饱和多孔介质三维时域粘弹性人工边界与动力反应分析的显式有限元法
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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。饱和多孔介质在自然界中普遍存在,如饱和土,它是由固体骨架和孔隙流体组成。当两者之间存在动力相互作用时,按饱和多孔介质分析比按单相固体介质分析更为科学、合理。由于饱和介质波动理论的复杂性及数学处理上的困难,有关饱和多孔介质动力响应研究的解析解仅局限于比较简单的边界情况。对于几何形状较复杂且存在近域介质非均匀和非线性的复杂情况,数值方法是有效的分析手段。在无限或半无限空间的饱和多孔介质动力有限元分析中,通常采用有限模型的数值方法进行分析。为反映无约束域能量辐射效应影响,需引入虚拟的人工边界条件。虚拟人工边界的处理方式和方法对模拟精度的影响较大。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,获得学术界和工程界的重视。本文主要完成了以下工作:
1.建立了饱和多孔介质的三维粘弹性人工边界
以外行的球面波为研究对象,分析人工边界应力,得到应力的表达式,固相介质在人工边界的法向应力与其在边界处的位移和速度分别成线性关系,固相介质在边界的切向应力也与其切向位移和速度成线性关系;本文假设孔隙流体无粘性,不承受剪力,只在人工边界的法向存在应力,其只与流体的法向速度成正比。因此,对固相介质,可以在人工边界的法向、切向设置连续分布的弹簧和粘滞阻尼器;对孔隙流体,只在人工边界的法向设置连续分布的粘滞阻尼器。地震波输入可以通过在人工边界上施加等效荷载来实现。
2.本文基于赵成刚教授的饱和多孔介质二维显式示有限元数值计算方法,建立了该理论的三维方法,并开发了实现该方法三维问题的有限元程序(PT-DEA)。
数值模拟算例表明本文提出的饱和多孔介质三维粘弹性动力人工边界具有很好的精度和稳定性。本文边界与解析解、扩展有限元结果很接近,精度高于现有粘性边界、一、二阶透射边界。
饱和多孔介质简谐动载动力响应分析表明,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 saturated porous media is general existed in the nature, for example, the saturated soil, which is made up of solid skeleton and pore liquid fluid medium. If the dynamic interacting between the previous two part is concerned, using the saturated soil theory analyzing is more appropriate and scientific than using the single medium analyzing. As the complex of the saturated porous media theory and the difficulties of the disposing in mathematics, the analytic solution of saturated porous medium dynamic response is topical the simple boundary condition. To the complex geometry and the near field heterogeneous and nonlinear condition, numerical method is the effective mean. In the dynamic finite element analysis of saturated porous media in whole space or half-space, a finite model numerical method is usually selected for computing. To reflect unrestricted region energy radiation effect, it is necessary to input the supposed artificial boundary and computing method. The artificial boundary disposing means and computing means is largely influenced the computing accuracy. As the decoupling of time and position, explicit finite element combining local artificial method is proper to the previous complex near field wave-motion problem's solving. The main studies in this dissertation are listed as follows:
1. An approximate spring-dashpot three-dimensional artificial boundary conditions of saturated porous media are presented. It is shown that the normal and tangential wave stresses of the solid phase on the boundary are proportional to displacement and velocity, and the pore fluid has only normal wave stress which is proportional to it's velocity on the artificial boundary. Therefore, the continuously distributed springs and dashpots can be set on the artificial boundaries in the normal and tangential directions to absorb the energy of outgoing waves. The wave motion input can be realized by applying equivalent loads on the artificial boundaries.
2. According to the dynamic analysis of saturated porous media by using explicit finite element method proposed by professor chenggang zhao, this paper develops finite element method which could solve the three-dimensional problems, and develops finite element programme named PT-DEA.
Numerical example indicate that the proposed three-dimensional viscous-spring artificial boundary enjoy good accuracy and good stability.
Numerical example indicates that the accuracy and good stability of the proposed three-dimensional viscous-spring artificial boundary correspond with that of existing analytical, extended finite element solution and the second-order transmitting boundary.
The analysis of the saturated porous medium dynamic response indicates that the combination method of the explicit finite element method and the three-dimensional viscous-spring boundary is effective.
