A state-of-the-art review of the numerical simulation of seismic wave motion based on spectral element method
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摘要:
基于谱元法（spectral element method，SEM）的地震波动数值模拟已被广泛用于地震震源破裂、大规模地震波传播、区域复杂场地及工程结构（群）地震反应、地震层析成像等重要问题的研究及应用当中，是目前地震工程学、地震学和地球物理学等地震科技领域共同关注的前沿热点技术。早期发展的Chebyshev谱元法（CSEM）和Legendre谱元法（LSEM）更接近谱方法的域分解思路，形式相对复杂且计算效率较低。目前广泛使用的是一种简洁形式的LSEM，其实施步骤和主要公式已经与有限元法完全一致，仅通过内置的Gauss-Lobatto-Legendre（GLL）高精度数值积分保留着与谱方法之间的联系。谱元法的巨大成功不仅源于算法本身的高精度、规整性和灵活性，更是得益于以SPECFEM2D/3D，SPECFEM_GLOBE，SPEED等为代表的开源软件集成了实现复杂模拟所需的三维复杂模型、不同地震震源模型或平面波输入、大规模并行计算、全球模拟、伴随方法（adjoint method）以及多尺度或不连续建模等关键技术。本文全面介绍CSEM、LSEM、间断伽辽金谱元法（discontinuous Galerkin，DG-SEM或DGM）、三角形单元谱元法、谱元法精度和稳定性方面的研究及应用进展，并详细阐述谱元法在我国的发展历程以及我国学者关于谱元法研究与工程应用的学术贡献。
Abstract:The spectral element method （SEM）-based numerical simulation of seismic wave motion has been widely applied in the study of earthquake source rupture process, large-scale seismic wave propagation, seismic response of regional complex sites without/with engineering structures （systems）, seismic tomography, and so forth. This technique is currently a frontier hotspot of common concern in the fields of earthquake science and technology including earthquake engineering, seismology, geophysics, et al. Spectral element method, which is sometimes also termed as spectral finite element method （SPECFEM） or spectral/hp element method, is a combination of spectral method and finite element method （FEM）. Hence, it shares the advantages of both the two methods, i.e., the high accuracy and fast convergence of spectral method, and the regularity and flexibility of FEM.
In early times, the Chebyshev spectral element method （CSEM） and Legendre spectral element method （LSEM） are originated from the domain decomposition of spectral methods, and therefore they inherit the complicated formulations of the latter, in which each of the interpolation basis functions is a linear combination of Chebyshev or Legendre orthogonal polynomials. Consequently, both the methods are as accurate as the spectral methods, but their applications are severely limited by the cumbersome and inefficient multi-layer nested computational structure that is resulted from those basis functions. Nowadays, the most frequently-used SEM is a concise form of LSEM developed by Komatitsch et al. In this method, the early-used complicated basis functions are simplified to the Lagrange shape functions that are commonly adopted in FEM, and those orthogonal polynomial-based analytical Gauss-Lobatto-Legendre （GLL） quadrature formulae are replaced by a simple numerical list of the GLL point coordinates and integration weights. Specifically, the non-equally distributed GLL points serve as the element nodes and the GLL high-precision numerical integration formula is applied to calculate the element mass, stiffness matrices and etc. This configuration makes the LSEM enjoy the same solution procedure and computational formulations as that of FEM, but avoid the significant defects of the classical high-order finite element method, including the intrinsic numerical error of the high-order polynomial interpolation based on equally-spaced grid and the lower computational efficiency due to the high-order consistent mass matrix. In a word, this LSEM has actually become a high-performance lumped-mass high-order finite element method. In addition to the above methods, the family of non-conforming spectral element methods has also been broadly studied and successfully applied in many problems, making themselves an important branch of the SEM. By introducing the so-called Lagrange multiplier or interior penalty term as a glue to effectively connect spectral elements with quite diffetent sizes, orders, shapes and so on, the non-conforming SEMs are more flexible and highly efficient in dealing with multi-scale or discontinuous problems, which apear frequently in large-scale or complicated seismic wave simulations.
The great success of SEM is not only due to the high accuracy, regularity and flexibility of the algorithm itself, but also attributed to those well-designed open-source SEM programs represented by SPECFEM2D/3D, SPECFEM_GLOBE, SPEED, etc. have incorporated a variety of key technologies that are required in complex simulations, such as three-dimensional complex models, different seismic source models or plane wave input method, large-scale parallel computing, global simulation, adjoint method, multi-scale or discontinuous modeling and so on. In the field of earthquake engineering, the SEM has been applied to perform physics-based deterministic numerical simulation of strong ground motion and to realize the “end-to-end” seismic response analysis that is from the source rupture to engineering structures or even engineering systems. The massive simulation data is a good supplement to the insufficient strong ground motion records, and the modeling of seismic wave propagation in actual geolocial structures can compensate for the weak physical background of traditional ground motion prediction equations （GMPEs） or stochastic methods. These simulations, which have reached a certain level of reliability, bring new vitality to earthquake engineering research and application. In the fields of seismology or geophysics, the highly-efficient forward simulation of SEM has been combined with the adjoint method, enabling a simultaneous modeling of the seismic wave fields generated from a number of observation stations, thus the number of forward simulations required for an inversion process can be reduced to an acceptable level. In this way, the advanced full wave inversion （FWI） or seismic tomography technique has been practically used to investigate seismic source mechanisms and to reveal regional or even global velocity structures. Finally, the development of SEM in China is elaborated. The SEM was introduced into China around the year of 2000, and the related studies mainly focused on the basic performance of the method and some preliminary applications until early 2010 s. In the past decade, the Chinese researchers have been conducting more and more innovative work on the SEM algorithms and various engineering applications, and some of the work has reached the research forefront of the world.
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图 11 西断层活化后的∆CFS和地震投影（a） 只有西断层；（b） 东西断层并存Figure 11. ∆CFS and aftershock projection after activation of the west fault（a） Case with only the west fault；（b） Case with co-existence of the east and west faults引言
近年来，与页岩气开发相关的水力压裂诱发地震案例在美国（Ellsworth，2013；Holland，2013；Skoumal et al，2015，2018）、加拿大（Atkinson et al，2016；Schultz et al，2016，2018）、欧洲（Clarke et al，2014；Galis et al，2017；Li et al，2019）、中国（Lei et al，2017；Meng et al，2019；Tan et al，2020）等多个国家和地区被广泛观测到。诱发地震现象最为突出的区域主要分布在北美的加拿大阿尔伯塔（Alberta）省福克斯溪（Fox Creek）地区（Wang et al，2016，2017）及不列颠哥伦比亚（British Columbia）省圣约翰堡（Fort St. John）和道森克里克（Dawson Creek）地区（Peña Castro et al，2020；Wang et al，2020，2021）、美国得克萨斯州（Texas）和美国俄克拉荷马州（Oklahoma）等地区，这些地区的页岩气开发直接造成附近地域地震频度和震级的显著提升，随之而来的地震危害性攀升不可忽视。
Ellsworth （2013）指出流体注入导致的孔隙压力和孔隙弹性应力变化是诱发地震的两种核心机制。流体注入使得断层的孔隙压力增加和有效正应力减小，摩尔圆左移相交于破裂包络线导致断层破裂失稳，是诱发地震的最主要和最常见的机制（Atkinson et al，2016；Schultz et al，2016）。惠钢等（2021）基于流固耦合理论对福克斯溪地区2015年2月8日发生的ML3.0地震事件展开数值模拟研究，结果表明压裂后断层周围的孔隙压力和局部应力场发生了显著变化，直接导致其活化，而此处孔隙压力的增加是断层活化的首要因素。另外，将作用到储层岩体的孔隙弹性应力耦合到流体计算中也是主要的研究方法。Deng等（2016）基于多孔线弹性理论和流固耦合模型计算了福克斯溪地区水力压裂作业引起的应力场演化过程。Segall和Lu （2015）及Chang和Segall （2016）采用类似方法建立多场耦合模型，研究了废水注入对不同类型断层的孔隙弹性应力扰动。本文拟借鉴上述经典模型及方法开展多孔弹性介质中的多场耦合数值模拟研究.
探索诱发地震的致灾机理是地质灾害预测与防治的重要基础工作，而断层受扰活化的动力学数值模拟则是重要的手段和方法。当受扰断层的剪应力小于接触面的摩擦力时，断层将保持静止状态；当二者相等时，断层将出现准动态的滑动趋势；当断层面某处的剪应力超过静摩擦力后，断层将产生局部破裂和活化，进而可能发生规模性滑动，导致诱发地震（Ellsworth，2013）。
加拿大西部盆地是全世界水力压裂诱发地震研究最为活跃的区域之一（Atkinson et al，2016；Schultz et al，2016），其中Duvernay地层曾发生水力压裂诱发地震的较大震级事件（Wang et al，2016，2017）。加拿大不列颠哥伦比亚地区的几次水力压裂诱发地震事件也引起了广泛关注。Wang等（2020）计算了2015年8月17日发生的MW4.6地震的应力降并对最大可能震级进行了详细分析，认为该次地震发生在一条先存断层上，其最大震级受控于地质构造条件。其后续数值模拟结果表明压裂注液区到目标断层之间很可能存在一条高渗透率的流体通道（Wang et al，2021），该结论与同一地区Montney地层的MW4.2水力压裂诱发地震类似（Peña Castro et al，2020）。Bao和Eaton （2016）详细论述了加西盆地福克斯溪地区6个压裂平台的注水作业与地震活动的时空关联性，发现震级最大事件（MW3.9）发生在一条看似从注水区域延伸到结晶基底的断层上。该断层附近陆续发生的地震事件表明其可能被工业注水持续激活，但是其具体活化过程并未在已有研究中进行严格的正演分析。Gao等（2022）认为上述MW3.9地震的触发是因为存在复杂的流体运移通道，首先是注入的液体沿Duvernay地层内部的水平向通道向东平移，到达东侧断层后使其发生无震滑移并触发较深部的一系列地震，精定位后的地震序列更是证明东断层到西断层之间在结晶基底附近可能存在一条近水平的流体通道，该通道促发了西断层末端MW3.9地震的发生。上述分析为复杂构造下的诱发地震活动成因提供了全新视角，但并未得到地震成像或应力计算的佐证。然而，较深部地震的延迟发生可能是压裂停止后流体缓慢扩散所直接导致。
本文拟基于加拿大福克斯溪地区已经观测到的地震目录和断层信息，拟开展断层受扰延迟活化的流固耦合数值模拟研究。模拟中将发震构造简化为两条主要断层，忽略已有研究中提出的含流体通道的复杂构造环境（Gao et al，2022）。在遵循实际施工进程和参数的前提下，重点分析页岩层内的压裂注水对储层下方主要断层的延迟活化过程。本模拟将结合福克斯溪地区的区域构造背景、地震分布、压裂注水工程和岩性参数，根据水力压裂原理、裂隙渗流理论、断层失稳准则和流固耦合理论详细分析水力压裂延迟活化断层的机制和动力学过程。
1. 研究区域与数据
1.1 研究区域及井位介绍
本文的研究区域位于加拿大阿尔伯塔省的福克斯溪地区（图1），在该地区曾观测到众多与页岩气开发有关的诱发地震（Schultz et al，2016）。重点关注的Duvernay地层在2014年12月至2015年4月水力压裂作业期间于6个不同井位发生了时空关系迥异的诱发地震序列，而2015年1月23日更是在1号平台附近发生了一次走滑-逆冲机制的MW3.9地震。诱发地震的丛集展布方位及震源机制解均为本文断层几何形态的模拟提供了依据（Bao，Eaton，2016；Hulls，2022）。
