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Schémas semi-implicites et de diffusion-redistanciation pour la dynamique des globules rouges

Abstract : In this work, we propose new semi-implicit schemes to improve the numerical simulations of the motion of an immersed red blood cell. We consider the levelset method where the interface is described as the 0 isoline of an auxiliary function and the fluid-structure coupling is done by adding a source term in the fluid equation.The idea of these semi-implicit scheme is to predict the position and the shape of the structure through a heat equation and to use this prediction to improve the accuracy of the source term in the fluid equation. This type of semi-implicit scheme has firstly been implemented in the case of a multiphase flow and a immersed elastic membrane and has shown better temporal stability than an explicit scheme, resulting in an improved global efficiency.In order extend this method to the case of a red blood cell, we propose an algorithm to compute the Willmore flow in dimenson 2 and 3. In the spirit of the diffusion generated motion methods, our method simulate a non linear four order flow by only solving heat equations. To ensure the conservation of the volume and area of the the vesicle, we add to the method a correction step that slightly moves the interface so that we recover the constraints.Combnation of these two steps allows to compute the behavior of a red blood cell left at rest. We validate this method by obtaining the convergence to an equilibrium shape in both 2D and 3D. Finaly we introduce a semi-implicit scheme in the case of a red blood cell that shows how we can use this method as a prediction in the complete coupling model.
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Submitted on : Thursday, October 31, 2019 - 2:39:14 PM
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Arnaud Sengers. Schémas semi-implicites et de diffusion-redistanciation pour la dynamique des globules rouges. Analyse fonctionnelle [math.FA]. Université Grenoble Alpes, 2019. Français. ⟨NNT : 2019GREAM032⟩. ⟨tel-02341602⟩

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