透射边界稳定性控制措施探讨
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摘要
多次透射边界是一种应用广泛的人工边界模拟方法,但类似于其它人工边界,它也存在计算稳定性问题。利用在透射边界区附加黏弹性元件的方案,探讨了在波动数值模拟中消除多次透射边界计算失稳的措施,该措施中弹簧和阻尼元件被附加在透射边界区内的有限元网格点上。数值计算分析表明,该措施是一种处理透射边界计算飘移失稳的有效措施,对透射边界的零频飘移失稳有较好的抑制作用,但该措施对抑制透射边界的高频失稳没有明显效果。
The multi-transmitting boundary is a local artificial boundary widely used in numerical simulation of the near-field wave motion in infinite media.But like other artificial boundaries,the multi-transmitting boundary has the instability of numerical simulation.The measure is studied for eliminating the instability of the transmitting boundary by means of adding visco-elastic elements in the artificial boundary area.In the measure,the spring-damper elements are emplaced at the finite element nodes next to the artificial boundary in the artificial boundary area.Validity of the suggested measure is tested by numerical analysis.The results show that the proposed measure is effective to solve the drift instability of multi-transmitting boundary,which is valid for eliminating the drift instability but no visible effects on the high-frequency instability.
The multi-transmitting boundary is a local artificial boundary widely used in numerical simulation of the near-field wave motion in infinite media.But like other artificial boundaries,the multi-transmitting boundary has the instability of numerical simulation.The measure is studied for eliminating the instability of the transmitting boundary by means of adding visco-elastic elements in the artificial boundary area.In the measure,the spring-damper elements are emplaced at the finite element nodes next to the artificial boundary in the artificial boundary area.Validity of the suggested measure is tested by numerical analysis.The results show that the proposed measure is effective to solve the drift instability of multi-transmitting boundary,which is valid for eliminating the drift instability but no visible effects on the high-frequency instability.
引文
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[3]WOLF P,SONG C.Finite-element modelling of unboundedmedia[C]//John Wiley,Sons,Chichester,1996.
[4]GIVOLI D.Numerical methods for problems in infinitedomain[C]//Elsevier,Amsterdam,1992.
[5]CHENG N,CHENG C H.Relationship between Liao andClayton-Engquist absorbing boundary condition:acousticcase[J].The Bulletin of the Seismological Society of America,1995,85(3):954–956.
[6]邵秀民,蓝志凌.各向异性弹性介质中波动方程的吸收边界条件[J].地球物理学报,1995,38(增刊1):56–72.(SHAO Xiu-min,LAN Zhi-ling.Absorbing boundaryconditions of equations of wave motion in anisotropic elasticmedia[J].Chinese Journal of Geophysics,1995,38(S1):56–72.(in Chinese))
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[9]廖振鹏,刘晶波.波动有限元模拟的基本问题[J].中国科学(B辑),1992,34(8):874–882.(LIAO Zheng-Peng,LIUJing-bo.Fundamental problems in finite element simulationof wave motion[J].Science in China(Series B),1992,34(8):874–882.(in Chinese))
[10]LIAO Zheng-peng,LIU Jing-bo.Numerical instabilities of alocal transmitting boundary[J].Earthquake Engineering andStructural Dynamics,1992,21(1):65–77.
