三角网格谱元法地震波场数值模拟

Numerical modeling of seismic wavefield with the SEM based on Triangles

谱元法结合了有限元法的灵活性和谱方法的指数收敛性,高效且高精度,是近年来发展的一种重要的地震波场数值模拟方法.经典的谱元法采用四边形(六面体)网格,利用一维Gauss-Legendre-Lobatto(GLL)积分的张量积得到对角的质量矩阵,以大大提高计算效率,但是四边形(六面体)网格不能够灵活地刻画复杂的几何模型的弯曲界面.为此,在谱元法中引入三角形(四面体)网格到二维(三维)是十分必要的.不同于经典的谱元法,在非结构化网格中不能使用GLL积分的张量积,使得非结构化网格的谱元法的实现存在着诸多的困难.目前,比较流行的三角网格谱元法,通过使用KoornwinderDubiner(KD)正交多项式,并正交化这些KD多项式构建基函数,同时利用重合的插值节点和积分节点以获取对角的质量矩阵;它所使用的积分点为优化的点集——Fekete点,且这些积分点能与四边形网格完全耦合.相比于四边形,三角网格谱元法能显著提高复杂模型的描述能力,对起伏地表模型有很大优势.本文引入高效的最佳匹配层(PML)吸收边界条件,并通过数值试验将三角网格谱元法与经典的谱元法进行对比研究.相比于经典的谱元法,三角网格谱元法显著缺点为较低的计算精度.对于7阶谱元,为了能够精确地模拟面波,三角网格谱元法需要在每个最短的面波波长内至少有11个采样点,然而经典的谱元法仅需4个采样点,并且前者所需的内存量约为后者的5.5倍.

The spectral element method（SEM）combines the geometrical flexibility of a finite element method with the exponential convergence rate associated with spectral method,which has being become an important method of seismic wavefiled modeling owing to its high efficiency and high accuracy in recent years.The classical SEM（i.e the SEM based on quadrilateral/hexahedral elements in 2-D/3-D）obtains a diagonal mass matrix by using the co-location of the interpolation and integration points,and tensor-product of 1-D GLL integration,which can decrease computational amount dramatically. It is necessary to introduce triangular（tetrahedron）elements into the SEM（TSEM）in 2-D（3-D）as the quadrilateral（hexahedral）elements can not represent the complex models with curved surfaces flexibly. Unlike the classical SEM,the tensor-product of 1-D GLL can not be used in unstructured meshes,which makes the SEM based on unstructured meshes become troublesome and difficult.In present,apopular TSEM can also obtain a diagonal mass matrix based upon the characteristic of the co-location of the interpolant points and integration points. The cardinal functions of the SEM are constructed by the orthogonal Koornwinder-Dubiner（KD）polynomials located at Fekete points.The optimal points set,Fekete points,enable a conforming matching between triangles and quadrilaterals while keeping the important properties of the classical SEM.The TSEM also has a high ability of representing complex geometries（especially for models with irregular topography）compared with the SEM.In this paper,we introduce the efficient perfectly matched layer（PML）boundary condition into the TSEM to suppress the spurious reflected waves from artificial boundaries.The compressed storage row（CSR）format is adopted to store the stiffness of the SEM and TSEM to save computer memory.The CSR format can also improve the efficiency of the SEM and TSEM as only the nonzeros of the stiffness matrix are involved in the computation of time updating.The Newmark time integration also applied in the time discretization of wave equation to improve the accuracy.A comparative study between the classical SEM and TSEM also been made in terms of computational accuracy,computational efficiency and computer memory occupation.The numerical tests demonstrate that a prominent disadvantage of TSEM over the classical SEM is its lower accuracy.In order to model the surface waves accurately,for the SEM with the polynomials of degree 7,the TSEM needs at least 11 sampling points in per minimal wavelength of the surface wave,while the classical SEM only need 4.Besides,the computer memories occupation of the TSEM is 5.5 times larger than that of the classical SEM.Then numerical tests also verify the effectiveness of the PML in the TSEM.The PML can give a favorable performance with 2 spectral elements.