粘弹性人工边界及其与透射人工边界的比较研究
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摘要
摘 要
在高层、高坝和核电站等重大复杂结构的动力反应分析中，波动能量向无限
域地基的逸散影响是重要的。由于这类问题的求解规模巨大，且可能包含多种介
质耦合、介质材料非线性及界面动接触非线性等因素影响，需要发展高效的数值
分析方法才能用于这些实际问题的解决。显式有限元结合局部人工边界的方法由
于其时空解耦的特点，很适宜于上述复杂的近场波动问题的求解，因而获得学术
界和工程界的重视。文中针对两种局部人工边界，粘弹性边界和透射边界，进行
了研究，获得了一些有参考意义的认识。
提出了一种改进的粘弹性人工边界，它与现有的粘弹性边界相比具有以下优
点：（1）物理模型简单，不含附加质量；（2）表达式与全部介质参数相关，物理
意义更加合理；（3）模拟精度高。
透射边界稳定性的算例分析表明：现有透射人工边界的稳定实施方法是经验
性的，缺乏可供参考的客观的参数选取依据，不便于工程人员操作；在某些情况
下会对计算精度产生不利影响。
精度比较分析表明：本文提出的广义粘弹性边界具有与二阶透射人工边界相
当的精度。有限元模型影响分析表明：（1）在满足波动有限元分析要求的前提下，
采用不同时空离散步距的有限元模型分析同一问题时，粘弹性边界结果稳定，透
射边界结果受不同模型影响较明显。（2）外源波动问题分析较相应的内源波动问
题需要更大的人工边界尺寸。
半球谷局部地形对入射地震动的影响问题和流-固介质分层耦合问题的分析
表明，显式有限元结合粘弹性人工边界的解耦时域波动分析方法是有效的近场波
动数值模拟方法。
For the dynamic analysis of some important complex structures such as high-rise,
high dam, nuclear power station and so on, the wave propagation from structure into
infinite foundation should be considered. The problem is huge for solution, and may
include mediums coupling, medium material nonlinearity and dynamic contact
interfaces etc. Therefore, an effective numerical method must be provided for the
dynamic analysis of this large complex system. The combination method of the
explicit finite element method and the local artificial boundary is applicable to solute
the above complicated problem because of its uncoupling characteristic. In the
dissertation, the studies on the viscous-spring boundary and the transmitting boundary
are carried out and some useful conclusions are achieved.
In the dissertation, a kind of new calculation formulation of viscous-spring
boundary is proposed. Comparing with the existing viscous-spring boundaries, it has
advantage as flowing: (1) the boundary model is simple and doesn’t include the added
mass. (2) The boundary expressions include all medium material constant. (3) The
method’s accuracy is high.
The numerical analysis of stability for transmitting boundary indicates that the
existing methods of control instability are experiential, be short of the objective basis
of parameter choosen and affect the accuracy in some conditions.
The analysis of accuracy indicates that the accuracy of viscous-spring boundary
in the dissertation correspond to that of second-order transmitting boundary. The
analysis of effect of finite element model indicate: (1) when the different finite
element models are used to analyze the same problem on the premise that the
accuracy of wave finite element is satisfied, the result of viscous-spring boundary for
different model is stabler than that of the transmitting boundary. (2) The analysis on
wave problem with load outside finite field needs bigger boundary size than that with
load inside finite field.
The analysis of the local landform problem of hemisphere canyon and the
coupling problem of layered medium of fluid and solid indicate that the combination
II
Abstract
method of the explicit finite element method and the viscous-spring boundary is
effective.
