成层半空间出平面自由波场的一维化时域算法
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摘要
提出了一种计算出平面SH波斜入射时弹性水平成层半空间中自由波场时域计算的一维化有限元方法．在进行有限元网格划分时,竖向单元取满足有限元模拟精度的任意尺寸,水平向网格尺寸由时间离散步长和水平视波速确定,并自动进行虚拟网格划分．基底设置人工边界,并将波动输入转化为等效荷载施加在边界节点上．然后将集中质量有限元法和中心差分法相结合建立节点运动方程,并将水平方向相邻节点的运动用该节点相邻时刻的运动表示,从而将求解节点运动的二维方程组转化为一维方程组．求解此方程组,即得到自由场中竖向一列节点的运动．最后根据行波传播的特点,可方便地确定全部自由波场．理论分析和数值算例表明,该方法具有较高的精度和良好的稳定性．
A 1-D finite element method in time domain is developed in this paper, which can be used to calculate the wave motion of free field in elastic layered half-space by antiplane SH wave oblique incidence. When the layered half-space is discretized, the vertical element size is determined conforming to the simulation accuracy; the horizontal element size is determined automatically by the horizontal apparent wave velocity and the discrete time step in the step-by-step calculation, and then the elements are divided virtually. Artificial boundary is constructed on the bottom of the computational area and the input wave motion is transformed into an equivalent load, which is applied on the nodes of the boundary. Then, the finite element method with lumped mass and the central difference method are combined to establish the wave motion equations in 2-D finite element model. Since the displacement of any node in the finite element model can be represented by that of the adjacent node in the horizontal direction, the 2-D wave motion equations can be transformed into 1-D equations. By solving the 1-D equations, the displacement of nodes in one vertical line can be obtained. Finally, the wave motion in elastic half-space is obtained based on the propagation characteristics of traveling wave. Both the theoretical analysis and the numerical results demonstrate that the proposed method features high accuracy and good stability.
A 1-D finite element method in time domain is developed in this paper, which can be used to calculate the wave motion of free field in elastic layered half-space by antiplane SH wave oblique incidence. When the layered half-space is discretized, the vertical element size is determined conforming to the simulation accuracy; the horizontal element size is determined automatically by the horizontal apparent wave velocity and the discrete time step in the step-by-step calculation, and then the elements are divided virtually. Artificial boundary is constructed on the bottom of the computational area and the input wave motion is transformed into an equivalent load, which is applied on the nodes of the boundary. Then, the finite element method with lumped mass and the central difference method are combined to establish the wave motion equations in 2-D finite element model. Since the displacement of any node in the finite element model can be represented by that of the adjacent node in the horizontal direction, the 2-D wave motion equations can be transformed into 1-D equations. By solving the 1-D equations, the displacement of nodes in one vertical line can be obtained. Finally, the wave motion in elastic half-space is obtained based on the propagation characteristics of traveling wave. Both the theoretical analysis and the numerical results demonstrate that the proposed method features high accuracy and good stability.
引文
1沈聚敏,周锡元,高小旺等．抗震工程学．北京:中国建筑工业出 版社,2000(Shen Jumin,Zhou Xiyuan,Gao Xiaowang,et al.Antiseismic Engineering.Beijing:China Architecture and Building Press,2000(in Chinese))
2廖振鹏．工程波动理论导论．北京:科学出版社(第二版), 2002(Liao Zhenpeng.Introduction to Wave Motion Theories in Engineering(2nd edition).Beijing:Science Press, 2002(in Chinese))
3潘旦光,楼梦麟,范立础．多点输入下大跨度结构地震反应分 析研究现状．同济大学学报, 2001,29(10):1213-1219(Pan Danguang,Lou Menglin,Fan Lichu.Status of seismic response analysis of long-span structures under multiple support excitations.Journal of Tongji University,2001, 29(10): 1213-1219(in Chinese))
4傅淑芳,刘宝诚．地震学教程．北京:地震出版社,1991(Fu Shufang, Liu Baocheng. Seismology Tutorial. Beijing: Seismic Press, 1991(in Chinese))
