有限元算法在声波方程数值模拟中的频散分析

印兴耀; 周建科; 吴国忱; 梁锴

doi:10.3969/j.issn.0253-3782.2014.05.017

有限元算法在声波方程数值模拟中的频散分析

中国山东青岛 266580 中国石油大学(华东)地球科学与技术学院

基金项目: 国家重点基础研究发展计划“973”项目(2013CB228604)资助.

详细信息

通讯作者:
印兴耀, e-mail: xyyin@upc.edu.cn

中图分类号: P315.3⁺1
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出版历程
- 收稿日期: 2014-02-22
- 修回日期: 2014-03-20
- 发布日期: 2014-08-31

Dispersion analysis for the finite element algorithm in acoustic wave equation numerical simulation

School of Geosciences, China University of Petroleum, Shandong Qingdao 266580, China

摘要

摘要: 针对有限元算法在地震波数值模拟中的数值频散问题，利用集中质量矩阵双线性插值有限元算法，推导了二维声波方程的频散函数.在此基础上采用定量分析方法，对比分析了网格纵横长度比变化时的入射方向、空间采样间隔、地震波频率以及地层速度对数值频散的影响.数值算例和模型正演结果表明：当采用集中质量矩阵双线性插值有限元算法时，为了有效地压制数值频散，在所使用震源子波的峰值频率对应的波长内，采样点数目应不少于20个；减小网格长度的纵横比可以有效地抑制入射角(波传播方向与z轴的夹角)较小的地震波的数值频散；地震波频率越高，传播速度越慢，频散越严重，尤其是当相速度与其所对应的频率比值小于2倍空间采样间隔时，不仅会出现严重的数值频散,还会出现假频现象.
- 双线性插值 /
- 有限元法 /
- 声波方程 /
- 集中质量矩阵 /
- 数值频散
Abstract: This paper focuses on the dispersion problems of finite element algorithm in numerical simulation of seismic wave, and the dispersion function of two-dimensional acoustic wave equation is derived by employing lumped mass matrix and bilinear interpolation finite element algorithm. And, we compared quantitatively the effect of incident direction with the variable ratio of vertical to horizontal grid, spatial sampling interval, seismic wave frequency, and formation velocity on numerical dispersion. The numerical examples and the forward modeling indicate, if we want to suppress the numerical dispersion effectively, it should not be less than 20 samples within the wavelength corresponding to peak frequency of source wavelet; reducing the ratio of vertical to horizontal grid can suppress the numerical dispersion with small incident angle (the angle between the direction of wave propagation and the z axis) remarkably; the slower the propagation velocity of the seismic wave with higher frequency, the more serious its dispersion is; when the ratio of phase velocity to the corresponding frequency is less than twice of spatial sampling interval, not only the numerical dispersion becomes very serious, but also the aliasing phenomenon will happen.
- bilinear interpolation /
- finite element method /
- acoustic wave equation /
- lumped mass matrix /
- numerical dispersion

HTML全文

图 1 无限大均匀网格的部分示意图

Figure 1. Sketch of a part of an unbounded uniform mesh

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图 2 频散c_p/c随平面波传播方向θ的变化

Figure 2. Variation of dispersion c_p/c with the propagation direction θ

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图 3 频数c_p/c随网格纵横长度比γ的变化

Figure 3. Variation of dispersion c_p/c with the ratio of vertical to horizontal γ

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图 4 θ取不同值时c_p/c随kΔx的变化(a)γ=1；(b)γ=0.4

Figure 4. Variation of c_p/c with kΔx for different values of θ(a)γ=1；(b)γ=0.4

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图 5 地层真实速度为2.0 km/s时的相速度c_p与角频率ω关系

Figure 5. Relationship between phase velocity c_p and angular frequency ω when the real formation velocity is 2.0 km/s

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图 6 不同介质速度时的c_p/c与ω关系

Figure 6. Variation of c_p/c with ωfor different medium velocities

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图 7 不同地层速度c时的不同频率分量波长与ω关系

Figure 7. Variation of wavelength with ω for different real formation velocities

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图 8 地震波在均匀介质中传播到0.46 s时的波场快照图(a)-(h)的网格依次为4m×4m、3m×3m、2.5m×2.5m、2m×2m、 4m×2m、4m×1.6m、4m×1m和4m×0.5m

Figure 8. Snapshots of acoustic wave propagation in the homogeneous media at 0.46 sIn Figs.(a)-(h)，the sizes of grid are 4m×4m，3m×3m，2.5m×2.5m，2m×2m，4m×2m，4m×1.6m，4m×1m，4m×0.5m，respectively

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图 9 震源主频为100 Hz时在0.46 s时的波场快照(a)网格为2 m×2 m；(b)网格为1 m×1 m

Figure 9. Snapshots of acoustic wave propagation at 0.46 s when the peak frequency of source wavelet is 100 Hz(a)The size of grid is 2 m×2 m；(b)The size of grid is 1 m×1 m

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图 10 含一低速夹层的速度模型

Figure 10. A velocity model with a low-velocity interlayer

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图 11 网格分别为3 m×3 m(a)、3 m×1.5 m(b)和1.5 m×1.5 m(c)情况下的单炮记录① 直达波; ② 薄层顶界面反射波; ③ 薄层底界面反射波; ④ 薄层内二次反射波; ⑤ 倾斜界面反射波

Figure 11. Shot records with grid sizes of 3 m×3 m(a)，3 m×1.5 m(b) and 1.5 m×1.5 m(c)① Direct wave; ② The reflection wave coming from the top of low velocity interlayer; ③ The reflection wave coming from the base of low velocity interlayer; ④ The re-reflection wave coming from low velocity interlayer; ⑤ The reflection wave coming from dipping interface

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S1 共点单元示意图带圈的数字为单元内编号，带下标大写字母为结构节点编号; a，b，c，d为单元编号

S1. Scheme of common-point element The circled numbers are the code of nodes in an element，the capital letters subscripted is the number of the node. a，b，c，d are number of elements

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施引文献(5)

期刊类型引用(2)

1.	武晔，赵晓燕，石砚斌，俞子钊，何欣娟，胡泊，熊仲华. 伪谱算法在声波数值模拟中的数值频散分析. 地震. 2017(02): 135-146 . 百度学术
2.	曹丹平，周建科，印兴耀. 三角网格有限元法波动模拟的数值频散及稳定性研究. 地球物理学报. 2015(05): 1717-1730 . 百度学术