基于非结构网格求解三维D'Alembert介质中声波方程的并行加权Runge-Kutta间断有限元方法

贺茜君, 杨顶辉, 仇楚钧, 周艳杰, 常芸凡. 2021. 基于非结构网格求解三维D'Alembert介质中声波方程的并行加权Runge-Kutta间断有限元方法. 地球物理学报, 64(3): 876-895, doi: 10.6038/cjg2021O0226
引用本文: 贺茜君, 杨顶辉, 仇楚钧, 周艳杰, 常芸凡. 2021. 基于非结构网格求解三维D'Alembert介质中声波方程的并行加权Runge-Kutta间断有限元方法. 地球物理学报, 64(3): 876-895, doi: 10.6038/cjg2021O0226
HE XiJun, YANG DingHui, QIU ChuJun, ZHOU YanJie, CHANG YunFan. 2021. A parallel weighted Runge-Kutta discontinuous galerkin method for solving acousitc wave equations in 3D D'Alembert media on unstructured meshes. Chinese Journal of Geophysics (in Chinese), 64(3): 876-895, doi: 10.6038/cjg2021O0226
Citation: HE XiJun, YANG DingHui, QIU ChuJun, ZHOU YanJie, CHANG YunFan. 2021. A parallel weighted Runge-Kutta discontinuous galerkin method for solving acousitc wave equations in 3D D'Alembert media on unstructured meshes. Chinese Journal of Geophysics (in Chinese), 64(3): 876-895, doi: 10.6038/cjg2021O0226

基于非结构网格求解三维D'Alembert介质中声波方程的并行加权Runge-Kutta间断有限元方法

  • 基金项目:

    本研究得到国家自然科学面上基金(41974114)及国家自然科学基金(地震联合基金)项目(U1839206)的联合资助

详细信息
    作者简介:

    贺茜君, 女, 1988年生, 副教授, 主要研究方向为地震波动方程的数值方法及波场模拟.E-mail: hexijun111@sina.com

    通讯作者: 杨顶辉, 教授, 主要从事计算地球物理、孔隙介质波传播理论、地震层析成像等研究.E-mail: ydh@mail.tsinghua.edu.cn
  • 中图分类号: P315;P631

A parallel weighted Runge-Kutta discontinuous galerkin method for solving acousitc wave equations in 3D D'Alembert media on unstructured meshes

More Information
  • 间断有限元方法(Discontinuous Galerkin method,简称DGM)在求解地震波动方程时具有低数值频散、网格剖分灵活等优点,因此,为适应数值模拟对模拟精度和复杂地质结构的要求,本文提出一种新的加权Runge-Kutta间断有限元(weighted Runge-Kutta discontinuous Galerkin,简称WRKDG)方法,用于求解三维D'Alembert介质中声波方程.本文不仅详细推导了其数值格式,特别地,根据常微分方程理论给出了满足数值稳定性条件的一般经验公式,并首次对该方法的数值频散和耗散进行了深入分析,且考虑了耗散参数对结果的影响.同时,我们也对该方法进行了精度测试,并分析了3D情形下WRKDG方法的并行加速比,结果表明3D WRKDG方法具有良好的并行性.最后,我们给出了包含均匀模型、非规则几何模型以及非均匀Marmousi模型在内的数值模拟算例.结果表明,该方法不仅计算准确,能与解析解很好地吻合,且能有效模拟包含球体在内的非规则模型及非均匀Marmousi模型中的衰减声波波场.数值模拟实验进一步验证了WRKDG方法在求解三维D'Alembert介质中声波方程时的正确性和有效性,并获得了对这种强衰减介质中波传播特征的规律性新认识.

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  • 图 1 

    一般四面体单元变换到参考单元示意图(Dumbser and KäserKäser, 2006),其中参考单元内1、2、3和4这四个顶点的坐标分别是(0, 0, 0)、(1, 0, 0)、(0, 1, 0)和(0, 0, 1)

    Figure 1. 

    Transformation from the general tetrahedron to the reference tetrahedron(Dumbser and KäserKäser, 2006)with four vertices (0, 0, 0), (1, 0, 0), (0, 1, 0), and (0, 0, 1)

    图 2 

    Von Neumann分析中用到的网格构型.在(a)中所示的规则六面体剖分的基础上,每个六面体中有图(b)中所示六个四面体网格(Mulder et al., 2014)

    Figure 2. 

