透射边界高频失稳机理及稳定实现——P-SV波动

章旭斌, 廖振鹏, 谢志南. 2021. 透射边界高频失稳机理及稳定实现——P-SV波动. 地球物理学报, 64(10): 3646-3656, doi: 10.6038/cjg2021O0420
引用本文: 章旭斌, 廖振鹏, 谢志南. 2021. 透射边界高频失稳机理及稳定实现——P-SV波动. 地球物理学报, 64(10): 3646-3656, doi: 10.6038/cjg2021O0420
ZHANG XuBin, LIAO ZhenPeng, XIE ZhiNan. 2021. Mechanism of high frequency instability and stable implementation for transmitting boundary—P-SV wave motion. Chinese Journal of Geophysics (in Chinese), 64(10): 3646-3656, doi: 10.6038/cjg2021O0420
Citation: ZHANG XuBin, LIAO ZhenPeng, XIE ZhiNan. 2021. Mechanism of high frequency instability and stable implementation for transmitting boundary—P-SV wave motion. Chinese Journal of Geophysics (in Chinese), 64(10): 3646-3656, doi: 10.6038/cjg2021O0420

透射边界高频失稳机理及稳定实现——P-SV波动

  • 基金项目:

    中央级公益性科研院所基本科研业务费专项(2017B08),国家自然科学基金(51808516,U2039209),黑龙江省"头雁"创新团队计划资助

详细信息
    作者简介:

    章旭斌, 男, 1986年生, 博士, 助理研究员, 主要研究方向为地震波动数值模拟.E-mail: zhangxvbin.hebi@163.com

  • 中图分类号: P315

Mechanism of high frequency instability and stable implementation for transmitting boundary—P-SV wave motion

  • 波动数值模拟的稳定性是获得可靠结果的前提.透射边界是一类具有高阶、高效等特点的人工边界,其引发的高频失稳是由内域格式和透射边界的不当耦合所致.本文针对P-SV波动有限元模拟中透射边界引发的失稳问题,基于GKS定理的群速度解释,通过对有限元和透射边界的频散分析揭示了数值失稳机理为透射边界和相邻内域格式支持了群速度指向内域的高频P波或SV波,波动能量将从边界进入内域引发数值失稳.同时,对比连续模型频散指出引发失稳的谐波是由有限元离散引入.本文采用修改的数值积分方法调整有限元刚度,以消除有限元中引发边界失稳的高频波动成分,从而稳定实现透射边界.理论分析和数值实验均表明本文稳定措施的有效性.

  • 加载中
  • 图 1 

    长方形单元和局部节点系

    Figure 1. 

    Rectangle element and local grid system

    图 2 

    正则模态示意图

    Figure 2. 

    Sketch of normal model

    图 3 

    连续模型和传统有限元的频散曲线,及在垂直x方向人工边界上的MTF频散曲线,其中参数ε=, β=1, Δτ=0.4

    Figure 3. 

    The dispersion curves of continuous model, classic FEM and MTF scheme when artificial boundary is perpendicular to x-axis with these parameters, ε=, β=1, Δτ=0.4

    图 4 

    修正有限元和在垂直x方向人工边界上的MTF频散曲线,其中参数ε=, β=1, Δτ=0.5

    Figure 4. 

    The dispersion curves of modified FEM and MTF scheme when artificial boundary is perpendicular to x-axis with these parameters, ε=, β=1, Δτ=0.5

    图 5 

    基岩半空间模型

    Figure 5. 

    Rock half-space model

    图 6 

    传统有限元结合MTF计算的接收点位移时程

    Figure 6. 

    The displacement time history of the receiver computed by classic FEM with MTF

    图 7 

    修正有限元结合MTF计算的接收点位移时程

    Figure 7. 

    The displacement time history of the receiver computed by modified FEM with MTF

    图 8 

    (a) 计算模型参数条件下,传统有限元和MTF的频散曲线,红线为一条水平频散曲线,红色方框为引发MTF失稳的谐波频率范围;(b) 截取的接收点失稳增长数值解;(c) 失稳增长数值解的频谱

    Figure 8. 

    (a) The dispersion curves of classic FEM and MTF at the given parameters of computational model, the red line is horizontal dispersion curve, and the red box indicates the harmonic frequency range that causes MTF instability; (b) Truncated numerical instability solution of receiver; (c) Spectrum of numerical instability solution

    图 9 

    盆地-基岩半空间模型

    Figure 9. 

    Basin-rock half-space model

    图 10 

    修正有限元结合MTF和黏弹性边界以及远置边界时计算的接收点位移时程

    Figure 10. 