5李山有,廖振鹏,周正华．大型结构地震反应数值模拟中的波动输 入．地震工程与工程振动,2001,6(2):1-5(Li Shanyou,Liao Zhenpeng,Zhou Zhenghua. Wave motion input in numerical simulation of seismic response for large-scale structure. Earthquake Engineering and Engineering Vibration, 2001, 6(2): 1-5 (in Chinese))
6黎在良,刘殿魁．固体中的波．北京:科学出版社,1995(Li Zailiang, Liu Diankui. Wave Motion in Solid. Beijing: Science Press, 1995(in Chinese))
7廖振鹏．近场波动问题的有限元解法．地震工程与工程振动, 1984, 4(2):1-14 (Liao Zhenpeng.The finite element solution of near-field wave motion problem. Earthquake Engineering and Engineering Vibration, 1984, 4(2): 1-14 (in Chinese))
8 Liao Zhenpeng,Huang Kongliang,Yang Baipo,et al. A transmitting boundary for transient wave analysis. Science in China (Series A), 1984, 27(10): 1063-1076
9 Lysmer J,Kulemeyer RL.Finite dynamic model for infinite media.Journal of Engineering Mechanics,ASCE, 1969, 95: 759-877
10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(Liu Jingbo,Lu Yandong.A direct method for analysis of dynamic soilstructure interaction.China Civil Engineering Journal, 1998, 31(3): 55-64 (in Chinese))
11刘晶波,王艳．SH波斜入射时弹性半空间自由波场的一维 算法．见:崔京浩．第14届全国结构工程学术会议论文集．第 14届全国结构工程学术会议,烟台,2005(Liu Jingbo,Wang Yan. The 1D computational method of free field in elastic half-space with obliquely incident SH wave.In:Cui Jinghao, ed. Proceedings of the 14th National Conferences on Structural Engineering. The 14th National Conference on Structural Engineering, Yantai, 2005(in Chinese))
12刘晶波,廖振鹏．有限元离散模型中的出平面波动．力学学报, 1992, 24(2): 207-215(Liu Jingbo,Liao Zhenpeng. The out-plane wave motion in finite element model. Acta Mechanica Sinica, 1992, 24(2): 207-215 (in Chinese))
2廖振鹏．工程波动理论导论．北京:科学出版社(第二版), 2002(Liao Zhenpeng.Introduction to Wave Motion Theories in Engineering(2nd edition).Beijing:Science Press, 2002(in Chinese))
3潘旦光,楼梦麟,范立础．多点输入下大跨度结构地震反应分 析研究现状．同济大学学报, 2001,29(10):1213-1219(Pan Danguang,Lou Menglin,Fan Lichu.Status of seismic response analysis of long-span structures under multiple support excitations.Journal of Tongji University,2001, 29(10): 1213-1219(in Chinese))
4傅淑芳,刘宝诚．地震学教程．北京:地震出版社,1991(Fu Shufang, Liu Baocheng. Seismology Tutorial. Beijing: Seismic Press, 1991(in Chinese))
5李山有,廖振鹏,周正华．大型结构地震反应数值模拟中的波动输 入．地震工程与工程振动,2001,6(2):1-5(Li Shanyou,Liao Zhenpeng,Zhou Zhenghua. Wave motion input in numerical simulation of seismic response for large-scale structure. Earthquake Engineering and Engineering Vibration, 2001, 6(2): 1-5 (in Chinese))
6黎在良,刘殿魁．固体中的波．北京:科学出版社,1995(Li Zailiang, Liu Diankui. Wave Motion in Solid. Beijing: Science Press, 1995(in Chinese))
7廖振鹏．近场波动问题的有限元解法．地震工程与工程振动, 1984, 4(2):1-14 (Liao Zhenpeng.The finite element solution of near-field wave motion problem. Earthquake Engineering and Engineering Vibration, 1984, 4(2): 1-14 (in Chinese))
8 Liao Zhenpeng,Huang Kongliang,Yang Baipo,et al. A transmitting boundary for transient wave analysis. Science in China (Series A), 1984, 27(10): 1063-1076
9 Lysmer J,Kulemeyer RL.Finite dynamic model for infinite media.Journal of Engineering Mechanics,ASCE, 1969, 95: 759-877
10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(Liu Jingbo,Lu Yandong.A direct method for analysis of dynamic soilstructure interaction.China Civil Engineering Journal, 1998, 31(3): 55-64 (in Chinese))
11刘晶波,王艳．SH波斜入射时弹性半空间自由波场的一维 算法．见:崔京浩．第14届全国结构工程学术会议论文集．第 14届全国结构工程学术会议,烟台,2005(Liu Jingbo,Wang Yan. The 1D computational method of free field in elastic half-space with obliquely incident SH wave.In:Cui Jinghao, ed. Proceedings of the 14th National Conferences on Structural Engineering. The 14th National Conference on Structural Engineering, Yantai, 2005(in Chinese))
12刘晶波,廖振鹏．有限元离散模型中的出平面波动．力学学报, 1992, 24(2): 207-215(Liu Jingbo,Liao Zhenpeng. The out-plane wave motion in finite element model. Acta Mechanica Sinica, 1992, 24(2): 207-215 (in Chinese))
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