A primal formulation for imposing periodic boundary conditions on conforming and nonconforming meshes

A variational multiscale Discontinuous Galerkin (VMDG) method is developed for microscale modeling of domains containing conforming and non-conforming meshes. Essentially, the product of the applied volume-average strain (or macro-strain) and the domain diameter acts as an imposed displacement jump within the VMDG terms. Hence, the method is suitable for modeling deformation of both block and truly (self) periodic representative volume elements (RVEs). The primal displacement field and macro-strain are the only unknowns because the method eliminates the Lagrange multiplier (LM) enforcement of the kinematic constraint. Rigorous derivation of the method provides a framework to accommodate either the macro-stress or macro-strain as the driver of the microscale boundary value problem. The method is developed first for finite deformations and then specialized to small deformation kinematics. Algorithmic modifications to the method are also studied for their effects on tangent symmetry and convergence rate. The results from numerical studies for isotropic and anisotropic materials show that the proposed method is robust, accurate, stable and variationally consistent for modeling complicated conforming and nonconforming RVEs.