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

Title
A Stable and Convergent Hodge Decomposition Method for Fluid–Solid Interaction
Authors
Yoon G.Min C.Kim S.
Ewha Authors
민조홍
SCOPUS Author ID
민조홍scopus
Issue Date
2018
Journal Title
Journal of Scientific Computing
ISSN
0885-7474JCR Link
Citation
Journal of Scientific Computing vol. 76, no. 2, pp. 727 - 758
Keywords
Extended Hodge decompositionFluid–solid interactionHelmholtz–Hodge decompositionNumerical analysis
Publisher
Springer New York LLC
Indexed
SCIE; SCOPUS WOS scopus
Document Type
Article
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. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.
DOI
10.1007/s10915-017-0638-x
Appears in Collections:
자연과학대학 > 수학전공 > Journal papers
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