山区场地地震动数值模拟方法研究

Both the geological structures and the general ground configurations on the mountain region are extremely complicated, so huge computational efforts have been required when the characteristics of earthquake induced ground motion during a mountian site are reseached by using the technique of numerical simulation for wave motion. It is impossible to avoid the fact that the analysis procedure with high efficiency have to be developed in the same requirements of numerical stability and accuracy during analysis of ground motion on the complicated site such as the mountain area. Based on the research achievements of spectral element simulation technique for wave motion on near fields, a method which may be used for complicated sites and probably with high efficiency will be developed in this project. The objective of this research is to enhance the ability that the complicated problems with large scale are solved when the numerical simulation technique for the wave motion is utilized. First, the strategy of the spectral element simulation for the area irregularly in geometric features and the medium nonhomogeneously and noncontinuously in physical features is researched, and the influence of selection of base functions for the spectral decomposition is analyzed, then the spectral element used in mountain sites is constructed and the equations of motion of the system are established. Secondly, a numerical time- stepping procedure is developed for the solution of the equations governing the motion of a MDOF system subjected to earthquake induced ground acceleration on the basis of the differential quadrture (DQ) rule. The DQ analysis procedure is employed to analyze the seismic response of the discrete site system, and the distribution characteristics of the strong ground motion in the local site are studied simultaneously. Finally, a real mountain site receiving the records from a strong motion instrument array is selected for the earthquake ground motion analysis so that the spectral element procedure developed in this project is examined.

山区场地的地质和地形条件均十分复杂，采用波动数值模拟方法研究山区场地地震动特征需要完成大规模的计算任务，在保证数值精度和稳定性的基础上提高计算效率是发展山区等复杂场地地震动分析方法时无法回避的现实。本项目拟在申请人前期关于近场波动谱元模拟方法研究的基础上，发展一种能够适应山区场地复杂特征的地震动数值模拟高效率计算方法，旨在提高波动数值模拟技术解决大型复杂实际问题的能力。首先，针对山区场地几何不规则和介质不均匀、不连续的特征，研究谱元离散的合理方式和策略。详细分析在有限单元上采用不同基函数进行谱分解对单元特性的影响，构造能够合理模拟山区场地的谱单元模型，建立场地运动微分方程。然后，改进申请人前期提出的地震反应逐步微分积分分析方法，采用该方法完成对谱元方程的求解，并研究局部场地条件对地震动的影响。最后，通过对一个获得强震观测记录的山区场地地震动的分析检验本项目方法的正确性和解决实际问题的效果。

多次透射公式（MTF）是一种“普适性”的人工边界条件，表现为时空解耦的显式一维形式，具有精度可控、施加方便的优点，能够在保证数值精度的条件下稳定地实现对近场波动的数值模拟。但现有的MTF数值格式是在有限元模型的基础上建立起来的，在谱元模型中不能适用。本项目的目标就是探索MTF与谱元模型结合的方法及其有效的数值格式，高效地实现近场波动问题的谱元模拟。首先，对MTF的数学本质进行了研究，发现MTF可描述为人工边界节点位移的N阶误差（N为透射阶次）为零的结果，它在本质上可认为是一种时-空域内沿一条斜向直线方向的N阶单边差分公式，其代数精度为N-1阶，从而也就揭示出高阶MTF能够提高模拟精度的原因。涉及到的MTF计算点（时-空差分节点）以及各个节点前的系数都可以通过二项式原理确定，这为构造MTF数值格式提供了依据。这些发现与一直以来认为MTF为“多次透射”物理机制的传统认识有所不同。其次，将基于三点抛物线内插公式和齐次内插递推方案的传统MTF有限元格式拓展至M阶多项式内插公式，研究了各插值阶次相应的MTF有限元格式的精度和稳定性问题，给出了它们的时域稳定性条件。关于MTF失稳临界值的结论深化了对MTF机制和特性的认识，对于统一人工波速、透射阶次等MTF重要参数的合理选取具有指导意义。再次，将上述研究成果推广到谱元模型，基于谱单元不等距节点分布特征建立了一组基于M次多项式插值的MTF谱元格式，并对该格式下边界反射误差开展时域分析。大量数值试验证实，这套谱元模型结合MTF边界的数值模拟方法十分有效且对复杂波动问题具有较强的适用性。最后，对具有一致质量矩阵的谱元方程（切比雪夫谱元模型）的时域积分方法进行了研究，指出采用中心差分方法的合理性和高效性，在谱元模型中依然保持了MTF能够实现时空解耦的优点。而传统的有限元模型结合MTF的波动模拟方法是基于集中质量矩阵才实现显式计算的。

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