在高层、高坝和核电站等重大复杂结构的动力反应分析中，波动能量向无限
域地基的逸散影响是重要的。由于这类问题的求解规模巨大，且可能包含多种介
质耦合、介质材料非线性及界面动接触非线性等因素影响，需要发展高效的数值
分析方法才能用于这些实际问题的解决。显式有限元结合局部人工边界的方法由
于其时空解耦的特点，很适宜于上述复杂的近场波动问题的求解，因而获得学术
界和工程界的重视。文中针对两种局部人工边界，粘弹性边界和透射边界，进行
了研究，获得了一些有参考意义的认识。
提出了一种改进的粘弹性人工边界，它与现有的粘弹性边界相比具有以下优
点：（1）物理模型简单，不含附加质量；（2）表达式与全部介质参数相关，物理
意义更加合理；（3）模拟精度高。
透射边界稳定性的算例分析表明：现有透射人工边界的稳定实施方法是经验
性的，缺乏可供参考的客观的参数选取依据，不便于工程人员操作；在某些情况
下会对计算精度产生不利影响。
精度比较分析表明：本文提出的广义粘弹性边界具有与二阶透射人工边界相
当的精度。有限元模型影响分析表明：（1）在满足波动有限元分析要求的前提下，
采用不同时空离散步距的有限元模型分析同一问题时，粘弹性边界结果稳定，透
射边界结果受不同模型影响较明显。（2）外源波动问题分析较相应的内源波动问
题需要更大的人工边界尺寸。
半球谷局部地形对入射地震动的影响问题和流-固介质分层耦合问题的分析
表明，显式有限元结合粘弹性人工边界的解耦时域波动分析方法是有效的近场波
动数值模拟方法。
For the dynamic analysis of some important complex structures such as high-rise,
high dam, nuclear power station and so on, the wave propagation from structure into
infinite foundation should be considered. The problem is huge for solution, and may
include mediums coupling, medium material nonlinearity and dynamic contact
interfaces etc. Therefore, an effective numerical method must be provided for the
dynamic analysis of this large complex system. The combination method of the
explicit finite element method and the local artificial boundary is applicable to solute
the above complicated problem because of its uncoupling characteristic. In the
dissertation, the studies on the viscous-spring boundary and the transmitting boundary
are carried out and some useful conclusions are achieved.
In the dissertation, a kind of new calculation formulation of viscous-spring
boundary is proposed. Comparing with the existing viscous-spring boundaries, it has
advantage as flowing: (1) the boundary model is simple and doesn’t include the added
mass. (2) The boundary expressions include all medium material constant. (3) The
method’s accuracy is high.
The numerical analysis of stability for transmitting boundary indicates that the
existing methods of control instability are experiential, be short of the objective basis
of parameter choosen and affect the accuracy in some conditions.
The analysis of accuracy indicates that the accuracy of viscous-spring boundary
in the dissertation correspond to that of second-order transmitting boundary. The
analysis of effect of finite element model indicate: (1) when the different finite
element models are used to analyze the same problem on the premise that the
accuracy of wave finite element is satisfied, the result of viscous-spring boundary for
different model is stabler than that of the transmitting boundary. (2) The analysis on
wave problem with load outside finite field needs bigger boundary size than that with
load inside finite field.
The analysis of the local landform problem of hemisphere canyon and the
coupling problem of layered medium of fluid and solid indicate that the combination
II
Abstract
method of the explicit finite element method and the viscous-spring boundary is
effective.
引文
参考文献：
[1] 廖振鹏. 工程波动理论导论. 科学出版社, 2002
[2] Z. S. Alterman and F. C. J. Karal. Propagation of Elastic Waves in Layered Media
by Finite-Difference Methods. Bulletin of the Seismological Society of America.
1968, 58: 367-398
[3] P. J. Wolf. A Comparison of Time-domain Transmitting Boundary. Earthquake
Engineering and Structural Dynamics. 1986, 14: 655-673
[4] E. Kausel. Local Transmitting Boundaries. Journal of Engineering Mechanics.
1988, 114(6): 1011-1027
[5] A. Sommerfeld. Lectures on Theoretical Physics. In the Series Partial Differential
Equations in Physics, Academic Press, New York. 1964, 6(28)
[6] J. Lysmer and R. L. Kulemeyer. Finite Dynamic Model for Infinite Media. Journal
of Engineering Mechanics Division, ASCE. 1969, 95(4): 759-877
[7] W. D. Smith. A Non-reflecting Plane Boundary for Wave Propagation Problems.
Journal of Computational Physics. 1974, 15: 492-503
[8] R. R. Kunar and J. Marti. A Non-reflecting Boundary for Explicit Calculations.
Computational Methods for Infinite Domain Media-Structure Interaction, ASME.
