The spectral element method: An efficient tool to simulate the seismic response of 2D and 3D geological structures
The spectral element method: An efficient tool to simulate the seismic response of 2D and 3D geological structures
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DOI：
10.1785/bssa0880020368
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发表时间：
1998-04
影响因子：
3
通讯作者：
D. Komatitsch;J. Vilotte
中科院分区：
文献类型：
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作者：
D. Komatitsch;J. Vilotte
我们提出谱元法来模拟弹性波在涉及复杂自由表面地形以及二维和三维几何形状的材料界面的实际地质结构中的传播。这里介绍的谱元法是一种用于弹性波方程空间近似的高阶变分方法。在这种方法中，质量矩阵通过构造是对角的，这大大降低了计算成本，并允许高效的并行实现。以变分形式引入吸收边界条件来模拟无界物理域。时间离散化基于一种能量 - 动量守恒格式，该格式可转化为经典的显式 - 隐式预测器/多校正器形式。阐述了长期能量守恒和稳定性特性以及吸收条件的效率。相关的库朗条件表现为ΔtC < O (nel−1/ndN−2)，其中nel是单元数量，nd是空间维度，N是多项式阶数。在实践中，当使用N = 8的多项式次数时，发现每波长大约5个点的空间采样是非常精确的。通过将谱元解与经典的二维（2D）兰姆（Lamb）和加文（Garvin）问题的解析解进行比较，展示了该方法的准确性。然后通过研究更实际的涉及实际几何形状和复杂自由边界条件的二维模型来说明该方法的灵活性。以较低的计算成本获得了瑞利波传播、表面衍射以及与自由表面曲率相关的瑞利波 - 体波模式转换的非常精确的建模。结果表明，该方法为研究三维（3D）表面地形对弹性波的衍射以及对强地面运动的相关局部影响提供了一种有效的工具。即使对于平缓的山丘地形，也显示出在空间和时间上都存在复杂的放大模式。考虑了对非均质山丘结构的扩展。在并行分布式存储架构上的高效实现将允许对三维放大现象进行实时可视化和交互式物理研究，以用于地震风险评估。
We present the spectral element method to simulate elastic-wave propagation in realistic geological structures involving complieated free-surface topography and material interfaces for two- and three-dimensional geometries. The spectral element method introduced here is a high-order variational method for the spatial approximation of elastic-wave equations. The mass matrix is diagonal by construction in this method, which drastically reduces the computational cost and allows an efficient parallel implementation. Absorbing boundary conditions are introduced in variational form to simulate unbounded physical domains. The time discretization is based on an energy-momentum conserving scheme that can be put into a classical explicit-implicit predictor/multi-corrector format. Long-term energy conservation and stability properties are illustrated as well as the efficiency of the absorbing conditions. The associated Courant condition behaves as ΔtC < O (nel−1/ndN−2), with nel the number of elements, nd the spatial dimension, and N the polynomial order. In practice, a spatial sampling of approximately 5 points per wavelength is found to be very accurate when working with a polynomial degree of N = 8. The accuracy of the method is shown by comparing the spectral element solution to analytical solutions of the classical two-dimensional (2D) problems of Lamb and Garvin. The flexibility of the method is then illustrated by studying more realistic 2D models involving realistic geometries and complex free-boundary conditions. Very accurate modeling of Rayleigh-wave propagation, surface diffraction, and Rayleigh-to-body-wave mode conversion associated with the free-surface curvature are obtained at low computational cost. The method is shown to provide an efficient tool to study the diffraction of elastic waves by three-dimensional (3D) surface topographies and the associated local effects on strong ground motion. Complex amplification patterns, both in space and time, are shown to occur even for a gentle hill topography. Extension to a heterogeneous hill structure is considered. The efficient implementation on parallel distributed memory architectures will allow to perform real-time visualization and interactive physical investigations of 3D amplification phenomena for seismic risk assessment.