On selection criteria for problems with moving inhomogeneities

We study mechanical problems with multiple solutions and introduce a thermodynamic framework to formulate two different selection criteria in terms of macroscopic energy productions and fluxes. Studying simple examples for lattice motion we then compare the implications for both resting and moving inhomogeneities.Comment: revised version contains new introduction, numerical simulations of Riemann problems, and a more detailed discussion of the causality principle; 18 pages, several figure

On weak convergence of locally periodic functions

We prove a generalization of the fact that periodic functions converge weakly to the mean value as the oscillation increases. Some convergence questions connected to locally periodic nonlinear boundary value problems are also considered.Comment: arxiv version is already officia

Spectral super-resolution in metamaterial composites

We investigate the optical properties of periodic composites containing metamaterial inclusions in a normal material matrix. We consider the case where these inclusions have sharp corners, and following Hetherington and Thorpe, use analytic results to argue that it is then possible to deduce the shape of the corner (its included angle) by measurements of the absorptance of such composites when the scale size of the inclusions and period cell is much finer than the wavelength. These analytic arguments are supported by highly accurate numerical results for the effective permittivity function of such composites as a function of the permittivity ratio of inclusions to matrix. The results show that this function has a continuous spectral component with limits independent of the area fraction of inclusions, and with the same limits for both square and staggered square arrays.Comment: 17 pages, 6 figure

Homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions

A microscopic heterogeneous system under random influence is considered. The randomness enters the system at physical boundary of small scale obstacles as well as at the interior of the physical medium. This system is modeled by a stochastic partial differential equation defined on a domain perforated with small holes (obstacles or heterogeneities), together with random dynamical boundary conditions on the boundaries of these small holes. A homogenized macroscopic model for this microscopic heterogeneous stochastic system is derived. This homogenized effective model is a new stochastic partial differential equation defined on a unified domain without small holes, with static boundary condition only. In fact, the random dynamical boundary conditions are homogenized out, but the impact of random forces on the small holes' boundaries is quantified as an extra stochastic term in the homogenized stochastic partial differential equation. Moreover, the validity of the homogenized model is justified by showing that the solutions of the microscopic model converge to those of the effective macroscopic model in probability distribution, as the size of small holes diminishes to zero.Comment: Communications in Mathematical Physics, to appear, 200

Inverse homogenization of a quasiperiodic composite

International audienc

Quasiperiodic Composites: Multiscale Reiterated Homogenization

International audienceWith recent technological advances, quasiperiodic and aperiodic materials present a novel class of metamaterials that possess very unusual, extraordinary properties such as superconductivity, unusual mechanical properties and diffraction patterns, extremely low thermal conductivity, etc. As all these properties critically depend on the microgeometry of the media, the methods that allow characterizing the effective properties of such materials are of paramount importance. In this paper, we analyze the effective properties of a class of multiscale composites consisting of periodic and quasiperiodic phases appearing at different scales. We derive homogenized equations for the effective behavior of the composite and discover a variety of new effects which could have interesting applications in the control of wave and diffusion phenomena

On the definition of marginal ice zone width

Sea ice features a dense inner pack ice zone surrounded by a marginal ice zone (MIZ) in which the sea ice properties are modified by interaction with the ice-free open ocean. The width of the MIZ is a fundamental length scale for polar physical and biological dynamics. Several different criteria for establishing MIZ boundaries have emerged in the literature-wave penetration, floe size, sea ice concentration, etc.-and a variety of definitions for the width between the MIZ boundaries have been published. Here, three desirable mathematical properties for defining MIZ width are proposed: invariance with respect to translation and rotation on the sphere; uniqueness at every point in the MIZ; and generality, including nonconvex shapes. The previously published streamline definition is shown to satisfy all three properties, where width is defined as the arc length of a streamline through the solution to Laplaces's equation within the MIZ boundaries, while other published definitions each satisfy only one of the desired properties. When defining MIZ spatial average width from streamline results, the rationale for averaging with respect to distance along both MIZ boundaries was left implicit in prior studies. Here it is made rigorous by developing and applying the mathematics of an analytically tractable idealization of MIZ geometry-the eccentric annulus. Finally, satellite-retrieved Arctic sea ice concentrations are used to investigate how well streamline-based MIZ spatial average width is approximated by alternative definitions that lack desirable mathematical properties or local width values but offer computational efficiency

On the definition of marginal ice zone width

Sea ice features a dense inner pack ice zone surrounded by a marginal ice zone (MIZ) in which the sea ice properties are modified by interaction with the ice-free open ocean. The width of the MIZ is a fundamental length scale for polar physical and biological dynamics. Several different criteria for establishing MIZ boundaries have emerged in the literature-wave penetration, floe size, sea ice concentration, etc.-and a variety of definitions for the width between the MIZ boundaries have been published. Here, three desirable mathematical properties for defining MIZ width are proposed: invariance with respect to translation and rotation on the sphere; uniqueness at every point in the MIZ; and generality, including nonconvex shapes. The previously published streamline definition is shown to satisfy all three properties, where width is defined as the arc length of a streamline through the solution to Laplaces's equation within the MIZ boundaries, while other published definitions each satisfy only one of the desired properties. When defining MIZ spatial average width from streamline results, the rationale for averaging with respect to distance along both MIZ boundaries was left implicit in prior studies. Here it is made rigorous by developing and applying the mathematics of an analytically tractable idealization of MIZ geometry-the eccentric annulus. Finally, satellite-retrieved Arctic sea ice concentrations are used to investigate how well streamline-based MIZ spatial average width is approximated by alternative definitions that lack desirable mathematical properties or local width values but offer computational efficiency