修正辛格式有限元法的地震波场模拟

苏波, 李怀良, 刘少林, 杨顶辉. 2019. 修正辛格式有限元法的地震波场模拟. 地球物理学报, 62(4): 1440-1452, doi: 10.6038/cjg2019M0538
引用本文: 苏波, 李怀良, 刘少林, 杨顶辉. 2019. 修正辛格式有限元法的地震波场模拟. 地球物理学报, 62(4): 1440-1452, doi: 10.6038/cjg2019M0538
SU Bo, LI HuaiLiang, LIU ShaoLin, YANG DingHui. 2019. Modified symplectic scheme with finite element method for seismic wavefield modeling. Chinese Journal of Geophysics (in Chinese), 62(4): 1440-1452, doi: 10.6038/cjg2019M0538
Citation: SU Bo, LI HuaiLiang, LIU ShaoLin, YANG DingHui. 2019. Modified symplectic scheme with finite element method for seismic wavefield modeling. Chinese Journal of Geophysics (in Chinese), 62(4): 1440-1452, doi: 10.6038/cjg2019M0538

修正辛格式有限元法的地震波场模拟

  • 基金项目:

    国家自然科学基金(41604034,41774118),国家重点研发计划项目课题(2017YFC150031)资助

详细信息
    作者简介:

    苏波, 男, 1977年生, 西南科技大学计算机科学与技术学院教师, 主要从事地震波正反演的研究工作.E-mail:bosu@foxmail.com

    刘少林, 男, 1988年生, 新加坡南洋理工大学、清华大学数学系博士后, 从事地震波传播与成像研究.E-mail:shaolinliu88@gamil.com

  • 中图分类号: P315

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-Nystr m(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.

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  • 图 1 

    等腰直角三角形单元网格

    Figure 1. 

    FEM element mesh constructed by isosceles right triangles

    图 2 

    三种辛格式的数值频散曲线

    Figure 2. 

    Numerical dispersion curves of three symplectic schemes

    图 3 

    二维均匀介质模型

    Figure 3. 

    2D homogeneous model for benchmark

    图 4 

    R1 (a)和R2 (b)处位移垂直分量合成波形与解析解波形对比; (c)和(d)分别对应于(a)和(b)图中的波形误差

    Figure 4. 

    Comparison of synthetic waveforms obtained by analytic method (black curve), FDM (red curve) and symplectic FEM (green curve). (a) and (b) show vertical component of synthetic waveform at R1 and R2, respectively. (c) and (d) show waveform misfits in (a) and (b), respectively

    图 5 

    R3处合成波形的水平分量(a)与垂直分量(b)和解析解波形对比

    Figure 5. 

    The synthetic waveforms of horizontal (a) and vertical (b) components at R3. Black curve is generated by analytic solution; red curve by second-order symplectic PRK; green curve by modified sympectic scheme; blue curve by third-order symplectic scheme

    图 6 

    Marmousi模型的P波速度结构

    Figure 6. 

    P-wave velocity structure of Marmousi model

    图 7 

    二阶PRK辛-FEM(a)与修正辛-FEM(b)得到的地表合成记录; (c)与(d)分别为二阶PRK辛-FEM与修正辛-FEM地表合成记录与参考解的残差

    Figure 7. 

    Synthetic seismograms generated by the second-order PRK symplectic FEM (a) and modified symplectic FEM (b); (c) difference between synthetic record in (a) and the reference solution; (d) difference between synthetic record in (b) and the reference solution

    图 附图 1 

    自由地表网格点示意图

    Figure 附图 1. 

    Schematic of grid points around free surface in FDM

    表 1 

    三种辛格式的| Δt2λi |max

    Table 1. 

    The values of | Δt2λi |max for three symplectic schemes

    方法 二阶PRK辛格式 修正辛格式 三阶辛格式
    t2λi|max 4 12 7.107
    下载: 导出CSV

    表 2 

    三种辛格式的稳定性范围b

    Table 2. 

    The stability parameter b for three temporal schemes

    有限元插值 二阶PRK辛格式 修正辛格式 三阶辛格式
    一阶插值 0.7071 1.2247 0.9425
    二阶插值 0.5869 1.0165 0.7823
    三阶插值 0.4463 0.7731 0.5950
    下载: 导出CSV

    表 3 

    修正辛-有限元和有限差分法计算效率和总体误差对比

    Table 3. 

    Computational efficiency and accuracy of the modified symplectic FEM and FDM

    参数 方法
    FDM 修正辛-FEM
    运行内存 100% 270.06%
    计算时间 100% 109.80%
    R1误差总量 28.12 4.82
    R2误差总量 10.83 6.31
    下载: 导出CSV

    表 4 

    1至3阶插值时不同存储策略内存消耗对比

    Table 4. 

    Memory requirements of two storage schemes with three interpolation orders

    刚度矩阵
    处理方法
    一阶插值 二阶插值 三阶插值
    刚度矩阵存储(MB) 实际总内存(MB) 刚度矩阵存储(MB) 实际总内存(MB) 刚度矩阵存储(MB) 实际总内存(MB)
    CSR存储 12.3963 277.6211 20.1325 104.5234 32.6123 135.3672
    IKMS 0.6868 66.0762 0.6872 34.5352 0.6882 35.3086
    下载: 导出CSV

    表 5 

    三种时间离散方法的计算效率和总体误差对比

    Table 5. 

    Computational efficiency and total error of three temporal methods

    参数 方法
    二阶PRK辛格式 修正辛格式 三阶辛格式
    运行内存 100% 100.19% 100.01%
    计算时间 100% 122.45% 125.05%
    X方向误差总量 13.62 6.33 6.78
    Z方向误差总量 15.28 10.22 10.26
    下载: 导出CSV
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收稿日期:  2018-09-03
修回日期:  2018-12-02
上线日期:  2019-04-05

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