间断Galerkin 有限元法在研究地震波传播和破裂动力学中的应用

地震学领域中，数值方法和计算机硬件共同促进了计算地震学的发展。随着计算机的性能大幅提升，计算地震学的关注重点逐渐转移到了数值方法的研究上。间断Galerkin 有限元法（Discontinuous Galerkin finite element method，DG-FEM）被引入到地震学中，为复杂地质模型和断层破裂动力学的数值模拟提供了新的求解方案。DG-FEM 继承了有限元类方法灵活的网格特性，能够对任意复杂几何结构的地质模型实现精确划分，同时DG-FEM 的基函数允许单元边界上不连续，特别适用于对高度非线性问题（破裂动力学）求解。基于上述背景，本文对DG-FEM 在地震波场传播和震源破裂动力学的数值模拟计算展开研究。
在截断计算域内，人工边界的虚假反射破坏了波场的真实信息。当DG-FEM在求解弹性波动方程时，由于复杂介质内含有多种类型和不同速度的波，且DG-FEM 单元间采用数值通量进行信息交换，完美匹配层（PML）吸收边界很难在长时间的计算中保持稳定。本文提出了辅助常微分方程复频移多轴PML，首先将辅助偏微分方程变为空间常微分方程，再加入多轴PML 技术提升了PML 吸收边界的稳定性，并用全空间模型和半空间模型进行了验证。不同网格大小和不同单元阶数的测试表明，本文PML 的计算效率较高。
和其他数值方法相比，常规的DG-FEM 计算效率低，但是DG-FEM 允许单元之间选择不同阶数的基函数，导致其质量矩阵、刚度矩阵都是局部的，有利于实现并行计算。一方面，本文利用METIS（对图形和网格分区开源软件）对网格进行分区实现负载均衡，再利用MPI 作为进程间的信息通信实现并行计算，加速比最高可达80%。另一方面，在保证精度的前提下，对非结构化网格采取𝑝 型自适应算法，通过对小尺寸网格进行单元降阶的方法来增加全局时间步长，有效降低了计算时间，且具有普适性。
在震源破裂动力学问题中，断层界面两侧的应力分量和速度分量不一定连续，DG-FEM 在原理上不需要相邻单元的变量在界面处连续（尽管在物理特性上是连续的），使DG-FEM 在求解断层破裂动力学上具备了理论优势，并且方便在断层边界面使用空间高阶形式。本文通过求解Riemann 问题获得断层两侧的应力和速度分量值，将摩擦准则和数值通量结合，成功施加了断层处的边界条件。通过对南加州地震中心的标准断层模型进行自发破裂模拟，与谱元法模拟结果对比，本文所提方法具有较高的准确性和可靠性，且方便施加其他显式时间积分方案。
综上，本文发展了一套用DG-FEM 求解地震波传播和震源破裂动力学的数值方法，对研究复杂几何介质的地震灾害具有重要参考价值，为后续发展复杂几何断层的破裂动力学数值方法奠定了基础。

In the field of seismology, numerical methods and computer hardware have jointly contributed to the development of computational seismology. With high performance of computer, the focus of computational seismology has gradually shifted to the study of numerical methods. The discontinuous Galerkin finite element method (DG-FEM) is introduced into seismology, which provides a new solution for the numerical simulation of complex geological models and fault rupture dynamics. DG-FEM inherits the flexible grid characteristics of finite element methods, and can accurately divide geological models with arbitrary complex structures. At the same time, DG-FEM introduces the concept of numerical flux in finite volume methods, which is especially suitable for highly nonlinear problems (rupture dynamics). At the same time, the basis function of DG-FEM allows discontinuities on the element boundary, which is especially suitable for solving highly nonlinear problems (rupture dynamics). Based on the above background, this dissertation studies the numerical simulation calculation of DG-FEM in seismic wavefield propagation and source rupture dynamics.
In the truncated computational domain, spurious reflections from artificial bound-aries affect the true wavefield. Since the complex medium contains various waves and different velocities, and DG-FEM uses the numerical flux to exchange information be-tween units, when DG-FEM solves the elastic wave equation, it is hard to remain stable in long-term calculations using the absorbing boundary of the perfectly matched layer. This dissertation proposes an auxiliary ordinary differential equation complex frequency-shifted multi-axis PML is proposed, First, the auxiliary partial differential equation is changed into a space ordinary differential equation, and then adding the multi-axis PML technology to improve the stability of the PML absorption boundary, Both full-space model and half-space model tests have verified its performance. Tests with different grid sizes and different element orders approve the high efficiency of the proposed PML.
Compared with other numerical methods, the conventional DG-FEM is computa-tionally inefficient. However, DG-FEM allows the selection of basis functions of different orders between elements, resulting in local mass and stiffness matrices, which is conducive to parallel computing. On the one hand, this dissertation uses METIS (open source software for graph and grid partitioning) to partition the grid to achieve load balancing. Then use MPI as the information communication between processes to realize parallel comput-ing, and the speedup ratio can reach up to 80 %. On the other hand, a 𝑝-type adaptive algorithm is adopted for unstructured grids, the global time step is increased by reducing the element order of small-sized grids, which effectively reduces the calculation time and is universal.