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Figure 1. Displacement and stress distribution of the numerical simulation process for activation of the west fault while the two faults are co-existedFigs. （a）−（f） show snapshots of the pore pressure field at the 6 corresponding time points；Figs. （g）−（l） present snapshots of the displacement field in the Y direction；Figs. （m）−（r） display snapshots of the ∆CFS （variation of Coulomb failure stress） field上述MW3.9地震及其前震和余震均发生在1号水平井附近，地震数据剖面图显示出两条明显的、急剧倾斜的地震活跃带，一直从Duvernay地层内的注入带延伸到结晶基底的上部（Bao，Eaton，2016；Wang et al，2016）。我们重新梳理了这些地震和井轨迹，如图2所示。基于时空关系聚合效应，能够判断出两个相对独立的丛集地震。浅绿色区域内的丛集地震主要发生于2014年12月23日至2015年1月15日之间，最大震级为发生在结晶基底之内的MW3.2地震，丛集地震的分布显示了其东侧断层的大致形态，估计倾角为75°—85°。浅红色区域内的丛集地震陆续发生于2015年1月10日至3月30日之间，最大震级为1月23日发生于结晶基底的MW3.9地震，丛集地震的分布显示其东侧断层的倾角约为80°—85°，地震丛集持续时间长、震级大，其时空关系存在显著的前震余震特征。MW3.9地震的破裂位于压裂井正下方、整个压裂段的末端，但是在压裂结束后15天才发生，伴随一定的前震和尾部较长的余震（图2c中粉红色和红色圆圈）。值得注意的是，图2a中红色线段表示的两口水平井均为自北向南实施压裂，北段压裂时触发绿色区域的丛集地震，但同时南部近井也有小震发生。考虑到1号平台施工过程中93%的压裂液未能返排，其附近断裂带可能发生了持续的增压和激活（Bao，Eaton，2016）。最大震级MW3.9地震位于前寒武纪结晶基底上部的深处，随后的序列则向较浅处靠近注入带的位置迁移。
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图 1 研究区位置（a）和1号平台两口水平井及附近地震事件的分布（Bao，Eaton，2016）（b）Figure 1. Study area （a） and distribution of earthquakes near the two horizontal wells of Platform 1 （Bao，Eaton，2016）（b）2. 研究方法与原理
本文的主要研究内容是数值模拟Duvernay地层内对应于2015年1月23日Mw3.9地震的西侧断层在压裂流体扰动下的活化过程，主要涉及水力裂缝扩展及其与断层的流固耦合、断层活化等关键步骤。本节将阐述数值模拟与模型构建的方法原理及实现路径。
2.1 人工压裂缝扩展模拟
水力压裂裂缝的尺度、扩展及闭合都对工程效能具有重大影响，本节首先使用经典的PKN模型来描述水力裂缝的扩展过程。PKN模型最初由Perkins和Kein提出（1961），后因Nordgren （1972）添加滤失项而改进，并因此得名。该模型基于垂直平面应变理论建立线弹性裂缝力学方程、缝内液体流动方程及连续方程，并在给定的边界条件下进行求解，是一个椭圆形等高裂缝模型，其表达式为：
$$ \begin{array}{c}L （ t ） ＝0.654{\left[\dfrac{G{q}_{0}^{3}}{ （ 1－\nu ） \mu {h}^{4}}\right]}^{1/5}{t}^{\,{4}/{5}} \text{，} \\ W （ t ） ＝2.5{\left[\dfrac{ （ 1－\nu ） \mu {q}_{0}^{2}}{Gh}\right]}^{1/5}{t}^{1/5} \text{，} \\ P （ t ） ＝2.5{\left[\dfrac{{G}^{6}\mu {q}_{0}^{2}}{ （ 1－\nu { ） }^{4}{h}^{6}}\right]}^{1/5}{t}^{1/5}\text{，} \end{array} $$ (1) 式中，$L （ t ） $为t时刻裂缝扩展长度，$ W （ t ） $为t时刻裂缝扩展宽度，$ P （ t ） $为t时刻流体压力，G为剪切模量，q0为注入速率，μ为注入流体的黏度，h为压裂缝缝高，v为泊松比。采用Python语言对式（1）进行模拟和可视化，得到裂缝扩展的缝长、缝宽和压力等关系（图3）。作为示例，这里实施的模拟计算时间为10 min。模拟计算采用加拿大Duvernay地层实际压裂注水参数作为输入（2 015 m3/d），裂缝的长宽高取值由PKN模型结果表达式计算求得，流体注入稳定后其孔隙压力均值约为（50±15） MPa （图3c）。据此，我们将该值作为后续多场耦合模拟的流体输入关键参数。
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图 2 地震丛集时空特征分布图两个相对独立的丛集地震分别用浅绿色和浅红色椭圆区域表示，最大震级事件的震源机制解绘制在图中（Wang et al，2016）。（a） 丛集地震的水平投影。红色线段代表1号平台的两口水平井，蓝色箭头指示出逐段压裂方向，蓝色倒三角形为压裂施工阶段P2的最后5天对应的压裂段位（Gao et al，2022）；（b） 垂向剖面。红色虚线表示两条估计断层轨迹，断层位置和倾角参考Bao和Eaton （2016）；（c） M-t图。图中标注了压裂施工阶段P1和P2的时间跨度，其中虚线框表示压裂施工阶段P2的最后5天，对应于图（a）中的蓝色倒三角形Figure 2. The spatiotemporal distribution of earthquake clustersTwo relatively independent clusters are represented by light green and light red elliptical regions，respectively. The focal mechanism solution of the largest seismic event is depicted in the figure （Wang et al，2016）. （a） The map view of the cluster earthquakes，with red lines representing two horizontal wells of Platform 1，blue arrows indicating the step-by-step fracturing direction，and a blue inverted triangle representing the fracturing segment corresponding to the last 5 days of fracturing construction stage P2 （Gao et al，2022）. （b） A vertical profile，with red dashed lines representing two estimated fault trajectories.The fault location and dip angle are referenced from Bao and Eaton （2016）. （c） The magnitude-time distribution chart，with fracturing construction stages P1 and P2 marked in the figure. The dashed box represents the last5 days of fracturing construction stage P2，corresponding to the blue inverted triangle in Fig.（a）2.2 多孔弹性介质流固耦合模拟理论
多孔弹性介质常用于研究流体和断层的交互力学作用（Segall，Lu，2015；Bao，Eaton，2016；Deng et al，2016；Goebel，Brodsky，2018）。假设压裂流体在介质中服从达西渗流定律，基于Biot方程描述固体孔隙弹性应力及流固耦合的质量守恒和力学平衡（Biot，1941；Rice，Cleary，1976），有如下多孔弹性介质流固耦合数学模型：
$$ \nabla \sigma ＋ {\rho }_{{{\mathrm{b}}}}g＝0 \text{，} $$ (2) $$ {\rho }_{{\mathrm{f}}}S\frac{ {\text{∂}} P}{ {\text{∂}} t} ＋ {\rho }_{{\mathrm{f}}}\alpha \frac{ {\text{∂}} {\varepsilon }_{v}}{ {\text{∂}} t} ＋ \nabla \cdot \left[-{\rho }_{\mathrm{f}}\frac{k}{{\eta }_{\mathrm{f}}} （ \nabla P-{\rho }_{\mathrm{f}}{{{g}}} ） \right]＝0 \text{，} $$ (3) $$ \varepsilon ＝\frac{1}{2} （ \nabla {\boldsymbol{u}} ＋ \nabla {{\boldsymbol{u}}}^{{\mathrm{T}}} ） \text{，} $$ (4) 式中，P为流体压力，ρf为流体密度，S为流体存储系数，k为渗透率，α为Biot系数，ηf为流体动态黏度，ρb为岩石体积密度，$\nabla {\boldsymbol{u}} $为位移场，ε为体积应变，g为重力加速度。
2.3 库仑应力计算
通过模拟断裂带周围孔隙压力和地应力的交互变化（Catalli et al，2013），基于摩尔-库仑准则计算断层附近的库仑应力改变量（Okada，1992）：
$$ \mathrm{\Delta }\mathrm{C}\mathrm{F}\mathrm{S}＝\Delta \tau －\mu （ \Delta {\sigma }_{{\mathrm{n}}}－\Delta P ） \text{，} $$ (5) 式中，∆P为孔隙压力变化量，∆τ为断层上的剪切应力变化量，∆σn为有效正应力变化量，∆CFS为库仑应力改变量，该值为正则表明断层更趋向于失稳。
2.4 数值计算步骤、模型构建及相关参数
本研究的模拟按照实际压裂井段的施工计划和断层活化进行计算，主要分为以下三个阶段：第一阶段为注水阶段，向距离目标断层物理位置相对接近的井段压裂注水，总注水时间折合为5天；第二阶段为停止注水阶段，此时流体将继续扩散，持续15天；第三阶段为断层活化及后续阶段，活化起始时间设为第二阶段结束时刻，断层活化后继续计算20天，总模拟时间共计40天。
基于Duvernay地层的实际构造、压裂参数和地震活动数据（图2），本文按照以下几何尺寸设置模型：边长为10 km，模型中包含了Duvernay页岩层、上下围岩、结晶基底等三类地层；西断层长600 m，厚10 m，东断层长400 m，厚10 m，东西断层倾角均约为83°；注入井位于西断层上方的储层内部，垂直展布于平面向内方向（图4）。采用PKN模型计算出的孔隙压力作为流体注入扰动。模型左、右、底边界都采用辊支撑（即第二类/自由边界条件），上界为自由表面、边界孔隙压力为0，其它边界无流体流动，在断层上下盘两侧分别设置监控线来观测相关参数在模拟过程中的变化，监测线与断层上下盘的距离接近于0。需要说明的是，本模型用二维垂直截面来清晰表达推测断层的线性特征（图2b），断层倾角的设定则综合考虑了地震丛集的分布特征和MW3.9最大震级事件的震源机制解。Zhang等（2016）计算得出MW3.9地震的震源特征为走滑逆冲机制类型，其滑动面（节面Ⅰ）的反演结果为走向176°，倾角83°，滑动角135°，滑动角135°表示其走滑分量和逆冲分量大约各占50%。二维截面模拟主要考虑YOZ剖面的逆冲机制部分（图2），因此无法体现垂直于模型平面向内的走滑分量，后续将走滑逆冲机制断层简称为“逆断层”。本文假设模拟平面（即YOZ面）内的断层尺度为“长度”，而垂直于YOZ面向内方向的断层尺度为“宽度”。
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图 3 裂缝扩展PKN模型的模拟计算结果（a） 裂缝长度和裂缝宽度的扩展演化；（b） 裂缝长度与裂缝高度的关系，红色表示裂缝高度的延伸方向，蓝色表示沿着裂缝长度方向的裂缝面；（c） 对应的裂缝流体压力演化图Figure 3. Simulation results of fracture propagation in PKN model（a） The evolution of fracture length and width；（b） The relationship between fracture length and height；（c） The corresponding evolution of fluid pressure within the fractures下面阐述本文模型及模拟设定的断层物理尺寸和位错距离的理论计算依据。Leonard （2010）在Wells和Coppersmith （1994）提出的“震级-破裂尺度”经验公式基础上，凭借更完备的数据集提出了隐伏逆断层的长度L与矩震级MW的经验关系式：
$$ {M}_{\mathrm{W}}＝1.67\mathrm{l}\mathrm{g}L ＋ 4.24 \text{.} $$ (6) 通过式（6）估算MW3.9地震所对应的断层长度约为625.8 m，证明本模型设置的断层近似长度600 m具有一定合理性。
进一步采用Leonard （2010）提出的中小震级断层位错关系分别计算上下盘的相对滑动距离。对于震级小于MW5.0的地震，地震矩与断层长度符合M0∝GL3，断层位错距离$ \overline {D} $为
$$ \overline {D} ＝C{L}^{2－\beta } \text{，} $$ (7) 式中：L为断层破裂长度，单位为m；C为经验常量，震级较小的逆断层一般取值为3.7×10−5 （Leonard，2010）。根据式（7），对于长度为600 m的断层，发生MW3.9地震所产生的滑动位移约为0.029 6 m。由于震源机制解中逆冲机制分量约占50%，因此在模拟中施加的断层位错距离为总滑移距离的50%，为0.014 8 m。数值模拟其它相关参数详见表1。
表 1 数值模拟的主要参数弹性
模量
E/GPa泊松比
v密度ρd
/（kg·m−3）孔隙度
ϕ渗透率
κBiot系数
α结晶基底 66 0.20 2 900 0.2 10−17 0.75 页岩层 25 0.25 2 500 0.2 10−15 0.75 上下围岩层 40 0.25 2 500 0.2 10−14 0.75 断层 14.4 0.20 2 500 0.02 10−12 0.80 3. 模拟结果及分析
本文的数值模拟工作分为两部分，首先分析Duvernay页岩层下方西断层独立存在时的活化过程，然后进一步研究东西两条断层同时存在时，西断层的活化过程，并讨论东断层对西断层活化的影响。数值计算主要分为三个阶段：一是压裂流体持续注入5天；二是停止注入15天，实际数据表明此时断层附近的∆CFS将达到临界值，因此模拟中断层活化的开启阀门受此时间控制；三是断层活化并进行有限滑动，直至20天后模拟结束。本节详细描述相关模拟结果。
3.1 独立西断层活化过程模拟
由于西断层位于Duvernay页岩层压裂注入点的正下方，并且延迟发生了一次位于结晶基底内的MW3.9最大震级事件，因此其活化过程可能与压裂流体的注入和持续渗透直接相关。
基于PKN模型计算结果和实际压裂参数（Bao，Eaton，2016），Duvernay页岩层的平均流体注入量为2 015 m3/d，经过换算，注入速率约为23.32 kg/s。本文使用该平均速率对两个压裂井进行源头设置，持续输入到附近地层之中。根据实际水平井压注施工日程和断层被激活的流体注入范围，将注水总时长设定为5天，停止注液15天后断层滑移，并继续计算观测20天。我们对整个模拟过程中的6个关键时间点的孔隙压力、Y方向位移和∆CFS进行了切片展示（图5）。流体注入一经开始（第1天，图5a，5g，5m），储层注液点附近就产生了高强度孔隙压力扩散、向上和向下的Y方向位移以及对应的∆CFS。上述变化发生在压裂注水点附近，最初注液阶段对下部断层的影响并不明显。注水持续5天后停止，此时孔隙压力已经扩散至更深部区域，围岩层下方与结晶基底边界上方出现明显的流体聚集（第5天，图5b的浅色区域）。对应的Y方向位移和∆CFS也出现进一步增强（图5h，5n）。并且，由于流体向下扩散，插入结晶基底的断层下部及端点附近出现明显的高值区（图5n）。此时结晶基底内的断层部分已经存在活化趋势，但断层面的剪切应力并未达到滑移临界值，处于蠕滑或者慢滑移状态。
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图 4 本文数值模拟的模型示意图及主要模型参数（a） 福克斯溪地区1号页岩气开发平台的水平井、水力裂缝、页岩储层及主要断层的构造位置示意图；（b） 二维数值模拟的模型构建、尺寸及边界条件；（c） 图（b）中关于压裂注入井与断层的放大表示图Figure 4. Illustration and key model parameters of the numerical simulation in this study（a） Schematic diagram of the structural positions of the horizontal wells，hydraulic fractures，shale reservoir，and major faults in the Fox Creek area，specifically highlighting Platform 1； （b） The construction，dimensions，and boundary conditions of the two-dimensional numerical model；（c） An enlarged representation of the hydraulic fracturing injection well and the fault within the diagram shown in Fig.（b）注水5天后则进入模拟的第二阶段，此时停止注液，持续15天。由于流体注入源头消失，水平井眼附近的孔隙压力因失去了持续增加的动力而逐渐减弱。但是，流体依然在重力作用下发生向下为主的扩散。由于结晶基底的渗透率更低，使得流体聚集到结晶基底上边界处（图5c，5d）。然而，延伸到基底的断层则具有相对较大的渗透率，因此流体持续灌注到断层下端，这一现象能够被清晰地观测到，且在本阶段最后一天（第20天）尤为明显（图5d）。上述流体渗透特性使得处于结晶基底内的断层内的孔隙压力逐渐增大，有效正应力逐渐减小，根据式（5），其对应的∆CFS也逐渐增大（图5o，5p）。当达到临界值后，断层则发生活化。上述流体的持续扩散及沿断层下端聚集，正是西断层延迟活化的关键原因，本模拟过程充分展示了这一点。
当模拟进行到第20天时，断层下端的∆CFS达到临界值，设定的断层阀门打开，此时断层打破之前的蠕滑状态，进入破裂和快速滑移状态（Zhu et al，2020）。模拟中假设断层的总滑动量和滑动方向受限于MW3.9最大震级事件，符合逆断层动力学特征。
由于逆断层活化，断层附近的位移方向发生了显著变化，图5k和5q分别展示了断层活化后Y方向位移变化与∆CFS的瞬时变化。断层活化后，随着断层滑动的停止和愈合，其附近的位移场和应力场基本维持既有分布和大小，趋于稳定（图5l，5r）。
为了更好地呈现断层活化前后其内部及附近区域的∆CFS和上下盘的位移情况，我们截取了相关模拟结果的切片继续进行展示（图6）。显然，断层活化前的位移场主要体现了压裂流体的注入和扩散对附近地层的形变影响（图6a−c），而活化后断层上下盘则体现出清晰的逆断层位移特征，即上盘向上移动、下盘向下移动（图6d）。此时，该逆断层的活化也导致∆CFS正值集中在断层附近（图6h），∆CFS的分布均表现出与断层活化前的注液阶段（图6e，6f）和停注扩散阶段（图6g）截然不同的辐射花样。位于结晶基底的断层内部的∆CFS也表现出对应的变化特征，从注液开始（图6i）到停止（图6j），再到流体扩散聚集（图6k），其正负值持续增加，直至达到临界值，断层活化。
