Random Checkerboard Research Articles

Electrostatic and magnetostatic properties of random materials.

Scale dependence of electrostatic and magnetostatic properties is investigated in the setting of spatially random linear lossless materials with statistically homogeneous and spatially ergodic random microstructures. First, from the Hill-Mandel homogenization conditions adapted to electric and magnetic fields, uniform boundary conditions are formulated for a statistical volume element (SVE). From these conditions, there follow upper and lower mesoscale bounds on the macroscale (effective) electrical permittivity and magnetic permeability. Using computational electromagnetics methods, these bounds are obtained through numerical simulations for composites of two types: (i) two-dimensional (2D) random checkerboard (two-phase) microstructures and (ii) analogous 3D random (two-phase) media. The simulation results demonstrate a scale-dependent trend of these bounds toward the properties of a representative volume element (RVE). This transition from SVE to RVE is described using a scaling function dependent on the mesoscale δ, the volume fraction v_{f}, and the property contrast k between two phases. The scaling function is calibrated through fitting the data obtained from extensive simulations (∼10000) conducted over the aforementioned parameter space. The RVE size of a given microstructure can be estimated down to within any desired accuracy using this scaling function as parametrized by the contrast and the volume fraction of two phases.

Does a Fractal Microstructure Require a Fractional Viscoelastic Model?

The question addressed by this paper is tackled through a continuum micromechanics model of a 2D random checkerboard, in which one phase is linear elastic and another linear viscoelastic of integer-order. The spatial homogeneity and ergodicity of the material statistics justify homogenization in the vein of the Hill–Mandel condition for viscoelastic media. Thus, uniform kinematic- or traction-controlled boundary conditions, applied to sufficiently large domains, provide macroscopic (RVE level) responses. With computational mechanics, this strategy is applied over the entire range of the relative content of both phases. Setting the volume fraction of either the elastic phase or the viscoelastic phase at the critical value (≃0.59) results in fractal patterns of site-percolation. Extensive simulations of boundary value problems show that, for a viscoelastic composite having such a fractal structure, the integer (not fractional) calculus model is adequate. In other words, the spatial randomness of the composite material—even in the fractal regime—is not necessarily the cause of the fractional order viscoelasticity.

Random checkerboard based homogenization for estimating effective thermal conductivity of fully saturated soils

This paper proposes homogenization scheme for estimating the effective thermal conductivity of fully saturated soils. This approach is based on the random checkerboard-like microstructure. Two modeling scales and two modeling approaches are distinguished and used, i.e. microscale and mesoscale and 1-step and 2-step homogenizations, respectively. The 2-step homogenization involves sequential averaging procedure, i.e. first, at microscale, a mineralogical composition of soil skeleton is considered and averaging process results in estimation of the skeleton effective thermal conductivity, and then, at mesoscale, a random spatial packing of solid skeleton and pores via random checkerboard microstructure is modeled and leads to evaluation of the soil overall thermal conductivity. The 1-step homogenization starts directly at the mesoscale and homogenization procedure yields evaluation of the overall soil thermal conductivity. At the mesoscale, the distinct nature of soil skeleton, as composed of soil separates, is considered and random variability of soil is modeled via enriched random checkerboard-like structure. Both approaches, i.e. 1-step and 2-step homogenizations, interrelate mineralogical composition with the soil texture characterized by the volume fractions of soil separates, i.e. sand, silt and clay. The probability density functions (PDFs) of thermal conductivity are assumed for each of the separates. The soil texture PDF of thermal conductivity is derived taking into consideration the aforementioned functions. Whenever the random checkerboard-like structure is used in averaging process, the Monte Carlo procedure is applied for estimation of homogenized thermal conductivity. Finally, the proposed methodology is tested against the laboratory data from our measurements as well as those available from literature.

