A free-surface boundary condition for including 3D topography in the finite-difference method
A free-surface boundary condition for including 3D topography in the finite-difference method
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DOI：
10.1785/bssa0870020494
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发表时间：
1997-04
影响因子：
3
通讯作者：
T. Ohminato;B. Chouet
中科院分区：
文献类型：
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作者：
T. Ohminato;B. Chouet
提出了一种引入无应力边界条件的灵活而简单的方法，以将三维 (3D) 地形纳入有限差分法中。通过以交错网格方案堆叠单位材料单元，将 3D 形貌离散化为楼梯。剪应力分布在单位材料单元的 12 个边缘上，因此只有剪应力出现在自由表面上，而法向应力始终嵌入固体区域内。这种配置使得可以通过将拉梅系数 λ 和 μ 设置为零来在自由表面处实现无应力边界条件，而不会产生任何物理上不合理的条件。通过改变计算域中 λ 和 μ 的分布来实现任意 3D 形貌。我们的方法使用简约的交错网格方案，仅需要传统交错网格方案中所用内存的 3/4，其中需要存储六个应力分量和三个速度分量。数值测试表明每个波长需要25个网格才能稳定计算。将有限差分结果与二维 (2D) 半圆形峡谷模型的边界元法结果进行比较。我们还介绍了一段半圆形峡谷和半球形空腔对垂直入射平面 P、SV 和 SH 波的响应，并讨论了高斯山对嵌入山中的各向同性点源的响应。在半圆形峡谷段中，合成物的后期部分的特征是从两个垂直侧壁散射的相。半球形空腔和二维半圆形峡谷都表现出能量在空腔底部的聚焦，尽管前者的聚焦效果更强。由于高斯山的强烈地形而产生的聚焦和散焦效应会在与源相对的侧面产生强烈的位移放大。山顶的后向散射也清晰可见。
A flexible and simple way of introducing stress-free boundary conditions for including three-dimensional (3D) topography in the finite-difference method is presented. The 3D topography is discretized in a staircase by stacking unit material cells in a staggered-grid scheme. The shear stresses are distributed on the 12 edges of the unit material cell so that only shear stresses appear on the free surface and normal stresses always remain embedded within the solid region. This configuration makes it possible to implement stress-free boundary conditions at the free surface by setting the Lamé coefficients λ and μ to zero without generating any physically unjustified condition. Arbitrary 3D topographies are realized by changing the distribution of λ and μ in the computational domain. Our method uses a parsimonious staggered-grid scheme that requires only 3/4 of the memory used in the conventional staggered-grid scheme in which six stress components and three velocity components need to be stored. Numerical tests indicate that 25 grids per wavelength are required for stable calculation. The finite-difference results are compared with those of the boundary-element method for the two-dimensional (2D) semi-circular canyon model. We also present the responses of a segment of semi-circular canyon and hemispherical cavity to vertically incident plane P, SV, and SH waves and discuss the response of a Gaussian hill to an isotropic point source embedded in the hill. In the segment of semi-circular canyon, the later portions of the synthetics are characterized by phases scattered from the two vertical side walls. The hemispherical cavity and 2D semi-circular canyon both show focusing of energy at the bottom of the cavity, although the focusing effect is stronger in the former geometry. Focusing and defocusing effects due to the strong topography of the Gaussian hill produce a strong amplification of displacements at a spot located on the flank opposite to the source. Backscattering from the top of the hill is also clearly seen.