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Fluid-structure interaction in blood flow capturing non-zero longitudinal structure displacement

Martina Bukac, Suncica Canic, Roland Glowinski, Josip Tambaca, Annalisa Quaini

arXiv:1207.0039v1math.NAcs.NA

TL;DR

Arterial-wall models had not captured both radial and longitudinal displacement, while classical loosely coupled blood-flow schemes can be unstable because of added-mass effects. The paper introduces a viscoelastic shell model and a modified loosely coupled splitting, whose accuracy is reported as comparable to a monolithic scheme.

  • Problem

    To the authors' knowledge, prior thin viscoelastic arterial-wall models had not captured both radial and longitudinal displacement, and classical loosely coupled schemes can be unstable in hemodynamics.

    Source: §1 · pp. 2, 3, 5

  • Method

    The authors model the arterial wall with a linearly viscoelastic Koiter shell and use a modified Lie splitting with implicit kinematic coupling for the fluid–structure problem.

    Source: §1 · p. 4, §6 · p. 43

  • Results

    The modified splitting significantly increases accuracy over the classical kinematically coupled scheme, making accuracy comparable to the monolithic scheme.

    Source: §1 · p. 4

  • Takeaways & Limitations

    The work is presented as a first step toward modeling multilayer arterial walls and their interaction with blood flow.

    Source: §6 · p. 44

    The coefficient-dependent choice of β that would provide the scheme's highest accuracy remains to be investigated.

    Source: §6 · p. 44

Abstract

from arXiv · show

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 advantages of partitioned schemes, such as modularity, simple implementation, and low computational costs.

1. Introduction

The paper introduces a viscoelastic arterial-wall model that captures both radial and longitudinal motion, and a loosely coupled scheme for solving its fluid–structure interaction with blood flow. Its modified splitting addresses added-mass difficulties and yields accuracy comparable to a monolithic scheme.

1. Introduction

  • Classical loosely coupled schemes can be intrinsically unstable in hemodynamics, while strongly coupled iterations that address energy mismatch carry high computational costs.Prior approaches also modify interface conditions or add stabilization to develop loosely coupled alternatives.Source: §1 · pp. 2–3
  • Implicit kinematic coupling places structure inertia and viscous effects in the fluid subproblem, while the remaining pressure explicitly loads the structure in the elastodynamics step.This modified Lie splitting increases accuracy over the classical kinematically coupled scheme and makes it comparable to the monolithic scheme.Source: §1 · pp. 3–4
  • The proposed arterial-wall model captures radial and longitudinal displacement, which measurements indicate can be comparable and, under adrenaline, longitudinal displacement can reach twice the radial magnitude.The model also includes wall viscoelasticity, identified as important in disease formation.Source: §1 · pp. 2, 5
  • The arterial-wall model is motivated by in vivo measurements indicating that longitudinal motion can be substantial, including a reported twofold radial displacement under adrenaline.The authors identify this work as the first, to their knowledge, to capture both displacement directions in a thin viscoelastic structure.Source: §1 · p. 5

2. The cylindrical linearly viscoelastic Koiter shell model

The shell model represents an axially symmetric, thin cylindrical arterial wall with axial and radial displacement. Its linearly viscoelastic Koiter formulation accounts for stretching, bending, and Kelvin–Voigt viscous response.

2. The cylindrical linearly viscoelastic Koiter shell model

  • The shell is assumed thin relative to its radius, homogeneous, linearly deforming, and axially symmetric, with axial and radial displacement components.The displacement is that of the shell's middle surface.Source: §2 · pp. 5–6
  • Koiter shell deformation comprises middle-surface stretching measured by metric change and flexure measured by curvature change.The weak formulation's membrane and flexural contributions are scaled by h/2 and h^3/24, respectively.Source: §2 · pp. 6, 8
  • The Kelvin–Voigt response combines stress proportional to strain with stress proportional to its time derivative.Elasticity and viscosity are represented by tensors A and B, with viscous counterparts to Young's modulus and Poisson's ratio.Source: §2 · pp. 6–7
  • The shell's axial and radial forcing components are surface force densities in the reference configuration, and the clamped model has prescribed boundary conditions.Source: §2 · p. 8

3. The fluid-structure interaction problem

The section formulates a pressure-driven fluid–structure interaction problem in a two-dimensional channel with a deformable wall and kinematic and dynamic coupling. It derives an energy equality for the coupled system, relating changes in kinetic, elastic, and viscous energies to inlet and outlet work.

