极坐标系下非分裂PML及时域有限元实现

周凤玺; 曹小林; M.B.贾克萨

doi:10.3879/j.issn.1000-0887.2015.09.007

极坐标系下非分裂PML及时域有限元实现

doi: 10.3879/j.issn.1000-0887.2015.09.007

¹兰州理工大学土木工程学院, 兰州 730050;²阿德莱德大学土木环境与采矿工程学院, 阿德莱德 5005, 澳大利亚

基金项目: 国家自然科学基金（11162008;51368038）；甘肃省环境保护厅科研基金(GSEP-2014-23)；甘肃省教育厅研究生导师基金(1103-07)

详细信息

作者简介:
周凤玺（1979—），男，甘肃人，副教授，博士，博士生导师(通讯作者. E-mail: geolut@163.com).

中图分类号: O347.4
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出版历程
- 收稿日期: 2015-04-09
- 修回日期: 2015-05-06
- 刊出日期: 2015-09-15

A Non-Splitting PML for Elastic Waves in Polar Coordinates and Its Finite Element Implementation

¹School of Civil Engineering, Lanzhou University of Technology, Lanzhou 730050, P.R.China;²School of Civil, Environmental and Mining Engineering, University of Adelaide, South Australia 5005, Australia

Funds: The National Natural Science Foundation of China(11162008;51368038)

摘要

摘要: 在弹性波传播的数值模拟中，吸收边界被广泛应用于截取有限空间进行无限空间问题的分析.完全匹配层（perfect matched layer, PML）吸收边界较其他吸收边界条件具有更优越的吸收性能，已被成功应用于直角坐标系下的弹性波方程正演模拟.考虑极坐标系下二阶弹性波动方程，通过采用辅助函数的方法，提出了一种非分裂格式的完全匹配层吸收边界条件.并且基于Galerkin近似技术，给出了非对称以及轴对称条件下的时域有限元计算格式.通过数值算例分析了该极坐标系下分裂格式的完全匹配层吸收边界的有效性.
- 完全匹配层 /
- 极坐标系 /
- 吸收边界条件 /
- 时域有限元
Abstract: In the solving of the elastic wave equations with the numerical approximation techniques, the absorbing boundary conditions had been widely used to truncate the infinite-space simulation to a finite-space one. The perfect matched layer (PML) technique as an absorbing boundary condition had exhibited excellent absorbing efficiency in the forward simulation of the elastic wave equation formulated in rectangular coordinates. Based on the stretched coordinate concept, an advanced non-splitting-field perfect matched layer (non-splitting PML) equation for elastic waves was formulated in the polar coordinate system. Through the introduction of integrated complex variables in the radial direction into the auxiliary functions, the PML formulation was extended in polar coordinates in view of the 2nd-order elastic wave equation with displacements as basic unknowns. In addition, aimed at the time-domain cases and with the finite-element method for space discretization, the finite-element time-domain (FETD) scheme in standard displacement-based formulation was presented. The scheme for the special cases in axisymmetric polar coordinates was also given. The effectiveness and validity of the present non-splitting PML formulation are demonstrated with several numerical examples.
- perfect matched layer /
- polar coordinate system /
- absorbing boundary condition /
- time-domain finite element

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参考文献(23)

