基于双二次插值的地震波场有限元法数值模拟
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摘要
有限元法是地震波场数值模拟最常用的一种方法,能够比较客观地反映地震波在复杂介质中的传播规律。为了提高有限元法数值模拟的计算效率和减少内存占用,采用双二次插值法实现了二维声波方程的有限元法数值模拟。在矩形网格剖分情况下,取每个单元的4个角点和4边中点为节点,在单元内采用双二次函数进行插值;根据质量守恒原则,将单元的质量分配到8个节点上,得到角节点质量非负的集中质量矩阵,避免矩阵的求逆运算;对结构刚度矩阵采用紧凑存储(只存储结构刚度矩阵下三角部分的非零元素),使得结构刚度矩阵每一行需存储的元素不超过11个;同时在时间循环过程中零元素不参与运算。模型算例的双二次插值有限元法数值模拟结果与双线性插值有限元法数值模拟结果对比表明,在无可见数值频散情况下,前者单步耗时更短,内存占用更少。
Finite element method is one of the most common methods for numerical simulation of seismic wavefield,it can objectively express the seismic wave propagation in complex medium.In this paper,we use quadratic interpolation in rectangular element to achieve the numerical simulation of seismic wavefield,the nodes of each element including the four corner nodes and the midpoints of four edges,and the interpolation is by biquadratic functions for each element.According to the law of mass conservation,we get the diagonal lumped mass matrix,whose corner nodes quality is non-negative and the inversion operation of mass matrix is avoided.Compacting memory is adopted to store structural stiffness matrix(only storing the none-zero elements in lower triangular matrix),the number of elements stored in each row of structural stiffness matrix is no more than 11,and the zero elements is not involved in computing.And by this method,not only reducing the memory occupation,but also improving the efficiency of calculation.Finally,through the comparison of biquadratic interpolation and bilinear interpolation in numerical simulation,the results show that without visible numerical dispersion,the time-consuming and memory occupation of single step of biquadratic interpolation is less than bilinear interpolation.
引文
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