非均匀层状介质一维波动方程精确解的有限差分算法
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摘要
平面波的传播问题通常可以归结为一维波动方程的定解问题。在非均匀介质中,即使简单的一维波动方程也需要借助于数值方法获得近似解。3层5点古典差分格式是计算偏微分方程一种常用算法,作为一种显式迭代格式,需要满足稳定性条件a v t/x≤1,其中v为波速,x为空间采样间隔,t为时间采样间隔。当a 1时,x v t,古典差分格式达到临界稳定状态。在这种情况下,平面波在t时间内的传播距离恰好等于空间采样间隔,差分格式真实地反映了平面波的传播原理,因而可以得到一维波动方程的精确解。但是,由于在非均匀介质中存在不连续的波阻抗界面,此方法不适于计算非均匀介质的波场。为了将临界稳定情况下的古典差分格式推广应用至非均匀层状介质,提出了一种能够处理波阻抗界面的有限差分格式,并应用傅里叶分析法得到其稳定性条件。模型算例验证了此算法的正确性。
The plane-wave propagation can be generalized as a definite-solution problem of one-dimensional wave equation.In spite of the simple formality,solutions of one-dimensional wave equation in inhomogeneous media have to be solved with the aid of numerical methods.The classic three-level five-point finite difference scheme is a usual numerical method to calculate partial differential equations,which must meet the stable condition as an explicit iteration method.The stable condition is a v t / x1≤,where v is wave velocity,tis time sample interval,and x is space sample interval.When a 1or x v t,the finite difference scheme is just up to the critical stable state.In such a case a space sample interval x just equals wave propagation distance in a time sample interval t,so the classic difference scheme exactly expresses plane-wave propagation theory and can be used to obtain exact solutions of one-dimensional wave equations.However,because of existence of wave impedance interfaces,the algorithm is unable to calculate wave fields in heterogeneous layer media.In order that the classic difference scheme in the critical stable state can be generalized to apply to heterogeneous layer media,an improved scheme is put forward,which can deal with impedance interfaces.Its stable condition is also given by Fourier transform analysis and the correctness is proved by some numerical model tests.
The plane-wave propagation can be generalized as a definite-solution problem of one-dimensional wave equation.In spite of the simple formality,solutions of one-dimensional wave equation in inhomogeneous media have to be solved with the aid of numerical methods.The classic three-level five-point finite difference scheme is a usual numerical method to calculate partial differential equations,which must meet the stable condition as an explicit iteration method.The stable condition is a v t / x1≤,where v is wave velocity,tis time sample interval,and x is space sample interval.When a 1or x v t,the finite difference scheme is just up to the critical stable state.In such a case a space sample interval x just equals wave propagation distance in a time sample interval t,so the classic difference scheme exactly expresses plane-wave propagation theory and can be used to obtain exact solutions of one-dimensional wave equations.However,because of existence of wave impedance interfaces,the algorithm is unable to calculate wave fields in heterogeneous layer media.In order that the classic difference scheme in the critical stable state can be generalized to apply to heterogeneous layer media,an improved scheme is put forward,which can deal with impedance interfaces.Its stable condition is also given by Fourier transform analysis and the correctness is proved by some numerical model tests.
引文
[1]金星,孔戈,丁海平.水平成层场地地震反应非线性分析[J].地震工程与工程振动,2004,24(3):38-43.JIN Xing,KONG Ge,DING Hai-ping.Nonlinear seismic response analysis of horizontal layered site[J].Earthquake Engineering and Engineering Vibration,2004,24(3):38-43.
[2]曾心传,龚平,王岚.关于利用一维波动理论计算土层地震反映传递函数的讨论[J].地壳变形与地震,1996,16(1):64-71.ZENG Xin-chuan,GONG Ping,WANG Lan.A discussion on using one-dimensional wave theory to calculate the transfer function[J].Crustal Deformation and Earthquake,1996,16(1):64-71.
[3]范留明,张镭于.成层土场地地震效应的时程算法研究[J].岩土力学,2009,30(9):2564-2568.FAN Liu-ming,ZHANG Lei-yu.Time-history algorithm for earthquake effect in layered-soil site[J].Rock and Soil Mechanics,2009,30(9):2564-2568.
