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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.
Spectral element method combines the flexibility of finite element method and the exponential convergence of spectral method with high efficiency and high precision, and is an important numerical simulation method of seismic wave field developed in recent years. The classical spectral element method uses a tetragonal (hexahedron) The diagonal mass matrix is obtained using the tensor product of one-dimensional Gauss-Legendre-Lobatto (GLL) integrals to greatly improve the computational efficiency, but the quadrilateral (hexahedral) grid can not flexibly describe the curved interface of complex geometric models For this reason, it is necessary to introduce a triangle (tetrahedron) mesh to two-dimensional (three-dimensional) in the spectral element method. Unlike the classical spectral method, tensor of GLL integral can not be used in unstructured grid Product, so that there are many difficulties in the realization of unstructured grid spectral element method.At present, the popular triangular grid spectral element method, through the use of KoornwinderDubiner (KD) orthogonal polynomials, and the orthogonalization of these KD polynomials Basis functions, while using coincident interpolation nodes and integration nodes to obtain diagonal mass matrices; the integration points used are the optimized set of points - the Finte points, and these integral points can be fully coupled with the quadrilateral mesh. in Triangular grid spectral element method can significantly improve the description ability of complex models, and has great advantages for ups and downs of surface model.In this paper, the introduction of efficient optimal matching layer (PML) absorption boundary conditions, and through numerical experiments triangular grid Compared with the classical spectral element method, the triangular lattice spectral element method has the obvious disadvantage of lower calculation accuracy. For the seventh-order spectral element, in order to be able to accurately simulate the surface wave , The triangular mesh method requires at least 11 sampling points for each shortest surface wave wavelength, whereas the classical spectral method requires only 4 sampling points, and the former requires approximately 5.5 Times