基于紧凑存储的任意四边形网格有限元法地震波场数值模拟

采用有限元法精确模拟复杂介质条件下的地震波场,一般采用三角单元,但在节点数相同的情况下,三角单元的计算精度不如矩形单元高,采用三角单元模拟复杂界面时,编制确定结构刚度矩阵非零元素位置的程序也较为麻烦。采用矩形单元离散含有倾斜或起伏界面的地质模型时,无法避免绕射噪声,若加密网格又会增加计算量。为此,本文基于任意四边形单元模拟声波的传播,在倾斜或起伏界面条件下,可以有效避免因离散引起的“阶梯状”界面,在不增加计算量以及内存占用的前提下,有效地消除离散绕射噪声。采用对角的集中质量矩阵代替一致质量矩阵,避免矩阵的求逆运算,从而提高显式有限元法的计算效率;对结构刚度矩阵采用紧凑存储格式,每一行需要存储的元素最多为5个,同时零元素不参与运算,既减少内存的占用,又极大地提高计算效率。 

Finite Element Method with Arbitrary Quadrilateral Meshes for Numerical Modeling of Seismic Wave Based on Compact Storage

In general, we apply finite element method with triangular meshes to accurately modeling wave field under complex structures; however, in the condition of the same number of nodes, computational accuracy of triangular element is lower than that of rectangular element. It is difficult to implement the computer program which locates the non-zero elements of global stiffness matrix when we adopt triangular elements to fit complex geometrical interface. Using rectangular elements to divide geological model containing dipping or rugged interface will result in diffraction noise, and infilling grid will increase computational amount. For these reasons, this paper applies finite element method to solve 2-D acoustic wave equation, arbitrary quadrilateral meshes is adopted to fit dipping or rugged interface, which can avoid introducing “ladder” interface resulted from discrete rectangular grid, and discrete diffraction noise is eliminated effectively in the condition of no increasing computational amount and memory occupation. Diagonal lumped mass matrix is used to replace consistent massmatrix to avoid matrix inversion and improve the computational efficiency of explicit finite element method. In addition, we employ compact storage format to store global stiffness matrix, and the elements need to be stored in each row is no more than 5, the zero elements are not involved in computing at the same time, by this method, not only reducing the memory occupation, but also improving the computational efficiency.