Benchmark solutions for transport in d-dimensional Markov binary mixtures

Abstract

Linear particle transport in stochastic media is key to such relevant applications as neutron diffusion in randomly mixed immiscible materials, light propagation through engineered optical materials, and inertial confinement fusion, only to name a few. We extend the pioneering work by Adams, Larsen and Pomraning Adams et al. (1989) (recently revisited by Brantley Brantley (2011)) by considering a series of benchmark configurations for mono-energetic and isotropic transport through Markov binary mixtures in dimension d. The stochastic media are generated by resorting to Poisson random tessellations in 1d slab, 2d extruded, and full 3d geometry. For each realization, particle transport is performed by resorting to the Monte Carlo simulation. The distributions of the transmission and reflection coefficients on the free surfaces of the geometry are subsequently estimated, and the average values over the ensemble of realizations are computed. Reference solutions for the benchmark have never been provided before for twoand three-dimensional Poisson tessellations, and the results presented in this paper might thus be useful in order to validate fast but approximated models for particle transport in Markov stochastic media, such as the celebrated Chord Length Sampling algorithm.

Monte Carlo particle transport in random media: The effects of mixing statistics

“…In such configurations, stochastic geometries with small Λ ∞ will induce low reflection probabilities and will further enhance the discrepancy with respect to the atomic mix case. This nontrivial behaviour, which stems from finite-size and interface effects dominating the transport process, has been previously observed for the benchmark configurations analysed in Larmier et al (2017) under similar conditions, i.e., small chunks of scattering material surrounded by an absorbing medium. The threshold behaviour of R at p > 0.7 might be subtly related to the percolation of the scattering material.…”

“…In this work, we have considered the effects of varying the stochastic tessellation model on the statistical properties of the resulting random media and on the transport-related physical observables, such as the reflection and the transmission probabilities. As such, this paper is a generalization of our previous findings (Larmier et al, 2017(Larmier et al, , 2016, and might be helpful for researchers interested in developing effective kernels for particle transport in disordered media. In order to single out the sensitivity of the simulation results to the various model parameters, we have proposed two benchmark configurations that are simple enough and yet retain the key physical ingredients.…”

“…1 1 0.6872 0.66667 0.5 2 1.3744 1.33333 0.1 10 6.872 6.66667 is assumed to be isotropic. Following Brantley (2011);Larmier et al (2017), the physical observables that we would like to determine are the ensemble-averaged reflection probability R on the face where the incident flux is imposed, the ensemble-averaged transmission probability T on the opposite face, and the ensembleaveraged scalar particle flux ϕ within the box. Observe that the flux ϕ integrated over the box has a clear physical interpretation: actually, the integral flux is equal to the average length V travelled by the particles within the geometry: see, e.g., the considerations in Blanco and Fournier (2003); Zoia et al (2012); Mazzolo et al (2014);de Mulatier et al (2014).…”

“…We will apply a normalized incident angular flux on the leakage surface at x = 0 (with zero interior sources). These specifications are inspired by our previous work Larmier et al (2017) on a d-dimensional generalization of the benchmark proposed by Adams, Larsen and Pomraning Adams et al (1989).…”

“…For this purpose, a number of benchmark problems for Markov mixing have been proposed in the literature so far Adams et al (1989); Zuchuat et al (1994); Brantley (2011); Brantley and Palmer (2009); Brantley (2009); Vasques (2005). In a previous work Larmier et al (2017), we have revisited the benchmark problem originally proposed by Adams, Larsen and Pomraning for transport in binary stochastic media with Markov mixing Adams et al (1989), and later extended in Brantley (2011); Brantley and Palmer (2009); Brantley (2009); Vasques (2005). In particular, while these authors had exclusively considered 1d slab or rod geometries, we have provided reference solutions obtained by Monte Carlo particle transport simulations through 2d extruded and 3d Markov tessellations, and discussed the effects of dimension on the physical observables.…”