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Bridging the gap between variational homogenization results and two-scale asymptotic averaging techniques on periodic network structures

  • * 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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  • In modern material sciences and multi-scale physics homogenization approaches provide a global characterization of physical systems that depend on the topology of the underlying microgeometry. Purely formal approaches such as averaging techniques can be applied for an identification of the averaged system. For models in variational form, two-scale convergence for network functions can be used to derive the homogenized model. The sequence of solutions of the variational microcsopic models and the corresponding sequence of tangential gradients converge toward limit functions that are characterized by the solution of the variational macroscopic model. Here, a further extension of this result is proved. The variational macroscopic model can be equivalently represented by a homogenized model on the superior domain and a certain number of reference cell problems. In this way, the results obtained by averaging strategies are supported by notions of convergence for network functions on varying domains.

    Mathematics Subject Classification: Primary: 4B45, 34E13, 34E05, 34E10.


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  • Figure 1.  The homogenization process: The sequence of solutions of the microscopic model as well as the corresponding sequence of tangential gradients are weakly two-scale convergent. The limits of these sequences can be represented by the solution of the homogenized model

    Figure 2.  Two-scale transform: The function $x:\Omega \times {\mathscr{Y}} \rightarrow {\cal{N}}^\Omega_\varepsilon$ is surjective, but not injective

    Figure 3.  Two-scale transform: Mapping from ${\cal{N}}^\Omega_\varepsilon$ to the product $\Omega \times {\mathscr{Y}}$

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