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

Gangjoon Yoon, Chohong Min, Seick Kim

Research output: Contribution to journalArticle

1 Citation (Scopus)

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.

Original languageEnglish
Pages (from-to)727-758
Number of pages32
JournalJournal of Scientific Computing
Volume76
Issue number2
DOIs
Publication statusPublished - 2018 Aug 1

Fingerprint

Fluid-solid Interaction
Hodge Decomposition
Decomposition Method
Poisson equation
Projection
Decomposition
Numerical methods
Fluids
Kinetic energy
Flow of fluids
Poisson's equation
Boundary conditions
Numerical Methods
Fluid
Robin Boundary Conditions
Numerical Instability
Nonlocal Boundary Conditions
Stability and Convergence
Incompressible Flow
Orthogonality

All Science Journal Classification (ASJC) codes

  • Software
  • Theoretical Computer Science
  • Engineering(all)
  • Computational Theory and Mathematics

Cite this

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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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A Stable and Convergent Hodge Decomposition Method for Fluid–Solid Interaction. / Yoon, Gangjoon; Min, Chohong; Kim, Seick.

In: Journal of Scientific Computing, Vol. 76, No. 2, 01.08.2018, p. 727-758.

Research output: Contribution to journalArticle

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