# Analysis of a linear 3D fluid–mesh–shell interaction problem

• Sunčica Čanić
• Marija Galić
• Matko Ljulj
• Boris Muha
• Josip Tambača
• Yifan Wang
Article

## Abstract

We study a linear fluid–structure interaction problem between an incompressible, viscous 3D fluid flow, a 2D linearly elastic Koiter shell, and an elastic 1D net of curved rods. This problem is motivated by studying fluid–structure interaction between blood flow through coronary arteries treated with metallic mesh-like devices called stents. The flow is assumed to be laminar, modeled by the time-dependent Stokes equations, and the structure displacement is assumed to be small, modeled by a system of linear Koiter shell equations allowing displacement in all three spatial directions. The fluid and the mesh-supported structure are coupled via the kinematic and dynamic coupling conditions describing continuity of velocity and balance of contact forces. The coupling conditions are evaluated along a linearized fluid–structure interface, which coincides with the fixed fluid domain boundary. No smallness on the structure velocity is assumed. We prove the existence of a weak solution to this linear fluid–composite structure interaction problem. This is the first result in the area of fluid–structure interaction that includes a 1D elastic mesh and takes into account structural displacements in all three spatial directions. Numerical simulations based on the finite element discretization of the coupled FSI problem are presented.

## Mathematics Subject Classification

74F10 76D 74K25 35D30 76M10

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## Authors and Affiliations

• Sunčica Čanić
• 1
• 2
• Marija Galić
• 3
• Matko Ljulj
• 3
• Boris Muha
• 3
• Josip Tambača
• 3
• Yifan Wang
• 2
1. 1.Department of MathematicsUniversity of HoustonHoustonUSA
2. 2.UC BerkeleyBerkeleyUSA
3. 3.Department of Mathematics, Faculty of ScienceUniversity of ZagrebZagrebCroatia