    The displacement time history of the receiver computed by modified FEM with MTF, viscous-spring boundary and extend boundary

  •  

    Bérenger J P. 2007. Perfectly Matched Layer (PML) for Computational Electromagnetics. Arcueil: Morgan & Claypool. http://science.ebookscanner.com/pdfbook/pe/perfectly-matched-layer-pml-for-computational-electromagnetics-free-download-online.pdf

     

    Bowden D C, Tsai V C. 2017. Earthquake ground motion amplification for surface waves. Geophysical Research Letters, 44(1): 121-127. doi: 10.1002/2016GL071885

     

    Chen H Q. 2006. Discussion on seismic input mechanism at dam site. Journal of Hydraulic Engineering (in Chinese), 37(12): 1417-1423. http://www.cnki.com.cn/Article/CJFDTotal-SLXB200612004.htm

     

    Duru K, Kozdon J E, Kreiss G. 2015. Boundary conditions and stability of a perfectly matched layer for the elastic wave equation in first order form. Journal of Computational Physics, 303: 372-395. doi: 10.1016/j.jcp.2015.09.048

     

    Gustafsson B, Kreiss H O, Sundström A. 1972. Stability theory of difference approximations for mixed initial boundary value problems. Ⅱ. Mathematics of Computation, 26(119): 649-686. doi: 10.1090/S0025-5718-1972-0341888-3

     

    Hagstrom T, Mar-Or A, Givoli D. 2008. High-order local absorbing conditions for the wave equation: Extensions and improvements. Journal of Computational Physics, 227(6): 3322-3357. doi: 10.1016/j.jcp.2007.11.040

     

    Higdon R L. 1987. Numerical absorbing boundary conditions for the wave equation. Mathematics of Computation, 49(179): 65-90. doi: 10.1090/S0025-5718-1987-0890254-1

     

    Jing L P, Liao Z P, Zou J X. 2002. A high-frequency instability mechanism in numerical realization of multi-transmitting formula. Earthquake Engineering and Engineering Vibration (in Chinese), 22(1): 7-13. http://en.cnki.com.cn/Article_en/ http://search.cnki.net/down/default.aspx?filename=DGGC200201002&dbcode=CJFD&year=2002&dflag=pdfdown

     

    Li D Z, Helmberger D, Clayton R W, et al. 2014. Global synthetic seismograms using a 2-D finite-difference method. Geophysical Journal International, 197(2): 1166-1183. doi: 10.1093/gji/ggu050

     

    Li X J, Yang Y. 2012. Measures for stability control of transmitting boundary. Chinese Journal of Geotechnical Engineering (in Chinese), 34(4): 641-645. http://www.cqvip.com/QK/95758X/20124/41500971.html

     

    Liao Z P, Liu J B. 1992. Numerical instabilities of a local transmitting boundary. Earthquake Engineering and Structural Dynamics, 21(1): 65-77. doi: 10.1002/eqe.4290210105

     

    Liao Z P. 2002. Introduction to Wave Motion Theories in Engineering (in Chinese). 2nd ed. Beijing: Science Press.

     

    Liao Z P, Zhou Z H, Zhang Y H. 2002. Stable implementation of transmitting boundary in numerical simulation of wave motion. Chinese Journal of Geophysics (in Chinese), 45(4): 533-545, doi:10.3321/j.issn:0001-5733.2002.04.011.

     

    Liu Y S, Xu T, Wang Y H, et al. 2019. An efficient source wavefield reconstruction scheme using single boundary layer values for the spectral element Method. Earth and Planetary Physics, 3(4): 342-357, doi:10.26464/epp2019035.

     

    Liu J B, Li B. 2005. A unified viscous-spring artificial boundary for 3-D static and dynamic applications. Science in China Series E Engineering & Materials Science, 48(5): 570-584. http://www.cnki.com.cn/Article/CJFDTotal-JEXK200509008.htm

     

    Liu Y, Sen M K. 2012. A hybrid absorbing boundary condition for elastic staggered-grid modelling. Geophysical Prospecting, 60(6): 1114-1132. doi: 10.1111/j.1365-2478.2011.01051.x

     

    Semblat J F, Lenti L, Gandomzadeh A. 2011. A simple multi-directional absorbing layer method to simulate elastic wave propagation in unbounded domains. International Journal for Numerical Methods in Engineering, 85(12): 1543-1563. doi: 10.1002/nme.3035

     

    Strikwerda J C. 2004. Finite Difference Schemes and Partial Differential Equations. 2nd ed. Philadelphia: SIAM.

