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This research is concerned with the self-adaptive numerical solution of the neutral particle radiation transport problem. Radiation transport is an extremely challenging computational problem since the governing equation is seven-dimensional (3 in space, 2 in direction, 1 in energy, and 1 in time) with a high degree of coupling between these variables. If not careful, this relatively large number of independent variables when discretized can potentially lead to sets of linear equations of intractable size. Though parallel computing has allowed the solution of very large problems, available computational resources will always be finite due to the fact that ever more sophisticated multiphysics models are being demanded by industry. There is thus the pressing requirement to optimize the discretizations so as to minimize the effort and maximize the accuracy. One way to achieve this goal is through adaptive phase-space refinement. Unfortunately, the quality of discretization (and its solution) is, in general, not known a priori; accurate error estimates can only be attained via the a posteriori error analysis. In particular, in the context of the finite element method, the a posteriori error analysis provides a rigorous error bound. The main difficulty in applying a well-established a posteriori error analysis and subsequent adaptive refinement in the context of radiation transport is the strong coupling between spatial and angular variables. This research attempts to address this issue within the context of the second-order, even-parity form of the transport equation discretized with the finite-element spherical harmonics method. The objective of this thesis is to develop a posteriori error analysis in a coupled space-angle framework and an efficient adaptive algorithm. Moreover, the mesh refinement strategy which is tuned for minimizing the error in the target engineering output has been developed by employing the dual argument of the problem. This numerical framework has been implemented in the general-purpose neutral particle code EVENT for assessment.

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L’objet de ce travail est de proposer un modèle adapté pour la simulation du remplissage de moules qui réponde au meilleur compromis entre temps de calcul et précision des résultats. La difficulté est double. Il faut prendre en compte le phénomène de remplissage qui est un problème complexe à frontières libres et les spécificités liées au Verre : viscosité fortement thermodépendante et température de fusion élevée qui nécessite de prendre en compte le rayonnement. Le Chapitre I est consacrée à la partie écoulement du Verre liquide. La bibliothèque numérique Aquilon/Thétis, adaptée pour traiter ce type de problèmes et les couplages thermique air/verre/parois, a été utilisée (Méthode V.O.F pour le suivi de l’interface, méthodes de type Lagrangien augmenté/Projection vectorielle pour le couplage Vitesse-Pression). Pour l’aspect radiatif, différentes approches sont proposées : conductivité radiative équivalente (Chapitre II), méthode explicite directe pour la validation (Chapitre III) et méthode d’harmoniques sphériques ou méthode PN (Chapitre IV). Dans le Chapitre V, la méthode PN retenue est validée dans des cas simples et est appliquée ensuite à des cas avec couplage convectif en géométries complexes et obstacles semi-transparents (1D, 2D et 3D, 2D axi-symétrique et milieu non gris). Une version P1 modifiée est présentée. Les résultats sont assez proches de ceux donnés par la méthode P3 avec des temps de calcul modestes. L’intérêt de ce modèle est qu’il est facilement intégrable dans des codes numériques existants : une seule équation différentielle du second ordre stationnaire à résoudre en 3D / The aim of this study is to propose an adapted model for the simulation of mould filling that must be a compromise solution between computational time and results accuracy. The double difficulty is to take into account the filling phenomenon that is a complex problem due to the presence of free boundaries and to the Glass specificities: viscosity that is highly thermal dependant and high melting temperature that requires taking into account radiation effects. Chapter I is devoted to the melting Glass flow. The numerical libraries Aquilon/Thétis, adapted for solving such type of problems and the thermal coupling between Air/Glass/Walls, has been used. (V.O.F method for front tracking, Augmented Lagrangian/Vector Projection methods for solving Pressure/Velocity coupling). For radiative aspect, different approaches are proposed: equivalent radiative conductivity (Chapter II), direct explicit method for validation (Chapter III) and spherical harmonics method or PN method (Chapter IV). In the Chapter V, the selected PN method is validated through simple cases and is then applied in other cases with convective coupling in complex geometries including semi-transparent inclusions (1D, 2D and 3D, 2D axi-symmetric and non grey medium). A P1 modified version is presented. The results are close to those given by P3 method but with reduced computational time. The main interest of this model is that it can be easily implemented in existing numerical codes: a single stationary second order partial differential equation to solve in 3D