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A Stable and Convergent Hodge Decomposition Method for Fluid–Solid Interaction

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Abstract

Fluid–solid interaction has been a challenging subject due to their strong nonlinearity and multidisciplinary nature. Many of the numerical methods for solving FSI problems have struggled with non-convergence and numerical instability. In spite of comprehensive studies, it has still been a challenge to develop a method that guarantees both convergence and stability. Our discussion in this work is restricted to the interaction of viscous incompressible fluid flow and a rigid body. We take the monolithic approach by Gibou and Min (J Comput Phys 231:3245–3263, 2012) that results in an augmented Hodge projection. The projection updates not only the fluid vector field but also the solid velocities. We derive the equivalence between the augmented Hodge projection and the Poisson equation with non-local Robin boundary condition. We prove the existence, uniqueness, and regularity for the weak solution of the Poisson equation, through which the Hodge projection is shown to be unique and orthogonal. We also show the stability of the projection in the sense that the projection does not increase the total kinetic energy of the fluid or the solid. Finally, we discuss a numerical method as a discrete analogue to the Hodge projection, then we show that the unique decomposition and orthogonality also hold in the discrete setting. As one of our main results, we prove that the numerical solution is convergent with at least first-order accuracy. We carry out numerical experiments in two and three dimensions, which validate our analysis and arguments.

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Acknowledgements

The authors would like to express deep gratitude to the reviewer for his/her helpful comments. G. Yoon was supported by National Institute for Mathematical Sciences (NIMS), and C. Min was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2009-0093827) and the Korea government (MSIT) (2017R1A2B1006688).

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Authors and Affiliations

  1. National Institute for Mathematical Sciences, Daejeon, 34047, Korea

    Gangjoon Yoon

  2. Department of Mathematics, Ewha Woman’s University, Seoul, 03760, Korea

    Chohong Min

  3. Department of Mathematics, Yonsei University, Seoul, 03722, Korea

    Seick Kim

Authors
  1. Gangjoon Yoon
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  2. Chohong Min
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  3. Seick Kim
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Corresponding author

Correspondence to Chohong Min.

Additional information

C. Min was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2009-0093827). S. Kim is supported by NRF-20151009350.

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Yoon, G., Min, C. & Kim, S. A Stable and Convergent Hodge Decomposition Method for Fluid–Solid Interaction. J Sci Comput 76, 727–758 (2018). https://doi.org/10.1007/s10915-017-0638-x

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  • DOI: https://doi.org/10.1007/s10915-017-0638-x

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