Binary Stochastic Mixture Research Articles

Publisher’s Note: “Benchmark simulations of radiative transfer in participating binary stochastic mixtures in two dimensions” [Matter Radiat. Extremes 9, 067802 (2024)

Publisher’s Note: “Benchmark simulations of radiative transfer in participating binary stochastic mixtures in two dimensions” [Matter Radiat. Extremes 9, 067802 (2024)

Benchmark simulations of radiative transfer in participating binary stochastic mixtures in two dimensions

We study radiative transfer in participating binary stochastic mixtures in two dimensions (2D) by developing an accurate and efficient simulation tool. For two different sets of physical parameters, 2D benchmark results are presented, and it is found that the influence of the stochastic mixture on radiative transfer is clearly parameter-dependent. Our results confirm that previous multidimensional results obtained in different studies are basically consistent, which is interpreted in terms of the relationship between the photon mean free path lp and the system size L. Nonlinear effects, including those due to scattering and radiation–material coupling, are also discussed. To further understand the particle size effect, we employ a dimensionless parameter lp/L, from which a critical particle size can be derived. On the basis of further 2D simulations, we find that an inhomogeneous mix is obtained for lp/L > 0.1. Furthermore, 2D material temperature distributions reveal that self-shielding and particle–particle shielding of radiation occur, and are enhanced when lp/L is increased. Our work is expected to provide benchmark results to verify proposed homogenized models and/or other codes for stochastic radiative transfer in realistic physical scenarios.

Cauchy formulas for linear transport in random media

In the context of linear transport in homogeneous non-stochastic media, the well-known Cauchy formulas express a remarkably elegant relation between the contributions to the angular flux within an arbitrary body due to particles starting from the body surface and those due to particles starting from the interior of the body. In this letter, we will extend these results to binary stochastic mixtures, a prototype class of random media that emerges in a broad spectrum of applications in physical and life sciences, ranging from hydrodynamical instabilities to tracer diffusion in biological tissues. In particular, for both quenched disorder and annealed disorder models we will establish generalized Cauchy-like formulas that appear to have a universal character and bear a close similarity to those previously derived for non-stochastic media.

A benchmark comparison of Monte Carlo particle transport algorithms for binary stochastic mixtures

We numerically investigate the accuracy of two Monte Carlo algorithms originally proposed by Zimmerman [1] and Zimmerman and Adams [2] for particle transport through binary stochastic mixtures. We assess the accuracy of these algorithms using a standard suite of planar geometry incident angular flux benchmark problems and a new suite of interior source benchmark problems. In addition to comparisons of the ensemble-averaged leakage values, we compare the ensemble-averaged material scalar flux distributions. Both Monte Carlo transport algorithms robustly produce physically realistic scalar flux distributions for the benchmark transport problems examined. The base Monte Carlo algorithm reproduces the standard Levermore–Pomraning model [3,4] results. The improved Monte Carlo algorithm generally produces significantly more accurate leakage values and also significantly more accurate material scalar flux distributions. We also present deterministic atomic mix solutions of the benchmark problems for comparison with the benchmark and the Monte Carlo solutions. Both Monte Carlo algorithms are generally significantly more accurate than the atomic mix approximation for the benchmark suites examined.

On the slowing down of charged particles in a binary stochastic mixture

A kinetic equation is addressed for the straight line slowing-down of charged particles, the geometrical domain consists of randomly distributed spherical grains of dense material imbedded in a light material. The dense material is assumed to be a Boolean medium (the sphere centers are sampled according to a Poisson random field). We focus on the fraction of particles $P$ which stop in the light medium. After setting some properties of the Boolean medium, we perform an asymptotic analysis in two extreme cases corresponding to grain radius very small and very large with respect to the stopping distance of the dense material. A fitted analytic formula is proposed for the quantity P and results of numerical simulations are presented in order to validate the proposed formula.

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.

A “two-grid” acceleration of binary stochastic mixture deterministic transport iterations in slab geometry

Particle transport in stochastic mixtures has been an area of active research during the last decade. The primary focus has been on the development of simple, efficient models that approximate the behavior of transport solutions which have been ensemble-averaged over a large number of realizations of the mixing statistics. These approximate models are often written as coupled systems of transport equations. The stochastic nature of this problem can be naturally incorporated into Monte Carlo transport codes, but there have also been efforts to solve these coupled equations deterministically. In this paper, we develop an efficient iterative technique for the deterministic solution of the coupled transport equations of the Levermore–Pomraning mix model. The basis for its development is the observation that these equations are similar in structure to traditional multigroup transport equations with upscattering. We provide the results of a Fourier analysis of this technique, and compare these with observed convergence rates from the implemented algorithm.

On the Linear Spectrum Mixing Model Over Discrete Random Media

The linear spectrum-mixing model for hyperspectral data analysis is evaluated in the context of the stochastic radiative transfer equations. The stochastic radiative transfer equations that treat media as binary stochastic mixtures are reformed and then solved within a discrete random layer over a rough interface. The layer constituents are randomly distributed within ellipsoidal spatial cells of various sizes. The solution of the stochastic radiative transfer equations is used to investigate the validity domain of the linear spectrum-mixing model. The solution is also used to estimate the deviations in the predictions of the linear spectrum-mixing model for the layer brightness temperature.

Application of Monte Carlo Chord-Length Sampling Algorithms to Transport through a Two-Dimensional Binary Stochastic Mixture

Monte Carlo algorithms are developed to calculate the ensemble-average particle leakage through the boundaries of a two-dimensional binary stochastic material. The mixture is specified within a rectangular area and consists of a fixed number of disks of constant radius randomly embedded in a matrix material. The algorithms are extensions of the proposal of Zimmerman et al., using chord-length sampling (CLS) to eliminate the need to explicitly model the geometry of the mixture. Two variations are considered. The first algorithm uses CLS for both material regions. The second algorithm employs limited CLS (LCLS), using only CLS in the matrix material. Ensemble-average leakage results are computed for a range of material interaction coefficients and compared against benchmark results for both accuracy and efficiency. Both algorithms are exact for purely absorbing materials and provide decreasing accuracy as scattering is increased in the matrix material. The LCLS algorithm shows a better accuracy than the CLS algorithm for all cases while maintaining an equivalent or better efficiency. Accuracy and efficiency problems with the CLS algorithm are due principally to assumptions made in determining the chord-length distribution within the disks.