AN EXPLICIT SPECTRAL-ELEMENT APPROACH TO FLUID-SOLID COUPLING PROBLEMS IN SEISMIC WAVE PROPAGATION
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摘要: 流固耦合地震波动问题主要研究由流体和固体构成的复杂系统中地震波传播特性及其规律. 传统模拟方法中一般以声波方程、弹性波方程的数值解分别描述理想流体和弹性固体中的波动, 并实时地处理两种不同性质介质之间的相互耦合作用, 数值格式复杂且限制数值模拟精度与计算效率. 本文采用谱元法结合多次透射公式人工边界条件实现了一种流固耦合地震波动问题的高阶显式数值计算方法. 该方法利用了流固耦合问题统一计算框架，可将饱和多孔介质的Biot波动方程分别退化为理想流体的声波方程和弹性固体的弹性波方程. 通过P波垂直入射的水平成层理想流体-饱和多孔介质-弹性固体场地模型、P波斜入射的不规则层状界面以及任意形状界面的理想流体-饱和多孔介质-弹性固体场地模型等三个算例, 与传递函数法解析解以及集中质量有限元法计算结果进行对比分析, 证明了本文方法的正确性与有效性. 数值模拟结果表明, 本文方法相较传统有限元法可以少得多的节点数量获得更高的数值精度, 并且在较宽的频率范围内都能可靠地模拟出流固耦合系统的动力响应, 充分体现出本文方法兼顾高精度、计算效率和复杂场地建模灵活的特点.Abstract: Fluid-solid coupling seismic wave motion problems are aimed at investigating the characteristics and laws of seismic wave propagation in the complex system composed of fluid and solid media. In traditional simulation methods, numerical solutions of acoustic and elastic wave equations are generally used to describe the waves in ideal fluid and elastic solid respectively, and the coupling between the two media with different properties is dealt with in real time. Consequently, traditional methods suffer from complex numerical schemes, relatively low numerical simulation accuracy and computational efficiency. Based on spectral element method and multi-transmitting formula artificial boundary condition, a high order explicit numerical method for fluid-solid coupling seismic wave motion problems is developed in this paper. This method uses a unified computational framework, in which Biot’s equations for saturated porous media can degenerate to acoustic and elastic wave equations for ideal fluid and elastic solid respectively. Three numerical examples of ideal fluid-saturated porous medium-elastic solid system are given: the horizontal layered site model with vertical incidence of P wave, the irregular layered interface model under obliquely incident P wave and arbitrary shape interface model under obliquely incident P wave. The accuracy and efficiency of the proposed method are verified in comparison with the results of transfer matrix method and lumped mass finite element method. The numerical simulation results show that compared with the traditional finite element method, this method can obtain higher numerical accuracy with much less nodes, and can reliably simulate the dynamic response of fluid-solid coupling problems in a wider frequency range. The proposed method fully represents the characteristics of high precision, high efficiency and flexibility to handle complex sites.
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图 1 流固耦合地震波动问题示意图
Figure 1. Schematic diagram of seismic wave motion problem in a fluid-poroelastic-solid model
图 2 流固耦合问题的不同介质交界面
Figure 2. Different interfaces in the fluid-poroelastic-solid problem
图 3 界面连续条件的谱元离散
Figure 3. Spectral-element discretization of the continuous condition on the interface
图 4 多次透射公式人工边界条件与地震动输入
Figure 4. Artificial boundary condition and seismic wave input based upon multi-transmitting formula
图 8 不同方案情况下C点的位移谱比
Figure 8. Spectral ratio of displacement at point C with different scheme
图 15 饱和多孔介质表面的位移幅值
Figure 15. Surface displacement amplitude of saturated porous medium
表 1 透射边界(MTF)谱元离散格式中插值点的局部坐标
Table 1. Local coordinates of the interpolation points in the spectral-element formulation of multi-transmitting formula
Number
(ngllξ)Local coordinates
(sk (k = 0, …, ngllξ−1))2 0, 2 3 0, 1, 2 4 0, 0.55278, 1.44721, 2 5 0, 0.34535, 1, 1.65465, 2 6 0, 0.23494, 0.71477, 1.28523, 1.76506, 2 
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表 2 介质参数表
Table 2. Parameters of medium
Media β μ0 ρs/kg·m−3 ρw/kg·m−3 ν μ/MPa Ew/GPa M/GPa α k0/m·s−1 ideal fluid 1 0 0 1000 0.49 0 2.25 2.25 1 1 saturated soil 0.26 0.001 2000 1000 0.49 83.2 2.25 4.78 0.697 10−7 bedrock 0 0 2500 0 0.2 480 0 - 0 0 $ \lambda=2 \mu i f(1-2 v) $
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