography on seismic ground motion in theYangminshan region of Taiwan based upon the spectral-elementmethod and LiDAR DTM[J].Bull.Seismol.Soc.Am.,99(2A):681-693.
Liu Q F,Yu Y Y,Zhang X B.2013.Three-dimensional groundmotion simulation for Shidian Basin[J].Journal of EarthquakeEngineering and Engineering Vibration(in Chinese),33(4):54-60.
Maday Y,Patera A T.1989.Spectral element methods for theincompressible Navier-Stokes equations[Z].New York:American Society of Mechanical Engineers,71-413.
Newmark N M.1959.A method for computation of structuraldynamics[C]//Engineering Mechanics Division:Proceedingsof the American Society of Civil Engineers,85(EM3):67-94
Patera A T.1984.A spectral element method for fluid dynamics:laminar flow in a channel expansion[J].Journal ofComputational Physics,54(3):468-488.
Peng H K.2007 Research on spectral element method-basedcharacteristics of guided wave propagation and damageidentification in structures[Ph.D.thesis].Shanghai:ShanghaiJiaotong University,1-140.
Peter D,Komatitsch D,Luo Y,et al.2011.Forward and adjointsimulations of seismic wave propagation on fully unstructuredhexahedral meshes[J].Geophys.J.Int.,186(2):721-739.
Priolo E.1999.2-D spectral element simulations of destructiveground shaking in Catania(Italy)[J].Journal of Seismology,3(3):289-309.
Priolo E.2001.Seismic ground motion simulation through the 2-Dspectral element method[J].Journal of ComputationalAcoustics,9(4):1561-1581.
Priolo E,Seriani G.1991.A numerical investigation of Chebyshevspectral element method for acoustic wave propagation[C]//Dublin:Proceedings of the 13th IMACS Conference onComparative Applied Mathematics.Dublin:Ireland,551-556.
Seriani G,Priolo E.1994.Spectral element method for acousticwave simulation in heterogeneous media[J].Finite Elements inAnalysis and Design,16(3-4):337-348
Seriani G,Priolo E,Carcione J M,et al.1992.High-order spectralelement method for elastic wave modeling[C]//62nd AnnualInternational Meeting,SEG,Expanded Abstracts:1285-1288.
Smerzini C,Villani M.2012.Broadband numerical simulations incomplex near-field geological configurations:the case of the2009 Mw 6.3L’Aquila earthquake[J].Bull.Seismol.Soc.Am.,102(6):2436-2451.
Stupazzini M,Paolucci R,Igel H.2009.Near-fault earthquakeground-motion simulation in the Grenoble valley by a highperformance spectral element code[J].Bull.Seism.Soc.Am.,99(1):286-301.
Taylor M A,Wingate B A.2000.A generalized diagonal massmatrix spectral element method for non-quadrilateral element[J].Applied Numerical Mathematics,33(1-4):259-265.
Taylor M A,Wingate B A,Vincent R E.2000.An algorithm forcomputing fekete points in the triangle[J].SIAM Journal onNumerical Analysis,38(5):1707-1720.
Tessmer E,Kosloff D.1994.3-D elastic modeling with surfacetopography by a Chebyshev spectral method[J].Geophysics,59(3):464-473.
Tromp J,Komatitsch D,Liu Q Y.2008.Spectral-element andadjoint methods in seismology[J].Commun.Comput.Phys.,3(1):1-32.
Wang W S,Li X F,Lu M W,et al.2012.Structure-preservingmodeling for seismic wavefields based upon a multisymplecticspectral element method[J].Chinese J.Geophys.(inChinese),55(10):3427-3439.
Wang X M,Seriani G,Lin W J.2007.Some theoretical problems ofcalculating elastic wave field by using the spectral elementmethod[J].Science China Physics,Mechanics&Astronomy(in Chinese),37(1):41-59.
Xing Y F,Guo J.2012.A self-adaptive Newmark method withparameters dependent upon structural dynamic characteristics[J].Chinese Journal of Theoretical and Applied Mechanics(inChinese),44(5):904-911.
Yan Z Z,Zhang H,Yang C C,et al.2009.The numericalsimulation of seismic wave propagation by the spectral elementmethod in Wenchuan great earthquake[J].Science China EarthSciences(in Chinese),39(4):393-402.
Zhou H,Gao M T,Yu Y X.2010.A study of topographical effecton SH waves[J].Progress in Geophys.(in Chinese),25(3):775-782.
安艺敬一,理查兹P G.1986.定量地震学——理论和方法(第一卷)[M].北京:地震出版社:41-130.
胡元鑫,刘新荣,罗建华,等.2011.汶川震区地震动三维地形效应的谱元法模拟[J].兰州大学学报(自然科学版),47(4):24-32.
刘启方,于彦彦,章旭斌.2013.施甸盆地三维地震动研究[J].地震工程与工程振动,33(4):54-60.
彭海阔.2007.基于谱元法的导波传播机理及结构损伤识别研究[博士论文].上海:上海交通大学:1-140.
