# Monte Carlo chord length sampling for d-dimensional Markov binary mixtures

## Abstract

The Chord Length Sampling (CLS) algorithm is a powerful Monte Carlo method that models the effects of stochastic media on particle transport by generating on-the-fly the material interfaces seen by the random walkers during their trajectories. This annealed disorder approach, which formally consists of solving the approximate Levermore–Pomraning equations for linear particle transport, enables a considerable speed-up with respect to transport in quenched disorder, where ensemble-averaging of the Boltzmann equation with respect to all possible realizations is needed. However, CLS intrinsically neglects the correlations induced by the spatial disorder, so that the accuracy of the solutions obtained by using this algorithm must be carefully verified with respect to reference solutions based on quenched disorder realizations. When the disorder is described by Markov mixing statistics, such comparisons have been attempted so far only for one-dimensional geometries, of the rod or slab type. In this work we extend these results to Markov media in two-dimensional (extruded) and three-dimensional geometries, by revisiting the classical set of benchmark configurations originally proposed by Adams, Larsen and Pomraning and extended by Brantley. In particular, we examine the discrepancies between CLS and reference solutions for scalar particle flux and transmission/reflection coefficients as a function of the materialmore »

- Authors:

- Univ. Paris-Saclay, Gif-sur-Yvette (France). Den-Service d'Etudes des Reacteurs et de Mathematiques Appliquees (SERMA), CEA
- Oregon State Univ., Corvallis, OR (United States). School of Nuclear Science & Engineering
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE; Electricite de France (EDF)

- OSTI Identifier:
- 1418910

- Report Number(s):
- LLNL-JRNL-735817

Journal ID: ISSN 0022-4073; TRN: US1801312

- Grant/Contract Number:
- AC52-07NA27344

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of Quantitative Spectroscopy and Radiative Transfer

- Additional Journal Information:
- Journal Volume: 204; Journal Issue: C; Journal ID: ISSN 0022-4073

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 22 GENERAL STUDIES OF NUCLEAR REACTORS; 97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; Chord Length Sampling; Markov geometries; Benchmark; Monte Carlo; Tripoli-4®; Mercury

### Citation Formats

```
Larmier, Coline, Lam, Adam, Brantley, Patrick, Malvagi, Fausto, Palmer, Todd, and Zoia, Andrea.
```*Monte Carlo chord length sampling for d-dimensional Markov binary mixtures*. United States: N. p., 2017.
Web. doi:10.1016/j.jqsrt.2017.09.014.

```
Larmier, Coline, Lam, Adam, Brantley, Patrick, Malvagi, Fausto, Palmer, Todd, & Zoia, Andrea.
```*Monte Carlo chord length sampling for d-dimensional Markov binary mixtures*. United States. doi:10.1016/j.jqsrt.2017.09.014.

```
Larmier, Coline, Lam, Adam, Brantley, Patrick, Malvagi, Fausto, Palmer, Todd, and Zoia, Andrea. Wed .
"Monte Carlo chord length sampling for d-dimensional Markov binary mixtures". United States.
doi:10.1016/j.jqsrt.2017.09.014. https://www.osti.gov/servlets/purl/1418910.
```

```
@article{osti_1418910,
```

title = {Monte Carlo chord length sampling for d-dimensional Markov binary mixtures},

author = {Larmier, Coline and Lam, Adam and Brantley, Patrick and Malvagi, Fausto and Palmer, Todd and Zoia, Andrea},

abstractNote = {The Chord Length Sampling (CLS) algorithm is a powerful Monte Carlo method that models the effects of stochastic media on particle transport by generating on-the-fly the material interfaces seen by the random walkers during their trajectories. This annealed disorder approach, which formally consists of solving the approximate Levermore–Pomraning equations for linear particle transport, enables a considerable speed-up with respect to transport in quenched disorder, where ensemble-averaging of the Boltzmann equation with respect to all possible realizations is needed. However, CLS intrinsically neglects the correlations induced by the spatial disorder, so that the accuracy of the solutions obtained by using this algorithm must be carefully verified with respect to reference solutions based on quenched disorder realizations. When the disorder is described by Markov mixing statistics, such comparisons have been attempted so far only for one-dimensional geometries, of the rod or slab type. In this work we extend these results to Markov media in two-dimensional (extruded) and three-dimensional geometries, by revisiting the classical set of benchmark configurations originally proposed by Adams, Larsen and Pomraning and extended by Brantley. In particular, we examine the discrepancies between CLS and reference solutions for scalar particle flux and transmission/reflection coefficients as a function of the material properties of the benchmark specifications and of the system dimensionality.},

doi = {10.1016/j.jqsrt.2017.09.014},

journal = {Journal of Quantitative Spectroscopy and Radiative Transfer},

number = C,

volume = 204,

place = {United States},

year = {Wed Sep 27 00:00:00 EDT 2017},

month = {Wed Sep 27 00:00:00 EDT 2017}

}

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