The Effective Modulus of Random Checkerboard Plates

We investigate the elastic effective modulus Eeff of two-dimensional checkerboard specimens in which square tiles are randomly assigned to one of two component phases. This is a model system for a wide class of multiphase polycrystalline materials such as granitic rocks and many ceramics. We study how the effective stiffness is affected by different characteristics of the specimen (size relative to the tiles, stiff fraction, and modulus contrast between the phases) and obtain analytical approximations to the probability distribution of Eeff as a function of these parameters. In particular, we examine the role of percolation of the soft and stiff phases, a phenomenon that is important in polycrystalline materials and composites with inclusions. In small specimens, we find that the onset of percolation causes significant discontinuities in the effective modulus, whereas in large specimens, the influence of percolation is smaller and gradual. The analysis is an extension of the elastic homogenization methodology of Dimas et al. (2015, “Random Bulk Properties of Heterogeneous Rectangular Blocks With Lognormal Young's Modulus: Effective Moduli,” ASME J. Appl. Mech., 82(1), p. 011003), which was devised for blocks with lognormal spatial variation of the modulus. Results are validated through Monte Carlo simulation. Compared with lognormal specimens with comparable first two moments, checkerboard plates have more variable effective modulus and are on average less compliant if there is prevalence of stiff tiles and more compliant if there is prevalence of soft tiles. These differences are linked to percolation.

Mesoscale bounds in viscoelasticity of random composites

Under consideration is the problem of size and response of the representative volume element (RVE) of spatially random linear viscoelastic materials. The model microstructure adopted here is the random checkerboard with one phase elastic and another viscoelastic, perfectly bonded everywhere. The method relies on the hierarchies of mesoscale bounds of relaxation moduli and creep compliances (Huet, 1995, 1999) obtained via solutions of two stochastic initial boundary value problems, respectively, under uniform kinematic and uniform stress boundary conditions. In general, the microscale viscoelasticity introduces larger discrepancy in the hierarchy of mesoscale bounds compared to elasticity, and this discrepancy grows as the time increases.

Electrical properties of random checkerboards at finite scales

Under investigation is the scale dependent electrical conductivity (and resistivity) of two-phase random checkerboards at arbitrary volume fractions and phase contrasts. Using variational principles, rigorous mesoscale bounds are obtained on the electrical properties at finite scales by imposing a boundary condition that is either uniform electric potential or uniform current density. We demonstrate the convergence of these bounds to the effective properties with increasing length scales. This convergence gives rise to the notion of a scalar-valued scaling function that accounts for the statistical nature of the mesoscale responses. A semi-analytical closed form solution for the scaling function is obtained as a function of phase contrast, volume fraction and the mesoscale. Finally, a material scaling diagram is constructed with which the convergence to the effective properties can be assessed for any random checkerboard with arbitrary phase contrast and volume fraction.

Randomized scalable checkerboard geometries: The electrostatic problem

Motivated by the spate of recent experimental and theoretical interest in effective dielectric properties of composite materials, we consider randomised scalable checkerboard geometries. Earlier calculations of their effective permittivity are subject to various numerical difficulties which give rise to significant errors, especially for high permittivity contrast ratio between the phases. We describe a numerical protocol for estimating the permittivity with attention to its application in the analysis of randomization of phases and scaling. We apply this protocol to different N × N checkerboard geometries and provide explicit calculations of the effective permittivity and field average. We also discuss, in detail, the influence that topological defects, i.e., phase contacts at corners, have on the effective dielectric properties. The associated statistics of the permittivity for the half-filling case is Gaussian for small permittivity ratios. Our results support the suggestion that the deviation function for the half filling case obeys a power-law scaling with N due to finite-size effects as we let the length scale of the stochastic inhomogeneities approach zero. We show how these results are generic for two-color random checkerboards. We argue that the effect of spatial fluctuations of permittivity is weak as N is increased. We also discuss the potential, the challenges, and the limitations of these calculations.