1.建立了饱和多孔介质的三维粘弹性人工边界
以外行的球面波为研究对象,分析人工边界应力,得到应力的表达式,固相介质在人工边界的法向应力与其在边界处的位移和速度分别成线性关系,固相介质在边界的切向应力也与其切向位移和速度成线性关系;本文假设孔隙流体无粘性,不承受剪力,只在人工边界的法向存在应力,其只与流体的法向速度成正比。因此,对固相介质,可以在人工边界的法向、切向设置连续分布的弹簧和粘滞阻尼器;对孔隙流体,只在人工边界的法向设置连续分布的粘滞阻尼器。地震波输入可以通过在人工边界上施加等效荷载来实现。
2.本文基于赵成刚教授的饱和多孔介质二维显式示有限元数值计算方法,建立了该理论的三维方法,并开发了实现该方法三维问题的有限元程序(PT-DEA)。
数值模拟算例表明本文提出的饱和多孔介质三维粘弹性动力人工边界具有很好的精度和稳定性。本文边界与解析解、扩展有限元结果很接近,精度高于现有粘性边界、一、二阶透射边界。
饱和多孔介质简谐动载动力响应分析表明,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 saturated porous media is general existed in the nature, for example, the saturated soil, which is made up of solid skeleton and pore liquid fluid medium. If the dynamic interacting between the previous two part is concerned, using the saturated soil theory analyzing is more appropriate and scientific than using the single medium analyzing. As the complex of the saturated porous media theory and the difficulties of the disposing in mathematics, the analytic solution of saturated porous medium dynamic response is topical the simple boundary condition. To the complex geometry and the near field heterogeneous and nonlinear condition, numerical method is the effective mean. In the dynamic finite element analysis of saturated porous media in whole space or half-space, a finite model numerical method is usually selected for computing. To reflect unrestricted region energy radiation effect, it is necessary to input the supposed artificial boundary and computing method. The artificial boundary disposing means and computing means is largely influenced the computing accuracy. As the decoupling of time and position, explicit finite element combining local artificial method is proper to the previous complex near field wave-motion problem's solving. The main studies in this dissertation are listed as follows:
1. An approximate spring-dashpot three-dimensional artificial boundary conditions of saturated porous media are presented. It is shown that the normal and tangential wave stresses of the solid phase on the boundary are proportional to displacement and velocity, and the pore fluid has only normal wave stress which is proportional to it's velocity on the artificial boundary. Therefore, the continuously distributed springs and dashpots can be set on the artificial boundaries in the normal and tangential directions to absorb the energy of outgoing waves. The wave motion input can be realized by applying equivalent loads on the artificial boundaries.
2. According to the dynamic analysis of saturated porous media by using explicit finite element method proposed by professor chenggang zhao, this paper develops finite element method which could solve the three-dimensional problems, and develops finite element programme named PT-DEA.
Numerical example indicate that the proposed three-dimensional viscous-spring artificial boundary enjoy good accuracy and good stability.
Numerical example indicates that the accuracy and good stability of the proposed three-dimensional viscous-spring artificial boundary correspond with that of existing analytical, extended finite element solution and the second-order transmitting boundary.
The analysis of the saturated porous medium dynamic response indicates that the combination method of the explicit finite element method and the three-dimensional viscous-spring boundary is effective.
引文
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[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]邱流潮,金峰.无限介质中波动分析的显式时域辐射边界.清华大学学报.2003,42(11):1530-1533.(Qiu Liuchao and Jin Feng.Explicit time-domain radiation boundaries for analysis of wave propagation in unbounded media.Journal of Tsinghua University,2003,42(11):1530-1533.(in Chinese))
[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]Du X L.Li L Y.A viscous2spring artificial boundary for near2field wave analysis in saturated porous media.Chinese J.Geophys.(in Chinese),2008,51(2):575-581
[8]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
[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]Liao Zhenpeng and Wong H L.A transmitting botmdary for the numerical simulation of elastic wave propagation.Soil Dynamics and Earthquake Engineering,1984,3(4):174-183.
[11]Chen Shaolin and Liao Zhenpeng.Multi-transmitting Formula for Attenuating Waves.Acta Seismologica Sinica.2003,16(3):283-291
[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
[14]R.L.Higdon.Absorbing Boundary Conditions for Acoustic and Elastic Waves in Stratified Media.Journal of Computational Physics.1992,101:386-418
[15]杜修力,李立云.饱和多孔介质近场波动分析的一种黏弹性人工边界.地球物理学报,2008,51(2):575-581
[16]A.J.Decks 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]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
[23]M.Novak.Vertical Vibration of Floating Piles.Journal of the Engineering Mechanics Division,ASCE.1977,103(1):153-168
[24]M.Novak and H.Mitwally.Transmitting Boundary for Axisymmetrical Dilation Problems.Journal of the Engineering Mechanics Division,ASCE.1988,114(1):181-187
[25]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.清华大学工学博士学位论文.2002
[26]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
[27]陈龙珠,吴世明,曾国熙.弹性波在饱和土层中的传播.力学学报,1987,19(3):276-283
[28]Modaressi H.“Absorbing boundary element for dynamic analysis of two-phase media.” Earthquake Engineerdng,10th World Conference,1157-1161.
[29]Degrande G,De Roeck G.Absorbing boundary condition for wave propagation in saturated poroelastic media-Part I:Formulation and efficiency evaluation[J].Soil Dynamics and Earthquake Engineering 1993,12:411-421.
[30]Akiyoshi T,Fuchida K,Fang H L.Absorbing boundary conditions for dynamic analysis of fluid-saturated porous media[J].Soil Dynamics and Earthquake Engineering 1994,13:387-397.
[31]Gajo A,Saetta A,Vitaliani R.Silent boundary conditions for wave propagation in saturated porous media[J].International Journal for Numerical and Analytical Methods in Geomechanics 1996.20:253-273.
[32]Akiyoshi T,Sun X,Fuchida K.General absorbing boundary conditions for dynamic analysis of fluid-saturated porous media[J].Soil Dynamics and Earthquake Engineering 1998.17:397-406.
[33]邵秀民,蓝志凌.流体饱和多孔介质波动方程的有限元解法[J].地球物理学报2000:264-278.
[34]Zerfa Z,Loret B.A viscous boundary for transient analyses of saturated porous media[J].Earthquake Engineering and Structural Dynamics 2004,33:89-110.
[35]Bowen R M.Incompressible porous media by use of the theory of mixtures[J].Internat J Engrg Sci 1980,18:19-45.
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