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Modal-based finite elements for efficient wave propagation analysis

机译:基于模态的有限元可进行有效的波传播分析

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We present an extension to the Geometric Multi-Scale Finite Element Method (GMsFEM) to better predict the dynamic response of heterogeneous materials and structures. The proposed method utilizes GMsFEM elements enriched by a number of vibration normal modes over the element domain. These modes can be calculated either numerically or analytically after imposing a proper set of boundary conditions at the element boundaries. In order to preserve many of the features of GMsFEM, including automatic enforcement of continuity across element boundaries, our enrichment functions are forced to be zero-valued at the boundaries. We applied our methodology to modeling one-dimensional stress wave propagation in smooth and notched bars, and two-dimensional stress wave propagation in a periodic elastic domain with a notch. For the problems of interest, we show that the enrichment functions provide a systematic way to increase the precision of GMsFEM while also leading to larger values of the stable time increment. In the traditional Finite Element Method (FEM), the stable time increment is dictated by the size of the smallest element in the domain. This can lead to simulations with a prohibitive computational cost when the FEM mesh is required to resolve small geometric features within a larger simulation domain of interest. In contrast, our method appears to be insensitive to small geometric features, with the stable time increment depending only on the chosen element nodes and the highest enrichment frequency.
机译:我们提出了几何多尺度有限元方法(GMsFEM)的扩展,以更好地预测异质材料和结构的动力响应。所提出的方法利用在元件域上被许多振动法向模式丰富的GMsFEM元件。在单元边界上施加一组适当的边界条件后,可以通过数值或分析方式计算这些模式。为了保留GMsFEM的许多功能,包括自动执行跨元素边界的连续性,我们的扩充功能被迫在边界处为零值。我们将我们的方法应用于对一维应力波在光滑和带缺口条形中的传播以及二维应力波在具有缺口的周期性弹性域中的传播进行建模的方法。对于感兴趣的问题,我们表明,富集函数提供了一种系统的方式来提高GMsFEM的精度,同时还导致稳定时间增量的较大值。在传统的有限元方法(FEM)中,稳定的时间增量取决于域中最小元素的大小。当需要FEM网格来解析较大的目标仿真域中的小几何特征时,这可能导致仿真的计算量过大。相反,我们的方法似乎对小的几何特征不敏感,其稳定的时间增量仅取决于所选的元素节点和最高的富集频率。

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