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Eulerian formulation and level set models for incompressible fluid-structure interaction

Published online by Cambridge University Press:  03 April 2008

Georges-Henri Cottet
Affiliation:
Laboratoire Jean Kuntzmann, Université de Grenoble and CNRS, BP 53, 8041 Grenoble Cedex 9, France. georges-henri.cottet@imap.fr
Emmanuel Maitre
Affiliation:
Laboratoire Jean Kuntzmann, Université de Grenoble and CNRS, BP 53, 8041 Grenoble Cedex 9, France. georges-henri.cottet@imap.fr
Thomas Milcent
Affiliation:
Laboratoire Jean Kuntzmann, Université de Grenoble and CNRS, BP 53, 8041 Grenoble Cedex 9, France. georges-henri.cottet@imap.fr
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Abstract

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.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2008

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References

M. Boulakia, Modélisation et analyse mathématique de problèmes d'interaction fluide-structure. Ph.D. thesis, Université de Versailles, France (2004).
Caffarelli, L., Kohn, R. and Nirenberg, L., Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. 35 (1982) 771. CrossRef
P.G. Ciarlet, Elasticité tridimensionnelle. Masson (1985).
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.
Cottet, G.-H. and Maitre, E., A level-set method for fluid-structure interactions with immersed surfaces. Math. Models Methods Appl. Sci. 16 (2006) 415438. CrossRef
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).
G.A. Holzapfel, Nonlinear solid mechanics: a continuum approach for engineering. Wiley (2000).
Lee, L. and Leveque, R.J., An immersed interface method for incompressible Navier-Stokes equations. SIAM J. Sci. Comp. 25 (2003) 832856. CrossRef
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).
E. Maitre, C. Misbah and A. Raoult, Comparison between advected-field and level-set methods in the study of vesicle dynamics. (In preparation).
Merodio, J. and Mechanical, R.W. Ogden response of fiber-reinforced incompressible non-linearly elastic solids. Int. J. Nonlinear Mech. 40 (2005) 213227. CrossRef
R.W. Ogden, Non-linear elastic deformations. Dover Publications (1984).
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.
S. Osher and R.P. Fedkiw, Level set methods and Dynamic Implicit Surfaces. Springer (2003).
Peskin, C.S., The immersed boundary method. Acta Numer. 11 (2002) 479517. CrossRef
Smereka, P., The numerical approximation of a delta function with application to level-set methods. J. Comp. Phys. 211 (2003) 7790. CrossRef
Solonikov, V.A., Estimates for solutions of nonstationary Navier-Stokes equations. J. Soviet Math. 8 (1977) 467529. CrossRef
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.
H. Triebel, Interpolation theory, function spaces, differential operators. North-Holland (1978).