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山东大学学报（理学版） ›› 2016, Vol. 51 ›› Issue (11): 99-106.doi: 10.6040/j.issn.1671-9352.0.2016.232

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弹性介质Hamilton正则方程与声波方程辛几何算法

方刚1,2,栾锡武1,2 *,方建会3   

  1. 1.国土资源部油气资源和环境地质重点实验室, 青岛海洋地质研究所, 山东 青岛 266071;2.海洋国家实验室海洋矿产资源评价与探测技术功能实验室, 山东 青岛 266071;3.中国石油大学(华东)理学院, 山东 青岛 266580
  • 收稿日期:2016-05-23 出版日期:2016-11-20 发布日期:2016-11-22
  • 通讯作者: 栾锡武(1966— ),男,博士,研究员,研究方向为综合地球物理勘探. E-mail:xluan@cgs.cn E-mail:fanggeo@163.com
  • 作者简介:方刚(1984— ),男,博士,助理研究员,研究方向为地震成像、地震波动力学. E-mail:fanggeo@163.com
  • 基金资助:
    国家自然科学基金资助项目(41504109);山东省自然科学基金资助项目(BS2015HZ008)

Hamiltonian canonical equations of elastic medium and symplectic geometric algorithm of acoustic wave equation

FANG Gang1,2, LUAN Xi-wu1,2 *, FANG Jian-hui3   

  1. 1. Key Laboratory of Marine Hydrocarbon Resources and Environmental Geology, Ministry of Land and Resources, Qingdao Institute of Marine Geology, Qingdao, 266071, Shandong, China;
    2. Laboratory for Marine Mineral Resources, Qingdao National Laboratory for Marine Science and Technology, Qingdao, 266071, Shandong, China;
    3. School of Science in China University of Petroleum, Qingdao 266580, Shandong, China
  • Received:2016-05-23 Online:2016-11-20 Published:2016-11-22

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摘要： 建立弹性介质的Hamilton正则方程,把声波介质视为特殊的弹性介质,由弹性介质Hamilton方程导出声波介质地震波方程,对声波方程Hamilton化后给出其蛙跳格式的辛差分算法。将声波方程辛算法应用于二维情况下的地震波场正演数值模拟计算,并与常规的有限差分算法进行比较。结果表明,在地震波场正演数值模拟计算中辛几何算法比常规有限差分算法更具优越性。

关键词: 弹性介质, 辛几何算法, 声波方程, Hamiltion正则方程

Abstract: The Hamilton canonical equations is built for elastic medium. Taking the acoustic medium as a special kind of elastic medium, the acoustic wave equation can be derivate from elastic Hamilton canonical equations. The symplectic geometric algorithm with leapfrog schemes can be obtained with the Hamiltonian of acoustic wave equation. The symplectic geometric algorithm of acoustic wave equation can be applied for 2D seismic wave fields numerical modeling. Comparing with the conventional finite differences algorithm, the results indicated that the symplectic geometric algorithm is more advantageous in seismic wave fields numerical modeling.

Key words: acoustic wave equation, Hamilton canonical equations, symplectic geometric algorithm, elastic medium

中图分类号: 