[4]胡元育,汤桃森,杨冰.桩基无损检测中几个问题的探讨[J].岩土力学,1988,9(2):65-70.HU Yuan-yu,TANG Tao-sen,YANG Bing.Some problem related to nondestructive inspection of pile quality[J].Rock and Soil Mechanics,1988,9(2):6570.
[5]刘润,禚瑞花,闫澍旺,等.动力打桩一维波动方程的改进及其工程应用[J].岩土力学,2004,25(增刊):383-387.LIU Run,ZHUO Rui-hua,YAN Shu-wang,et al.Apply of improved wave equation method to practical ocean piling engineering[J].Rock and Soil Mechanics,2004,25(Supp.):383-387.
[6]王健,周风华.采用Laplace变换分析桩基应力波传播问题[J].岩土力学,2011,32(1):179-185.WANG Jian,ZHUO Feng-hua.Analyzing stress wave propagations in a pile foundation using Laplace transform[J].Rock and Soil Mechanics,2011,32(1):179-185.
[7]张丽琴,王家映,严德天.一维波动方程波阻抗反演的同伦方法[J].地球物理学报,2004,47(6):1111-1117.ZHANG Li-qin,WANG Jia-ying,YAN De-tian.A homotopy method for the impedance inversion of 1-D wave equation[J].Chinese Journal of Geophysics,2004,47(6):1111-1117.
[8]张宇,张关泉.求解一维波动方程反问题的消除多次波方法[J].计算数学,1997,(3):293-304.ZHANG Yu,ZHANG Guan-quan.The multi-reflection elimination method to solve inverse problem of 1-D wave equation[J].Mathematica Numberica Sinica,1997,(3):293-304.
[9]成谷,张宝金,马在田,等.积分道与基于Born散射的一维声波方程速度反演[J].地球物理学进展,2003,18(1):122-127.CHENG Gu,ZHANG Bao-jin,MA Zai-tian,et al.Integral trace and 1-D velocity inversion of acoustic wave equation based on Born scattering[J].Progress in Geophysics,2003,18(1):122-127.
[10]马在田.地震成像技术-有限差分偏移[M].北京:石油工业出版社,1989.
[11]袁修贵,李远禄.基于小波的一维线性波动方程的数值解法[J].数学理论与应用,2003,23(1):11-14.YUAN Xiu-gui,LI Yuan-lu.Wavelet-based numerical solution of 1-D linear wave equation[J].Mathematical Theory and Applications,2003,23(1):11-14.
[12]张钦礼,牛莉.一维波动方程的小波解法[J].华北航天工业学院学报,2001,11(1):45-48.ZHANG Qin-li,NIU Li.Wavelet solution of one-dimensional wave equation[J].Journal of North China Institute of Astronautic Engineering,2001,11(1):45-48.
[13]廖振鹏,刘恒,谢志南.波动数值模拟的一种显式方法——一维波动[J].力学学报,2009,41(3):350-360.LIAO Zhen-peng,LIU Heng,XIE Zhi-nan.An explicit method for numerical simulation of wave motion—1-D wave motion[J].Chinese Journal of Theoretical and Applied Mechanics,2009,41(3):350-360.
[14]邓定文.一维电磁波方程的四阶紧致差分格式[J].应用数学,2012,25(4):917-922.DENG Ding-wen.Fourth-order compact difference scheme for one-dimensional telegraph equation[J].Mathematica Applicata,2012,25(4):917-922.
[15]李晓江,张文飞.三维波动方程的一维化有限元法[J].石油地球物理勘探,1996,31(1):21-27.LI Xiao-jiang,ZHANG Wen-fei.One-dimensional finite element method for solving 3-D wave equation[J].Oil Geophysical Prospecting,1996,31(1):21-27.
[16]刘晶波,王艳.成层半空间出平面自由波场的一维化时域算法[J].力学学报,2006,38(2):219-225.LIU Jing-bo,WANG Yan.A 1-D time-domain method for 2-D wave motion in elastic layered half-space by antiplane wave oblique incidence[J].Chinese Journal of Theoretical and Applied Mechanics,2006,38(2):219-225.
[2]曾心传,龚平,王岚.关于利用一维波动理论计算土层地震反映传递函数的讨论[J].地壳变形与地震,1996,16(1):64-71.ZENG Xin-chuan,GONG Ping,WANG Lan.A discussion on using one-dimensional wave theory to calculate the transfer function[J].Crustal Deformation and Earthquake,1996,16(1):64-71.
