Modal-based finite elements for efficient wave propagation analysis

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.