Variational multiscale enrichment method with mixed boundary conditions for modeling diffusion and deformation problems

This manuscript presents the formulation and implementation of the variational multiscale enrichment (VME) method using canopy-shaped microscale enrichment functions obtained through the use of a new family of microscale boundary conditions. The purpose of the new enrichment functions and the new boundary condition is to relax the overconstraint imposed by the homogeneous microscale boundary condition (e.g., residual free bubbles) commonly employed in the variational multiscale literature. The formulation and implementation of the method are presented for diffusion and elasticity problems. The performance of the proposed method is assessed by comparing with direct numerical simulations on diffusion and deformation problems. A boundary parameter identification approach is proposed to obtain near-optimal boundary conditions. The identification approach is verified in the context of the deformation response of particle-reinforced composites.