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Efficient finite element strategies for solving the radiative transfer equation

Abstract : The discrete ordinate method coupled with the finite element method is often used for numerically solving the radiative transfer equation. The main goal of this thesis is to improve upon such numerical technique. Instead of using standard finite elements, this thesis reformulates the radiative transfer equation using vectorial finite elements. In comparison to standard finite elements, this reformulation yields faster timings for the linear system assemblies, as well as for the solution phase when solving scattering media problems. Preconditioned Krylov subspace methods like the GMRES and the BiCGSTAB are employed for solving the linear systems arising from the proposed vectorial finite element discretization. The developed methods are validated against benchmark problems available in literature. In addition, the method of manufactured solutions is used for verifying the proposed method. For solving large problems of radiation on parallel computers, the vectorial finite element method is parallelized using domain decomposition and angular decomposition approaches. The proposed parallel methods possess quasi-linear scaling capabilities on a large number of processes. The developed parallel solvers are used to perform large scale numerical simulations with billions of unknowns. Overall it is shown how to perform complex numerical simulations of radiation at scales that were previously unattainable by standard radiative transfer equation solvers.
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Mohd Afeef Badri. Efficient finite element strategies for solving the radiative transfer equation. Thermics [physics.class-ph]. Université de Nantes, 2018. English. ⟨tel-02520959⟩

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