no. 3
Eulerian formulation and level set models for incompressible fluid-structure interaction
ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 3, pp. 471-492.

This paper is devoted to eulerian models for incompressible fluid-structure systems. These models are primarily derived for computational purposes as they allow to simulate in a rather straightforward way complex 3D systems. We first analyze the level set model of immersed membranes proposed in [Cottet and Maitre, Math. Models Methods Appl. Sci. 16 (2006) 415-438]. We in particular show that this model can be interpreted as a generalization of so-called Korteweg fluids. We then extend this model to more generic fluid-structure systems. In this framework, assuming anisotropy, the membrane model appears as a formal limit system when the elastic body width vanishes. We finally provide some numerical experiments which illustrate this claim.

DOI : 10.1051/m2an:2008013
Classification : 76D05, 74B20, 74F10
Mots clés : fluid structure interaction, elastic membrane, eulerian method, level set method, Korteweg fluid, Navier-Stokes equations 
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     title = {Eulerian formulation and level set models for incompressible fluid-structure interaction},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
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Cottet, Georges-Henri; Maitre, Emmanuel; Milcent, Thomas. Eulerian formulation and level set models for incompressible fluid-structure interaction. ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 3, pp. 471-492. doi : 10.1051/m2an:2008013. http://archive.numdam.org/articles/10.1051/m2an:2008013/

[1] M. Boulakia, Modélisation et analyse mathématique de problèmes d'interaction fluide-structure. Ph.D. thesis, Université de Versailles, France (2004).

[2] L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. 35 (1982) 771. | MR | Zbl

[3] P.G. Ciarlet, Elasticité tridimensionnelle. Masson (1985). | MR | Zbl

[4] G.-H. Cottet and E. Maitre, A level-set formulation of immersed boundary methods for fluid-structure interaction problems. C. R. Acad. Sci. Paris, Ser. I 338 (2004) 581-586. | MR | Zbl

[5] G.-H. Cottet and E. Maitre, A level-set method for fluid-structure interactions with immersed surfaces. Math. Models Methods Appl. Sci. 16 (2006) 415-438. | MR | Zbl

[6] G.-H. Cottet, E. Maitre and T. Milcent, An Eulerian method for fluid-structure interaction with biophysical applications, in European Conference on Computational Fluid Dynamics, ECCOMAS CFD 2006, P. Wesseling, E. Oñate and J. Périaux Eds., TU Delft, The Netherlands (2006).

[7] G.A. Holzapfel, Nonlinear solid mechanics: a continuum approach for engineering. Wiley (2000). | MR | Zbl

[8] L. Lee and R.J. Leveque, An immersed interface method for incompressible Navier-Stokes equations. SIAM J. Sci. Comp. 25 (2003) 832-856. | MR

[9] E. Maitre, T. Milcent, G.-H. Cottet, A. Raoult and Y. Usson, Applications of level set methods in computational biophysics. Math. Comput. Model. (to appear).

[10] E. Maitre, C. Misbah and A. Raoult, Comparison between advected-field and level-set methods in the study of vesicle dynamics. (In preparation).

[11] J. Merodio and R.W. Ogden, Mechanical response of fiber-reinforced incompressible non-linearly elastic solids. Int. J. Nonlinear Mech. 40 (2005) 213-227.

[12] R.W. Ogden, Non-linear elastic deformations. Dover Publications (1984).

[13] R.W. Ogden, Nonlinear elasticity, anisoptropy, material staility and residual stresses in soft tissue, in Biomechanics of Soft Tissue in Cardiovascular Systems, G.A. Holzapfel and R.W. Ogden Eds., CISM Course and Lectures Series 441, Springer, Wien (2003) 65-108. | Zbl

[14] S. Osher and R.P. Fedkiw, Level set methods and Dynamic Implicit Surfaces. Springer (2003). | MR | Zbl

[15] C.S. Peskin, The immersed boundary method. Acta Numer. 11 (2002) 479-517. | MR | Zbl

[16] P. Smereka, The numerical approximation of a delta function with application to level-set methods. J. Comp. Phys. 211 (2003) 77-90. | MR | Zbl

[17] V.A. Solonikov, Estimates for solutions of nonstationary Navier-Stokes equations. J. Soviet Math. 8 (1977) 467-529. | Zbl

[18] M. Sy, D. Bresch, F. Guillén-González, J. Lemoine and M.A. Rodríguez-Bellido, Local strong solution for the incompressible Korteweg model. C. R. Acad. Sci. Paris, Ser. I 342 (2006) 169-174. | MR | Zbl

[19] H. Triebel, Interpolation theory, function spaces, differential operators. North-Holland (1978). | MR | Zbl

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