A nonlinear, 3D fluid-structure interaction problem driven by the time-dependent dynamic pressure data: a constructive existence proof

Abstract

We study a 3D fluid-structure interaction (FSI) problem between an incompressible, viscous fluid modeled by the Navier-Stokes equations, and the motion of an elastic structure, modeled by the linearly elastic cylindrical Koiter shell equations, allowing structure displacements that are not necessarily radially symmetric. The problem is set on a cylindrical domain in 3D, and is driven by the time-dependent inlet and outlet dynamic pressure data. The coupling between the fluid and the structure is fully nonlinear (2-way coupling), giving rise to a nonlinear, moving-boundary problem in 3D. We prove the existence of a weak solution to this 3D FSI problem by using an operator splitting approach in combination with the Arbitrary Lagrangian Eulerian mapping, which satisfies a geometric conservation law property. We effectively prove that the resulting computational scheme converges to a weak solution of the full, nonlinear 3D FSI problem.

Existence of a weak solution to a fluid–elastic structure interaction problem with the Navier slip boundary condition

“…This is obtained by considering (43) and by using formula (42) to express J η (∇ η · w η ) = ∂ t J η . Furthermore, in the third and fourth integral we replaced J n with J n+1 for higher accuracy.…”

On the convergence of fixed point iterations for the moving geometry in a fluid-structure interaction problem

“…which proves (25). Now, for the difference of the viscous terms on the right-hand side of (24) 22 = 0 (see (19)) and Lemma 3.1, one has…”

“…compare also (19) for h 1 ≡ h, h 2 ≡ 1. Replacing v(x, t) by u(y, t) we rewrite the Piola transformation as v P (y, t) = JJ −1 (y, t)u(y, t) = R(y, t)u(y, t),…”

Existence of a weak solution to the fluid-structure interaction problem in 3D

“…Since our space regularity on the fixed domain for u is less than H 1 , this limit had to be passed on the moving domain. It was proved in [23,24] that: ∇η ∆t,k u ∆t,k → ∇ η u, weakly in L 2 (0, T ; L 2 (Ω)), as k → ∞. Now, by using the weak and strong convergences obtained in the section 4, we are going to prove the convergences for all the terms I i 1.…”

“…In [23 -27] Muha and Čanić obtained the existence of weak solutions to several FSI problems by using the time discretization via operator splitting method. In [26] they studied the interaction in 2D case, and in [23,27] the 3D cylindrical case where the structure is described by linear and nonlinear Koiter shell equations, respectively. In [24], they observed a multi-layer structure of blood vessel in 2D, and in [25], they studied the 2D model with Navier-slip condition on the fluid-structure interface, where some additional difficulties due to the loss of the trace regularity of unknowns were successfully tackled.…”