# Specular reflection treatment for the 3D radiative transfer equation solved with the discrete ordinates method

## Abstract

The contribution of this paper relies in the development of numerical algorithms for the mathematical treatment of specular reflection on borders when dealing with the numerical solution of radiative transfer problems. The radiative transfer equation being integro-differential, the discrete ordinates method allows to write down a set of semi-discrete equations in which weights are to be calculated. The calculation of these weights is well known to be based on either a quadrature or on angular discretization, making the use of such method straightforward for the state equation. Also, the diffuse contribution of reflection on borders is usually well taken into account. However, the calculation of accurate partition ratio coefficients is much more tricky for the specular condition applied on arbitrary geometrical borders. This paper presents algorithms that calculate analytically partition ratio coefficients needed in numerical treatments. The developed algorithms, combined with a decentered finite element scheme, are validated with the help of comparisons with analytical solutions before being applied on complex geometries.

- Authors:

- Université de Nantes, LTN UMR CNRS 6607 (France)
- Sorbonne Universités, UPMC Université Paris 06, UMR 7598, inria de Paris, Laboratoire Jacques-Louis Lions, F-75005, Paris (France)

- Publication Date:

- OSTI Identifier:
- 22622276

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 334; Other Information: Copyright (c) 2017 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; ALGORITHMS; ANALYTICAL SOLUTION; COMPARATIVE EVALUATIONS; COMPUTER CALCULATIONS; DISCRETE ORDINATE METHOD; EQUATIONS; FINITE ELEMENT METHOD; PARTITION; QUADRATURES; RADIANT HEAT TRANSFER; REFLECTION

### Citation Formats

```
Le Hardy, D., Favennec, Y., E-mail: yann.favennec@univ-nantes.fr, Rousseau, B., and Hecht, F.
```*Specular reflection treatment for the 3D radiative transfer equation solved with the discrete ordinates method*. United States: N. p., 2017.
Web. doi:10.1016/J.JCP.2017.01.019.

```
Le Hardy, D., Favennec, Y., E-mail: yann.favennec@univ-nantes.fr, Rousseau, B., & Hecht, F.
```*Specular reflection treatment for the 3D radiative transfer equation solved with the discrete ordinates method*. United States. doi:10.1016/J.JCP.2017.01.019.

```
Le Hardy, D., Favennec, Y., E-mail: yann.favennec@univ-nantes.fr, Rousseau, B., and Hecht, F. Sat .
"Specular reflection treatment for the 3D radiative transfer equation solved with the discrete ordinates method". United States.
doi:10.1016/J.JCP.2017.01.019.
```

```
@article{osti_22622276,
```

title = {Specular reflection treatment for the 3D radiative transfer equation solved with the discrete ordinates method},

author = {Le Hardy, D. and Favennec, Y., E-mail: yann.favennec@univ-nantes.fr and Rousseau, B. and Hecht, F.},

abstractNote = {The contribution of this paper relies in the development of numerical algorithms for the mathematical treatment of specular reflection on borders when dealing with the numerical solution of radiative transfer problems. The radiative transfer equation being integro-differential, the discrete ordinates method allows to write down a set of semi-discrete equations in which weights are to be calculated. The calculation of these weights is well known to be based on either a quadrature or on angular discretization, making the use of such method straightforward for the state equation. Also, the diffuse contribution of reflection on borders is usually well taken into account. However, the calculation of accurate partition ratio coefficients is much more tricky for the specular condition applied on arbitrary geometrical borders. This paper presents algorithms that calculate analytically partition ratio coefficients needed in numerical treatments. The developed algorithms, combined with a decentered finite element scheme, are validated with the help of comparisons with analytical solutions before being applied on complex geometries.},

doi = {10.1016/J.JCP.2017.01.019},

journal = {Journal of Computational Physics},

number = ,

volume = 334,

place = {United States},

year = {Sat Apr 01 00:00:00 EDT 2017},

month = {Sat Apr 01 00:00:00 EDT 2017}

}