Computational networks and systems - homogenization of variational problems on micro-architectured networks and devices

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


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Boundary value problems on large periodic networks arise in many applications such as soil mechanics in geophysics or the analysis of photonic crystals in nanotechnology. As a model example, singularly perturbed elliptic differential equations of second order are addressed. Typically, the length of periodicity is very small compared to the size of the covered region. The overall complexity of the networks raises serious problems on the computational side. The high density of the graph, the huge number of ed...
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Kropat, Erik; Meyer-Nieberg, Silja; Weber, Gerhard Wilhelm (American Institute of Mathematical Sciences (AIMS), 2017)
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 correspondi...
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Citation Formats
E. Kropat, S. Meyer-Nieberg, and G. W. Weber, “Computational networks and systems - homogenization of variational problems on micro-architectured networks and devices,” OPTIMIZATION METHODS & SOFTWARE, pp. 586–611, 2019, Accessed: 00, 2020. [Online]. Available: