Random Radius Research Articles

A generalized spatial-temporal model of rainfall based on a clustered point process

The objective in this paper is to present and fit a relatively simple stochastic spatial-temporal model of rainfall in which the arrival times of rain cells occur in a clustered point process. In the x - y plane, rain cells are represented as discs; each disc having a random radius; the locations of the disc centres being given by a two-dimensional Poisson process. The intensity of each cell is a random variable that remains constant over the area of the disc and throughout the lifetime of the cell, the lifetime being an exponential random variable. The cells are randomly classified from 1 to n with different parameters for the different cell types, so that the random variables of an arbitrary cell, e. g. radius and intensity, are correlated. Multi-site second-order properties are derived and used to fit the model to hourly rainfall data taken from six sites in the Thames basin, UK.

A New Constitutive Model for Several Metals Under Arbitrary Temperature and Loading Conditions

A new theory of viscoplasticity is described that models yielding as a random phenomenon. A circle on the deviatoric stress plane represents the intensity of yielding with the radius equal to the random yielding microstress. This random model does not utilize a yield surface; yielding intensity is quantified by expected values defined in the deviatoric stress plane. The circle in the deviatoric stress plane with a random radius is a simple way to model multi-axial loading. Approximations for stress, strain energy density, and plastic strain energy density are used to improve the computational efficiency of parameter selection and to quantify the flow criterion. The exact state equations are derived, which can be manipulated to describe a wide variety of loading conditions for a broad temperature range. Reversed loading, stress relaxation, creep, and nonproportional loading are all natural properties of the model, which require little additional elaboration. Material properties were specified for five metals, three at room temperature and two over a wide temperature range.

Uniqueness of Unbounded Occupied and Vacant Components in Boolean Models

We consider Boolean models in $d$-dimensional Euclidean space. Each point of a stationary, ergodic point process is the center of a ball with random radius. In this way, the space is partitioned into an occupied and a vacant region. We are interested in the number of unbounded occupied or vacant components that can coexist. We show that under very general conditions on the distribution of the radius random variable, there can be at most one unbounded component of each type. In case the point process is Poisson, we obtain uniqueness of the unbounded components without imposing any condition at all. Although we do not prove the necessity of the conditions to prove uniqueness, we obtain examples of stationary, ergodic point processes where the unbounded components are not unique when the conditions are violated. Finally, we discuss more general random shapes than just balls which are centered at the points of the point process.

On grey scaling and continuum percolation

A theorem is given to establish the equality of the critical exponents β, v,γ in continuum and lattice percolation. A model is given to simulate electrical transport properties of continuum percolation systems in two and three dimensions where the holes have identical or random radii. The results for constant radii agree with those obtained by other methods. The results for random radii are new. The conductivity exponent is found and is shown to satisfy the rigorous bounds derived using variational methods by Halperin et al.

ON THE PROBLEM OF HEAT CONDUCTION FOR RANDOM DISPERSIONS OF SPHERES ALLOWED TO OVERLAP

We consider a random two-phase medium which represents a matrix containing an array of allowed to overlap spherical inclusions with random radii. A statistical theory of transport phenomena in the medium, on the example of heat propagation, is constructed by means of the functional (Volterra-Wiener) series approach. The functional series for the temperature is rendered virial in the sense that its truncation after the p-tuple term yields results for all multipoint correlation functions of the temperature field that are asymptotically correct to the order np, where n is the mean number of spheres per unit volume. The case p=2 is considered in detail and the needed kernels of the factorial series are found to the order n2. In this way not only the effective conductivity, but also the full statistical solution, i.e., all needed correlation functions, can be expressed in a closed form, correct to the said order.

A Geometric Model of Marbling in Beef Longissimus Dorsi

The Boolean random set model with isotropic, convex primary grain was investigated as a means of describing the spatial distribution of fat in beef longissimus dorsi. Estimates of contact distribution functions (CDF) for 10 images were compared to theoretical values. Results indicated that CDF estimates were not statistically different from what was expected given the assumption of the Boolean model was correct. Estimates of model parameters assuming circular primary grains with random radius tended to be different from those estimated without constraining their shape. The assumption of circular primary grains was concluded to be incorrect.