汪文帅,李小凡,鲁明文,等.2012.基于多辛结构谱元法的保结构地震波场模拟[J].地球物理学报,55(10):3427-3439.
王秀明,Seriani G,林伟军.2007.利用谱元法计算弹性波场的若干理论问题[J].中国科学(G辑):物理学力学天文学,37(1):41-59.
邢誉峰,郭静.2012.与结构动特性协同的自适应Newmark方法[J].力学学报,44(5):904-911.
严珍珍,张怀,杨长春,等.2009.汶川大地震地震波传播的谱元法数值模拟研究[J].中国科学(D辑):地球科学,39(4):393-402.
周红,高孟潭,俞言祥.2010.SH波地形效应特征的研究[J].地球物理学进展,25(3):775-782.
Chaljub E,Komatitisch D,Vilotte J P,et al.2007.Spectralelement analysis in seismology[J].Advances in Geophysics,48(7):365-419.
Cupillard P,Delavaud E,Burgos G,et al.2012.RegSEM:aversatile code based on the spectral element method to computeseismic wave propagation at the regional scale[J].Geophys.J.Int.,188(3):1203-1220.
De Martin F.2011.Verification of a spectral-element method codefor the Southern California earthquake center LOH.3viscoelastic case[J].Bull.Seismol.Soc.Am.,101(6):2855-2865.
De Basabe J D,Sen M.2010.Stability of the high-order finiteelements for acoustic or elastic wave propagation with highorder time stepping[J].Geophys.J.Int.,181(1):577-590.
Dezfulian H,Seed H B.1970.Seismic response of soil depositsunderlain by sloping rock boundaries[J].Journal of the SoilMechanics and Foundations Division,96(6):1893-1916.
Dezfulian H,Seed H B.1971.Response of nonuniform soil depositsto travelling seismic waves[J].Journal of the Soil Mechanicsand Foundation Division,97(SM1):27-46.
Faccioli E,Maggio F,Paolucci1P,et al.1997.2Dand 3Delasticwave propagation by apseudo-spectral domain decompositionmethod[J].Journal of Seismology,1(3):237-251.
Hu Y X,Liu X R,Luo J H,et al.2011.Simulation of the threedimensional topographic effects on seismic ground motion inWenchuan earthquake region based upon the spectral-elementmethod[J].Journal of Lanzhou University(Natural Sciences)(in Chinese),47(4):24-32.
Komatitsch D,Barnes C,Tromp J.2000.Simulation of anisotropicwave propagation based upon a spectral element method[J].Geophysics,65(4):1251-1260.
Komatitsch D,Liu Q Y,Tromp J,et al.2004.Simulations ofground motion in the Los Angeles basin based upon the spectralelement method[J].Bull.Seismol.Soc.Am.,94(1):187-206.
Komatitsch D,Martin R,Tromp J,et al.2001.Wave propagationin 2-D elastic media using a spectral element method withtriangles and quadrangles[J].Journal of ComputationalAcoustics,9(2):703-718
Komatitsch D,Tromp J.1999.Introduction to the spectral elementmethod for three-dimensional seismic wave propagation[J].Geophys.J.Int.,139(3):806-822.
Komatitsch D,Tromp J.2002.Spectral-element simulations ofglobal seismic wave propagationⅡ.Three-dimensional models,oceans,rotation and self-gravitation[J].Geophys.J.Int.,150(1):303-318.
Komatitsch D,Vilotte J P.1998.The spectral-element method:anefficient tool to simulate the seismic response of 2D and 3Dgeological structures[J].Bull.Seismol.Soc.Am.,88(2):368-392.
Laurenzano G,Priolo E,Tondi E.2008.2Dnumerical simulationsof earthquake ground motion:examples from the Marcheregion,Italy[J].Journal of Seismology,12(3):395-412.
Lee S J,Chen H W,Liu Q Y,et al.2008.Three dimensionalsimulations of seismic-wave propagation in the Taipei basin withrealistic topography based upon the spectral-element method[J].Bull.Seismol.Soc.Am.,98(1):253-264.
Lee S J,Chan Y C,Komatitsch D,et al.2009.Effects of realisticsurface topography on seismic ground motion in theYangminshan region of Taiwan based upon the spectral-elementmethod and LiDAR DTM[J].Bull.Seismol.Soc.Am.,99(2A):681-693.
Liu Q F,Yu Y Y,Zhang X B.2013.Three-dimensional groundmotion simulation for Shidian Basin[J].Journal of EarthquakeEngineering and Engineering Vibration(in Chinese),33(4):54-60.
Maday Y,Patera A T.1989.Spectral element methods for theincompressible Navier-Stokes equations[Z].New York:American Society of Mechanical Engineers,71-413.
Newmark N M.1959.A method for computation of structuraldynamics[C]//Engineering Mechanics Division:Proceedingsof the American Society of Civil Engineers,85(EM3):67-94
Patera A T.1984.A spectral element method for fluid dynamics:laminar flow in a channel expansion[J].Journal ofComputational Physics,54(3):468-488.
