A high-order generalized Finite Element Method for multiscale structural dynamics and wave propagation

This paper introduces a high-order multiscale Generalized/eXtended Finite Element Method (GFEM) tailored for the solution of structural dynamics and wave propagation problems exhibiting fine-scale and/or localized solution features such as singularities and discontinuities. The proposed method uses an explicit time-marching scheme together with a block-diagonal lumped mass matrix applicable to arbitrary patch approximation spaces and adopts shape functions computed numerically on-the-fly through the solution of local initial/boundary value problems. Numerical results in one-, two-, and three-dimensions for problems exhibiting singularities in the spatial gradient show that the proposed GFEM is able to capture relevant features of the response using structural-scale meshes that are much coarser than those required by Direct Generalized Finite Element Analyses (DGFEAs) of comparable accuracy. In addition, a detailed study of the critical time step size of the method is presented and compared against the one for DGFEA discretizations that provide similar levels of accuracy. It shows that the proposed method has considerably looser time step size restrictions than available DGFEAs.