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An Energy Stable Monolithic Eulerian Fluid-Structure Numerical Scheme

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Abstract

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.

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Acknowledgements

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

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Authors and Affiliations

  1. UPMC (Paris VI) Laboratoire Jacques-Louis Lions Place Jussieu, Sorbonne Université, Boite 187, Paris, 75252, France

    Olivier Pironneau

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  1. Olivier Pironneau
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Correspondence to Olivier Pironneau.

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Dedicated to Philippe G. Ciarlet on the occasion of his 80th birthday

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Pironneau, O. An Energy Stable Monolithic Eulerian Fluid-Structure Numerical Scheme. Chin. Ann. Math. Ser. B 39, 213–232 (2018). https://doi.org/10.1007/s11401-018-1061-9

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  • DOI: https://doi.org/10.1007/s11401-018-1061-9

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