Instability of 2D random gravitational packings of identical hard discs

The authors consider 2D random packings of hard discs under gravity. They address the question of the nature of the typical packing of identical discs in the thermodynamic limit. They present numerical results on packings built by gravitational deposition of discs with random radii. Their results suggest that, in the limit of zero randomness: (i) the convergence of the system towards its thermodynamic limit becomes extremely slow; (ii) the limit packing fraction seems to increase drastically. They show in particular that if round-off errors are the only source of randomness of the system, the asymptotic regime cannot be reached because of the huge number of discs needed (at least 1016 discs using single precision floating-point arithmetic).

Probability-type strength criteria for a body having stochastically distributed disk-type cracks subject to axisymmetric loading

Equations have been derived and diagrams have been constructed for the limiting state in axisymmetric loading for a body containing internal diskoid cracks having random radius and orientation. The diagrams correspond to a given failure probabilitv and mean values for the failure stress. The most likely values for the failure stresses have been determined for several cases. A study is made of how the failure criterion is affected by the structural inhomogeneity, the volume, and the form (complexity) of the state of stress.

Diffusion of large molecules in porous media.

The results of the first computer simulations of transport of macromolecules in a disordered porous medium, represented by a network of capillary tubes with random radii, are reported. We develop a method for determining the macroscopic diffusivity D which, in principle, is exact. The effects of both convection and diffusion are taken into account. We find D/D/sub infinity//similar to/exp(-a/r/sub m/), independent of the morphology of the medium or the microscopic transport laws, where a is the effective size of the molecules, D/sub infinity/ the diffusivity in an unbounded solvent, and r/sub m/ the mean effective size of the pores, consistent with experimental data.

Scaling structure of viscous fingering.

When a fluid in a porous medium attempts to displace one which is more viscous, an initially flat boundary between them is unstable and fingers of the inviscid liquid penetrate the other. We model the medium numerically using a lattice of capillary tubes of random radii. Previous studies by one of the authors (P.R.K.) have already shown that the displacement is compact, but that the boundary between the two fluids is fractal. In this paper we study the distribution of velocities normal to the interface. We find that the distribution is consistent with the hypothesis that, for a system of size L, ${L}^{f(\ensuremath{\alpha})}$ sites have velocities scaling as ${L}^{\mathrm{\ensuremath{-}}\ensuremath{\alpha}}$. The scaling function f(\ensuremath{\alpha}) is measured and its variation with the viscosity ratio and randomness of the medium is found.

The fractal nature of viscous fingering in porous media

Viscous fingering in porous media is an instability which occurs when a less viscous fluid displaces a more viscous one. An interface between the fluids is unstable against small perturbations and gives rise to a fingered configuration. In the oil industry viscous fingering can be a serious problem when displacing viscous oil by a more mobile fluid because it leads to poor recovery of the hydrocarbon. Recent work suggests an analogy between viscous fingering at an infinite viscosity ratio and diffusion-limited aggregation (DLA) and hence that the fingers may be fractal with a fractal dimension of around 1.7 (in two dimensions). This leaves unanswered the question of the nature of the fingered patterns at a finite viscosity ratio. To answer this a network model of the porous medium has been used. The rock is modelled as a lattice of capillary tubes of random radius through which miscible displacement occurs. At a high viscosity ratio and in the presence of a large amount of disorder the model reproduces DLA fingering patterns. The results of this model provide evidence that at a finite viscosity ratio the displaced area is compact with a surface fractal dimension between 1 and a DLA result of 1.7 with increasing viscosity ratio.

An efficient sampling method for probability of failure calculation

A sampling method is developed for failure probability calculations which requires that the following assumptions are fullfilled: 1. (i) the limit state function, z, defining the combinations of values of the basic variables for which failure will occur, is known; 2. (ii) the basic variables are stochastically independent, and normally distributed, with mean value 0 and variance 1; 3. (iii) the reliability index, β, i.e. the distance from the origin in the k-dimensional basic variable space to the point (design point) on the failure surface which is closest to the origin, is known. The main idea is to restrict the sampling domain in the basic variable space, Ω, to the tail part of the joint distribution of the basic variables. The sampling domain is restricted to values outside the k-dimensional ß-sphere in Ω, since no failure values occur inside the ß-sphere. A random value is sampled by sampling a random direction in Ω and a random radius, R > ß. The R-sampling is performed by an appropriate variable transformation and use of the acceptance sampling technique. The method outlined is especially useful when the failure surface has a dangerous curvature, i.e. when the P f-value is significantly greater than the value calculated by first order reliability technique approximating the failure surface with the hyperplane tangential to the failure surface at the design point. A remarkable increase in efficiency is obtained compared to the simple Monte Carlo technique which includes the ß-sphere in the sampling domain. The number of simulations necessary in order to obtain the same accuracy as by the simple Monte Carlo technique is reduced by a factor of about 1 divided by the probability content outside the ß-sphere, ( 1 − Γ k(ß 2) ), where Γ k denotes the cumulative chisquare distribution with k degrees of freedom. Numerical examples are given for a k-dimensional hyperparaboloid with k = 4, 6, 8, 10. Reasonable results are obtained based on only 100 simulations. The corresponding number of simulations necessary to obtain the same accuracy by use of the simple Monte Carlo technique are in some of the cases more than one million.

