中文摘要：

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。

英文摘要：

Based on strong and weak forms of elastic wave equations, a Chebyshev spectral element method （SEM） using the Galerkin variational principle is developed by discretizing the wave equation in the spatial and time domains and introducing the preconditioned conjugate gradient （PCG）-element by element （EBE） method in the spatial domain and the staggered predictor/corrector method in the time domain. The accuracy of our proposed method is verified by comparing it with a finite-difference method （FDM） for a homogeneous solid medium and a double layered solid medium with an inclined interface. The modeling results using the two methods are in good agreement with each other. Meanwhile, to show the algorithm capability, the suggested method is used to simulate the wave propagation in a layered medium with a topographic traction free surface. By introducing the EBE algorithm with an optimized tensor product technique, the proposed SEM is especially suitable for numerical simulation of wave propagations in complex models with irregularly free surfaces at a fast convergence rate, while keeping the advantage of the finite element method.