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

  • * Corresponding author: Erik Kropat

    * 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

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  • 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.

    Mathematics Subject Classification: Primary: 34B45, 34E13, 34E05, 34E10.


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  • Figure 1.  Homogenization theory: The limit process

    Figure 2.  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 $[0, 1)^d$

    Figure 3.  Covering with cells: The $\varepsilon$-cells and the corresponding graphs

    Figure 4.  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 $\Omega$

    Figure 5.  Two-scale transform: The function $x:\Omega \times \mathbb{G} \rightarrow \mathcal{N}^\Omega_\varepsilon$ is surjective, but not injective

    Figure 6.  Two-scale transform: Mapping from $\mathcal{N}^\Omega_\varepsilon$ to the product $\Omega \times \mathbb{G}$

    Figure 7.  Illustrative example: The reference graph $\mathbb{G}$

    Figure 8.  Illustrative example: Diffusion coefficient and source term

    Figure 9.  Illustrative example: The periodic network $\mathcal{N}^\Omega_1$

    Figure 10.  Illustrative example: Solutions of the microscopic model for different lengths of periodicity $\varepsilon$

    Figure 11.  Illustrative example: The approximate homogenized model and the solution for $\varepsilon=0.2$

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