[11]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.(LIAOZheng-peng,ZHOU Zheng-hua,ZHANG Yan-hong.Stableimplementation of transmitting boundary in numericalsimulation of wave motion[J].Chinese Journal of Geophysics,2002,45(4):533–545.(in Chinese))
[12]李小军,唐晖.结构体系动力方程求解的显式积分格式的能耗特征[J].工程力学,2007,24(2):28–33.(LI Xiao-jun,TANG Hui.Numerical dissipation property of an explicitintegration scheme for dynamic equation of structuralsystem[J].Engineering Mechanics,2007,24(2):28–33.(inChinese))
[13]李小军.非线性场地地震反应分析方法研究[D].哈尔滨:国家地震局工程力学研究所,1993.(LI Xiao-jun.Study onthe method for analyzing the earthquake response ofnonlinear site[D].Harbin Institute of Engineering Mechanics,State Seismological Bureau,1993.(in Chinese))
[14]李小军,廖振鹏.时域局部透射边界的计算飘移失稳[J].力学学报,1996,28(5):627–632.(LI Xiao-jun,LIAOZhen-peng.The drift instability of local transmittingboundary in time domain[J].Acta Mechanica Sinica,1996,28(5):627–632.(in Chinese))
[15]WAGNER R L,CHEW W C.An analysis of Liao's absorbingboundary condition[J].Journal of Electromagnetic Waves andApplications,1995,9(7-8):993–1009.
[16]周正华,廖振鹏.消除多次透射公式飘移失稳的措施[J].力学学报,2001,33(4):550–554.(ZHOU Zheng-hua,LIAOZhen-peng.A measure for eliminating drift instability of themulti-transmitting formula[J].Acta Mechanica Sinica,2001,33(4):550–554.(in Chinese))
[17]周正华,廖振鹏.修正算子0 0γB的物理含义解释[J].地震工程与工程振动,2004,24(5):17–19.(ZHOU Zheng-hua,LIAO Zhen-peng.The interpretation of physical implication of modified operator0 0γB[J].Earthquake Engineering and Engineering Vibration,2004,24(5):17–19.(in Chinese))
[18]刘晶波,谷音,杜义欣.一致黏弹性人工边界及黏弹性边界元[J].岩土工程学报,2006,28(9):1070–1075.(LIU Jing-bo,GU Yin,DU Yi-xin.Consistent viscous-spring artificial boundaries and viscous-spring boundary elements[J].Chinese Journal of Geotechnical Engineering,2006,28(9):1070–1075.(in Chinese))
[19]LI Xiao-Jun,LIAO Zhen-peng,GUAN Hui-min.An explicit finite element-finite difference method for analyzing the effect of visco-elastic local topography on the earthquake motion[J].ACTA Seismologica Sinica,1995,8(3):447–456.
[20]李小军,廖振鹏,杜修力.有阻尼体系动力问题的一种显式差分解法[J].地震工程与工程振动,1992,12(4):74–80.(LI Xiao-jun,LIAO Zhen-peng,DU Xiu-li.An explicit finite difference method for viscoelastic dynamic problem[J].Earthquake Engineering and Engineering Vibration,1992,12(4):74–80.(in Chinese))
[2]WOLF P.Soil-structure dynamic interaction analysis in timedomain[C]//Prentice-Hall,Englewood Cliffs.NJ,1988.
[3]WOLF P,SONG C.Finite-element modelling of unboundedmedia[C]//John Wiley,Sons,Chichester,1996.
[4]GIVOLI D.Numerical methods for problems in infinitedomain[C]//Elsevier,Amsterdam,1992.
[5]CHENG N,CHENG C H.Relationship between Liao andClayton-Engquist absorbing boundary condition:acousticcase[J].The Bulletin of the Seismological Society of America,1995,85(3):954–956.
[6]邵秀民,蓝志凌.各向异性弹性介质中波动方程的吸收边界条件[J].地球物理学报,1995,38(增刊1):56–72.(SHAO Xiu-min,LAN Zhi-ling.Absorbing boundaryconditions of equations of wave motion in anisotropic elasticmedia[J].Chinese Journal of Geophysics,1995,38(S1):56–72.(in Chinese))
[7]邵振海,洪伟,周健义.透射边界条件的统一理论[J].中国科学(E),2000,30(1):64–69.(SHAO Zhen-hai,HONGWei,ZHOU Jian-yi.A unified theory of transmittingboundary conditions[J].Science in China(E),2000,30(1):64–69.(in Chinese))
[8]廖振鹏.工程波动理论导论[M].第2版.北京:科学出版社,2002.(LIAO Zheng-Peng.Introduction to wave motiontheory in engineering[M].2nd ed.Beijing:Science Press,2002.(in Chinese))
[9]廖振鹏,刘晶波.波动有限元模拟的基本问题[J].中国科学(B辑),1992,34(8):874–882.(LIAO Zheng-Peng,LIUJing-bo.Fundamental problems in finite element simulationof wave motion[J].Science in China(Series B),1992,34(8):874–882.(in Chinese))
[10]LIAO Zheng-peng,LIU Jing-bo.Numerical instabilities of alocal transmitting boundary[J].Earthquake Engineering andStructural Dynamics,1992,21(1):65–77.