In the problem of rupture dynamics, the stress components and velocity components on both sides of the fault interface are not necessarily continuous. DG-FEM does not require the variables of adjacent elements to be continuous at the interface (although physically continuous), this gives DG-FEM a theoretical advantage in solving fault rupture dynamics. And it is convenient to use the spatial high-order form at the fault boundary surface. In this dissertation, the stress and velocity components on both sides of the fault are obtained by solving the Riemann problem, and the boundary conditions at the fault are successfully applied by combining the friction criterion and the numerical flux. By simulating the spontaneous rupture of the standard fault model of the Southern California Earthquake Center, compared with the simulation results of the spectral element method, the method proposed in this dissertation has high accuracy and reliability, and it is convenient to apply other explicit time integration schemes.
In conclusion, this dissertation develops a theory of seismic wave propagation and source rupture dynamics using DG-FEM. It has important reference value for studying earthquake disasters in complex geometric media. It lays a foundation for the subsequent development of numerical methods for rupture dynamics of complex geometrical faults.

[1] ZHANG Z, ZHANG W, CHEN X F, et al. Rupture dynamics and ground motion from potential earthquakes around taiyuan, china[J]. Bulletin of the Seismological Society of America, 2017, 107(3):1201-1212.
[2] OLSEN K B, ARCHULETA R J. Three-dimensional simulation of earthquakes on the los angeles fault system[J]. Bulletin of the Seismological Society of America, 1996, 86(3):575-596.
[3] ZHANG Z, ZHANG W, CHEN X F. Three-dimensional curved grid finitedifference modelling for non-planar rupture dynamics[J]. Geophysical Journal International, 2014, 199(2):860-879.
[4] HUANG H, ZHANG Z, CHEN X F. Investigation of topographical effects on rupture dynamics and resultant ground motions[J]. Geophysical Journal International, 2018, 212(1):311-323.
[5] COCKBURN B, KARNIADAKIS G, SHU C. Discontinuous galerkin methods: Theory, computation and applications[R]. [S.l.]: Brown University, Providence, RI (US), 2000.
[6] HESTHAVEN J S, WARBURTON T. Nodal discontinuous galerkin methods: algorithms, analysis, and applications[M]. [S.l.]: Springer Science & Business Media, 2007.
[7] MOCZO P, KRISTEK J, GALIS M. The finite-difference modelling of earthquake motions: Waves and ruptures[M]. [S.l.]: Cambridge University Press, 2014.
[8] IGEL H. Computational seismology: a practical introduction[M]. [S.l.]: Oxford University Press, 2017.
[9] ALTERMAN Z, ABOUDI J, KARAL F. Pulse propagation in a laterally heterogeneous solid elastic sphere[J]. Geophysical Journal International, 1970, 21(3): 243-260.
[10] BOORE D M. Love waves in nonuniform wave guides: Finite difference calculations[J]. Journal of Geophysical Research, 1970, 75(8):1512-1527.
[11] ALFORD R, KELLY K, BOORE D M. Accuracy of finite-difference modeling of the acoustic wave equation[J]. Geophysics, 1974, 39(6):834-842.
[12] MADARIAGA R. Dynamics of an expanding circular fault[J]. Bulletin of the Seismological Society of America, 1976, 66(3):639-666.
[13] VIRIEUX J, MADARIAGA R. Dynamic faulting studied by a finite difference method[J]. Bulletin of the Seismological Society of America, 1982, 72(2):345-369.
[14] VIRIEUX J. Sh-wave propagation in heterogeneous media: Velocity-stress finitedifference method[J]. Geophysics, 1984, 49(11):1933-1942.
[15] VIRIEUX J. P-sv wave propagation in heterogeneous media: Velocity-stress finitedifference method[J]. Geophysics, 1986, 51(4):889-901.
[16] FRANKEL A. A review of numerical experiments on seismic wave scattering[J]. Scattering and attenuation of seismic waves, Part II, 1989:639-685.
[17] GRAVES R W. Modeling three-dimensional site response effects in the marina district basin, san francisco, california[J]. Bulletin of the Seismological Society of America, 1993, 83(4):1042-1063.
[18] PITARKA A. 3d elastic finite-difference modeling of seismic motion using staggered grids with nonuniform spacing[J]. Bulletin of the Seismological Society of America, 1999, 89(1):54-68.
[19] 张伟. 含起伏地形的三维非均匀介质中地震波传播的有限差分算法及其在强地面震动模拟中的应用[D]. 北京: 北京大学, 2006.
[20] REN H X, CHEN X F, HUANG Q H. Numerical simulation of coseismic electromagnetic fields associated with seismic waves due to finite faulting in porous media[J]. Geophysical Journal International, 2012, 188(3):925-944.
[21] ZHANG W, ZHANG Z, CHEN X F. Three dimensional elastic wave numerical modelling in the presence of surface topography by a collocated-grid finitedifference method on curvilinear grids[J]. Geophysical Journal International, 2012,190(1):358-378.
[22] SUN Y C, ZHANG W, CHEN X F. Seismic-wave modeling in the presence of surface topography in 2d general anisotropic media by a curvilinear grid finitedifference method[J]. Bulletin of the Seismological Society of America, 2016, 106 (3):1036-1054.
[23] SUN Y C, ZHANG W, XU J K, et al. Numerical simulation of 2-d seismic wave propagation in the presence of a topographic fluid–solid interface at the sea bottom by the curvilinear grid finite-difference method[J]. Geophysical Journal International, 2017, 210(3):1721-1738.