3. The fluid-structure interaction problem

  • The fluid occupies the upper half of a two-dimensional channel, with a deformable lateral boundary and symmetry conditions at the axis.The boundary is described in Lagrangian coordinates by the wall displacement and in Eulerian coordinates by its height function.Source: §3 · pp. 8–9
  • The model uses Navier–Stokes equations for an incompressible, viscous, Newtonian fluid and a deformable wall governed by the Koiter shell equations.The fluid and structure are coupled through velocity continuity and balance of contact forces.Source: §3 · pp. 10–11

3.1. The energy of the coupled FSI problem

  • The energy derivation combines the structure and fluid momentum balances, using the coupling conditions to relate interface forces and velocities.The structure balance includes kinetic, membrane and flexural elastic, and viscous energy contributions.Source: §3.1 · pp. 12–13
  • The coupled energy equality balances changes in fluid and structure kinetic energy, structure elastic and viscous energy, and fluid viscous energy against work by inlet and outlet data.The interface contributions cancel when the fluid and structure energy relations are added.Source: §3.1 · pp. 13–14

4. The numerical scheme

The scheme extends the kinematically coupled approach to capture both radial and longitudinal shell motion while improving accuracy through a modified Lie splitting. It formulates the moving-domain problem in an ALE framework and advances it through three loosely coupled sub-problems.

4. The numerical scheme

  • The extended kinematically coupled scheme targets both radial and longitudinal shell displacement and introduces a modified Lie splitting to increase accuracy.The scheme retains the original approach's advantages, including avoiding sub-iterations for stability and allowing modular fluid and structure solvers.Source: §4 · pp. 14–15

4.1. The Lie scheme

  • The Lie scheme is first-order accurate under the stated finite-dimensional smoothness conditions and divides the problem into Stokes, fluid-advection, and structural-elastodynamics sub-problems.These sub-problems remain linked through the kinematic coupling condition and fluid pressure in the elastodynamics problem.Source: §4.1 · p. 16

4.2. The first-order system in ALE framework

  • The ALE mapping takes the fixed reference domain to the moving fluid domain and is defined as a harmonic extension of the boundary mapping.The mapping satisfies a Laplace equation in the reference domain, with prescribed boundary values.Source: §4.2 · p. 17
  • The first-order formulation uses the kinematic condition to express the viscous structure equations through the fluid-velocity trace on the interface.The authors state that this rewriting is used in the splitting algorithm.Source: §4.2 · pp. 18–19

4.3. Details of the operator-splitting scheme

  • The modified split assigns fractions of the normal stress to the viscous and elastic structure parts; β = 1 gave the highest accuracy in Examples 1 and 2.β = 0 corresponds to the splitting introduced in.Source: §4 · p. 15, §4.3 · pp. 19–20
  • Each time step solves a Stokes problem with viscous structure coupling, a fluid-and-ALE advection problem, then elastodynamics loaded by the remaining normal-stress part.The final structural step also uses the kinematic boundary condition to relate structure velocity to the fluid-velocity trace.Source: §4.3 · pp. 19–21
  • The pressure trace used to load the structure is justified under corner compatibility conditions that give the pressure sufficient regularity.For this problem, the authors state that pressure belongs to W 1,8/7(Ω) under those conditions, sufficient for a trace on Γ(tn).Source: §4.3 · pp. 22–23

5. Numerical results

The numerical results assess accuracy across radial-only, extended, and full Koiter-shell models and report a common-carotid simulation compared with experimental data. The simulation yields flow and wall-displacement values within reported physiological ranges and captures viscoelastic hysteresis.