[11]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.(LIAOZheng-peng,ZHOU Zheng-hua,ZHANG Yan-hong.Stableimplementation of transmitting boundary in numericalsimulation of wave motion[J].Chinese Journal of Geophysics,2002,45(4):533–545.(in Chinese))
[12]李小军,唐晖.结构体系动力方程求解的显式积分格式的能耗特征[J].工程力学,2007,24(2):28–33.(LI Xiao-jun,TANG Hui.Numerical dissipation property of an explicitintegration scheme for dynamic equation of structuralsystem[J].Engineering Mechanics,2007,24(2):28–33.(inChinese))
[13]李小军.非线性场地地震反应分析方法研究[D].哈尔滨:国家地震局工程力学研究所,1993.(LI Xiao-jun.Study onthe method for analyzing the earthquake response ofnonlinear site[D].Harbin Institute of Engineering Mechanics,State Seismological Bureau,1993.(in Chinese))
[14]李小军,廖振鹏.时域局部透射边界的计算飘移失稳[J].力学学报,1996,28(5):627–632.(LI Xiao-jun,LIAOZhen-peng.The drift instability of local transmittingboundary in time domain[J].Acta Mechanica Sinica,1996,28(5):627–632.(in Chinese))
[15]WAGNER R L,CHEW W C.An analysis of Liao's absorbingboundary condition[J].Journal of Electromagnetic Waves andApplications,1995,9(7-8):993–1009.
[16]周正华,廖振鹏.消除多次透射公式飘移失稳的措施[J].力学学报,2001,33(4):550–554.(ZHOU Zheng-hua,LIAOZhen-peng.A measure for eliminating drift instability of themulti-transmitting formula[J].Acta Mechanica Sinica,2001,33(4):550–554.(in Chinese))
[17]周正华,廖振鹏.修正算子0 0γB的物理含义解释[J].地震工程与工程振动,2004,24(5):17–19.(ZHOU Zheng-hua,LIAO Zhen-peng.The interpretation of physical implication of modified operator0 0γB[J].Earthquake Engineering and Engineering Vibration,2004,24(5):17–19.(in Chinese))
[18]刘晶波,谷音,杜义欣.一致黏弹性人工边界及黏弹性边界元[J].岩土工程学报,2006,28(9):1070–1075.(LIU Jing-bo,GU Yin,DU Yi-xin.Consistent viscous-spring artificial boundaries and viscous-spring boundary elements[J].Chinese Journal of Geotechnical Engineering,2006,28(9):1070–1075.(in Chinese))
[19]LI Xiao-Jun,LIAO Zhen-peng,GUAN Hui-min.An explicit finite element-finite difference method for analyzing the effect of visco-elastic local topography on the earthquake motion[J].ACTA Seismologica Sinica,1995,8(3):447–456.
[20]李小军,廖振鹏,杜修力.有阻尼体系动力问题的一种显式差分解法[J].地震工程与工程振动,1992,12(4):74–80.(LI Xiao-jun,LIAO Zhen-peng,DU Xiu-li.An explicit finite difference method for viscoelastic dynamic problem[J].Earthquake Engineering and Engineering Vibration,1992,12(4):74–80.(in Chinese))
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