AMD46, 1981, 183-204
[9] R. Clayton and B. Engquist. Absorbing Boundary Condition for Acoustic and
Elastic Wave Equations. Bulletin of the Seismological Society of America. 1977,
67(6): 1529-1540
[10]B. Engquist and A. Maja. Absorbing Boundary Condition for the Simulation of
Waves. Mathematics of Computation. 1977, 31(139): 629-651
[11]R. Clayton and B. Engquist. Absorbing Boundary Conditions for Wave-Equation
Migration. Geophysics. 1980, 45(5): 895-904
[12]R. L. Higdon. Absorbing Boundary Conditions for Difference Approximations to
The Multi-Dimensional Wave Equation. Mathematics of Computation. 1986,
47(176): 437-459
[13]R. L. Higdon. Numerical Absorbing Boundary Conditions for the Wave Equation.
Mathematics of Computation. 1987, 49(179): 65-90
95
北京工业大学工学硕士学位论文
[14]R. L. Higdon. Absorbing Boundary Conditions for Acoustic and Elastic Waves in
Stratified Media. Journal of Computational Physics. 1992, 101: 386-418
[15]John P. Wolf and Chongming Song. Unit-impulse Response Matrix of Unbounded
Medium by Infinitesimal Finite-element Cell Method. Computer Methods in
Applied Mechanics and Engineering. 1995, 122: 251-272
[16]A. J. Deeks and M. F. Randolph. Axisymmetric Time-Domain Transmitting
Boundaries. Journal of Engineering Mechanics. 1994, 120(1): 25-42
[17]刘晶波, 吕彦东. 结构-地基动力相互作用问题分析的一种直接方法. 土木工
程学报. 1998, 31(3): 55-64
[18]杜修力. 局部解耦的时域波分析方法. 世界地震工程. 2000, 16(3): 22-26
[19]Liao Zhenpeng, Yang Baipo and Yuan Yifan. Feedback Effects of Low-rise
Building on Vertecal Earthquake Ground Montion and Application of Transmitting
Boundaries for Transient Wave Analysis. Institute of Engineering Mechanics,
Academia Sinica, Research Report. 1978, 062
[20]廖振鹏, 杨柏坡, 袁一凡. 暂态弹性波分析中人工边界的研究. 地震工程与
工程振动. 1982, 2(1): 1-11
[21]廖振鹏, 黄孔亮, 杨柏坡, 袁一凡. 暂态波透射边界. 中国科学, A 辑. 1984,
27(6): 556-564
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1997
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17(5): 37-43
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109-112
[25]李小军, 廖振鹏, 杜修力. 有阻尼体系动力问题的一种显式解法. 地震工程
与工程振动. 1992, 13(4): 74-79
[26]周正华, 李山有和侯兴民. 阻尼振动方程的一种显式直接积分方法. 世界地
震工程. 1999, 15(1): 41-44
[27]张晓志, 程岩, 谢礼立. 结构动力反应分析的三阶显式方法. 地震工程与工
96
参 考 文 献
程振动. 2002, 22(3): 1-8
[28]陈学良, 袁一凡. 求解振动方程的一种显式积分格式及其精度与稳定性. 地
震工程与工程振动. 2002, 22(3): 9-14
[29]Chen Shaolin and Liao Zhenpeng. Multi-transmitting Formula for Attenuating
Waves. Acta Seismologica Sinica. 2003, 16(3): 283-291
[30]廖振鹏. 无限弹性介质中暂态标量波问题的有限元模型. 地震工程与工程振
动. 1982, 2(4): 38-52
[31]关慧敏, 廖振鹏. 局部透射边界和叠加边界的精度分析与比较. 力学学报.
