Integration of transmitting boundary and spectral-element method and improvement on the accuracy of wave motion simulation
YU Yanyan1,DING Haiping1,2,LIU Qifang1
1. Institute of Engineering Mechanics,China Earthquake Administration,Harbin 150080,China；
2. Key Laboratory of Structure Engineering of Jiangsu Province,Suzhou University of Science and Technology,Suzhou 215011,China
The implementation of accuracy-matched artificial boundary condition to match the high-order spectral element method (SEM) has great significance for wave motion simulations based on SEM. A method was introduced,integrating the multi-transmitting formula (MTF) with multi-order accuracy and the spectral-element method characterized by nonuniformly distributed nodes. The technique was implemented in the widely used software packages called SPECFEM2D and SPECFEM3D that adopt the explicit Newmark time scheme and parallel computing technique. The responses of 2D and 3D wave source problems were simulated by using the modified SEM packages. By comparing the results with that based on viscous boundary and the extended solution,it is shown that SEM combined with MTF significantly improves the absorbing efficiency of large-angle incidence body waves and surface waves at the boundary,and removes the drifting phenomenon of the viscous boundary. High-order MTF is a proper artificial boundary condition which can match well with the high accuracy high-order SEM.
于彦彦,丁海平1，2，刘启方1. 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 Yanyan1,DING Haiping1,2,LIU Qifang1. Integration of transmitting boundary and spectral-element method and improvement on the accuracy of wave motion simulation. JOURNAL OF VIBRATION AND SHOCK, 2017, 36(2): 13-22.
[1] Virieux J. SH-wave propagation in heterogeneous media: velocity-stress finite-difference method[J]. Geophysics, 1984, 49(11): 1933-1942.
[2] Virieux J. P-SV wave propagation in heterogeneous media: Velocity-stress finite-difference method[J]. Geophysics, 1986, 51(4): 889-901.
[3] Graves R W. Simulating seismic wave propagation in 3D elastic media using staggered-grid finite differences[J]. Bull. Seism. Soc. Am., 1996, 86(4): 1091-1106.
[4] Lee S J, Chen H W, Huang B S. Simulations of strong ground motion and 3D amplification effect in the Taipei basin by using a composite grid finite-difference method[J]. Bull. Seism. Soc. Am., 2008, 98(3): 1229-1242.
[5] Bao H, Bielak J, Ghattas O, et al. Large-scale simulation of elastic wave propagation in heterogeneous media on parallel computers[J]. Comp. Meth. appl. Mech. Eng., 1998, 152(1): 85-102.
[6] Tessmer E, Kosloff D. 3-D elastic modeling with surface topo- graphy by a Chebychev spectral method[J]. Geophysics, 1994, 59(3): 464-473.
[7] Furumura T, Kennett B L N, Furumura M. Seismic wave-field calculation for laterally heterogeneous whole earth models using the pseudo spectral method[J]. Geophys. J. Int., 1998, 135(3): 845-860.
[8] Patera A T. A spectral element method for fluid dynamics: laminar flow in a channel expansion[J]. J. Comput. Phys., 1984, 54(3): 468-488.
[9] Maday Y, Patera A T. Spectral element methods for the incompressible Navier-Stokes equations, in State-of-the-art surveys on computational mechanics[M]. ASME, New York, 1989, 71-143.
[10] Komatitsch D, Vilotte J P. The spectral element method: An efficient tool to simulate the seismic response of 2D and 3D geological structure[J]. Bull. Seism. Soc.Am., 1998, 88(2): 368-392.
[11] Lee S J, Chen H W, Liu Q, et al. Three-dimensional simulations of seismic- wave propagation in the Taipei Basin with realistic topography based upon the spectral-element method[J]. Bull. Seism. Soc. Am., 1998, 88(2): 368-392.
[12] Komatitsch D, Liu Q, Tromp J, et al. Simulations of ground motion in the Los Angeles basin based upon the spectral- element method[J]. Bull. Seism. Soc. Am., 2004, 94(1): 187-206.
[13] 严珍珍，张怀，杨长春，等．汶川大地震地震波传播的谱元法数值模拟研究[J]．中国科学（D 辑）：地球科学，2009, 39(4): 393-402.
YAN Zhen-zhen, ZHANG Huai, YANG Chang-chun, et al. The numerical simulation of seismic wave propagation in Wenchuan great earthquake by the spectral element method[J]. Scientia Sinica Terrae, 2009, 39 (4): 393- 402.
[14] 胡元鑫，刘新荣，罗建华，等.汶川震区地震动三维地形效应的谱元法模拟[J]. 兰州大学学报（自然科学版）, 2011, 47(4): 24-32.