    Grid configuration used in Von Neumann analysis. Based on the (a) regular hexahedral division, each hexahedron has (b) six tetrahedrons (Mulder et al., 2014)

    图 3 

    θ=π/2时,P1阶和P2阶WRKDG方法取φ=0, π/12, π/4时数值频散R随采样率δ的变化情况.(a-b)P1阶WRKDG方法,(c-d)P2阶WRKDG方法,(a)、(c)对应耗散参数r=0,而(b)、(d)对应耗散参数r=10

    Figure 3. 

    Numerical dispersion R as a function of sampling rate δ for P1 and P2 WRKDG methods when θ=π/2 and φ=0, π/12, π/4. (a-b) are for P1 WRKDG method, while (c-d) are for P2 WRKDG method. Panels (a) and (c) show the cases for r=0; panels (b) and (d) show the cases for r=10

    图 4 

    θ=π/2时,P1阶和P2阶WRKDG方法取φ=0, π/12, π/4时数值耗散S随采样率δ的变化情况.(a-b)P1阶WRKDG方法,(c-d)P2阶WRKDG方法,(a)、(c)对应耗散参数r=0,而(b)、(d)对应耗散参数r=10

    Figure 4. 

    Numerical dissipation S as a function of sampling rate δ for P1 and P2 WRKDG methods when θ=π/2 and φ=0, π/12, π/4. (a-b) are for P1 WRKDG method, while (c-d) are for P2 WRKDG method. Panels (a) and (c) show the cases for r=0; panels (b) and (d) show the cases for r=10

    图 5 

    (a) 6000个四面体网格;(b)利用Metis将(a)中所有网格分给5个进程

    Figure 5. 

    (a) 6000 tetrahedrons; (b) Metis divides all tetrahedrons in (a) into 5 processors

    图 6 

    并行算法流程图

    Figure 6. 

    Flow chart of the parallel algorithm

    图 7 

    (a) 所考虑进程的辅助网格示意图,即图中绿色网格部分,这类网格属于其他进程的内网格,位于该进程所有网格的边界处;(b) 图a的侧面图;(c) 属于该进程内、作为其他进程的辅助网格,即图中红色网格部分

    Figure 7. 

    (a) Illustration of the auxiliary grids of this processor (the green part in the figure). This type of grids belongs to the internal grids of other processors, and is located at the boundary of this process; (b) the side view of figure a; (c) the auxiliary grids of other processors which belongs to this processor (the red part in the figure)

    图 8 

    3D WRKDG算法的并行加速比(图中线型“-o”表示).其中横轴表示进程数,纵轴表示并行加速比.图中线型“-*”表示理想情形下的并行加速比

    Figure 8. 

    Parallel speed-ups of the 3D WRKDG algorithm (see the line type "-o"). The horizontal axis is the number of processors, and the vertical axis is the parallel speed-ups. The line type "-*" in the figure represents the parallel speed-up in the ideal case

    图 9 

    在耗散参数r=0的3D均匀介质模型中,T=0.5 s时刻接收点处的归一化的波形记录图,图中红色实线代表解析解,蓝色虚线及黑色实线分别表示利用P2P1方法得到的数值解

    Figure 9. 

    Comparisons of normalized waveforms at time T=0.5 s for the 3D homogeneous model with dissipation parameter r=0, in which the red solid line in the figure represents the analytical solution, and the blue dotted line and black solid line represent the numerical solution computed by the P2 and P1 methods, respectively

    图 10 

    在耗散参数r=0的3D均匀介质模型中,使用P2方法计算得到的T=0.5 s时刻的波场快照图

    Figure 10. 

    Snapshots of the acoustic wave fields computed by the P2 method at time T=0.5 s for the 3D homogeneous model with dissipation parameter r=0

    图 11 

    在均匀介质模型中,不同耗散参数r=0、2、4、8和16对应的接收器处的声波波形记录

    Figure 11. 

    Waveform records at the receiver with different dissipation parameters r=0, 2, 4, 8 and 16 for the homogeneous model

    图 12 

    非规则几何模型示意图,在计算区域0≤x, y, z≤2中,有一个球状区域,球中心坐标(1, 1, 0.5)km,半径0.2 km

    Figure 12. 

    Illustration of the irregular geometric model. In the computational domain 0≤x, y, z≤2 km, there is a spherical area with spherical center coordinates (1, 1, 0.5) km and a radius of 0.2 km

    图 13 

    (a) 球体部分四面体网格的3D示意图;(b) 二维剖面y=0处的网格剖分示意图

    Figure 13. 

