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Implicit finite element methodology for the numerical modeling of incompressible two-fluid flows with moving hyperelastic interface

Aymen Laadhari
Applied Mathematics and Computation
Vol. 333, pp. 376-400, September 2018


This work, at the interface between the Applied Mathematics and Physics is connected about the numerical modelisation of biological vesicles, a pattern for the red blood cells. For this reason, the pattern of Canham and Helfrich is adopted to describe the behaviour of the vesicles. The numerical modelisation uses the Level Set method in finite element framework. A new algorithm of numerical resolution combining one technique of Lagrange multipliers with an automatic mesh adaptation ensures the accurate conservation of volumes and surfaces. Thus this algorithm enables to exceed an existing crucial restriction of the Level Set method, that’s to say, the wastes of mass usually noticed in this kind of problems. Moreover, the proprieties of convergence of the Level Set method are thus much more improved, as shown in many numerical tests. Those tests chiefly include elementary problems of advection, motions by mean curvature just as motions by spread of surface. Concerning the static equilibrum of the vesicles, a mechanical equilibrum equation (Euler- Lagrange equation) of a vesicle membrane under a generalized elastic bending energy is obtained and the approach is based on shape optimization tools. In dynamics, the motion of a vesicle under the effect of a shear flow is elaborated with high Reynolds numbers. The effect of confinement is respected, and the standard regimes of tank-treading and of tumbling motion are found again. Finally, for the first time, the effect of the inertia terms is elaborated and we show that beyond a critical value of Reynolds number the vesicle passes from a tumbling motion to a tank-treading motion.

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  author = {Aymen Laadhari},
  title = {Implicit finite element methodology for the numerical modeling of incompressible two-fluid flows with moving hyperelastic interface},
  journal = {Applied Mathematics and Computation},
  year = {2018},
  month = {September},
  pages = {376-400},
  volume = {333},
  number = {},
  keywords = {Fluid-structure interaction; Hyperelastic membrane; Incompressible flow; Newton; Monolithic; Finite element method}