[24] SUN Y C, ZHANG W, CHEN X F. 3d seismic wavefield modeling in generally anisotropic media with a topographic free surface by the curvilinear grid finitedifference method[J]. Bulletin of the Seismological Society of America, 2018, 108 (3A):1287-1301.
[25] SUN Y C, REN H, ZHENG X Z, et al. 2-d poroelastic wave modelling with a topographic free surface by the curvilinear grid finite-difference method[J]. Geophysical Journal International, 2019, 218(3):1961-1982.
[26] ZANG N, ZHANG W, CHEN X F. An overset-grid finite-difference algorithm for simulating elasticwave propagation in media with complex free-surface topography [J]. Geophysics, 2021, 86(4):T277-T292.
[27] WANG W, ZHANG Z, ZHANG W, et al. Cgfdm3d-eqr: A platform for rapid response to earthquake disasters in 3d complex media[J]. Seismological Research Letters, 2022.
[28] VIRIEUX J, OPERTO S. An overview of full-waveform inversion in exploration geophysics[J]. Geophysics, 2009, 74(6):WCC1-WCC26.
[29] GAZDAG J. Modeling of the acoustic wave equation with transform methods[J]. Geophysics, 1981, 46(6):854-859.
[30] KOSLOFF D D, BAYSAL E. Forward modeling by a fourier method[J]. Geophysics, 1982, 47(10):1402-1412.
[31] KOSLOFF D, RESHEF M, LOEWENTHAL D. Elastic wave calculations by the fourier method[J]. Bulletin of the Seismological Society of America, 1984, 74(3):875-891.
[32] CARCIONE J M, WANG P J. A chebyshev collocation method for the wave equation in generalized coordinates[J]. geophysics, 1993.
[33] FURUMURA T, KENNETT B, FURUMURA M. Seismic wavefield calculation for laterally heterogeneous whole earth models using the pseudospectral method [J]. Geophysical Journal International, 1998, 135(3):845-860.
[34] FURUMURA T, KENNETT B, TAKENAKA H. Parallel 3-d pseudospectral simulation of seismic wave propagation[J]. Geophysics, 1998, 63(1):279-288.
[35] FURUMURA T, KOKETSU K, WEN K L. Parallel psm/fdm hybrid simulation of ground motions from the 1999 chi-chi, taiwan, earthquake[J]. Pure and Applied Geophysics, 2002, 159(9):2133-2146.
[36] ZHAO G, LIU Q H. The 3-d multidomain pseudospectral time-domain algorithm for inhomogeneous conductive media[J]. IEEE Transactions on Antennas and Propagation, 2004, 52(3):742-749.
[37] LYAKHOVSKY V, BEN-ZION Y. Evolving geometrical and material properties of fault zones in a damage rheology model[J]. Geochemistry, Geophysics, Geosystems, 2009, 10(11).
[38] SCHLUE J. Finite element matrices for seismic surfacewaves in three-dimensional structures[J]. Bulletin of the Seismological Society of America, 1979, 69(5):1425-1438.
[39] MARFURT K J. Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equations[J]. Geophysics, 1984, 49(5):533-549.
[40] SERON F, SANZ F, KINDELAN M, et al. Finite-element method for elastic wave propagation[J]. Communications in applied numerical methods, 1990, 6(5):359-368.
[41] ZIENKIEWICZ O C, TAYLOR R L, ZHU J Z. The finite element method: its basis and fundamentals[M]. [S.l.]: Elsevier, 2005.
[42] BIELAK J. Ground motion modeling using 3d finite element methods[J]. Proc. 2nd Intl. Sympo. Effects of Surface Geology on Seismic Motion, 1998, 1998, 1:121-133.
[43] BIELAK J,XUJ,GHATTASO. Earthquake ground motion and structural response in alluvial valleys[J]. Journal of Geotechnical and Geoenvironmental Engineering, 1999, 125(5):413-423.
[44] ASKAN A, BIELAK J. Full anelastic waveform tomography including model uncertainty[J]. Bulletin of the Seismological Society of America, 2008, 98(6):2975-2989.
[45] EPANOMERITAKIS I, AKCELIKV,GHATTAS O, et al. Anewton-cg method for large-scale three-dimensional elastic full-waveform seismic inversion[J]. Inverse Problems, 2008, 24(3):034015.
[46] 杨顶辉. 双相各向异性介质中弹性波方程的有限元解法及波场模拟[J]. 地球物理学报, 2002, 45(4):575-583.
[47] MOCZO P, KRISTEK J, GALIS M, et al. On accuracy of the finite-difference and finite-element schemes with respect to p-wave to s-wave speed ratio[J]. Geophysical Journal International, 2010, 182(1):493-510.
[48] HUANG X, DI B, WEI J, et al. Application of generalized stiffness reduction method in solving biot’s poroelastic wave equations using finite-element method [C]//82nd EAGE Annual Conference & Exhibition: volume 2020. [S.l.]: European Association of Geoscientists & Engineers, 2020: 1-5.
[49] PATERA A T. A spectral element method for fluid dynamics: laminar flow in a channel expansion[J]. Journal of computational Physics, 1984, 54(3):468-488.
[50] MADAYY, PATERAAT. Spectral element methods for the incompressible navierstokes equations[J]. IN: State-of-the-art surveys on computational mechanics (A90-47176 21-64). New York, 1989:71-143.
[51] PRIOLO E, CARCIONE J M, SERIANI G. Numerical simulation of interface waves by high-order spectral modeling techniques[J]. The Journal of theAcoustical Society of America, 1994, 95(2):681-693.
