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

印兴耀, 周建科, 吴国忱, 梁锴. 2014: 有限元算法在声波方程数值模拟中的频散分析. 地震学报, 36(5): 944-1898. DOI: 10.3969/j.issn.0253-3782.2014.05.017
引用本文: 印兴耀, 周建科, 吴国忱, 梁锴. 2014: 有限元算法在声波方程数值模拟中的频散分析. 地震学报, 36(5): 944-1898. DOI: 10.3969/j.issn.0253-3782.2014.05.017
Yin Xingyao, Zhou Jianke, Wu Guochen, Liang Kai. 2014: Dispersion analysis for the finite element algorithm in acoustic wave equation numerical simulation. Acta Seismologica Sinica, 36(5): 944-1898. DOI: 10.3969/j.issn.0253-3782.2014.05.017
Citation: Yin Xingyao, Zhou Jianke, Wu Guochen, Liang Kai. 2014: Dispersion analysis for the finite element algorithm in acoustic wave equation numerical simulation. Acta Seismologica Sinica, 36(5): 944-1898. 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

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

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

  • 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.
    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.
  • 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

    图  2   频散cp/c随平面波传播方向θ的变化

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

    图  3   频数cp/c随网格纵横长度比γ的变化

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

    图  4   θ取不同值时cp/ckΔx的变化(a)γ=1;(b)γ=0.4

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

    图  5   地层真实速度为2.0 km/s时的相速度cp与角频率ω关系

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

    图  6   不同介质速度时的cp/cω关系

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

    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

    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

    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

    图  10   含一低速夹层的速度模型

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

    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

    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

  • 董良国, 李培明. 2004. 地震波传播数值模拟中的频散问题[J]. 天然气工业, 24(6): 53-56.

    Dong L G, Li P M. 2004. Dispersive problem in seismic wave propagation numerical modeling[J]. Natural Gas Industry, 24(6): 53-56 (in Chinese).

    杜世通. 1982. 变速不均匀介质中波动方程的有限元法数值解[J]. 华东石油学院学报, 6(2): 1-20.

    Du S T. 1982. Finite element numerical solution of wave propagation in non-homogeneous medium with variable velocities[J]. Journal of East China Petroleum Institute, 6(2): 1-20 (in Chinese).

    房营光, 莫海鸿. 2000. 有限元网格中波动的频散与稳定性的一种改进方法[J]. 地震工程与工程振动, 20(1): 21-26.

    Fang Y G, Mo H H. 2000. An improved method for dispersion and stability of wave motion in finite element meshes[J]. Journal of Earthquake Engineering and Engineering Vibration, 20(1): 21-26 (in Chinese).

    孙成禹. 2007. 地震波理论与方法[M]. 东营: 中国石油大学出版社: 31-37.

    Sun C Y. 2007. Theory and Methods of Seismic Waves[M]. Dongying: China University of Petroleum Press: 31-37 (in Chinese).

    孙成禹, 宫同举, 张玉亮, 张文颖. 2009. 波动方程有限差分法中的频散与假频分析[J]. 石油地球物理勘探, 44(1): 43-48.

    Sun C Y, Gong T J, Zhang Y L, Zhang W Y. 2009. Analysis on dispersion and alias in finite-difference solution of wave equation[J]. Oil Geophysical Prospecting, 44(1): 43-48 (in Chinese).

    徐世浙. 1994. 地球物理中的有限元法[M]. 北京: 科学出版社: 135-146.

    Xu S Z. 1994. Finite Element Method for Geophysics[M]. Beijing: Science Press: 135-146 (in Chinese).

    薛东川, 王尚旭. 2008. 利用组合质量矩阵压制数值频散[J]. 石油地球物理勘探, 43(3): 318-320.

    Xue D C, Wang S X. 2008. Using combined mass matrix to suppress numerical dispersion[J]. Oil Geophysical Prospecting, 43(3): 318-320 (in Chinese).

    Abboud N N, Pinsky P M. 1992. Finite element dispersion analysis for the three-dimensional second-order scalar wave equation[J]. Int J Numer Meth Engng, 35(6): 1183-1218.

    Christon M A. 1999. The influence of the mass matrix on the dispersive nature of the semi-discrete, second-order wave equation[J]. Comput Methods Appl Mech Engng, 173(1): 147-166.

    De Basabe J D, Sen M K. 2007. Grid dispersion and stability criteria of some common finite-element methods for acoustic and elastic wave equations[J]. Geophysics, 72(6): T81-T95.

    Liu J B, Sharan S K, Yao L. 1994. Wave motion and its dispersive properties in a finite element model with distortional elements[J]. Comput Struct, 52(2): 205-214.

    Liu Y, Sen M K. 2009. A new time-space domain high-order finite-difference method for the acoustic wave equation[J].J Comput Phys, 228(23): 8779-8806.

    Mullen R, Belytschko T. 1982. Dispersion analysis of finite element semidiscretizations of the two-dimensional wave equation[J]. Int J Numer Meth Engng, 18(1): 11-29.

    Seriani G, Oliveira S P. 2008. Dispersion analysis of spectral element methods for elastic wave propagation[J]. Wave Motion, 45(6): 729-744.

  • 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

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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

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