Analysis of a 3D nonlinear, moving boundary problem describing fluid-mesh-shell interaction

Abstract

We consider a nonlinear, moving boundary, fluid-structure interaction problem between a time-dependent incompressible, viscous fluid flow, and an elastic structure composed of a cylindrical shell supported by a mesh of elastic rods. The fluid flow is modeled by the time-dependent Navier-Stokes equations in a three-dimensional cylindrical domain, while the lateral wall of the cylinder is modeled by the two-dimensional linearly elastic Koiter shell equations coupled to a one-dimensional system of conservation laws defined on a graph domain, describing a mesh of curved rods. The mesh-supported shell allows displacements in all three spatial directions. Two-way coupling based on kinematic and dynamic coupling conditions is assumed between the fluid and composite structure, and between the mesh of curved rods and Koiter shell. Problems of this type arise in many applications, including blood flow through arteries treated with vascular prostheses called stents. We prove the existence of a weak solution to this nonlinear, moving boundary problem by using the time discretization via a Lie operator splitting method combined with an Arbitrary Lagrangian-Eulerian approach, and a nontrivial extension of the Aubin-Lions-Simon compactness result to problems on moving domains.

Moving boundary problems

“…In most literature involving FSI with thin structures (except for the recent results in [23,106]), the lateral boundary is assumed to displace only in the vertical (normal, transverse) direction, rendering longitudinal displacement negligible. By using η to denote the vertical component of displacement, the fluid domain Ω η (t) can be described by…”

“…ALE mappings have been extensively used in numerical simulations of moving boundary problems; see, e.g., [41,42,131]. Recently, they have proven to be useful in mathematical analysis as well [23,102,103]. In numerics, one of the reasons for the introduction of ALE mappings is the calculation of the discretized time derivative ∂ t u since a finite difference approximation of the time derivative (e.g., (u n+1 − u n )/Δt) contains the functions u n+1 and u n which are defined on two different domains, one corresponding to the time t n+1 = (n + 1)Δt, and the other to t n = nΔt.…”

“…In addition to numerical solvers, the ALE approach has recently been used to study the existence of solutions to moving boundary problems by either mapping the entire fluid problem onto the fixed domain Ω via an ALE mapping and analyzing the problem there (as in [102][103][104]) or by mimicking the approach described above, used in numerical simulations, and working on the current domain Ω n (as in [23]). In the former case, an additional set of nonlinearities is introduced because the gradient operator ∇ in physical space is mapped into an operator ∇ η , defined on the fixed, reference domain.…”

“…These problems are particularly evident when elastic structures are thin (modeled by the reduced membrane or shell equations) and the structure model accounts for both transverse and tangential components of displacement, and the structure is interacting in three dimensions with the flow of an incompressible viscous fluid. In those cases the weak solution techniques based on finite energy spaces are often times insufficient to guarantee even the Lipschitz regularity of the fluid-structure interface; see [23]. This is one of the reasons why most literature on the existence of (weak) solutions to moving boundary problems involving thin elastic structures assumes only the transverse component of displacement to be different from zero [14,25,68,70,95,96,102,103].…”

“…This is one of the reasons why most literature on the existence of (weak) solutions to moving boundary problems involving thin elastic structures assumes only the transverse component of displacement to be different from zero [14,25,68,70,95,96,102,103]. Recently, a three-dimensional FSI problem allowing transverse and tangential displacements of a mesh-supported shell was studied in [23] where an existence of a weak solution in three dimensions was obtained under an extra assumption on the uniform Lipschitz property for the fluid-structure interface. FSI problems with elastic structures that are slightly more regular than the Koiter shell (such as, e.g., tripolar materials studied in [16,119]) do not suffer from this difficulty.…”