ON THE PRIMAL AND MIXED DUAL FORMATS IN VARIATIONALLY CONSISTENT COMPUTATIONAL HOMOGENIZATION WITH EMPHASIS ON FLUX BOUNDARY CONDITIONS

Abstract

In this paper, we view homogenization within the framework of variational multiscale methods. The standard (primal) variational format lends itself naturally to the choice of Dirichlet boundary conditions on the Representative Volume Element (RVE). However, how to impose flux boundary conditions, treated as Neumann conditions in the standard variational format, is less obvious. Therefore, in this paper we propose and investigate a novel mixed variational setting, where the fluxes are treated as additional primary fields, in order to provide the natural variational environment for such flux boundary conditions. This mixed dual formulation allows for a conforming implementation of (lower bound) flux boundary conditions in the framework of discretization-based homogenization. To focus on essential features, a very simple problem is studied: the classical stationary linear heat equation. Furthermore, we consider the standard context of model-based homogenization (without loss of generality), since we are only concerned with the RVE problem and merely assume that the relevant macroscale fields are properly prolonged. Numerical results from the primary and the mixed dual variational formats are compared and their convergence properties for mesh finite element (FE) refinement and RVE size are assessed.