[1]	Lysmer J, Kuhlemeyer R L. Finite dynamic model for infinite media[J]. Journal of the Engineering Mechanics Division,1969,95(4): 869-878.
[2]	White W, Valliappan S, Lee I K. Unified boundary for finite dynamic model[J]. Journal of the Engineering Mechanics Division,1977,103(5): 949-964.
[3]	Marfurt K J. Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equations[J]. Geophysics,1984,49(5): 533-549.
[4]	Clayton R, Engquist B. Absorbing boundary conditions for acoustic and elastic wave equations[J]. Bulletin of the Seismological Society of America,1977,67(6): 1529-1540.
[5]	Engquist B, Majda A. Absorbing boundary conditions for numerical simulation of waves[J]. Proceedings of the National Academy of Sciences,1977,74(5): 1765-1766.
[6]	LIAO Zhen-feng, HUANG Kong-liang, YANG Bai-po, YUAN Yi-fan. A transmitting boundary for transient wave analyses[J]. Science China: Mathematics,1984,27(10): 1063-1076.
[7]	Higdon R L. Numerical absorbing boundary conditions for the wave equation[J]. Mathematics of computation,1987,49(179): 65-90.
[8]	熊章强, 唐圣松, 张大洲. 瑞利面波数值模拟中的PML吸收边界条件[J]. 物探与化探, 2009,33(4): 453-457.(XIONG Zhang-qiang, TANG Sheng-song, ZHANG Da-zhou. PML absorbing boundary condition for numerical modeling of Rayleigh wave[J]. Geophysical and Geochemical Exploration,2009,33(4): 453-457.(in Chinese))
[9]	Berenger J P. Three-dimensional perfectly matched layer for the absorption of electromagnetic waves[J]. Journal of Computational Physics,1996,127(2): 363-379.
[10]	Udagedara I, Premaratne M, Rukhlenko I D, Hattori H T, Agrawal G P. Unified perfectly matched layer for finite-difference time-domain modeling of dispersive optical materials[J]. Optics Express,2009,17(23): 21179-21190.
[11]	MA You-neng, YU Jin-hua, WANG Yuan-yuan. A novel unsplit perfectly matched layer for the second-order acoustic wave equation[J]. Ultrasonics,2014,54(6): 1568-1574.
[12]	Komatitsch D, Tromp J. A perfectly matched layer absorbing boundary condition for the second-order seismic wave equation[J]. Geophysical Journal International,2003,154(1): 146-153.
[13]	PING Ping, ZHANG Yu, XU Yi-xian. A multiaxial perfectly matched layer (M-PML) for the long-time simulation of elastic wave propagation in the second-order equations[J]. Journal of Applied Geophysics,2014,101: 124-135.
[14]	张洪欣, 吕英华, 黄永明. 柱坐标系下PML-FDTD及总场-散射场区连接条件[J]. 重庆邮电学院学报, 2003,15(3): 4-8.(ZHANG Hong-xin, LU Ying-hua, HUANG Yong-ming. The study on PML-FDTD and boundary connecting conditions of total-scattering fields in 3-D cylindrical coordinates[J]. Journal of Chongqing University of Posts and Telecommunications,2003,15(3): 4-8.(in Chinese))
[15]	Teixeira F L, Chew W C. PML-FDTD in cylindrical and spherical grids[J]. Microwave and Guided Wave Letters,IEEE,1997,7(9): 285-287.
[16]	Teixeira F L, Chew W C. Finite-difference computation of transient electromagnetic waves for cylindrical geometries in complex media[J]. Geoscience and Remote Sensing, IEEE Transactions on,2000,38(4): 1530-1543.
[17]	HE Jiang-qi, LIU Qing-huo. A nonuniform cylindrical FDTD algorithm with improved PML and quasi-PML absorbing boundary conditions[J]. Geoscience and Remote Sensing, IEEE Transactions on,1999,37(2): 1066-1072.
[18]	LIU Qing-huo, HE Jiang-qi. An efficient PSTD algorithm for cylindrical coordinates[J].IEEE transactions on Antennas and Propagation,2001,49(9): 1349-1351.
[19]	LIU Qing-huo. Perfectly matched layers for elastic waves in cylindrical and spherical coordinates[J]. The Journal of the Acoustical Society of America,1999,105(4): 2075-2084.
[20]	ZHENG Yi-bing, HUANG Xiao-jun. Anisotropic perfectly matched layers for elastic waves in Cartesian and curvilinear coordinates[R]. Earth Resources Laboratory Industry Consortia Annual Report. Earth Resources Laboratory, Massachusetts Institute of Technology, 2002: 1-18.
[21]	SHI Dong-yang, LIAO Xin, TANG Qi-li. Highly efficient H¹-Galerkin mixed finite element method(MFEM) for parabolic integro-differential equation[J]. Applied Mathematics and Mechanics(English Edition),2014,35(7): 897-912.
[22]	Collino F, Tsogka C. Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media[J]. Geophysics,2001,66(1): 294-307.
[23]	Matzen R. An efficient finite element time-domain formulation for the elastic second-order wave equation: a non-split complex frequency shifted convolutional PML[J].International Journal for Numerical Methods in Engineering,2011,88(10): 951-973.