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

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

doi:10.3934/naco.2017010

Abstract

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.

Highlights

This research is motivated by our studies on flow and transport through extremely large capillary systems in the fields of groundmotion prediction in geo-engineering and groundwater contamination monitoring in environmental2010 Mathematics Subject Classification
The microscopic problem is the variational form of a model with second-order differential equations on edge of the periodic network NεΩ in the macroscopic domain Ω = (0, 10)2
The present study is a part of our investigations on mathematical problems in spatial data analysis related to large scale flow and transport models for capillary systems in the soil

Summary

Introduction

This research is motivated by our studies on flow and transport through extremely large capillary systems in the fields of groundmotion prediction in geo-engineering and groundwater contamination monitoring in environmental2010 Mathematics Subject Classification. 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. Huge capillary networks can be represented by periodic graphs with a very small length of periodicity when compared to the total extension of the domain under consideration. The physical process on the periodic networked is modeled by a system of second order differential equations that define a self-adjoint operator. The solution of this diffusion-reaction system describes for example the spatial distribution of a certain substance

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