Binary Stochastic Media Research Articles

On the Chord Length Sampling in 1-D Binary Stochastic Media

When the chord length sampling method is applied to solve radiation transport problems in random media where inclusions are randomly distributed in a background region, inaccuracy occurs due to two major factors: memory effect and boundary effect. In this article, by applying chord length sampling to solve fixed source and eigenvalue problems in 1-D binary stochastic media, an investigation on how and why these two effects give rise to the inaccuracy in the final solutions is performed. The investigation is based on a series of radiation transport simulations for the calculation of reflection rate, flux distribution, and effective multiplication factor.

Chord length distributions between hard disks and spheres in regular, semi-regular, and quasi-random structures

In binary stochastic media in two- and three-dimensions consisting of randomly placed impenetrable disks or spheres, the chord lengths in the background material between disks and spheres closely follow exponential distributions if the disks and spheres occupy less than 10% of the medium. This work demonstrates that for regular spatial structures of disks and spheres, the tails of the chord length distributions (CLDs) follow power laws rather than exponentials. In dilute media, when the disks and spheres are widely spaced, the slope of the power law seems to be independent of the details of the structure. When approaching a close-packed arrangement, the exact placement of the spheres can make a significant difference. When regular structures are perturbed by small random displacements, the CLDs become power laws with steeper slopes. An example CLD from a quasi-random distribution of spheres in clusters shows a modified exponential distribution.

A Coupled Diffusion Synthetic Acceleration for Binary Stochastic Mixture Transport Iterations

The standard model for transport through binary stochastic media involves two coupled transport equations. Previous research has shown that several types of source iterations applied to the solution of these equations can converge arbitrarily slowly when one or both of the materials is optically thick and diffusive. In this work, we derive, analyze, and implement an acceleration scheme for binary stochastic mixture transport iterations. The equations are derived using the modified four-step method and take the form of discretized coupled diffusion equations. A Fourier analysis indicates that for a wide variety of physical problems and spatial mesh sizes, the scheme is rapidly convergent. Spectral radii measured during these accelerated iterations compare very well with Fourier analysis predictions.

Chord length distributions in binary stochastic media in two and three dimensions

Simple models for transport through stochastic media usually assume that the chord lengths of materials are distributed exponentially. Theory predicts that, in a medium consisting of disks/spheres that can interpenetrate, chord lengths in the background material (between the disks/spheres) should exactly follow an exponential. In a medium with impenetrable (non-overlapping) disks/spheres, the distribution is only approximately exponential. This paper demonstrates, through direct numerical simulations, that for randomly distributed disks in 2D and spheres in 3D, with distributions of radii, chord lengths in the background material (between the disks/spheres) are accurately described by exponentials over five orders of magnitude when the material is dilute. The chord lengths inside the disks and spheres are not exponentially distributed, but those distributions can be calculated. A scaling relationship between the mean chord lengths in the two materials is presented for an infinite medium. By knowing the mean properties of the disks/spheres in a medium, this relationship allows one to accurately describe the statistical properties of the background material. The stochastic simulations are validated by this infinite medium relationship. When the fraction of space occupied by the disks or spheres becomes large, the distributions are no longer accurately described by an exponential.

Benchmarks and models for time-dependent grey radiation transport with material temperature in binary stochastic media

We present benchmark calculations for radiation transport coupled to a material temperature equation in a slab geometry binary random media. The mixing statistics are taken to be homogeneous Markov statistics where the material chunk sizes are described by Poisson distribution functions. The material opacities are taken to be constant. Benchmark values for time evolution of the ensemble average values of material temperature energy density and radiation transmission are computed via a Monte Carlo-type method. These benchmarks are used as a basis for comparison with three other approximate methods of solution. One of these approximate methods is simple atomic mix which is seen to consistently over absorb resulting in lower steady-state radiation transmission and material temperature. The second approximate model is an adaptation of what is commonly called the Levermore–Pomraning model and which we refer to as the standard model. It is shown to consistently under absorb resulting in higher steady-state radiation transmission and material temperature. We show that recasting the temperature coupling as a type of effective scattering can be useful in formulating the third approximate model, an adaptation of a model due to Su and Pomraning which attempts to account for the effects of scattering in a stochastic context. We show this last adaptation shows consistent improvement over both the atomic mix and standard models.

Modification to a previous higher order model for particle transport in binary stochastic media

The problem of particle transport and radiative transfer through a stochastic mixture of two immiscible materials is considered. When the material mixing process is assumed to obey Markovian statistics, an ensemble average of this stochastic transport equation leads to two coupled transport equations containing four different ensemble-averaged unknowns. Modification to a previous higher order closure involving additional transport equations is suggested as a new closure. The closure philosophy is to demand that the closed set of equations reproduces certain exact limiting behaviors in rod geometry. To assess the accuracy of the proposed modification, numerical results predicted by this new higher order closure, for both the Markovian and the non-Markovian mixing statistics, are given and compared with exact benchmark results as well as with results from the standard closure available in the literature.