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图 5 西断层活化数值模拟全过程的位移场及应力场分布图（a）−（f），（g）−（l），（m）−（r）分别为6个时间点对应的孔隙压力场分布、Y方向位移场及∆CFS应力场的切片Figure 5. Distribution of displacement field and stress field during the numerical simulation process of activation for the west faultFigs.（a）−（f），（g）−（l） and （m）−（r） show snapshots of the pore pressure field，the displacement field in the Y direction and the ∆CFS （Coulomb failure stress） at the corresponding six time points在流固耦合模拟中，我们分别在断层上下盘设置了监控线，便于对应力应变等关键参数进行过程分析。图7绘制了断层上盘的孔隙压力变化曲线，横坐标对应从上至下沿断层上盘的距离，曲线颜色对应于不同采样时刻。由于断层下盘的孔隙压力变化曲线在形态和数值上都与上盘相似，故只展示上盘结果。图中由深到浅的蓝色曲线对应于0—20 d的断层上盘孔隙压力值，由浅到深的红色曲线则对应于20—40 d的孔隙压力值。结果表明，断层上盘活化前后的孔隙压力分布具有明显的规律性：围岩和结晶基底界面存集了大量向下扩散渗流的流体，使其孔隙压力值始终处于高位；断层活化前，位于围岩上半部分的孔隙压力值呈U形分布，靠近注液点的上端和渗透率变化界面的下端具有更高的孔隙压力值，而位于结晶基底的下半部分的孔隙压力则快速降低；对应于前5天注液阶段的断层上端孔隙压力值（前5条蓝色曲线左端）逐渐上升，停止注液后的孔隙压力值则逐渐降低；断层活化后，其上端到中部的孔隙压力值呈线性增加特征，下半部分趋势与活化前的快速降低吻合；总体上围岩中的断层面孔隙压力高于结晶基底，与渗透率减小具有直接关系。
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图 6 西断层活化前后的位移场及断层库仑应力分布图（a）−（d）为断层活化前后的位移场，图（e）−（h）为断层外部区域的∆CFS应力场切片，图（i）−（l）为断层内部的∆CFS等值线切片。图（g）和（k）标注了断层活化前∆CFS高值区的等值线值Figure 6. Distribution of displacement field and Coulomb stress of the western fault before and after activationFigs. （a）−（d） show the displacement fields before and after fault activation；Figs. （e）−（h） show the ∆CFS stress field snapshots in the external area of the fault；Figs. （i）−（l） show the ∆CFS contour snapshots in the internal area of the fault. Figs. （g） and （k） mark the contour value of high ∆CFS value area before fault reactive图8给出了断层上下盘的∆CFS值分布曲线，图中的浅蓝色条带表示接近断层上下端点的区域，浅绿色条带表示结晶基底界面附近区域。从图8a中可知断层活化前上下盘的∆CFS值随着流体的注入和渗透而逐渐增加，特别是结晶基底部分较为明显。断层上下盘的∆CFS值由红色和蓝色实线绘制，高值区主要集中在断层两端，且正负有分。断层中段的上盘∆CFS为正，下盘为负，值域区间主要在±0.5 MPa之间，断层活化后其上下盘的∆CFS值则几乎保持稳定，体现于不同时间采样点的曲线基本重复（图8b）。断层两端的∆CFS值达到5—6 MPa，这与断层模型两端的几何形态和约束有关，其值大小不能完全反映实际情况。活化前的孕震阶段结晶基底以下的断层∆CFS值更高，体现出更容易活化的特征，事实上后续的MW3.9地震正发生于这一区域。
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图 7 沿断层上盘的孔隙压力变化曲线曲线颜色对应于不同采样时刻的断层上盘孔隙压力值，蓝色为断层活化前0—20天，红色为模拟的20—40 天，断层活化后Figure 7. The curve of pore pressure variation along the upper plate of the fault is shownThe curve colors correspond to the pore pressure values of the hanging wall of the fault at different sampling times，with blue representing 0−20 days before reactivation and red representing 20−40 days after reactivation3.2 东断层存在时西断层的活化过程模拟
本节进一步分析东西两条断层同时存在时，西断层的活化过程以及东断层对西断层活化的影响。数值模拟采用完全相同的步骤计算三个阶段的孔隙压力、Y方向位移和∆CFS值，并进行6个关键时间点的切片展示（图9）。整个模拟过程中流体的注入、扩散、聚集模式与仅有西断层的情况类似，但是部分流体也渗透到了东断层之中（图9c−f）。这一流体渗透模式也使得东断层下部（结晶基底部分）出现了∆CFS的显著变化，但其集中程度和数值显然小于西断层（图9m−p）。实际观测表明东断层对流体的响应更为迅速，率先活化（Bao，Eaton，2016；Gao et al，2022），这与东断层的岩石物理性质及强度有关。本模拟虽然得出流固耦合作用下东断层的活化趋势，但其活化过程本文不作分析，重点关注的仍是西断层在流体作用下的延迟活化过程。西断层活化后东断层下端点的辐射花样也发生了细微变化，体现出断层之间的相互触发作用，尽管后期并未观测到东断层附近的新发地震。
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图 8 断层活化前（a）、后（b）沿断层上下盘的∆CFS变化曲线上盘∆CFS对应浅红色曲线，下盘∆CFS为深灰色曲线，每条曲线代表一个模拟采样间隔单位（d）蓝色条带对应断层两侧端部，绿色条带对应代表断层与结晶基底交界区域Figure 8. The ∆CFS variation curves before （a） and after （b） fault activation along the fault planeThe shallow red curve corresponds to the ∆CFS of hanging wall，while the deep gray curve corresponds to the∆CFS of footwall. Each curve represents a simulated sampling interval （in days）. The blue bands correspond to the ends of both sides of the fault, and the green bands correspond to the area where the fault meets the crystalline basement本节的模拟结果表明西断层附近区域流固耦合场的演化模式与存在东断层的情况大体类似。但是东断层的存在是否会使得西断层更易活化呢?已有研究表明可能存在两条断层之间的流体通道，使流体通过东断层传导到西断层并使其成核活化（Gao et al，2022）。实际观测中东断层先被及时激活，而西断层则是延迟活化，其机制可能是东断层已有位错、流体扩散和流体通道的共同作用。基于建模的复杂度和可靠性原因，本研究暂不考虑潜在流体通道，只从流体注入和扩散角度分析两条断层共存时的相互影响，且重点关注西断层的活化过程，因为其对应该时段该区域最大震级事件。因此，我们将西断层活化前一天的库仑应力场进行对比分析，并标注出∆CFS集中点的数值（图10）。结果表明，在仅有西断层的情况下，经过5天注水和15天扩散的流体作用，西断层端点出现2.56 MPa的∆CFS高值区，MW3.9地震破裂点附近出现1.62 MPa的∆CFS高值区。而在两条断层并存的情况下，上述∆CFS高值区分别增加至2.98 MPa和2.11 MPa。这表明东断层在结晶基底中的∆CFS集中区一定程度上增加了西断层的库仑应力，使得西断层更容易活化。注意到东西断层之间在结晶基底内形成了一个∆CFS高值区联通带（图10b），这是邻近断层间应力相互扰动的结果，当某一断层应力达到其临界值后将会率先活化。
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图 9 东西两条断层共存时西断层活化数值模拟全过程的位移场及应力场分布图（a）−（f），（g）−（l），（m）−（r）分别为6个时间点对应的孔隙压力场分布、Y方向位移场及∆CFS应力场的切片Figure 9. Distribution of displacement field and stress field of the numerical simulation process for activation of the west fault while the east and west faults are co-existedFigs.（a）−（f），（g）−（l） and （m）−（r） show snapshots of the pore pressure field，the displacement field in the Y direction and the ∆CFS （Coulomb failure stress） field at the corresponding six time points4. 讨论
本文模拟结果表明西断层在活化过程中存在三处变化显著的位置：第一处是靠近注液点的断层近端（沿断层走向0—50 m处），此处流体扩散到达的时间更短、孔隙压力更高；第二处是断层远端部分（沿断层走向550—600 m处），其机理是断层尖端的应变累积和应力集中；第三处是沿断层400—450 m处，此厚度非常小的矩形面，端点对应矩形短边，上下盘对应矩形的长边。断层贯穿两个不同岩石物理属性的地层，周围应力场和形变值集中分布。本研究假设断层是一个矩形，当断层上下盘（矩形长边）发生相对滑动时，短边会出现局部变形以达到符合震级的相对滑动距离，但是有限元方法的连续性使其仍然保持连接，而无法实现上下盘的实际分离。这种约束使得图8b中断层左右两端出现了库仑应力变化的异常高值区。第一处和第三处断层位置是断层的上下端点，其变化会受到模型本身设置的限制。此外，活化前位于结晶基底内的断层附近出现更为集中的库仑应力增加，这是因为模型中结晶基底具有更高的弹性模量，即该结晶基底的刚度和脆性更强，更容易引起相对更大震级的地震。这与本文研究区的实际观测相符合，东断层和西断层附近发生的较大震级地震均发生于结晶基底内部。
本文将注水速率设定为23.32 kg/s，这与加拿大福克斯溪地区Duvernay地层的实际平均施工强度相符。由于该段压裂时的返排率低至7%，说明大量流体存留在地下构造缝隙之中，因此并未在模拟中考虑流体的返排影响。经过测试，发现改变流体注入速率也会显著影响断层附近的库仑应力变化值，从而改变活化时间。当增加或减小注入量时，断层被激活的时间也会相应地减小或增加。这一分析具有一定的实际意义。如果能够事先计算出“安全可控诱发震级范围”的流体注入量或速率，就能为压裂施工提供有价值的建议。同理，这一规律可以应用到风险发生后，调整施工参数以调控地震风险当中。
将西断层附近发生的地震事件投影到其活化后库仑应力变化∆CFS的场图中，得到图11中的结果。图11a和11b分别展示了只有西断层以及东西断层并存的两种情况，图中灰点表示2015年1月23日MW3.9事件之前的地震，浅绿点表示该事件及其之后的地震。显然，MW3.9地震及其余震和后续诱发地震几乎完全位于西断层活化产生的库仑应力增加区，且震源深度与震级存在正相关关系（假设深度为正值）。该结果表明本文的多场耦合数值模拟结果具有合理性和可解释性。这里的库仑应力改变量∆CFS主要通过式（5）计算，除了表达断层位错产生的本身应力场外，还包含了孔隙压力的影响。模型中两个注液点附近的库仑应力呈现出集中趋势，并且随流体扩散到断层附近，这使得库仑应力场变得复杂。另外，东侧灰点所指示的诱发地震也位于东侧断层的库仑应力增加区（图11b）。
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图 10 断层活化前的∆CFS分布（a） 只有西断层；（b） 东西断层并存Figure 10. Distribution of ∆CFS before fault activation（a） Case with only the west fault；（b） Case with co-existence of the east and west faults最后需要指出本研究存在的局限性。文中模拟计算结果显示断层上下盘位移和库仑应力集中区能够与实测地震相吻合，虽然能够体现出模拟的合理性和主要的断层活化机制，但是仍有较多细节需要进一步探索：第一，本文仅从压裂注水和流体扩散的角度分析西断层活化过程，并未考虑前震与MW3.9主震的成核关系，后续需要结合已有研究（Gao et al，2022），从时空关系和机理上进行更多更细致的分析；第二，本文用于计算的模型为二维模型，且只考虑了断层的逆冲机制部分，这样不免忽略掉了一些三维空间计算应该考虑的走滑分量；第三，两条断层的实际活化顺序是先东断层而后西断层（图2），而东断层具有远场触发的特点，西断层则具有延迟触发的特点，其特殊性还需要在进一步搜集相关数据的基础上继续研究；第四，断层的临界滑动特征还需要进一步分析，其孕震、成核、破裂、滑动、愈合过程仍要结合速率状态方程等手段进行深入讨论。上述问题值得在今后的进一步工作中系统研究。
5. 结论
本文针对加拿大阿尔伯塔省Duvernay地层附近两条明显的、高度倾斜的地震活跃带（断层）展开数值模拟研究。根据丛集地震展布和震源机制信息，在模型中假设两条断层的产状和基本性质。在PKN裂缝扩展模型和多孔弹性理论的基础上，依据当地地质地层构造数据建立模型，并进行多物理场耦合运算。结果表明，在仅有西断层的情况下，经过5天注水和15天扩散的流体作用，西断层附近出现了2.56 MPa的∆CFS高值区，而在两条断层并存的情况下，∆CFS高值区增加至2.98 MPa。这说明东断层在结晶基底中的∆CFS集中区一定程度上增加了西断层的库仑应力，使得西断层更容易活化。因此，我们认为东断层的存在会改变西断层被激活的时间和强度，这与性质相似断层之间的相互触发有关。同时，模拟计算出西断层活化产生的库仑应力增加区覆盖了MW3.9地震及其余震，表明实际地震空间分布与模型中的断层设定及应力演化结果相符。本研究能够一定程度上解答中强型诱发地震的“尾随效应”难题，为注液诱发地震的危害性预测提供新的物理视角。最后，对本研究存在的不足进行了总结。
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图 1 标量波情形下1—10阶谱元法的网格频散曲线（De Basabe，Sen，2007）
Figure 1. Grid dispersion curves of 1st- to 10th-order spectral elements in acoustic case （De Basabe，Sen，2007）
1 Gauss-Lobatto-Legendre高精度数值积分的节点和权系数
1 The nodes and weights of Gauss-Lobatto-Legendre high-precision numerical integration
GLL节点数 积分节点 积分权系数 2 ±1 1 3 0 1.333 333 333 333 333 3 ±1 0.333 333 333 333 333 3 4 ±0.447 213 595 499 957 9 0.833 333 333 333 333 4 ±1 0.166 666 666 666 666 7 5 0 0.711 111 111 111 111 1 ±0.654 653 670 707 977 2 0.544 444 444 444 444 5 ±1 0.100 000 000 000 000 0 6 ±0.285 231 516 480 645 1 0.554 858 377 035 486 2 ±0.765 055 323 929 464 7 0.378 474 956 297 847 0 ±1 0.066 666 666 666 666 7 7 0 0.487 619 047 619 047 6 ±0.468 848 793 470 714 2 0.431 745 381 209 862 7 ±0.830 223 896 278 567 0 0.276 826 047 361 565 9 ±1 0.047 619 047 619 047 6 8 ±0.209 299 217 902 478 9 0.412 458 794 658 703 8 ±0.591 700 181 433 142 3 0.341 122 692 483 504 4 ±0.871 740 148 509 606 6 0.210 704 227 143 506 1 ±1 0.035 714 285 714 285 7 9 0 0.371 519 274 376 417 2 ±0.363 117 463 826 178 2 0.346 428 510 973 046 3 ±0.677 186 279 510 737 7 0.274 538 712 500 161 7 ±0.899 757 995 411 460 2 0.165 495 361 560 805 5 ±1 0.027 777 777 777 777 8 10 ±0.165 278 957 666 387 0 0.327 539 761 183 897 6 ±0.477 924 949 810 444 5 0.292 042 683 679 683 8 ±0.738 773 865 105 505 0 0.224 889 342 063 126 4 ±0.919 533 908 166 458 9 0.133 305 990 851 070 1 ±1 0.022 222 222 222 222 2 注：积分节点和权系数的数目与GLL节点数相对应，数值相同、正负号不同的积分节点对应相同的积分权系数，为简化起见，表中只列出一个。 
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表 1 谱元法结合二阶中心差分求解格式的稳定条件（De Basabe，Sen，2010）
Table 1 Stability criteria for spectral element method solved by classical second-order centered-difference time integration algorithm （De Basabe，Sen，2010）
谱单元阶次 1 2 3 4 5 6 7 8 标量波情形
（SH波动）qEmax 0.709 0.288 0.164 0.104 0.0714 0.0516 0.039 0 0.030 4 qdmax 0.709 0.577 0.593 0.604 0.608 0.608 0.608 0.607 弹性波情形
（P-SV波动）qEmax 0.816 0.333 0.189 0.120 0.082 3 0.059 5 0.044 9 0.035 0 qdmax 0.816 0.666 0.684 0.697 0.700 0.700 0.700 0.699
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巴振宁,赵靖轩,桑巧稚,梁建文. 2024. 基于Davidenkov本构模型的三维沉积盆地非线性地震动谱元法模拟[J]. 岩土工程学报,46(7):1387–1397. doi: 10.11779/CJGE20230582 Ba Z N,Zhao J X,Sang Q Z,Liang J W. 2024. Nonlinear ground motion simulation of three-dimensional sedimentary basin based on Davidenkov constitutive model and spectral element method[J]. Chinese Journal of Geotechnical Engineering,46(7):1387–1397 (in Chinese).
