SUMMARY

The discontinuous Galerkin finite element method (DGM) is a promising algorithm for modelling wave propagation in fractured media. It allows for discontinuities in the displacement field to simulate fractures or faults in a model. Our approach is based on the interior-penalty formulation of DGM, and the fractures are simulated using the linear-slip model, which is incorporated into the weak formulation. On the other hand, the spectral element method (SEM) can be used to simulate elastic wave propagation in non-fractured media. SEM uses continuous basis functions which do not allow for discontinuities in the displacement field. However, the computation cost of DGM is significantly larger than SEM due primarily to increase in the number of degrees of freedom. Here we propose a hybrid Galerkin method (HGM) for elastic wave propagation in fractured media that combines the salient features of each of the algorithm resulting in significant reduction in computational cost compared to DGM. We use DGM in areas containing fractures and SEM in regions without fractures. The coupling between the domains at the interfaces is satisfied in the weak form through interface conditions. The degree of reduction in computation time depends primarily on the density of fractures in the medium. In this paper, we formulate and implement HGM for seismic wave propagation in fractured media. Using realistic 2-D/3-D numerical examples, we show that our proposed HGM outperforms DGM with reduced computation cost and memory requirement while maintaining the same level of accuracy.

中文翻译：

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裂隙介质中地震波传播的混合Galerkin有限元方法

概要

不连续Galerkin有限元方法（DGM）是一种用于模拟裂缝介质中波传播的有前途的算法。它允许位移场中的不连续性来模拟模型中的裂缝或断层。我们的方法基于DGM的内部惩罚公式，并使用线性滑移模型模拟裂缝，该模型已纳入弱公式中。另一方面，光谱元素法（SEM）可用于模拟弹性波在非破裂介质中的传播。SEM使用连续基函数，不允许位移场不连续。但是，DGM的计算成本明显大于SEM，这主要是由于自由度数量的增加。在这里，我们提出了一种混合Galerkin方法（HGM），用于在破裂介质中传播弹性波，该方法结合了每种算法的显着特征，与DGM相比，可显着降低计算成本。我们在有裂缝的区域使用DGM，在无裂缝的区域使用SEM。通过接口条件以弱形式满足接口处域之间的耦合。计算时间的减少程度主要取决于介质中的裂缝密度。本文中，我们为裂缝介质中的地震波传播制定并实现了HGM。通过使用实际的2-D / 3-D数值示例，我们证明了我们提出的HGM在降低了计算成本和内存需求的同时，还优于DGM，同时保持了相同的准确性。