Dispersion in flow through porous media—II. Two-phase flow

In this paper we extend our previous study (Sahimi et al., 1986, Chem. Engng Sci. 41, 2103–2122) of dispersion processes in porous media occupied by two fluid phases. We report results of Monte Carlo investigations of dispersion in two-phase flow through disordered porous media represented by square and simple cubic networks of pores of random radii. The percolation theory of Heiba et al. (1982, SPE 11015, 57th Annual Fall Meeting of the Soc. Petrol. Engrs) is used to determine the statistical distribution of phases in the porespace. One of the phases is assumed to be strongly wetting on the porewall in the presence of the other phase. A pore size distribution is chosen which yields through the percolation theory of Heiba et al. network relative permeabilities that are in agreement with the available experimental data. As in one-phase flow dispersion is diffusive in the cases simulated, i.e. it can be described by the convective-diffusion equation. Longitudinal dispersivity in a given phase rises greatly as the saturation of that phase approaches residual (i.e. its percolation threshold); transverse dispersivity also increases, but more slowly. As residual saturation of a phase is neared, the backbone of the subnetwork occupied by the phase becomes increasingly tortuous, with local mazes spotted along it that are highly effective dispersers. Dispersivities are found to be phase, saturation and saturation history dependent. Some limited Monte Carlo experiments with a residence time representation of the effects of deadend paths within a phase or reversible adsorption on the pore walls demonstrate that the approach developed can be extended to study the influence of such delay mechanisms on the dispersion process.

Microdeformation losses of single-mode fibers

In this paper, we consider the scattering losses of single-mode fibers that are caused by microdeformations such as microbends of the fiber axis and random fluctuations of the fiber core diameter. Since very little is known about the statistics of microdeformations of actual fibers, we assume that the autocorrelation functions of random bends and random core diameter fluctuations are Gaussian, characterized by the rms deviation and the correlation length of the random function. We consider single-mode fibers with step, parabolic, and triangular (linear) refractive-index profiles and reach the following conclusions: (1) Whereas for equal (large) mode radii the microbending losses of all three fiber types are the same, losses due to random core diameter fluctuations can be three times as high in step-index fibers as in triangular-index fibers. Since triangular-index fibers have sometimes been observed to have lower scattering losses than step-index fibers, one might conclude that, in these cases, excess losses may be caused by random radius fluctuations rather than by microbends. (2) Radial refractive-index ripples, which tend to be present in the deposited claddings of single-mode fibers, seem unlikely to be a major source of microdeformation losses. (3) The wavelength dependence of microdeformation losses depends strongly on the value of the correlation length of the Gaussian autocorrelation function of the fiber deformations. If the correlation length is of the same order of magnitude as the fiber radius, the losses are only slightly wavelength dependent. For very long correlation lengths the losses are very much smaller (for the same rms variation of the random functions), but they become strongly wavelength dependent, increasing sharply with increasing wavelength.

DISPERSION IN DISORDERED POROUS MEDIA

We report results of Monte Carlo investigations of dispersion in one- and two-phase flow through disordered porous media represented by square and simple cubic networks of pores of random radii. Dispersion results from the different flow paths and consequent different transit limes available to tracer particles crossing from one plane to another in a porous medium. Dispersion is found to be diffusive for the process simulated, i.e., a concentration front of solute particles can be described macroscopically by a convective diffusion equation. Dispersivity in the direction of mean flow, i.e., longitudinal dispersivity, is found to be an order of magnitude larger than dispersivity transverse to the direction of mean flow. In two-phase flow, longitudinal dispersivity in a given phase increases greatly as the saturation of that phase approaches its percolation threshold; transverse dispersivity also increases, but more slowly. As the percolation threshold is neared, the backbone of the sublatlice occupied by t...