巴振宁,赵靖轩,张郁山,梁建文,张玉洁. 2023. 基于GP14.3运动学混合震源模型和SPECFEM3D谱元法的宽频地震动模拟[J]. 地球物理学报,66(3):1125–1138. doi: 10.6038/cjg2022Q0181 Ba Z N,Zhao J X,Zhang Y S,Liang J W,Zhang Y J. 2023. Broadband ground motion spectral element simulation based on GP14.3 kinematic hybrid source model and SPECFEM3D[J]. Chinese Journal of Geophysics,66(3):1125–1138 (in Chinese).
曹丹平,周建科,印兴耀. 2015. 三角网格有限元法波动模拟的数值频散及稳定性研究[J]. 地球物理学报,58(5):1717–1730. doi: 10.6038/cjg20150522 Cao D P,Zhou J K,Yin X Y. 2015. The study for numerical dispersion and stability of wave motion with triangle-based finite element algorithm[J]. Chinese Journal of Geophysics,58(5):1717–1730 (in Chinese).
车承轩. 2007. 谱元法模拟起伏自由表面地层中的弹性波传播[D]. 大庆:大庆石油学院:1−49. Che C X. 2007. The Spectral Element Method for Elastic Wave Simulation in a Formation with a Topographic Traction Free Surface[D]. Daqing:Daqing Petroleum Institute:1−49 (in Chinese).
陈少林,程书林,柯小飞. 2019b. 海洋地震工程流固耦合问题统一计算框架——不规则界面情形[J]. 力学学报,51(5):1517–1529. Chen S L,Cheng S L,Ke X F. 2019b. A unified computational framework for fluid-solid coupling in marine earthquake engineering:Irregular interface case[J]. Chinese Journal of Theoretical and Applied Mechanics,51(5):1517–1529.
陈少林,柯小飞,张洪翔. 2019a. 海洋地震工程流固耦合问题统一计算框架[J]. 力学学报,51(2):594–606. Chen S L,Ke X F,Zhang H X. 2019a. A unified computational framework for fluid-solid coupling in marine earthquake engineering[J]. Chinese Journal of Theoretical and Applied Mechanics,51(2):594–606.
戴志军,李小军,侯春林. 2015. 谱元法与透射边界的配合使用及其稳定性研究[J]. 工程力学,32(11):40–50. doi: 10.6052/j.issn.1000-4750.2014.03.0196 Dai Z J,Li X J,Hou C L. 2015. A combination usage of transmitting formula and spectral element method and the study of its stability[J]. Engineering Mechanics,32(11):40–50 (in Chinese).
董兴朋,杨顶辉. 2017. 球坐标系下谱元法三维地震波场模拟[J]. 地球物理学报,60(12):4671–4680. doi: 10.6038/cjg20171211 Dong X P,Yang D H. 2017. Numerical modeling of the 3-D seismic wave field with the spectral element method in spherical coordinates[J]. Chinese Journal of Geophysics,60(12):4671–4680 (in Chinese).
贺春晖,王进廷,张楚汉. 2017. 基于震源-河谷波场数值模拟的坝址地震动参数确定方法[J]. 地球物理学报,60(2):585–592. doi: 10.6038/cjg20170213 He C H,Wang J T,Zhang C H. 2017. Determination of seismic parameters for dam sites by numerical simulation of the rupture-canyon wave field[J]. Chinese Journal of Geophysics,60(2):585–592 (in Chinese).
胡元鑫,刘新荣,罗建华,张梁,葛华. 2011. 汶川震区地震动三维地形效应的谱元法模拟[J]. 兰州大学学报(自然科学版),47(4):24–32. Hu Y X,Liu X R,Luo J H,Zhang L,Ge H. 2011. Simulation of three-dimensional topographic effects on seismic ground motion in Wenchuan earthquake region based upon the spectral-element method[J]. Journal of Lanzhou University (Natural Sciences),47(4):24–32 (in Chinese).
蒋涵,周红,高孟潭. 2015. 山脊线与坡度和峰值速度放大系数的相关性研究[J]. 地球物理学报,58(1):229–237. doi: 10.6038/cjg20150120 Jiang H,Zhou H,Gao M T. 2015. A study on the correlation of the ridge line and slope with peak ground velocity amplification factor[J]. Chinese Journal of Geophysics,58(1):229–237 (in Chinese).
孔曦骏,邢浩洁,李鸿晶. 2022. 流固耦合地震波动问题的显式谱元模拟方法[J]. 力学学报,54(9):2513–2528. doi: 10.6052/0459-1879-22-068 Kong X J,Xing H J,Li H J. 2022. An explicit spectral-element approach to fluid-solid coupling problems in seismic wave propagation[J]. Chinese Journal of Theoretical and Applied Mechanics,54(9):2513–2528 (in Chinese).
李冰非,董兴朋,李小凡,司洁戈. 2019. 基于辛-谱元-FK 混合方法的保结构远震波场模拟[J]. 地球物理学报,62(11):4339–4352. doi: 10.6038/cjg2019M0688 Li B F,Dong X P,Li X F,Si J G. 2019. Structure-preserving modeling of teleseismic wavefield using symplectic SEM-FK hybrid method[J]. Chinese Journal of Geophysics,62(11):4339–4352 (in Chiese).
李冰非,李小凡,李峰,龚飞. 2021. 基于辛-谱元方法的地球自由振荡保弥散衰减数值模拟[J]. 地球物理学报,64(11):4022–4030. doi: 10.6038/cjg2021P0019 Li B F,Li X F,Li F,Gong F. 2021. Dissipation preserving simulation for Earth’s free oscillations based on symplectic spectral element method[J]. Chinese Journal of Geophysics,64(11):4022–4030 (in Chinese).
李昊臻,刘少林,董兴朋,蒙伟娟,杨顶辉. 2024. 基于逐元和轴对称谱元的混合方法及远震波场模拟[J]. 地球物理学报,67(5):1819–1831. doi: 10.6038/cjg2023R0180 Li H Z,Liu S L,Dong X P,Meng W J,Yang D H. 2024. Hybrid method based on element-by-element and axisymmetric spectral element method for teleseismic wavefield simulation[J]. Chinese Journal of Geophysics,67(5):1819–1831 (in Chinese).
李鸿晶,王竞雄. 2022. 时域谱元法的质量特性模型及其构建方法[J]. 地震学报,44(1):60–75. doi: 10.11939/jass.20210117 Li H J,Wang J X. 2022. The mass property model and its implementation in the time-domain spectral element method[J]. Acta Seismologica Sinica,44(1):60–75 (in Chinese).
李琳,刘韬,胡天跃. 2014. 三角谱元法及其在地震正演模拟中的应用[J]. 地球物理学报,57(4):1224–1234. doi: 10.6038/cjg20140419 Li L,Liu T,Hu T Y. 2014. Spectra element method with triangular mesh and its application in seismic modeling[J]. Chinese Journal of Geophysics,57(4):1224–1234 (in Chinese).
李孝波. 2014. 基于谱元法的玉田震害异常研究[D]. 哈尔滨:中国地震局工程力学研究所:1−136. Li X B. 2014. The Investigation of Seismic Damage Anomaly in Yutian based on Spectral Element Method[D]. Harbin:Institute of Engineering Mechanics,China Earthquake Administration:1−136 (in Chinese).
林伟军,苏畅,Seriani G. 2018. 多网格谱元法及其在高性能计算中的应用[J]. 应用声学,37(1):42–52. doi: 10.11684/j.issn.1000-310X.2018.01.007 Lin W J,Su C,Seriani G. 2018. The poly-grid spectral element method and its application in high performance computing[J]. Journal of Applied Acoustics,37(1):42–52 (in Chinese).
林伟军,王秀明,张海澜. 2005. 用于弹性波方程模拟的基于逐元技术的谱元法[J]. 自然科学进展,15(9):1048–1057. doi: 10.3321/j.issn:1002-008X.2005.09.004 Lin W J,Wang X M,Zhang H L. 2005. An element-by-element spectral element method for the modeling of elastic wave equation[J]. Progress in Natural Science,15(9):1048–1057 (in Chinese).
林伟军. 2007. 弹性波传播模拟的Chebyshev谱元法[J]. 声学学报,32(6):525–533. doi: 10.3321/j.issn:0371-0025.2007.06.007 Lin W J. 2007. A Chebyshev spectral element method for elastic wave modeling[J]. Acta Acustica,32(6):525–533 (in Chinese).
刘晶波,廖振鹏. 1989. 离散网格中的弹性波动(Ⅱ)--几种有限元离散模型的对比分析[J]. 地震工程与工程振动, 9 (2):1−11. Liu J B,Liao Z P. 1989. Elastic wave motion in discrete grids (II) – Comparison of common finite element models[J]. Earthquake Engineering and Engineering Vibration,9(2):1–11 (in Chinese).
刘晶波,廖振鹏. 1990. 离散网格中的弹性波动(Ⅲ)--时域离散化对波传播规律的影响[J]. 地震工程与工程振动, 10 (2):1−10. Liu J B,Liao Z P. 1990. Elastic wave motion in discrete grids (III) – The effect of discretization in time domain on wave motion[J]. Earthquake Engineering and Engineering Vibration,10(2):1–10 (in Chinese).
刘启方,于彦彦,章旭斌. 2013. 施甸盆地三维地震动研究[J]. 地震工程与工程振动,33(4):54–60. Liu Q F,Yu Y Y,Zhang X B. 2013. Three-dimensional ground motion simulation of Shidian basin[J]. Journal of Earthquake Engineering and Engineering Vibration,33(4):54–60 (in Chinese).
刘少林,李小凡,刘有山,朱童,张美根. 2014. 三角网格有限元法声波与弹性波模拟频散分析[J]. 地球物理学报,57(8):2620–2630. doi: 10.6038/cjg20140821 Liu S L,Li X F,Liu Y S,Zhu T,Zhang M G. 2014. Dispersion analysis of triangle-based finite element method for acoustic and elastic wave simulations[J]. Chinese Journal of Geophysics,57(8):2620–2630 (in Chinese).
刘少林,杨顶辉,孟雪莉,汪文帅,徐锡伟,李小凡. 2022. 模拟地震波传播的优化质量矩阵Legendre谱元法[J]. 地球物理学报,65(12):4802–4815. doi: 10.6038/cjg2022Q0145 Liu S L,Yang D H,Meng X L,Wang W S,Xu X W,Li X F. 2022. A Legendre spectral element method with optimal mass matrix for seismic wave modeling[J]. Chinese Journal of Geophysics,65(12):4802–4815 (in Chinese).
刘少林,杨顶辉,徐锡伟,李小凡,申文豪,刘有山. 2021. 模拟地震波传播的三维逐元并行谱元法[J]. 地球物理学报,64(3):993–1005. doi: 10.6038/cjg2021O0405 Liu S L,Yang D H,Xu X W,Li X F,Shen W H,Liu Y S. 2021. Three-dimensional element-by-element parallel spectral-element method for seismic wave modeling[J]. Chinese Journal of Geophysics,64(3):993–1005 (in Chinese).
刘有山,刘少林,张美根,马德堂. 2012. 一种改进的二阶弹性波动方程的最佳匹配层吸收边界条件[J]. 地球物理学进展,27(5):2113. doi: 10.6038/j.issn.1004-2903.2012.05.036 Liu Y S,Liu S L,Zhang M G,Ma D T. 2012. An improved perfectly matched layer absorbing boundary condition for second order elastic wave equation[J]. Progress in Geophysics,27(5):2113 (in Chinese).
刘有山,滕吉文,徐涛,刘少林,司芗,马学英. 2014. 三角网格谱元法地震波场数值模拟[J]. 地球物理学进展,29(4):1715–1726. doi: 10.6038/pg20140430 Liu Y S,Teng J W,Xu T,Liu S L,Si X,Ma X Y. 2014. Numerical modeling of seismic wavefield with the SEM based on triangles[J]. Progress in Geophysics,29(4):1715–1726 (in Chinese).
陆新征,田源,许镇,熊琛. 2021. 城市抗震弹塑性分析[M]. 北京:清华大学出版社:237−317. Lu X Z,Tian Y,Xu Z,Xiong C. 2021. Elastoplastic Analysis of Urban Seismic Resistance[M]. Beijing:Tsinghua University Press:237−317 (in Chinese).
孟雪莉,刘少林,杨顶辉,汪文帅,徐锡伟,李小凡. 2022. 基于优化数值积分的谱元法模拟地震波传播[J]. 石油地球物理勘探,57(3):602–612. Meng X L,Liu S L,Yang D H,Wang W S,Xu X W,Li X F. 2022. Simulating seismic wave propagation using spectral element method based on optimized numerical integration[J]. Geophysical Exploration of Petroleum,57(3):602–612 (in Chinese).
秦国良,徐忠. 2000. 谱元方法求解二维不可压缩Navier-Stokes方程[J]. 应用力学学报,17(4):20–25. doi: 10.3969/j.issn.1000-4939.2000.04.004 Qin G L,Xu Z. 2000. A spectral element method for incompressible Navier-Stokes equations[J]. Chinese Journal of Applied Mechanics,17(4):20–25 (in Chinese).
