Reflection characteristics and parameter optimization of a local transmitting boundary for the modeling of elastic waves

PDF(2893 KB)

Journal of Vibration and Shock ›› 2022, Vol. 41 ›› Issue (12) : 301-312.

XING Haojie1, LI Xiaojun2, LIU Aiwen1, LI Hongjing3, ZHOU Zhenghua4, CHEN Su1

Author information +

History +

Abstract

In order to improve the boundary accuracy in the simulation of complex wave problems, the reflection characteristics of a new local transmitting boundary on elastic waves were thoroughly studied. In this case there are two groups (or 4 kinds) of reflection modes namely P-P, P-S and S-P, S-S, respectively. Reflection coefficients are defined on the basis of the potential functions of P and S waves. After transforming the wave field expressions of potential functions into that of u and w components and substituting the results into the boundary formulas, the expressions of reflection coefficients Rpp, Rps, Rsp and Rss were derived. Combining theoretical analyses with numerical experiments, the following results were obtained: Benefitted from the multiple computational wave velocities contained in the new boundary, a good simultaneous absorption of P and S waves with quite different velocities can be achieved; the property of elastic medium itself generates additional zero-reflection angles in reflection coefficients, which is advantageous to the boundary accuracy; sometimes the S-P reflection may suffer from abnormal large amplitude, which is unacceptable in practical simulations, fortunately, this can be avoided when most of (means 1/1, 2/2, 2/3, 3/4, 3/5, …) the artificial wave velocities are strictly set to be the S-wave velocity.

Key words

modeling of elastic waves / a local transmitting boundary / reflection coefficients / simultaneous absorption of P and S waves / additional zero-reflection angles / the S-P large reflection

Cite this article

EndNote

Ris (Procite)

Bibtex

Download Citations

XING Haojie1, LI Xiaojun2, LIU Aiwen1, LI Hongjing3, ZHOU Zhenghua4, CHEN Su1. Reflection characteristics and parameter optimization of a local transmitting boundary for the modeling of elastic waves[J]. Journal of Vibration and Shock, 2022, 41(12): 301-312

