Computational networks and systems - Homogenization of self-adjoint differential operators in variational form on periodic networks and micro-architectured systems

Abstract

Micro-architectured systems and periodic network structures play an import role in multi-scale physics and material sciences. Mathematical modeling leads to challenging problems on the analytical and the numerical side. Previous studies focused on averaging techniques that can be used to reveal the corresponding macroscopic model describing the effective behavior. This study aims at a mathematical rigorous proof within the framework of homogenization theory. As a model example, the variational form of a self-adjoint operator on a large periodic network is considered. A notion of two-scale convergence for network functions based on a so-called two-scale transform is applied. It is shown that the sequence of solutions of the variational microscopic model on varying networked domains converges towards the solution of the macroscopic model. A similar result is achieved for the corresponding sequence of tangential gradients. The resulting homogenized variational model can be easily solved with standard PDE-solvers. In addition, the homogenized coefficients provide a characterization of the physical system on a global scale. In this way, a mathematically rigorous concept for the homogenization of self-adjoint operators on periodic manifolds is achieved. Numerical results illustrate the effectiveness of the presented approach.