Ad hoc angular discretization of the radiative transfer equation. (March 2019) - Favennec, Y.; Mathew, T.; Badri, M.A.; Jolivet, P.; Rousseau, B.; Lemonnier, D.; Coelho, P.J.
Highlights: For efficiently solving the radiative transfer equation with finite element methods, we develop a novel algorithm of angular mesh adaptation based on a posteriori calculation. Our mesh adaptation algorithm leads to ad hoc angular discretizations which are optimized in such a way that they suit the particular physics under consideration. The algorithm is drafted in such a way that it could be easily implemented using per-existing open-source mathematical libraries. Finally, by means of numerical testing we prove that ad hoc angular discretizations with lesser number of directions out performers an ordinary uniform discretization with high number of direction. Consequently, we are able to reduce the computational cost of a radiative transfer simulation, while maintaining the accuracy of the solution. novelties: We approximate the angular functional space using vertex-based piece-wise constant functions, while continuous piece-wise linear functions are chosen for the spatial functional space. We developed an angular adaptivity algorithm based on an a posteriori error estimation using vertex-based P0 Lagrange functions built on a triangulation of a unit sphere without any other predefined rule. The angular ad hoc discretization is performed on physical problems that include specular reflection on boundaries. Abstract: Solving the radiative transfer equation (RTE) with the finite element method requires both angular and spatial discretizations. We develop a novel algorithm of angular mesh discretization based on a posteriori calculations. The obtained angular discretizations are optimized in such a way that they suit the particular physics under consideration. Such ad hoc angular discretizations are implemented to increase the finite element solution efficiency. Using comparative numerical tests based on the solution accuracy, the solutions provided by the obtained ad hoc discretizations are proven to be better than the ones provided by the standard uniform angular discretizations. The algorithm is drafted in such a way that it could be easily implemented using pre-existing open-source mathematical libraries. This paper is an extended version of a paper presented at the CTRPM-VI conference [1]. As an extension to the two-dimensional version presented at the conference, a three-dimensional adaptive ad hoc angular discretization of the RTE is addressed here.