TY - JOUR
T1 - A Stable and Convergent Hodge Decomposition Method for Fluid–Solid Interaction
AU - Yoon, Gangjoon
AU - Min, Chohong
AU - Kim, Seick
N1 - Funding Information:
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). 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.
Funding 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-20151 009350.
Funding Information:
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).
Publisher Copyright:
© 2018, Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2018/8/1
Y1 - 2018/8/1
N2 - 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.
AB - 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.
KW - Extended Hodge decomposition
KW - Fluid–solid interaction
KW - Helmholtz–Hodge decomposition
KW - Numerical analysis
UR - http://www.scopus.com/inward/record.url?scp=85040863335&partnerID=8YFLogxK
U2 - 10.1007/s10915-017-0638-x
DO - 10.1007/s10915-017-0638-x
M3 - Article
AN - SCOPUS:85040863335
VL - 76
SP - 727
EP - 758
JO - Journal of Scientific Computing
JF - Journal of Scientific Computing
SN - 0885-7474
IS - 2
ER -