Scaling function in conductivity of planar random checkerboards

Under investigation is the finite-size scaling of the Fourier thermal conductivity in two-phase planar random checkerboard microstructures at 50% nominal volume fraction. Examples considered include Aluminum–Copper, Constantan–Lead, Stainless Steel–Gold, Inconel X-750–Aluminum, Titanium Dioxide–Gold, Carbon Steel–Diamond, Lead–Diamond, Boron–Diamond, Molybdenum–Test, Constantan–Diamond. Mesoscale bounds are obtained using an approach consistent with the Hill–Mandel homogenization condition. Extensive numerical simulations are conducted on 10 types of microstructures with the contrast (k) ranging from 1.54 to 100. The effects of mesoscale (δ) and phases’ contrast are evaluated and generic scaling laws are established quantitatively. This is accomplished using a non-dimensional scaling function derived by contracting the mesoscale conductivity and resistivity tensors. The scaling function very closely fits all the material combinations and is given by g(δ,k)=12(k-1/k)2exp[-0.53(δ-1)0.69]. As a verification of our procedure, it is observed that, with increasing domain size, the mesoscale conductivity tends to the exact theoretical result for macroscopic conductivity of random checkerboards: being the geometric mean of two phases. By choosing an appropriate functional form of the scaling function, a material scaling diagram is constructed with which one can rapidly estimate the size of representative volume element for a given contrast within acceptable accuracy.

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.

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 104 squares with conductivity ratios of up to 106. The achievable relative precision in these examples is on the order of 10−11.

Laminar and Orientation-Dependent Characteristics of Spatial Nonlinearities: Implications for the Computational Architecture of Visual Cortex

A full understanding of the computations performed in primary visual cortex is an important yet elusive goal. Receptive field models consisting of cascades of linear filters and static nonlinearities may be adequate to account for responses to simple stimuli such as gratings and random checkerboards, but their predictions of responses to complex stimuli such as natural scenes are only approximately correct. It is unclear whether these discrepancies are limited to quantitative inaccuracies that reflect well-recognized mechanisms such as response normalization, gain controls, and cross-orientation suppression or, alternatively, imply additional qualitative features of the underlying computations. To address this question, we examined responses of V1 and V2 neurons in the monkey and area 17 neurons in the cat to two-dimensional Hermite functions (TDHs). TDHs are intermediate in complexity between traditional analytic stimuli and natural scenes and have mathematical properties that facilitate their use to test candidate models. By exploiting these properties, along with the laminar organization of V1, we identify qualitative aspects of neural computations beyond those anticipated from the above-cited model framework. Specifically, we find that V1 neurons receive signals from orientation-selective mechanisms that are highly nonlinear: they are sensitive to phase correlations, not just spatial frequency content. That is, the behavior of V1 neurons departs from that of linear-nonlinear cascades with standard modulatory mechanisms in a qualitative manner: even relatively simple stimuli evoke responses that imply complex spatial nonlinearities. The presence of these findings in the input layers suggests that these nonlinearities act in a feedback fashion.

Effective dielectric tensor for electromagnetic wave propagation in random media

We derive exact strong-contrast expansions for the effective dielectric tensor εe of electromagnetic waves propagating in a two-phase composite random medium with isotropic components explicitly in terms of certain integrals over the n-point correlation functions of the medium. Our focus is the long-wavelength regime, i.e., when the wavelength is much larger than the scale of inhomogeneities in the medium. Lower-order truncations of these expansions lead to approximations for the effective dielectric constant that depend upon whether the medium is below or above the percolation threshold. In particular, we apply two- and three-point approximations for εe to a variety of different three-dimensional model microstructures, including dispersions of hard spheres, hard oriented spheroids, and fully penetrable spheres as well as Debye random media, the random checkerboard, and power-law-correlated materials. We demonstrate the importance of employing n-point correlation functions of order higher than two for high dielectric-phase-contrast ratio. We show that disorder in the microstructure results in an imaginary component of the effective dielectric tensor that is directly related to the coarseness of the composite, i.e., local-volume-fraction fluctuations for infinitely large windows. The source of this imaginary component is the attenuation of the coherent homogenized wave due to scattering. We also remark on whether there is such attenuation in the case of a two-phase medium with a quasiperiodic structure.

