I am trying to learn Gmsh and Fenics and was looking at an example which shows the application of Boundary conditions on a simple Poisson problem.

Here is the link:


The example defines the domain Ω to be a model of blood vessels with an aneurysm. It has one inlet vessel and two outlet vessels. They define no slip boundary conditions on the walls of the vessels and aneurysm, that is u=u0=0.0. We let u=u1=1.0 be the Dirichlet condition on the inlet, the outlets will have the prescribed values u=u2=2.0 and u=u3=3.0. In summary, we have:

u=u0=0.0 on ΓD,0 (noslip boundary);
u=u1=1.0 on ΓD,1 (inlet);
u=u2=2.0 on ΓD,2 (outlet 1) ;
u=u3=3.0 on ΓD,3 (outlet 2) ;
f=0.0 (source term)

I can understand the code overall, but I am unable to understand the meaning of these Boundary conditions, i.e., unable to visualize them in a physical sense, intuitively. I' ll appreciate if you can post any other link or better example too which explains the BC phenomena better.

Thank you!

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    $\begingroup$ It seems like u is probably a velocity field. I am no expert though. Here is a link to a physical interpretation of the no-slip boundary condition. In oversimplified terms it just means that on average fluid particles along the vessel walls won't be moving (zero velocity). It seems like the inlet and inlet/outlet terms are just specifying that there is a constant velocity at said openings. Perhaps you are looking for a deeper explanation than this and personally I can't offer much more. $\endgroup$ – wgwz Jun 22 '15 at 18:04
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    $\begingroup$ This would be a good question to ask the Fenics folks. $\endgroup$ – Bill Barth Jun 22 '15 at 19:14

If you look at the documentation,1 $u$ is described as solving the Poisson equation $(\nabla u,\nabla v) = (f,v)$ for $v\in H^1(\Omega)$. This means that it is unlikely to be a velocity field (it's scalar, after all). I would interpret the documentation insofar as only the geometric domain is meant to model a blood vessel, while the PDE is only a very academic example that isn't supposed to model anything. Hence, there is no further interpretation than that the value of $u$ is prescribed (as different values at different parts of the boundary).

1. which is outdated, by the way -- current is 1.5, and 1.6 is coming up soon

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  • $\begingroup$ I'm still voting to close (and the reason is as good as "too localized", which doesn't exist anymore), but this got too long for a comment. $\endgroup$ – Christian Clason Jun 22 '15 at 20:53

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