[52] SERIANI G, PRIOLO E. Spectral element method for acoustic wave simulation in heterogeneous media[J]. Finite elements in analysis and design, 1994, 16(3-4):337-348.
[53] CAPDEVILLE Y. Methode couplee elements spectraux-solution modale pour la propagation d’ondes dans la terre a l’echelle globale[D]. [S.l.]: Paris 7, 2000.
[54] KOMATITSCH D, TROMP J. Spectral-element simulations of global seismic wave propagation—i. validation[J]. Geophysical Journal International, 2002, 149 (2):390-412.
[55] KOMATITSCH D,TROMPJ. Spectral-element simulations of global seismic wave propagation—ii. three-dimensional models, oceans, rotation and self-gravitation[J].Geophysical Journal International, 2002, 150(1):303-318.
[56] PETER D, KOMATITSCH D, LUO Y, et al. Forward and adjoint simulations of seismic wave propagation on fully unstructured hexahedral meshes[J]. Geophysical Journal International, 2011, 186(2):721-739.
[57] 刘有山, 滕吉文, 徐涛, 等. 三角网格谱元法地震波场数值模拟[J]. 地球物理学进展, 2014, 29(4):1715-1726.
[58] 董兴朋, 杨顶辉. 球坐标系下谱元法三维地震波场模拟[J]. 地球物理学报, 2017, 60(12):4671-4680.
[59] 张平, 韩立国. 基于谱元法的各向异性黏弹性介质地震波场模拟和分析[J].世界地质, 2017, 36(1):255-265.
[60] DORMY E, TARANTOLA A. Numerical simulation of elastic wave propagation using a finite volume method[J]. Journal of Geophysical Research: Solid Earth, 1995, 100(B2):2123-2133.
[61] LEVEQUE R J, et al. Finite volume methods for hyperbolic problems: volume 31 [M]. [S.l.]: Cambridge university press, 2002.
[62] DUMBSER M, KASER M, DE LA PUENTE J. Arbitrary high-order finite volume schemes for seismic wave propagation on unstructured meshes in 2d and 3d[J]. Geophysical Journal International, 2007, 171(2):665-694.
[63] BENJEMAA M, GLINSKY-OLIVIER N, CRUZ-ATIENZA V, et al. Dynamic non-planar crack rupture by a finite volume method[J]. Geophysical Journal International, 2007, 171(1):271-285.
[64] ZHANG W, ZHUANG Y, ZHANG L. A new high-order finite volume method for 3d elastic wave simulation on unstructured meshes[J]. Journal of Computational Physics, 2017, 340:534-555.
[65] SANCHEZ-SESMA F J, CAMPILLO M. Diffraction of p, sv, and rayleigh waves by topographic features: A boundary integral formulation[J]. Bulletin of the seismological Society of America, 1991, 81(6):2234-2253.
[66] CHAILLAT S, SEMBLAT J F, BONNET M. A preconditioned 3-d multi-region fast multipole solver for seismic wave propagation in complex geometries[J]. Communications in computational physics, 2012, 11(2):594-609.
[67] GRASSO E, CHAILLAT S, BONNET M, et al. Application of the multi-level time-harmonic fast multipole bem to 3-d visco-elastodynamics[J]. Engineering Analysis with Boundary Elements, 2012, 36(5):744-758.
[68] SANCHEZ-SESMA F, RAMOS-MARTINEZ J, CAMPILLO M. An indirectboundary element method applied to simulate the seismic response of alluvial valleys for incident p, s and rayleigh waves[J]. Earthquake engineering&structural dynamics, 1993, 22(4):279-295.
[69] SANCHEZ-SESMAF J,LUZONF. Seismic response of three-dimensional alluvial valleys for incident p, s, and rayleighwaves[J]. Bulletin of the Seismological Society of America, 1995, 85(1):269-284.
[70] LIANGJ, LIU Z,HUANGL, et al. The indirect boundary integral equation method for the broadband scattering of plane p, sv and rayleigh waves by a hill topography [J]. Engineering Analysis with Boundary Elements, 2019, 98:184-202.
[71] KASER M, IGEL H. Numerical simulation of 2dwave propagation on unstructured grids using explicit differential operators[J]. Geophysical Prospecting, 2001, 49(5):607-619.
[72] KASER M, IGEL H, SAMBRIDGE M, et al. A comparative study of explicit differential operators on arbitrary grids[J]. Journal of Computational Acoustics, 2001, 9(03):1111-1125.
[73] KASER M,DUMBSERM. An arbitrary high-order discontinuous galerkin method for elastic waves on unstructured meshes—i. the two-dimensional isotropic case with external source terms[J]. Geophysical Journal International, 2006, 166(2):855-877.
[74] DUMBSERM, KASER M. An arbitrary high-order discontinuous galerkin method for elastic waves on unstructured meshes—ii. the three-dimensional isotropic case [J]. Geophysical Journal International, 2006, 167(1):319-336.
[75] KASER M, DUMBSER M, DE LA PUENTE J, et al. An arbitrary high-order discontinuous galerkin method for elastic waves on unstructured meshes—iii. Viscoelastic attenuation[J]. Geophysical Journal International, 2007, 168(1):224-242.
[76] DE LA PUENTE J, KASER M, DUMBSER M, et al. An arbitrary high-order discontinuous galerkin method for elasticwaves on unstructured meshes-iv. Anisotropy [J]. Geophysical Journal International, 2007, 169(3):1210-1228.
