[sita]

Hello everyone,

At the moment I am implementing a one-dimensional FEM solver for blood flow, that is, 1D Navier-Stokes flow in a network of elastic tubes. Considering the large number of publications on this subject, I have a feeling I'm sort of reinventing the wheel, writing my own code from scratch. Is anyone here aware of the existence of freeware/commercial code for this purpose?

Thanks in advance!
Sita

P.S. I'm already using SimVascular for 3D simulations, but I'm looking for something more lightweight to use alongside of that, which is why I started implementing this one-dimensional solver.

[FMDenaro]

Originally Posted by sita

Hello everyone,

At the moment I am implementing a one-dimensional FEM solver for blood flow, that is, 1D Navier-Stokes flow in a network of elastic tubes. Considering the large number of publications on this subject, I have a feeling I'm sort of reinventing the wheel, writing my own code from scratch. Is anyone here aware of the existence of freeware/commercial code for this purpose?

Thanks in advance!
Sita

What are you doing exactly? The 1D constraint drives the continuity equation (for incompressible flow) to give you only u=constant...

[sita]

Hi,

Thanks for getting back so soon! Sorry, apparently I wasn't quite clear: we're looking at flow in an elastic tube here, so cross-sectional area is one of the variables. The equations are:

dA/dt + dQ/dx = 0,
dQ/dt + d/dx(alpha*Q^2/A) + A/rho dp/dx + k*Q/A = 0,

where alpha and k are constants, related to the shape of the velocity profile and viscous loss, respectively. This system is complemented with a constitutive relation between p and A. Alternatively these equations can be written in terms of A and u, p and Q, or p and u.

Sita

[FMDenaro]

Originally Posted by sita

Hi,

Thanks for getting back so soon! Sorry, apparently I wasn't quite clear: we're looking at flow in an elastic tube here, so cross-sectional area is one of the variables. The equations are:

dA/dt + dQ/dx = 0,
dQ/dt + d/dx(alpha*Q^2/A) + A/rho dp/dx + k*Q/A = 0,

where alpha and k are constants, related to the shape of the velocity profile and viscous loss, respectively. This system is complemented with a constitutive relation between p and A. Alternatively these equations can be written in terms of A and u, p and Q, or p and u.

Sita

In this case you will get practically the Euler equations not the NSE. The flow coupled with the elastic tube will show the character of hyperbolicity and the wave will travel at supersonic velocity

[sita]

Hi Filippo,

Thanks, that's at least partly right (the waves won't travel at supersonic speed though, see e.g. refs below), but that's not my question. I have this system of equations, which I'm trying to solve numerically using FEM, and since I'm not the first person doing this, I was hoping there might be existing code somewhere out there, saving me the effort of coding from scratch. Are you aware of any?

Sita

J.C. Stettler, P. Niederer, and M. Anliker, Theoretical analysis of arterial hemodynamics including the influence of bifurcations. part i: Mathematical model and prediction of normal pulse patterns. Ann. Biomed. Eng., 9 (1981) 145–164.

J.C. Stettler, P. Niederer, and M. Anliker, Theoretical analysis of arterial hemodynamics including the influence of bifurcations, part ii: Critical evaluation of theoretical model and comparison with noninvasive measurements of flow patterns in normal and pathological cases. Ann. Biomed. Eng., 9 (1981) 165–175.

[FMDenaro]

Originally Posted by sita

Hi Filippo,

Thanks, that's at least partly right (the waves won't travel at supersonic speed though, see e.g. refs below), but that's not my question. I have this system of equations, which I'm trying to solve numerically using FEM, and since I'm not the first person doing this, I was hoping there might be existing code somewhere out there, saving me the effort of coding from scratch. Are you aware of any?

Sita

J.C. Stettler, P. Niederer, and M. Anliker, Theoretical analysis of arterial hemodynamics including the influence of bifurcations. part i: Mathematical model and prediction of normal pulse patterns. Ann. Biomed. Eng., 9 (1981) 145–164.

J.C. Stettler, P. Niederer, and M. Anliker, Theoretical analysis of arterial hemodynamics including the influence of bifurcations, part ii: Critical evaluation of theoretical model and comparison with noninvasive measurements of flow patterns in normal and pathological cases. Ann. Biomed. Eng., 9 (1981) 165–175.

No, I am not aware of FEM software for hyperbolic equations. At the best of my knowledge, the elastic waves travel at supersonic velocity (of course, you need to define the sound velocity of the liquid and that of the elastic material separately)

[sita]

Ah, that's too bad, hopefully someone else knows.

For physiological conditions the flow is subcritical, with the characteristic system having one positive and one negative eigenvalue. Lucky for us humans

[sita]

In the meantime someone tipped me about Firedrake, a package for the solution of PDEs using FEM (similar to FreeFEM++). This is very helpful, so I thought I'd post it here, in case others are looking for something like this too.

Other tips are still most welcome though.

[duri]

At 1D level FEM and FDM are same. Why are you looking for 1D FEM tool. It is easy to develop a simple 1D FDM code which can give same accuracy.

[sita]

Thanks, yes, I'm aware of that. The main reason that I'm looking for a FEM solver is that for my particular application (cardiovascular flow) there are lots of publications using FEM. So I thought, before I go into coding and debugging and the whole lot, there might actually be code available somewhere, either freeware or commercial.

[sita]

Sorry, this is a rather old post, but I'm posting here anyway, so that hopefully others looking for 1D cardiovascular flow solvers may find this link faster than I did:

http://haemod.uk/nektar