秦国良,徐忠. 2001. 谱元方法求解正方形封闭空腔内的自然对流换热[J]. 计算物理,18(2):119–124. doi: 10.3969/j.issn.1001-246X.2001.02.005 Qin G L,Xu Z. 2001. Computation of natural convection in two-dimensional cavity using spectral element method[J]. Chinese Journal of Computational Physics,18(2):119–124 (in Chinese).
任骏声,张怀,周元泽,张振,石耀霖. 2024. 基于卷积滤波的谱元法在长时程波场模拟中的应用[J]. 地球物理学报,67(5):1832–1838. doi: 10.6038/cjg2022Q0160 Ren J S,Zhang H,Zhou Y Z,Zhang Z,Shi Y L. 2024. Application of spectral element method based on convolving filtering in long-term wavefield modeling[J]. Chinese Journal of Geophysics,67(5):1832–1838 (in Chinese).
唐杰. 2011. 气枪激发信号传播的谱元法数值模拟研究[J]. 地球物理学报,54(9):2348–2356. doi: 10.3969/j.issn.0001-5733.2011.09.018 Tang J. 2011. Study on SEM numerical simulation of airgun signal transition[J]. Chinese Journal of Geophysics,54(9):2348–2356 (in Chinese).
汪文帅,李小凡,鲁明文,张美根. 2012. 基于多辛结构谱元法的保结构地震波场模拟[J]. 地球物理学报,55(10):3427–3439. doi: 10.6038/j.issn.0001-5733.2012.10.026 Wang W S,Li X F,Lu M W,Zhang M G. 2012. Structure-preserving modeling for seismic wave fields based on a multisymplectic spectral element method[J]. Chinese Journal of Geophysics,55(10):3427–3439 (in Chinese).
汪文帅,李小凡. 2013. 基于辛格式的谱元法及其在横向各向同性介质波场模拟中的应用[J]. 数值计算与计算机应用,34(1):20–30. doi: 10.3969/j.issn.1000-3266.2013.01.003 Wang W S,Li X F. 2013. The SEM based on symplectical schemes and its application in modeling the wave propagation in transversely isotropic media[J]. Journal on Numerical Methods and Computer Applications,34(1):20–30 (in Chinese).
汪文帅,张怀,李小凡. 2013. 间断的 Galerkin 方法在地震波场数值模拟中的应用概述[J]. 地球物理学进展,28(1):171–179. doi: 10.6038/pg20130118 Wang W S,Zhang H,Li X F. 2013. Review on application of the discontinuous Galerkin method for modeling of the seismic wavefield[J]. Progress in Geophysics,28(1):171–179 (in Chinese).
王竞雄,李鸿晶,邢浩洁. 2022. 水平成层场地地震反应的集中质量切比雪夫谱元分析方法[J]. 地震学报,44(1):76–86. doi: 10.11939/jass.20210091 Wang J X,Li H J,Xing H J. 2022. The Lumped mass Chebyshev spectral element method for seismic response analysis of horizontally layered soil sites[J]. Acta Seismologica Sinica,44(1):76–86 (in Chinese).
王童奎,李瑞华,李小凡,张美根,龙桂华. 2007. 横向各向同性介质中地震波场谱元法数值模拟[J]. 地球物理学进展,22(3):778–784. doi: 10.3969/j.issn.1004-2903.2007.03.018 Wang T K,Li R H,Li X F,Zhang M G,Long G H. 2007. Numerical spectral element modeling for seismic wave propagation in transversely isotropic medium[J]. Progress in Geophysics,22(3):778–784 (in Chinese).
王童奎,谢占安,付兴深,高文中,刘萱. 2009. 弹性介质中谱元法叠后逆时偏移方法研究[J]. 石油物探,48(4):354–358. doi: 10.3969/j.issn.1000-1441.2009.04.006 Wang T K,Xie Z A,Fu X S,Gao W Z,Liu X. 2009. Research on post-stack reverse-time migration in elastic media based on spectral element method[J]. Geophysical Prospecting for Petroleum,48(4):354–358 (in Chinese).
王秀明,Seriani G,林伟军. 2007. 利用谱元法计算弹性波场的若干理论问题[J]. 中国科学:G辑,37(1):41–59. Wang X M,Seriani G,Lin W J. 2007. Several theoretical issues on the spectral-element calculation of elastic wave field[J]. Science China Series G,37(1):41–59 (in Chinese).
向新民. 2000. 谱方法的数值分析[M]. 北京:科学出版社:1−327. Xiang X M. 2000. Numerical Analysis of Spectral Methods[M]. Beijing:Science Press:1−327 (in Chinese).
谢志南,章旭斌. 2017,弱形式时域完美匹配层[J]. 地球物理学报, 60 (10):3823−3831. Xie Z N,Zhang X B. 2017. Weak-form time-domain perfectly matched layer[J]. Chinese Journal of Geophysics,60(10):3823–3831 (in Chinese).
谢志南,郑永路,章旭斌,唐丽华. 2019. 弱形式时域完美匹配层——滞弹性近场波动数值模拟[J]. 地球物理学报,62(8):3140–3154. doi: 10.6038/cjg2019M0425 Xie Z N,Zheng Y L,Zhang X B,Tang L H. 2019. Weak-form time-domain perfectly matched layer for numerical simulation of viscoelastic wave propagation in infinite domain[J]. Chinese Journal of Geophysics,62(8):3140–3154 (in Chinese).
谢志南,郑永路,章旭斌. 2018. 常Q滞弹性介质地震波动数值模拟——时域本构优化逼近[J]. 地球物理学报,61(10):4007–4020. doi: 10.6038/cjg2018L0704 Xie Z N,Zheng Y L,Zhang X B. 2018. Optimized approximation for constitution of constant Q viscoelastic media in time domain seismic wave simulation[J]. Chinese Journal of Geophysics,61(10):4007–4020 (in Chinese).
邢浩洁,李鸿晶,李小军. 2021a. 一维波动有限元模拟中透射边界的时域稳定条件[J]. 应用基础与工程科学学报,29(3):617–632. Xing H J,Li H J,Li X J. 2021a. Time-domain stability conditions of multi-transmitting formula in one-dimensional finite element simulation of wave motion[J]. Journal of Basic Science and Engineering,29(3):617–632 (in Chinese).
邢浩洁,李鸿晶,杨笑梅. 2017. 基于切比雪夫谱元模型的成层场地地震反应分析[J]. 岩土力学,38(2):593–600. Xing H J,Li H J,Yang X M. 2017. Seismic response analysis of horizontal layered soil sites based on Chebyshev spectral element model[J]. Rock and Soil Mechanics,2017, 38 (2):593−600 (in Chinese).
邢浩洁,李鸿晶. 2017a. 透射边界条件在波动谱元模拟中的实现:一维波动[J]. 力学学报,49(2):367–379. Xing H J,Li H J. 2017b. Implementation of multi-transmitting boundary condition for wave motion simulation by spectral element method:One dimension case[J]. Chinese Journal of Theoretical and Applied Mechanics,49(2):367–379 (in Chinese).
邢浩洁,李鸿晶. 2017b. 透射边界条件在波动谱元模拟中的实现:二维波动[J]. 力学学报,49(4):894–906. Xing H J,Li H J. 2017b. Implementation of multi-transmitting boundary condition for wave motion simulation by spectral element method:Two dimension case[J]. Chinese Journal of Theoretical and Applied Mechanics,49(4):894–906 (in Chinese).
邢浩洁,李鸿晶. 2017c. 波动切比雪夫谱元模拟的时间积分方法研究[J]. 南京工业大学学报(自然科学版),39(2):70–76. Xing H J,Li H J. 2017c. Investigation of time integration method for Chebyshev spectral element simulation of wave motion[J]. Journal of Nanjing Tech University (Natural Science Edition),39(2):70–76 (in Chinese).
邢浩洁,李小军,刘爱文,李鸿晶,周正华,陈苏. 2021b. 波动数值模拟中的外推型人工边界条件[J]. 力学学报,53(5):1480–1495. Xing H J,Li X J,Liu A W,Li H J,Zhou Z H,Chen S. 2021. Extrapolation-type artificial boundary conditions in the numerical simulation of wave motion[J]. Chinese Journal of Theoretical and Applied Mechanics,53(5):1480–1495 (in Chinese).
邢浩洁,刘爱文,李小军,陈苏,傅磊. 2022. 多人工波速优化透射边界在谱元法地震波动模拟中的应用[J]. 地震学报,44(1):26–39. doi: 10.11939/jass.20210090 Xing H J,Liu A W,Li X J,Chen S,Fu L. 2022. Application of an optimized transmitting boundary with multiple artificial wave velocities in spectral-element simulation of seismic wave propagation[J]. Acta Seismologica Sinica,44(1):26–39 (in Chinese).
许传炬,林玉闽. 2000. Poiseuille-Bénard流的出口边界条件及其谱元法计算[J]. 力学学报,32(1):1–10. doi: 10.3321/j.issn:0459-1879.2000.01.001 Xu C J,Lin Y M. 2000. Open boundary conditions in simulation by spectral element methods of Poiseuille-Bénard channel flow[J]. Chinese Journal of Theoretical and Applied Mechanics,32(1):1–10 (in Chinese).
严珍珍,张怀,杨长春,石耀霖. 2009. 汶川大地震地震波传播的谱元法数值模拟研究[J]. 中国科学:D辑,39(4):393–402. Yan Z Z,Zhang H,Yang C C,Shi Y L. 2009. Study on the seismic wave propagation of the great Wenchuan earthquake by using the numerical simulation of spectral element method[J]. Science China Series D,39(4):393–402 (in Chinese).
于彦彦,丁海平,刘启方. 2017. 透射边界与谱元法的结合及对波动模拟精度的改进[J]. 振动与冲击, 36 (2):13−22. Yu Y Y,Ding H P,Liu Q F. 2017. Integration of transmitting boundary and spectral element method and improvement on the accuracy of wave motion simulation[J]. Journal of Vibration and Shock, 36 (2):13−22 (in Chinese).
于彦彦,芮志良,丁海平. 2023. 三维局部场地地震波散射问题谱元并行模拟方法[J]. 力学学报,55(6):1342–1354. doi: 10.6052/0459-1879-23-052 Yu Y Y,Rui Z L,Ding H P. 2023. Parallel spectral element method for 3d local-site ground motion simulations of wave scattering problem[J]. Chinese Journal of Theoretical and Applied Mechanics,55(6):1342–1354 (in Chinese).
章旭斌,谢志南. 2022. 波动谱元模拟中透射边界稳定性分析[J]. 工程力学,39(10):26–35. doi: 10.6052/j.issn.1000-4750.2021.06.0428 Zhang X B,Xie Z N. 2022. Stability analysis of transmitting boundary in wave spectral element simulation[J]. Engineering Mechanics,39(10):26–35 (in Chinese).
赵靖轩,巴振宁,阔晨阳,刘博佳. 2023. 2022年9月5日泸定MS6.8地震宽频带地震动谱元法模拟[J]. 地震学报,45(2):179–195. doi: 10.11939/jass.20220190 Zhao J X,Ba Z N,Kuo C Y,Liu B J. 2023. Broadband ground motion simulations applied to the Luding MS6.8 earthquake on September 5,2022 based on spectral element method[J]. Acta Seismologica Sinica,45(2):179–195 (in Chinese).
周红,高孟潭,俞言祥. 2010. SH 波地形效应特征的研究[J]. 地球物理学进展,25(3):775–782. doi: 10.3969/j.issn.1004-2903.2010.03.005 Zhou H,Gao M T,Yu Y X. 2010. A study of topographical effect on SH waves[J]. Progress in Geophysics,25(3):775–782 (in Chinese).
周红. 2018. 九寨沟7.0级地震地表地震动位移及静态位移的模拟研究[J]. 地球物理学报,61(12):4851–4861. doi: 10.6038/cjg2018M0010 Zhou H. 2018. Research on ground motion displacement and static displacement near the fault of Jiuzhaigou MS7.0 earthquake[J]. Chinese Journal of Geophysics,61(12):4851–4861 (in Chinese).
朱伯芳. 1998. 有限单元法原理与应用[M]. 第二版. 北京:水利水电出版社:1−176. Zhu B F. 1998. The Finite Element Method Theory and Applications[M]. 2nd ed. Beijing:China Water&Power Press:1−176 (in Chinese).
Abraham J R,Smerzini C,Paolucci R,Lai C G. 2016. Numerical study on basin-edge effects in the seismic response of the Gubbio valley,Central Italy[J]. B Earthq Eng,14:1437–1459. doi: 10.1007/s10518-016-9890-y
Aguirre V M H,Paolucci R,Sánchez-Sesma F J,Mazzieri I. 2023. Three-dimensional numerical modeling of ground motion in the Valley of Mexico:A case study from the MW3. 2 earthquake of July 17,2019[J]. Earthq Spectra, 39 (4):2323−2351.
Alford R M,Kelly K R,Boore D M. 1974. Accuracy of finite-difference modeling of the acoustic wave equation[J]. Geophysics,39(6):834–842. doi: 10.1190/1.1440470
Antonietti P F,Mazzieri I,Quarteroni A,Rapetti F. 2012. Non-conforming high order approximations of the elastodynamics equation[J]. Comput Method Appl M,209:212–238.
Asmar A H. 2005. Partial Differential Equations with Fourier Series and Boundary Value Problems[M]. 2nd ed. USA:Courier Dover Publications:227−321.
Ba Z N,Fu J S,Wang F B,Liang J W,Zhang B,Zhang L. 2024a. Physics-based seismic analysis of ancient wood structure:fault-to-structure simulation[J]. Earthq Eng Eng Vib,23(3):727–740. doi: 10.1007/s11803-024-2268-2
Ba Z N,Sang Q Z,Wu M T,Liang J W. 2021. The revised direct stiffness matrix method for seismogram synthesis due to dislocations:from crustal to geotechnical scale[J]. Geophys J Int,227(1):717–734. doi: 10.1093/gji/ggab248
Ba Z N,Wu M T,Liang J W,Zhao J X,Lee V W. 2022. A two-step approach combining FK with SE for simulating ground motion due to point dislocation sources[J]. Soil Dyn Earthq Eng,57:107224.
Ba Z N,Zhao J X,Sang Q Z,Liang J W. 2024b. Nonlinear seismic response of an alluvial basin modelled by spectral element method:Implementation of a Davidenkov constitutive model[J]. J Earthq Eng,1−30.
Ba Z N,Zhao J X,Wang Y. 2024c. GA-BPNN prediction model of broadband ground motion parameters in Tianjin area driven by synthetic database based on hybrid simulated method[J]. Pure Appl Geophys,181:1195–1220. doi: 10.1007/s00024-024-03431-1
Ba Z N,Zhao J X,Zhu Z H,Zhou G Y. 2023. 3D physics-based ground motion simulation and topography effects of the 05 September 2022 MW6. 6 Luding earthquake,China[J]. Soil Dyn Earthq Eng, 172 :108048.