    Illustration of (a) tetrahedrons in the ball and (b) the grid division at the cross section y=0

    图 14 

    T=0.3 s时刻的波场快照图,其中(a)对应于无耗散情形r=0,而(b)对应于耗散参数r=4

    Figure 14. 

    Snapshots of the seismic waves at T=0.3 s with dissipation parameters (a) r =0 and (b) r=4

    图 15 

    3D Marmousi模型

    Figure 15. 

    3D Marmousi model

    图 16 

    对于3D Marmousi模型,T=1.0 s时刻的波场快照图,其中耗散参数r=2

    Figure 16. 

    Snapshots at T=1.0 s for the 3D Marmousi model with dissipation parameter r=2

    表 1 

    3D D′ Alembert介质中P1阶和P2阶WRKDG方法的真实最大库朗数(αmax)D′Ale与从经验公式(28)获得的值(αmax)′D′Ale的对比

    Table 1. 

    Comparisons of the actual maximum Courant numbers (αmax)D′Aleand the values (αmax)′D′Ale given by empirical formula (28) for 3D P1 and P2 WRKDG methods in D′ Alembert media

    P1 P2
    hr (αmax)D′Ale (αmax)′D′Ale hr (αmax)D′Ale (αmax)′D′Ale
    100 0.049 0.049 100 0.043 0.043
    50 0.082 0.082 50 0.067 0.066
    10 0.175 0.175 10 0.120 0.117
    5 0.205 0.204 5 0.133 0.129
    下载: 导出CSV

    表 2 

    3D D′ Alembert介质中P2阶WRKDG的误差及收敛阶

    Table 2. 

    Convergence rates of u for the acoustic wave in 3D D′Alembert medium

    h(km) L2 error order L1 error order
    r=1 5.00×10-1 1.430×10-2 - 2.978×10-2 -
    2.50×10-1 1.728×10-3 3.049 3.770×10-3 2.982
    2.00×10-1 8.127×10-4 3.381 1.714×10-3 3.533
    1.25×10-1 1.925×10-4 3.064 4.085×10-4 3.051
    1.00×10-1 1.009×10-4 2.893 2.137×10-4 2.904
    r=10 5.00×10-1 9.249×10-3 - 2.091×10-2 -
    2.50×10-1 1.063×10-3 3.121 2.386×10-3 3.132
    2.00×10-1 5.200×10-4 3.205 1.133×10-3 3.338
    1.25×10-1 1.221×10-4 3.082 2.675×10-4 3.071
    1.00×10-1 6.345×10-5 2.934 1.388×10-4 2.941
    下载: 导出CSV

    表 3 

    3D D′ Alembert介质中波形记录的波谷处的振幅和衰减系数

    Table 3. 

    The amplitudes and attenuation ratios at the trough at the receiver for the acoustic wave in D′Alembert medium

    r Amplitude at trough Attenuation ratio
    at trough
    Theoretical attenuation
    ratio erT/2
    0 -2.647×10-3 - -
    2 -2.159×10-3 0.816 0.817
    4 -1.759×10-3 0.664 0.668
    8 -1.172×10-3 0.443 0.446
    16 -5.234×10-4 0.198 0.199
    下载: 导出CSV

    表 A1 

    四面体单元所属四个面的定义顺序(Dumbser and KäserKäser, 2006)

    Table A1. 

    Face Definition on tetrahedrons (Dumbser and KäserKäser, 2006)

    Face Vertices
    1 1 3 2
    2 1 2 4
    3 1 4 3
    4 2 3 4
    下载: 导出CSV

    表 A2 

    (a) 三维坐标轴ξη,和ζ与面积分用到的参变量χτ之间的对应关系;(b)对于不同的h,在Ωi中的参变量χτ与相邻单元Ωj中参变量χ′和τ′的对应关系(Dumbser and KäserKäser, 2006)

    Table A2. 

    (a) Relationship between the three-dimensional coordinate axes ξ, η, and ζ and the face parameters χ and τ used in the area integrals; (b)Relationship between the face parameters χ and τ in the tetrahedron Ωi and the face parameters χ′ and τ′ in the adjacent tetrahedron Ωj (Dumbser and KäserKäser, 2006)

    Face 1 2 3 4
    ξ τ χ 0 1-χ-τ
    η χ 0 τ χ
    ζ 0 τ χ τ
    (a)
    h 1 2 3
    χ τ 1-χ-τ χ
    τ′ χ τ 1-χ-τ
    (b)
    下载: 导出CSV
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出版历程
收稿日期:  2020-06-17
修回日期:  2020-12-29
上线日期:  2021-03-10

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