5.3. The Common Carotid Artery (CCA) Example

  • The left common carotid artery simulation’s numerical results showed very good agreement with experimental data.The model considered a straight arterial segment, with geometric parameters taken from reported measurements.Source: §5 · p. 23, §5.3 · p. 39

5. Numerical results

  • For Examples 1 and 2, β = 1 gave the highest accuracy; unlike stability, accuracy varies with β.At β = 1, Step 3 loads the structure elastodynamics problem with the entire fluid-pressure contribution; the optimal β may depend on problem coefficients.Source: §5 · p. 24, Table 7 · p. 38, §5.2 · p. 39

5.1. Example 1: The benchmark problem with only radial displacement.

  • In the reported comparison, our method’s error was noticeably smaller than that of the classical kinematically coupled scheme and comparable to the monolithic scheme; all used △t = 10−4.Source: §5.1 · pp. 26, 30

5.1.1. Homogeneous Dirichlet vs. absorbing boundary conditions

  • Absorbing structural boundary conditions reduce reflected-wave formation, while homogeneous Dirichlet conditions produce a boundary layer near the inlet that decays exponentially away from the boundary.The solutions differ most near the inlet in the boundary-layer comparison; at later times, the reported displacement solutions differ throughout the domain.Source: §5.1.1 · pp. 30–31, Figure 12 · p. 32, Figure 13 · p. 32

5.1.2. Example 1b.

  • Example 1b extends the Formaggia benchmark with longitudinal displacement, whose magnitude is the same as the radial displacement despite the degenerate, uncoupled longitudinal equation.The longitudinal equation has no spatial derivatives, and the radial and longitudinal displacement equations have no coupling terms.Source: §5 · p. 23, §5.1.2 · pp. 32, 34, 35

5.2. Example 2.

  • Example 2 uses the full linearly viscoelastic Koiter shell model, including coupling between radial and longitudinal displacement, as a benchmark for FSI methods.The simulation omits the negligible fourth- and fifth-order derivative terms.Source: §5 · p. 23, §5.2 · pp. 35–36
  • In Example 2, longitudinal displacement was the same order of magnitude as radial displacement, and inlet flow-rate oscillations were attributed to small C1 and damped by increased viscoelasticity.The scheme with β = 1 also gained accuracy over the classical β = 0 scheme in this example.Source: Figure 17 · p. 36, §5.2 · pp. 37, 39, Figure 18 · p. 37

5.3. The Common Carotid Artery (CCA) Example

  • Simulated CCA blood velocity ranged from 22 to 97 cm/s, within expected values for the artery.The pressure drop was about 0.0673 mmHg per centimeter, associated with a local Reynolds number of about 1000 and maximum flow velocity around 100 cm/s.Source: §5.3 · p. 41, Figure 23 · p. 42
  • Simulated radial displacement reached about 6% of vessel radius, within the reported normal range of 3%–13%.Figure 23 shows the radial displacement over time.Source: §5.3 · p. 41, Figure 23 · p. 42
  • The CCA simulation produced total longitudinal displacement of 0.15 mm (ηz from −0.1 to 0.05 mm), within the experimental 0.052–0.302 mm range [80, 1, 3, 4].Figure 23 shows the longitudinal displacement over time.Source: §5.3 · pp. 41–42, Figure 23 · p. 42
  • The diameter-pressure loop showed viscoelastic hysteresis, with an EDR of 8.5%, comparable to the 7.8% reported for a young subject.The energy dissipation ratio quantifies the area inside the diameter-pressure loop relative to the combined areas inside and under it.Source: §5.3 · pp. 42–43, Figure 24 · p. 43

6. Conclusions

The conclusions pair a radial-and-longitudinal Koiter-shell model with a modified, loosely coupled scheme for blood-flow FSI. The β-scheme is reported unconditionally stable and monolithic-comparable while retaining partitioned-scheme advantages.

6. Conclusions

  • Two test problems show monolithic-comparable accuracy while the scheme retains modularity, easy implementation, and low cost without sub-iterations.The kinematically coupled β-scheme is first-order accurate in time, second-order in space, and unconditionally stable for 0 ≤ β ≤ 1.Source: §6 · p. 43
  • The proposed thin, linearly viscoelastic cylindrical Koiter-shell model captures both radial and longitudinal arterial-wall displacement.The fluid and structure are fully coupled using kinematic and dynamic conditions.Source: §6 · p. 43
  • The accuracy-maximizing choice of β remains to be determined as a function of the problem coefficients.Source: §6 · p. 44
  • The time-splitting methodology extends to three dimensions, although implementing the algorithm there would be more complex.Source: §6 · p. 44
  • Extensions to thick arterial walls modeled with 2D/3D elasticity are under way.Source: §6 · p. 44
  • The work is a first step toward modeling multilayered arterial walls by coupling a thin shell with a thick media/adventitia structure.Source: §6 · p. 44
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