[77] S H J, WARBURTON, 李继春, 等. 交点间断Galerkin 方法: 算法, 分析和应 用[M]. 北京: 科学出版社, 2011.
[78] WILCOX L C, STADLER G, BURSTEDDE C, et al. A high-order discontinuous galerkin method for wave propagation through coupled elastic–acoustic media[J]. Journal of Computational Physics, 2010, 229(24):9373-9396.
[79] HERMANN V, KASER M, CASTRO C E. Non-conforming hybrid meshes for efficient 2-d wave propagation using the discontinuous galerkin method[J]. Geophysical Journal International, 2011, 184(2):746-758.
[80] ETIENNE V, CHALJUB E, VIRIEUX J, et al. An hp-adaptive discontinuous galerkin finite-element method for 3-d elastic wave modelling[J]. Geophysical Journal International, 2010, 183(2):941-962.
[81] LISITSA V, TCHEVERDA V, BOTTER C. Combination of the discontinuous galerkin method with finite differences for simulation of seismic wave propagation [J]. Journal of Computational Physics, 2016, 311:142-157.
[82] 薛昭, 董良国, 李晓波, 等. 起伏地表弹性波传播的间断Galerkin 有限元数值模拟方法[J]. 地球物理学报, 2014, 57(4):15.
[83] 贺茜君, 杨顶辉, 吴昊. 间断有限元方法的数值频散分析及其波场模拟[J].地球物理学报, 2014, 57(3):906-917.
[84] 贺茜君, 杨顶辉, 仇楚钧, 等. 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[J]. 地球物理学报, 2021.
[85] ZHAN Q, SUN Q, REN Q, et al. A discontinuous galerkin method for simulating the effects of arbitrary discrete fractures on elastic wave propagation[J]. Geophysical Journal International, 2017, 210(2):1219-1230.
[86] ZHANG Y, GAO J, HAN W, et al. A discontinuous galerkin method for seis-mic wave propagation in coupled elastic and poroelastic media[J]. Geophysical Prospecting, 2019, 67(5):1392-1403.
[87] ZHANG Y, ZHAO H, YAN W, et al. A unified numerical scheme for coupled multiphysics model[J]. IEEE Transactions on Geoscience and Remote Sensing, 2020, 59(10):8228-8240.
[88] MARUYAMA T. On the force equivalents of dynamically elastic dislocations with reference to the earthquake mechanism[J]. Bull.ssm.soc.Am, 1963, 41.
[89] BURRIDGE R, KNOPOFF L. Body force equivalents for seismic dislocations[J]. Bulletin of the Seismological Society of America, 1964, 54(6A):1875-1888.
[90] BYERLEE J. Friction of rocks[M]//Rock friction and earthquake prediction. [S.l.]: Springer, 1978: 615-626.
[91] DIETERICH J H. Time-dependent friction as a possible mechanism for aftershocks [J]. Journal of Geophysical Research, 1972, 77(20):3771-3781.
[92] DIETERICH J H. Time-dependent friction in rocks[J]. Journal of Geophysical Research, 1972, 77(20):3690-3697.
[93] DIETERICH J H. Time-dependent friction and the mechanics of stick-slip[M]//Rock friction and earthquake prediction. [S.l.]: Springer, 1978: 790-806.
[94] DIETERICH J H. Modeling of rock friction: 1. experimental results and consti-tutive equations[J]. Journal of Geophysical Research: Solid Earth, 1979, 84(B5): 2161-2168.
[95] SCHOLZ C, MOLNAR P, JOHNSON T. Detailed studies of frictional sliding of granite and implications for the earthquake mechanism[J]. Journal of geophysical research, 1972, 77(32):6392-6406.
[96] SCHOLZ C, ENGELDER J T. The role of asperity indentation and ploughing in rock friction—i: Asperity creep and stick-slip[C]//International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts: volume 13. [S.l.]: Elsevier, 1976: 149-154.
[97] KOSLOFF D D, LIU H P. Reformulation and discussion of mechanical behavior of the velocity-dependent friction law proposed by dieterich[J]. Geophysical Research Letters, 1980, 7(11):913-916.
[98] RUINA A L. Friction laws and instabilities: A quasistatic analysis of some dry frictional behavior[D]. [S.l.]: Brown University, 1981.
[99] RUINA A. Slip instability and state variable friction laws[J]. Journal of Geophys-ical Research: Solid Earth, 1983, 88(B12):10359-10370.
[100] ANDREWS D. Rupture propagation with finite stress in antiplane strain[J]. Journal of Geophysical Research, 1976, 81(20):3575-3582.
[101] IDA Y. Cohesive force across the tip of a longitudinal-shear crack and griffith’s specific surface energy[J]. Journal of Geophysical Research, 1972, 77(20):3796-3805.
[102] ANDREWS D. A numerical study of tectonic stress release by underground explosions[J]. Bulletin of the Seismological Society of America, 1973, 63(4): 1375-1391.
[103] ANDREWS D. Rupture velocity of plane strain shear cracks[J]. Journal of Geo-physical Research, 1976, 81(32):5679-5687.
[104] DAY S M, MINSTER J B. Numerical simulation of attenuated wavefields using a padé approximant method[J]. Geophysical Journal International, 1984, 78:105-118.