Baker J W,Luco N,Abrahamson N A,Graves R W,Maechling P J,Olsen K B. 2014. Engineering uses of physics-based ground motion simulations[C]//Proceedings of the Tenth US Conference on Earthquake Engineering,Anchorage,Alaska:1−11.
Bradley B A. 2019. On-going challenges in physics-based ground motion prediction and insights from the 2010–2011 Canterbury and 2016 Kaikoura,New Zealand earthquakes[J]. Soil Dyn Earthq Eng,124:354–364. doi: 10.1016/j.soildyn.2018.04.042
Briani M,Sommariva A,Vianello M. 2012. Computing Fekete and Lebesgue points:simplex,square,disk[J]. J Comput Appl Math,236(9):2477–2486. doi: 10.1016/j.cam.2011.12.006
Canuto C,Hussaini M Y,Quarteroni A,Zang T A. 1988. Spectral Methods in Fluid Dynamics[M]. Berlin:Springer-Verlag:1−550.
Canuto C,Hussaini M Y,Quarteroni A,Zang T A. 2006. Spectral Methods – Fundamentals in Single Domains[M]. Berlin:Springer-Verlag:1−552.
Capdeville Y,Gung Y,Romanowicz B. 2005. Towards global earth tomography using the spectral element method:a technique based on source stacking[J]. Geophys J Int,162(2):541–554. doi: 10.1111/j.1365-246X.2005.02689.x
Carmona A E,Peter D B,Parisi L,Mai P M. 2024. Anelastic tomography of the Arabian plate[J]. B Seismol Soc Am,114(3):1347–1364.
Castelli F,Cavallaro A,Grasso S,Lentini V. 2016. Seismic microzoning from synthetic ground motion earthquake scenarios parameters:the case study of the City of Catania (Italy)[J]. Soil Dyn Earthq Eng,88:307–327. doi: 10.1016/j.soildyn.2016.07.010
Chaljub E,Komatitsch D,Vilotte J P,Capdeville Y,Valette B,Festa G. 2007. Spectral-element analysis in seismology[J]. Adv Geophys,48:365–419.
Chaljub E,Maufroy E,Moczo P,Kristek J,Hollender F,Bard P Y,Priolo E,Klin P,Martin F,Zhang Z G,Zhang W,Chen X F. 2015. 3-D numerical simulations of earthquake ground motion in sedimentary basins:testing accuracy through stringent models[J]. Geophys J Int,201(1):90–111. doi: 10.1093/gji/ggu472
Chaljub E,Moczo P,Tsuno S,Bard P Y,Kristek J,Käser M,Stupazzini M,Kristekova M. 2010. Quantitative comparison of four numerical predictions of 3D ground motion in the Grenoble Valley,France[J]. B Seismol Soc Am,100(4):1427–1455. doi: 10.1785/0120090052
Che C X,Wang X M,Lin W J. 2010. The Chebyshev spectral element method using staggered predictor and corrector for elastic wave simulations[J]. Appl Geophys,7(2):174–184. doi: 10.1007/s11770-010-0242-9
Chen M,Niu F L,Liu Q Y,Tromp J,Zheng X F. 2015. Multiparameter adjoint tomography of the crust and upper mantle beneath East Asia:1. Model construction and comparisons[J]. J Geophys Res-Sol Ea,120(3):1762–1786. doi: 10.1002/2014JB011638
Chen M,Niu F L,Tromp J,Lenardic A,Lee C T A,Cao W R,Ribeiro J. 2017. Lithospheric foundering and underthrusting imaged beneath Tibet[J]. Nat Commun,8(1):15659. doi: 10.1038/ncomms15659
Chen Q,Babuška I. 1995. Approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle[J]. Comput Method Appl M, 128 (3−4):405−417.
Chen Z W,Huang D R,Wang G. 2023a. Large‐scale ground motion simulation of the 2016 Kumamoto earthquake incorporating soil nonlinearity and topographic effects[J]. Earthq Eng Struct D,52(4):956–978. doi: 10.1002/eqe.3795
Chen Z W,Huang D R,Wang G. 2023b. A regional scale coseismic landslide analysis framework:Integrating physics-based simulation with flexible sliding analysis[J]. Eng Geol,315:107040. doi: 10.1016/j.enggeo.2023.107040
Cohen G C. 2002. Higher-Order Numerical Methods for Transient Wave Equations[M]. Berlin:Springer:1−346.
Cohen G,Joly P,Roberts J E,Tordjman N. 2001. Higher order triangular finite elements with mass lumping for the wave equation[J]. SIAM J Numer Anal,38(6):2047–2078. doi: 10.1137/S0036142997329554
Cui Y,Poyraz E,Olsen K B,Zhu J,Withers K,Callaghan S,Larkin J,Guest C,Choi D,Chourasia A,Shi Z,Day S M,Maechling P J,Jordan T H. 2013. Physics-based seismic hazard analysis on petascale heterogeneous supercomputers[C]//Proceedings of the International Conference on High Performance Computing,Networking,Storage and Analysis:1−12.
Dauksher W,Emery A F. 1997. Accuracy in modeling the acoustic wave equation with Chebyshev spectral finite elements[J]. Finite Elem Anal Des,26(2):115–128. doi: 10.1016/S0168-874X(96)00075-3
De Basabe J D,Sen M K,Wheeler M F. 2008. The interior penalty discontinuous Galerkin method for elastic wave propagation:Grid dispersion[J]. Geophys J Int,175(1):83–93. doi: 10.1111/j.1365-246X.2008.03915.x
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. doi: 10.1190/1.2785046
De Basabe J D,Sen M K. 2009. New developments in the finite-element method for seismic modeling[J]. The Leading Edge,28(5):562–567. doi: 10.1190/1.3124931
De Basabe J D,Sen M K. 2010. Stability of the high-order finite elements for acoustic or elastic wave propagation with high-order time stepping[J]. Geophys J Int,181(1):577–590. doi: 10.1111/j.1365-246X.2010.04536.x
De Basabe J D,Sen M K. 2015. A comparison of finite-difference and spectral-element methods for elastic wave propagation in media with a fluid-solid interface[J]. Geophys J Int,200(1):278–298. doi: 10.1093/gji/ggu389
De Basabe J D. 2009. High-Order Finite Element Methods for Seismic Wave Propagation[D]. Austin:The University of Texas at Austin:1−128.
Dubiner M. 1991. Spectral methods on triangles and other domains[J]. J Sci Comput,6:345–390. doi: 10.1007/BF01060030
Dumbser M,Käser M,Toro E F. 2007. An arbitrary high-order Discontinuous Galerkin method for elastic waves on unstructured meshes-V. Local time stepping and p-adaptivity[J]. Geophys J Int,171(2):695–717. doi: 10.1111/j.1365-246X.2007.03427.x
Dumbser M,Käser M. 2006. An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes—II. The three-dimensional isotropic case[J]. Geophys J Int,167(1):319–336. doi: 10.1111/j.1365-246X.2006.03120.x
Evangelista L,Del Gaudio S,Smerzini C,d’Onofrio A,Festa G,Iervolino I,Landolfi L,Paolucci R,Santo A,Silvestri F. 2017. Physics-based seismic input for engineering applications:a case study in the Aterno river valley,Central Italy[J]. B Earthq Eng,15:2645–2671. doi: 10.1007/s10518-017-0089-7
Faccioli E,Maggio F,Paolucci R,Quarteroni A. 1997. 2D and 3D elastic wave propagation by a pseudo-spectral domain decomposition method[J]. J Seismol,1:237–251. doi: 10.1023/A:1009758820546
Faccioli E,Quarteroni A. 1999. Comment on “The spectral element method:An efficient tool to simulate the seismic response of 2D and 3D geological structures,” by D. Komatitsch and J. -P. Vilotte[J]. B Seismol Soc Am, 89 (1):331−331.
Feng K W,Huang D R,Wang G,Jin F,Chen Z W. 2022. Physics-based large-deformation analysis of coseismic landslides:A multiscale 3D SEM-MPM framework with application to the Hongshiyan landslide[J]. Eng Geol,297:106487. doi: 10.1016/j.enggeo.2021.106487
Fichtner A,Simutė S. 2018. Hamiltonian Monte Carlo inversion of seismic sources in complex media[J]. J Geophys Res-Sol Ea,123(4):2984–2999. doi: 10.1002/2017JB015249
French S W,Romanowicz B A. 2014. Whole-mantle radially anisotropic shear velocity structure from spectral-element waveform tomography[J]. Geophys J Int,199(3):1303–1327. doi: 10.1093/gji/ggu334
Fu J S,Ba Z N,Wang F B. 2024. A simulation approach for site-city interaction in basin under oblique incident waves and its applications[J]. Soil Dyn Earthq Eng,177:108407. doi: 10.1016/j.soildyn.2023.108407
Gatti F,Touhami S,Lopez-Caballero F,Paolucci R,Clouteau D,Fernandes V A,Kham M,Voldoire F. 2018. Broad-band 3-D earthquake simulation at nuclear site by an all-embracing source-to-structure approach[J]. Soil Dyn Earthq Eng,115:263–280. doi: 10.1016/j.soildyn.2018.08.028
Geng Y H,Qin G L,Wang Y,He W. 2016. The research of space-time coupled spectral element method for acoustic wave equations[J]. Chinese Journal of Acoustics,35(1):29–47.
Giraldo F X,Taylor M A. 2006. A diagonal-mass-matrix triangular-spectral-element method based on cubature points[J]. J Eng Math,56:307–322.
Gopalakrishnan S,Chakraborty A,Mahapatra D R. 2008. Spectral Finite Element Method:Wave Propagation,Diagnostics and Control in Anisotropic and Inhomogeneous Structures[M]. London:Springer-Verlag:1−22.
Gottlieb D,Orszag S A. 1977. Numerical Analysis of Spectral Methods:Theory and Applications[M]. Philadelphia:Society for Industrial and Applied Mathematics:1−167.
Grasso S,Maugeri M. 2014. Seismic microzonation studies for the city of Ragusa (Italy)[J]. Soil Dyn Earthq Eng,56:86–97. doi: 10.1016/j.soildyn.2013.10.004
Graves R,Jordan T H,Callaghan S,Deelman E,Field E,Juve G,Kesselman C,Maechling P,Mehta G,Milner K,Okaya D,Small P,Vahi K. 2011. CyberShake:A physics-based seismic hazard model for southern California[J]. Pure Appl Geophys,168:367–381. doi: 10.1007/s00024-010-0161-6
Graves R,Pitarka A. 2015. Refinements to the Graves and Pitarka (2010) broadband ground‐motion simulation method[J]. Seismol Res Lett,86(1):75–80. doi: 10.1785/0220140101
Han L,Wang J X,Li H J,Sun G J. 2020. A time-domain spectral element method with C1 continuity for static and dynamic analysis of frame structures[J]. Structures,28:604–613. doi: 10.1016/j.istruc.2020.08.074
He C H,Wang J T,Zhang C H,Jin F. 2015. Simulation of broadband seismic ground motions at dam canyons by using a deterministic numerical approach[J]. Soil Dyn Earthq Eng,76:136–144. doi: 10.1016/j.soildyn.2014.12.004
He C H,Wang J T,Zhang C H. 2016. Nonlinear spectral‐element method for 3D seismic‐wave propagation[J]. B Seismol Soc Am,106(3):1074–1087. doi: 10.1785/0120150341
Hermann V,Käser M,Castro C E. 2011. Non-conforming hybrid meshes for efficient 2-D wave propagation using the discontinuous Galerkin method[J]. Geophys J Int,184(2):746–758. doi: 10.1111/j.1365-246X.2010.04858.x
Ho L W,Patera A T. 1990. A Legendre spectral element method for simulation of unsteady incompressible viscous free-surface flows[J]. Comput Method Appl M,80:355–366. doi: 10.1016/0045-7825(90)90040-S
Huang D R,Wang G,Du C Y,Jin F,Feng K W,Chen Z W. 2020. An integrated SEM-Newmark model for physics-based regional coseismic landslide assessment[J]. Soil Dyn Earthq Eng,132:106066. doi: 10.1016/j.soildyn.2020.106066
Huang D R,Wang G,Du C Y,Jin F. 2021. Seismic amplification of soil ground with spatially varying shear wave velocity using 2D spectral element method[J]. J Earthq Eng,25(14):2834–2849. doi: 10.1080/13632469.2019.1654946
Infantino M,Mazzieri I,Özcebe A G,Paolucci R,Stupazzini M. 2020. 3D physics‐based numerical simulations of ground motion in Istanbul from earthquakes along the Marmara segment of the north Anatolian fault[J]. B Seismol Soc Am,110(5):2559–2576. doi: 10.1785/0120190235
Infantino M,Smerzini C,Lin J. 2021. Spatial correlation of broadband ground motions from physics‐based numerical simulations[J]. Earthq Eng Struct D,50(10):2575–2594. doi: 10.1002/eqe.3461
Karaoğlu H,Romanowicz B. 2018. Inferring global upper-mantle shear attenuation structure by waveform tomography using the spectral element method[J]. Geophys J Int,213(3):1536–1558. doi: 10.1093/gji/ggy030
Karniadakis G,Sherwin S J. 2005. Spectral/hp Element Methods for Computational Fluid Dynamics[M]. New York:Oxford University Press:1−652.
Käser M,Dumbser M,Puente J,Igel H. 2007. An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes—III. Viscoelastic attenuation[J]. Geophys J Int,168(1):224–242. doi: 10.1111/j.1365-246X.2006.03193.x
Käser M,Dumbser M. 2006. An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes—I. The two-dimensional isotropic case with external source terms[J]. Geophys J Int,166(2):855–877. doi: 10.1111/j.1365-246X.2006.03051.x
Käser M,Dumbser M. 2008. A highly accurate discontinuous Galerkin method for complex interfaces between solids and moving fluids[J]. Geophysics,73(3):T23–T35. doi: 10.1190/1.2870081
Käser M,Hermann V,Puente J. 2008. Quantitative accuracy analysis of the discontinuous Galerkin method for seismic wave propagation[J]. Geophys J Int,173(3):990–999. doi: 10.1111/j.1365-246X.2008.03781.x
Kato B,Wang G. 2021. Regional seismic responses of shallow basins incorporating site‐city interaction analyses on high‐rise building clusters[J]. Earthq Eng Struct D,50(1):214–236. doi: 10.1002/eqe.3363
Kato B,Wang G. 2022. Seismic site–city interaction analysis of super-tall buildings surrounding an underground station:A case study in Hong Kong[J]. B Earthq Eng,20(3):1431–1454. doi: 10.1007/s10518-021-01295-7
Komatitsch D,Barnes C,Tromp J. 2000a. Wave propagation near a fluid-solid interface:A spectral-element approach[J]. Geophysics,65(2):623–631. doi: 10.1190/1.1444758
Komatitsch D,Barnes C,Tromp J. 2000b. Simulation of anisotropic wave propagation based upon a spectral element method[J]. Geophysics,65(4):1251–1260. doi: 10.1190/1.1444816
Komatitsch D,Erlebacher G,Göddeke D,Michéa D. 2010. High-order finite-element seismic wave propagation modeling with MPI on a large GPU cluster[J]. J Comput Phys,229(20):7692–7714.
