Bridging Scales with a Generalized Finite Element Method

The generalized FEM (GFEM) has been successfully applied to the simulation of dynamic propagating fractures, polycrystalline and fiber-reinforced microstructures, porous materials, etc. A-priori knowledge about the solution of these problems are used in the definition of their GFEM approximation spaces. This leads to more accurate and robust simulations than available finite element methods while relaxing some meshing requirements. This is demonstrated in a simulation of intergranular crack propagation in a brittle polycrystal using simple background meshes. For many classes of problems – like those with material non-linearities or involvingmultiscale phenomena – a-priori knowledge of the solution behavior is limited. In this paper, we present a GFEMbased on the solution of interdependent global (structural) and fine-scale or local problems. The local problems focus on the resolution of fine-scale features of the solution in the vicinity of, e.g., evolving fracture process zones while the global problem addresses the macroscale structural behavior. Fine-scale solutions are accurately solved using an hp-adaptive GFEM and thus the proposed method does not rely on analytical solutions. These solutions are embedded into the global solution space using the partition of unity method. This GFEM enables accurate modeling of problems involving multiple scales of interest using meshes with elements that are orders of magnitude larger than those required by the FEM. Numerical examples illustrating the application of this class of GFEM to high-cycle fatigue crack growth of small cracks and to problems exhibiting localized non-linear material responses are presented