Peng H K.2007 Research on spectral element method-basedcharacteristics of guided wave propagation and damageidentification in structures[Ph.D.thesis].Shanghai:ShanghaiJiaotong University,1-140.
Peter D,Komatitsch D,Luo Y,et al.2011.Forward and adjointsimulations of seismic wave propagation on fully unstructuredhexahedral meshes[J].Geophys.J.Int.,186(2):721-739.
Priolo E.1999.2-D spectral element simulations of destructiveground shaking in Catania(Italy)[J].Journal of Seismology,3(3):289-309.
Priolo E.2001.Seismic ground motion simulation through the 2-Dspectral element method[J].Journal of ComputationalAcoustics,9(4):1561-1581.
Priolo E,Seriani G.1991.A numerical investigation of Chebyshevspectral element method for acoustic wave propagation[C]//Dublin:Proceedings of the 13th IMACS Conference onComparative Applied Mathematics.Dublin:Ireland,551-556.
Seriani G,Priolo E.1994.Spectral element method for acousticwave simulation in heterogeneous media[J].Finite Elements inAnalysis and Design,16(3-4):337-348
Seriani G,Priolo E,Carcione J M,et al.1992.High-order spectralelement method for elastic wave modeling[C]//62nd AnnualInternational Meeting,SEG,Expanded Abstracts:1285-1288.
Smerzini C,Villani M.2012.Broadband numerical simulations incomplex near-field geological configurations:the case of the2009 Mw 6.3L’Aquila earthquake[J].Bull.Seismol.Soc.Am.,102(6):2436-2451.
Stupazzini M,Paolucci R,Igel H.2009.Near-fault earthquakeground-motion simulation in the Grenoble valley by a highperformance spectral element code[J].Bull.Seism.Soc.Am.,99(1):286-301.
Taylor M A,Wingate B A.2000.A generalized diagonal massmatrix spectral element method for non-quadrilateral element[J].Applied Numerical Mathematics,33(1-4):259-265.
Taylor M A,Wingate B A,Vincent R E.2000.An algorithm forcomputing fekete points in the triangle[J].SIAM Journal onNumerical Analysis,38(5):1707-1720.
Tessmer E,Kosloff D.1994.3-D elastic modeling with surfacetopography by a Chebyshev spectral method[J].Geophysics,59(3):464-473.
Tromp J,Komatitsch D,Liu Q Y.2008.Spectral-element andadjoint methods in seismology[J].Commun.Comput.Phys.,3(1):1-32.
Wang W S,Li X F,Lu M W,et al.2012.Structure-preservingmodeling for seismic wavefields based upon a multisymplecticspectral element method[J].Chinese J.Geophys.(inChinese),55(10):3427-3439.
Wang X M,Seriani G,Lin W J.2007.Some theoretical problems ofcalculating elastic wave field by using the spectral elementmethod[J].Science China Physics,Mechanics&Astronomy(in Chinese),37(1):41-59.
Xing Y F,Guo J.2012.A self-adaptive Newmark method withparameters dependent upon structural dynamic characteristics[J].Chinese Journal of Theoretical and Applied Mechanics(inChinese),44(5):904-911.
Yan Z Z,Zhang H,Yang C C,et al.2009.The numericalsimulation of seismic wave propagation by the spectral elementmethod in Wenchuan great earthquake[J].Science China EarthSciences(in Chinese),39(4):393-402.
Zhou H,Gao M T,Yu Y X.2010.A study of topographical effecton SH waves[J].Progress in Geophys.(in Chinese),25(3):775-782.
安艺敬一,理查兹P G.1986.定量地震学——理论和方法(第一卷)[M].北京:地震出版社:41-130.
胡元鑫,刘新荣,罗建华,等.2011.汶川震区地震动三维地形效应的谱元法模拟[J].兰州大学学报(自然科学版),47(4):24-32.
刘启方,于彦彦,章旭斌.2013.施甸盆地三维地震动研究[J].地震工程与工程振动,33(4):54-60.
彭海阔.2007.基于谱元法的导波传播机理及结构损伤识别研究[博士论文].上海:上海交通大学:1-140.
汪文帅,李小凡,鲁明文,等.2012.基于多辛结构谱元法的保结构地震波场模拟[J].地球物理学报,55(10):3427-3439.
王秀明,Seriani G,林伟军.2007.利用谱元法计算弹性波场的若干理论问题[J].中国科学(G辑):物理学力学天文学,37(1):41-59.
邢誉峰,郭静.2012.与结构动特性协同的自适应Newmark方法[J].力学学报,44(5):904-911.
严珍珍,张怀,杨长春,等.2009.汶川大地震地震波传播的谱元法数值模拟研究[J].中国科学(D辑):地球科学,39(4):393-402.
周红,高孟潭,俞言祥.2010.SH波地形效应特征的研究[J].地球物理学进展,25(3):775-782.
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