[105] OLSEN K B, ARCHULETA R J, MATARESE J R. Three-dimensional simulation of a magnitude 7.75 earthquake on the san andreas fault[J]. Science, 1995, 270 (5242):1628-1632.
[106] OLSEN K, MADARIAGA R, ARCHULETA R J. Three-dimensional dynamic simulation of the 1992 landers earthquake[J]. Science, 1997, 278(5339):834-838.
[107] MADARIAGA R, OLSEN K, ARCHULETA R. Modeling dynamic rupture in a 3d earthquake fault model[J]. Bulletin of the Seismological Society of America, 1998, 88(5):1182-1197.
[108] OLSEN K, DAY S, MINSTER J, et al. Terashake2: Spontaneous rupture simula-tions of m w 7.7 earthquakes on the southern san andreas fault[J]. Bulletin of the Seismological Society of America, 2008, 98(3):1162-1185.
[109] BRIETZKE G B, COCHARD A, IGEL H. Importance of bimaterial interfaces for earthquake dynamics and strong ground motion[J]. Geophysical Journal Interna-tional, 2009, 178(2):921-938.
[110] LOZOS J C, OLSEN K B, BRUNE J N, et al. Broadband ground motions from dynamic models of rupture on the northern san jacinto fault, and comparison with precariously balanced rocks[J]. Bulletin of the Seismological Society of America, 2015, 105(4):1947-1960.
[111] PREMUS J, GALLOVIČ F, HANYK L, et al. Fd3d_tsn: A fast and simple code for dynamic rupture simulations with gpu acceleration[J]. Seismological Research Letters, 2020, 91(5):2881-2889.
[112] GALLOVIČ F, VALENTOVÁ L, AMPUERO J P, et al. Bayesian dynamic finite-fault inversion: 1. method and synthetic test[J]. Journal of Geophysical Research: Solid Earth, 2019, 124(7):6949-6969.
[113] GALLOVIČ F, VALENTOVÁ L, AMPUERO J P, et al. Bayesian dynamic finite-fault inversion: 2. application to the 2016 mw 6.2 amatrice, italy, earthquake[J]. Journal of Geophysical Research: Solid Earth, 2019, 124(7):6970-6988.
[114] 罗扬. 用有限差分方法求解断层动力学问题[D]. 北京: 北京大学, 2007.
[115] 祝贺君. 使用有限差分方法研究地震波传播和震源动力学问题[D]. 北京:北京大学, 2008.
[116] 朱德瀚, 张伟, 祝贺君, 等. 利用曲线网格有限差分方法研究三维倾斜断层的破裂动力学[J]. 地震學報, 2010, 32(4):401-411.
[117] 张振国. 三维非平面断层破裂动力学研究[D]. 合肥: 中国科学技术大学, 2014.
[118] ZHANG Z, ZHANG W, XIN D. A dynamic-rupture model of the 2019 mw 7.1 ridgecrest earthquake being compatible with the observations[J]. Seismological Research Letters, 2020.
[119] ZHANG W, ZHANG Z, FU H, et al. Importance of spatial resolution in ground motion simulations with 3-d basins: An example using the tangshan earthquake[J]. Geophysical Research Letters, 2019, 46(4).
[120] 张文强. 破裂动力学的曲线网格有限差分方法研究及高性能计算[D]. 合肥: 中国科学技术大学, 2020.
[121] ZHANG W, ZHANG Z, LI M, et al. Gpu implementation of curved-grid finite-difference modelling for non-planar rupture dynamics[J]. Geophysical Journal International, 2020, 222(3):2121-2135.
[122] DAS S. A numerical study of rupture propagation and earthquake source mecha-nism.[D]. America: Massachusetts Institute of Technology, 1976.
[123] DAY S M. Three-dimensional finite difference simulation of fault dynamics: rectangular faults with fixed rupture velocity[J]. Bulletin of the Seismological Society of America, 1982, 72(3):705-727.
[124] 张海明. 半无限空间中平面断层的三维自发破裂传播的理论研究[D]. 北京: 北京大学, 2004.
[125] ZHANG H, CHEN X F. Dynamic rupture on a planar fault in three-dimensional half space—i. theory[J]. Geophysical Journal International, 2006, 164(3):633-652.
[126] ZHANG H, CHEN X F. Dynamic rupture on a planar fault in three-dimensional half-space–ii. validations and numerical experiments[J]. Geophysical Journal In-ternational, 2006, 167(2):917-932.
[127] 钱峰, 吴葆宁, 冯禧, 等. 基于非结构化网格的边界积分方程方法的断层自发破裂模拟[J]. 地球物理学报, 2019, 62(9):11.
[128] QIAN F, ZHANG H. Effects of initial stress field and critical slip-weakening distance on rupture selectivity of 3d buried branched faults[J]. Bulletin of the Seismological Society of America, 2020, 110(2):816-824.
[129] ANDO R. Fast domain partitioning method for dynamic boundary integral equa-tions applicable to non-planar faults dipping in 3-d elastic half-space[J]. Geophys-ical Supplements to the Monthly Notices of the Royal Astronomical Society, 2016, 207(2):833-847.
[130] KANEKO Y, LAPUSTA N, AMPUERO J P. Spectral element modeling of spon-taneous earthquake rupture on rate and state faults: Effect of velocity-strengthening friction at shallow depths[J]. Journal of Geophysical Research: Solid Earth, 2008, 113(B9).
