We present a new model and a novel loosely coupled partitioned numerical scheme modeling fluid-structure interaction (FSI) in blood flow allowing non-zero longitudinal displacement. Arterial walls are modeled by a linearly viscoelastic, cylindrical Koiter shell model capturing both radial and longitudinal displacement. Fluid flow is modeled by the Navier-Stokes equations for an incompressible, viscous fluid. The two are fully coupled via kinematic and dynamic coupling conditions. Our numerical scheme is based on a new modified Lie operator splitting that decouples the fluid and structure sub-problems in a way that leads to a loosely coupled scheme which is unconditionally stable. This was achieved by a clever use of the kinematic coupling condition at the fluid and structure sub-problems, leading to an implicit coupling between the fluid and structure velocities. The proposed scheme is a modification of the recently introduced "kinematically coupled scheme" for which the newly proposed modified Lie splitting significantly increases the accuracy. The performance and accuracy of the scheme were studied on a couple of instructive examples including a comparison with a monolithic scheme. It was shown that the accuracy of our scheme was comparable to that of the monolithic scheme, while our scheme retains all the main ad-$ vantages of partitioned schemes, such as modularity, simple implementation, and low computational costs.

In this paper we derive and mathematically justify models of micropolar rods and plates from the equations of linearized micropolar elasticity. Derivation is based on the asymptotic techniques with respect to the small parameter being the thickness of the elastic body we consider. Justification of the models is obtained through the convergence result for the displacement and microrotation fields when the thickness tends to zero. The limiting microrotation is then related to the macrorotation of the cross-section (transversal segment) and the model is rewritten in terms of macroscopic unknowns. The obtained models are recognized as being either the Reissner-Mindlin plate or the Timoshenko beam type. Mathematics Subject Classification (2000)74K20 · 74K10 · 74A35

Abstract. Fluid-structure interaction describing wave propagation in arteries driven by the pulsatile blood flow is a complex problem. Whenever possible, simplified models are called for. One-dimensional models are typically used in arterial sections that can be approximated by the cylindrical geometry allowing axially symmetric flows. Although a good first approximation to the underlying problem, the one-dimensional model suffers from several drawbacks: the model is not closed (an ad hoc velocity profile needs to be prescribed to obtain a closed system) and the model equations are quasi-linear hyperbolic (oversimplifying the viscous fluid dissipation), typically producing shock wave solutions not observed in healthy humans. In this manuscript we derived a simple, closed reduced model that accounts for the viscous fluid dissipation to the leading order. The resulting fluid-structure interaction system is of hyperbolic-parabolic type. Arterial walls were modeled by a novel, linearly viscoelastic cylindrical Koiter shell model and the flow of blood by the incompressible, viscous Navier-Stokes equations. Kelvin-Voigt-type viscoelasticity was used to capture the hysteresis behavior observed in the measurements of the arterial stress-strain response. Using the a priori estimates obtained from an energy inequality, together with the asymptotic analysis and ideas from homogenization theory for porous media flows, we derived an effective model which is an 2 -approximation to the three-dimensional axially symmetric problem, where is the aspect ratio of the cylindrical arterial section. Our model shows two interesting features of the underlying problem: bending rigidity, often times neglected in the arterial wall models, plays a nonnegligible role in the 2 -approximation of the original problem, and the viscous fluid dissipation imparts long-term viscoelastic memory effects on the motion of the arterial walls. This does not, to the leading order, influence the hysteresis behavior of arterial walls. The resulting model, although two-dimensional, is in the form that allows the use of one-dimensional finite element method techniques producing fast numerical solutions. We devised a version of the Douglas-Rachford time-splitting algorithm to solve the underlying hyperbolic-parabolic problem. The results of the numerical simulations were compared with the experimental flow measurements performed at the Texas Heart Institute, and with the data corresponding to the hysteresis of the human femoral artery and the canine abdominal aorta. Excellent agreement was observed. 1. Introduction. The study of flow of a viscous incompressible fluid through a compliant tube is of interest to many applications. A major application is blood flow through human arteries. Understanding wave propagation in arterial walls, local hemodynamics, and temporal wall shear stress gradient is important in understanding the mechanisms leading to various complications in the cardiovascular function. Many clinical treatments can be studied in detail only if a r...

We present a mathematical model for a study of the mechanical properties of endovascular stents in their expanded state. The model is based on the theory of slender curved rods. Stent struts are modeled as linearly elastic curved rods that satisfy the kinematic and dynamic contact conditions at the "vertices" where the struts meet. This defines a stent as a mesh of curved rods. A weak formulation for the stent problem is defined and a finite element method for a numerical computation of its solution was developed. Numerical simulations showing the pressure-displacement (axial and radial) relationship for the entire stent are presented. From the numerical data and from the energy of the problem we deduced an "effective" pressure-displacement relationship of the law of Laplace-type for the mechanical behavior of stents, where the proportionality constant in the Laplace law was expressed explicitly in terms of the geometric and mechanic properties of a stent.

A one-dimensional model of a curved rod is derived from the three-dimensional linearized elasticity. No positivity assumption on the curvature of the central line of the curved rod is made. The model is obtained by taking the limit in the equilibrium equation of the three-dimensional elastic rod when the thickness of the rod goes to zero. The appropriate convergence result is proved.