A theory for near‐normal incidence microwave scattering from first‐year sea ice

A theory is presented to explain near‐normal incidence microwave scattering from first‐year sea ice. The theory is based upon the hypothesis of an abnormal number of relatively large (in terms of wavelength) flat areas on the surface which are all at the same elevation relative to mean sea level. These flat areas are assumed to be randomly located circular spots with random radii. The incoherent power scattered by these equiheight flat areas is a result of their random transverse location while the coherent scattered power is a consequence of their equal elevation. The incoherent power is found to vary as the average number of flat spots per unit area while the coherent power varies as the second moment of the number per unit area. Application of this theory to typical radar altimeter data produces parameter estimates which are in reasonable agreement with the available surface truth. Finally, it is conjectured that this model may also explain certain anomalous scattering data obtained from other types of terrain and planetary surfaces.

Estimation of stature from parts of bones--lower end of femur and upper end of radius.

A hundred and one right and 109 left dry, fully ossified, human femora have been studied to derive regression formulae for the establishment of the total length of the femur from the length of the lower end taken from the adductor tubercle. The formulae are statistically significant and their validity has been tested on 62 right and 59 left femora. A hundred and fifteen right and 106 left dry, fully ossified radii have been studied to derive regression formulae for establishment of the total length of the radius from the length of the upper end taken to the lower level of insertion of the biceps brachii muscle on the radial tuberosity. The validity of the formulae which are statistically significant has been tested on 51 right and 56 left dry and fully ossified random radii. All the regression formulae have a high degree of prediction, and are valuable in establishing the stature of an individual.

Parameter estimation in the Poisson field of discs

SUMMARY In the plane, a Poisson field of discs with random radii is considered. A method of estimating the Poisson intensity and the expectation and variance of the radii is proposed, and then investigated in detail for normally distributed radii. Foils consisting of a fission material and an isolation material are exposed to neutron radiation. Tracks caused by fission products in the isolation material are enlarged by a chemical etching process and made observable under an optical microscope as dark spots on the surface of the foil. Up to a clumping effect, the number of spots is proportional to the measured neutron flux, the shape and size of the spots depending mainly on the process of etching. In what follows, a mathematical model of the observed pattern of spots is given and a method for its statistical analysis proposed. Discs of random size are placed at random on the plane. More precisely, we suppose that the disc centres are distributed in a Poisson field of a constant intensity A and that the disc radii r are independent, identically distributed according to a probability density f(r)

Aspects of Necrophoric Behavior in the Red Imported Fire Ant, Solenopsis Invicta

AbstractRemoval of dead ants from the nest (necrophoric behavior) is released solely by contact chemical cues in the fire ant, Solenopsis invicta Buren. Exhaustively extracted corpses do not release necrophoric behavior, but the extracts do when applied to filter paper bits. The necrophoric releaser is absent at death but appears rapidly and reaches a plateau within about an hour. The rate of signal appearance is identical in heat and freeze killed workers, implying a non-enzymatic origin. There is no specialized caste or size of worker which carries out necrophoric labor. In the field, in the absence of slope, corpse-bearing workers head outward from the nest on random radii and drop their corpses at unpredictable distances, making refuse piles rare. There is a positive relationship between slope and the presence of refuse piles, and these are located downhill from the mound. When the headings of necrophoric workers were measured in circular arenas in the lab, the only potential orientational cue (tested : landmarks, light, 5°, 10°, 15° slope) which resulted in non-random distribution of headings was slope. The concentration of the headings was a direct function of the slope and seemed to plateau at about 15°. Corpse-bearing ants show stereotyped behavior upon encountering refuse piles and adding their burden to it. Chemical stimuli probably issuing from the feces in the refuse pile bring about the end of necrophoric behavior and maintain the refuse pile. These chemical cues, as also those initiating necrophoric behavior, must be contacted to be effective.

Scattering of plane wave by infinite semicylindrical bosses with random radii on a plane surface

This paper covers both theoretical and experimental phases of the subject of scattering of a plane wave incident on infinite semicylindrical bosses of random radii with periodic spacing. The probability density function of the radius of a semicylinder is assumed to be uniform, and the spacing between the centers of adjacent bosses is maintained constant. The spacing is so selected as to make the shadow effects negligible for the angles of incidence considered in this paper. The scalar form of the Helmholtz integral formulation was employed, although some of the inherent assumptions are invalid, owing to discontinuities in the surface. A comparison of the lobe structure (scintillation) of the experimental and theoretical angular acoustic-intensity scatter patterns, in particular, the main specular lobe, shows a reasonable agreement with the additional inference about the weakness of the assumption of the model material being rigid.