A Stable and Convergent Hodge Decomposition Method for Fluid-Solid Interaction
- 주제(키워드) Fluid-solid interaction , Helmholtz-Hodge decomposition , Extended Hodge decomposition , Numerical analysis
- 주제(기타) Mathematics, Applied
- 설명문(일반) [Yoon, Gangjoon] Natl Inst Math Sci, Daejeon 34047, South Korea; [Min, Chohong] Ewha Womans Univ, Dept Math, Seoul 03760, South Korea; [Kim, Seick] Yonsei Univ, Dept Math, Seoul 03722, South Korea
- 관리정보기술 faculty
- 등재 SCIE, SCOPUS
- 발행기관 SPRINGER/PLENUM PUBLISHERS
- 발행년도 2018
- URI http://www.dcollection.net/handler/ewha/000000151714
- 본문언어 영어
- Published As http://dx.doi.org/10.1007/s10915-017-0638-x
초록/요약
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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