A modular, operator‐splitting scheme for fluid–structure interaction problems with thick structures

Abstract

SUMMARY We present an operator‐splitting scheme for fluid–structure interaction (FSI) problems in hemodynamics, where the thickness of the structural wall is comparable to the radius of the cylindrical fluid domain. The equations of linear elasticity are used to model the structure, while the Navier–Stokes equations for an incompressible viscous fluid are used to model the fluid. The operator‐splitting scheme, based on the Lie splitting, separates the elastodynamics structure problem from a fluid problem in which structure inertia is included to achieve unconditional stability. We prove energy estimates associated with unconditional stability of this modular scheme for the full nonlinear FSI problem defined on a moving domain, without requiring any sub‐iterations within time steps. Two numerical examples are presented, showing excellent agreement with the results of monolithic schemes. First‐order convergence in time is shown numerically. Modularity, unconditional stability without temporal sub‐iterations, and simple implementation are the features that make this operator‐splitting scheme particularly appealing for multi‐physics problems involving FSI. Copyright © 2013 John Wiley & Sons, Ltd.

Stability and Convergence Analysis of the Extensions of the Kinematically Coupled Scheme for the Fluid-Structure Interaction

“…We assume that the thick structure elastodynamics is described by a ts (η, ξ) = 2µ s Ω S D(η) : D(ξ) + λ s Ω S (∇ · η)(∇ · ξ) + C as Ω S η · ξ, (7.2) where Ω S = (0, L) × (0, H), with L = 5 cm, H = 0.1 cm. The last term in the thick structure model is obtained from the axially symmetric model, and it represents a spring keeping the top and bottom boundaries connected [9]. The thick structure physical parameters are ρ s = 1.1 g/cm 3 , µ s = 2.586 · 10 5 dyne/cm 2 , λ s = 2.328 · 10 6 dyne/cm 2 and C as = 4 · 10 6 dyne/cm 4 .…”

“…The generalized Robin-Neumann explicit coupling scheme for the fluid-thick structure interaction problem was analyzed in [24] where it was proved that it is convergent, with the order of convergence of O( ∆t √ h ). We consider the β -scheme for the fluid-thick structure interaction problem presented in [9]. A basic stability estimate for the case β = 0 is derived in [9] where convergence of the β -scheme was proved numerically.…”

“…We denote by Ω h F , Ω h S , the strip in the fluid and the structure domain, respectively, that consists of all the elements that have at least one vertex on the interface. The width of Ω h F and Ω h S is of order h. The partitioned numerical scheme for the interaction between a fluid and a thick structure presented in [9], based on the kinematically coupled scheme, reads as follows…”

Coupling schemes for the FSI forward prediction challenge: Comparative study and validation

“…Computational complexity can be significantly reduced via an explicit–implicit discretization of the kinematic–dynamic interface coupling. These approaches, often referred to in the literature as semi‐implicit coupling schemes, use a fractional step time discretization in either the fluid (see, e.g., ) or the solid (see, e.g., ) subsystems. The explicit part of the coupling reduces computational complexity, whereas the implicit one guarantees stability.…”

A simplified fluid–structure model for arterial flow. Application to retinal hemodynamics

“…This is, for example, the case for valve simulation [1, 2,3,4] or in the aorta [5,6,7]. It is well-known that these simulations are very demanding, and in spite of the progress achieved in recent years ( [8,9,10] to name but a few), they remain challenging and the subject of active research.…”