1994, 26(3): 303-311
[32]艾龙根, 舒胡毕. 弹性动力学, 第二卷 线性理论. 石油工业出版社. 1984
[33]关慧敏, 廖振鹏. 局部人工边界稳定性的一种分析方法. 力学学报. 1996,
28(3): 376-380
[34]关慧敏, 廖振鹏. 一种改善多次透射边界稳定性的措施. 地震工程与工程振
动. 1997, 17(4): 1-8
[35]景立平, 廖振鹏, 邹经相. 多次透射公式一种高频失稳机制. 地震工程与工
程振动. 2002, 22(1): 7-13
[36]李小军, 廖振鹏. 时域局部透射边界的计算飘移失稳. 力学学报. 1996, 28(5):
627-632
[37]周正华, 廖振鹏. 消除多次透射公式飘移失稳的措施. 力学学报. 2001, 33(4):
550-554
[38]M. Novak, T. Nogami and F. Aboul-Ella. Dynamic Soil Reactions for Plane Strain
Case. Journal of the Engineering Mechanics Division, ASCE. 1978, 104(4):
953-959
[39]M. Novak. Vertical Vibration of Floating Piles. Journal of the Engineering
Mechanics Division, ASCE. 1977, 103(1): 153-168
[40]M. Novak and H. Mitwally. Transmitting Boundary for Axisymmetrical Dilation
Problems. Journal of the Engineering Mechanics Division, ASCE. 1988, 114(1):
181-187
[41]王振宇. 大型结构-地基系统动力反应计算理论及其应用研究. 清华大学工学
97
北京工业大学工学硕士学位论文
博士学位论文. 2002
[42]刘晶波, 王振宇, 张克峰, 裴欲晓. 考虑土-结构相互作用大型动力机器基础
三维有限元分析. 工程力学. 2002, 19(3): 34-49
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水利水电科学研究院工学博士学位论文. 2001
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subjected to incident plane waves. Report No. 78-16, U.S.C. Department of Civil
Engineering. 1978
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hemispherical alluvial valley. Soil Dynamics and Earthquake Engineering. 1984,
3(3): 133-144
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cylindrical canyons with variable depth-to-width ratio. Soil Dynamics and
Earthquake Engineering. 1990, 9(3): 141-150
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程与工程振动. 1993, 13(2): 1-11
[50]潘忠诚. 数学物理方法教程. 南开大学出版社. 1993
[51]王进廷. 多层多相介质动力分析研究. 清华大学博士后研究报告. 2003
[1] 廖振鹏. 工程波动理论导论. 科学出版社, 2002
[2] Z. S. Alterman and F. C. J. Karal. Propagation of Elastic Waves in Layered Media
by Finite-Difference Methods. Bulletin of the Seismological Society of America.
1968, 58: 367-398
[3] P. J. Wolf. A Comparison of Time-domain Transmitting Boundary. Earthquake
Engineering and Structural Dynamics. 1986, 14: 655-673
[4] E. Kausel. Local Transmitting Boundaries. Journal of Engineering Mechanics.
1988, 114(6): 1011-1027
[5] A. Sommerfeld. Lectures on Theoretical Physics. In the Series Partial Differential
Equations in Physics, Academic Press, New York. 1964, 6(28)
[6] J. Lysmer and R. L. Kulemeyer. Finite Dynamic Model for Infinite Media. Journal
of Engineering Mechanics Division, ASCE. 1969, 95(4): 759-877
[7] W. D. Smith. A Non-reflecting Plane Boundary for Wave Propagation Problems.
Journal of Computational Physics. 1974, 15: 492-503
[8] R. R. Kunar and J. Marti. A Non-reflecting Boundary for Explicit Calculations.
Computational Methods for Infinite Domain Media-Structure Interaction, ASME.
AMD46, 1981, 183-204
[9] R. Clayton and B. Engquist. Absorbing Boundary Condition for Acoustic and
Elastic Wave Equations. Bulletin of the Seismological Society of America. 1977,
67(6): 1529-1540
[10]B. Engquist and A. Maja. Absorbing Boundary Condition for the Simulation of
Waves. Mathematics of Computation. 1977, 31(139): 629-651
[11]R. Clayton and B. Engquist. Absorbing Boundary Conditions for Wave-Equation
Migration. Geophysics. 1980, 45(5): 895-904
[12]R. L. Higdon. Absorbing Boundary Conditions for Difference Approximations to
The Multi-Dimensional Wave Equation. Mathematics of Computation. 1986,
47(176): 437-459
[13]R. L. Higdon. Numerical Absorbing Boundary Conditions for the Wave Equation.