Komatitsch D,Liu Q,Tromp J,Süss P,Stidham C,Shaw J H. 2004. Simulations of ground motion in the Los Angeles basin based upon the spectral-element method[J]. B Seismol Soc Am,94(1):187–206.
Komatitsch D,Martin R,Tromp J,Taylor M A,Wingate B A. 2001. Wave propagation in 2-D elastic media using a spectral element method with triangles and quadrangles[J]. J Comput Acoust,9(2):703–718.
Komatitsch D,Martin R. 2007. An unsplit convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation[J]. Geophysics,72(5):SM155–SM167. doi: 10.1190/1.2757586
Komatitsch D,Ritsema J,Tromp J. 2002. The spectral-element method,Beowulf computing,and global seismology[J]. Science,298(5599):1737–1742. doi: 10.1126/science.1076024
Komatitsch D,Tromp J. 1999. Introduction to the spectral element method for three-dimensional seismic wave propagation[J]. Geophys J Int,139(3):806–822. doi: 10.1046/j.1365-246x.1999.00967.x
Komatitsch D,Tromp J. 2002a. Spectral-element simulations of global seismic wave propagation—I. Validation[J]. Geophys J Int,149(2):390–412. doi: 10.1046/j.1365-246X.2002.01653.x
Komatitsch D,Tromp J. 2002b. Spectral-element simulations of global seismic wave propagation—II. Three-dimensional models,oceans,rotation and self-gravitation[J]. Geophys J Int,150(1):303–318. doi: 10.1046/j.1365-246X.2002.01716.x
Komatitsch D,Tromp J. 2003. A perfectly matched layer absorbing boundary condition for the second-order seismic wave equation[J]. Geophys J Int,154(1):146–153. doi: 10.1046/j.1365-246X.2003.01950.x
Komatitsch D,Vilotte J P,Vai R,Castillo-Covarrubias J M,Sánchez-Sesma F J. 1999. The spectral element method for elastic wave equations — Application to 2‐D and 3‐D seismic problems[J]. Int J Numer Meth Eng,45(9):1139–1164. doi: 10.1002/(SICI)1097-0207(19990730)45:9<1139::AID-NME617>3.0.CO;2-T
Komatitsch D,Vilotte J P. 1998. The spectral element method:An efficient tool to simulate the seismic response of 2D and 3D geological structures[J]. B Seismol Soc Am,88(2):368–392. doi: 10.1785/BSSA0880020368
Komatitsch D,Vilotte J P. 1999. Reply to comment by E. Faccioli and A. Quarteroni on “The spectral element method:An efficient tool to simulate the seismic response of 2D and 3D geological structures,” by D. Komatitsch and J. -P. Vilotte[J]. B Seismol Soc Am, 89 (1):332−334.
Krishnan S,Ji C,Komatitsch D,Tromp J. 2006. Performance of two 18-story steel moment-frame buildings in southern California during two large simulated San Andreas earthquakes[J]. Earthq Spectra,22(4):1035–1061. doi: 10.1193/1.2360698
Laurenzano G,Priolo E,Tondi E. 2008. 2D numerical simulations of earthquake ground motion:examples from the Marche Region,Italy[J]. J Seismol,12:395–412. doi: 10.1007/s10950-008-9095-1
Laurenzano G,Priolo E. 2005. Numerical modeling of the 13 December 1990 M 5.8 east Sicily earthquake at the Catania accelerometric station[J]. B Seismol Soc Am,95(1):241–251. doi: 10.1785/0120030126
Lee S J,Chen H W,Liu Q,Komatitsch D,Huang B S,Tromp J. 2008. Three-dimensional simulations of seismic-wave propagation in the Taipei basin with realistic topography based upon the spectral-element method[J]. B Seismol Soc Am,98(1):253–264. doi: 10.1785/0120070033
Lee S J,Komatitsch D,Huang B S,Tromp J. 2009. Effects of topography on seismic-wave propagation:An example from northern Taiwan[J]. B Seismol Soc Am,99(1):314–325. doi: 10.1785/0120080020
Lee U. 2009. Spectral Element Method in Structural Dynamics[M]. Singapore:John Wiley & Sons (Asia) Pte Ltd:1−448.
Liang J W,Wu M T,Ba Z N,Liu Y. 2021. A hybrid method for modeling broadband seismic wave propagation in 3D localized regions to incident P,SV,and SH waves[J]. Int J Appl Mech,13(10):2150119. doi: 10.1142/S1758825121501192
Liu Q F,Yu Y Y,Yin D Y,Zhang X B. 2018. Simulations of strong motion in the Weihe basin during the Wenchuan earthquake by spectral element method[J]. Geophys J Int,215(2):978–995. doi: 10.1093/gji/ggy320
Liu Q F,Yu Y Y,Zhang X B. 2015. Three-dimensional simulations of strong ground motion in the Shidian basin based upon the spectral-element method[J]. Earthq Eng Eng Vib,14(3):385–398. doi: 10.1007/s11803-015-0031-4
Liu Q Y,Gu Y J. 2012. Seismic imaging:From classical to adjoint tomography[J]. Tectonophysics,566:31–66.
Liu Q Y,Polet J,Komatitsch D,Tromp J. 2004. Spectral-element moment tensor inversions for earthquakes in southern California[J]. B Seismol Soc Am,94(5):1748–1761. doi: 10.1785/012004038
Liu Q Y,Tromp J. 2006. Finite-frequency kernels based on adjoint methods[J]. B Seismol Soc Am,96(6):2383–2397. doi: 10.1785/0120060041
Liu Q Y,Tromp J. 2008. Finite-frequency sensitivity kernels for global seismic wave propagation based upon adjoint methods[J]. Geophys J Int,174(1):265–286. doi: 10.1111/j.1365-246X.2008.03798.x
Liu S L,Yang D H,Dong X P,Liu Q C,Zheng Y C. 2017. Element-by-element parallel spectral-element methods for 3-D teleseismic wave modeling[J]. Solid Earth,8(5):969–986. doi: 10.5194/se-8-969-2017
Liu T,Sen M K,Hu T Y,De Basabe J D,Li L. 2012. Dispersion analysis of the spectral element method using a triangular mesh[J]. Wave Motion,49:474–483. doi: 10.1016/j.wavemoti.2012.01.003
Liu Y S,Teng J W,Lan H Q,Si X,Ma X Y. 2014. A comparative study of finite element and spectral element methods in seismic wavefield modeling[J]. Geophysics,79(2):T91–T104. doi: 10.1190/geo2013-0018.1
Liu Y S,Teng J W,Xu T,Badal J. 2017. Higher-order triangular spectral element method with optimized cubature points for seismic wavefield modeling[J]. J Comput Phys,336:458–480. doi: 10.1016/j.jcp.2017.01.069
Lloyd A J,Wiens D A,Zhu H,Tromop J,Nyblade A A,Aster R C,Hansen S E,Dalziel I W D,Wilson T J,Ivins E R,O’Donnell J P. 2020. Seismic structure of the Antarctic upper mantle imaged with adjoint tomography[J]. J Geophys Res-Sol Ea, 125 (3).
Lu X Z,Tian Y,Wang G,Huang D R. 2018. A numerical coupling scheme for nonlinear time history analysis of buildings on a regional scale considering site‐city interaction effects[J]. Earthq Eng Struct D,47(13):2708–2725. doi: 10.1002/eqe.3108
Ma H. 1993. A spectral element basin model for the shallow water equations[J]. J Comput Phys,109:133–149. doi: 10.1006/jcph.1993.1205
Magnoni F,Casarotti E,Komatitsch D,Stefano R D,Ciaccio M G,Tape C,Melini D,Michelini A,Piersanti A,Tromp J. 2022. Adjoint tomography of the Italian lithosphere[J]. Commun Earth Environ,3(1):69. doi: 10.1038/s43247-022-00397-7
Marfurt K J. 1984. Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equations[J]. Geophysics,49(5):533–549. doi: 10.1190/1.1441689
Martin R,Komatitsch D,Gedney S D,Bruthiaux E. 2010. A high-order time and space formulation of the unsplit perfectly matched layer for the seismic wave equation using Auxiliary Differential Equations (ADE-PML)[J]. CMES-Comp Model Eng,56(1):17–40.
Martin R,Komatitsch D,Gedney S D. 2008. A variational formulation of a stabilized unsplit convolutional perfectly matched layer for the isotropic or anisotropic seismic wave equation[J]. CMES-Comp Model Eng,37(3):274–304.
Mazzieri I,Stupazzini M,Guidotti R,Smerzini C. 2013. SPEED:SPectral Elements in Elastodynamics with Discontinuous Galerkin:A non‐conforming approach for 3D multi‐scale problems[J]. Int J Numer Meth Eng,95(12):991–1010. doi: 10.1002/nme.4532
Michéa D,Komatitsch D. 2010. Accelerating a three-dimensional finite-difference wave propagation code using GPU graphics cards[J]. Geophys J Int,182(1):389–402.
Moczo P,Kristek J,Halada L. 2000. 3D fourth-order staggered-grid finite-difference schemes:Stability and grid dispersion[J]. B Seismol Soc Am,90(3):587–603. doi: 10.1785/0119990119
Morency C,Tromp J. 2008. Spectral-element simulations of wave propagation in porous media[J]. Geophys J Int,175(1):301–345. doi: 10.1111/j.1365-246X.2008.03907.x
Mulder W A,Zhebel E,Minisini S. 2014. Time-stepping stability of continuous and discontinuous finite-element methods for 3-D wave propagation[J]. Geophys J Int,196(2):1123–1133. doi: 10.1093/gji/ggt446
Mulder W A. 1999. Spurious modes in finite-element discretizations of the wave equation may not be all that bad[J]. Appl Numer Math,30(4):425–445. doi: 10.1016/S0168-9274(98)00078-6
Mulder W A. 2001. Higher-order mass-lumped finite elements for the wave equation[J]. J Comput Acoust,9(2):671–680. doi: 10.1142/S0218396X0100067X
Mulder W A. 2013. New triangular mass-lumped finite elements of degree six for wave propagation[J]. Prog Electromagn Res,141:671–692. doi: 10.2528/PIER13051308
Mullen R,Belytschko T. 1982. Dispersion analysis of finite element semidiscretizations of the two‐dimensional wave equation[J]. Int J Numer Meth Eng,18(1):11–29. doi: 10.1002/nme.1620180103
Oliveira S P,Seriani G. 2011. Effect of element distortion on the numerical dispersion of spectral element methods[J]. Commun Comput Phys,9(4):937–958. doi: 10.4208/cicp.071109.080710a
Ostachowicz W,Kudela P,Krawczuk M,Zak A. 2012. Guided Waves in Structures for SHM:The Time-Domain Spectral Element Method[M]. London:John Wiley & Sons:1−334.
Padovani E,Priolo E,Seriani G. 1994. Low and high order finite element method:experience in seismic modeling[J]. J Comput Acoust,2(4):371–422. doi: 10.1142/S0218396X94000233
Paolucci R,Evangelista L,Mazzieri I,Schiappapietra E. 2016. The 3D numerical simulation of near-source ground motion during the Marsica earthquake,central Italy,100 years later[J]. Soil Dyn Earthq Eng,91:39–52. doi: 10.1016/j.soildyn.2016.09.023
Paolucci R,Gatti F,Infantino M,Smerzini C,Özcebe A G,Stupazzini M. 2018. Broadband ground motions from 3D physics‐based numerical simulations using artificial neural networks[J]. B Seismol Soc Am,108(3A):1272–1286. doi: 10.1785/0120170293
Paolucci R,Mazzieri I,Smerzini C. 2015. Anatomy of strong ground motion:near-source records and three-dimensional physics-based numerical simulations of the MW6.0 2012 May 29 Po Plain earthquake,Italy[J]. Geophys J Int,203(3):2001–2020. doi: 10.1093/gji/ggv405
Paolucci R,Smerzini C,Vanini M. 2021. BB‐SPEEDset:A validated dataset of broadband near‐source earthquake ground motions from 3D physics‐based numerical simulations[J]. B Seismol Soc Am,111(5):2527–2545. doi: 10.1785/0120210089
Pasquetti R,Rapetti F. 2004. Spectral element methods on triangles and quadrilaterals:comparisons and applications[J]. J Comput Phys,198(1):349–362. doi: 10.1016/j.jcp.2004.01.010
Patera A T. 1984. A spectral element method for fluid dynamics:Laminar flow in a channel expansion[J]. J Comput Phys,54:468–488. doi: 10.1016/0021-9991(84)90128-1
Pelties C,Käser M,Hermann V,Castro C E. 2010. Regular versus irregular meshing for complicated models and their effect on synthetic seismograms[J]. Geophys J Int,183(2):1031–1051. doi: 10.1111/j.1365-246X.2010.04777.x
Pelties C,Puente J,Ampuero J P,Brietzke G B,Käser M. 2012. Three‐dimensional dynamic rupture simulation with a high‐order discontinuous Galerkin method on unstructured tetrahedral meshes[J]. J Geophys Res-Sol Ea,117:B02309.
Pilz M,Parolai S,Stupazzini M,Paolucci R,Zschau J. 2011. Modelling basin effects on earthquake ground motion in the Santiago de Chile basin by a spectral element code[J]. Geophys J Int,187(2):929–945. doi: 10.1111/j.1365-246X.2011.05183.x
Pozrikidis C. 2014. Introduction to Finite and Spectral Element Methods Using MATLAB[M]. 2nd ed. USA:University of Massachusetts,CRC Press:1−793.
Priolo E,Carcione J M,Seriani G. 1994. Numerical simulation of interface waves by high‐order spectral modeling techniques[J]. J Acoust Soc Am,95(2):681–693. doi: 10.1121/1.408428
Priolo E,Seriani G. 1991. A numerical investigation of Chebyshev SEM for acoustic wave propagation[C]//Proceedings of 13th World Congress on Computation and Applied Mathematics,Ireland:Trinity College Dublin:551−556.
Priolo E. 1999. 2-D spectral element simulations of destructive ground shaking in Catania (Italy)[J]. J Seismol,3:289–309. doi: 10.1023/A:1009838325266
Priolo E. 2001. Earthquake ground motion simulation through the 2-D spectral element method[J]. J Comput Acoust,9(4):1561–1581. doi: 10.1142/S0218396X01001522
Puente J,Ampuero J P,Käser M. 2009. Dynamic rupture modeling on unstructured meshes using a discontinuous Galerkin method[J]. J Geophys Res-Sol Ea,114:B10302.
