An adaptive concurrent multiscale method for microstructured elastic solids

The present paper introduces a concurrent adaptive multiscale methodology for elasticity problems in which macroscopic deformation strongly interact with microscopic deformation fields at the scale of the microstructure. This situation occurs in a variety of important situations such as ductile fracture, shear banding and when sharp discontinuities are present in the material domain. Our analysis starts from the idea that numerical simulations must ensures that numerical accuracy is maximum while the error from homogenization is minimum. While the first usually implies that element size must be refined, the second implies that elements may not be smaller than the representative volume elements (RVEs) of the microstructure. To accommodate these two conditions, a finite element method is introduced such that continuum elements can be replaced by explicit RVEs through properly defined kinetic conditions reminiscent to those used in classical homogenization. When combined with adaptive refinement, the methodology provides a flexible numerical method in which both continuum and microstructural descriptions can naturally coexist within a single simulation. We show, through various examples that the proposed framework addresses the important issue of reaching the optimal modeling accuracy for a minimal computational cost.