[3]范留明,张镭于.成层土场地地震效应的时程算法研究[J].岩土力学,2009,30(9):2564-2568.FAN Liu-ming,ZHANG Lei-yu.Time-history algorithm for earthquake effect in layered-soil site[J].Rock and Soil Mechanics,2009,30(9):2564-2568.
[4]胡元育,汤桃森,杨冰.桩基无损检测中几个问题的探讨[J].岩土力学,1988,9(2):65-70.HU Yuan-yu,TANG Tao-sen,YANG Bing.Some problem related to nondestructive inspection of pile quality[J].Rock and Soil Mechanics,1988,9(2):6570.
[5]刘润,禚瑞花,闫澍旺,等.动力打桩一维波动方程的改进及其工程应用[J].岩土力学,2004,25(增刊):383-387.LIU Run,ZHUO Rui-hua,YAN Shu-wang,et al.Apply of improved wave equation method to practical ocean piling engineering[J].Rock and Soil Mechanics,2004,25(Supp.):383-387.
[6]王健,周风华.采用Laplace变换分析桩基应力波传播问题[J].岩土力学,2011,32(1):179-185.WANG Jian,ZHUO Feng-hua.Analyzing stress wave propagations in a pile foundation using Laplace transform[J].Rock and Soil Mechanics,2011,32(1):179-185.
[7]张丽琴,王家映,严德天.一维波动方程波阻抗反演的同伦方法[J].地球物理学报,2004,47(6):1111-1117.ZHANG Li-qin,WANG Jia-ying,YAN De-tian.A homotopy method for the impedance inversion of 1-D wave equation[J].Chinese Journal of Geophysics,2004,47(6):1111-1117.
[8]张宇,张关泉.求解一维波动方程反问题的消除多次波方法[J].计算数学,1997,(3):293-304.ZHANG Yu,ZHANG Guan-quan.The multi-reflection elimination method to solve inverse problem of 1-D wave equation[J].Mathematica Numberica Sinica,1997,(3):293-304.
[9]成谷,张宝金,马在田,等.积分道与基于Born散射的一维声波方程速度反演[J].地球物理学进展,2003,18(1):122-127.CHENG Gu,ZHANG Bao-jin,MA Zai-tian,et al.Integral trace and 1-D velocity inversion of acoustic wave equation based on Born scattering[J].Progress in Geophysics,2003,18(1):122-127.
[10]马在田.地震成像技术-有限差分偏移[M].北京:石油工业出版社,1989.
[11]袁修贵,李远禄.基于小波的一维线性波动方程的数值解法[J].数学理论与应用,2003,23(1):11-14.YUAN Xiu-gui,LI Yuan-lu.Wavelet-based numerical solution of 1-D linear wave equation[J].Mathematical Theory and Applications,2003,23(1):11-14.
[12]张钦礼,牛莉.一维波动方程的小波解法[J].华北航天工业学院学报,2001,11(1):45-48.ZHANG Qin-li,NIU Li.Wavelet solution of one-dimensional wave equation[J].Journal of North China Institute of Astronautic Engineering,2001,11(1):45-48.
[13]廖振鹏,刘恒,谢志南.波动数值模拟的一种显式方法——一维波动[J].力学学报,2009,41(3):350-360.LIAO Zhen-peng,LIU Heng,XIE Zhi-nan.An explicit method for numerical simulation of wave motion—1-D wave motion[J].Chinese Journal of Theoretical and Applied Mechanics,2009,41(3):350-360.
[14]邓定文.一维电磁波方程的四阶紧致差分格式[J].应用数学,2012,25(4):917-922.DENG Ding-wen.Fourth-order compact difference scheme for one-dimensional telegraph equation[J].Mathematica Applicata,2012,25(4):917-922.
[15]李晓江,张文飞.三维波动方程的一维化有限元法[J].石油地球物理勘探,1996,31(1):21-27.LI Xiao-jiang,ZHANG Wen-fei.One-dimensional finite element method for solving 3-D wave equation[J].Oil Geophysical Prospecting,1996,31(1):21-27.
[16]刘晶波,王艳.成层半空间出平面自由波场的一维化时域算法[J].力学学报,2006,38(2):219-225.LIU Jing-bo,WANG Yan.A 1-D time-domain method for 2-D wave motion in elastic layered half-space by antiplane wave oblique incidence[J].Chinese Journal of Theoretical and Applied Mechanics,2006,38(2):219-225.
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