Modified symplectic scheme with finite element method for seismic wavefield modeling
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摘要:
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.
Abstract: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 added to the second-order explicit Runge-Kutta-Nyström (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.
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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 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
表 1
三种辛格式的| Δt2λi |max值
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 |Δt2λi|max 4 12 7.107 
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表 2
三种辛格式的稳定性范围b值
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
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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
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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
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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
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