# American Institute of Mathematical Sciences

June  2017, 7(2): 139-169. doi: 10.3934/naco.2017010

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

 1 University of the Bundeswehr Munich, Faculty of Informatics, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany 2 Middle East Technical University, Institute of Applied Mathematics, 06531 Ankara, Turkey

* Corresponding author: Erik Kropat

+ Honorary positions: Faculty of Economics, Business and Law, University of Siegen, Germany; Center for Research on Optimization and Control, University of Aveiro, Portugal; University of North Sumatra, Indonesia

Received  July 2016 Revised  April 2017 Published  June 2017

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.

Citation: Erik Kropat, Silja Meyer-Nieberg, Gerhard-Wilhelm Weber. Computational networks and systems-homogenization of self-adjoint differential operators in variational form on periodic networks and micro-architectured systems. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 139-169. doi: 10.3934/naco.2017010
##### References:

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##### References:
Homogenization theory: The limit process
Periodic networks: The network $\mathcal{N}^\Omega_\varepsilon$ is the segment of the infinite network $\mathcal{N}_\varepsilon$ contained in the polyhedral domain $\Omega$, that is obtained by copying and scaling from the reference graph $\mathbb{G}$ in the unit cell $[0, 1)^d$
Covering with cells: The $\varepsilon$-cells and the corresponding graphs
Feasible networks: The network $\mathcal{N}^\Omega_1$ (left figure) and the network $\mathcal{N}^\Omega_{\frac{{1}}{{2}}}$ are feasible networks where the corresponding $\varepsilon$-cells cover the domain $\Omega$
Two-scale transform: The function $x:\Omega \times \mathbb{G} \rightarrow \mathcal{N}^\Omega_\varepsilon$ is surjective, but not injective
Two-scale transform: Mapping from $\mathcal{N}^\Omega_\varepsilon$ to the product $\Omega \times \mathbb{G}$
Illustrative example: The reference graph $\mathbb{G}$
Illustrative example: Diffusion coefficient and source term
Illustrative example: The periodic network $\mathcal{N}^\Omega_1$
Illustrative example: Solutions of the microscopic model for different lengths of periodicity $\varepsilon$
Illustrative example: The approximate homogenized model and the solution for $\varepsilon=0.2$
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