TY - JOUR

T1 - Finite element approach to immersed boundary method with different fluid and solid densities

AU - Boffi, Daniele

AU - Cavallini, Nicola

AU - Gastaldi, Lucia

N1 - Generated from Scopus record by KAUST IRTS on 2020-05-05

PY - 2011/12/1

Y1 - 2011/12/1

N2 - The Immersed Boundary Method (IBM) has been designed by Peskin for the modeling and the numerical approximation of fluid-structure interaction problems, where flexible structures are immersed in a fluid. In this approach, the NavierStokes equations are considered everywhere and the presence of the structure is taken into account by means of a source term which depends on the unknown position of the structure. These equations are coupled with the condition that the structure moves at the same velocity of the underlying fluid. Recently, a finite element version of the IBM has been developed, which offers interesting features for both the analysis of the problem under consideration and the robustness and flexibility of the numerical scheme. Initially, we considered structure and fluid with the same density, as it often happens when dealing with biological tissues. Here we study the case of a structure which can have a density higher than that of the fluid. The higher density of the structure is taken into account as an excess of Lagrangian mass located along the structure, and can be dealt with in a variational way in the finite element approach. The numerical procedure to compute the solution is based on a semi-implicit scheme. In fluid-structure simulations, nonimplicit schemes often produce instabilities when the density of the structure is close to that of the fluid. This is not the case for the IBM approach. In fact, we show that the scheme enjoys the same stability properties as in the case of equal densities. © 2011 World Scientific Publishing Company.

AB - The Immersed Boundary Method (IBM) has been designed by Peskin for the modeling and the numerical approximation of fluid-structure interaction problems, where flexible structures are immersed in a fluid. In this approach, the NavierStokes equations are considered everywhere and the presence of the structure is taken into account by means of a source term which depends on the unknown position of the structure. These equations are coupled with the condition that the structure moves at the same velocity of the underlying fluid. Recently, a finite element version of the IBM has been developed, which offers interesting features for both the analysis of the problem under consideration and the robustness and flexibility of the numerical scheme. Initially, we considered structure and fluid with the same density, as it often happens when dealing with biological tissues. Here we study the case of a structure which can have a density higher than that of the fluid. The higher density of the structure is taken into account as an excess of Lagrangian mass located along the structure, and can be dealt with in a variational way in the finite element approach. The numerical procedure to compute the solution is based on a semi-implicit scheme. In fluid-structure simulations, nonimplicit schemes often produce instabilities when the density of the structure is close to that of the fluid. This is not the case for the IBM approach. In fact, we show that the scheme enjoys the same stability properties as in the case of equal densities. © 2011 World Scientific Publishing Company.

UR - https://www.worldscientific.com/doi/abs/10.1142/S0218202511005829

UR - http://www.scopus.com/inward/record.url?scp=84255191221&partnerID=8YFLogxK

U2 - 10.1142/S0218202511005829

DO - 10.1142/S0218202511005829

M3 - Article

VL - 21

SP - 2523

EP - 2550

JO - Mathematical Models and Methods in Applied Sciences

JF - Mathematical Models and Methods in Applied Sciences

SN - 0218-2025

IS - 12

ER -