Bridging multi-scale method for localization problems

A bridging multi-scale method is proposed for the analysis of a class of localization problems in which the micropolar-continuum model is used to describe the localized deformation and the dynamic failure occurs only in a small number of localized regions. It starts with a concurrent discretization of the entire domain with both coarse- and fine-scale finite-element meshes. The coarse-scale mesh is employed to capture the nonlinear response with long wavelength outside the localized regions, whereas the fine-scale mesh captures the detailed physics of the localized deformation. For both the coarse and fine-scale meshes to coexist, a bridging scale term is constructed so that the information common to both scales is correctly subtracted. To achieve computational efficiency, the localized regions are first identified by a preliminary calculation and the fine-scale degrees of freedom (DOFs) outside the localized regions are mathematically represented by the construction of dynamic interface conditions applied to the edges of these regions. Hence, a large portion of the fine-scale DOFs are eliminated and the fine-scale equations are reduced to a much smaller set with the added dynamic interface conditions. The two-way coupled coarse-scale and reduced fine-scale equations are then solved by a mixed time integration procedure. Applications to one-dimensional and two-dimensional dynamic shear localization problems are presented.