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

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2017-06-01

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

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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.

Subject Keywords

Computational networks and systems, Micro-architectured systems, Homogenization theory, Self-adjoint operators, Periodic microstructures, Two-scale convergence, Variational problems on graphs and networks, Diffusion-reaction systems, Periodic networks, Singular perturbations

URI

https://hdl.handle.net/11511/51130

Journal

NUMERICAL ALGEBRA CONTROL AND OPTIMIZATION

DOI

https://doi.org/10.3934/naco.2017010

Collections

Graduate School of Applied Mathematics, Article

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Citation Formats

E. Kropat, S. Meyer-Nieberg, and G. W. Weber, “Computational networks and systems - Homogenization of self-adjoint differential operators in variational form on periodic networks and micro-architectured systems,” NUMERICAL ALGEBRA CONTROL AND OPTIMIZATION, pp. 139–169, 2017, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/51130.