  • O316
[1] 梅凤翔. 李群和李代数对约束力学系统的应用[M]. 北京: 科学出版社, 1999. MEI Fengxiang. Applications of Lie Groups and Lie Algebras to constrained mechanical systems[M]. Beijing: Science Press, 1999.
[2] CHEN Xiangwei, ZHANG Ruichao, MEI Fengxiang. Perturbation of symmetries of birkhoff system and adiabatic invariants[J]. Acta Mechanica Sinica, 2000, 16(3):282-288.
[3] 罗绍凯. Hamilton系统的Mei对称性、Noether对称性和Lie对称性[J]. 物理学报, 2003, 52(12):2941-2944. LOU Shaokai. Mei symmetry, Noether symmetry and Lie symmetry of Hamiltonian system [J]. Acta Phys Sin, 2003, 52(12):2941-2944.
[4] 梅凤翔. 约束力学系统的对称性与守恒量[M]. 北京: 北京理工大学出版社, 2004. MEI Fengxiang. Symmetries and conserved quantities of constrained mechanical systems[M]. Beijing: Beijing Institute of Technology Press, 2004.
[5] FU Jingli, CHEN Liqun. Form invariance, Noether symmetry and Lie symmetry of Hamiltonian systems in phase space[J]. Mechanics Research Communications, 2004, 31(1):9-19.
[6] TARASOV V E, ZASLAVSKY G M. Nonholonomic constraints with fractional derivatives[J]. Journal of Physics A: Mathematical and General, 2006, 39(21):9797-9815.
[7] MEI Fengxiang, WU Huibin. Symmetry of Lagrangians of nonholonomic systems[J]. Physics Letters A, 2008, 372(13):2141-2147.
[8] CAI Jianle, LOU Shaokai, MEI Fengxiang. Conformal invariance and conserved quantity of Hamilton systems[J]. Chin Phys B, 2008, 17(9):3170-3174.
[9] 梅凤翔. 经典约束力学系统对称性与守恒量研究进展[J]. 力学进展, 2009, 39(1):37-43. MEI Fengxiang. Advances in the symmetries and conserved quantities of classical constrained systems[J]. Advances in Mechanics, 2009, 39(1), 37-43.
[10] DORODNITSYN V, KOZLOV R. Invariance and first integrals of continuous and discrete Hamiltonian equations[J]. Journal of Engineering Mathematics, 2010, 66(1):253-270.
[11] AGRAWAL O P. Generalized variational problems and Euler-Lagrange equations[J]. Comput Math Appl, 2010, 59(5):1852-1864.
[12] GOU Yongxin, LIU Chang, LIU Shixing. Generalized Birkhoffian realization of nonholonomic systems[J]. Communications in Mathematics, 2010, 18(1):21-35
[13] FANG Jianhui, ZHANG Mingjiang, ZHANG Weiwei. A new type of conserved quantity induced by symmetries of Lagrange system[J]. Physics Letters A, 2010, 374(17):1806-1811.
[14] NUCCI M C. Many conserved quantities induced by Lie symmetries of a Lagrangian system[J]. Physics Letters A, 2011, 375(11):1375-1377.
[15] WU Huibin, MEI Fengxiang. Symmetry of Lagrangians of a holonomic variable mass system[J]. Chin Phys B, 2012, 21(6):064501.1-064501.4.
[16] LOU Shaokai, LI Lin. Fractional generalized Hamiltonian equations and its integral invariants[J] , Nonlinear Dynamacs, 2013, 73(1):339-346.
[17] 张毅. 非保守动力学系统Noether对称性的摄动与绝热不变量[J]. 物理学报, 2013, 60(16):164501.1-164501.5. ZHANG Yi. Perturbation to Noether symmetries and adiabatic invariants for nonconservative dynamic systems[J]. Acta Phys Sin, 2013, 62(16):164501.1-164501.5.
[18] 李彦敏, 陈向炜, 吴惠彬, 等. 广义Birkhoff系统的两类广义梯度表示[J]. 物理学报, 2016, 65(8):080201.1-080201.4. LI Yanmin, CHEN Xiangwei, WU Huibin, et al. Two kinds of generalized gradient representations for generalized Birkhoff system[J]. Acta Phys Sin, 2016, 65(8):080201.1-080201.4.
[19] 刘世兴, 郭永新, 刘畅. 一类特殊非完整力学系统的辛算法计算[J]. 物理学报, 2008, 57(3):1311-1315. LIU Shixing, GUO Yongxin, LIU Chang. A special nonholonomic mechanical system calculated by symplectic method[J]. Acta Phys Sin, 2008, 57(3):1311-1315.
[20] 刘世兴, 刘畅, 常鹏, 等. 辛几何算法在特殊Chaplygin系统中的应用研究[J]. 物理学报, 2011, 60(3):034501.1-034501.4. LIU Shixing, LIU Chang, CHANG Peng, et al. The application of symplectic geometric algorithm in a special Chaplygin system[J]. Acta Phys Sin, 2011, 60(3):034501.1-034501.4.
[21] 冯康, 秦孟兆. 哈密顿系统的辛几何算法[M]. 杭州: 浙江科学技术出版社, 2003. FENG Kang, QIN Mengzhao. Symplectic geometric algorithms for hmiltonian systems[M]. Hangzhou: Zhejiang Science and Technology Press, 2003.
[22] 刘洪, 罗明秋, 李幼铭, 等. 共反射点轨迹的Hamilton方法[J]. 地球物理学报, 1999, 42(5):685-694. LIU Hong, LOU Mingqiu, LI Youming, et al. Hamilton method in describing common reflection point trajectory[J]. Chinese Journal of Geophysics, 1999, 42(5):685-694.
[23] 罗明秋, 刘洪, 李幼铭. 地震波传播的哈密顿表述及辛几何算法[J]. 地球物理学报, 2001, 44(1):120-128. LOU Mingqiu, LIU Hong, LI Youming. Hamiltonian description and symplectic method of seismic wave propagation[J]. Chinese Journal of Geophysics, 2001, 44(1):120-128.
[24] 曹禹, 杨孔庆. 对声波和弹性波传播模拟的Hamilton系统方法[J]. 物理学报, 2003, 52(8):1984-1992. CAO Yu, YANG Kongqing. Hamiltonian system approach for simulation of acoustic and elastic wave propagation[J]. Acta Phys Sin, 2003, 52(8):1984-1992.
[25] 方刚, 张斌. 弹性介质的Lagrange动力学与地震波方程[J]. 物理学报, 2013, 62(15):154502.1-154502.6. FANG Gang, ZHANG Bin. Lagrangian dynamics and seismic wave align of elastic medium[J]. Acta Phys Sin, 2013, 62(15):154502.1-154502.6.
[26] 朱滨. 弹性力学[M]. 合肥: 中国科学技术大学出版社, 2008. ZHU Bin. Elastic Mechanics[M]. Hefei: Press of University of Science and Technology of China, 2008.
[27] BONOMI E, BRIEGER L, NARDONE C, et al. 3D spectral reverse time migration with no-wraparound absorbing conditions[C] //SEG International Exposition and 68th Annual Meeting. New Orleans Louisiana: Society of Exploration Geophysists, 1998: 1925-1928.
[1] 张兴刚,孔维姝 . 两端固定弹性弦的理论研究与数值模拟[J]. J4, 2008, 43(10): 71-76 .
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