     

    Taflove A, Hagness S C. 2005. Computational Electrodynamics: The Finite-Difference Time-Domain Method. 3rd ed. Norwood, MA: Artech House, Inc.

     

    Trefethen L N. 1983. Group velocity interpretation of the stability theory of Gustafsson, Kreiss, and Sundström. Journal of Computational Physics, 49(2): 199-217. doi: 10.1016/0021-9991(83)90123-7

     

    Xie Z N, Liao Z P. 2012. Mechanism of high frequency instability caused by transmitting boundary and method of its elimination——SH wave. Chinese Journal of Theoretical and Applied Mechanics (in Chinese), 44(4): 745-752. http://d.wanfangdata.com.cn/Periodical/lxxb201204011

     

    Xu G, Hamouda A M S, Khoo B C. 2016. Time-domain simulation of second-order irregular wave diffraction based on a hybrid water wave radiation condition. Applied Mathematical Modelling, 40(7-8): 4451-4467. doi: 10.1016/j.apm.2015.11.034

     

    Yue B, Guddati M N. 2005. Dispersion-reducing finite elements for transient acoustics. The Journal of the Acoustical Society of America, 118(4): 2132-2141. doi: 10.1121/1.2011149

     

    Zhang X B, Liao Z P, Xie Z N. 2015. Mechanism of high frequency coupling instability and stable implementation for transmitting boundary-SH wave motion. Chinese Journal of Geophysics (in Chinese), 58(10): 197-206, doi: 10.6038/cjg20151017.

     

    Zhang X B, Xie Z N, Liao Z P. 2019. Reflection amplification instability of the transmitting boundary in SH wave simulation. Journal of Harbin Engineering University (in Chinese), 40(6): 1031-1035.

     

    Zhao M, Du X L, Liu J B, et al. 2011. Explicit finite element artificial boundary scheme for transient scalar waves in two-dimensional unbounded waveguide. International Journal for Numerical Methods in Engineering, 87(11): 1074-1104. doi: 10.1002/nme.3147

     

    Zhou Z H, Liao Z P. 2001. A measure for eliminating drift instability of the multi-transmitting formula. Acta Mechanica Sinica (in Chinese), 33(4): 550-554. http://www.cqvip.com/qk/91029x/2001004/6008125.html

     

    陈厚群. 2006. 坝址地震动输入机制探讨. 水利学报, 37(12): 1417-1423. doi: 10.3321/j.issn:0559-9350.2006.12.004

     

    景立平, 廖振鹏, 邹经湘. 2002. 多次透射公式的一种高频失稳机制. 地震工程与工程振动, 22(1): 7-13. doi: 10.3969/j.issn.1000-1301.2002.01.002

     

    李小军, 杨宇. 2012. 透射边界稳定性控制措施探讨. 岩土工程学报, 34(4): 641-645. https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC201204010.htm

     

    廖振鹏. 2002. 工程波动理论导论. 2版. 北京: 科学出版社.

     

    廖振鹏, 周正华, 张艳红. 2002. 波动数值模拟中透射边界的稳定实现. 地球物理学报, 45(4): 533-545, doi:10.3321/j.issn:0001-5733.2002.04.011. http://www.geophy.cn//CN/abstract/abstract3509.shtml

     

    谢志南, 廖振鹏. 2012. 透射边界高频失稳机理及其消除方法——SH波动. 力学学报, 44(4): 745-752. doi: 10.6052/0459-1879-11-312

     

    章旭斌, 廖振鹏, 谢志南. 2015. 透射边界高频耦合失稳机理及稳定实现——SH波动. 地球物理学报, 58(10): 197-206, doi:10.6038/cjg20151017. http://www.geophy.cn//CN/abstract/abstract11850.shtml

     

    章旭斌, 谢志南, 廖振鹏. 2019. SH波动模拟中透射边界反射放大失稳研究. 哈尔滨工程大学学报, 40(6): 1031-1035. https://www.cnki.com.cn/Article/CJFDTOTAL-HEBG201906001.htm

     

    周正华, 廖振鹏. 2001. 消除多次透射公式飘移失稳的措施. 力学学报, 33(4): 550-554. doi: 10.3321/j.issn:0459-1879.2001.04.015

  • 加载中

(10)

计量
  • 文章访问数:  343
  • PDF下载数:  72
  • 施引文献:  0
出版历程
收稿日期:  2020-11-03
修回日期:  2021-06-28
上线日期:  2021-10-10

目录