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Networks & Heterogeneous Media

March 2020 , Volume 15 , Issue 1

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Matrix valued inverse problems on graphs with application to mass-spring-damper systems
Travis G. Draper, Fernando Guevara Vasquez, Justin Cheuk-Lum Tse, Toren E. Wallengren and Kenneth Zheng
2020, 15(1): 1-28 doi: 10.3934/nhm.2020001 +[Abstract](715) +[HTML](146) +[PDF](531.98KB)
Abstract:

We consider the inverse problem of finding matrix valued edge or node quantities in a graph from measurements made at a few boundary nodes. This is a generalization of the problem of finding resistors in a resistor network from voltage and current measurements at a few nodes, but where the voltages and currents are vector valued. The measurements come from solving a series of Dirichlet problems, i.e. finding vector valued voltages at some interior nodes from voltages prescribed at the boundary nodes. We give conditions under which the Dirichlet problem admits a unique solution and study the degenerate case where the edge weights are rank deficient. Under mild conditions, the map that associates the matrix valued parameters to boundary data is analytic. This has practical consequences to iterative methods for solving the inverse problem numerically and to local uniqueness of the inverse problem. Our results allow for complex valued weights and give also explicit formulas for the Jacobian of the parameter to data map in terms of certain products of Dirichlet problem solutions. An application to inverse problems arising in networks of springs, masses and dampers is presented.

Mathematical analysis of transmission properties of electromagnetic meta-materials
Mario Ohlberger, Ben Schweizer, Maik Urban and Barbara Verfürth
2020, 15(1): 29-56 doi: 10.3934/nhm.2020002 +[Abstract](317) +[HTML](135) +[PDF](5846.15KB)
Abstract:

We study time-harmonic Maxwell's equations in meta-materials that use either perfect conductors or high-contrast materials. Based on known effective equations for perfectly conducting inclusions, we calculate the transmission and reflection coefficients for four different geometries. For high-contrast materials and essentially two-dimensional geometries, we analyze parallel electric and parallel magnetic fields and discuss their potential to exhibit transmission through a sample of meta-material. For a numerical study, one often needs a method that is adapted to heterogeneous media; we consider here a Heterogeneous Multiscale Method for high contrast materials. The qualitative transmission properties, as predicted by the analysis, are confirmed with numerical experiments. The numerical results also underline the applicability of the multiscale method.

Incompressible limit of a continuum model of tissue growth for two cell populations
Pierre Degond, Sophie Hecht and Nicolas Vauchelet
2020, 15(1): 57-85 doi: 10.3934/nhm.2020003 +[Abstract](305) +[HTML](127) +[PDF](719.27KB)
Abstract:

This paper investigates the incompressible limit of a system modelling the growth of two cells population. The model describes the dynamics of cell densities, driven by pressure exclusion and cell proliferation. It has been shown that solutions to this system of partial differential equations have the segregation property, meaning that two population initially segregated remain segregated. This work is devoted to the incompressible limit of such system towards a free boundary Hele Shaw type model for two cell populations.

Homogenization of Bingham flow in thin porous media
María Anguiano and Renata Bunoiu
2020, 15(1): 87-110 doi: 10.3934/nhm.2020004 +[Abstract](331) +[HTML](116) +[PDF](483.66KB)
Abstract:

By using dimension reduction and homogenization techniques, we study the steady flow of an incompresible viscoplastic Bingham fluid in a thin porous medium. A main feature of our study is the dependence of the yield stress of the Bingham fluid on the small parameters describing the geometry of the thin porous medium under consideration. Three different problems are obtained in the limit when the small parameter \begin{document}$ \varepsilon $\end{document} tends to zero, following the ratio between the height \begin{document}$ \varepsilon $\end{document} of the porous medium and the relative dimension \begin{document}$ a_\varepsilon $\end{document} of its periodically distributed pores. We conclude with the interpretation of these limit problems, which all preserve the nonlinear character of the flow.

