[Dune] Dune Digest, Vol 84, Issue 22

[Carsten Gräser]

Hi,
Am 12.06.2012 10:19, schrieb Marco Cisternino:
> Hi,
> thanks for all the advices.
> Andreas, in order to redefine my problem: I want to manage curvilinear
> grid, then during refinement I would like to move new nodes on the
> curvilinear parametrization to get a smoother grid. To make this
> possible I should be able to move new vertices on the parametrization,
> but this would mean getting 5-vertices element as the first neighbours
> to the locally refined region.
> o-----------o
> |              |
> |              |
> |              |
> o----m----o
> |      |       |
> n----n-----n
> |      |       |
> o----n---.-o
> This would be the standard refinement, but I want to move the m-vertex
> on a parametrization of o----m----o, like this
> o-----------o
> |              |
> |      m     |
> |   /   |   \  |
> o      |      o
> |       |      |
> n----n-----n
> |      |       |
> o----n---.-o
> But now the element o-m-o-o-o has 5 vertices.
> I don't want to avoid hanging nodes as you can see. But I think I can't
> manage 5-vertices elements with ALUGrid, isn't it? And with UG??
> I cannot understand why this kind of element is hard to be managed. When
> an hanging node is present, let me say in the first configuration I
> plotted, I have 5 intersections anyway!
the problem is that moving m would require to change the geometry of the
neighboring coarse grid element. Before moving m it's a bilinearly
transformed quad. After moving m its quad transformed by a piecewise
bilinear transformation. It will be complicated to implement this
and I doubt that any present grid implementation does this.
(@Robert: What does ALUGrid for surface grids?)

But there's a natural solution: For the moment assume that you
have piecewise linear conforming finite elements. Then there
would be no dof for m since the value would be constrained to
be the linear interpolation of the neighboring vertices. If you
want a dof for m you must refine all neighboring elements.
That's quite natural since elements with hanging nodes play
the role of the closures in red/green refinement: They allow
to have conforming ansatz space that is coarse on one side
and fine on the other, but the approximation on the closure
is not as good as on a classically refined element.

Transfered to the parametrized grid this means that you should
simply not move m in the situation you described. If you want
to have a better approximation of the geometry at m you must
refine all neighboring elements. In the following situation
if you want a better approximation on the a-b edge you have
to refine A and B. Then you have a non-hanging dof on a-b
with a better approximation and but hanging nodes on c-d
that still provide the old approximation quality.

o---o
| A |
a---b
| B |
c---d
|   |
o---o

Regards,
Carsten

> Does only UG allow me to move vertices?
> What happen if 4 elements are coarsened? Do I get a 5-vertex element???
>         o
>     /   |   \
> o      |      o
> |       |      |
> o----o-----o
> |      |       |
> o----o---.-o
> 
> Thanks a lot again!!
> Cheers,
> Marco
> 
> 
> 
> 
> Il 11/06/2012 21:20, dune-request at dune-project.org ha scritto:
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-- 
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Dr. Carsten Gräser       | phone: +49-30 / 838-75349
Freie Universität Berlin | fax  : +49-30 / 838-54977
Institut für Mathematik  | email: graeser at math.fu-berlin.de
Arnimallee 6             |
14195 Berlin, Germany    | URL  : http://page.mi.fu-berlin.de/graeser
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