# A Nonlinear Fluid-Structure Interaction Problem in Compliant Arteries Treated with Vascular Stents

A Nonlinear Fluid-Structure Interaction Problem in Compliant Arteries Treated with Vascular Stents We study a nonlinear fluid-structure interaction problem between an incompressible, viscous fluid in 3D and an elastic structure whose Lamé elastic parameters, thickness and density are all functions of space allowing jump discontinuities. This problem is motivated by studying the interaction between blood flow and arterial walls treated with vascular prostheses called stents. A stent is a metallic mesh-like tube used to prop the clogged arteries open. The Navier–Stokes equations for an incompressible, viscous fluid are used to model blood flow, and the cylindrical Koiter shell equations with discontinuous coefficients are used to model the elastic properties of arterial walls treated with stents. The fluid and structure are coupled via two coupling conditions evaluated at the moving fluid-structure interface. No assumption on axial symmetry is used in the model. We prove the existence of a weak solution to the underlying nonlinear 3D moving-boundary problem, and design a loosely-coupled partitioned scheme ( $$\beta$$ β -scheme) for its solution. The numerical scheme was motivated by the main steps in the constructive existence proof. The existence proof shows that the proposed numerical $$\beta$$ β -scheme converges to a weak solution of the nonlinear problem. This is the first convergence result for the proposed partitioned $$\beta$$ β -scheme. Several numerical examples are presented where different stent configurations are considered. The numerical fluid-structure interaction solutions clearly show that the presence of a stent induces wave reflections in arterial walls, and significant flow disturbances, especially near the proximal site of the stent. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# A Nonlinear Fluid-Structure Interaction Problem in Compliant Arteries Treated with Vascular Stents

, Volume 73 (3) – Jun 1, 2016
41 pages

/lp/springer_journal/a-nonlinear-fluid-structure-interaction-problem-in-compliant-arteries-0pNCQxpEt0
Publisher
Springer Journals
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-016-9343-7
Publisher site
See Article on Publisher Site

### Abstract

We study a nonlinear fluid-structure interaction problem between an incompressible, viscous fluid in 3D and an elastic structure whose Lamé elastic parameters, thickness and density are all functions of space allowing jump discontinuities. This problem is motivated by studying the interaction between blood flow and arterial walls treated with vascular prostheses called stents. A stent is a metallic mesh-like tube used to prop the clogged arteries open. The Navier–Stokes equations for an incompressible, viscous fluid are used to model blood flow, and the cylindrical Koiter shell equations with discontinuous coefficients are used to model the elastic properties of arterial walls treated with stents. The fluid and structure are coupled via two coupling conditions evaluated at the moving fluid-structure interface. No assumption on axial symmetry is used in the model. We prove the existence of a weak solution to the underlying nonlinear 3D moving-boundary problem, and design a loosely-coupled partitioned scheme ( $$\beta$$ β -scheme) for its solution. The numerical scheme was motivated by the main steps in the constructive existence proof. The existence proof shows that the proposed numerical $$\beta$$ β -scheme converges to a weak solution of the nonlinear problem. This is the first convergence result for the proposed partitioned $$\beta$$ β -scheme. Several numerical examples are presented where different stent configurations are considered. The numerical fluid-structure interaction solutions clearly show that the presence of a stent induces wave reflections in arterial walls, and significant flow disturbances, especially near the proximal site of the stent.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Jun 1, 2016

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