The General Non-Abelian Kuramoto Model on the 3-sphere
Vladimir Jaćimović and Aladin Crnkić
2020, 15(1): 111-124 doi: 10.3934/nhm.2020005 +[Abstract](368) +[HTML](182) +[PDF](691.7KB)
Abstract:

We introduce non-Abelian Kuramoto model on \begin{document}$ S^3 $\end{document} in the most general form. Following an analogy with the classical Kuramoto model (on the circle \begin{document}$ S^1 $\end{document}), we study some interesting variations of the model on \begin{document}$ S^3 $\end{document} that are obtained for particular coupling functions. As a partial case, by choosing "standard" coupling function one obtains a previously known model, that is referred to as Kuramoto-Lohe model on \begin{document}$ S^3 $\end{document}.

We briefly address two particular models: Kuramoto models on \begin{document}$ S^3 $\end{document} with frustration and with external forcing. These models on higher dimensional manifolds have not been studied so far. By choosing suitable values of parameters we observe different nontrivial dynamical regimes even in the simplest setup of globally coupled homogeneous population.

Although non-Abelian Kuramoto models can be introduced on various symmetric spaces, we restrict our analysis to the case when underlying manifold is the 3-sphere. Due to geometric and algebraic properties of this specific manifold, variations of this model are meaningful and geometrically well justified.

A remark about the periodic homogenization of certain composite fibered media
François Murat and Ali Sili
2020, 15(1): 125-142 doi: 10.3934/nhm.2020006 +[Abstract](317) +[HTML](120) +[PDF](578.14KB)
Abstract:

We explain in this paper the similarity arising in the homogenization process of some composite fibered media with the problem of the reduction of dimension \begin{document}$ 3d-1d $\end{document}. More precisely, we highlight the fact that when the homogenization process leads to a local reduction of dimension, studying the homogenization problem in the reference configuration domain of the composite amounts to the study of the corresponding reduction of dimension in the reference cell. We give two examples in the framework of the thermal conduction problem: the first one concerns the reduction of dimension in a thin parallelepiped of size \begin{document}$ \varepsilon $\end{document} containing another thinner parallelepiped of size \begin{document}$ r_ \varepsilon \ll \varepsilon $\end{document} playing a role of a "hole". As in the homogenization, the one-dimensional limit problem involves a "strange term". In addition both limit problems have the same structure. In the second example, the geometry is similar but now we assume a high contrast between the conductivity (of order \begin{document}$ 1 $\end{document}) in the small parallelepiped of size \begin{document}$ r_ \varepsilon : = r \varepsilon $\end{document}, for some fixed \begin{document}$ r $\end{document} (\begin{document}$ 0 < r < \frac{1}{2} $\end{document}) and the conductivity (of order \begin{document}$ \varepsilon^2 $\end{document}) in the big parallelepiped of size \begin{document}$ \varepsilon $\end{document}. We prove that the limit problem is a nonlocal problem and that it has the same structure as the corresponding periodic homogenized problem.

The Lax-Friedrichs scheme for interaction between the inviscid Burgers equation and multiple particles
John D. Towers
2020, 15(1): 143-169 doi: 10.3934/nhm.2020007 +[Abstract](215) +[HTML](120) +[PDF](639.29KB)
Abstract:

We propose a finite difference method based on the Lax-Friedrichs scheme for a model of interaction between multiple solid particles and an inviscid fluid. The single-particle version has been studied extensively during the past decade. The model studied here consists of the inviscid Burgers equation with multiple nonconservative moving source terms that are singular and account for drag force interaction between the fluid and the particles. Each particle trajectory satisfies a differential equation that ensures conservation of momentum of the entire system. To deal with the singular source terms we discretize a model that associates with each particle an advection PDE whose solution is a shifted Heaviside function. This alternative model is well known but has not previously been used in numerical methods. We propose a definition of entropy solution which directly generalizes the previously defined single-particle notion of entropy solution. We prove convergence (along a subsequence) of the Lax-Friedrichs approximations, and also prove that if the set of times where the particle paths intersect has Lebesgue measure zero, then the limit is an entropy solution. We also propose a higher resolution version of the scheme, based on MUSCL processing, and present the results of numerical experiments.

2018  Impact Factor: 0.871

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