[131] KANEKO Y, AMPUERO J P, LAPUSTA N. Spectral-element simulations of long-term fault slip: Effect of low-rigidity layers on earthquake-cycle dynamics[J]. Journal of Geophysical Research: Solid Earth, 2011, 116(B10).
[132] FESTA G, VILOTTE J P. Influence of the rupture initiation on the intersonic transition: Crack-like versus pulse-like modes[J]. Geophysical Research Letters, 2016, 33(15):5-11(7).
[133] GALVEZ P, AMPUERO J P, DALGUER L A, et al. Dynamic earthquake rupture modelled with an unstructured 3-d spectral element method applied to the 2011 m 9 tohoku earthquake[J]. Geophysical Journal International, 2014, 198(2):1222-1240.
[134] BENJEMAA M, GLINSKY-OLIVIER N, CRUZ-ATIENZA V M, et al. 3-d dy-namic rupture simulations by a finite volume method[J]. Geophysical Journal International, 2009, 178(1):541-560.
[135] LI Q, LIU M, ZHANG H. A 3-d viscoelastoplastic model for simulating long-term slip on non-planar faults[J]. Geophysical Journal International, 2009, 176(1):293- 306.
[136] WANG H, LIU M, DUAN B, et al. Rupture propagation along stepovers of strike-slip faults: Effects of initial stress and fault geometry[J]. Bulletin of the Seismo-logical Society of America, 2020, 110(3):1011-1024.
[137] DE LA PUENTE J, AMPUERO J P, KÄSER M. Dynamic rupture modeling on unstructured meshes using a discontinuous galerkin method[J]. Journal of Geophysical Research: Solid Earth, 2009, 114(B10).
[138] PELTIES C, GABRIEL A A, AMPUERO J P. Verification of an ader-dg method for complex dynamic rupture problems[J]. Geoscientific Model Development, 2014, 7(3):847-866.
[139] PELTIES C, HUANG Y, AMPUERO J P. Pulse-like rupture induced by three-dimensional fault zone flower structures[J]. Pure and Applied Geophysics, 2015, 172(5):1229-1241.
[140] TAGO J, CRUZ-ATIENZA V M, VIRIEUX J, et al. A 3d hp-adaptive discontinuous galerkin method for modeling earthquake dynamics[J]. Journal of Geophysical Research: Solid Earth, 2012, 117(B9).
[141] WOLLHERR S, GABRIEL A A, UPHOFF C. Off-fault plasticity in three-dimensional dynamic rupture simulations using a modal discontinuous galerkin method on unstructured meshes: implementation, verification and application[J]. Geophysical Journal International, 2018, 214(3):1556-1584.
[142] YE R, KUMAR K, MAARTEN V, et al. A multi-rate iterative coupling scheme for simulating dynamic ruptures and seismic waves generation in the prestressed earth[J]. Journal of Computational Physics, 2020, 405:109098.
[143] ZHANG W, SHEN Y. Unsplit complex frequency-shifted pml implementation using auxiliary differential equations for seismic wave modeling[J]. Geophysics, 2010, 75(4):T141-T154.
[144] AKI K, RICHARDS P G. Quantitative seismology[M/OL]. 2nd ed. University Science Books, 2002. http://www.worldcat.org/isbn/0935702962.
[145] BEDFORD A, DRUMHELLER D. Elastic wave propagation[J]. John Wileg g Sons, 1994:151-165.
[146] DUBINER M. Spectral methods on triangles and other domains[J]. Journal of Scientific Computing, 1991, 6(4):345-390.
[147] TORO E F. Riemann solvers and numerical methods for fluid dynamics: a practical introduction[M]. [S.l.]: Springer Science & Business Media, 2013.
[148] VERSTEEG H, MALALASEKERA W. An introdution to computational fluid dynamics[J]. The finite volume method, 1995.
[149] SHU C W. Total-variation-diminishing time discretizations[J]. SIAM Journal on Scientific and Statistical Computing, 1988, 9(6):1073-1084.
[150] SPITERI R J, RUUTH S J. A new class of optimal high-order strong-stability-preserving time discretization methods[J]. SIAM Journal on Numerical Analysis, 2002, 40(2):469-491.
[151] COURANT R, FRIEDRICHS K, LEWY H. Über die partiellen Differenzengle-ichungen der mathematischen Physik[J/OL]. Mathematische Annalen, 1928, 100 (1):32-74. https://doi.org/10.1007/BF01448839.
[152] ENGQUIST B, MAJDA A. Absorbing boundary conditions for numerical simula-tion of waves[J]. Proceedings of the National Academy of Sciences, 1977, 74(5): 1765-1766.
[153] GIVOLI D. High-order local non-reflecting boundary conditions: a review[J]. Wave motion, 2004, 39(4):319-326.
[154] DURU K. Perfectly matched layers and high order difference methods for wave equations[D]. [S.l.]: Acta Universitatis Upsaliensis, 2012.
[155] BERENGER J P. A perfectly matched layer for the absorption of electromagnetic waves[J]. Journal of computational physics, 1994, 114(2):185-200.
[156] CHEW W C, LIU Q. Perfectly matched layers for elastodynamics: a new absorbing boundary condition[J]. Journal of computational acoustics, 1996, 4(04):341-359.
[157] BÉCACHE E, FAUQUEUX S, JOLY P. Stability of perfectly matched layers, group velocities and anisotropic waves[J]. Journal of Computational Physics, 2003, 188 (2):399-433.
