Partitioned Algorithms for Fluid-Structure Interaction Problems in Haemodynamics

Question

1. What contributions have the authors mentioned in the paper "Partitioned algorithms for fluid-structure interaction problems in haemodynamics" ?

2. What is the correct framework to describe its dynamics?

3. What are the effects of the partitioned algorithms?

4. What are the two matching conditions enforced at the FS interface?

Accepted Answer

The authors consider the fluid-structure interaction problem arising in haemodynamic applications.. The authors review several strategies that all rely on the exchange of Robin interface conditions and review their performances reported recently in the literature.. The authors also analyze the stability of explicit partitioned procedures and convergence of iterative implicit partitioned procedures on a simple linear FSI problem for a general BDFp temporal discretizations.

Accepted Answer

Since the deformation of large arteries during a cardiac beat is quite large, thecorrect framework to describe its dynamics is given by the finite elasticity equations.

Accepted Answer

Implicit partitioned algorithms are also affected by the added mass effect as they feature very slow convergence, unless special treatments of the interface conditions are considered.

Accepted Answer

The two matching conditions enforced at the FS interface are the continuityof velocities (3.1)4 and the continuity of normal stresses (3.1)5 (physical interface conditions), while condition (3.2)2 enforces that the fluid domain remains at all time in contact with the solid (geometrical interface condition).

Accepted Answer

Another nice feature of RR procedures is that they allow to solve without any complication an enclosed fluid problem, that is a FSI problem where Dirichlet or flow rate boundary conditions are enforced on all portions of the fluid domain except the FS interface.

Accepted Answer

The fluid-structure interaction (FSI) problem in large vessels haemodynamics is characterized by a considerable amount of energy exchanged between blood and arterial wall in each cardiac beat [36, 6, 13, 39, 12, 2, 14].

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The intuitive reason why algorithm (5.5) fails to be stable for large added mass is that the presence of the fluid on the structure appears as an extra inertia term.

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The good convergence property of the RN scheme with (5.15) for FSI problems has been confirmed by the analysis provided in [2], which highlights that for a model problem as the one presented in (5.1), but with a one-dimensional elastic structure described by the generalized string model, convergence is achieved without any relaxation (ω = 1) and with asymptotic convergence factor σ(1) 1 very insensitive to the ratio ρf/ρs (i.e. to the added mass effect).

Accepted Answer

In haemodynamics, the use of explicit partitioned algorithms turns out to beextremely problematic for stability reasons, because of the large added-mass of the fluid on the structure.

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From the mechanical perspective, the media is the most significant layer in healthy arteries and is made primarily by elastin and collagen fibers.

Accepted Answer

By applying the divergence operator to the fluid momentum equation, it is possible to write an equivalent coupled problem involving just the fluid pressure and the piston displacement, as follows⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ pf = 0 in Ωf , ∂pf ∂n = 0 on Γwall, pf = g on Γin, ∂pf ∂n = −ρf η̈s on Γp, mη̈s + cη̇s + kηs = ∫

Accepted Answer

By exploiting the linearity of the fluid problem, the authors have that the force exerted by the fluid on the piston can be written as∫Γppf dγ = ∫

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Such problems are coupled through the physical interface conditions, which guarantee the continuity of the velocity and of the normal stresses between fluid and structure, and the geometrical interface condition, which guarantees the continuity of displacements between the fluid and the structure domains.

Accepted Answer

In the case of the double loop algorithm, the authors review also a strategy to solve the internal linearized FSI problem, by reformulating it as an interface equation [3, 9].

Accepted Answer

in this case, the geometrical and physical interface conditions and the constitutive non-linearities are all treated in the same loop.

Accepted Answer

The first strategy corresponds to simple iterations at each time step between the fluid and the structure subproblems with Robin boundary conditions (see [27, 29, 26] for the DN case).

Partitioned Algorithms for Fluid-Structure Interaction Problems in Haemodynamics

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Most frequently asked questions

1. What contributions have the authors mentioned in the paper "Partitioned algorithms for fluid-structure interaction problems in haemodynamics" ?

2. What is the correct framework to describe its dynamics?

3. What are the effects of the partitioned algorithms?

4. What are the two matching conditions enforced at the FS interface?

5. What is the way to solve a FSI problem?

6. What is the FSI problem in large vessels?

7. What is the reason why the algorithm fails to be stable for large added mass?

8. What is the good convergence property of the RN scheme for FSI problems?

9. What is the main reason for the use of explicit partitioned algorithms?

10. What is the important layer in healthy arteries?

11. How can the authors write a coupled equation involving just the fluid pressure and the piston displacement?

12. How can the authors write a coupled equation involving the fluid pressure and the piston displacement?

13. What are the main problems that are coupled through the physical interface?

14. How do the authors solve the internal linearized FSI problem?

15. In what case is the structure subproblem treated in the same loop?

16. What is the first strategy for solving the structure and fluid subproblems?

Figures

Citations

The cardiovascular system: Mathematical modelling, numerical algorithms and clinical applications *

Geometric multiscale modeling of the cardiovascular system, between theory and practice

An analysis of a new stable partitioned algorithm for FSI problems. Part I: Incompressible flow and elastic solids

Time accurate partitioned algorithms for the solution of fluid–structure interaction problems in haemodynamics

Generalized Robin-Neumann explicit coupling schemes for incompressible fluid-structure interaction: Stability analysis and numerics

References

Biomechanics: Mechanical Properties of Living Tissues

Solving Ordinary Differential Equations I: Nonstiff Problems

A new constitutive framework for arterial wall mechanics and a comparative study of material models

Numerical Approximation of Partial Differential Equations

Lagrangian-Eulerian finite element formulation for incompressible viscous flows☆

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