The effective conductivity of random checkerboards

An algorithm is presented for the fast and accurate solution of the electrostatic equation on multi-component random checkerboards. It relies on a particular choice of integral equation, extended as to separate ill-conditioning due to singular fields in corners from ill-conditioning due to interaction of clusters of well-conducting squares at large distances. Two separate preconditioners take care of the two separate phenomena. In a series of numerical examples, effective conductivities are computed for random checkerboards containing up to 10^4 squares with conductivity ratios of up to 10^6. The achievable relative precision in these examples is on the order of 10^{−11}

The effective conductivity of arrays of squares: large random unit cells and extreme contrast ratios

An integral equation based scheme is presented for the fast and accurate computation of effective conductivities of two-component checkerboard-like composites with complicated unit cells at very high contrast ratios. The scheme extends recent work on multi-component checkerboards at medium contrast ratios. General improvement include the simplification of a long-range preconditioner, the use of a banded solver, and a more efficient placement of quadrature points. This, together with a reduction in the number of unknowns, allows for a substantial increase in achievable accuracy as well as in tractable system size. Results, accurate to at least nine digits, are obtained for random checkerboards with over a million squares in the unit cell at contrast ratio 10^6. Furthermore, the scheme is flexible enough to handle complex valued conductivities and, using a homotopy method, purely negative contrast ratios. Examples of the accurate computation of resonant spectra are given.Comment: 28 pages, 11 figures, submitted to J. Comput. Phy

Generalized two-body self-consistent theory of random linear dielectric composites: an effective-medium approach to clustering in highly-disordered media

Effects of two-body dipolar interactions on the effective permittivity/conductivity of a binary, symmetric, random dielectric composite are investigated in a self-consistent framework. By arbitrarily splitting the singularity of the Green tensor of the electric field, we introduce an additional degree of freedom into the problem, in the form of an unknown "inner" depolarization constant. Two coupled self-consistent equations determine the latter and the permittivity in terms of the dielectric contrast and the volume fractions. One of them generalizes the usual Coherent Potential condition to many-body interactions between single-phase clusters of polarizable matter elements, while the other one determines the effective medium in which clusters are embedded. The latter is in general different from the overall permittivity. The proposed approach allows for many-body corrections to the Bruggeman-Landauer (BL) scheme to be handled in a multiple-scattering framework. Four parameters are used to adjust the degree of self-consistency and to characterize clusters in a schematic geometrical way. Given these parameters, the resulting theory is "exact" to second order in the volume fractions. For suitable parameter values, reasonable to excellent agreement is found between theory and simulations of random-resistor networks and pixelwise-disordered arrays in two and tree dimensions, over the whole range of volume fractions. Comparisons with simulation data are made using an "effective" scalar depolarization constant that constitutes a very sensitive indicator of deviations from the BL theory.Comment: 14 pages, 7 figure

Exact result for the effective conductivity of a continuum percolation model

Journal ArticleA random two-dimensional checkerboard of squares of conductivities 1 and 8 in proportions p and 1 - p is considered. Classical duality implies that the effective conductivity obeys o* = V8 at p = 1/2. It is rigorously found here that to leading order as 8--0, this exact result holds for all p in the interval (1- pc,pc), where pc=0.59 is the site percolation probability, not just at p = 1/2. In particular, o*(p,8)=78+O (8), as 8 -- 0. which is argued to hold for complex 8 as well. The analysis is based on the identification of a "symmetric" backbone, which is statistically invariant under interchange of the components for any pE(1--pc,pc), like the entire checkerboard at p =1/2. This backbone is defined in terms of "choke points" for the current, which have been observed in an experiment

Comparing Virtual Reality to Conventional Simulator Visuals: Effects of Peripheral Visual Cues in Roll-Axis Tracking Tasks

This paper compares the effects of peripheral visual cues on manual control between a conventional fixed-base simulator and virtual reality. The results were also compared with those from a previous experiment conducted in a motion-base simulator. Fifteen participants controlled a system with second-order dynamics in a disturbance-rejection task. Tracking performance, control activity, simulator sickness questionnaire answers, and biometrics were collected. Manual control behavior was modeled for the first time in a virtual reality environment. Virtual reality did not degrade participants manual control performance or alter their control behavior. However, peripheral cues were significantly more effective in virtual reality. Control activity decreased for all conditions with peripheral cues. The trends introduced by the peripheral visual cues from the previous experiment were replicated. Finally, VR was not more nauseogenic than the conventional simulator. These results suggest that virtual reality might be a good alternative to conventional fixed-base simulators for training manual control skills

A new method for numerical solution of checkerboard fields

We consider a generalized version of the standard checkerboard and discuss the difficulties of finding the corresponding field by standard numerical treatment. A new numerical method is presented which converges independently of the local conductivities

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

Anomalous phase separation dynamics in a correlated electron system: machine-learning enabled large-scale kinetic Monte Carlo simulations

Phase separation plays a central role in the emergence of novel functionalities of correlated electron materials. The structure of the mixed-phase states depends strongly on the nonequilibrium phase-separation dynamics, which has so far yet to be systematically investigated, especially on the theoretical side. With the aid of modern machine learning methods, we demonstrate the first-ever large-scale kinetic Monte Carlo simulations of the phase separation process for the Falicov-Kimball model, which is one of the canonical strongly correlated electron systems. We uncover an unusual phase-separation scenario where domain coarsening occurs simultaneously at two different scales: the growth of checkerboard clusters at smaller length scales and the expansion of super-clusters, which are aggregates of the checkerboard clusters of the same sign, at a larger scale. We show that the emergence of super-clusters is due to a hidden dynamical breaking of the sublattice symmetry. Arrested growth of the checkerboard patterns and of the super-clusters is shown to result from a correlation-induced self-trapping mechanism. Glassy behaviors similar to the one reported in this work could be generic for other correlated electron systems.Comment: 11 pages, 11 figure