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本文基于弹性波动方程,从其弱形式出发,利用Galerkin变分原理,通过对方程进行空间和时间上的离散,在空间域中引入预条件共轭梯度的逐元算法,在时间域中引入时间积分的交错网格预处理/多次校正算法,发展了弹性波模拟的Chebyshev谱元算法。针对均匀固体介质和具有倾斜分层的分区均匀固体介质模型,通过与有限差分算法结果相比较验证其精度的可信性,同时利用该算法模拟了弹性波在具有水平分层的任意起伏自由表面模型中的传播,并分析了其传播特点。研究表明,我们提出的交错网格预处理/多次校正算法的Chebyshev谱元算法,保留了有限元法的优势,并且采用了具有最优张量乘积技术的元到元的算法,能够处理带有起伏自由表面的复杂介质模型,它具有比有限元法收敛快,计算效率较高等优点,特别适合于复杂结构和复杂介质中的弹性波传播的数值模拟。
Based on the elastic wave equation, based on the weak form, this paper uses the Galerkin variational principle, through the spatial and temporal discretization of the equation, introduces the preconditioned conjugate gradient algorithm in the space domain, introduces the time in the time domain Integral staggered grid pretreatment / multiple correction algorithm, the development of elastic wave simulation Chebyshev spectral element algorithm. Aiming at homogeneous solid medium and partitioned homogeneous solid medium model with inclined stratification, the accuracy of the model is verified by comparison with the results of finite difference method. At the same time, this algorithm is used to simulate the elastic wave propagation on any undulating free surface with horizontal stratification Propagation in the model and its spread characteristics. The results show that the proposed Chebyshev spectral element algorithm of staggered grid preprocessing / multiple correction algorithm retains the advantages of the finite element method and adopts the element-to-element algorithm with the optimal tensor product technique, which can process the band The complex medium model with undulating free surface has the advantages of faster convergence and higher computational efficiency than the finite element method and is especially suitable for the numerical simulation of elastic wave propagation in complex structures and complex media.