Multiscale analysis of wave propagation in heterogeneous structures

The analysis of wave propagation in solids with complex microstructures, and local heterogeneities finds extensive applications in areas such as material characterization, structural health monitoring (SHM), and metamaterial design. Within continuum mechanics, sources of heterogeneities are typically associated to localized defects in structural components, or to periodic microstructures in phononic crystals and acoustic metamaterials. Numerical analysis of this class of solids often requires computational meshes which are refined enough to resolve the wavelengths of propagating waves and to properly capture the fine geometrical features of the heterogeneities. It is common for the size of the microstructure to be small compared to the dimensions of the structural component under investigation, which suggests multiscale analysis as an effective approach to minimize computational costs while retaining predictive accuracy.;The objective of this research is to develop a multiscale framework for the efficient analysis of the dynamic behavior of heterogeneous solids. The proposed method, called Geometric Multiscale Finite Element Method (GMsFEM), is based on the formulation of multi-node elements with numerically computed shape functions. Such shape functions are capable of explicitly modeling the geometry of heterogeneities at sub-elemental length scales, and are computed to automatically satisfy compatibility of the solution across boundaries of adjacent elements. Numerical examples first illustrate the approach and validate it through comparison with available analytical and numerical solutions. The developed methodology is then applied to the analysis of periodic media, structural lattices, and phononic crystal structures. Numerical predictions for the latter are validated with experimental measurements. Finally, GMsFEM is exploited to study the interaction of guided elastic waves and defects in plate structures. Both two- and three-dimensional analyses correlate the computed scattering patterns to the relevant geometrical feature of damage. The broad range of considered applications show that the developed multiscale strategy is systematic, self-consistent and easy to implement. Such characteristics are extremely attractive to expand the method to nonlinear and multi-physics problems, and make GMsFEM a possible candidate for integration as part of commercial finite element softwares.