Re: Fwd: [ff3d-users] Inner wall definitions

[Stephane Del Pino]

Le lundi 5 février 2007 12:43, Emilio Melero García a écrit :
> In the figure you include, what would happen if the domain i am interested
> in were to be the 32 squares that surround the four "central" ones. Then
> the approximation would be exact, would it not??
If the domain is *exactly* the blue zone, then the short answer is yes. For a 
more precise answer, you should look more carefully to the real 
discretization. I think that here we can consider that it is true since you 
use Neumann boundary conditions.

> Also, since the fundament of the fictitious domain is an intersection with
> a geometry using whole number of elements, the error that one makes with a
> domain that does not conform to the discretisation of the background mesh
> consists in using elements that are not completely contained in the domain,
> thus, is always in the direction of increasing the volume of the domain. Am
> i right?
Yes, for the *particular* fictitious domain methods that are used in ff3d, 
this is true.

> Another question, when i build the tetrahedral mesh out of the background
> mesh, am i right if i consider that every element of the background mesh
> contains the same number of tetrahedra, and is completely filled with those
> tetrahedra (Vol of element= sum of vol of tetrahedra of each element?)
Yes. The tetraedrization always cut hexahedra into 5 tetrahedra.

> Do you know why this could be happenning?
Ok. You are facing the most difficult part of the fictitious domain 
approximation: the post-processing. Let me clarify a few things:
- the tetrahedization in fact works as follows: each hexahedron is broken into 
5 tetrahedra in order to build a conforming mesh. *Then* if a domain is 
specified only the tetrahedra that intersect the domain are kept (ie: one of 
there vertices is inside the domain). This is enough to represent the 
computed solution. This explain that you do not see all the hexahedra at the 
boundary.
- the error you see is due to arithmetic error: you try to put the exact 
number of hexahedra, so you are playing with the machine precision...
- when you use only Neumann or Robin boundary conditions, the fictitious 
domain approximation is almost optimal: this means that the error is almost 
the same as for the finite element discretization. So you do not need to try 
to have your mesh to fit exactly the boundaries of your capillaries. 

If you think this is too confuzing you could also try to solve your problem 
using an unstructured 3d mesh. ff3d can work with such meshes. For instance, 
you can build a mesh with gmsh (http://www.geuz.org/gmsh/) and use ff3d to 
solve your problem on it...

Best regards,
Stéphane.