基于快速多极子间接边界元法的复杂场地三维地震动模拟及其应用研究

There is a great need to synthetically consider the site conditions, the propagation path and the source characteristics for accurate determination of earthquake ground motion. In order to improve the precision and effective frequency of earthquake ground motion simulation under complexity site conditions, existing calculation models and simulation methods are required to be further improved and developed. Firstly, considering the bedding characteristics of the natural stratum structure, a new fast multipole indirect boundary element method is established based on dynamic Green's function of the slant disc surface load in layered half-space. This method has the advantage of dimensionality reduction, automatic satisfaction of the radiation conditions and avoiding the high-frequency dispersion. Additionally, by implying a multipole expansion and local expansion algorithms, there is a significant reduction of the storage and computation requirement for solving large-scale problem of wave motion comparing with the conventional boundary element method. Based on the FMM-IBEM, three-dimensional earthquake ground motion of Urumqi foreland basin is simulated as an application case. Through detailed parametric analysis, this project strives to reveal the influence mechanism of the sedimentary basins and the nearby mountain on the ground motion and the dynamic interaction between them in the region, and to explore the effects of the source characteristics and inhomogeneous stratum structure on the ground motion. Finally, a reasonable forecast is obtained for the distribution characteristics of the earthquake ground motion in space. This Research will provide a new method for solving the large-scale elastic wave motion problems in complex media, and offer a theoretical basis for the seismic design of large engineering structures, seismic zoning, urban planning and other related works in Urumqi and other basin-mountain coupling zones.

地震动场的精细确定需综合考虑场地条件、传播途径和震源特性。为提高复杂场地条件下地震动模拟的定量化水平和有效频率，尚须进一步发展和完善现有计算模型和模拟方法。项目首先针对天然地层结构的层理特征，基于层状半空间斜面圆盘荷载动力格林函数，建立一种新的快速多极子间接边界元法。该方法具有降维、自动满足辐射条件和无高频数值弥散的优点，且结合多极展开和局部展开技术，可显著降低传统边界元法求解大规模波动问题所需的存储量和计算量。进而以乌鲁木齐山前盆地为例，进行三维地震动快速多极边界元模拟。通过详细的参数分析，揭示该区域沉积盆地、山体对地震动的不同影响机理及其动力相互作用，探究震源特性、不均匀地层对地震动的影响规律，合理预测该区域未来地震动场空间分布特征。研究成果可为复杂介质中大规模弹性波动问题求解提供新的方法思路，为乌鲁木齐及类似盆山耦合区域的大型工程结构抗震设计、地震区划、城市规划等工作提供理论依据。

局部场地效应本质是不均质体（地形））对地震波的散射问题。地震波动模拟数值方法主要包括有限元法、有限差分法、边界元方法等。其中边界元具有降低问题求解维数、自动满足无限远辐射条件和无高频数值弥散的优点，因此特别适合复杂三维地震波动问题模拟。但传统边界元方法直接存储格林函数矩阵并采用Gauss消元法求解矩阵方程，对于大尺度、高频或者多散射体等复杂问题，存储量和运算量巨大。为提高复杂场地条件下地震动模拟的定量化水平和有效频率，项目首先基于基于间接边界方法和快速多极展开技术，建立一种新的快速多极子间接边界元法。通过多极展开和局部展开技术，显著降低了传统边界元法求解大规模波动问题所需的存储量和计算量。方法要点：对于低频多散射体问题和高频大尺度散射问题，分别通过对斜面圆盘荷载动力格林的函数泰勒级数展开和平面波展开技术进行边界方程的加速求解。对于方程的GMRES求解过程，通过采用对角矩阵进行预处理技术大幅度提高了收敛速度。在算法优化和稳定性检验基础上，基于FM-IBEM，分别研究了数千空隙群和夹杂体对地震波的散射，模拟了三维凹陷地形、均匀沉积盆地及复杂层状沉积盆地地震响应宽频精细化模拟。通过详细的参数分析，揭示了三维沉积盆地、山体地形对地震动的不同影响机理及其动力相互作用，考察了震源特性、不均匀地层对地震动的影响规律。总体上，研究成果为复杂介质中大规模弹性波动问题求解提供了新的方法思路，为山体、盆地区域的大型工程结构抗震设计、地震区划、城市规划等工作提供了部分科学依据。

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