A comparative study of finite element and spectral element methods in seismic wavefield modeling
A comparative study of finite element and spectral element methods in seismic wavefield modeling
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DOI：
10.1190/geo2013-0018.1
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发表时间：
2014-03
期刊：
影响因子：
3.3
通讯作者：
Youshan Liu;J. Teng;Haiqiang Lan;Xiang Si;Xueying Ma
中科院分区：
文献类型：
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作者：
Youshan Liu;J. Teng;Haiqiang Lan;Xiang Si;Xueying Ma
对有限元法和谱元法在时域弹性波场模拟中的计算精度和效率进行了深入研究。充分利用质量集中技术和求积法则，对无矩阵求逆的有限元法和扫描电镜法的对角质量矩阵进行了综合比较。我们研究了有限元和扫描电镜的基础上，分别与一次和二次多项式的四边形，三角形网格与一次和三次多项式。一般情况下，当多项式阶数相同时，四边形网格上的数值解比三角形网格上的数值解具有更高的精度。在插值点数相同的情况下，有限元法与扫描电镜法具有相当的精度。考虑到三角形和四边形的SEM，前者遭受较大的计算成本和相对较低的精度相比，后者。此外，收敛性研究证明，三角形SEM产生一致的更大的误差比四边形SEM的任何阶数和单元尺寸。然而，三角形SEM可以有效地适应任意复杂的几何形状。在插值多项式阶数相同的条件下，有限元法的计算效率与扫描电镜法相当。此外，推导了变分形式的完全匹配层（PML）边界条件。通过在频域中引入四个中间变量，PML避免了卷积计算，在时域中通过逆傅里叶变换得到精确解。数值算例验证了PML的正确性和有效性。
The computational accuracy and efficiency of finite element method and spectral element method (SEM) are investigated thoroughly in time-domain elastic wavefield modeling. The diagonal mass matrices of the FEM and SEM free from matrix inversion are compared comprehensively by making full use of the mass-lumped technique and quadrature rules. We investigate the FEM and SEM based, respectively, on quadrilateral with the polynomials of degrees one and two, and on triangular grids with the polynomials of degrees one and three. Generally, the numerical solutions based on quadrilateral grids have a higher precision than those computed on triangular grids when the same order of polynomials is used. The FEM has a comparable accuracy to the SEM with the same number of interpolant points. In view of the triangular and quadrilateral SEMs, the former suffers from larger computational costs and relatively lower accuracy compared with the latter. Furthermore, the convergence study proves that the triangular SEM produces consistently larger errors than the quadrilateral SEM for any order and element sizes. However, the triangular SEM can adapt to arbitrary complex geometries effectively. In terms of efficiency, the FEM has an efficiency comparable with the SEM on condition that the order of interpolation polynomials is identical. In addition, a perfectly matched layer (PML) boundary condition in variational form is deduced. By introducing four intermediate variables in frequency domain, the PML avoids convolution calculation and obtains an exact solution through inverse Fourier transform in time domain. The numerical examples verify the validity and effectiveness of the PML.