A Generalized Finite Element Method for Multiscale Simulations

Abstract : This report focuses on recent advances of the Generalized Finite Element Method (GFEM) for multiscale simulations. This method is based on the solution of interdependent global and local scale problems, and can be applied to a broad class of multiscale problems of relevance to the United States Air Force. The local problems focus on the resolution of fine scale features of the solution while the global problem addresses the macro-scale structural behavior. The local solutions are embedded into the global solution space using the partition of unity method. A rigorous a-priori error estimate for the method is presented along with numerical verification of convergence properties predicted by the estimate. The convergence analysis shows optimal convergence of the method on problems with strong singularities. It also shows that the method can deliver the same accuracy as direct numerical simulations (DNS) while using meshes with elements that are orders of magnitude larger than in the DNS case.