Smooth pursuit eye tracking over a structured background in first-episode schizophrenic patients

Whilst most laboratory smooth pursuit tasks are performed in the dark, in everyday life pursuit commonly occurs over a structured background. This background provides a powerful stimulus to the optokinetic reflex (OKR), inducing a background "drag" on pursuit eye movements. An inability to inhibit the influence of the OKR may be a contributing factor to the dysfunctional pursuit performance observed in many schizophrenic patients. Smooth pursuit performance was measured in 23 first-episode schizophrenic patients and 23 healthy controls matched for age and estimated IQ, both in the dark and over a structured background (a random checkerboard of black and white squares). Velocity gain was measured, as well as the number and size of corrective saccades (catch-up saccades) and intrusive saccades (anticipatory saccades and square wave jerks). Overall, schizophrenic patients had lower velocity gain and made more catch-up saccades than controls. The effect of the background was to lower velocity gain and increase the number of catch-up saccades to the same extent in schizophrenic patients and controls. There were no significant interactions between group and background effect. These results suggest that, although their overall level of performance was worse, the schizophrenic patients were as able as controls to inhibit the effect of the OKR. Since lesion studies show that inhibition of the OKR requires intact inferior parietal regions in man (Lawden et al., 1995), one hypothesis is that the parietal component of smooth pursuit may be intact in schizophrenia.

Effective conductivity, dielectric constant, and diffusion coefficient of digitized composite media via first-passage-time equations

We generalize the Brownian motion simulation method of Kim and Torquato [J. Appl. Phys. 68, 3892 (1990)] to compute the effective conductivity, dielectric constant and diffusion coefficient of digitized composite media. This is accomplished by first generalizing the first-passage-time equations to treat first-passage regions of arbitrary shape. We then develop the appropriate first-passage-time equations for digitized media: first-passage squares in two dimensions and first-passage cubes in three dimensions. A severe test case to prove the accuracy of the method is the two-phase periodic checkerboard in which conduction, for sufficiently large phase contrasts, is dominated by corners that join two conducting-phase pixels. Conventional numerical techniques (such as finite differences or elements) do not accurately capture the local fields here for reasonable grid resolution and hence lead to inaccurate estimates of the effective conductivity. By contrast, we show that our algorithm yields accurate estimates of the effective conductivity of the periodic checkerboard for widely different phase conductivities. Finally, we illustrate our method by computing the effective conductivity of the random checkerboard for a wide range of volume fractions and several phase contrast ratios. These results always lie within rigorous four-point bounds on the effective conductivity.

Exact result for the effective conductivity of a continuum percolation model.

A random two-dimensional checkerboard of squares of conductivities 1 and \ensuremath{\delta} in proportions p and 1-p is considered. Classical duality implies that the effective conductivity obeys ${\mathrm{\ensuremath{\sigma}}}^{\mathrm{*}}$= \ensuremath{\surd}\ensuremath{\delta} at p=1/2. It is rigorously found here that to leading order as \ensuremath{\delta}\ensuremath{\rightarrow}0, this exact result holds for all p in the interval (1-${\mathit{p}}_{\mathit{c}}$,${\mathit{p}}_{\mathit{c}}$), where ${\mathit{p}}_{\mathit{c}}$\ensuremath{\approxeq}0.59 is the site percolation probability, not just at p=1/2. In particular, ${\mathrm{\ensuremath{\sigma}}}^{\mathrm{*}}$(p,\ensuremath{\delta})= \ensuremath{\surd}\ensuremath{\delta} +O(\ensuremath{\delta}), as \ensuremath{\delta}\ensuremath{\rightarrow}0, which is argued to hold for complex \ensuremath{\delta} as well. The analysis is based on the identification of a ``symmetric'' backbone, which is statistically invariant under interchange of the components for any p\ensuremath{\in}(1-${\mathit{p}}_{\mathit{c}}$,${\mathit{p}}_{\mathit{c}}$), 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.

Scaling law for conduction in partially connected systems

The electrical transport properties of systems of conducting particles embedded in an insulator are considered. For low volume fractions of the particles, the conducting matrix may only be “partially” connected, as particles may only touch at corners or edges. As a model where these connectedness questions can be precisely formulated, we consider a random checkerboard in dimensions d = 2 and 3, where the squares in d = 2 or cubes in d = 3 are randomly assigned the conductivities 1 with probability p or 0 < δ ⪢ 1 with probability 1 − p. To analyze connectedness, we introduce a new parameter, d m, called the minimal dimension, which measures connectedness of the conducting matrix via the dimension of the dominant contacts between particles. Based on analysis of the checkerboards, we propose a general scaling law for the effective conductivity σ ∗ as δ → 0, namely σ ∗∼δ q, where q = 1 2 (d − d m for 0 ⩽ d − d m ⩽ 2 and q =1 for d − d m ⩾ 2. The applicability of this law to situations where d m is non-integral, such as the checkerboards at criticality, is discussed in detail.

