Chinese Annals of Mathematics, Series B

, Volume 39, Issue 2, pp 213–232 | Cite as

An Energy Stable Monolithic Eulerian Fluid-Structure Numerical Scheme

  • Olivier Pironneau


The conservation laws of continuum mechanics, written in an Eulerian frame, do not distinguish fluids and solids, except in the expression of the stress tensors, usually with Newton’s hypothesis for the fluids and Helmholtz potentials of energy for hyperelastic solids. By taking the velocities as unknown monolithic methods for fluid structure interactions (FSI for short) are built. In this paper such a formulation is analysed when the solid is compressible and the fluid is incompressible. The idea is not new but the progress of mesh generators and numerical schemes like the Characteristics-Galerkin method render this approach feasible and reasonably robust. In this paper the method and its discretisation are presented, stability is discussed through an energy estimate. A numerical section discusses implementation issues and presents a few simple tests.


Fluid-Structure interactions Numerical method Energy stability Finite element method 

2000 MR Subject Classification

65M60 74F10 74S30 76D05 76M25 


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The author thanks Frédéric Hecht for very valuable discussions and comments.


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Copyright information

© Fudan University and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.UPMC (Paris VI) Laboratoire Jacques-Louis Lions Place JussieuSorbonne UniversitéParisFrance

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