References

[1] 廖振鹏.工程波动理论导论(第二版). 北京：科学出版社, 2002.
LIAO Zhenpeng. An Introduction to Wave Motion Theories in Engineering (Second Edition)[M]. Beijing: Science Press, 2002.
[2] 徐世刚, 刘洋. 基于优化有限差分和混合吸收边界条件的三维VTI介质声波和弹性波数值模拟[J]. 地球物理学报, 2018, 61(7): 2950-2968.
XU Shigang, LIU Yang. 3D acoustic and elastic VTI modeling with optimal finite-difference schemes and hybrid absorbing boundary conditions[J]. Chinese Journal of Geophysics, 2018, 61(7): 2950-2968.
[3] LIAO Zhenpeng, WONG Hongliang. A transmitting boundary for the numerical simulation of elastic wave propagation[J]. Soil Dynamics and Earthquake Engineering, 1984, 3(4): 174-183.
[4] LIAO Zhenpeng, WONG Hongliang, YANG Baipo, et al. A transmitting boundary for transient wave analyses[J]. Scientia Sinica (Series A), 1984, XXVII(10): 1063-1076.
[5] 陈少林, 孙杰, 柯小飞. 平面波输入下海水-海床-结构动力相互作用分析[J]. 力学学报, 2020, 52(2): 578-590.
CHEN Shaolin, SUN Jie, KE Xiaofei. Analysis of water-seabed-structure dynamic interaction excited by plane waves[J]. Chinese Journal of Theoreical and Applied Mechanics, 2020, 52(2): 578-590.
[6] 陈少林, 柯小飞, 张洪翔. 海洋地震工程流固耦合问题统一计算框架[J]. 力学学报, 2019, 51(2): 594-606.
CHEN Shaolin, KE Xiaofei, ZHANG Hongxiang. A unified computational framework for fluid-solid coupling in marine earthquake engineering[J]. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(2): 594-606.
[7] SHI Li, WANG Peng, CAI Yuanqiang, et al. Multi-transmitting formula for finite-element modeling of wave propagation in a saturated poroelastic medium[J]. Soil Dynamics and Earthquake Engineering, 2016, 80: 11-24.
[8] 陈少林, 张莉莉, 李山有. 半圆柱形沉积盆地对SH波散射的数值分析[J]. 工程力学, 2014, 31(4): 218-224.
CHEN Shaolin, ZHANG Lili, LI Shanyou. Numerical analysis of the plane SH waves scattering by semi-cylindrical alluvial valley[J]. Engineering Mechanics, 2014, 31(4): 218-224.
[9] HUANG Junjie. An incrementation-adaptive multi-transmitting boundary for seismic fracture analysis of concrete gravity dams[J]. Soil Dynamics and Earthquake Engineering, 2018, 110: 145-158.
[10] 杜修力, 陈厚群, 侯顺载. 拱坝-地基系统的三维非线性地震反应分析[J]. 水利学报, 1997, 8: 7-14.
DU Xiuli, CHEN Houqun, HOU Shunzai. Three dimensional non-linear earthquake response analysis of arch dam-foundation system[J]. Shuili Xuebao, 1997, 8: 7-14.
[11] 李建波, 梅雨润, 林皋等. 基于人工透射边界的核电厂结构抗震分析[J]. 核动力工程, 2016, 37(5): 24-28.
LI Jianbo, MEI Yurun, LIN Gao, et al. Seismic analysis of nuclear power plant structure based on multi-transmission formula boundary model[J]. Nuclear Power Engineering, 2016, 37(5): 24-28.
[12] 李忠诚, 凡红, 李建波. 基于动力人工边界的核电工程地震响应分析[J]. 核动力工程, 2016, 37(3): 47-50.
LI Zhongcheng, FAN Hong, LI Jianbo. Seismic response analysis based on dynamic artificial boundaries for nuclear power engineering[J]. Nuclear Power Engineering, 2016, 37(5): 24-28.
[13] XING Haojie, LI Xiaojun, LI Hongjing, et al. Spectral-element formulation of multi-transmitting formula and its accuracy and stability in 1D and 2D seismic wave modeling[J]. Soil Dynamics and Earthquake Engineering, 2021, 140. https://doi.org/10.1016/j.soildyn.2020.106218
[14] 邢浩洁, 李鸿晶. 透射边界条件在波动谱元模拟中的实现: 二维波动[J]. 力学学报, 2017, 49(4): 894-906.
XING Haojie, LI Hongjing. Implemention of multi-transmitting boundary condition for wave motion simulation by spectral element method: two dimensional case[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(4): 894-906.
[15] 于彦彦, 丁海平, 刘启方. 透射边界与谱元法的结合及对波动模拟精度的改进[J]. 振动与冲击, 2017, 36(2): 13-22.
YU Yanyan, DING Haiping, LIU Qifang. Integration of transmitting boundary and spectral-element method and improvement on the accuracy of wave motion simulation[J]. Journal of Vibration and Shock, 2017, 36(2): 13-22.
[16] 戴志军, 李小军, 侯春林. 谱元法与透射边界的配合使用及其稳定性研究[J]. 工程力学, 2015, 32(11): 40-50.
DAI Zhijun, LI Xiaojun, HOU Chunlin. A combination usage of transmitting formula and spectral element method and the study of its stability[J]. Engineering Mechanics, 2015, 32(11): 40-50.
[17] 廖振鹏, 周正华, 张艳红. 波动数值模拟中透射边界的稳定实现[J]. 地球物理学报, 2002, 45(4): 533-545.
LIAO Zhenpeng, ZHOU Zhenghua, ZHANG Yanhong. Stable implementation of transmitting boundary in numerical simulation of wave motion[J]. Chinese Journal of Geophysics, 2002, 45(4): 533-545.
[18] 景立平. 多次透射公式实用形式稳定性分析[J]. 地震工程与工程振动, 2004, 24(4): 20-24.
JING Liping. Stability analysis of practical formula for multi-transmitting boundary[J]. Earthquake Engineering and Engineering Vibration, 2004, 24(4): 20-24.
[19] 李小军, 杨宇.透射边界稳定性控制措施探讨[J]. 岩土工程学报, 2012, 34(4): 641-645.
LI Xiaojun, YANG Yu. Measures for stability control of transmitting boundary[J]. Chinese Journal of Geotechnical Engineering, 2012, 34(4): 641-645.
[20] 章旭斌, 廖振鹏, 谢志南.透射边界高频耦合失稳机理及稳定实现——SH波动[J]. 地球物理学报, 2015, 58(10): 3639-3648.
ZHANG Xubin, LIAO Zhenpeng, XIE Zhinan. Mechanism of high frequency coupling instability and stable implementation for transmitting boundary —— SH wave motion[J]. Chinese Journal of Geophysics, 2015, 58(10): 3639-3648.
[21] XIE Zhinan, ZHANG Xubin. Analysis of high-frequency local coupling instability induced by multi-transmitting formula – P-SV wave simulation in a 2D waveguide[J]. Earthquake Engineering and Engineering Vibration, 2017, 16(1): 1-10.
[22] LIAO Zhenpeng, YANG Guang. Multi-directional transmitting boundaries for steady-state SH waves[J]. Earthquake Engineering and Structural Dynamics, 1995, 24: 361-371.
[23] 廖振鹏, 李小军. 推广的多次透射边界: 标量波情形[J]. 力学学报, 1995, 27(1): 69-78.
LIAO Zhenpeng, LI Xiaojun. Generalized multi-transmitting boundary: scalar wave case[J]. Acta Mechanica Sinica, 1995, 27(1): 69-78.
[24] XING Haojie, LI Xiaojun, LI Hongjing, et al. The theory and new unified formulas of displacement-type local absorbing boundary conditions[J]. Bulletin of the Seismological Society of America, 2021, xx: 1-23. doi: 10.1785/0120200155
[25] 邢浩洁, 李小军, 刘爱文等. 波动数值模拟中的外推型人工边界条件[J]. 力学学报, 2021, 53(5): 1480-1495.
Xing Haojie，Li Xiaojun，Liu Aiwen,et al. Extrapolation-type artificial boundary conditions in the numerical simulation of wave motion[J]. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(5): 1480-1495.
[26] 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.
[27] Higdon R L. Radiation boundary conditions for elastic wave propagation[J]. SIAM Journal on Numerical Analysis, 1990, 27(4): 831-870.
[28] PENG Chengbin, Toksöz MN. An optimal absorbing boundary condition for finite difference modeling of acoustic and elastic wave propagation[J]. Journal of the Acoustical Society of America, 1994, 95(2): 733-745.