Date

2019-05-04

Author

Kropat, Erik
Meyer-Nieberg, Silja
Weber, Gerhard Wilhelm

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Networked materials and micro-architectured systems gain increasingly importance in multi-scale physics and engineering sciences. Typically, computational intractable microscopic models have to be applied to capture the physical processes and numerous transmission conditions at singularities, interfaces and borders. The topology of the periodic microstructure governs the effective behaviour of such networked systems. A mathematical concept for the analysis of microscopic models on extremely large periodic networks is developed. We consider microscopic models for diffusion-advection-reaction systems in variational form on periodic manifolds. The global characteristics are identified by a homogenization approach for singularly perturbed networks with a periodic topology. We prove that the solutions of the variational models on varying networks converge to a two-scale limit function. In addition, the corresponding tangential gradients converge to a two-scale limit function for vanishing lengths of branches. We identify the variational homogenized model. Complex network models, previously considered as completely intractable, can now be solved by standard PDE-solvers in nearly no time. Furthermore, the homogenized coefficients provide an effective characterization of the global behaviour of the variational system.

Subject Keywords

Computational networks and systems, Micro-architectured systems, Microstructures, Two-scale convergence, Homogenization theory, Variational problems on graphs and networks, Diffusion-advection-reaction systems, Singular perturbations, Emerging applications

URI

https://hdl.handle.net/11511/57919

Collections

Graduate School of Applied Mathematics, Article