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 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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