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Discrétisation du problème de couplage instationnaire des équations de Navier-Stokes avec l'équation de la chaleur

Abstract : The analytical solutions of the majority of partial differential equations are difficult to calculate, hence, numerical methods are employed. This work is divided into two parts. First, we study the time dependent Navier-Stokes equations coupled with the heat equation with nonlinear viscosity depending on the temperature known as the Boussinesq (buoyancy) model . Then, numerical experiments are presented to confirm the theoretical accuracy of the discretization using the Freefem++ software. In the first part, we propose first order numerical schemes based on the finite element method for the space discretization and the semi-implicit Euler method for the time discretization. In order to gain time and order of convergence, we study a second order scheme in time and space by using respectively the second order BDF method "Backward Differentiation Formula" and the finite element method. An optimal a priori error estimate is then derived for each numerical scheme. Finally, numerical experiments are presented to confirm the theoretical results. The second part is dedicated to the modeling of the thermal instability that appears from time to time while printing using a 3D printer. Our purpose is to build a reliable scheme for the 3D simulation. For this reason, we propose a trivial parallel algorithm based on the domain decomposition method. The numerical results show that this method is not efficient in terms of scalability. Therefore, it is important to use a one-level preconditioning method "ORAS". When using a large number of subdomains, the numerical test shows a slow convergence. In addition, we noticed that the iteration number depends on the physical model. A coarse space correction is required to obtain a better convergence and to be able to model in three dimensions.
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https://tel.archives-ouvertes.fr/tel-02935878
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Submitted on : Thursday, September 10, 2020 - 5:18:18 PM
Last modification on : Saturday, September 12, 2020 - 3:36:18 AM

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  • HAL Id : tel-02935878, version 1

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Rim Aldbaissy. Discrétisation du problème de couplage instationnaire des équations de Navier-Stokes avec l'équation de la chaleur. Analyse numérique [math.NA]. Sorbonne Université; Université Saint-Joseph (Beyrouth), 2019. Français. ⟨NNT : 2019SORUS013⟩. ⟨tel-02935878⟩

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