Mathematics of Computation. 1987, 49(179): 65-90
95
北京工业大学工学硕士学位论文
[14]R. L. Higdon. Absorbing Boundary Conditions for Acoustic and Elastic Waves in
Stratified Media. Journal of Computational Physics. 1992, 101: 386-418
[15]John P. Wolf and Chongming Song. Unit-impulse Response Matrix of Unbounded
Medium by Infinitesimal Finite-element Cell Method. Computer Methods in
Applied Mechanics and Engineering. 1995, 122: 251-272
[16]A. J. Deeks and M. F. Randolph. Axisymmetric Time-Domain Transmitting
Boundaries. Journal of Engineering Mechanics. 1994, 120(1): 25-42
[17]刘晶波, 吕彦东. 结构-地基动力相互作用问题分析的一种直接方法. 土木工
程学报. 1998, 31(3): 55-64
[18]杜修力. 局部解耦的时域波分析方法. 世界地震工程. 2000, 16(3): 22-26
[19]Liao Zhenpeng, Yang Baipo and Yuan Yifan. Feedback Effects of Low-rise
Building on Vertecal Earthquake Ground Montion and Application of Transmitting
Boundaries for Transient Wave Analysis. Institute of Engineering Mechanics,
Academia Sinica, Research Report. 1978, 062
[20]廖振鹏, 杨柏坡, 袁一凡. 暂态弹性波分析中人工边界的研究. 地震工程与
工程振动. 1982, 2(1): 1-11
[21]廖振鹏, 黄孔亮, 杨柏坡, 袁一凡. 暂态波透射边界. 中国科学, A 辑. 1984,
27(6): 556-564
[22]王勖成, 邵敏. 有限元法基本原理和数值方法. 第 2 版. 清华大学出版社,
1997
[23]杜修力, 王进廷. 阻尼弹性结构动力计算的显式差分法. 工程力学. 2000,
17(5): 37-43
[24]王进廷, 杜修力. 有阻尼动力分析的一种显式差分法. 工程力学. 2002, 19(3):
109-112
[25]李小军, 廖振鹏, 杜修力. 有阻尼体系动力问题的一种显式解法. 地震工程
与工程振动. 1992, 13(4): 74-79
[26]周正华, 李山有和侯兴民. 阻尼振动方程的一种显式直接积分方法. 世界地
震工程. 1999, 15(1): 41-44
[27]张晓志, 程岩, 谢礼立. 结构动力反应分析的三阶显式方法. 地震工程与工
96
参 考 文 献
程振动. 2002, 22(3): 1-8
[28]陈学良, 袁一凡. 求解振动方程的一种显式积分格式及其精度与稳定性. 地
震工程与工程振动. 2002, 22(3): 9-14
[29]Chen Shaolin and Liao Zhenpeng. Multi-transmitting Formula for Attenuating
Waves. Acta Seismologica Sinica. 2003, 16(3): 283-291
[30]廖振鹏. 无限弹性介质中暂态标量波问题的有限元模型. 地震工程与工程振
动. 1982, 2(4): 38-52
[31]关慧敏, 廖振鹏. 局部透射边界和叠加边界的精度分析与比较. 力学学报.
1994, 26(3): 303-311
[32]艾龙根, 舒胡毕. 弹性动力学, 第二卷 线性理论. 石油工业出版社. 1984
[33]关慧敏, 廖振鹏. 局部人工边界稳定性的一种分析方法. 力学学报. 1996,
28(3): 376-380
[34]关慧敏, 廖振鹏. 一种改善多次透射边界稳定性的措施. 地震工程与工程振
动. 1997, 17(4): 1-8
[35]景立平, 廖振鹏, 邹经相. 多次透射公式一种高频失稳机制. 地震工程与工
程振动. 2002, 22(1): 7-13
[36]李小军, 廖振鹏. 时域局部透射边界的计算飘移失稳. 力学学报. 1996, 28(5):
627-632
[37]周正华, 廖振鹏. 消除多次透射公式飘移失稳的措施. 力学学报. 2001, 33(4):
550-554
[38]M. Novak, T. Nogami and F. Aboul-Ella. Dynamic Soil Reactions for Plane Strain
Case. Journal of the Engineering Mechanics Division, ASCE. 1978, 104(4):
953-959
[39]M. Novak. Vertical Vibration of Floating Piles. Journal of the Engineering
Mechanics Division, ASCE. 1977, 103(1): 153-168
[40]M. Novak and H. Mitwally. Transmitting Boundary for Axisymmetrical Dilation
Problems. Journal of the Engineering Mechanics Division, ASCE. 1988, 114(1):
181-187
[41]王振宇. 大型结构-地基系统动力反应计算理论及其应用研究. 清华大学工学
97
北京工业大学工学硕士学位论文
博士学位论文. 2002
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