HU Yuan-xin, LIU Xing-rong, LUO Jian-hua, et al. Simulation of three- dimensional topographic effects on seismic ground motion in Wenchuan earthquake region based upon the spectral-element method[J]. Journal of Lanzhou University (Natural Sciences), 2011, 47(4): 24-32.
[15] 刘启方，于彦彦，章旭斌. 施甸盆地三维地震动研究[J]. 地震工程与工程振动，2013, 33(4): 54-60.
LIU Qi-fang, YU Yan-yan, ZHANG Xv-bin. Three-dimensional ground motion simulation for Shidian Basin[J]. Journal of Earthquake Engineering and Engineering Vibration, 2013, 33(4): 54-60.
[16] Komatitsch D, Tromp J. Introduction to the spectral element method for three-dimensional seismic wave propagation[J]. Geophys. J. Int., 1999, 139: 806-822.
[17] Xie Z N, Komatitsch D, Martin R, et al. Improved forward wave propag- ation and adjoint-based sensitivity kernel calculations using a numerically stable finite-element PML[J]. Geophys. J. Int., 2014, 198(3): 1714-1747.
[18] Komatitsch D, Tromp J. A perfectly matched layer absorbing boundary condition for the second-order seismic wave equation[J]. Geophys. J. Int., 2003, 154:146–153.
[19] Liao Z P, Wong H L. A transmitting boundary for the numerical simulation of elastic wave propagation[J]. Soil Dyn. Earthq. Eng., 1984, 3:174-183.
[20] Liao Z P, Wong H L, Yang B P, et al. A transmitting boundary for transient wave analyses[J]. Scientia Sinica (Series A), 1984, 27(10): 1063 -1076.
[21] 廖振鹏, 黄孔亮, 杨柏坡等. 暂态波透射边界[J]. 中国科学(A辑), 1984, 26(6):50-56 .
LIAO Zhen-peng, WONG Hong-liang, YANG Bai-po, et al. A transmitting boundary condition for transient wave[J]. Scientia Sinica Mathematica, 1984, 26(6): 50-56 .
[22] Wolf J P. A comparison of time-domain transmitting boundaries[J]. Earthquake Engng. Struct. Dyn., 1986, 14(4): 655-673.
[23] Kausel E. Local transmitting boundaries[J]. J. Eng. Mech., 1988, 114(6): 1011-1027.
[24] Chew W C, Wagner R L. A modified form of Liao's absorbing boundary condition[C]. IEEE Antennas and Propagation Society International Symposium, Chicago, IL, 1992, 536-539.
[25] Cheng N, Cheng C H. Relationship between Liao and Clayton-Engquist absorbing boundary conditions: Acoustic case[J]. Bull. Seism. Soc. Am., 1995, 85(3): 954-956.
[26] Wagner R L, Chew W C. An analysis of Liao's absorbing boundary condition[J]. J. Electromagn. Waves Appl., 1995, 9: 993- 1009.
[27] Yeo T S, Kooi P S, Leong M S, et al. A performance assessment of Liao's absorbing boundary conditions for FDTD method[J]. Microwave and Optical Technology Letters, 1997, 16(3): 186-197.
[28] Zhang L, Yu T. A method of improving the stability of Liao's higher-order absorbing boundary condition[J]. Progress In Electromagnetics Research M, 2012, 27: 167-178.
[29] 陈少林，廖振鹏.多次透射公式在衰减波场中的实现[J]. 地震学报, 2003, 25(3):272-279.
Chen S L, Liao Z P. Multi-transmitting formula for attenuation waves[J]. Acta Seismologica Sinica, 2003, 25(3):272-279.
[30] 丁海平，宋贞霞. 震源参数对地震动相干函数的影响[J]. 振动与冲击，2010, 29(7): 16-19.
DING Hai-ping, SONG Zhen-xia. Effect of source parameter on seismic coherency function[J]. Journal of Vibration and Shock, 2010, 29(7): 16- 19.
[31] 刘启方，金星，丁海平. 复杂场地条件下震源参数对断层附近长周期地震动的影响[J]. 地球物理学报，2008, 51(1): 186-196.
LIU Qi-fang, JIN Xing, DING Hai-ping. Effects of the source parameters on long period near-fault ground motion in the case of complex site condition[J]. Chinese J. Geophys., 2008, 51(1): 186-196.
[32] Hughes T J R. The finite-element method, linear static and dynamic finite element analysis[M]. Prentice-Hall International, 1987.
[33] 廖振鹏.工程波动理论导论[M]. 北京：科学出版社, 2002.
LIAO Zhen-peng. Introduction to wave motion theories for engineering[M]. Beijing: Science Press, 2002.
