Abstract We study and use overlapping triangular finite elements enriched by trigonometric functions and implicit time integration to solve transient wave propagations in inhomogeneous media. We show explicitly that the total dispersion error of the calculated solutions can be split into two parts, the spatial error and temporal error. The study of the spatial dispersion error shows the effectiveness of the enriched overlapping finite elements compared to the traditional finite elements and the overlapping finite elements without enrichment. The study of the temporal error of the Bathe time integration scheme shows monotonic convergence to zero with decreasing time step size. The result is that we see monotonic convergence to exact solutions as the mesh of enriched overlapping finite elements is refined and the time step is decreased. We demonstrate the effectiveness of using the proposed scheme in the solution of waves traveling in inhomogeneous media at different speeds, where reflected and transmitted waves are predicted accurately by “simply” using a fine enough mesh and small enough time step.

中文翻译：

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具有丰富重叠三角形元素的非均匀介质中的瞬态波传播

摘要 我们研究并使用由三角函数和隐式时间积分丰富的重叠三角形有限元来解决非均匀介质中的瞬态波传播问题。我们明确地表明，计算解决方案的总色散误差可以分为两部分，空间误差和时间误差。空间弥散误差的研究表明，与传统有限元和未富集的重叠有限元相比，富集重叠有限元的有效性。Bathe 时间积分方案的时间误差研究表明，随着时间步长的减小，单调收敛到零。结果是，随着丰富的重叠有限元的网格被细化并且时间步长被减少，我们看到单调收敛到精确解。