Puente J,Dumbser M,Käser M,Igel H. 2008. Discontinuous Galerkin methods for wave propagation in poroelastic media[J]. Geophysics,73(5):T77–T97. doi: 10.1190/1.2965027
Puente J,Käser M,Dumbser M,Igel H. 2007. An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes-IV. Anisotropy[J]. Geophys J Int,169(3):1210–1228. doi: 10.1111/j.1365-246X.2007.03381.x
Rønquist E M,Patera A T. 1987. A Legendre spectral element method for the Stefan problem[J]. Int J Numer Meth Eng,24(12):2273–2299.
Sawade L,Beller S,Lei W,Tromp J. 2022. Global centroid moment tensor solutions in a heterogeneous earth:the CMT3D catalogue[J]. Geophys J Int,231(3):1727–1738. doi: 10.1093/gji/ggac280
Schuberth B. 2003. The Spectral Element Method for Seismic Wave Propagation:Theory,Implementation and Comparison to Finite Difference Methods [D]. München:Ludwig Maximilians Universität:1−163.
Seriani G,Oliveira S P. 2007. Optimal blended spectral-element operators for acoustic wave modeling[J]. Geophysics,72(5):SM95–SM106. doi: 10.1190/1.2750715
Seriani G,Oliveira S P. 2008a. Dispersion analysis of spectral element methods for elastic wave propagation[J]. Wave Motion,45(6):729–744. doi: 10.1016/j.wavemoti.2007.11.007
Seriani G,Oliveira S P. 2008b. DFT modal analysis of spectral element methods for acoustic wave propagation[J]. J Comput Acoust,16(4):531–561. doi: 10.1142/S0218396X08003774
Seriani G,Priolo E,Carcione J,Padovani E,Geofisica O. 1992. High-order spectral element method for elastic wave modeling[C]// Seg Technical Program Expanded Abstracts 1992. Society of Exploration Geophysicists:1285−1288.
Seriani G,Priolo E. 1991. High-order spectral element method for acoustic wave modeling[C]//Expanded Abstracts of the Society of Exploration Geophysicists,61st International Meeting of the SEG,Houston,Texas:1561−1564.
Seriani G,Priolo E. 1994. Spectral element method for acoustic wave simulation in heterogeneous media[J]. Finite Elem Anal Des,16:337–348. doi: 10.1016/0168-874X(94)90076-0
Seriani G,Su C. 2012. Wave propagation modeling in highly heterogeneous media by a poly-grid Chebyshev spectral element method[J]. J Comput Acoust,20(2):1240004. doi: 10.1142/S0218396X12400048
Seriani G. 1997. A parallel spectral element method for acoustic wave modeling[J]. J Comput Acoust,5(1):53–69. doi: 10.1142/S0218396X97000058
Seriani G. 1998. 3-D large-scale wave propagation modeling by spectral element method on Cray T3E multiprocessor[J]. Comput Method Appl M,164:235–247. doi: 10.1016/S0045-7825(98)00057-7
Seriani G. 2004. Double-grid Chebyshev spectral elements for acoustic wave modeling[J]. Wave Motion,39(4):351–360. doi: 10.1016/j.wavemoti.2003.12.008
Sherwin S J,Karniadakis G E. 1995. A triangular spectral element method; applications to the incompressible Navier-Stokes equations[J]. Comput Method Appl M, 123 (1−4):189−229.
Smerzini C,Amendola C,Paolucci R,Bazrafshan A. 2024. Engineering validation of BB-SPEEDset,a data set of near-source physics-based simulated accelerograms[J]. Earthq Spectra,40(1):420–445. doi: 10.1177/87552930231206766
Smerzini C,Paolucci R,Stupazzini M. 2011. Comparison of 3D,2D and 1D numerical approaches to predict long period earthquake ground motion in the Gubbio plain,Central Italy[J]. B Earthq Eng,9:2007–2029. doi: 10.1007/s10518-011-9289-8
Smerzini C,Pitilakis K,Hashemi K. 2017. Evaluation of earthquake ground motion and site effects in the Thessaloniki urban area by 3D finite-fault numerical simulations[J]. B Earthq Eng,15:787–812. doi: 10.1007/s10518-016-9977-5
Smerzini C,Pitilakis K. 2018. Seismic risk assessment at urban scale from 3D physics-based numerical modeling:the case of Thessaloniki[J]. B Earthq Eng,16:2609–2631. doi: 10.1007/s10518-017-0287-3
Soto V,Sáez E,Magna-Verdugo C. 2020. Numerical modeling of 3D site-city effects including partially embedded buildings using spectral element methods. Application to the case of Viña del Mar city,Chile[J]. Eng Struct,223:111188. doi: 10.1016/j.engstruct.2020.111188
Stich D,Martín R,Morales J. 2010. Moment tensor inversion for Iberia–Maghreb earthquakes 2005–2008[J]. Tectonophysics,483:390–398. doi: 10.1016/j.tecto.2009.11.006
Stupazzini M,Infantino M,Allmann A,Paolucci R. 2020. Physics‐based probabilistic seismic hazard and loss assessment in large urban areas:A simplified application to Istanbul[J]. Earthq Eng Struct D,50(1):99–115.
Stupazzini M,Paolucci R,Igel H. 2009. Near-fault earthquake ground-motion simulation in the Grenoble valley by a high-performance spectral element code[J]. B Seismol Soc Am,99(1):286–301. doi: 10.1785/0120080274
Su C,Seriani G. 2023. Poly-grid spectral element modeling for wave propagation in complex elastic media[J]. J Theor Comput Acous,31(1):2350003. doi: 10.1142/S2591728523500032
Sun P G,Huang D R. 2023. Regional-scale assessment of earthquake-induced slope displacement considering uncertainties in subsurface soils and hydrogeological condition[J]. Soil Dyn Earthq Eng,164:107593. doi: 10.1016/j.soildyn.2022.107593
Tape C,Liu Q,Maggi A,Tromp J. 2009. Adjoint tomography of the southern California crust[J]. Science,325(5943):988–992. doi: 10.1126/science.1175298
Tape C,Liu Q,Maggi A,Tromp J. 2010. Seismic tomography of the southern California crust based on spectral-element and adjoint methods[J]. Geophys J Int,180(1):433–462. doi: 10.1111/j.1365-246X.2009.04429.x
Tape C,Liu Q,Tromp J. 2007. Finite‐frequency tomography using adjoint methods — Methodology and examples using membrane surface waves[J]. Geophys J Int,168(3):1105–1129. doi: 10.1111/j.1365-246X.2006.03191.x
Taylor M A,Wingate B A,Vincent R E. 2000. An algorithm for computing Fekete points in the triangle[J]. SIAM J Numer Anal,38(5):1707–1720. doi: 10.1137/S0036142998337247
Tian Y,Chen S Y,Liu S M,Lu X Z. 2023. Influence of tall buildings on city-scale seismic response analysis:A case study of Shanghai CBD[J]. Soil Dyn Earthq Eng,173:108063. doi: 10.1016/j.soildyn.2023.108063
Tian Y,Lu X Z,Huang D R,Wang T. 2022. SCI effects under complex terrains:Shaking table tests and numerical simulation[J]. J Earthq Eng,27(5):1237–1260.
Tian Y,Sun C J,Lu X Z,Huang Y L. 2020. Quantitative analysis of site-city interaction effects on regional seismic damage of buildings[J]. J Earthq Eng,26(8):4365–4385.
Trefethen L N. 2000. Spectral Methods in MATLAB[M]. Philadelphia:Society for Industrial and Applied Mathematics:1−160.
Tromp J,Komatitsch D,Liu Q. 2008. Spectral-element and adjoint methods in seismology[J]. Commun Comput Phys,3(1):1–32.
Tromp J,Tape C,Liu Q. 2005. Seismic tomography,adjoint methods,time reversal and banana-doughnut kernels[J]. Geophys J Int,160(1):195–216.
Virieux J. 1986. P-SV wave propagation in heterogeneous media:Velocity-stress finite-difference method[J]. Geophysics,51(4):889–901. doi: 10.1190/1.1442147
Vosse van de F N,Minev P D. 1996. Spectral Element Methods :Theory and Applications[R]. EUT report. Netherlands:Eindhoven University of Technology:1−117.
Wang G,Du C Y,Huang D R,Jin F,Koo R C H,Kwan J S H. 2018. Parametric models for 3D topographic amplification of ground motions considering subsurface soils[J]. Soil Dyn Earthq Eng,115:41–54. doi: 10.1016/j.soildyn.2018.07.018
Wang J X,Li H J,Sun G J,Han L. 2022a. Free vibration analysis of rectangular thin plates with corner and inner cutouts using C1 Chebyshev spectral element method[J]. Thin Wall Struct,181:110031. doi: 10.1016/j.tws.2022.110031
Wang J X,Li H J,Xing H J. 2022b. A lumped mass Chebyshev spectral element method and its application to structural dynamic problems[J]. Earthq Eng Eng Vib,21(3):843–859. doi: 10.1007/s11803-022-2117-0
Wang X C,Wang J T,Zhang C H. 2022. A broadband kinematic source inversion method considering realistic Earth models and its application to the 1992 Landers earthquake[J]. J Geophys Res-Sol Ea,127(3):e2021JB023216. doi: 10.1029/2021JB023216
Wang X C,Wang J T,Zhang C H. 2023. Deterministic full-scenario analysis for maximum credible earthquake hazards[J]. Nat Commun,14(1):6600. doi: 10.1038/s41467-023-42410-3
Wang X C,Wang J T,Zhang L,He C H. 2021a. Broadband ground-motion simulations by coupling regional velocity structures with the geophysical information of specific sites[J]. Soil Dyn Earthq Eng,145:106695. doi: 10.1016/j.soildyn.2021.106695
Wang X C,Wang J T,Zhang L,Li S,Zhang C H. 2021b. A multidimension source model for generating broadband ground motions with deterministic 3D numerical simulations[J]. B Seismol Soc Am,111(2):989–1013. doi: 10.1785/0120200221
Wang X C,Wang J T. 2023. A physics‐based spectral matching (PBSM) method for generating fully site‐related ground motions[J]. Earthq Eng Struct D,52(9):2812–2829. doi: 10.1002/eqe.3897
Wu M T,Ba Z N,Liang J W. 2022. A procedure for 3D simulation of seismic wave propagation considering source‐path‐site effects:Theory,verification and application[J]. Earthq Eng Struct D,51(12):2925–2955. doi: 10.1002/eqe.3708
Xie Z N,Komatitsch D,Martin R,Matzen R. 2014. Improved forward wave propagation and adjoint-based sensitivity kernel calculations using a numerically stable finite-element PML[J]. Geophys J Int,198(3):1714–1747. doi: 10.1093/gji/ggu219
Xie Z N,Matzen R,Cristini P,Komatitsch D,Marin R. 2016. A perfectly matched layer for fluid-solid problems:Application to ocean-acoustics simulations with solid ocean bottoms[J]. J Acoust Soc Am,140(1):165–175. doi: 10.1121/1.4954736
Xie Z N,Zheng Y L,Cristini P,Zhang X B. 2023. Multi-axial unsplit frequency-shifted perfectly matched layer for displacement-based anisotropic wave simulation in infinite domain[J]. Earthq Eng Eng Vib,22(2):407–421. doi: 10.1007/s11803-023-2170-3
Xing H J,Li X J,Li H J,Liu A W. 2021a. Spectral-element formulation of multi-transmitting formula and its accuracy and stability in 1D and 2D seismic wave modeling[J]. Soil Dyn Earthq Eng,140:1–15.
Xing H J,Li X J,Li H J,Xie Z N,Chen S L,Zhou Z H. 2021b. The theory and new unified formulas of displacement-type local absorbing boundary conditions[J]. Bull Seismol Soc Am,111(2):801–824. doi: 10.1785/0120200155
Yu Y Y,Ding H P,Liu Q F. 2017. Three-dimensional simulations of strong ground motion in the Sichuan basin during the Wenchuan earthquake[J]. B Earthq Eng,15:4661–4679. doi: 10.1007/s10518-017-0154-2
Yu Y Y,Ding H P,Zhang X B. 2021. Simulations of ground motions under plane wave incidence in 2D complex site based on the spectral element method (SEM) and multi-transmitting formula (MTF):SH problem[J]. J Seismol,25:967–985. doi: 10.1007/s10950-021-09995-y
Yu Y Y,Ding H P,Zhang X B. 2024. Formulation and performance of multi-transmitting formula with spectral element method in 2D ground motion simulations under plane-wave incidence:SV wave problem[J]. J Earthq Eng,28(7):1837–1860. doi: 10.1080/13632469.2023.2268748
Zhang L,Wang J T,Xu Y J,He C H,Zhang C H. 2020. A procedure for 3D seismic simulation from rupture to structures by coupling SEM and FEM[J]. B Seismol Soc Am,110(3):1134–1148. doi: 10.1785/0120190289
Zhang M Z,Zhang L,Wang X C,Su W,Qiu Y X,Wang J T,Zhang C H. 2023. A framework for seismic response analysis of dams using numerical source‐to‐structure simulation[J]. Earthq Eng Struct D,52(3):593–608. doi: 10.1002/eqe.3774
Zhou H,Chen X F. 2010. A new technique to synthesize seismography with more flexibility:the Legendre spectral element method with overlapped elements[J]. Pure Appl Geophys,167:1365–1376. doi: 10.1007/s00024-010-0106-0
Zhou H,Jiang H. 2015. A new time-marching scheme that suppresses spurious oscillations in the dynamic rupture problem of the spectral element method:the weighted velocity Newmark scheme[J]. Geophys J Int,203(2):927–942. doi: 10.1093/gji/ggv341
Zhou H,Li J T,Chen X F. 2020. Establishment of a seismic topographic effect prediction model in the Lushan MS7.0 earthquake area[J]. Geophys J Int,221(1):273–288. doi: 10.1093/gji/ggaa003
Zhu C Y,Qin G L,Zhang J Z. 2011. Implicit Chebyshev spectral element method for acoustics wave equations[J]. Finite Elem Anal Des,47(2):184–194. doi: 10.1016/j.finel.2010.09.004
Zhu H J,Bozdağ E,Tromp J. 2015. Seismic structure of the European upper mantle based on adjoint tomography[J]. Geophys J Int,201(1):18–52. doi: 10.1093/gji/ggu492
Zhu H J,Komatitsch D,Tromp J. 2017. Radial anisotropy of the North American upper mantle based on adjoint tomography with USArray[J]. Geophys J Int,211(1):349–377. doi: 10.1093/gji/ggx305
Zienkiewicz O C,Taylor R L,Zhu J Z. 2013. The Finite Element Method:Its Basis and Fundamentals[M]. 7th ed. UK:Butterworth-Heinemann,Elsevier:257−460.
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