修正辛格式有限元法的地震波场模拟

Modified symplectic scheme with finite element method for seismic wavefield modeling

三角网格有限元法具有网格剖分的灵活性,能有效模拟地震波在复杂介质中的传播.但传统有限元法用于地震波场模拟时计算效率较低,消耗较大计算资源.本文采用改进的核矩阵存储(IKMS)策略以提高有限元法的计算效率,该方法不用组合总体刚度矩阵,且相比于常规有限元法节省成倍的内存.对于时间离散,将有限元离散后的地震波运动方程变换至Hamilton体系,在显式二阶辛Runge-Kutta-Nystr9m(RKN)格式的基础之上加入额外空间离散算子构造修正辛差分格式,通过Taylor展开式得到具有四阶时间精度时间格式,且辛系数全为正数.本文从理论上分析了时空改进方法相比传统辛-有限元方法在频散压制、稳定性提升等方面的优势.数值算例进一步证实本方法具有内存消耗少、稳定性强和数值频散弱等优点.

Finite element method(FEM)with triangular mesh has the properties of flexibility and adaptability in complex medium.However,the traditional finite element method is inefficient in seismic wavefield modeling because this method commonly requires a large amount of computation resources.Here we propose a so-called improved kernel matrix storage(IKMS)strategy to improve the computation efficiency of FEM.The improved FEM does not need to assemble global stiffness matrix and the memory requirement is many times less than that of the conventional FEM.In terms of temporal discretization,the elastic wave equation after FEM discretization is first transformed into a Hamilton system.Then,an additional spatial discretization term is addedto the second-order explicit Runge-Kutta-Nystr9 m(RKN)scheme.Based on Taylor series expansion,we obtain a fourth-order symplectic scheme with all positive symplectic coefficients.Theoretical analysis shows that the temporal-spatial modified numerical scheme is superior over the traditional symplectic FEM in suppressing numerical dispersion and increasing numerical stability.Furthermore,numerical results verify that this method requires less computer resource and achieves higher numerical accuracy.