[158] KÜPERKOCH L, MEIER T, LEE J, et al. Automated determination of p-phase arrival times at regional and local distances using higher order statistics[J]. Geo-physical Journal International, 2010, 181(2):1159-1170.
[159] HERMANN V. Ader-dg - analysis, further development and applications[M/OL]. Ludwig-Maximilians-Universität München, 2011. http://nbn-resolving.de/urn:nb n:de:bvb:19-125403.
[160] XUE Z, LI X, DONG L, et al. A new perfectly matched layer (cfs-npml) in elastic wave modeling with discontinuous galerkin finite-element method[C]//2013 SEG Annual Meeting. [S.l.]: OnePetro, 2013.
[161] ZHAN* Q, REN Q, SUN Q, et al. Discontinuous galerkin pseudospectral time domain algorithm (dg-pstd) with auxiliary ordinary differential equations perfectly matched layer (aode-pml) for 3d seismic modelling[M]//SEG Technical Program Expanded Abstracts 2015. [S.l.]: Society of Exploration Geophysicists, 2015: 3633-3638.
[162] ABARBANEL S, GOTTLIEB D. A mathematical analysis of the pml method[J]. Journal of Computational Physics, 1997, 134(2):357-363.
[163] MEZA-FAJARDO K C, PAPAGEORGIOU A S. A nonconvolutional, split-field, perfectly matched layer for wave propagation in isotropic and anisotropic elastic media: Stability analysis[J]. Bulletin of the Seismological Society of America, 2008, 98(4):1811-1836.
[164] ZHANG Z, ZHANG W, CHEN X F. Complex frequency-shifted multi-axial per-fectly matched layer for elastic wave modelling on curvilinear grids[J]. Geophysical Journal International, 2014, 198(1):140-153.
[165] CHEN X F. Seismogram synthesis for multi-layered media with irregular interfaces by global generalized reflection/transmission matrices method. i. theory of two-dimensional sh case[J]. Bulletin of the Seismological Society of America, 1990, 80(6A):1696-1724.
[166] CHEN X F. A systematic and efficient method of computing normal modes for multilayered half-space[J]. Geophysical Journal International, 1993, 115(2):391-409.
[167] CHEN X F. Seismogram synthesis for multi-layered media with irregular interfaces by global generalized reflection/transmission matrices method. ii. applications for 2d sh case[J]. Bulletin of the Seismological Society of America, 1995, 85(4): 1094-1106.
[168] CHEN X F. Seismogram synthesis for multi-layered media with irregular interfaces by global generalized reflection/transmission matrices method. iii. theory of 2d p-sv case[J]. Bulletin of the Seismological Society of America, 1996, 86(2):389-405.
[169] CHEN X F. Generation and propagation of seismic sh waves in multi-layered media with irregular interfaces[J]. Advances in geophysics, 2007, 48:191-264.
[170] BREUER A, HEINECKE A, RETTENBERGER S, et al. Sustained petascale performance of seismic simulations with seissol on supermuc[C]//KUNKEL J M, LUDWIG T, MEUER H W. Supercomputing. Cham: Springer International Pub-lishing, 2014: 1-18.
[171] FU H, HE C, CHEN B, et al. 18.9-pflops nonlinear earthquake simulation on sunway taihulight: Enabling depiction of 18-hz and 8-meter scenarios[C/OL]//SC ’17: Proceedings of the International Conference for High Performance Comput-ing, Networking, Storage and Analysis. New York, NY, USA: Association for Computing Machinery, 2017. https://doi.org/10.1145/3126908.3126910.
[172] BOHLEN T. Parallel 3-d viscoelastic finite difference seismic modelling[J]. Com-puters & Geosciences, 2002, 28(8):887-899.
[173] GURNIS M, KELLOGG L, BLOXHAM J, et al. Computational infrastructure for geodynamics (cig)[C]//AGU Fall Meeting Abstracts: volume 2004. [S.l.: s.n.], 2004: SF31B-04.
[174] GRAY S H, MARFURT K J. Migration from topography: Improving the near-surface image[J]. Canadian Journal of Exploration Geophysics, 1995, 31(1-2):18- 24.
[175] DAY S M, DALGUER L A, LAPUSTA N, et al. Comparison of finite difference and boundary integral solutions to three-dimensional spontaneous rupture[J]. Journal of Geophysical Research: Solid Earth, 2005, 110(B12).
[176] PELTIES C, DE LA PUENTE J, AMPUERO J P, et al. Three-dimensional dynamic rupture simulation with a high-order discontinuous galerkin method on unstructured tetrahedral meshes[J]. Journal of Geophysical Research: Solid Earth, 2012, 117 (B2).
[177] TORO E F. Riemann solvers and numerical methods for fluid dynamics[M]. [S.l.]: Springer Berlin Heidelberg, 2013: 87-114.
[178] HARRIS R, ARCHULETA R, AAGAARD B, et al. The source physics of large earthquakes-validating spontaneous rupture methods[C]//AGU Fall Meeting Ab-stracts: volume 2004. [S.l.: s.n.], 2004: S12A-05.
[179] ROJAS O, DAY S, CASTILLO J, et al. Modelling of rupture propagation using high-order mimetic finite differences[J]. Geophysical Journal International, 2008, 172(2):631-650.
[180] PRESS W H, TEUKOLSKY S A, VETTERLING W T, et al. Numerical recipes 3rd edition: The art of scientific computing[M]. [S.l.]: Cambridge university press, 2007.