A new class of edge-preserving smoothing filters

A new class of edge-preserving noise-cleaning filters are introduced which use both spatial and nearest-neighbor constraints on image pixels to smooth an image. They are simple, fast and good at preserving edges and thin structural details in images. Edges and corners of varying contrasts that may be present in real images have been simulated in synthesized random checkerboards. The performance of the new smoothing filters on the blurred and noisy checkerboards has been quantitatively compared with that of four other filters.

The distribution of human motion detector properties in the monocular visual field

The detection of coherent movement in stroboscopically (100 Hz) displayed moving random checkerboard (“Julesz-”) patterns is studied psychophysically for eccentricities up to 48° in the temporal visual field. Starting from the assumption that the studied visual subsystem consists of ensembles of ‘bilocal’ movement detectors (“Reichardt-detectors”), the parameters of these elementary detectors are deduced from the experimental results. This leads to the following interesting insights into the functional architecture of the system. At any eccentricity there is a critical velocity value V c (near the center of the range of detectable velocities) at which both the spans and the delays reach their minimum value. Thus V c can be defined as the ratio of the minimum span to the minimum delay values. At velocities below V c the spans are constant and the delays are inversely proportional to V. At velocities above V c the delays are constant and the spans increase proportional to V. The critical velocity V c at any given eccentricity equals N times V co , where V co , is the critical velocity for foveal vision and N an eccentricity scaling factor. ( N is the inverse normalized “cortical magnification factor”). Thus there is a complete structural invariance in terms of eccentricity-scaled units. Given the eccentricity scaling factor, the determination of two subject dependent constants of foveal vision, the minimum span and minimum delay, suffices to predict the main properties of the motion detection system at any eccentricity.

Direction-selective units in the rabbit's nucleus of the optic tract

A class of direction-selective (DS) units, histologically localized within the nucleus of the optic tract (NOT) was isolated in the rabbit's pretectum. These units typically had a maintained discharge of 25–50 action potentials/sec and large receptive fields (up to 40 × 150°) in the visual streak area of the contralateral eye. They were excited by a visual pattern moving in one direction and inhibited by motion in the opposite direction. The reactions were sustained. Excitatory and inhibitory acceptance angles were each 180°. Most units were excited by anterior motion of the stimulus and reacted to a wide range of velocities (0.01–20°/sec). Random checkerboard patterns (elements 0.8°), grids of black and white stripes (1, 2 and 4° wide) and single black and white edges were all effective, with a decreasing response magnitude in roughly this order. A stimulus area of 2 × 2° was already effective; response increased with area and was maximal for 15 × 15° and larger stimuli. Latency for visual stimulation was 60±10 (S.D.) msec, for electrical stimulation of the chiasm 2.2 ± 0.3 (S.D.) msec. Synaptic latency and presynaptic conduction velocity were estimated at 0.7 msec and 13 m/sec, respectively. A strong convergence of retinal DS fibers upon NOT units is postulated. Since most properties of NOT units are compatible with those of optokinetic nystagmus, and electrical stimulation of the NOT elicits vigorous nystagmus, these data suggest that these cells are the essential afferent link in the optokinetic reflex arc.

Variance Measurements by a Coherent Optical Method, with Application to Image Processing

A procedure for measuring the point-to-point local variance of a spatial random scene is discussed. The relations that are required between the second-order statistics of the random process and the parameters of the optical configuration to furnish accurate and stable variance estimates are established. The procedure is experimentally verified by using calibrated samples of the random checkerboard and the overlapping circular grain models. A modification of the procedure enables the generation of spatial masks that are useful in automatic image processing. The spatial masks are used as binary spatial gates to isolate certain classes of objects within a scene and reject all others. Typical masks are synthesized and their applications illustrated.