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

 From: Emilio Melero García Subject: Fwd: [ff3d-users] Inner wall definitions Date: Mon, 5 Feb 2007 12:43:24 +0100

Hello Stephane,

First of all i want to heartly thank you for you e-mail. I understand that is not your work to be teaching people out and for that i thank you...

However, despite the risk of being a pain in the neck for you i would like to discuss a little here. I have read your mail and the figure, and i think i understand the concept of fictitious domain, but i have some questions.

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??

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?

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?)

If the answers to the preceeding questions are yes, then there is definitely something odd that i can not understand with what i am obtaining for my problem.

In the first figure i attach you can see a general view of how my problem is configured. You can see the box with the two capillaries inside. The capillaries are built as differences of coaxial cilinders of different radius, and placed so that they coincide on the lower base but not on the upper. The box has 6.6 cm in the z dimension (upwards in the image) and the background mesh has 67 vertices. Therefore the z dimension is discretized in 66 elements that have 0.1 cm each. Let's consider now that the capillaries have 5.0 cm long, and that both begin at 1.0 cm and end at 6.0 cm.

In Figure 2 you can see a view of the generated visualization tetrahedral mesh (see the .ff file attached) in which one can see that the capillaries do not begin at 1.0 cm (i.e. 10 elements from the bottom) but do so at 1.1 cm (11 elements). This also happens if i use only the outer cylinders instead to define the domain (domain Omega = domain(S,outside(<1,0,2>) and outside(<1,1,2>)); instead of domain Omega = domain(S,outside(<1,0,0>) and outside(<1,1,0>));) as seen in figure 3.

If one now uses the attached pov file, in which i have set different ending and beginnings for the outer and inner cylinders of the capillaries, one can see in figures 4 and 5 that the mesh show different limits of the cylinders. The outer ones start both at 0.9 cm elements when one should do it at 0.9 and the other at 0.8 cm, and finish one at 6.0 cm and the other at 5.8 cm instead of at 6.0 and 5.9 cm. Since the length of the cylinders is a integer number of elements of the background mesh, this effect can not be due to the fictitious domain approx, which in any case would add extra length to the cylinders, not cut them. The inner cylinders also show discrepancies on their endings (the only limit we can visualize), ending at 5.9 and 5.7 cm instead of 5.8 and 5.7 cm as written in the pov file.

Do you know why this could be happenning?

Best regards

Emilio

2007/2/1, Stephane Del Pino < address@hidden>:
Hello Emilio.

I think you need some enlightenment about the fictitious domain method that
you are using in the ff3d:

A fictitious domain method consists in using a mesh that does not match the
geometry of the problem that you want to solve.

For instance look at the attached picture. Suppose that you want to solve a
problem in the blue domain (left part). The fictitious domain method will use
a cartesian mesh to solve the problem. With ff3d's approach, only the blue
cells will be used for the computation: some vertices (outside of the domain)
are needed to build a good approximation of the solution inside the domain.

So (to simplify, in fact it is a bit more tricky) you can consider that when
you use the tetrahedrize keyword, you do not see the geometry of the domain,
but all the cells that are used to compute the solution.

Post-processing of fictitious domain approximation is not an easy task: often,
the solution is better than it looks like...

Best regards,
Stéphane.

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--
###############################################
Dr. Emilio Melero García
Laboratorio de Estudios Cristalográficos, IACT
Instituto de parasitología Lopez Neyra,
P.T. Ciencias de la Salud, E-18100 Armilla,

Correo elec. :emilio_arroba_lec_._csic_._es
Tfno: 958181621, lab 107
###############################################

--
###############################################
Dr. Emilio Melero García
Laboratorio de Estudios Cristalográficos, IACT
Instituto de parasitología Lopez Neyra,
P.T. Ciencias de la Salud, E-18100 Armilla,

Correo elec. :emilio_arroba_lec_._csic_._es
Tfno: 958181621, lab 107
###############################################

figures.zip
Description: Zip archive

files.zip
Description: Zip archive